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https://arxiv.org/abs/alg-geom/9511003

alg-geom/9511003

Dr. P. E. Newstead

L. Brambila Paz, I. Grzegorczyk, and P. E. Newstead

Geography of Brill-Noether loci for small slopes

34pp and 3 figures. Figures (not included in e-print) may be obtained by sending your mailing address to [email protected]; complete hard copy also available. LaTeX 2.09

Let $X$ be a non-singular projective curve of genus $g\ge2$ over an algebraically closed field of characteristic zero. Let $\mo$ denote the moduli space of stable bundles of rank $n$ and degree $d$ on $X$ and $\wn $ the Brill-Noether loci in $\mo .$ We prove that, if $0\leq d \leq n $ and $\wn $ is non-empty, then it is irreducible of the expected dimension and smooth outside $\wnn$. We prove further that in this range $\wn$ is non-empty if and only if $d>0$, $n\leq d+(n-k)g$ and $(n,d,k) \not= (n,n,n)$. We also prove irreducibility and non-emptiness for the semistable Brill-Noether loci.

alg-geom

\section*{Introduction} \bigskip The moduli spaces of stable vector bundles over an algebraic curve have been extensively studied from many points of view since they were first constructed more than 30 years ago, and much is now known about their detailed structure, in particular in terms of their topology. Except in the classical case of line bundles, however, relatively little is known about their geometry in terms, for example, of the existence and structure of their subvarieties. In the case of line bundles, where the moduli spaces are all isomorphic to the Jacobian, Brill-Noether theory has long provided a basic source of geometrical information. This theory, which originated in the last century, is concerned with the subvarieties of the moduli spaces determined by bundles having at least a specified number of independent sections. Basic questions, concerning non-emptiness, connectedness, irreducibility, dimension, singularities, cohomology classes, etc., have been completely answered when the underlying curve is generic, and departures from the generic behaviour are indeed used to describe curves with special properties. The definitions can easily be extended to bundles of any rank, but the basic questions are then far from being answered even for a generic curve. In particular, given integers $n$, $d$, $k$ with $n\geq2$ and $k\geq1$, one would like to know when there exist stable (or semistable) bundles of rank $n$ and degree $d$ having at least $k$ independent sections. We use the term \lq\lq geography'' to refer to the study of this problem (and related questions such as irreducibility and dimension of the corresponding loci) by analogy with a similar use of the term in the theory of algebraic surfaces; we shall see indeed that much of the data obtained by ourselves and others can be conveniently summarised in graphical form (see \S2 and particularly Figures 1 and 2). In order to describe some of these ideas in more detail and to state our own results, we introduce some further notation. Let $X$ be a non-singular projective curve of genus $ g\geq 2 $ defined over an algebraically closed field of characteristic $0$ and let ${\cal M}(n,d) $ denote the moduli space of stable vector bundles over $X$ of rank $n$ and degree $d$. For any integer $k\geq1$, the {\it Brill-Noether locus} ${\cal W}^{k-1}_{n,d}$ is the set of stable bundles in ${\cal M}(n,d)$ having at least $k$ independent global sections; this is in fact a subvariety of ${\cal M}(n,d)$ (see 1.1). (The superscript $k-1$ is used here rather than $k$ largely for historical reasons, since projective dimension rather than vector space dimension was regarded classically as the important notion.) Associated with this locus is the number $$\rho^{k-1}_{n,d} = n^2(g-1)+1-k(k-d+n(g- 1)).$$ This is called the {\it Brill-Noether number}, and is the \lq\lq expected'' dimension of ${\cal W}^{k-1}_{n,d}$. In a similar way, we denote by $\mot$ the moduli space of S-equivalence classes of semistable vector bundles over $X$ and by ${\wnt} $ the corresponding Brill-Noether locus; again this is a subvariety of $\mot$ (see 1.2 for full definitions in this case). In the case $n=1$, the Brill-Noether loci (as remarked above) have been well known since the last century. In fact the variety ${\cal W}^{k-1}_{1,d}$ is always non-empty if $\rho^{k-1}_{1,d}\geq0$, and connected if $\rho^{k-1}_{1,d}>0$. The variety may be reducible, but each component has dimension at least $\rho^{k-1}_{1,d} $. For a generic curve $X$, ${\cal W}^{k-1}_{1,d}$ is empty if $\rho^{k-1}_{1,d}<0$ and is irreducible of dimension $\rho^{k-1}_{1,d}$ with singular locus ${\cal W}^{k}_{1,d}$ if $g>\rho^{k-1}_{1,d}>0$. Modern proofs of these results have been given by Kempf, Kleiman and Laksov, Fulton and Lazarsfeld, Griffiths and Harris, and Gieseker. A full treatment of this case is contained in [ACGH]. For higher rank and $X$ generic, it is known that, for $0<d\leq n(g-1)$, ${\cal W}^0_{n,d}$ is irreducible of dimension $\rho^0_{n,d}$ [Su] and ${\rm Sing}\,{\cal W}^0_{n,d}={\cal W}^1_{n,d}$ [L]. The most extensive results to date are those of Teixidor [Te2]; these describe many cases when $\rho^{k-1}_{n,d} \geq 0$ and ${\cal W}^{k-1}_{n,d}$ is non-empty, as expected (see 2.5, where we shall use these results to draw our \lq\lq map''). Further results on non-emptiness and irreducibility are known when $n=2$ and $k=2,3$ [Su, T, Te1, Te3], while ${\cal W}^{k-1}_{3,1}$ and ${\cal W}^{k-1}_{3,2}$ are described in [NB]. On the other hand, even for $X$ generic, ${\cal W}^{k-1}_{n,d}$ may have components of dimension greater than $\rho^{k-1}_{n,d}$ [BF] and the singular set of ${\cal W}^{k-1}_{n,d}$ may be strictly larger than ${\cal W}^{k}_{n,d}$ [Te2]. In this paper we consider the case when $n\geq 2$ and $0\leq d\leq n$ and study the varieties ${\cal W}^{k-1}_{n,d} $ and $\wnt$. Our main results, which provide a complete answer to the basic questions in this case, are: \bigskip {\bf THEOREM A :} {\it If ${\cal W}^{k-1}_{n,d}$ is non-empty, then it is irreducible, of dimension $\rho^{k-1}_{n,d} $ and ${\rm Sing}\, {\cal W}^{k-1}_{n,d} ={\cal W}^k_{n,d} $.} \bigskip {\bf THEOREM $\tilde{{\rm {\bf A}}}$ :} {\it If $\wnt$ is non-empty, then it is irreducible.} \bigskip {\bf THEOREM B :} {\it ${\cal W}^{k-1}_{n,d}$ is non-empty if and only if $$d>0,\ \ n\leq d+(n-k)g\ \ and\ \ (n,d,k)\neq(n,n,n).$$ } {\bf THEOREM $\tilde{{\rm {\bf B}}}$:} {\it $\wnt $ is non-empty if and only if either $$d=0\ \ and\ \ k\leq n$$or $$d>0\ \ and\ \ n\leq d+(n-k)g.$$} Our results give partial answers to questions 1 and 3 on the VBAC Problems List [VBAC], and are valid for all non-singular curves, not just generic ones. Note that, in the case $k\leq d<n$, Theorems B and $\widetilde{{\rm B}}$ follow from Teixidor's results [Te2]. Note also that the condition $n\leq d+(n-k)g $ implies that $\rho^{k-1}_{n,d} \geq 1,$ so in particular ${\cal W}^{k-1}_{n,d} $ is empty when $\rho^{k-1}_{n,d} =0$ for $ 0\leq d \leq n$; this gives another example where the results in higher rank differ from those in rank 1. After the work for this paper was completed, an alternative proof of Theorem $\widetilde{{\rm B}}$, using variational methods based on the Yang-Mills-Higgs functional, was announced by G. Daskalopoulos and R. Wentworth [DW]. In proving our theorems, we shall distinguish the three cases $0<d<n$, $d=0$ and $d=n$, although all three will depend on the use of extensions of the form $$0\rightarrow {{\cal O}^k}\rightarrow E \rightarrow F \rightarrow 0.$$ In \S 1 we fix notation and give the basic definitions. In \S 2 we give a proof (due to G. Xiao) of Clifford's Theorem for semistable bundles (Theorem 2.1) and explain the geography of the Brill-Noether loci. In \S3, we introduce the use of extensions (Proposition 3.1) and prove the necessary condition $n\leq d+(n-k)g$ in Theorems B and $\widetilde{{\rm B}}$ (Theorem 3.3). In \S 4 we prove Theorems A and $\widetilde{{\rm A}}$ when $0<d<n$ (Theorems 4.3, 4.4). \S5 provides the setting for the proofs of Theorems B and $\widetilde{{\rm B}}$ which are completed in \S 6 (Theorem 6.3). Finally, in \S\S 7, 8, we prove all four theorems for $d = 0 $ (Theorems 7.1, 7.3) and $d=n$ (Theorems 8.2, 8.5). Our methods yield some information on the more detailed geometry of the Brill- Noether loci (see, for example, Corollary 4.5, Theorem 7.2, Theorem 8.3). These varieties are also closely connected with various types of augmented bundle for which moduli spaces have recently been constructed. These include $k$-pairs [BeDW], coherent systems [LeP1, 2] (also discussed as \lq\lq Brill-Noether pairs'' in [KN], and just \lq\lq pairs'' in [Be, RV]) and extensions [BG]; for a general survey, see [BDGW]. We propose to return to these questions in future papers. {\bf Acknowledgement.} The work for this paper was completed during a visit by the first two authors to Liverpool. They wish to acknowledge the generous hospitality of the University of Liverpool. The second author would like to thank Institut Henri Poincar\'{e} for support and hospitality. All the authors wish to thank A.~D. King for many useful discussions; the introductory material in \S\S1, 2 (and in particular the map of \S2) owe a great deal to unpublished notes of King. We also wish to thank R. Morris for designing Figures 1 and 2, and M. Tapia (CIMAT) for help with computer calculations which refuted an earlier conjecture and helped to lead us to a correct statement of Proposition 6.1. \bigskip \renewcommand{\thesection}{\S \arabic{section}} \section{Notation and definitions}\renewcommand{\thesection}{\arabic{section}} In this section, we give some basic notations and definitions. We denote by $X$ a non-singular projective curve of genus $g\geq 2$, fixed throughout the paper, and write ${{\cal O}^k }={\cal O }^k_X$ for the trivial bundle of rank $k$ over $X$. For any integers $n$ and $d$ with $n\geq1$, let ${{\cal M}(n,d)}$ denote the moduli space of stable vector bundles of rank $n$ and degree $d$ over $X$. We write $\mu(E)=\deg E/{\rm rk}\, E$ for the {\it slope} of a bundle $E$. We make no distinction between locally free sheaves and vector bundles over $X$. However a subsheaf of a vector bundle is called a subbundle only if the quotient is itself a vector bundle. \bigskip {\bf 1.1.} {\it Brill-Noether loci for stable bundles. } As a set of points, ${{\cal W}^{k-1}_{n,d}}$ can be defined by $$ {{\cal W}^{k-1}_{n,d}}=\{ E\in {{\cal M}(n,d)}| h^0(E)\geq k\} .$$ Suppose first $(n,d)=1$. To obtain a scheme structure on ${{\cal W}^{k-1}_{n,d}}$, let ${\cal U}$ be a universal bundle over $ X\times {{\cal M}(n,d)}$. Choose an effective divisor $D$ of a sufficiently large degree that $H^1(E\otimes L(D))=0$ for all $E\in {{\cal M}(n,d)}$. (Here $L(D)$ is the line bundle associated to $D$.) Then, in the exact sequence $$0{\rightarrow}H^0(E){\rightarrow}H^0(E\otimes L(D)){\rightarrow}H^0(E|_D){\rightarrow}H^1(E){\rightarrow}0,$$ the middle two terms have dimensions independent of $E$. Globalising this, we obtain $$0\rightarrow\pi_*{\cal U}\rightarrow\pi_*({\cal U}\otimes p^*_XL(D))\stackrel{\phi}{\rightarrow} \pi_*\left({\cal U}|_{D\times{\cal M}(n,d)}\right)\rightarrow R^1_{\pi}{\cal U}\rightarrow 0,$$where $\pi:X\times {{\cal M}(n,d)}\rightarrow{\cal M}(n,d)$ is the projection map. The middle two terms of this sequence are vector bundles. We can now define ${\cal W}^{k-1}_{n,d}$ as the determinantal locus where $\phi$ drops rank by at least $k$. The \lq\lq expected'' dimension of ${\cal W}^{k-1}_{n,d}$ is given by $$\rho^{k-1}_{n,d}=n^2(g-1)+1-k(k-d+n(g-1)),$$ which is called the {\it Brill-Noether number} associated to ${\cal W}^{k-1}_{n,d}$. It follows from the theory of determinantal varieties (see [ACGH] for further details) that, if ${\cal W}^{k-1}_{n,d}$ is non-empty and ${\cal W}^{k-1}_{n,d}\neq{\cal M}(n,d)$, then $\dim{\cal W}^{k-1}_{n,d}\geq\rho^{k-1}_{n,d}$. For $(n,d)\not=1$, there is no universal bundle over $X\times {{\cal M}(n,d)}$. However the above construction works for any locally universal family (for instance, over a Quot scheme); we can then define ${\cal W}^{k-1}_{n,d}$ to be the image of the variety so obtained under the natural morphism to ${\cal M}(n,d)$. It follows from geometric invariant theory that this is a closed subvariety of ${\cal M}(n,d)$. \bigskip {\bf 1.2. }{\it Brill-Noether loci for semistable bundles.} Let $E$ be a semistable bundle of rank $n$. Then there exists a filtration $$0=E_0\subset E_1\subset E_2......\subset E_r=E$$ such that $E_i/E_{i- 1}$ is a stable bundle with $\mu (E_i/ E_{i-1})= \mu (E)$ for $0<i\leq r$. The associated graded bundle $\bigoplus_i (E_i/ E_{i-1})$ depends only on $E$ and is denoted by ${\rm gr}\, E$. We say that two semistable bundles are {\it S-equivalent} if ${\rm gr}\, E\cong{\rm gr}\, F$. There exists a moduli space $\widetilde {\cal M}(n,d)$ of S-equivalence classes of semistable vector bundles of rank $n$ and degree $d$, which is an irreducible projective variety and is a natural compactification of ${\cal M}(n,d)$ (see [S]). Writing $[E]$ for the S-equivalence class of $E$, we now define $$ \wnt=\{ [E]\in {{\cal M}(n,d)}| h^0({\rm gr}\, E)\geq k\} .$$ Since $h^0({\rm gr}\, E)\geq h^0(E)$ for all $E$, we can also define $\wnt$ as the set of S-equivalence classes which contain a bundle $E$ with $h^0(E)\geq k$. We can give $\wnt$ a structure of variety by using a locally universal family as above. Note that this variety does not have to be the closure of ${{\cal W}^{k-1}_{n,d}}$, as there may exist components containing semi-stable bundles only. These components may have dimension smaller then $\rho^{k-1}_{n,d}$. For examples where this occurs, see \S7. \bigskip {\bf 1.3. }{\it Petri map}. If $h^0(E)=k$, the tangent space to ${{\cal W}^{k-1}_{n,d}}$ at $E$ is the kernel of the map $$ p^*: {\rm Ext^1} (E,E){\rightarrow }H^0(E)^*\otimes H^1(E)$$ which is dual to the {\it Petri map} $$ p: H^0(E)\otimes H^0(E^*\otimes K){\rightarrow } H^0({\rm End}(E)\otimes K)$$ defined by multiplication of sections. It follows easily that ${{\cal W}^{k-1}_{n,d}}$ is smooth of dimension $\rho^{k-1}_{n,d}$ at $E$ if and only if the Petri map is injective. (Incidentally there exist bundles $E$ for which the Petri map is not injective [Te2, \S5].) Note also that ${\cal W}^{k}_{n,d}\subset{\rm Sing}\,{\cal W}^{k-1}_{n,d}$ whenever ${\cal W}^{k-1}_{n,d}\neq{\cal M}(n,d)$ (see [ACGH, Chapter II \S2 and p.~189). \bigskip \renewcommand{\thesection}{\S\arabic{section}}\section{ \bf Brill-Noether geography of vector bundles of higher ranks} \renewcommand{\thesection}{\arabic{section}} Our main object in this section is to produce a \lq\lq map'' on which we can display the results of Brill-Noether theory for bundles of arbitrary rank. Before doing this, however, we shall state and prove a simple but fundamental result, which is a direct generalisation of Clifford's Theorem for line bundles. \noindent{\bf Theorem 2.1 (Clifford's Theorem). } {\it Let $E$ be a semistable bundle of rank $n$ and degree $d$ with $0\leq\mu(E)\leq 2g-2$. Then $$h^0(E)\leq n+{d\over2}.$$} {\it Proof:} (As far as we are aware, no complete proof of this result has appeared in the literature. The following is due to G. Xiao.) The proof is by induction on $n$, the case $n=1$ being the classical theorem. For $E$ a semistable bundle of rank $n\geq2$, note first that we can assume that $h^0(E)>0$ and $h^1(E)>0$ (i.e. $E$ is {\it special}), for otherwise the result follows at once from Riemann-Roch. Now let $E_1$ be a proper subbundle of $E$ of maximal slope and let $E_2=E/E_1$. Certainly $E_1$ and $E_2$ are both semistable. By semistability of $E$, we have $\mu(E_1)\leq 2g-2$ and $\mu(E_2)\geq0$. On the other hand, since $h^0(E)>0$, $E$ possesses a subbundle of non-negative degree; so $\mu(E_1)\geq0$. Similarly, since $h^1(E)>0$, $E$ possesses a quotient line bundle of degree $\leq 2g-2$; by comparing the slope of the kernel of this quotient with that of $E_1$, one sees easily that $\mu(E_2)\leq 2g-2$. The result now follows at once by induction.\hfill$\diamondsuit$ \bigskip To construct our map, we first associate with ${\cal W}^{k-1}_{n,d}$ and $\wnt$ the rational numbers $$\lambda={k\over n},\ \ \mu={d\over n}.$$If $d<0$ and $k>0$, then $\wnt$ is empty, while obviously ${\cal W}^{k-1}_{n,d}={\cal M}(n,d)$ if $k\leq 0$. We can therefore plot $\mu$ against $\lambda$ in the first quadrant of the standard coordinate system (see Figure 1). Advantages of plotting things in this way are that every point with rational coordinates can in principle support bundles and that all ranks are represented in the same diagram. In the remainder of the section, we describe some important features of the map. \bigskip {\bf 2.2} {\it Riemann-Roch line} $\mu= \lambda+g-1$. By the Riemann- Roch Theorem $$h^0(E)-h^1(E)=d-n(g-1).$$ Therefore for $\mu \geq \lambda +g-1$, i.e. above the Riemann-Roch line, ${{\cal W}^{k-1}_{n,d}}$ is the whole moduli space. Note also that any semistable bundle $E$ with $\mu(E)>2g-2$ has $h^1(E)=0$; so, for $\mu>2g-2$, $\wnt$ is empty below the Riemann-Roch line. \bigskip {\bf 2.3.} {\it Clifford line} $\mu=2\lambda-2$. By Theorem 2.1 every ${{\cal W}^{k-1}_{n,d}}$ below this line is empty. The interesting part of the map is therefore the pentagonal region bounded by the axes, the Riemann-Roch line, the Clifford line and the line $\mu=2g-2$. This corresponds to the region in which there may exist special semistable bundles. \bigskip{\bf 2.4.} {\it Brill-Noether curve}. Define $$\tilde{\rho} ={1\over n^2}(\rho^{k-1}_{n,d}- 1)=(g-1)-\lambda(\lambda-\mu+(g-1)).$$We call the curve $\tilde{\rho}=0$ the {\it Brill-Noether curve}. The curve is a branch of a hyperbola, below which the expectation is that the Brill-Noether loci will be finite. \bigskip {\bf 2.5.} {\it Teixidor parallelograms}. In [Te2], Teixidor defines ranges of values for $n$, $d$, $k$ such that for generic curves the ${{\cal W}^{k-1}_{n,d}}$ are non-empty and have a component of the expected dimension. These ranges correspond to points $(\lambda, \mu )$ lying in or on one of the parallelograms marked T on the map. These parallelograms have vertices at integer points, sides parallel to $\lambda=0$ and $\mu=\lambda$ and have all their vertices on or above the Brill-Noether curve 2.4. If all the vertices lie above $\tilde{\rho} = 0$, then ${\cal W}^{k-1}_{n,d}$ is non-empty whenever $(\lambda, \mu )$ lies in or on the parallelogram. If the lower right vertex of the parallelogram lies on $\tilde{\rho} = 0$, this still holds with the possible exception of those points of the parallelogram with the same $\mu$-coordinate as this vertex; for such points, Teixidor shows only that $\wnt$ is non-empty. \bigskip\bigskip\centerline{FIGURE 1} \bigskip\bigskip In this paper we are concerned with the region $ {0\leq \mu \leq 1}$ of the map (see Figure 2). The subregion $\lambda\leq\mu<1$ lies in a Teixidor parallelogram, but the remainder of the region does not. In any case, Teixidor proves only non-emptiness (and, for $X$ generic, the existence of a component of the correct dimension), whereas we shall solve the non-emptiness, irreducibility and singularity problems for the entire region. A key r\^{o}le in this is played by the tangent line at $(1,1)$ to $\tilde{\rho}=0$. This is given by $\mu+(1-\lambda)g=1$ or equivalently $n=d+(n-k)g$. Thus the inequality $n\leq d+(n-k)g$ in Theorems B and $\widetilde{{\rm B}}$ describes the area on or above this tangent line. Theorem B therefore states that for $ \mu \leq 1$, ${\cal W}^{k-1}_{n,d}$ is empty below this line, while Theorem $\widetilde{{\rm B}}$ says that the same is true for $\wnt$ except on $\mu=0$. On the other hand, the Brill-Noether number $\rho^{k-1}_{n,d}$ can be positive below the line, so this is not a sufficient condition for the non-emptiness of ${\cal W}^{k-1}_{n,d}$. This phenomenon can be compared with the \lq\lq fractal mountain range'' of Drezet and Le Potier, which excludes the existence of some stable bundles on ${\bf P}^2$, which should exist for purely dimensional reasons [DL]. \bigskip\bigskip \centerline{FIGURE 2}\bigskip \renewcommand{\thesection}{\S\arabic{section}} \section{Emptiness of Brill-Noether loci}\renewcommand{\thesection}{\arabic{section}} In this section, we assume that $E$ has rank $n\geq 2$ and that either $E$ is stable and $\mu (E)\leq 1$ or $E$ is semistable and $\mu (E)<1$. Our main purpose is to prove the necessity of the conditions in Theorems B and $\widetilde{{\rm B}}$ (see Theorem 3.3). We begin with the following proposition, which will be used many times in the paper. \bigskip \noindent{\bf Proposition 3.1.} {\it Let $E$ be a stable bundle of degree $d$, $0\leq d \leq n$ (or a semistable bundle with $0\leq d < n$), and $h^0(E) \geq k > 0$. Let $V$ be a subbundle of $E$ generated by $k$ independent global sections of $E$. Then $V$ is a trivial bundle of rank $k$. } \bigskip {\it Proof:} We have the exact sequence $$0\rightarrow V\rightarrow E \rightarrow F \rightarrow 0.$$ If $V$ is non-trivial, there exists a section $s\in h^0(E) $ such that $\deg D > 0, $ where $D$ is the divisor of zeros of $s$. Then $\deg L(D) > 0$ and $\mu (L(D))\geq 1.$ But $L(D)$ is a subbundle of a stable (resp. semistable) bundle $E$ and $\mu (E) \leq1 (resp. <1)$. This is a contradiction, so $V\cong {\cal O}^k$.\hfill$\diamondsuit$ \bigskip \noindent{\bf Remark 3.2.} i) The above implies that for any $E\in {\cal W}^{k-1}_{n,d} $, \ $0\leq d \leq n$, $E$ can be presented as an extension of the form $$0\rightarrow {{\cal O}^k}\rightarrow E \rightarrow F \rightarrow 0.\eqno(1)$$Similarly, every point of $\wnt$, $0\leq d<n$, has a representative $E$ which can be presented in this form. \bigskip ii) Note that, if $d\geq0$ and $E$ is stable, or $d>0$ and $E$ is semistable, then $h^0(E^*)=0$. Except in the case $d=0$, $E$ semistable, we may therefore assume that $h^0(F^*)=0$ in the above sequence, as $F^*$ is a subbundle of $E^*$. \bigskip \noindent{\bf Theorem 3.3. } {\it ${\cal W}^{k-1}_{n,d}$ is empty for $d>0$, $n>d+(n-k)g$ and for $d=0$. $\widetilde {\cal W}^{k-1}_{n,d}$ is empty for $d>0$, $n>d+(n-k)g$ and for $d=0$, $k>n$.} \bigskip Theorem 3.3 has also been proved in the case $d>0$ by Anne Maisani.\bigskip {\it Proof.}:\ By Remark 3.2(i), every point of $\wnt$ can be represented by a bundle $E$ of the form (1). It follows at once that $\wnt$ is empty if $k>n$ or if $k=n$ and $d>0$. Moreover, if $d=0$, (1) contradicts the stability of $E$; so ${\cal W}^{k-1}_{n,d}$ is empty. We can therefore suppose that $k<n$ and $d>0$. In this case, the extensions (1) are classified by the elements of the vector space $H=\bigoplus^k H^1(F^*)$, i.e. by $k$-tuples $(e_1,\ldots,e_k)$ with $e_i\in H^1(F^*)$. Moreover two extensions are isomorphic if the corresponding points are in the same orbit of the natural action of $GL(k)$ on $H$. Thus, if $e_1,\ldots,e_k$ are linearly dependent, we can suppose (using this action) that $e_k=0$; hence the extension has a partial splitting to give ${\cal O}} \newcommand{\qa}{{\cal Q}$ as a direct summand of $E$, contradicting the stability hypothesis. Now, since $h^0(F^*)=0$ by Remark 3.2(ii), we have $$h^1(F^*)=d+(n- k)(g-1).$$So $e_1,\ldots,e_k$ are necessarily linearly dependent if $k>d+(n- k)(g-1)$, or equivalently $n>d+(n-k)g$.\hfill$\diamondsuit$ \bigskip For future convenience we finish this section with the following proposition. \bigskip \noindent{\bf Proposition 3.4.} {\it Let $F$ be a fixed bundle of rank $n-k$ and degree $d$ with $h^0(F^*)=0$. Then, if $n\leq d+(n-k)g$, the extensions $$0\rightarrow {{\cal O}^k}\rightarrow E \rightarrow F \rightarrow 0$$ with no trivial summands are classified up to automorphism of ${\cal O}} \newcommand{\qa}{{\cal Q}^k$ by a variety of dimension $k(d+(n-k)g-n)$.} \bigskip {\it Proof :} The extensions of this form are classified by the linearly independent $k$-tuples of elements of $ H^1(F^*)$ modulo the linear action of $GL(k)$, in other words by the Grassmannian $Grass_k (H^1(F^*))$. Now\begin{eqnarray*}\dim Grass_k(H^1(F^*)) = k(h^1(F^*)-k)&=& k(d+(n-k)(g- 1)-k)\\&=&k(d+(n-k)g-n).\end{eqnarray*}\hfill$\diamondsuit$ \bigskip \renewcommand{\thesection}{\S\arabic{section}} \section{Irreducibility} \renewcommand{\thesection}{\arabic{section}}In this section we shall use Proposition 3.1 and Remark 3.2 to prove Theorems A and $\tilde{{\rm A}}$ when $0<d<n$. We begin with a lemma which is probably well known (and certainly frequently assumed), but which we could not find in a suitable form in the literature (see [Ty, Theorem 2.5.1] for a similar result). Let ${\cal F} $ be a bounded set of non-stable bundles of rank $n$ and degree $d$. Then there exists a finite number of families of bundles of rank $n$ over $X$, parametrised by varieties $V_{\alpha}$, including representatives of all bundles in the given set (up to isomorphism). For $v\in V_{\alpha}$, let $E_v$ denote the corresponding bundle over $X$, and let $n_{\alpha,v}$ denote the dimension of the closure of the set $\{ w\in V_{\alpha}|E_w\cong E_v\}$ in $V_{\alpha}$. Write$$m_{\alpha}=\min\{n_{\alpha,v}|v\in V_{\alpha}\},\ \ \ \ p=\max_{\alpha}\{\dim V_{\alpha}-m_{\alpha}\}.$$In these circumstances, we shall say that ${\cal F}} \newcommand{\cg}{{\cal G}$ {\it depends on at most $p$ parameters}. \begin{lema}$\!\!\!${\bf .}~ \label{311} {\it Any bounded set ${\cal F}$ of non-stable bundles of rank $n$ depends on at most $n^2(g-1)$ parameters.} \end{lema} \begin{rema}$\!\!\!${\bf .}~\rm \label{32} i) Since stable bundles of rank $n$ and degree $d$ depend on precisely $n^2(g-1)+1$ parameters, this means that for counting problems we can assume that the dimension of any bounded family of vector bundles of rank $n$ is at most $n^2(g-1)+1$. ii) If $g=1$, Lemma 4.1 is not true. Actually there are \lq\lq more'' unstable than stable bundles in this case (see [A]). \end{rema} {\it Proof of Lemma 4.1:} If $E$ is a non-stable vector bundle of rank $n$ then there exists a filtration $$ 0=E_0\subset E_1\subset E_2 \subset ...\subset E_r=E $$ with $E_i/E_{i-1} $ stable and $\mu (E_i/E_{i-1} ) \leq \mu (E_{i- 1}/E_{i-2})$. For $E\in{\cal F}} \newcommand{\cg}{{\cal G}$, the ranks and degrees of the $E_i$ can take only finitely many values, so we can suppose these ranks and degrees are all fixed. Let $${\rm rk}\,(E_i/E_{i-1}) = n_i, \ \ \ \ \deg (Hom(E_j/E_ {j- 1},E_i/E_{i-1})) = d_{j,i}$$ and let $\beta _i$ be the minimum number of parameters on which the set of bundles which can occur as $E_i$ in the above filtration depends. From the exact sequence $$ 0\rightarrow E_1\rightarrow E_2\rightarrow E_2/E_1\rightarrow 0 $$ we have that $$\beta _2 \leq n^2_1(g-1)+1 +n^2_2(g-1)+1$$if $h^1(Hom(E_2/E_1,E_1))=0$ for all $E_1$, $E_2/E_1$ and $$\beta _2 \leq n^2_1(g- 1)+1 +n^2_2(g-1)+1 +\max \{h^1(Hom(E_2/E_1,E_1))\}-1$$otherwise, since $E_1$ and $E_2/E_1$ are stable. In the first case, clearly$$\beta _2 \leq (n_1+n_2)^2(g-1).$$In the second, since $Hom(E_2/E_1,E_1)$ is semistable and $d_{2,1}\geq0$, we have by Clifford's theorem $$h^0(Hom(E_2/E_1, E_1)) \leq \frac{d_{2,1}}{2} + n_1n_2.\eqno(2)$$ So by Riemann-Roch $h^1(Hom(E_2/E_1,E_1)) \leq n_1n_2g - \frac{d_{2,1}}{2}$. Therefore $$ \begin{array}{lll} \beta _2& \leq &(n_1^2+n_2^2)(g-1)+1 + n_1n_2g - \frac{d_{2,1}}{2} \\ &=&(n_1+n_2)^2(g-1)+1 -n_1n_2(g-2) -\frac{d_{2,1}}{2}\\ &\leq &(n_1+n_2)^2(g-1) \end{array} $$unless $g=2$ and $d_{2,1}=0$. In the exceptional case, the left-hand side of (2) is $0$ unless $E_2/E_1\cong E_1$, when it is $1$; so the inequality can be improved unless $E_1$ and $E_2/E_1$ are isomorphic line bundles. But the extensions of the required form in which $E_1$ and $E_2/E_1$ are isomorphic line bundles depend on$$2g-1\leq4(g-1)$$parameters. This completes the proof for $r=2$. For $r\geq3$, we proceed by induction on $r$. The same argument as above gives $$\beta _r \leq\beta_{r-1}+n_r^2(g-1)+1 +\max\{ h^1(Hom(E_r/E_{r-1},E_{r-1}))\} -1$$(unless $h^1(Hom(E_r/E_{r-1},E_{r-1}))$ is always zero, in which case there is a better estimate as above). Now \begin{eqnarray*}h^1(Hom(E_r/E_{r-1},E_{r- 1}))&\leq&\sum_{i=1}^{r- 1}h^1(Hom(E_r/E_{r-1},E_i/E_{i- 1}))\\&\leq&\sum_{i=1}^{r-1}(n_in_rg- \frac{d_{r,i}}{2})\end{eqnarray*} by Clifford's Theorem and Riemann-Roch. So$$\beta _r \leq\beta_{r-1}+n_r^2(g- 1)+2\sum_{i=1}^{r-1}n_in_r(g-1),$$and the result follows from the inductive hypothesis.\hfill$\diamondsuit$ \bigskip We are now ready to prove Theorem $\tilde{{\rm A}}$ when $0<d<n$. \begin{guess}$\!\!\!${\bf .}~ \label{33} If $0<d<n$ and $\wnt$ is non-empty, then it is irreducible. \end{guess} {\it Proof:} By Proposition 3.1 and Remark 3.2, any point of $\wnt $ has a representative $E$ of the form (1) with $h^0(F^*)=0$. For fixed rank and degree, the set $ \{ F | h^0(F^*) =0\}$ is bounded. It follows by a standard argument (due originally to Serre, see for example [A, Theorem 2]) that there is an irreducible family which includes representatives of all such bundles. The condition $h^0(F^*)=0$ defines an open subfamily, parametrised by an irreducible variety $Y$. The required extensions are then parametrised by a projective bundle over $Y$, and those for which $E$ is semistable by an open subset of the total space of this bundle. This subset is again irreducible and maps onto $\wnt$. So $\wnt$ is irreducible.\hfill$\diamondsuit$ \bigskip Next we prove Theorem A for $0<d<n$. \begin{guess}$\!\!\!${\bf .}~ \label{34} If $0<d<n$ and ${\cal W}^{k-1}_{n,d}$ is non-empty, then it is irreducible of dimension $\rho^{k-1}_{n,d}$. Moreover ${\rm Sing}\,{\cal W}^{k-1}_{n,d} ={\cal W}^{k}_{n,d}$. \end{guess} {\it Proof :} Suppose ${\cal W}^{k-1}_{n,d} $ is not empty. Since ${\cal W}^{k-1}_{n,d}$ is an open subset of $\wnt$, it is irreducible. From 1.1 we know that $\rho^{k-1}_{n,d} \leq $dim ${\cal W}^{k-1}_{n,d}$. Given $n$, $d$, $k$, let ${\cal S} $ be the set of all possible extensions $$0\rightarrow {\cal O}^k \rightarrow E \rightarrow F \rightarrow 0$$with $h^0(F^*)=0$, ${\rm rk}\, F=n-k$ and $\deg F=d$, such that $E$ does not have trivial summands. From Remark 4.2(i) and Propositions 3.1 and 3.4 we obtain \begin{eqnarray*} \dim{\cal W}^{k-1}_{n,d} & \leq &{\rm the\ number\ of\ parameters\ on\ which\ }{\cal S}\ {\rm depends}\\ &\leq&(n-k)^2(g- 1)+1 +k(d+(n-k)g-n)\\ &=&\rho^{k-1}_{n,d} \end{eqnarray*} Therefore, $\dim{\cal W}^{k-1}_{n,d} = \rho^{k-1}_{n,d}$. To see that ${\rm Sing}\, {\cal W}^{k-1}_{n,d} = {\cal W}^{k}_{n,d}$, note first that, since $\rho^{k-1}_{n,d}>\rho_{n,d}^k$, $${\cal W}^{k-1}_{n,d}\neq{\cal W}^{k}_{n,d}.$$ Now let $E \in {\cal W}^{k-1}_{n,d} -{\cal W}^{k}_{n,d}$, so that $H^0(E) \cong H^0({\cal O}^k)$. Since $$ \begin{array}{lll} H^0(E)\otimes H^0(E^*\otimes K)& \cong &H^0({\cal O}^k)\otimes H^0(E^*\otimes K)\\ &\cong &H^0({\cal O}^k \otimes E^*\otimes K)\\ &\hookrightarrow& H^0(E\otimes E^*\otimes K), \end{array} $$ the Petri map is injective. So, by 1.3, ${\cal W}^{k-1}_{n,d}$ is smooth at $E$. Since ${\cal W}^{k}_{n,d}\subset{\rm Sing}\,{\cal W}^{k-1}_{n,d}$ by 1.3, we have ${\rm Sing}\,{\cal W}^{k-1}_{n,d} = {\cal W}^{k}_{n,d}$ as required.\hfill$\diamondsuit$ Theorems 4.3 and 4.4 complete the proof of Theorems A and $\tilde{{\rm A}}$ when $0<d<n$. The cases $d=0$ and $d=n$ will be covered in \S\S 7, 8. We finish this section with \begin{coro}$\!\!\!${\bf .}~ If $(n-k,d)=1$ and ${\cal W}^{k-1}_{n,d}$ is non-empty, then there is a dominant rational map $g: Grass_k({\cal R}^1_p({\cal U}^*)) --\rightarrow {\cal W}^{k-1}_{n,d},$ where ${\cal U} $ is the universal bundle over $X\times {\cal M}(n-k,d)$ and $p$ the projection to ${\cal M}(n- k,d)$. \end{coro} {\it Proof :} By Lemma 4.1 and the proof of Theorem 4.4, the stable bundles $E$ constructed from non-stable $F$ belong to a proper subvariety of ${\cal W}^{k-1}_{n,d}$. The corollary now follows from the proofs of Proposition 3.4 and Theorem 4.4. \hfill$\diamondsuit$ \bigskip \renewcommand{\thesection}{\S\arabic{section}}\section{A criterion for non-emptiness}\renewcommand{\thesection} {\arabic{section}} In this section we will give the setting that we need to prove Theorems B and $\tilde{\rm B}$ for $0<d<n$. More precisely, we shall give a criterion for the non-emptiness of ${\cal W}^{k-1}_{n,d}$ by estimating the number of conditions on an extension (1) which are required for $E$ to be non-stable. Assume that $0<d<n$ and let $F$ be a stable bundle of rank $i$ and degree $d$. Let $$ \xi : 0\rightarrow {\cal O}^k \rightarrow E \rightarrow F \rightarrow 0$$ be an extension of $F$ by ${\cal O}^k$ such that $E$ does not have trivial summands. By the proof of Theorem 3.3, such $\xi$ exist if and only if $k+i=n\leq d+ig.$ If $E$ is non-stable, then it has a stable quotient bundle $H$ of rank $s<n$ and degree $d'$ such that $$\mu (H) \leq \mu (E). \eqno(3)$$ This fits in the following diagram: $$ \begin{array}{ccccccccc} 0&\rightarrow &{\cal O}^k&\rightarrow &E&\rightarrow &F&\rightarrow &0 \\ &&\downarrow&&\downarrow&&\downarrow&&\\ 0&\rightarrow &M&\rightarrow &H&\rightarrow &H_1&\rightarrow &0 \\ &&\downarrow&&\downarrow&&\downarrow&&\\ &&0&&0&&0&& \end{array}\eqno(4) $$ where $M$ is the image of ${\cal O}^k \rightarrow E\rightarrow H$. Note that $M\not=0$; otherwise there would exist a non-zero homomorphism $g:F\rightarrow H$. Since both are stable, $\mu (F) \leq \mu (H)$. Since $\mu (E) < \mu (F)$, this contradicts (3). Moreover, since $M$ has non-negative degree and $H$ is stable, $\deg H\geq 0$ Since $M$ is generated by its global sections, it must be trivial. Otherwise there would exist a section of $H$ generating a line bundle of positive degree; in conjunction with (3), this contradicts the stability of $H$. For the same reason, $H_1$ must be torsion-free (and hence locally free). One can now complete diagram (4) as follows $$ \begin{array}{ccccccccc} &&0&&0&&0&&\\ &&\downarrow&&\downarrow&&\downarrow &&\\ 0&\rightarrow &{\cal O}^{n-s-l}&\rightarrow &G&\rightarrow &G_1^l&\rightarrow &0 \\ &&\downarrow&&\downarrow&&\downarrow&&\\ 0&\rightarrow &{\cal O}^k&\rightarrow &E^n&\rightarrow &F^i&\rightarrow &0 \\ &&\downarrow&&\downarrow&&\downarrow&&\\ 0&\rightarrow &{\cal O}^{s-m}&\rightarrow &H^s&\rightarrow &H_1^m&\rightarrow &0 \\ &&\downarrow&&\downarrow&&\downarrow&&\\ &&0&&0&&0&& \end{array}\eqno(5) $$(where the superscripts denote the ranks of the various bundles). Note that $m>0$; otherwise $H$ would be trivial and $E$ would have a trivial summand. Also $l>0$; otherwise$$\deg H=\deg H_1=\deg F=d,$$contradicting (3). Note that the existence of the top sequence in (5) implies that the $k$-tuple of elements of $H^1(F^*)$ defining $\xi$ maps under the surjective homomorphism $H^1(F^*) \rightarrow H^1(G_1)$ to a $k$-tuple of which at most $(n-s-l) $ components are linearly independent. Since, by Riemann-Roch,$$h^1(G_1^*)\geq d-d'+l(g- 1),$$this rank condition defines a subvariety $Z$ in $\bigoplus^k H^1(F^*)$ with $$ {\rm codim}\,Z \geq (s-m)(d-d'+lg -n+s).\eqno(6)$$ On the other hand, the stability of $F$ implies that every quotient bundle of $H_1$ has slope greater than every subbundle of $G_1$, and hence that $h^0(H_1^*\otimes G_1)=0$. So$$h^1(H_1^*\otimes G_1)=ld'-m(d-d')+lm(g-1).$$Since $F$ varies in a bounded set, so do $G_1$ and $H_1$ (note that $d-d'>0$ by (3)); so, by Lemma 4.1, the non-trivial extensions occuring in the right-hand column of (5) depend on at most $ l^2(g-1) +1 +m^2(g-1) +1 + ld'-m(d-d')+lm(g-1) -1$ \hfill$=\dim{\cal M}(i,d) + ld' -m(d-d')-lm(g-1)$\ \ \ \ (7) \noindent parameters. \begin{propo}$\!\!\!${\bf .}~ \label{41} Let $0<d<n$. If $$(s-m)(d-d'+lg -n+s) > ld'-m(d-d')-lm(g- 1)$$for all possible choices of $s$, $d'$, $m$ and $l$, then ${\cal W}^{k-1}_{n,d} $ is non-empty. \end{propo} {\it Proof:} By (7), the general element $F$ of $\cm(i,d)$ admits families of extensions $0\rightarrow G_1\rightarrow F\rightarrow H_1\ra0$ as above depending on at most$$ld'-m(d- d')-lm(g-1)$$parameters. If the inequality holds, it follows from (6) that there exists a non-empty open set of extensions $\xi$ for which no diagram (5) exists. If this holds for all possible choices of $s$, $d'$, $m$ and $l$ (of which there are finitely many), then the general extension $\xi$ must define a stable bundle $E$.\hfill$\diamondsuit$ \bigskip We will use Proposition 5.1 to prove Theorems B and $\tilde{{\rm B}}$ for $0<d<n$. In view of Theorem 3.3, it is sufficient to show that the inequality of Proposition 5.1 holds whenever the numerical conditions needed for (5) to exist hold. For convenience, we restate these conditions now. In the first place, (3) can be stated as $$sd-nd' \geq 0. \eqno(a)$$ The stability of $F$ implies that $$ (l+m)d' -md >0. \eqno(b)$$ Since $H$ is stable, we have from the proof of Theorem 3.3 $$d' -s +mg \geq 0. \eqno(c)$$ Finally the inequality in Proposition 5.1 can be written as $$m(n-s-l)-d'(l+s)+s(d+lg- n+s)>0. \eqno(d)$$ In the next section, we shall prove that $(a)$, $(b)$ and $(c)$ imply $(d)$, thus completing the proofs. \begin{rema}$\!\!\!${\bf .}~\rm The necessary condition $$n\leq d+(n-k)g $$ of Theorems B and $\tilde{\rm B}$ does not enter the calculation explicitly. In fact this inequality is a consequence of the hypotheses of Proposition 6.1 below. \end{rema} \bigskip \renewcommand{\thesection}{\S\arabic{section}}\section{Proof of the inequality}\renewcommand{\thesection}{\arabic{section}} Our object in this section is to prove \begin{propo}$\!\!\!${\bf .}~ Suppose $(a)$, $(b)$ and $(c)$ hold with $$0<d<n,\ \ 0<s\leq n-l\ \ {\rm and}\ \ l>0.$$ Then $(d)$ holds. \end{propo} It will be helpful for the proof to represent some of the data in a geometrical form. We do this as follows: \bigskip\bigskip \centerline{FIGURE 3} \bigskip \bigskip In this figure, we regard $n$, $d$, $s$ and $l$ as fixed and $m$, $d'$ as variables. The curve $q$ is (a branch of ) the hyperbola $$(l+m)d'-md=0$$ defining the inequality $(b)$ and has the form indicated since $l>0.$ The lines $\ell _a $ and $\ell _c $ defining the inequalities $(a)$ and $(c)$ depend on $s$, but the line $\ell $ joining $C$ (the intersection of $\ell_a$ and $\ell_c$) to $(0,0)$ has equation $$(n-d)d' =dmg$$ which is independent of $s$. The shaded region is the region where $(a),(b)$ and $(c)$ are all satisfied. \begin{lema}$\!\!\!${\bf .}~ Suppose the hypotheses of Proposition 6.1 hold. Then $$(n-d)s\geq n(n-d- lg).\eqno{(*)}$$\end{lema} {\it Proof:} Note that, for any $s$, the line $\ell _c$ has slope $-g <0.$ It follows that, if $(a),(b)$ and $(c)$ hold, then the point $C$ must lie above $q$. Now $\ell $ meets $q$ at $(0,0)$ and the point $D$ with coordinates $$m=\frac{n-d-lg}{g} , \ \ \ \ d'=\frac{d(n-d-lg)}{n-d}.$$ Since $\ell $ has positive slope, the $m$-coordinate of $C$ must be at least as great as that of $D$, i.e. $$\frac{(n-d)s}{ng} \geq \frac{n-d-lg}{g}.$$ Clearing denominators, this gives $(*)$. (Note that the coordinates of $D$ in this proof could be negative; this does not affect the argument.) \hfill$\diamondsuit$ {\it Proof of Proposition 6.1:} The value of the LHS of $(d)$ at $C$ is $$\frac{(n-d)s}{ng}(n-s-l)- \frac{ds}{n}(l+s) +s(d+lg-n+s).$$ A simple calculation shows that this is equal to $$\frac{s(g-1)}{ng}[s(n-d)-n(n-d-lg)+l(n-d)].$$ It follows at once from Lemma 6.2 that this is positive. In other words, $C$ lies below the line defining the inequality $(d)$, which has non-negative slope. So the whole region in which $(a)$, $(b)$ and $(c)$ all hold also lies below this line. \hfill$\diamondsuit$ We are now ready to state\begin{guess}$\!\!\!${\bf .}~ Theorems {\rm B} and $\widetilde{{\rm B}}$ hold for $0<d<n$.\end{guess} {\it Proof:} This follows from Theorem 3.3 and Propositions 5.1 and 6.1.\hfill$\diamondsuit$ \bigskip \renewcommand{\thesection}{\S\arabic{section}} \section{The case $\mu = 0 $}\renewcommand{\thesection} {\arabic{section}} \begin{guess}$\!\!\!${\bf .}~ Theorems {\rm A} and {\rm B} hold for $d=0$.\end{guess} {\it Proof:} For bundles of degree $0$, the existence of a section contradicts stability; so ${\cal W}^{k-1}_{n,d}$ is always empty. This gives Theorem B, and Theorem A holds trivially.\hfill$\diamondsuit$ \bigskip On the other hand, we have \begin{guess}$\!\!\!${\bf .}~ For $1\leq k\leq n,$ there exists a bijective morphism $$\widetilde{\cm}(n-k,0)\rightarrow \widetilde{{\cal W}} \newcommand{\cp}{{\cal P} }^{k- 1}_{n,0}.$$\end{guess} {\it Proof:} If $E$ is a semistable bundle of degree $0$ with $k$ independent sections, then by Proposition 3.1 we have an extension $$0\rightarrow {\cal O}^k \rightarrow E\rightarrow F\rightarrow 0.$$ So $E$ is S-equivalent to ${\cal O}^k \oplus F$ for some semistable bundle $F$ of rank $n-k$ and degree $0$. Hence the formula $[F] \rightarrow [{\cal O}^k \oplus F] $ defines a bijection from $\widetilde{\cm} (n-k,0)$ to $\widetilde{{\cal W}} \newcommand{\cp}{{\cal P} }^{k-1}_{n,0}.$ Since $\widetilde{\cm} (n-k,0)$ is a coarse moduli space, this is a morphism.\hfill$\diamondsuit$ \begin{guess}$\!\!\!${\bf .}~ Theorems $\widetilde{{\rm A}}$ and $\widetilde{{\rm B}}$ hold for $d=0$.\end{guess} {\it Proof:} By Theorem 3.3, $\widetilde{{\cal W}} \newcommand{\cp}{{\cal P}}^{k-1}_{n,0} $ is empty if $k>n$. The rest of Theorem $\widetilde{{\rm B}}$ now follows from Theorem 7.2, as does Theorem $\widetilde{{\rm A}}$ when we recall that $\widetilde{\cm} (n-k,0)$ is irreducible.\hfill$\diamondsuit$ \begin{rema}$\!\!\!${\bf .}~\rm Note that $$ \dim\widetilde{\cm} (n-k,0) =(n-k)^2 (g-1)+1 < \rho ^{k- 1}_{n,0}$$ if $n<(n-k)g$. This is no contradiction since the points of $\widetilde{{\cal W}} \newcommand{\cp}{{\cal P}}^{k-1}_{n,0} $ correspond to S-equivalence classes of bundles, not isomorphism classes. \end{rema} \bigskip\renewcommand{\thesection}{\S\arabic{section}}\section{The case $\mu = 1 $}\renewcommand{\thesection} {\arabic{section}} In this final section we prove our theorems for the case $d=n$. For stable bundles the key result is \begin{propo}$\!\!\!${\bf .}~ ${\cal W}} \newcommand{\cp}{{\cal P} ^{n-2}_ {n,n} $ is non-empty. \end{propo} {\it Proof:} Consider the extensions $$0\rightarrow {\cal O}} \newcommand{\qa}{{\cal Q} ^{n-1} \rightarrow E \rightarrow F \rightarrow 0,$$ where $F$ is a line bundle of degree $n$. Since $n\leq n+g,$ there exist extensions of this form for which $E$ has no trivial summands. If $E$ is non-stable, we have as in \S 5 a diagram $$ \begin{array}{ccccccccc} 0&\rightarrow &{\cal O}} \newcommand{\qa}{{\cal Q} ^{n-1}&\rightarrow &E&\rightarrow &F&\rightarrow &0 \\ &&\downarrow&&\downarrow&&\downarrow&&\\ 0&\rightarrow &M&\rightarrow &H&\rightarrow &H_1&\rightarrow &0 \\ &&\downarrow&&\downarrow&&\downarrow&&\\ &&0&&0&&0&& \end{array} $$ with $H$ stable and $\mu (H) \leq \mu (E) .$ If $H_1$ is a line bundle, then $H_1 \cong F$; so $$\deg H=\deg F +\deg M \geq d$$ and $\mu (H) > \mu (E)$, which is a contradiction. It follows that $H_1$ must be a torsion sheaf. If $\mu (H) <1,$ this contradicts the stability of $H$ just as in \S 5. However, if $\mu (H) =1$, it is possible for $H$ to have a section with a zero. This can happen only if $H={\cal O}} \newcommand{\qa}{{\cal Q} (x) $ for some $x \in X.$ Moreover, in this case, we cannot have $M={\cal O}} \newcommand{\qa}{{\cal Q} (x),$ since ${\cal O}} \newcommand{\qa}{{\cal Q} (x) $ is not generated by global sections, so our diagram must become $$ \begin{array}{ccccccccc} &&0&&0&&0&&\\ &&\downarrow&&\downarrow&&\downarrow&&\\ 0&\rightarrow &{\cal O}^{n- 2}&\rightarrow &G&\rightarrow &F(-x)&\rightarrow &0 \\ &&\downarrow&&\downarrow&&\downarrow&&\\ 0&\rightarrow &{\cal O}} \newcommand{\qa}{{\cal Q} ^{n-1}&\rightarrow &E&\rightarrow &F&\rightarrow &0 \\ &&\downarrow&&\downarrow&&\downarrow&&\\ 0&\rightarrow &{\cal O}&\rightarrow &{\cal O}} \newcommand{\qa}{{\cal Q} (x)&\rightarrow &{\cal O}} \newcommand{\qa}{{\cal Q} _x&\rightarrow &0 \\ &&\downarrow&&\downarrow&&\downarrow&&\\ &&0&&0&&0&& \end{array} $$ The existence of this diagram implies that the $(n-1)$-tuple of elements of $H^1(F^*)$ corresponding to the extension $$0\rightarrow {\cal O}} \newcommand{\qa}{{\cal Q} ^{n-1} \rightarrow E \rightarrow F \rightarrow 0$$ must become dependent in $H^1(F(-x)^*)$. Now $$h^1(F(-x)^*) = n+g-2 \geq n-1;$$ so this condition defines a subvariety of $\bigoplus^{n-1} H^1(F^*)$ of codimension $g>1$. Since $F(-x)$ depends on only one parameter, we can find an extension for which no such diagram exists.\hfill$\diamondsuit$ \begin{guess}$\!\!\!${\bf .}~ Theorems {\rm A} and {\rm B} hold for $d=n$.\end{guess} {\it Proof:} Proposition 3.1 remains true for stable bundles when $d=n$. The arguments of \S 4 therefore apply to prove Theorem A in this case. For Theorem B, it follows from Proposition 8.1 that ${\cal W}} \newcommand{\cp}{{\cal P} ^{k-1}_{n,n}$ is non- empty for $k\leq n-1$. On the other hand ${\cal W}} \newcommand{\cp}{{\cal P} ^{n-1}_{n,n}$ is certainly empty, since a bundle with $n$ independent sections is either trivial or has a section with a zero; when $d=n$, either possibility contradicts stability. Thus Theorem B holds when $d=n$.\hfill$\diamondsuit$ \bigskip We now turn to the semistable case. As in the case $d=0$, we obtain a result which is interesting in its own right. \begin{guess}$\!\!\!${\bf .}~ Let $S^nX$ denote the $n$th symmetric power of $X$. Then there exists a bijective morphism $S^nX \rightarrow \widetilde{{\cal W}} \newcommand{\cp}{{\cal P}}^{n- 1}_{n,n}.$ \end{guess} {\it Proof:} Let $E$ be a semistable bundle of rank and degree $n$ with $n$ independent sections. Since $E\not\cong {\cal O}} \newcommand{\qa}{{\cal Q} ^n,$ it must possess a section with a zero. Semistability then gives an extension $$0\rightarrow {\cal O}} \newcommand{\qa}{{\cal Q} (x) \rightarrow E \rightarrow E' \rightarrow 0,$$ where $E'$ is semistable of rank and degree $n-1$ and has $n- 1$ independent sections. It follows by induction that $E$ is S-equivalent to a bundle of the form ${\cal O}} \newcommand{\qa}{{\cal Q} (x_1)\oplus \dots \oplus {\cal O}} \newcommand{\qa}{{\cal Q} (x_n).$ The existence of the required morphism follows from the universal properties of $S^nX$ and $\cm (n,n)$.\hfill$\diamondsuit$ We need also \begin{propo}$\!\!\!${\bf .}~ For $k<n$, the point $[E] \in \widetilde{{\cal W}} \newcommand{\cp}{{\cal P}}^{k-1}_{n,n} $ determined by a semistable bundle $E$ lies in the closure of ${\cal W}} \newcommand{\cp}{{\cal P} ^{k-1}_{n,n}.$ \end{propo} {\it Proof:} For those bundles $E$ which can be expressed as extensions $$0\rightarrow {\cal O}^k \rightarrow E\rightarrow F\rightarrow 0,$$ we argue exactly as in Theorem 4.3. The remaining bundles are those which possess a section with a zero. We then have an extension $$0\rightarrow {\cal O}} \newcommand{\qa}{{\cal Q} (x) \rightarrow E\rightarrow F\rightarrow 0,$$ so that $E$ is S-equivalent to ${\cal O}} \newcommand{\qa}{{\cal Q} (x) \oplus F.$ We can suppose inductively that $[F]$ belongs to the closure of ${\cal W}} \newcommand{\cp}{{\cal P} ^{k-2}_{n-1,n-1}$. (Note that, in the case $n=2$, $F $ is a line bundle, so this is trivial. More generally, it is trivial whenever $k=1$, since then ${\cal W}} \newcommand{\cp}{{\cal P} ^{k-2}_{n-1,n-1}={\cm}(n-1,n-1)$.) It is therefore sufficient to prove the proposition when $E\cong {\cal O}} \newcommand{\qa}{{\cal Q} (x) \oplus F$ and $F \in {\cal W}} \newcommand{\cp}{{\cal P} ^{k-2}_{n-1,n-1}$. For this, we consider extensions $$0\rightarrow F \rightarrow E'\rightarrow {\cal O}} \newcommand{\qa}{{\cal Q} (x)\rightarrow 0,$$ Note that we have an inclusion ${\cal O}} \newcommand{\qa}{{\cal Q} \subset {\cal O}} \newcommand{\qa}{{\cal Q} (x) $ and that this section of ${\cal O}} \newcommand{\qa}{{\cal Q} (x)$ lifts to $E'$ if and only if the pull-back of the extension by this inclusion is trivial. We therefore have a family of such extensions parametrised by $$V=Ker [H^1({\cal O}} \newcommand{\qa}{{\cal Q} (x)^* \otimes F) \rightarrow H^1(F)].$$ Now, since $F$ is stable with $\mu(F) =1,$ $$h^1({\cal O}} \newcommand{\qa}{{\cal Q} (x)^* \otimes F) = (n-1)(g-1)$$and $$h^1(F)=h^0(F) -(n-1) +(n-1)(g-1) <(n-1)(g-1).$$ So $\dim V\geq1$ and there exist non-trivial extensions $$0\rightarrow F \rightarrow E'\rightarrow {\cal O}} \newcommand{\qa}{{\cal Q} (x)\rightarrow 0$$ with $h^0(E') \geq k$. Now suppose $E'$ is such an extension, and that it possesses a section with a zero. Since $F$ is stable, this cannot be a section of $F$, so it maps to a section of ${\cal O}} \newcommand{\qa}{{\cal Q} (x).$ The corresponding subbundle must map isomorphically to ${\cal O}} \newcommand{\qa}{{\cal Q} (x),$ splitting the extension. It follows that $V$ parametrises a family such that the general member has a subbundle ${\cal O}^k $ and therefore defines a point in the closure of ${\cal W}} \newcommand{\cp}{{\cal P} ^{k- 1}_{n,n} $, while the special member corresponding to $0 \in V $ is ${\cal O}} \newcommand{\qa}{{\cal Q} (x) \oplus F.$ Hence $[{\cal O}} \newcommand{\qa}{{\cal Q} (x) \oplus F]$ is in the closure of ${\cal W}} \newcommand{\cp}{{\cal P} ^{k-1}_{n,n} $ as required.\hfill$\diamondsuit$ We now have finally \begin{guess}$\!\!\!${\bf .}~ Theorems $\widetilde{{\rm A}}$ and $\widetilde{{\rm B}}$ hold for $d=n$.\end{guess} {\it Proof:} $\widetilde{{\cal W}} \newcommand{\cp}{{\cal P}} ^{k-1}_{n,n} $ is irreducible for $k=n$ by Theorem 8.3 and for $k<n$ by Theorem 8.2 and Proposition 8.4. On the other hand, the proof of Theorem 8.3 shows that no semistable bundle with $d=n$ can have more than $n$ independent sections.\hfill$\diamondsuit$

9511

alg-geom/9511018

en

https://arxiv.org/abs/alg-geom/9511018

alg-geom/9511018

Alexander Polishchuk

Alexander Polishchuk

Symplectic biextensions and a generalization of the Fourier-Mukai transform

AMSLaTeX. Missing package "comm" is inserted

A generalization of the Fourier-Mukai transform is proposed. The construction is based on analogy with the classical picture of representations of the Heisenberg group.

alg-geom

\section{Symplectic biextensions} Let $X$ be an abelian variety. A {\it biextension} of $X^2$ is a line bundle $L$ on $X^2$ together with isomorphisms \begin{align*} &L_{x+x',y}\simeq L_{x,y}\otimes L_{x',y},\\ &L_{x,y+y'}\simeq L_{x,y}\otimes L_{x,y'} \end{align*} --- this is a symbolic notation for isomorphisms $(p_1+p_2,p_3)^*L\simeq p_{13}^*L\otimes p_{23}^*L$ and $(p_1,p_2+p_3)^*L\simeq p_{12}^*L\otimes p_{13}^*L$ on $X^3$, satisfying some natural cocycle conditions (see e.g. \cite{Breen}). A {\it skew-symmetric biextension} of $X^2$ is a biextension $L$ of $X^2$ together with an isomorphism of biextensions $\phi:\sigma^*L\widetilde{\rightarrow} L^{-1}$, where $\sigma:X^2\rightarrow X^2$ is the permutation of factors, and a trivialization $\Delta^*L\simeq\O_X$ of $L$ over the diagonal $\Delta:X\rightarrow X^2$ compatible with $\phi$. Every biextension $L$ of $X^2$ induces a homomorphism $\psi_L:X\rightarrow\hat{X}$ which is given on the level of points by $x\mapsto L_{x\times X}$. If $L$ is skew-symmetric, then $\widehat{\psi_L}=-\psi_L$. It is easy to see that $\psi_L=\psi_{L'}$ if and only if $L$ and $L'$ are isomorphic. Moreover, any skew-symmetric homomorphism $\psi:X\rightarrow\hat{X}$ defines a skew-symmetric biextension by the formula $L(\psi)=(\psi\times\operatorname{id})^*\cal P$, where $\cal P$ is the normalized Poincar\'e line bundle on $\hat{X}\times X$, such that $\psi_{L(\psi)}=\psi$. A skew-symmetric biextension $L$ is called {\it symplectic} if $\psi_L$ is an isomorphism. Let $Y\subset X$ be an abelian subvariety. Then $Y$ is called {\it isotropic} with respect to $L$ if there is an isomorphism of skew-symmetric biextensions $L|_{Y\times Y}\simeq\O_{Y\times Y}$. This is equivalent to the condition that the composition $$Y\stackrel{i}{\rightarrow} X\stackrel{\psi_L}{\rightarrow}\hat{X} \stackrel{\hat{i}}{\rightarrow} \hat{Y}$$ is zero. An isotropic subvariety $Y\subset X$ is called {\it lagrangian} if the morphism $Y\rightarrow \widehat{X/Y}$ induced by $\psi_L$ is an isomorphism. A skew-symmetric biextension $L$ of $X^2$ is called {\it quasi-split} if there exists a lagrangian subvariety in $X$. One can see easily that such a biextension is necessarily symplectic. The simplest example of an abelian variety with a symplectic biextension is $X=\hat{A}\times A$ for any abelian variety $A$ with the biextension $L_A=P\otimes\sigma^*P^{-1}$ where $$P=p_{14}^*\cal P\in\operatorname{Pic}(\hat{A}\times A\times \hat{A}\times A),$$ $\cal P$ is the Poincar\'e line bundle on $A\times\hat{A}$. A symplectic biextension is called {\it split} if it is isomorphic to this one. Below we show how to construct all quasi-split symplectic biextensions. Let $Y\subset X$ be a lagrangian subvariety. Let us denote $A=X/Y\simeq\hat{Y}$ so that there is an exact sequence $$0\rightarrow \hat{A}\rightarrow X\stackrel{p}{\rightarrow} A\rightarrow 0,$$ such that $\psi_L|_{\hat{A}}=\hat{p}$. The projection $p$ splits up to isogeny, that is there exists a homomorphism $s:A\rightarrow X$ such that $ps=n\operatorname{id}_A$. \begin{lem} One can always choose a section $s:A\rightarrow X$ as above such that \break $\hat{s}\psi_L s=0$. \end{lem} \noindent {\it Proof}. Start with any $s$ as above and then replace $n$ by $2n^2$, and $s$ by $2ns-\hat{s}\psi_Ls$. \qed\vspace{3mm} Choose $s:A\rightarrow X$ as in lemma, and let $\pi=(\operatorname{id},s):\hat{A}\times A\rightarrow X$ be the corresponding isogeny. Then since $L|_{\hat{A}^2}$ and $(s\times s)^*L$ are trivial, it is easy to see that $\pi^*L\simeq L_A^{\otimes n}=p_{14}^*\cal P^{\otimes n}\otimes p_{23}^*\cal P^{\otimes -n}$. Thus, every quasi-split symplectic biextension descends from the power of the split one. It remains to determine which subgroups $\ker(\pi)\subset\hat{A}\times A$ can occur. Let $P$ be any biextension of $X'\times X''$. Then the restrictions of $P^{\otimes n}$ on $X'_n\times X''$ and $X'\times X''_n$ are canonically trivialized but these trivializations differ over $X'_n\times X''_n$ by a bilinear morphism $e_n(P):X'_n\times X''_n\rightarrow{\Bbb G}_m$. In the case of the Poincar\'e line bundle $\cal P$ over $\hat{A}\times A$ this construction gives a canonical perfect pairing $e_n:\hat{A}_n\times A_n\rightarrow{\Bbb G}_m$. In our situation the canonical trivializations of $L_A^{\otimes n}$ over $(\hat{A}_n\times A_n)\times (\hat{A}\times A)$ and $(\hat{A}\times A)\times (\hat{A}_n\times A_n)$ differ over $(\hat{A}_n\times A_n)^2$ by a bilinear morphism $e_n(L_A):(\hat{A}_n\times A_n)^2\rightarrow{\Bbb G}_m$ which is given by the formula \begin{equation} e_n(L_A)((\xi,x),(\xi',x'))=e_n(\xi,x')e_n(\xi',x) \end{equation} where $x,x'\in A_n$, $\xi,\xi'\in\hat{A}_n$. By definition $\ker{\pi}$ is the graph of a morphism $\phi:A_n\rightarrow \hat{A}_n$ induced by $s$. Now the biextension $L_A^{\otimes n}$ descends to $X$ if and only if there exist trivializations (as a biextension) of $L_A^{\otimes n}$ over $\ker{\pi}\times (\hat{A}\times A)$ and $(\hat{A}\times A)\times\ker{\pi}$ which are compatible over $(\ker{\pi})^2$. Since such trivializations are unique they coincide with the restrictions of the canonical trivializations above. Hence, the descent condition is that $\ker(\pi)$ is isotropic with respect to $e_n(L_A)$ which means that $\phi:A_n\rightarrow\hat{A}_n$ is skew-symmetric with respect to $e_n$, that is $\widehat{\phi}=-\phi$. Thus, any quasi-split symplectic biextension arises from a pair $(A,\phi)$, where $\phi:A_n\rightarrow\hat{A}_n$ is a skew-symmetric morphism, as described above. It is easy to see that if we change $\phi$ by $\phi+f_n$ where $f_n$ is the restriction of a symmetric homomorphism $f:A\rightarrow\hat{A}$ to $A_n$ (then $f_n$ is automatically skew-symmetric), then we get isomorphic symplectic biextensions---this corresponds to a change of an isotropic morphism $s:A\rightarrow X$. Also, one can change $n$ by $nm$ and $\phi$ by the composition $$A_{nm}\stackrel{m}{\rightarrow}A_n\stackrel{\phi}{\rightarrow}\hat{A}_n\rightarrow \hat{A}_{nm},$$ so that the corresponding symplectic biextension will be the same. However, this doesn't exhaust examples of pairs $(A,\phi)$ giving isomorphic biextensions. For example, it is easy to see that $A/\ker(\phi)=s(A)\subset X$ is a lagrangian subvariety in $X$, $X/s(A)\simeq \hat{A}/\phi(A_n)$, and the biextension associated with the pair $(A,\phi)$ corresponds also to the pair $(\hat{A}/\phi(A_n),\psi)$ where $\psi$ is the composition $$\psi:(\hat{A}/\phi(A_n))_n\rightarrow \phi(A_n)\stackrel{\phi^{-1}}{\rightarrow} A_n/\ker(\phi)\rightarrow (A/\ker(\phi)_n).$$ These considerations lead to the following theorem. \begin{thm}\label{pairs} Let $L$ be a symplectic biextension of $X^2$. For any lagrangian subvariety $Y\subset X$ there exists a lagrangian subvariety $Z\subset X$ such that $Y\cap Z$ is finite. Any pair of lagrangian subvarieties $(Y,Z)$ in $X$ such that $Y\cap Z$ is finite, is isomorphic to the pair $(\hat{A},A/\ker(\phi))$ in $\hat{A}\times A/(\phi,\operatorname{id})(A_n)$ with its canonical symplectic biextension for some abelian variety $A$ and a skew-symmetric homomorphism $\phi:A_n\rightarrow\hat{A}_n$. \end{thm} \noindent {\it Proof} . The first assertion is clear. To prove the second we should start with the lagrangian subspace $Y\subset X$ in the above argument and choose a splitting of $p:X\rightarrow X/Y$ up to isogeny which factors through $Z$. More precisely, let $f:Z\rightarrow X/Y$ be the restriction of $p$ to $Z$. Choose an isogeny $g:X/Y\rightarrow Z$ such that $fg=n\operatorname{id}_{X/Y}$. Then the composition of $g$ with the embedding of $Z$ in $X$ gives a lagrangian morphism $s:X/Y\rightarrow X$ such that $ps=n\operatorname{id}_{X/Y}$. Now we get an isogeny $\hat{A}\times A\rightarrow X$ as above (where $A=X/Y$) such that $Y$ and $Z$ are the images of $\hat{A}$ and $A$ respectively, which finishes the proof. \qed\vspace{3mm} Let us give an example of a quasi-split symplectic biextension which is not split. Let $A$ be a principally polarized abelian variety with $\operatorname{End}(A)={\Bbb Z}$. Then there is a symplectic isomorphism $\phi_n:A_n\rightarrow\hat{A}_n$ such that for every symmetric morphism $f:A\rightarrow \hat{A}$ the corresponding morphism $f|_{A_n}$ is proportional to $\phi_n$. Now if $\dim(A)>1$ we can choose a symplectic morphism $\phi:A_n\rightarrow\hat{A}_n$ which is not proportional to $\phi_n$. It is easy to see that the corresponding symplectic biextension of $X^2$, where $X=\hat{A}\times A/(\phi,\operatorname{id})(A_n)$, is not split. \section{Representations of the Heisenberg groupoid} Let $X$ be an abelian variety, $L$ be a symplectic biextension of $X^2$. Throughout this section we assume that there exists a biextension $P$ of $X^2$ such that $L\simeq P\otimes\sigma^*P^{-1}$ (an isomorphism of skew-symmetric biextensions). This is equivalent to the condition $\psi_L=f-\hat{f}$ for some $f:X\rightarrow \hat{X}$. For example, the quasi-split biextension associated with a pair $(A,\phi)$, where $\phi:A_n\rightarrow\hat{A}_n$ and $n$ is odd, satisfies this condition. \begin{defi} The Heisenberg groupoid $H(X)=H(X,P)$ is the stack of monoidal groupoids such that $H(X)(S)$ for a scheme $S$ over $k$ is the monoidal groupoid generated by the central subgroupoid ${\cal P}ic(S)$ of ${\Bbb G}_m$-torsors on $S$ and the symbols $T_x$, $x\in X(S)$ with the composition law $$T_x\circ T_{x'}= P_{x,x'} T_{x+x'}.$$ In other words, objects of $H(X)(S)$ are pairs $(M,x)$ where $M$ is a line bundle over $S$, $x\in X(S)$. A morphism $(M,x)\rightarrow(M',x')$ exists only if $x=x'$ and is given by an isomorphism $M\rightarrow M'$. The composition law is defined by the formula $$(M,x)\circ (M',x')=(P_{x,x'}\otimes M\otimes M', x+x').$$ Denoting $T_x=(\O_S,x)$ we recover the above relation. \end{defi} If we replace $P$ by $P'=P\otimes\Lambda(M)$ for some line bundle $M$ on $X$ trivialized along the zero section, where $\Lambda(M)=(p_1+p_2)^*M\otimes p_1^*M^{-1}\otimes p_2^*M^{-1}$ (see e.g. \cite{Breen}) we get an equivalent Heisenberg groupoid. The equivalence $H(X,P)\rightarrow H(X,P')$ is defined by the functor which is the identity on ${\cal P}ic(S)$ and sends $T_x$ to $M_x^{-1}T_x$. Since any symmetric biextension of $X^2$ has form $\Lambda(M)$ this shows that up to a non-unique equivalence the Heisenberg groupoid doesn't depend on a choice of $P$ such that $L=P\otimes\sigma^*P$. \begin{rem} One can see easily that the Heisenberg groupoid can be considered as an extension of the group scheme $X$ by the stack of line bundles in the sense of Deligne (see \cite{Des}), namely, we associate to each point $x\in X(S)$ the trivial gerb of line bundles, and the composition is given by the formula above. \end{rem} The Heisenberg groupoid $H(\hat{A}\times A)$ corresponding to a split biextension is generated by the Picard subgroupoid ${\cal P}ic$ and symbols $T_x$, $T_y$ where $x\in \hat{A}$, $y\in A$ with the following defining relations: \begin{align*} &T_xT_{x'}=T_{x+x'},\\ &T_yT_{y'}=T_{y+y'},\\ &T_y T_x=\langle x,y\rangle T_x T_y. \end{align*} Is is easy to see (see e.g. \cite{Weilrep}) that the map $T_y\mapsto t_y^*$, $T_x\mapsto \cdot\otimes\cal P_x$ defines an action of $H(\hat{A}\times A)$ on ${\cal D}^b(A)$, where $t_y:A\rightarrow A$ is the translation by $y\in A$, $\cal P_x=\cal P|_{x\times A}$ for $x\in\hat{A}$. Below we construct an analogous action for an arbitrary isotropic subvariety of an abelian variety with a symplectic biextension. Let $Y\subset X$ be an isotropic subvariety. Then $P|_{Y\times Y}$ has a natural structure of a symmetric biextension. \begin{defi} A pair $(Y,\a)$, where $Y$ is an isotropic abelian subscheme of $X$ with respect to $L$ and $\a$ is a line bundle on $Y$ with fixed trivialization along the zero section, is called isotropic if an isomorphism of symmetric biextensions of $Y\times Y$ is given: $$\Lambda(\a)\simeq P|_{Y\times Y}$$ which can be written symbolically as $P_{y,y'}=\a_{y+y'}\a_y^{-1}\a_{y'}^{-1}$ for $y,y'\in Y$. \end{defi} For any isotropic subvariety $Y\subset X$ there exists $\a$ such that the pair $(Y,\a)$ is isotropic. \begin{defi} For an isotropic pair $(Y,\a)$ we define ${\cal F}(Y,\a)$ as the category of pairs $(\AA,a)$ where $\AA\in{\cal D}^b(X)$, $a$ is an isomorphism in ${\cal D}^b(Y\times X)$: \begin{equation}\label{Schrsp} a:(ip_1+p_2)^*\AA\widetilde{\rightarrow} P^{-1}|_{Y\times X} \otimes p_1^*\a^{-1}\otimes p_2^*\AA \end{equation} where $i:Y\hookrightarrow X$ is the embedding, such that $(e\times\operatorname{id})^*a=\operatorname{id}$. This isomorphism can be written symbolically as follows: $$a_{y,x}:\AA_{y+x}\widetilde{\rightarrow}P^{-1}_{y,x}\a^{-1}_y\AA_x$$ where $y\in Y$, $x\in X$. These data should satisfy the following cocycle condition: $$a_{y_1+y_2,x}=a_{y_2,x}\circ a_{y_1,y_2+x}:\AA_{y_1+y_2+x} \rightarrow P^{-1}_{y_1,x+y_2} P^{-1}_{y_2,x}\a^{-1}_{y_1}\a^{-1}_{y_2}\AA_x \simeq P^{-1}_{y_1+y_2,x}\a^{-1}_{y_1+y_2}\AA_x,$$ or in standard notation $$(p_1+p_2,p_3)^*a=(p_2,p_3)^*a\circ (p_1,ip_2+p_3)^*a$$ in ${\cal D}^b(Y\times Y\times X)$. The morphisms between such pairs are morphisms between the corresponding objects in ${\cal D}^b(X)$ commuting with the isomorphisms in (\ref{Schrsp}). \end{defi} It is easy to see that the category ${\cal F}(Y,\a)$ is equivalent to ${\cal D}^b(X/Y)$ provided the projection $p:X\rightarrow X/Y$ has a section $s:X/Y\rightarrow X$. However, in general this is not true: one encounters some twisted versions of ${\cal D}^b(X/Y)$ considered in the next section. There is a natural action of the Heisenberg groupoid $H(X)$ on the category ${\cal F}(Y,\a)$ such that an object $(M,x)$ acts by the functor $$\AA\mapsto M\otimes P|_{X\times x}\otimes t_x^*(\AA).$$ In the case $X=\hat{A}\times A$, $Y=\hat{A}\subset X$ this action coincides with the action of $H(\hat{A}\times A)$ on ${\cal D}^b(A)\simeq{\cal F}(\hat{A})$ mentioned above. By analogy with the classical Heisenberg group it is natural to ask when these representations are irreducible in some sense. More precisely, for the construction of Weil representation it is relevant to know that all intertwining operators from Schr\"odinger representation to itself are proportional to the identity. As shown in \cite{Weilrep} certain analogue of this property holds for the action of $H(\hat{A}\times A)$ on ${\cal D}^b(A)$. One can treat the case of an arbitrary lagrangian subvariety similarly, however, we don't need this result. \section{Modules over Azumaya algebras} We begin this section by briefly recalling the various ways to speak about the category of coherent modules over a scheme $S$ "twisted" by an element $e\in H^2(S,{\Bbb G}_m)$: via Cech cocycles, gerbs, and Azumaya algebras. The simplest way to define such a category is to fix an open covering $(U_i)$ of $S$ (say, in flat topology) such that $e$ is represented by a Cech cocycle $\a_{ijk}\in\O^*(U_{ijk})$ where $U_{ijk}=U_i\times_S U_j\times_S U_k$. Then we define $\operatorname{Coh}(S,\a)$ as the category of collections $({\cal F}_i)$ of coherent sheaves on $U_i$ together with a system of isomorphisms $f_{ij}:{\cal F}_i\rightarrow{\cal F}_j$ over $U_{ij}=U_i\times_S U_j$ (such that $f_{ji}=f_{ij}^{-1}$) satisfying the twisted cocycle condition: $f_{jk}f_{ij}=\a_{ijk}f_{ik}$ over $U_{ijk}$. It is easy to see that up to equivalence this category depends only on the cohomology class of $\a$. The more abstract way to define this category (which doesn't involve a choice of covering) is to represent $e$ by a ${\Bbb G}_m$-{\it gerb}. Recall that a ${\Bbb G}_m$-gerb is a stack of groupoids ${\cal G}$ such that locally there is a unique isomorphism class of objects of ${\cal G}$ and the automorphism group of any object is ${\Bbb G}_m$. Equivalence classes of ${\Bbb G}_m$-gerbs over $S$ are in bijective correspondence with $H^2(S,{\Bbb G}_m)$. Now consider the category of representations of ${\cal G}$, i.e. the category of functors of stacks ${\cal G}\rightarrow\operatorname{Coh}(S)$ where $\operatorname{Coh}(S)$ is the stack of coherent sheaves. Choosing an open covering and a collection of objects $V_i\in{\cal G}(U_i)$ we arrive to the Cech description above. Sometimes $e$ is represented by a sheaf of Azumaya algebras $\AA$ over $S$. Then locally $\AA$ is isomorphic to a matrix algebra of rank $n^2$ over $S$. Now let ${\cal G}(\AA)$ be the ${\Bbb G}_m$-gerb of representations of $\AA$ in locally free $\O_S$-modules of rank $n$. Then it is easy to see that ${\cal G}(\AA)$ represents the same cohomology class $e\in H^2(S,{\Bbb G}_m)$ and the categories of representations of ${\cal G}(\AA)$ and $\AA$ in coherent sheaves on $S$ are equivalent. By abuse of notation we denote all these equivalent categories by $\operatorname{Coh}(S,e)$. Let $E\rightarrow S$ be a $K$-torsor where $K$ is a finite flat commutative group scheme over $S$, let $0\rightarrow{\Bbb G}_m\rightarrow G\rightarrow K\rightarrow 0$ be a central extension of $K$. Then it defines an element $e(G,E)\in H^2(S,e)$ such that the category of $G$-equivariant coherent sheaves on $E$ of weight 1 is equivalent to $\operatorname{Coh}(S,e(G,E))$. Here a weight of a $G$-equivariant coherent sheaf is defined as the weight of the induced ${\Bbb G}_m$-equivariant sheaf. Indeed, consider the gerb ${\cal G}(G,E)$ of liftings of $E$ to $G$-torsors (an object of ${\cal G}(G,E)$ over $U\rightarrow S$ is a $G$-torsor $\widetilde{E}$ over $U$ together with an isomorphism of $K$-torsors $\widetilde{E}/{\Bbb G}_m\simeq E$). Then we claim that the category of weight-1 $G$-equivariant sheaves is equivalent to the category of representations of ${\cal G}(G,E)^{op}$ which is $\operatorname{Coh}(S,e)$ where $e$ is the inverse of the cohomology class of ${\cal G}(G,E)$. To see this note that a lifting of $E$ to a $G$-torsor can be considered as a weight-1 $G$-equivariant line bundle $L$ over $E$. A choice of such bundle over $E_U$ defines the equivalence ${\cal F}\mapsto {\cal F}\otimes L^{-1}$ of the category of weight-1 $G$-equivariant sheaves with the category of $K$-equivariant sheaves on $E_U$, hence with $\operatorname{Coh}(U)$ depending contravariantly on $L$, whereas the assertion. The class $e(G,E)$ is trivial if and only if there is a global object of ${\cal G}(G,E)$, i.e. a global lifting of $E$ to a $G$-torsor. Also it is easy to see that $e(G,E)$ depends biadditively on the pair of classes $[G]\in H^2(K,{\Bbb G}_m)$, $[E]\in H^1(S,K)$. We'll apply this in the particular case when $S=A$ is an abelian variety, $p:E\rightarrow A$ is an isogeny of abelian varieties, so that $E$ can be considered as a $K$-torsor where $K=\ker(p)$. Then for any central extension $\pi:G\rightarrow K$ by ${\Bbb G}_m$ the previous construction gives a class $e(G,E)\in H^2(A,{\Bbb G}_m)$ which is an obstruction for existence of a line bundle $M$ over $E$ such that $K\subset K(M)$ and $G$ is the restriction of Mumford's extension $G(M)\rightarrow K(M)$ to $K$ (see \cite{Mum}). Let $\rho:G\rightarrow\operatorname{GL}_n$ be a weight-1 representation of $G$, $\overline{\rho}:K\rightarrow\operatorname{PGL}_n$ be the corresponding projective representation of $K$. Then the $PGL_n$-torsor $E_{\overline{\rho}}$ on $A$ obtained as the push-forward of $E$ by $\overline{\rho}$ gives rise to an Azumaya algebra with the class $e(G,E)$. Consider $G$ as a ${\Bbb G}_m$-torsor over $K$ so that $G_u=\pi^{-1}(u)$ for $u\in K$. Let us denote by $\O_K(G)$ the corresponding line bundle over $K$. Then a weight-1 $G$-equivariant sheaf on $E$ can be described by the following data: a coherent sheaf ${\cal F}$ on $E$ and an isomorphism over $K\times E$: \begin{equation}\label{equiv} p_1^*\O_K(G)\otimes p_2^*{\cal F}\widetilde{\rightarrow} (ip_1+p_2)^*{\cal F} \end{equation} where $i:K\rightarrow E$ is the inclusion, satisfying the natural cocycle condition. The above construction gives an equivalence of this category with $\operatorname{Coh}(A,e(G,E))$. We need also a derived category version of this equivalence. The slight difficulty is that derived categories of coherent sheaves don't glue well in any of standard topologies. However, as shown in the Appendix, the descent formalism for finite flat morphisms extends to derived categories. This allows to rephrase the definition of $\operatorname{Coh}(S,e)$ (e.g. in Cech version) for a class $e$ which is killed by a finite flat morphism $S'\rightarrow S$ into a description of the corresponding derived category ${\cal D}^b(S,e)$. Similarly, one can describe the derived category of weight-1 $G$-equivariant sheaves above as the category of objects ${\cal F}\in{\cal D}^b(E)$ with isomorphisms (\ref{equiv}) satisfying the cocycle condition and to show that it is equivalent to ${\cal D}^b(A,e(G,E))$. Let $X$ be an abelian variety, $L=P\otimes\sigma^*P^{-1}$ be a symplectic biextension of $X^2$, $(Y,\a)$ be an isotropic pair. \begin{prop} There is a canonical class $e(Y)\in H^2(X/Y,{\Bbb G}_m)$ such that the category ${\cal F}(Y,\a)$ defined in the previous section is equivalent to ${\cal D}^b(X/Y,e(Y))$. \end{prop} \noindent {\it Proof} . Choose a homomorphism of abelian varieties $s:Z\rightarrow X$ and a line bundle $\b$ on $Z$ such that the restriction of the composition $ps:Z\rightarrow X/Y$ is an isogeny and there is an isomorphism of biextensions of $s^{-1}(Y)\times Z$ \begin{equation}\label{inters} (s\times s)^*P|_{s^{-1}(Y)\times Z}\simeq \Lambda(\b)|_{s^{-1}(Y)\times Z}. \end{equation} For example, one can take $Z=X/Y$ and $s':Z\rightarrow X$ such that $ps'=n\operatorname{id}_{X/Y}$, then $s^{\prime -1}(Y)=(X/Y)_n$ and $(ns'\times ns')^*P|_{(X/Y)_{n^2}\times X/Y}$ is a trivial biextension (see \cite{Breen}, 4.2), hence we can take $s=ns'$ and $\b=\O_Z$. Then $\Lambda((s|_{s^{-1}(Y)})^*\a\otimes \b^{-1}|_{s^{-1}(Y)})$ is a trivial biextension, hence the ${\Bbb G}_m$-torsor $\b|_{s^{-1}(Y)}\otimes (s|_{s^{-1}(Y)})^*\a^{-1}$ defines a central extension $G$ of $s^{-1}(Y)$ by ${\Bbb G}_m$. It is easy to see that the class $e(G,Z)\in H^2(X/Y,{\Bbb G}_m)$ defined above doesn't depend on a choice of $\a$ such that the pair $(Y,\a)$ is isotropic. We claim also that it doesn't depend on a choice of $Z$ and $\b$. Indeed, if we change $\b$ by $\a'_Z=\b\otimes\gamma$ where $\Lambda(\gamma)|_{s^{-1}(Y)\times Z}$ is trivial, then the new central extension is the sum of $G$ and the restriction of Mumford's extension $G(\gamma)\rightarrow K(\gamma)$ to $s^{-1}(Y)$. But $\gamma$ has a natural structure of $G(\gamma)$-equivariant line bundle, hence $e(G(\gamma),Z)=0$. Also, it is easy to see that $e(G,Z)$ doesn't change if we replace $s:Z\rightarrow X$ by the composition of $s$ with an isogeny $Z'\rightarrow Z$. It remains to check that $e(G,Z)$ is invariant under the change $s'=s+f$ where $f:Z\rightarrow Y$ is any homomorphism. In this case $s^{\prime -1}(Y)=s^{-1}(Y)$ and $$(s'\times s')^*P|_{s^{-1}(Y)\times Z}\simeq \Lambda(\a'_Z)|_{s^{-1}(Y)\times Z}$$ where $\a'_Z=\b\otimes f^*\a\otimes (f,s)^*P$. On the other hand $$(s'|_{s^{-1}(Y)})^*\a\simeq (s|_{s^{-1}(Y)})^*\a\otimes (f^*\a\otimes (f,s)^*P)|_{s^{-1}(Y)}$$ so that the corresponding central extension of $s^{-1}(Y)$ is the same. Given an object $(\AA,a)$ of ${\cal F}(Y,\a)$ where $\AA\in{\cal D}^b(X)$, $a$ is an isomorphism (\ref{Schrsp}), we can consider $s^*\AA\in{\cal D}^b(Z)$. Then $a$ induces an isomorphism $$a_{s(u),s(z)}:s^*\AA_{u+z}\widetilde{\rightarrow}(s\times s)^*P^{-1}_{u,z} \a^{-1}_{s(u)} s^*\AA_z$$ where $u\in s^{-1}(Y)$, $z\in Z$, satisfying the usual cocycle condition. Let $F(\AA)=s^*\AA\otimes\b$, then $a_{s(u),s(z)}$ together with (\ref{inters}) gives an isomorphism \begin{equation} F(a):F(\AA)_{u+z}\widetilde{\rightarrow}\b_u\otimes \a^{-1}_{s(u)}\otimes F(\AA)_z \end{equation} where $u\in s^{-1}(Y)$, $z\in Z$. In other words, $(F(\AA),F(a))$ can be considered as a weight-1 $G$-equivariant object of ${\cal D}^b(Z)$. This gives the required equivalence as one can check applying Theorem A of Appendix to the morphism $Y\times Z\rightarrow X$ and the trivial descent for the projection $Y\times Z\rightarrow Z$. \qed\vspace{3mm} \begin{rem} If $(Y,\a)$ and $(Z,\b)$ are isotropic pairs the ${\Bbb G}_m$-torsor $\b|_{Y\cap Z}\otimes\a|^{-1}_{Y\cap Z}$ defines a central extension $G$ of $Y\cap Z$ by ${\Bbb G}_m$, which in case when $Y$ and $Z$ are lagrangian and $Y\cap Z$ is finite gives the class $e_Y\in H^2(X/Y,{\Bbb G}_m)$. The corresponding commutator form on $Y\cap Z$ measures the difference between the symmetric structures on $P|_{(Y\cap Z)^2}$ restricted from $Y^2$ and $Z^2$. In other words, this is the standard symplectic form associated with the biextension $L|_{Y\times Z}$ measuring the difference between two trivializations of $L|_{(Y\cap Z)^2}$ restricted from $Y\times (Y\cap Z)$ and $(Y\cap Z)\times Z$. \end{rem} By definition the class $e(Y)$ vanishes if the projection $X\rightarrow X/Y$ splits. It turns out that if $Y$ is lagrangian then the converse is also true. \begin{prop} Let $Y\subset X$ be a lagrangian subvariety. If $e_Y=0$ then the projection $X\rightarrow X/Y$ splits. \end{prop} \noindent {\it Proof} . According to Theorem \ref{pairs} we can assume that $X=\hat{A}\times A/(\phi,\operatorname{id})(A_n)$, $Y=A/\ker(\phi)\subset X$ for an abelian variety $A$ and a skew-symmetric homomorphism $\phi:A_n\rightarrow\hat{A}_n$. Now we can take $Z=\hat{A}\subset X$ in the definition of $e_Y$. The kernel of the projection $Z\rightarrow X/Y$ is $\phi(A_n)\subset\hat{A}$ and the commutator form of its central extension considered above is (up to sign) $$e(\phi(x),\phi(y))=e_n(\phi(x),y)$$ where $x,y\in A_n$. The triviality of $e_Y$ implies that there exists a symmetric homomorphism $g:\hat{A}\rightarrow A$ such that $\phi(A_n)\subset\ker(g)$ and $e=e^g|_{\phi(A_n)^2}$ where $e^g$ is the standard symplectic form on $\ker(g)$. In other words, the following equality holds: $$e_n(\phi(x),y)=e_n(\phi(x),g(n^{-1}\phi(y)))$$ for all $x,y\in A_n$, which implies that $y-g(n^{-1}\phi(y))\in\ker(\phi)$ for $y\in A_n$. Note that $x\mapsto g(n^{-1}x)\mod(\ker(\phi))$ is a well-defined homomorphism $\hat{A}\rightarrow A/\ker(\phi)$ since $g(A_n)\subset \ker(\phi)$ (which is obtained from $\phi(A_n)\subset\ker(g)$ by duality). Thus, a homomorphism $$\hat{A}\times A\rightarrow A/\ker(\phi):(x,y)\mapsto y-g(n^{-1}x)\mod(\ker(\phi))$$ descends to a homomorphism $X=\hat{A}\times A/(\phi,\operatorname{id})(A_n)\rightarrow A/\ker(\phi)=Y$ splitting the embedding $Y\rightarrow X$. \qed\vspace{3mm} \section{Intertwining functors} Let $X$ be an abelian variety with a symplectic biextension $L$ of $X^2$. In this section we construct an equivalence of $H(X)$-representations ${\cal F}(Y,\a)\simeq{\cal F}(Z,\b)$ for isotropic pairs $(Y,\a)$ and $(Z,\b)$ such that $Y$ and $Z$ are lagrangian. The idea is to mimic the classical construction. Namely, consider the functor of ``integration over $Z$'' $$R:{\cal F}(Y,\a)\rightarrow{\cal F}(Z,\b): \AA\mapsto p_{2*}(P|_{Z\times X}\otimes p_1^*\b\otimes (ip_1+p_2)^*\AA).$$ The following symbolic notation stresses the analogy with the classical case: $$R(\AA)_x=\int_{Z} P_{z,x}\b_z\AA_{z+x}dz.$$ It easy to check that $R(\AA)$ has a natural structure of an object of ${\cal F}(Z,\b)$: \begin{align*} &R(\AA)_{z'+x}=\int_{Z} P_{z,z'+x}\b_z\AA_{z+z'+x}dz\simeq \int_{Z} P_{z,x}\b_{z+z'}\b_{z'}^{-1}\AA_{z+z'+x}dz\simeq\\ &\int_{Z} P_{z-z',x}\b_z\b_{z'}^{-1}\AA_{z+x}dz\simeq P^{-1}_{z',x}\b_{z'}^{-1}R(\AA)_x \end{align*} --- here we used the isomorphism $P|_{Z^2}\simeq\Lambda(\b)$ and the change of variable $z\mapsto z-z'$. It is also clear that $R$ commutes with the action of $H(X)$. In the classical theory in order to get an invertible intertwining operator one should replace the integration over $Z$ by the integration over $Z/Y\cap Z$ in the above formula. This doesn't work literally in our context--- it turns out that in the correct definition one eliminates the "excess" integration over the connected component of $Y\cap Z$, and over a "largangian half" of the group of connected components of $Y\cap Z$. Instead of working out the case when $\dim(Y\cap Z)>0$ we use the following simple lemma which allows to avoid it. \begin{lem} For any pair $Y$ and $Z$ of lagrangian subvarieties of $X$ there exists a lagrangian subvariety $T\subset X$ such that the intersections $Y\cap T$ and $Z\cap T$ are finite. \end{lem} \noindent {\it Proof} . We can work in the category of abelian varieties up to isogeny. We have an isogeny $X\sim Y\times \hat{Y}$ and $Z/Y\cap Z\subset Y\times \hat{Y}$ is isogenic to the graph of a symmetric morphism $$g: Z/Y\cap Z\rightarrow Y/Y\cap Z\sim \widehat{Z/Y\cap Z}.$$ Let $K\sim\ker(g)$. We have a decomposition $Z/Y\cap Z\sim K\times K'$ such that $g$ is given by a symmetric isogeny $K'\rightarrow\hat{K'}$. Now let $\hat{Y}\sim Z/Y\cap Z\times K''\sim K\times K'\times K''$. Let us define a symmetric morphism $f:\hat{Y}\rightarrow Y$ to be a symmetric isogeny on $K$ and zero on two other factors. Then we can take the graph of $f$ to be $T$. \qed\vspace{3mm} Thus, we may assume that $Y\cap Z$ is finite. We have a natural central extension $G$ of $Y\cap Z$ by ${\Bbb G}_m$ given by the ${\Bbb G}_m$-torsor $\b|_{Y\cap Z}\otimes\a^{-1}|_{Y\cap Z}$ such that ${\cal F}(Y,\a)$ is equivalent to the category of weight-1 $G$-equivariant objects of ${\cal D}^b(Z)$, while ${\cal F}(Z,\b)$---to that of weight-1 $G^{-1}$-equivariant objects of ${\cal D}^b(Y)$, where $G^{-1}$ is the inverse central extension of $Y\cap Z$ (given by the inverse ${\Bbb G}_m$-torsor). Let $e$ be the commutator form of $G$. Choose a lagrangian subgroup $H\subset Y\cap Z$ with respect to $e$ and a trivialization of the central extension $G$ over $H$ (which is the same as a lifting of $H$ to a subgroup in $G$). Then we can define the reduced functor $$\overline{R}:{\cal F}(Y,\a)\rightarrow{\cal F}(Z,\b): \overline{R}(\AA)_x=\int_{Z/H} P_{z,x}\b_z\AA_{z+x}dz.$$ To give a meaning to this notice that an object $P_{z,x}\b_z\AA_{z+x}\in{\cal D}^b(Z\times X)$ descends canonically to an object of ${\cal D}^b(Z/H\times X)$ (use the additional data on $\AA\in{\cal F}(Y,\a)$ and the isomorphism $\a|_H\simeq\b|_H$). As above it is easy to check that $\overline{R}(\AA)$ has a natural structure of an object of ${\cal F}(Z,\b)$ and $\overline{R}$ commutes with $H(X)$-action. \begin{thm}\label{main} The functor $\overline{R}$ is an equivalence of categories. \end{thm} \noindent {\it Proof} . First let us rewrite $\overline{R}$ as the functor from the category of weight-1 $G$-equivariant objects of ${\cal D}^b(Z)$ to that of weight-1 $G^{-1}$-equivariant objects of ${\cal D}^b(Y)$ using the equivalences defined above. Recall that an equivalence of the first category with ${\cal F}(Y,\a)$ is given by a functor $F_Y$ which associates to $\AA\in{\cal F}(Y,\a)$ the $G$-equivariant object $\AA|_Y\otimes\b\in{\cal D}^b(Z)$, while the second equivalence is induced by $F_Z:{\cal F}(Z,\b)\rightarrow{\cal D}^b(Y):\AA'\mapsto\AA'|_Y\otimes\a$. Now for $\AA\in{\cal F}(Y,\a)$ we have \begin{align*} F_Z(\overline{R}(\AA))_y &= \a_y\int_{Z/H} P_{z,y}\b_z\AA_{z+y}dz\simeq \a_y\int_{Z/H} P_{z,y}\b_z P_{y,z}^{-1}\a_y^{-1}\AA_z dz\simeq\\ &\simeq\int_{Z/H} L_{z,y} F_Y(\AA)_z dz \end{align*} The latter integral should be understood in the same sense as above: the $G$-equivariance data on $F_Y(\AA)$ allow to descend $L_{z,y} F_Y(\AA)_z$ to an object of ${\cal D}^b(Z/H\times Y)$. Notice that $G$-equivariance data on an object ${\cal G}\in{\cal D}^b(Z)$ includes the descent data for the projection $Z\rightarrow Z/H$, so that $G$-equivariant objects of ${\cal D}^b(Z)$ can be considered as objects of ${\cal D}^b(Z/H)$ with some additional data. More precisely, the isomorphism $${\cal G}_{z+u}\simeq G_u L_{z,u}{\cal G}_z$$ where $u\in Y\cap Z$, induced by the $G$-equivariance data and the trivialization of $L_{z,u}$ commutes with the descent data for $Z\rightarrow Z/H$, so it induces an isomorphism of descended objects on $Z/H\times (Y\cap Z)$ $$\overline{{\cal G}}_{z+u}\simeq G_u \overline{L}_{z,u}\overline{{\cal G}}_z$$ --- these are the additional data for $\overline{{\cal G}}\in{\cal D}^b(Z)$. Similar, we can consider $G^{-1}$-equivariant objects of ${\cal D}^b(Y)$ as objects of ${\cal D}^b(Y/H)$ with additional data. It is easy to see that biextension $L_{z,y}$ of $Z\times Y$ descends to a biextension $\overline{L}$ of $Z/H\times Y/H$ which induces an isomorphism $Z/H\widetilde{\rightarrow}\widehat{Y/H}$. Thus, $\overline{R}$ is compatible with the Fourier-Mukai transform ${\cal D}^b(Z/H)\rightarrow{\cal D}^b(Y/H)$ via the "forgetting" functors ${\cal F}(Y,\a)\rightarrow{\cal D}^b(Z/H)$ and ${\cal F}(Z,\b)\rightarrow{\cal D}^b(Y/H)$ described above. Let $\overline{Q}:{\cal F}(Z,\b)\rightarrow{\cal F}(Y,\a)$ be the functor defined in the same way as $\overline{R}$ but with $Y$ and $Z$ interchanged. Then it is compatible with the ``inverse'' Fourier transform ${\cal D}^b(Y/H)\rightarrow{\cal D}^b(Z/H)$ given by the kernel $\overline{L}_{y,z}\simeq\overline{L}_{z,y}^{-1}$. Its composition with the direct Fourier transform is isomorphic to a shift in the derived category and it is easy to see that this isomorphism extends to our additional data, so that $\overline{Q}$ is quasi-inverse to $\overline{R}$ up to shift. \qed\vspace{3mm} Consider the following example. Let $X=\hat{A}\times A$, $L=L_A$ be the standard split symplectic biextension, $Y=\hat{A}\subset\hat{A}\times A$ be its standard lagrangian subvariety. Let $f:A\rightarrow \hat{A}$ be a symmetric morphism, $Z=A/\ker(f_n)\simeq (f,n\operatorname{id}_A)(A)\subset\hat{A}\times A$ where $f_n=f|_{A_n}$. Then it is easy to see that $Z$ is lagrangian. Assume in addition that $mn\ker(f)=0$ for some $m$ relatively prime to $n$. Then we claim that the projection $X\rightarrow X/Z$ splits. Indeed, changing $m$ if necessary we may assume that $m+kn=1$ for some integer $k$. Notice that we have an isomorphism $X/Z\simeq \hat{A}/f(A_n)\simeq A/(n\ker(f))$. Now we can define the splitting morphism $$X/Z\simeq A/(n\ker(f))\stackrel{(k,m)}{\rightarrow} A/\ker(f)\times A\simeq X.$$ Hence, $e_Y=e_Z=0$ and we get an equivalence of derived categories $${\cal D}^b(A)={\cal D}^b(X/Y)\simeq{\cal D}^b(X/Z)\simeq{\cal D}^b(Z)= {\cal D}^b(A/ker(f_n))$$ --- here we used the Fourier-Mukai equivalence for $Z$ and $X/Z\simeq\hat{Z}$. The proof of Theorem \ref{main} shows that we can eliminate the assumption that there exists a biextension $P$ of $X^2$ such that $L\simeq P\otimes\sigma^*P^{-1}$ once we can define the categories in question without it. In fact, if the characteristic of the ground field is not equal to 2, we can do it as follows. Let $Y\subset X$ be a lagrangian subvariety. Choose another lagrangian subvariety $Z\subset X$ such that $Y\cap Z$ is finite. Then we have a ${\Bbb G}_m$-valued symplectic form on $Y\cap Z$ defined by the canonical duality between $Y\cap Z=\ker(Y\rightarrow X/Z)=\ker(Y\rightarrow\hat{Z})$ and $\ker(Z\rightarrow\hat{Y})=\ker(Z\rightarrow X/Y)=Y\cap Z$. Since the characteristic is different from 2 there exists a central extension $G$ of $Y\cap Z$ by ${\Bbb G}_m$ with such commutator form (unique up to an isomorphism), so we have the corresponding class $e_Y\in H^2(X/Y,{\Bbb G}_m)$ which doesn't depend on the choices made (and coincides with the one defined previously using $P$). \begin{thm} Assume that the characteristic of the ground field is different from 2. Then for every pair of lagrangian subvarieties $Y$ and $Z$ the categories ${\cal D}^b(X/Y,e_Y)$ and ${\cal D}^b(X/Z, e_Z)$ are equivalent. \end{thm} \noindent {\it Proof} . As before we may assume that $Y\cap Z$ is finite. Now choose a central extension $G$ of $Y\cap Z$ inducing the canonical symplectic form on it and define the functor from the category of weight-1 $G$-equivariant objects of ${\cal D}^b(Z)$ to that of weight-1 $G^{-1}$-equivariant objects of ${\cal D}^b(Y)$ by the formula $$\overline{R}({\cal F})_y=\int_{Z/H} L_{z,y} {\cal F}_z dz$$ where $H\subset Y\cap Z$ is lagrangian. As above it is easy to check that this is an equivalence. \qed\vspace{3mm} \begin{rem} The constructed equivalences are not canonical and they don't agree for triples of lagrangian subvarieties. The corresponding analogue of Maslov index and a partial generalization of this theory to abelian schemes will be discussed in a forthcoming paper. \end{rem} \bigskip \centerline{\bf Appendix. Descent for derived categories} \bigskip An unpleasant property of the derived category of coherent sheaves of $\O_S$-modules on a scheme $S$ is that one can not glue this category from its counterparts over open parts of $S$. However, the following descent result holds. \vspace{3mm} \noindent {\bf Theorem A.} {\it Let $p:S'\rightarrow S$ be a finite flat morphism. Then the category ${\cal D}^b(S)$ is equivalent to the the following category ${\cal D}^b(S',p)$: its objects are pairs $({\cal F},f)$ where ${\cal F}\in{\cal D}^b(S')$, $f:p_1^*{\cal F}\widetilde{\rightarrow} p_2^*{\cal F}$ is an isomorphism in ${\cal D}^b(S'\times_S S')$ (where \break $p_i:S'\times_S S'\rightarrow S'$, $i=1,2$ are the projections) satisfying the following cocycle condition $p_{23}^*f\circ p_{12}^*f=p_{13}^*f$ over $S'\times_S S'\times_S S'$.} \vspace{3mm} \noindent {\it Proof} . Let $p^*:{\cal D}^b(S)\rightarrow{\cal D}^b(S',p)$ be the natural functor. Let us check first that $p^*$ is fully faithful. Assume that we have a morphism $f:p^*{\cal F}\rightarrow p^*{\cal G}$ in ${\cal D}^b(S')$ such that the following diagram is commutative: \begin{equation} \setlength{\unitlength}{0.20mm} \begin{array}{ccccc} p_1^*p^*{\cal F} & \lrar{} & p_2^*p^*{\cal F} \\ \ldar{p_1^*(f)} & & \ldar{p_2^*(f)} \\ p_1^*p^*{\cal G} & \lrar{} & p_2^*p^*{\cal G} \end{array} \end{equation} Applying the functor $p_{1*}$ to this diagram and composing it with the adjunction morphism $p^*{\cal F}\rightarrow p_{1*}p_1^*p^*{\cal F}$ we get the following diagram \begin{equation} \setlength{\unitlength}{0.20mm} \begin{array}{ccccc} p^*{\cal F} & & \\ \ldar{f} & \ldrar{} & \\ p^*{\cal G} &\lrar{} & p^*p_*p^*{\cal G} \end{array} \end{equation} where the diagonal morphism is $p^*f'$, $f':{\cal F}\rightarrow p_*p^*{\cal G}$ is obtained from $f$ by adjunction. Let us denote by $f''$ the composition $${\cal F}\stackrel{f'}{\rightarrow}p_*p^*{\cal G}\simeq p_*\O_{S'}\otimes{\cal G}\rightarrow (p_*\O_{S'}/\O_S)\otimes{\cal G}.$$ Then it follows from the diagram above that $p^*(f'')=0$. Since $p_*p^*(?)\simeq p_*\O_{S'}\otimes ?$ it follows that $f''=0$, hence $f'$ factors through a morphism $\overline{f}:{\cal F}\rightarrow{\cal G}$ and $f=p^*(\overline{f})$. Thus, the functor $p^*:{\cal D}^b(S)\rightarrow{\cal D}^b(S',p)$ is full and faithful. It remains to check that any object of ${\cal D}^b(S',p)$ belongs to its essential image. This is easy to prove by devissage with respect to the standard $t$-structure on ${\cal D}^b(S')$ since the corresponding truncation functors are compatible with descent data (the base of induction is provided by the classical descent for coherent sheaves). \qed\vspace{3mm}

9407

alg-geom/9407004

en

https://arxiv.org/abs/alg-geom/9407004

alg-geom/9407004

Andreas Steffens

Andreas Steffens

On the stability of the tangent bundle of Fano manifolds

9 pages, LaTeX, to appear in Math. Ann

By using the classification of Fano 3-folds we prove: Let $X$ be Fano 3-fold. Assume that the tangent bundle $T_X$ of $X$ is not stable (i.e. semi-stable or unstable). Then $b_2\geq 2$ and a relative tangent sheaf $T_{X/Y}$ of a contraction $f:X\longrightarrow Y$ of an extremal face on $X$ is a destabilising subsheaf of $T_X$.

alg-geom

\section*{Introduction} A smooth variety $X$ over the field of complex numbers ${\Bbb C}$ is called {\em Fano} if its anticanonical divisor $-\!K_X$ is ample. Stability (in the sense of Mumford and Takemoto) with respect to $-K_X$ of the tangent bundle $T_X$ can be considered as an algebraic analogue to the existence of a K\"ahler-Einstein metric on $X$, since the result of Kobayashi \cite{K} and L\"ubke \cite{Lubke} shows that the existence of a K\"ahler-Einstein metric implies the stability of the tangent bundle. But the converse is not true, e.g.~ ${\Bbb P}^2$ blown up in two points has stable tangent bundle, which do not admit a K\"ahler-Einstein metric, cf.~ \cite{Ma}. By Tian's solution of Calabi's conjecture for Del-Pezzo surfaces \cite{Ti} and by \cite{Fahlaoui} we have a complete picture in dimension 2: If $X$ is a Del-Pezzo surface, then $X$ has stable tangent bundle $T_X$, unless $X$ is isomorphic to ${\Bbb P}^1\times{\Bbb P}^1$, or ${\Bbb P}^2$ blown-up in a point. In both cases the relative tangent bundle $T_{X/{\Bbb P}^1}$ of a canonical projection to ${\Bbb P}^1$ is a destabilising subsheaf of $T_X$. If the dimension of $X$ is $\geq 3$, then the existence of a K\"ahler-Einstein metric remains an open question. In this article, our main results are as follows: \begin{theorem*}\label{dim 3 main 1} Let $X$ be a Fano 3-fold with $b_2\geq 2$. Assume that the tangent bundle $T_X$ of $X$ is not stable. Then the relative tangent sheaf $T_{X/Y}$ of a contraction $f:X\longrightarrow Y$ of an extremal face on $X$ is a destabilising subsheaf of $T_X$. \end{theorem*} \begin{theorem*}\label{dim 3 main 2} {}From the 87 deformation classes of Fano 3-folds with $b_2\geq 2$, cf.~ \cite{Mori-Mukai.1,Mori-Mukai.2} the members of \begin{center} \begin{tabular}{cl} 68 & deformation classes have stable tangent bundle,\\ 12 & deformation classes have semistable (but not stable) tangent bundle and\\ 7 & deformation classes have unstable tangent bundle. \end{tabular} \end{center} \end{theorem*} For a detailed description of the deformation classes whose members have semistable or unstable tangent bundle see theorem \ref{dim 3 b2 groesser 1} below. Our main tool in proving theorem \ref{dim 3 main 2} is Mori theory. By Mori theory we understand the results and techniques concerning the cone of curves on a manifold $X$ whose canonical divisor $K_X$ is not numerically effective. The proofs of theorem 1 and 2 use the classification of Fano-3-folds. If $\dim X\geq 4$, then the problem of the stability of the tangent bundle seems hopeless, if one wants to use classification. But one may expect that theorem 1 holds in any dimension. \section{Preliminaries} A smooth connected variety $X$ over the field of complex numbers ${\Bbb C}$ is called simply a {\em manifold}. All manifolds are assumed to be projective, unless otherwise stated. $K_X$ denotes the canonical divisor of a normal variety $X$. Assume $X$ smooth and set $n=\dim X$. Let $H$ be an ample line bundle on $X$. If ${\cal F}$ is a torsion free coherent sheaf on $X$ we define $\mu({\cal F})$ to be $c_1({\cal F}).H^{n-1}/rk({\cal F})$. We call ${\cal F}$ {\em semistable} (resp. {\em stable}) if for all proper subsheaves ${\cal F}^\prime$ of ${\cal F}$ with $0\le rk({\cal F}^\prime) \le rk({\cal F})$ we have $\mu({\cal F}^\prime)\leq\mu({\cal F})$ (resp. $\mu({\cal F}^\prime)<\mu({\cal F})$). Let $X$ be a normal variety of dimension $n$. We use the following notation: \begin{enumerate} \item[] $N^1(X):=(\{ \mbox{ Cartier divisors on }X\}/\equiv)\otimes{\Bbb R}$ \item[] $N_1(X):=(\{ \mbox{ 1-cycles on }X\}/\equiv)\otimes{\Bbb R}$ \item[] $\overline{N\!E}(X):=$ the closure of the convex cone generated by effective 1-cycles in $N_1(X)$. \end{enumerate} Here the symbol $\equiv$ means numerical equivalence and the symbol $\sim$ will denoted linear equivalence. \begin{Def} (1) A curve $C$ on $X$ is called {\em extremal} if \begin{enumerate} \item[(a)] $(K_X.C)<0$, \item[(b)] given $u,v\in\overline{N\!E}(X)$ then $u,v\in{\Bbb R}_+[C]$ if $u+v\in{\Bbb R}_+[C]$. \end{enumerate} If $C$ is an extremal curve on $X$, then the set $R={\Bbb R}_+[C]$ is called an {\em extremal ray} on $X$.\\ 2) Let $H$ be a nef Cartier divisor on $X$. The set $F:= H^\perp\cap\overline{N\!E}\setminus\{0\}$ is called an {\em extremal face} if $F$ is entirely contained in the set $\{z\in N_1(X)\ |\ \ (K_X.z)<0\ \}$. \end{Def} \begin{theorem}[Cone theorem (Mori, Kawamata, Koll\'ar \protect{\cite{Mori.1,KMM}})] \hfill\\ Assume that $X$ has only canonical singularities. Fix an ample divisor ${\cal L}$. Then for any $\varepsilon >0$, there exist extremal curves $\ell_1,\ldots,\ell_r$ such that $$\overline{N\!E}(X)=\sum_{i=1}^r {\Bbb R}_+[\ell_i] +\overline{N\!E_\varepsilon}(X).$$ Here $\overline{N\!E_\varepsilon}(X):=\{\ z\in\overline{N\!E}(X)\ | \ (K_X.z)> -\varepsilon ({\cal L}.z)\ \}$. \end{theorem} \begin{theorem}[Contraction theorem (Shokurov, Kawamata \cite{KMM})]\hfill\\ Let $F$ be an extremal face of $\overline{N\!E}(X)$. Assume that $X$ has only canonical singularities. Then there exists a morphism $\varphi=\mbox{cont}_F:X\longrightarrow Y$ onto a normal projective variety Y, such that: For any irreducible curve $C$ on $X$ the image $\varphi(C)$ is a point if and only if $[C]\in F$. \end{theorem} \section{Fano varieties with $b_2=1$} Fano 3-folds with $b_2=1$ are classified by Iskovskih \cite{Iskovskih.1,Iskovskih.2}. There are 18 classes of Fano 3-folds with $b_2=1$ up to deformation. \begin{remark} Let $X$ be a Fano manifold of dimension $n$. By a criterion for stability \cite{Hoppe}, the tangent bundle $T_X$ of $X$ is stable with respect to $(-K_X)$, if one of the following equivalent conditions is fulfilled: \begin{enumerate} \item[$(A_i)$] $H^0(X,{\Omega^i \otimes L^{-1}}) = 0$ for all $L\in Pic(X)$ with $L.(-K_X)^{n-1} \geq -\frac{i}{n}(-K_X)^n$. \item[$(B_i)$] $H^0(X,{\bigwedge^i T_X \otimes L^{-1}}) = 0$ for all $L\in Pic(X)$ with $L.(-K_X)^{n-1} \geq \frac{i}{n}(-K_X)^n$. \end{enumerate} Stability is granted when all conditions $(A_i)$ or $(B_{n-i})$, $1\leq i \leq n-1$, hold. \end{remark} {}From now on assume that $b_2(X)=1$. Let $L$ be the ample generator of $Pic(X) \simeq {\Bbb Z}$. Then we have that $-K_X=rL$ with $1\leq r\leq n+1$, where the integer $r$ is called the {\em index} of $X$. By the Kobayashi-Ochiai characterisation of projective space and hyperquadrics \cite{KoOc}, we have: $$ r =n+1 \ \Leftrightarrow \ X \simeq {\Bbb P}^n \mbox{ and} \ \ r = n \ \Leftrightarrow \ X \simeq Q_n \subset {\Bbb P}^{n+1} $$ \begin{remark} If $X$ is ${\Bbb P}^n$ or $Q_n$, then on may verify the conditions $(A_i)$ directly. \end{remark} \begin{remark} Let $X$ be a Fano manifold with $b_2=1$ and $L$ the ample generator of $Pic(X)$. Then we have: \begin{enumerate} \item If the index $r$ of $X$ is $1$, then the conditions $(A_i)$ are fulfilled (cf. \cite[Theorem 3]{Reid.2}). \item $H^0(X,{\Omega^1_X \otimes L^{m}})=0$ for $m \leq 0$. In particular the condition $(A_1)$ is fulfilled in any case. \item If the index $r$ of $X$ is $\leq n$, then the condition $(A_{n-1})$ is fulfilled. \end{enumerate} \end{remark} \begin{pf} 1),2) Since $1\!\leq\! i \!\leq\! n-1$, we have $i\frac{1}{n} < 1$. If $m < 0$, then we have $H^0(X,{\Omega^i\! \otimes\! L^m})\!=\!0$ by Kodaira-Nakano vanishing theorem. Since $-K_X$ is ample, it follows by the Kodaira vanishing theorem that $h^0\!(\Omega^i_X)\!=\!h^i\!({\cal O}_X)\!=\!h^{n-i}\!(\omega_X)=0$. \noindent 3) Since the condition $(A_{n-1})$ is equivalent to $(B_1)$, it suffices to show that $H^0(X,{T_X \otimes L^{-m}}) = 0$, for $m \geq 1$. But this is a consequence of \cite[Theorem 1]{Wahl}, because the index of $X$ is different from $n+1$. \end{pf} \begin{corollary} Let $X$ be a Fano 3-fold with $b_2=1$. Then the tangent bundle of $X$ is stable. \end{corollary} \section{Fano 3-folds with $b_2\geq 2$} The proof of theorem \ref{dim 3 main 1} and theorem \ref{dim 3 main 2} is a by-product of the proof of the following: \begin{theorem}\label{dim 3 b2 groesser 1} Let $X$ be a Fano 3-fold.\\ i) The members of the deformation classes in the following list have semistable tangent bundle. \begin{enumerate} \item[(1)] the blow-up of ${\Bbb P}^3$ with center a line. \item[(2)] the blow-up of $Y$ with center two exceptional fibers $\ell$ and $\ell^\prime$ of the blow-up $\Phi : Y\to {\Bbb P}^3$ such that $\ell$ and $\ell^\prime$ lie on the same irreducible component of the exceptional set of $\Phi$. Here $\Phi : Y\to {\Bbb P}^3$ is the blow-up of ${\Bbb P}^3$ with center two disjoint lines in ${\Bbb P}^3$. \item[(3)] the product of a Del-Pezzo surface (i.e Fano 2-fold) with ${\Bbb P}^1$. \end{enumerate} ii) The members of the deformation classes in the following list have unstable tangent bundle. \begin{enumerate} \item[(1)] $V_7$, that is, the ${\Bbb P}^1$-bundle ${\Bbb P}({\cal O}\oplus{\cal O}(1))$ over ${\Bbb P}^2$. \item[(2)] the blow-up of the Veronese cone $W_4\subset{\Bbb P}^6$ with center the vertex, that is ${\Bbb P}({\cal O}\oplus{\cal O}(2))$ over ${\Bbb P}^2$. \item[(3)] the blow-up of $V_7$ with center a line on the exceptional divisor $D\simeq{\Bbb P}^2$ of the blow-up $V_7\to{\Bbb P}^3$. \item[(4)] the blow-up of $V_7$ with center the strict transform of a line passing through the center of the blow-up $V_7\to{\Bbb P}^3$. \item[(5)] the blow-up of the cone over a smooth quadric surface in ${\Bbb P}^3$ with center the vertex, that is, the ${\Bbb P}^1$-bundle ${\Bbb P}({\cal O}\oplus{\cal O}(1,1))$ over ${\Bbb P}^1\times{\Bbb P}^1$. \item[(6)] the blow-up of ${\Bbb P}^1\times{\Bbb F}_1$ with center $\{t\}\times e$, where $t\in{\Bbb P}^1$ and $e$ is an exceptional curve of the first kind on ${\Bbb F}_1$. \item[(7)] the blow-up of $\widetilde{{\Bbb P}}(L)$ with center two exceptional lines of the blow-up $\widetilde{{\Bbb P}}(L)\to{\Bbb P}^3$. Here $\widetilde{{\Bbb P}}(L)\to {\Bbb P}^3$ is the blow-up of ${\Bbb P}^3$ with center a line in ${\Bbb P}^3$. \end{enumerate} iii) If $X$ is not contained in a deformation class listed as above, then $T_X$ is stable with respect to the anticanonical divisor $-K_X$. \end{theorem} \begin{pf} Instead of presenting here the long proof of theorem \ref{dim 3 b2 groesser 1}, we will treat some special cases and examples. For the proof of theorem \ref{dim 3 b2 groesser 1}, we will refer the reader to \cite{St}. The plan of the proof is as follows: \noindent Step 1. Vanishing results for $H^0(X,T_X\otimes{\cal L}^{-1})$ and $H^0(X,\Omega^1_X\otimes{\cal L}^{-1})$. \noindent Step 2. Direct check of stability for the list of families with $b_2=2$. \noindent Step 3. Reduction of the cases with $b_2\geq 3$ to those studied at Step 2, or to lower dimensional vanishing statements. \end{pf} \noindent Products of Fano manifolds\\ Let $Y_1, Y_2$ be two Fano manifolds of dimension $n_1$ and $n_2$ respectively. Then $X\!=\!Y_1\!\times\!Y_2$ is a Fano manifold of dimension $n=n_1 + n_2$. By an easy computation, one gets $\mu(T_X)= \mu({\pi_1^*}T_{Y_1}) = \mu({\pi_2^*}T_{Y_2})$. It is a well known fact that a vector bundle $E_1\oplus E_2$ is semistable if and only if $E_1$ and $E_2$ are semistable vector bundles with $\mu(E_1)=\mu(E_2)$. Thus, we have proved: \begin{proposition} Let $Y_1$, $Y_2$ be two Fano manifolds with semistable tangent bundle. Then the Fano manifold $X\!=\!Y_1\!\times\!Y_2$ has semistable tangent bundle. \end{proposition} \begin{corollary} Let $X$ be isomorphic to a product of a Del-Pezzo surface with ${\Bbb P}^1$. Then $T_X$ is semistable. \end{corollary} \begin{pf} By \cite{Fahlaoui} the tangent bundle of a Del-Pezzo surface is a semistable vector bundle. \end{pf} \begin{example}% Let $X$ be the blow-up of $\widetilde{{\Bbb P}^3}(L)$ with center two exceptional lines of the blow-up $\widetilde{{\Bbb P}^3}(L)\to{\Bbb P}^3$. Here $\widetilde{{\Bbb P}^3}(L)$ is the blow-up of ${\Bbb P}^3$ with center a line $L$. We will show that $X$ has unstable tangent bundle. Consider the following diagram \begin{center} \begin{picture}(250,100)(50,0) \put(200,90){$\bf X$} \put(20,80){${\Bbb P}^1$} \put(20,40){${\Bbb F}_1$} \put(100,40){$\widetilde{V_7}(L\ni p_1)$} \put(185,40){$\widetilde{{\Bbb P}^3}\!(L)$} \put(270,40){$\widetilde{V_7}(L\ni p_2)$} \put(100,0){$V_7$} \put(20,0){${\Bbb P}^2$} \put(200,0){${\Bbb P}^3$} \put(195,85){\vector(-3,-2){50}} \put(155,70){$f\!_1$} \put(205,85){\vector(0,-1){30}} \put(207,65){$f\!_2$} \put(215,85){\vector(3,-2){50}} \put(245,70){$f\!_3$} \put(105,35){\vector(0,-1){20}} \put(110,20){$f\!_{1,1}$} \put(205,35){\vector(0,-1){20}} \put(265,35){\vector(-3,-2){45}} \put(95,45){\vector(-1,0){60}} \put(60,50){$f\!_{1,2}$} \put(25,55){\vector(0,1){20}} \put(30,60){$f\!_{1,2,2}$} \put(25,35){\vector(0,-1){20}} \put(30,20){$f\!_{1,2,1}$} \put(115,5){\vector(1,0){80}} \put(150,10){$f\!_{1,1,2}$} \put(95,5){\vector(-1,0){60}} \put(60,10){$f\!_{1,1,1}$} \end{picture} \end{center} where \begin{itemize} \item $V_7\simeq{\Bbb P}({\cal O}_{{\Bbb P}^2}\oplus{\cal O}_{{\Bbb P}^2}(1))$, the map $f\!_{1,1,2}$ is the blow-up of ${\Bbb P}^3$ in a point $p_1\in L$ and $f\!_{1,1,1}$ is the canonical projection $V_7\to{\Bbb P}^2$. \item $f\!_{1,1}:\widetilde{V_7}(L\ni p_1)\to V_7$ is the blow-up of $V_7$ with center the strict transform of $L$ in the blow-up of $V_7\to{\Bbb P}^3$. The projection $f\!_{1,2}:\widetilde{V_7}(L\ni p_1)\to{\Bbb F}_1$ is a ${\Bbb P}^1$-bundle. \item $f_1:X\to\widetilde{V_7}(L\ni p_1)$ is the blow-up of $\widetilde{V_7}(L\ni p_1)$ with center an exceptional line of the blow-up $\widetilde{V_7}(L\ni p_1)\to V_7$. \end{itemize} Let $H_1\!=\!f_1^*f_{1,1}^*f_{1,1,1}^*{\cal O}_{{\Bbb P}^2}(1),$ $H_2\!=\!f_1^*f_{1,1}^*f_{1,1,2}^*{\cal O}_{{\Bbb P}^3}(1),$ $H_3\!=\!f_1^*f_{1,2}^*f_{1,2,2}^*{\cal O}_{{\Bbb P}^1}(1)$ and $D_{f_{1,1}}$, $D_{f_1}$ the pull-backs of the exceptional divisors of the blow-ups $f_{1,1}$ resp. $f_1$ on $X$. Then we have \begin{eqnarray*} && -K_X=2H_1+2H_2-D_{f_{1,1}}-D_{f_1}, \ H_3\sim H_1-D_{f_{1,1}}\\ &&\mbox{\hspace{-20pt}} (a_1H_1\!+\!a_2H_2\!-\!a_3D_{f_{1,1}}\!-\!a_4D_{f_1}).(-\!K_X)^2= 12a_1+15a_2-5a_3-3a_4,\ (-K_X)^3=46. \end{eqnarray*} Let $g:=f_{1,2,2}\circ f_{1,2} \circ f_1$. Since $g^*\Omega^1_{{\Bbb P}^1}\simeq {\cal O}_X(-2H_1+2D_{f_{1,1}})$, it follows that $g^*\Omega^1_{{\Bbb P}^1} \subset \Omega^1_X$ is a $(-K_X)$- destabilising subsheaf, with $\mu(g^*\Omega^1_{{\Bbb P}^1})\!=\!-14 > -\frac{46}{3}\!=\!\mu(\Omega^1_X).$ \ENDBOX \end{example} \noindent Before we go to the next example we shall collect some auxiliary lemmas. \begin{Lemma}\label{Conic-bundel.1} Let $S$ be a smooth projective surface, $f: X \longrightarrow S$ a conic bundle and $\Delta \subset S$ the discriminant locus. Then we have an exact sequence $$ 0 \longrightarrow f^*\Omega^1_S \stackrel{\delta}{\longrightarrow} \Omega^1_X \longrightarrow \Omega^1_{X/S} \longrightarrow 0 $$ and $\Omega^1_{X/S}\simeq{\cal I}_\Gamma\otimes\omega_X\otimes f^*\omega^{-1}_S$, where $\Gamma$ is a closed Cohen-Macaulay subscheme of $X$ of pure dimension 1 with $f(\Gamma)=\Delta$. The restriction $f|_{\Gamma\!\setminus\! f^{-1}(\Delta_{sing})}: \Gamma\!\setminus\! f^{-1}(\Delta_{sing})\longrightarrow \Delta_{reg}$ is an isomorphism and $\Gamma \cap X_s=(X_s)_{red}$ for all $s\in \Delta_{sing}$. \end{Lemma} \begin{pf} $f^*\Omega^1_S \longrightarrow \Omega^1_X$ drops rank in codimension 2, whence the first three assertions follow from the theory of the Eagon-Northcott complex \cite{Eagon-Northcott}. It is also clear that $f(\Gamma)=\Delta$. $S$ can be covered by affine open sets $U_\alpha$ such that $f^{-1}(U_\alpha)$ is isomorphic over $U_\alpha$ to the closed subsheme of $U_\alpha\times{\Bbb P}^2$ given by a quadratic equation: $$ g_\alpha:=\sum_{0\leq i \leq j \leq 2} A_{ij}X_iX_j ,\ \ \ A_{ij} \in H^0(U_\alpha,{\cal O}_{U_\alpha}).$$ Using the diagram $$\begin{array}{ccccccc} & & & 0 & & \\ & & & \downarrow & & \\ 0 \longrightarrow & {\cal O}_{U_\alpha\times{\Bbb P}^2}(-2)|_{f^{-1}(U_\alpha)} & \longrightarrow & \Omega^1_{U_\alpha\times{\Bbb P}^2/U_\alpha}|_{f^{-1}(U_\alpha)} & \longrightarrow & \Omega^1_{X/S}|_{f^{-1}(U_\alpha)} & \longrightarrow 0 \\ & \multicolumn{2}{r}{(\frac{\partial g_\alpha}{\partial X_0}, \frac{\partial g_\alpha}{\partial X_1},\frac{\partial g_\alpha}{\partial X_2})\searrow}& \downarrow & \\ & & &3{\cal O}_{U_\alpha\times{\Bbb P}^2}(-1)|_{f^{-1}(U_\alpha)} & & & \\ & & & \multicolumn{2}{r}{\downarrow\ \ \ (X_0,X_1,X_2)} & \\ & & & {\cal O}_{U_\alpha\times{\Bbb P}^2}|_{f^{-1}(U_\alpha)} & & \\ & & & \downarrow & & \\ & & & 0 & & \\ \end{array}$$ one can deduce that $\Gamma\cap f^{-1}(U_\alpha)$ is the closed subscheme of $U_\alpha\times{\Bbb P}^2$ given by the equations: $$\frac{\partial g_\alpha}{\partial X_0}=\frac{\partial g_\alpha}{\partial X_1}= \frac{\partial g_\alpha}{\partial X_2}=0 $$ (in fact, the three equations are enough by Euler's identity). Now the last 2 assertions are clear. \end{pf} \begin{Lemma}\label{Cbundel.Aufblasung.T} Let $Y \stackrel{\pi}{\longrightarrow} X$ be the blow-up of a conic bundle $ X \stackrel{f}{\longrightarrow} S$ with center a smooth irreducible subsection $C$ over $S$ (i.e.~ $f|_C:C\to S$ is an embedding). Let $L \in Pic(Y)$, such that $(\pi_*L)^{**}$ is a $f$-ample line bundle on $X$. Then: \begin{eqnarray*} &(i)& H^0(Y,T_Y \otimes L^{-1})=0, \mbox{ if } \mu(L) > \mu(T_{Y/S}) \mbox{ and}\\ &(ii)& H^0(Y,\Omega^1_Y \otimes L^{-1})=0. \end{eqnarray*} \end{Lemma} \begin{pf} Straightforward and left to the reader. \end{pf} \begin{example} Let $X$ be the blow-up of ${\Bbb P}^3$ with center a union of a cubic $C$ in a plane $S$ and a point $p$ not in $S$. We will prove that $X$ has stable tangent bundle. For this we make use of the following diagram\\ \begin{picture}(300,150)(0,0) \put(145,70){$\bf X$} \put(0,70){${\Bbb P}({\cal O}_{{\Bbb P}^2} \oplus {\cal O}_{{\Bbb P}^2}(1))$} \put(110,0){${\Bbb P}({\cal O}_{{\Bbb P}^2} \oplus {\cal O}_{{\Bbb P}^2}(2))$} \put(260,70){$Y_3$} \put(145,130){$Y_4$} \put(42,130){${\Bbb P}^3$} \put(42,0){${\Bbb P}^2 $} \put(260,0){$W_4$} \put(140,73){\vector(-1,0){45}} \put(115,79){$f_1$} \put(150,65){\vector(0,-1){55}} \put(154,35){$f_2$} \put(157,73){\vector(1,0){100}} \put(198,79){$f_3$} \put(150,80){\vector(0,1){45}} \put(154,100){$f_4$} \put(50,65){\vector(0,-1){55}} \put(54,35){$\alpha$} \put(105,5){\vector(-1,0){45}} \put(85,9){$\beta$} \put(195,5){\vector(1,0){60}} \put(210,10){$\gamma$} \put(263,65){\vector(0,-1){55}} \put(267,35){$\delta$} \put(140,133){\vector(-1,0){81}} \put(95,137){$\varepsilon$} \put(50,83){\vector(0,1){45}} \put(54,100){$\zeta$} \end{picture}\\ \mbox{} Define $\zeta:V_7={\Bbb P}({\cal O}_{{\Bbb P}^2}\oplus{\cal O}_{{\Bbb P}^2}(1))\to {\Bbb P}^3$ to be the blow-up of ${\Bbb P}^3$ in $p$ and denote again by $C\subset V_7$ the proper transform of $C\subset S \subset {\Bbb P}^3$. Define also $f_2$ to by the elementary transformation of $f_1$ along C, and $W_4$ to be the cone over the Veronese surface in ${\Bbb P}^5$ and $\gamma$ the blow-up of the vertex. Let $H_1:=f_1^*\alpha^*{\cal O}_{{\Bbb P}^2}(1)$, $H_2:=f_1^*\zeta^*{\cal O}_{{\Bbb P}^3}(1)$, $H_3:=f_2^*{\cal O}_{{\Bbb P}({\cal O}_{{\Bbb P}^2}\!\oplus{\cal O}_{{\Bbb P}^2}(2))}(1)$ and $D_{f_1}$, $D_{f_2} $ the exceptional divisors of $f_1$ resp. $f_2$. Then it follows that we have \begin{eqnarray*} && D_{f_2} \sim 3H_1 -D_{f_1},\ \ H_2-H_1 \sim H_3-D_{f_2} \\ && -K_X\sim 2H_1+2H_2-D_{f_1}\sim H_1+2H_3-D_{f_2}\\ && \Rightarrow\ -K_X\sim H_2+H_3\\ &&(a_1H_1+a_2H_2+a_3H_3).(-K_X)^2= 9a_1+13a_2+19a_3,\ (-K_X)^3=32.\\ && a_1H_1+a_2H_2+a_3H_3\sim\\ && \sim (a_1+2a_3)H_1+(a_2+a_3)H_2-a_3D_{f_1}\\ && \sim (a_1+a_2)H_1+(a_2+a_3)H_3-a_2D_{f_2}. \end{eqnarray*} Since $f_2:X\longrightarrow{\Bbb P}({\cal O}\oplus{\cal O}(2))$ is the elementary transformation of $f_1:X\longrightarrow{\Bbb P}({\cal O}\oplus{\cal O}(1))$ the blow-up of ${\Bbb P}({\cal O}\oplus{\cal O}(1))$ with center a smooth subsection $C$ over ${\Bbb P}^2$, we have an isomorphism $D_{f_2}\stackrel{f_1}{\simeq}\alpha^{-1}(C^\prime)\simeq Z:={\Bbb P}({\cal O}_{C^\prime}\oplus{\cal O}_{C^\prime}(1)) \stackrel{\rho}{\longrightarrow} C^\prime :=\alpha(C).$ By this isomorphism, the curve $C\subset\alpha^{-1}(C^\prime)$ corresponds to a curve $C^{\prime\prime}$ on $D_{f_2}$. The map $g:=\alpha\circ f_1:X\longrightarrow {\Bbb P}^2$ is a conic bundle with discriminate locus $C^\prime$. Since $C\subset Z$ is linear equivalent to $H_2|Z$ and $D_{f_1}|D_{f_2}=C^{\prime\prime}\stackrel{f_1}{\simeq} C$, it follows that ${\cal O}_X(D_{f_1})|D_{f_2}\simeq{\cal O}_{D_{f_2}}(C^{\prime\prime})\simeq {\cal O}_Z(C)={\cal O}_Z(1)$. Furthermore, we have: ${\cal O}_X(H_1)|D_{f_2} \simeq \rho^*{\cal O}_{C^\prime}(1)$ and ${\cal O}_X(H_2)|D_{f_2} \simeq {\cal O}_Z(1)$. Thus, we have ${\cal O}_X(H_3)|D_{f_2}\simeq\rho^*{\cal O}_{C^\prime}(2)$ and ${\cal O}_X(a_1H_1+a_2H_2+a_3H_3)|Z\simeq \rho^*{\cal O}_{C^\prime}(a_1+2a_3)\otimes{\cal O}_Z(a_2).$ Let ${\cal L}={\cal O}_X(a_1H_1+a_2H_2+a_3H_3) \subset T_X$. Consider the exact sequence $$0\longrightarrow T_{X/{\Bbb P}^2}\longrightarrow T_X \longrightarrow g^*T_{{\Bbb P}^2}.$$ Using lemma \ref{Cbundel.Aufblasung.T}, it follows that $a_2+a_3\leq 0$, if $9a_1+13a_2+19a_3 > 5=\mu(T_{X/{\Bbb P}^2}).$\\ Let ${\cal L} \subset g^*T_{{\Bbb P}^2}$ be a line bundle with maximal $\mu({\cal L})$. Using \begin{eqnarray*} && 0\not=H^0(X,g^*T_{{\Bbb P}^2} \otimes {\cal L}^{-1}) \subset H^0(D_{f_2},g^*T_{{\Bbb P}^2}\otimes {\cal L}^{-1}|D_{f_2})\simeq\\ &&\simeq H^0(C^\prime,T_{{\Bbb P}^2}|_{C^\prime}\otimes{\cal O}_{C^\prime}(-a_1-2a_3) \otimes S^{-a_2}({\cal O}_{C^\prime}\oplus{\cal O}_{C^\prime}(1))),\mbox{ it follows} \\ && a_2\leq 0 \mbox{ and } a_1+a_2+2a_3\leq 1, \mbox{ because } \deg C^\prime=3 \end{eqnarray*} Hence $\mu({\cal L})=9a_1+13a_2+19a_3=9(a_1+a_2+2a_3)+(a_2+a_3)+3a_2\leq 9$. Now let ${\cal L} \subset \Omega^1_X$. By lemma \ref{Conic-bundel.1} we have an exact sequence: \begin{eqnarray*} && \mbox{\hspace{-30pt}} 0\longrightarrow g^*\Omega^1_{{\Bbb P}^2} \longrightarrow \Omega^1_X \longrightarrow {\cal I}_{C^{\prime\prime}}\otimes\omega_{X/{\Bbb P}^2} \longrightarrow 0\ (\mbox{ with } \omega_{X/{\Bbb P}^2}\simeq{\cal O}_X(3H_1-H_2-H_3)\ ).\\ && \mbox{If }H^0(X,\Omega^1_X\otimes{\cal L}^{-1})\not=0 \mbox{ then } \left\{ \begin{array}{cl} (1) & H^0(X,g^*\Omega^1_{{\Bbb P}^2}\otimes{\cal L}^{-1})\not=0\mbox{ or}\\ (2) & H^0(X,{\cal I}_{C^{\prime\prime}} \otimes\omega_{X/{\Bbb P}^2}\otimes{\cal L}^{-1})\not=0. \end{array} \right. \end{eqnarray*} (1) $g_*(g^*\Omega^1_{{\Bbb P}^2}\otimes{\cal L}^{-1})\subset \Omega^1_{{\Bbb P}^2}(-a_1-2a_3)\otimes S^{-a_2-a_3}({\cal O}_{C^\prime}\oplus{\cal O}_{C^\prime}(1))$ $$\Rightarrow a_2+a_3\leq 0\mbox{ and }a_1+a_2+3a_3\leq -2;$$ $\beta_*f_{2*}(g^*\Omega^1_{{\Bbb P}^2}\otimes{\cal L}^{-1})\subset \Omega^1_{{\Bbb P}^2}(-a_1-a_2)\otimes S^{-a_2-a_3}({\cal O}_{C^\prime}\oplus{\cal O}_{C^\prime}(2))$ $$\Rightarrow a_2+a_3\leq 0 \mbox{ and }a_1+3a_2+2a_3\leq -2.$$ If ${\cal L}\subset g^*\Omega^1_{{\Bbb P}^2}$ has maximal $\mu({\cal L})$, then we have: \begin{eqnarray*} && 0\not=H^0(X,g^*\Omega^1_{{\Bbb P}^2}\otimes{\cal L}^{-1}) \subset H^0(D_{f_2},g^*\Omega^1_{{\Bbb P}^2}\otimes{\cal L}^{-1}|D_{f_2})\simeq\\ && H^0(C^\prime,\Omega^1_{{\Bbb P}^2}|_{C^\prime}\otimes{\cal O}_{C^\prime}(-a_1-2a_3) \otimes S^{-a_2}({\cal O}_{C^\prime}\oplus{\cal O}_{C^\prime}(1)))\\ && \Rightarrow\ a_2\leq 0 \mbox{ and } a_1+a_2+2a_3\leq -2. \end{eqnarray*} (2) ${\cal I}_{C^{\prime\prime}}\otimes\omega_{X/{\Bbb P}^2}\otimes{\cal L}^{-1}\subset {\cal O}_X((3-a_1)H_1+(-1-a_2)H_2+(-1-a_3)H_3)$ $$\Rightarrow \left\{ \begin{array}{clcl} (\alpha\circ f_1)_*: & a_2+a_3\leq -2 & \mbox{and} & a_1+a_2+3a_3\leq -1;\\ (\beta\circ f_2)_* : & a_2+a_3\leq -2 & \mbox{and} & a_1+4a_2+2a_3\leq -2. \end{array} \right. $$ If $0\not=H^0(X,{\cal I}_{C^{\prime\prime}}\otimes {\cal O}_X((3-a_1)H_1+(-1-a_2)H_2+(-1-a_3)H_3))$, then $\omega_{X/{\Bbb P}^2}\otimes{\cal L}^{-1}$ has a global section vanishing on $C^{\prime\prime}$. It follows that $\omega_{X/{\Bbb P}^2}\otimes{\cal L}^{-1}|D_{f_2}$ has a global section vanishing on $C^{\prime\prime}$. Therefore \begin{eqnarray*} && 0\not=H^0(D_{f_2},{\cal O}_X((3-a_1)H_1+(-1-a_2)H_2+(-1-a_3)H_3) \otimes{\cal O}_{D_{f_2}}(-C^{\prime\prime}))\\ && \simeq H^0(Z,\rho^*{\cal O}_{C^\prime}(1-a_1-2a_3)\otimes{\cal O}_Z(-2-a_2)), \mbox{ implies that}\\ && a_2\leq -2 \mbox{ and } a_1+a_2+2a_3\leq -1. \end{eqnarray*} Hence $\mu({\cal L})=9a_1+13a_2+19a_3=9(a_1+a_2+2a_3)+(a_2+a_3)+3a_2\leq -17$. \ENDBOX \end{example} \addcontentsline{toc}{section}{Literaturverzeichnis}

9407

alg-geom/9407010

en

https://arxiv.org/abs/alg-geom/9407010

alg-geom/9407010

Richard Hain

Richard Hain and Jun Yang

Real Grassmann Polylogarithms and Chern Classes

42 pages, amslatex

In this paper we define real grassmann polylogarithms, which are real single valued analogues of the grassmann polylogarithms (or higher logarithms) defined by Hain and MacPherson. We prove the existence of all such real grassmann polylogs, at least generically. We also prove that the canonical choice of such an m-polylogarithm represents the Beilinson Chern class on the rank m part of the algebraic K-theory of the generic point of every complex algebraic variety. One part of each such grassmann m-polylogarithm is a real, single-valued function defined generically on the grassmannian of m planes in C^{2m}. We prove that this function represents the Borel regulator (up to a factor of 2) on K_{2m-1} of all number fields.

alg-geom

\section{Introduction}\label{intro} In this paper we define and prove the existence of real Grassmann polylogarithms which are the real single-valued analogues of the Grassmann polylogarithms defined in \cite{hain-macp} and constructed in \cite{hain-macp,hana-macp,hana-macp_2,hain:generic}. We prove that if $\eta_X$ is the generic point of a complex algebraic variety $X$, then the $m$th such polylogarithm represents the Beilinson Chern class $$ c_m^B : r_m K_n(\eta_X) \to \Hd^{2m-m}(\eta_X,{\Bbb R}(m)), $$ where $r_m K_n$ denotes the part of the algebraic $K$-theory coming from $GL_m$. In particular, we show that there is a {Bloch-Wigner function} $D_m$ defined on a Zariski open subset of the grassmannian of $m$ planes in ${\Bbb C}^{2m}$ which satisfies a canonical $(2m+1)$-term functional equation. In order to describe our results in more detail, we recall the definition of Bloch-Wigner functions from \cite[\S 11]{hain-macp}. Denote the algebra of global holomorphic forms with logarithmic singularities at infinity on a smooth complex algebraic variety $X$ by $\Omega^\bullet(X)$. A {\it Bloch-Wigner function} on $X$ is simply a real single-valued function on $X$ that is a polynomial in real and imaginary parts of multivalued functions on $X$ of the form $$ x \mapsto \sum_I\int_{x_0}^x w_{i_1}\dots w_{i_r} $$ where each $w_{i_j}$ is in $\Omega^1(X)$. For example, the logarithm and classical dilogarithm $\ln_2(x)$ can be expressed as iterated integrals, $$ \log x = \int_1^x \frac{dz}{z},\quad \ln_2(x) = \int_0^x \frac{dz}{1-z}\frac{dz}{z}, $$ so the single valued logarithm $$ D_1(x) = \log|x| $$ and the Bloch-Wigner function $$ D_2(x) = \Im \ln_2(x) + \log|x|\Arg(1-x) $$ are both Bloch-Wigner functions. More generally, Ramakrishnan's single-valued cousins of the classical polylogarithms $\ln_m(x)$ are Bloch-Wigner functions. The set of Bloch-Wigner functions on $X$ is an ${\Bbb R}$ algebra, which will be denoted by ${\cal BW}(X)$. Define the irregularity $q(X)$ of $X$ to be half the first Betti number of any smooth compactification of $X$. When $q(X)=0$, the elements of ${\cal BW}(X)$ have a more intrinsic description --- they are precisely the functions that occur as the matrix entries of period maps of real variations of mixed Hodge structure over $X$, all of whose weight graded quotients are constant variations of type $(p,p)$. The complex $\Omega_{\cal BW}^\bullet(X)$ of Bloch-Wigner forms on $X$ is defined to be the subcomplex of the real de~Rham complex of $X$ that is generated by ${\cal BW}(X)$ and the real and imaginary parts of elements of $\Omega^\bullet(X)$ (cf.\ \cite{yang:thesis}.) It is closed under exterior differentiation, and is therefore a differential graded algebra. It has a natural weight filtration $W_\bullet$ which comes from the usual weight filtrations on $\Omega^\bullet(X)$ and on iterated integrals. It is easy to describe the weight filtration on ${\cal BW}(X)$ when $q(X)=0$: the weight of an iterated integral of length $l$ is simply $2l$. This filtration induces one on ${\cal BW}(X)$ in the natural way, so that $$ D_1 \in W_2{\cal BW}({\Bbb C}^\ast)\quad\text{and}\quad D_2 \in W_4{\cal BW}({\Bbb C}-\{0,1\}), $$ for example. The Bloch-Wigner (or BW-)cohomology of $X$ is the analogue of the Deligne cohomology of $X$ constructed using Bloch-Wigner forms in place of usual forms. Specifically, the Bloch-Wigner cohomology of $X$ is the cohomology of the complex $$ {\Bbb R}(m)_{\cal BW}\supdot(X) := \cone[F^pW_{2m}\Omega\supdot(X) \rightarrow W_{2m}\Omega_{\cal BW}\supdot(X)\otimes {\Bbb R}(m-1)][-1], $$ where the map $\Omega^\bullet(X) \to \Omega_{\cal BW}^\bullet(X)\otimes{\Bbb R}(m-1)$ is defined by taking a complex valued form to its reduction mod ${\Bbb R}(m)$.% \footnote{Recall that ${\Bbb R}(m)$ is the subgroup $(2\pi i)^m{\Bbb R}$ of ${\Bbb C}$. There is a standard identification ${\Bbb C}/{\Bbb R}(m)\cong {\Bbb R}(m-1)$.} The inclusion of $\Omega_{\cal BW}^\bullet(X)$ into the real log complex associated to $X$ induces a natural map $$ H\supdot_{\cal BW}(X,{\Bbb R}(m))\to \Hd\supdot(X,{\Bbb R}(m)). $$ Grassmann polylogarithms are specific elements of the BW-cohomology of certain simplicial varieties $G^m_\bullet$. Recall from \cite{hain-macp} that the variety $G^m_n$ is defined to be the Zariski open subset of the grassmannian of $n$ dimensional linear subspaces of ${\Bbb P}^{m+n}$ which consists of those $n$ planes that are transverse to each stratum of the union of the coordinate hyperplanes. Intersecting elements of $G^m_n$ with the $j$th coordinate hyperplane defines ``face maps'' $$ A_j : G^m_n \to G^m_{n-1}, \qquad j=0,\dots,m+n $$ These satisfy the usual simplicial identities. It is natural to place $G^m_n$ in dimension $m+n$ as there are $m+n+1$ face maps emanating from it. The collection of the $G^m_n$ with $0\le n \le m$ and the face maps $A_j$ will be denoted by $G^m_\bullet$. It is a truncated simplicial variety. We can apply the functor ${\Bbb R}(m)_{\cal BW}\supdot({\phantom{X}})$ to a simplicial variety $X_\bullet$ to obtain a double complex ${\Bbb R}(m)_{\cal BW}\supdot(X_\bullet)$. The homology of the associated total complex will be denoted by $H_{\cal BW}^\bullet(X_\bullet,{\Bbb R}(m))$. There is a natural map $$ \delta : H^{2m}_{\cal BW}(G^m\subdot,{\Bbb R}(m)) \to \Omega^m(G^m_0). $$ A {\it real Grassmann $m$-logarithm} is an element $L_m$ of $H^{2m}_{\cal BW}(G^m\subdot,{\Bbb R}(m))$ whose image under $\delta$ is the ``volume form'' $$ \mbox{vol}_m = \frac{dx_1}{x_1}\wedge \dots \wedge \frac{dx_m}{x_m} $$ on $G^m_0 \cong ({\Bbb C}^\ast)^m$. This is the real analogue of the definition of a generalized $m$-logarithm given in \cite[\S 12]{hain-macp}. Since Bloch-Wigner functions are single valued, many of the technical problems one encounters when working with multivalued are not present. Our first result is: \medskip \noindent{\bf Theorem A.} {\it For each $m \le 4$ there exists a real Grassmann $m$-logarithm.} \medskip The real Grassmann trilogarithm was first constructed as a real Deligne cohomology class in \cite{yang:thesis}. The symmetric group $\Sigma_{m+n+1}$ acts on $G^m_n$. If we insist that each component of a representative of $L_m$ span a copy of the alternating representation, then $L_m$ is unique. A {\it generic real Grassmann $m$-logarithm} is an element of $H_{\cal BW}^{2m}(U^m_\bullet,{\Bbb R}(m))$, where $U^m_\bullet$ is a Zariski open subset of $G^m_\bullet$ satisfying $U^m_0 = G^m_0$, whose image under $$ \delta : H^{2m}_{\cal BW}(U^m\subdot,{\Bbb R}(m)) \to \Omega^m(G^m_0) $$ is the volume form $\mbox{vol}_m$. Observe that every real Grassmann $m$-logarithm is a generic real Grassmann $m$-logarithm. In the general case, we prove the following result, which is the analogue for single valued polylogarithms of the main result of \cite{hain:generic}. \medskip \noindent{\bf Theorem B.} {\it For each $m \ge 1$ there exists a canonical generic real Grassmann $m$-logarithm.} \medskip The truncated simplicial variety $G^m_\bullet$ can be viewed as a ``quotient'' of the classifying space $B_\bullet GL_m(\C)$ of rank $m$ vector bundles. One of the key points in the construction of real Grassmann polylogarithms and in relating them to Chern classes is to show that the ``alternating part'' of the universal Chern class $$ c_m \in \Hd^{2m}(B_\bullet GL_m(\C),{\Bbb R}(m)) $$ descends, along the ``quotient map'', to a class in $\Hd^{2m}(G^m_\bullet,{\Bbb R}(m))$. This approach is hinted at in the survey \cite[p.~107]{bryl-zucker} of Brylinski and Zucker, although the existence of this descended class is not evident. The descent of the universal Chern class established in Section \ref{descent}. The part of the cocycle of a (generic) real Grassmann $m$-logarithm is a Bloch-Wigner function $D_m$ defined (generically) on $G^m_{m-1}$ which satisfies the $(2m+1)$-term functional equation \begin{equation}\label{funct_eqn} \sum_{j=0}^{2m}(-1)^j A_j^\ast D_m = 0. \end{equation} The function $D_1$ is simply $\log|{\phantom{X}}|$, the second function $D_2$ is the pullback of the Bloch-Wigner dilogarithm along the ``cross-ratio map'' $G^2_1 \to {\Bbb P}^1 - \{0,1,\infty\}$ (cf. \cite[p.~403]{hain-macp}). The functional equation (\ref{funct_eqn}) is the standard 5-term equation. The function $D_3$ is the single-valued trilogarithm whose existence was established in \cite{hain-macp} and for which Goncharov remarkably expressed in terms of the single-valued classical trilogarithm in \cite{goncharov:trilog}. Recall that the {\it rank} filtration $$ 0 = r_0 K_m(R) \subseteq r_1 K_m(R) \subseteq r_2 K_m(R) \subseteq \dots \subseteq K_m(R)_{\Bbb Q} := K_m(R)\otimes {\Bbb Q} $$ of the rational $K$-groups of a ring $R$ is defined by $$ r_j K_m(R) = \im \{H_m(GL_j(R),{\Bbb Q})\rightarrow H_m(GL(R),{\Bbb Q})\}\cap K_m(R)_{\Bbb Q}. $$ Here $K_m(R)_{\Bbb Q}$ is identified with its image in $H_m(GL(R),{\Bbb Q})$ under the Hurewicz homomorphism. By Suslin's stability theorem \cite{suslin}, $r_mK_m(F) = K_m(F)_{\Bbb Q}$ whenever $F$ is an infinite field. In Section \ref{homology} we show that a Bloch-Wigner function defined generically on $G^m_{m-1}$ and which satisfies the functional equation (\ref{funct_eqn}) defines an element of $$ H^{2m-1}(GL_m({\Bbb C})^\delta,{\Bbb R}), $$ where $GL_m({\Bbb C})^\delta$ denotes the general linear group with the discrete topology. Such a function therefore defines a mapping $$ r_mK_{2m-1}(\spec {\Bbb C}) \to {\Bbb R}. $$ \medskip \noindent{\bf Theorem C.} {\it If $D_m$ is a Bloch-Wigner function defined generically on $G^m_{m-1}$ associated to the canonical choice of a generic real Grassmann $m$-logarithm, then the associated map $r_mK_{2m-1}({\Bbb C}) \to {\Bbb R}$ is equals the restriction of the Beilinson-Chern class $$ c_m^B : K_{2m-1}(\spec{\Bbb C}) \to \Hd^1(\spec{\Bbb C},{\Bbb R}(m))\cong {\Bbb R} $$ to $r_mK_{2m-1}(\spec{\Bbb C})$.} \medskip If $k$ is a number field then, by \cite{yang:rank,borel-yang}, $$ r_mK_{2m-1}(\spec k)=K_{2m-1}(\spec k) $$ It follows that the {\it regulator mapping} $$ c_p^B : K_{2m-1}(\spec k) \to \Hd^1(\spec k,{\Bbb R}(m)) \approx {\Bbb R}^{d_m}, $$ where $d_m$ is $r_1 + r_2$ or $r_2$ according to whether $m$ is odd or even, can be expressed in terms of a generic real $m$-logarithm function. It then follows by standard arguments, using fundamental results of Borel, that when $m>1$, the value $\zeta_k(m)$ of the Dedekind zeta function of $k$ can be expressed, up to the product of a suitable power of $\pi$, a non-zero rational number, and the square root of the discriminant of $k$, as a determinant of values of $D_m$ evaluated at certain $k$-rational points of $G^m_{m-1}$. This generalizes the classical theorem of Dedekind for the residue at $s=1$ of $\zeta_k(s)$ and similar formulas for the values at $s=2$ due to Bloch and Suslin, and at $s=3$ due to Goncharov \cite{goncharov:trilog} and Yang \cite{yang:annoc,yang:thesis}. An ultimate goal is to express a generic real Bloch-Wigner $m$-logarithm function in terms of the single-valued classical $m$-logarithm. In this case, the Borel regulator would be expressed in terms of a determinant of values of the single-valued classical $m$-logarithm at $k$ rational points of ${\Bbb P}^1-\{0,1,\infty\}$.\footnote{ This is a weak statement of Zagier's conjecture.} More importantly, it would show how to use the single-valued classical $m$-logarithm to define the $m$th regulator. To date this has only been done when $m\le3$: $m=2$ by Bloch and Suslin \cite{bloch}, $m=3$ by Goncharov \cite{goncharov:trilog}. More generally, a generic real Grassmann $m$-logarithm defines a function $$ r_mK_n(\eta_X) \to \Hd^{2m-n}(\eta_X,{\Bbb R}(m)) $$ where $\eta_X$ denotes the generic point of the complex algebraic variety $X$. \medskip \noindent{\bf Theorem D.} {\it The function $$ r_mK_n(\eta_X) \to \Hd^{2m-n}(\eta_X,{\Bbb R}(m)) $$ associated to the canonical choice of a generic real Grassmann polylogarithm is the restriction of the Beilinson-Chern class $c_m^B$ to $r_mK_n(\eta_X)$.} \begin{remark} One should be able to write down the Chern class on all of $K\subdot(\eta_X)$ using Goncharov's work \cite{goncharov:chern} by constructing ``generic bi-Grassmann polylogarithms'', but we have not yet done this. \end{remark} \noindent{\it Conventions.} In this paper, all simplicial objects are strict --- that is, they are functors from the category $\Delta$ of finite ordinals and {\it strictly} order preserving maps to, say, the category of algebraic varieties. As is standard, the finite set $\{0,1,\dots,n\}$ with its natural ordering will be denoted by $[n]$. Let $r$ and $s$ be positive integers with $r \le s$. Denote the full subcategory of $\Delta$ whose objects are the ordinals $[n]$ with $r\le n \le s$ by $\Delta[r,s]$. An $(r,s)$ {\it truncated} simplicial object of a category ${\cal C}$ is a contravariant functor from $\Delta[r,s]$ to ${\cal C}$. The word {\it simplicial} will be used generically to refer to both simplicial objects and truncated simplicial objects. However, we will use the word truncated when we do want to emphasize the difference. The distinction will be made in \S \ref{descent} where is will be significant. By Deligne cohomology, we shall mean Beilinson's refined version of Deligne cohomology as defined in \cite{beilinson:hodge} which is sometimes called absolute Hodge cohomology. It can be expressed as an extension $$ 0 \to \Ext^1_{\cal H}({\Bbb Q},H^{k-1}(X,{\Bbb Q}(p))) \to \Hd^k(X,{\Bbb Q}(p)) \to \Hom_{\cal H}({\Bbb Q},H^k(X,{\Bbb Q}(p)))\to 0 $$ where ${\cal H}$ denotes the category of ${\Bbb Q}$ mixed Hodge structures. To avoid confusion between, say, the $K$-theory of the ring ${\Bbb C}$ and the variety ${\Bbb C}$, we shall view $K$ as a functor on schemes. We shall therefore denote the $K$-theory of a ring $R$ by $K_\bullet(\spec R)$. \section{Bloch-Wigner Forms} In this section, we introduce the complex of Bloch-Wigner forms on a smooth complex algebraic variety. Let $X$ be a smooth variety. Choose any smooth compactification ${\overline{X}}$ of $X$ where $D:={\overline{X}} - X$ is a normal crossings divisor. Denote the complex of global meromorphic forms on ${\overline{X}}$ which are holomorphic on $X$ and have logarithmic singularities along $D$ by $\Omega^\bullet({\overline{X}} \log D)$. This maps injectively to the complex of holomorphic forms on $X$. Its image does not depend on the choice of ${\overline{X}}$, \cite[(3.3)]{hain-macp}, and will be denoted by $\Omega^\bullet(X)$. We now recall the definition of Bloch-Wigner functions on $X$ from \cite[\S 11]{hain-macp}. Denote the algebra of all iterated integrals of elements of $\Omega^1(X)$ by $A(X)$, and those that are relatively closed by $H^0(A(X))$. Fix a base point $x\in X$. Taking a relatively closed iterated integral $$ \sum \int \omega_{i_1}\omega_{i_2}\dots\omega_{i_r} $$ to the function $$ z \mapsto \int_x^z \omega_{i_1}\omega_{i_2}\dots\omega_{i_r} $$ defines an injective algebra homomorphism $$ H^0(A(X))\rightarrow \widetilde{E}^0(X,x) $$ where $\widetilde{E}^0(X,x)$ denotes the multivalued differentiable functions on $X$ (see \cite[\S 2]{hain-macp}). The image of the above map will be denoted by $\widetilde{\O}(X,x)$. Let $\widetilde{\O}_{\Bbb R}(X,x)$ denote the subalgebra of the algebra of multivalued, real valued functions on $X$ generated (as an ${\Bbb R}$-algebra) by the real and imaginary parts of elements of $\widetilde{\O}(X,x)$. The algebra ${\cal BW}(X)$ is defined to be the subalgebra of $\widetilde{\O}_{\Bbb R}(X,x)$ consisting of single valued functions. Equivalently, ${\cal BW}(X)$ is the subalgebra of $\widetilde{\O}_{\Bbb R}(X,x)$ invariant under monodromy: $$ {\cal BW}(X)=\widetilde{\O}_{\Bbb R}(X,x)^{\pi_1(X,x)}. $$ Although this construction makes use of a base point, the ring ${\cal BW}(X)$ itself depends only on $X$. The assignment of ${\cal BW}(X)$ to $X$ is a contravariant functor from the category of smooth complex algebraic varieties to the category of ${\Bbb R}$-algebras. We call ${\cal BW}(X)$ {\it the ring of Bloch-Wigner functions on $X$}. There is a natural weight filtration on $\Omega^\bullet(X)$. It induces one on $A(X)$ by linear algebra, and one on $H^0(A(X))$ by restriction. This weight filtration passes to $\widetilde{\O}(X,x)$ and eventually to a weight filtration on ${\cal BW}(X)$. This filtration is independent of all choices (cf. \cite[\S 11]{hain-macp}). Denote the subalgebra of the real de~Rham complex of $X$ generated by the real and imaginary parts of elements of $\Omega^\bullet(X)$ by $\Omega^\bullet_{\Bbb R}(X)$. \begin{definition} The {\it complex of Bloch-Wigner forms on $X$\/} is defined to be the sub-algebra of the de~Rham complex $$ \Omega_{\cal BW}^\bullet(X) = {\cal BW}(X)\cdot\Omega^\bullet_{\Bbb R}(X), $$ generated by ${\cal BW}(X)$ and $\Omega^\bullet_{\Bbb R}(X)$. \end{definition} It is not difficult to see that the image of the natural inclusion $$ \Omega_{\cal BW}^\bullet (X)\hookrightarrow E^\bullet(X) $$ is closed under $d$, so that $\Omega_{\cal BW}^\bullet(X)$ is a d.g.\ algebra. The weight filtration of $\Omega^\bullet(X)$ induces a weight filtration on $\Omega_{\Bbb R}^\bullet(X)$. Taking the convolution of the weight filtrations of ${\cal BW}(X)$ and $\Omega^\bullet(X)$ we obtain a natural weight filtration on $\Omega_{\cal BW}^\bullet(X)$. \section{Bloch-Wigner cohomology} \label{bw-coho} We introduce a natural analogue of Deligne (or more accurately, absolute Hodge) cohomology with coefficients in ${\Bbb R}(m)$ which is defined using Bloch-Wigner forms. For a real vector space $V$, we denote $V\otimes {\Bbb R}(m)$ by $V(m)$, and we identify the quotient $V_{\Bbb C}/V(m)$ with $V(m-1)$ using the natural projection associated to the decomposition $V_{\Bbb C} = V_{\Bbb R}(m-1) \oplus V_{\Bbb R}(m)$. Suppose that $X$ is a smooth complex algebraic manifold. Taking the value of an element of $\Omega^\bullet(X)$ mod ${\Bbb R}(m)$ defines a map $$ \Omega^\bullet(X) \to \Omega^\bullet_{\Bbb R}(m-1) $$ which preserves the weight filtrations. Composing with the canonical inclusion $\Omega^\bullet_{\Bbb R}(X) \hookrightarrow \Omega_{\cal BW}\supdot(X)$ twisted by ${\Bbb R}(m-1)$, we obtain a natural weight filtration preserving map $$ \Omega\supdot(X) \rightarrow \Omega_{\cal BW}\supdot(X)(m-1). $$ For each $m\ge 0$, define a complex ${\Bbb R}(m)_{\cal BW}\supdot(X)$ by $$ {\Bbb R}(m)_{\cal BW}\supdot(X) := \cone[F^mW_{2m}\Omega\supdot(X) \rightarrow W_{2m}\Omega_{\cal BW}\supdot(X)(m-1)][-1]. $$ Define $H\supdot_{\cal BW}(X,{\Bbb R}(m))$ to be the cohomology of this complex. We shall call it the {\it ${\cal BW}$-cohomology of $X$ with coefficients in ${\Bbb R}(m)$}. It is clearly functorial in $X$. Denote by $\Hd^\bullet(X,{\Bbb R}(m))$ the absolute Hodge cohomology of $X$ with coefficients in ${\Bbb R}(m)$. This is Beilinson's refined version of Deligne-Beilinson cohomology \cite{beilinson:hodge}. It is defined as the cohomology of the complex $$ {\Bbb R}(m)_{\cal D}^\bullet(X) := \cone[F^mW_{2m}A_{\Bbb C}\supdot(X) \rightarrow W_{2m}A_{\Bbb R}\supdot(X)(m-1)][-1], $$ where $$ A = ((A_{\Bbb R}^\bullet,W_\bullet),(A_{\Bbb C}^\bullet,W_\bullet, F^\bullet)) $$ is a real mixed Hodge complex for $X$. (Note that the weight filtration used is the {\it filtration decal\'ee} of Deligne \cite[p.~15]{deligne:II}.) \begin{proposition} There is a natural map $H^\bullet_{\cal BW}(X,{\Bbb R}(m)) \to \Hd^\bullet(X,{\Bbb R}(m))$. \end{proposition} \begin{pf} We use the real mixed Hodge complex described in \cite[p.~73]{durfee-hain}. Denote it by $$ \left( (\widetilde{A}_{\Bbb R}^\bullet(X),W_\bullet), (\widetilde{A}_{\Bbb C}^\bullet(X),W_\bullet,F^\bullet)\right). $$ The mixed Hodge complex with the {\it filtration decal\'ee} used to compute Deligne cohomology is $$ \left( (A_{\Bbb R}^\bullet(X),W_\bullet),(A_{\Bbb C}^\bullet(X),W_\bullet,F^\bullet)\right), $$ where $$ W_lA^k = \left\{ a \in W_{l-k}\widetilde{A}^k : da \in W_{l-k-1}\widetilde{A}^{k+1}\right\} $$ and $$ A^\bullet = \bigcup_{l \ge 0} W_lA^\bullet $$ (cf. \cite[p.~145]{morgan}.) It is straightforward to show that the image of the inclusion of $\Omega_{\cal BW}^\bullet(X)$ into $\widetilde{A}_{\Bbb R}^\bullet(X)$ is contained in $A^\bullet(X)$ and that the corresponding map $$ \Omega_{\cal BW}^\bullet(X) \to A_{\Bbb R}^\bullet(X) $$ preserves $W_\bullet$. The result follows as the diagram $$ \begin{matrix} W_{2m}\Omega^\bullet(X) & \to & W_{2m}\Omega^\bullet_{\cal BW}(X)(m-1) \cr \downarrow & & \downarrow \cr W_{2m}A^\bullet_{\Bbb C}(X) & \to & W_{2m}A^\bullet_{\Bbb R}(X)(m-1)\cr \end{matrix} $$ in which the horizontal maps are reduction of values mod ${\Bbb R}(m)$, commutes. \end{pf} We shall need the simplicial analogue of this result. The proof is similar to that of the previous result. \begin{proposition} For each smooth simplicial variety $X_\bullet$, there is a natural map $$ H_{\cal BW}^\bullet(X_\bullet,{\Bbb R}(m)) \to \Hd^\bullet(X_\bullet,{\Bbb R}(m)). \qed $$ \end{proposition} Our polylogarithms lie in the ${\cal BW}$-cohomology of certain simplicial varieties. In order to construct polylogarithms, we shall need to compare the ${\cal BW}$- and Deligne cohomologies of these simplicial varieties. Recall that the irregularity $q(X)$ of a smooth complex algebraic variety $X$ is defined by $$ q(X) := \dim W_1H^1(X;{\Bbb Q})/2 = \dim H^{1,0}({\overline{X}}), $$ where ${\overline{X}}$ is any smooth completion of $X$. Most varieties in this paper will satisfy $q(X)=0$. This condition is satisfied by all Zariski open subsets of simply connected smooth varieties. In particular, it is satisfied by each $G^m_n({\Bbb C})$. Suppose that $X$ is a topological space with finitely generated fundamental group. Denote the ${\Bbb C}$-form of the Malcev Lie algebra associated to $\pi_1(X,x)$ by ${\frak g}(X,x)$. This is a topological Lie algebra. There is a canonical homomorphism $$ \Hcts^\bullet({\frak g}(X,x)) \to H^\bullet(X,{\Bbb C}), $$ induced, for example, by the homomorphism $$ \Lambda^\bullet({\frak g}^\ast)\rightarrow \Omega^\bullet(X)\hookrightarrow E^\bullet(X), $$ (see \cite[\S 7]{hain-macp}). Recall from \cite[(8.3)]{hain-macp} that $X$ is a rational $n$-$K(\pi,1)$ if this map is an isomorphism in degrees $\le n$ and injective in degree $n+1$. \begin{theorem}\label{summand} Suppose that $X_\bullet$ is a simplicial variety where each $X_m$ is smooth and has $q=0$. If, for each $m$, $X_m$ is a rational $(n-m)$-% $K(\pi,1)$, then the natural map $$ H_{\cal BW}^t(X_\bullet,{\Bbb R}(m)) \to \Hd^t(X_\bullet,{\Bbb R}(m)) $$ has a canonical splitting whenever $t\le n$. In particular, this map is surjective in degrees $\le n$. \end{theorem} \section{Simplicial Spaces with Symmetric Group Actions} \label{S-variety} Certain simplicial spaces come equipped with actions of symmetric groups on their spaces of simplices. This symmetric group action gives an extra algebraic structure to the cohomology of complexes associated to such simplicial spaces. This was first explored in \cite[\S9]{hain-macp}. Here we develop those ideas a little further. \begin{definition} A simplicial topological space $X\subdot$ is called a $\Sigma\subdot$-space if there exits, on each $X_n$, a continuous action of the symmetric group $\Sigma_{n+1}$ on $n+1$ letters which is compatible with the simplicial structure of $X\subdot$ in the sense that the face maps $$ A_j:X_{n+1}\rightarrow X_n, $$ satisfy the conditions \begin{equation*} A_j\circ (i-1,i) = \begin{cases} (i-1, i)\circ A_j & j> i;\\ A_{i-1} & j=i;\\ A_i & j=i-1;\\ (i-2,i-1)\circ A_j & j< i. \end{cases} \end{equation*} \end{definition} When the simplicial space $X\subdot$ has extra structure, (examples being when $X\subdot$ is a simplicial manifold or a simplicial variety), we require the $\Sigma\subdot$ action to preserve this additional structure. \begin{example} Let $X$ be an arbitrary topological space. Consider the simplicial space $X_\bullet$ where $X_n=X^{n+1}$ with the obvious face maps. The symmetric group $\Sigma_{n+1}$ acts on $X_n$ by permuting the factors. It is routine to check that $X\subdot$ is a $\Sigma_\bullet$-space. \end{example} \begin{example} This type of simplicial space arises in the construction of classifying spaces of principal $G$ bundles. Let $X$ be a principal $G$-space, where $G$ is a topological group. Define a simplicial space $(X/G)_\bullet$ by letting $$ (X/G)_n= X^{n+1}/G, $$ where $G$ acts diagonally of $X^{n+1}$. Since the $\Sigma_{n+1}$-action, permuting the factors of $X^{n+1}$, commutes with the $G$-action, $(X/G)_n$ inherits a $\Sigma_{n+1}$-action which gives it the structure of a $\Sigma_\bullet$-space. Note that $(X/G)_\bullet$ is just the standard simplicial model of the classifying space $B_\bullet G$ of principal $G$-bundles. When $X$ is an algebraic variety and $G$ is an algebraic group which acts on $X$ algebraically and where each $X^n/G$ is an algebraic variety, $(X/G)_\bullet$ is a simplicial variety. \end{example} We now describe several $\Sigma_\bullet$-varieties to be used in the paper. The scheme $G^m_n$ introduced in the introduction has an alternative description (see \cite[(5.6)]{hain-macp}). We say that $m+n$ vectors in a vector space $k^m$ are in {\it general position} if each $m$ of them are linearly independent. The alternative description of $G^m_n$ is: $$ G^m_n(k)=\{(n+m+1)\text{-tuples}\; (v_0,\dots,v_{m+n}) \; \text{in $k^m$ in general position}\}/GL_m(k). $$ We will denote the point of $G^m_n$ corresponding to the orbit of $(v_0,\dots, v_{m+n})$ by $[v_0,\dots, v_{m+n}]$. The face map $$ A_i: G^{m+1}_n \rightarrow G^m_n $$ is defined by $$ [v_0,\dots,v_{m+n+1}] \mapsto [v_0,\dots,\hat{v}_i,\dots,v_{m+n+1}]. $$ It is clear that $G^m_n$ has a $\Sigma_{m+n+1}$ action. If we place $G^m_n$ in dimension $n+m$, then the $G^m_n$ with $0\le n \le m$, together with the face maps form an $(m,2m)$ truncated $\Sigma\subdot$-variety $G^m\subdot$. It is convenient to complete the truncated variety $G^m\subdot$ to a simplicial variety $B_\bullet G^m$ by adding a point in each degree less than $m$. Observe that $B_\bullet G^m$ is also a $\Sigma\subdot$-variety. Similarly, each Zariski open subset $U^m_\bullet$ of $G^m_\bullet$ can be completed to a simplicial variety $B_\bullet U^m$. If $U^m\subdot$ is invariant under the symmetric group actions on $G^m\subdot$, then $B_\bullet U^m$ is also a $\Sigma\subdot$-variety. Define $$ E_nG^m(k) = \{(n+m+1)\text{-tuples}\; (v_0,\dots,v_{m+n}) \; \text{in $k^m$ in general position}\}, $$ and let $E_\bullet G^m$ be the set of $E_nG^m$ with $n\ge 0$. Then with the obvious face maps, with the obvious face maps, $E_\bullet G^m$ forms a $\Sigma_\bullet$-variety. The group scheme $GL_m$ acts on $E_\bullet G^m$ via the diagonal action. Observe that the quotient is $B_\bullet G^m$. Denote the projection map by $$ \pi_m: E_\bullet G^m\rightarrow B_\bullet G^m. $$ For a Zariski open subset $U^m_\bullet$ of $G^m_\bullet$, let $E_\bullet U^m = \pi_m^{-1}(B_\bullet U^m)$. Then $E_\bullet U^m$ is a simplicial open subvariety of $E_\bullet G^m$, which is a $\Sigma\subdot$-variety when $U^m_\bullet$ is. Let $V$ be a $k$-module with a $\Sigma_n$-action, where $k$ is a field of characteristic 0. Define the {\it alternating operator} $$ \Alt_n: V\rightarrow V $$ by \begin{equation*} \Alt_n(v) = \frac{1}{n!}\sum_{\sigma\in \Sigma_n}\sgn(\sigma)\sigma(v). \end{equation*} When $n>1$, an element $v\in V$ is called an {\it alternating element of $V$} if $\Alt_n(v)=v$. Let $sV$ denote the submodule of $V$ consisting of all of its alternating elements. (This will be called the {\it alternating part\/} of $V$ in the sequel.) The {\it sign decomposition\/} of $V$ is the decomposition $$ V= sV\oplus rV $$ where $rV= \ker\; \Alt_n$ is the unique $\Sigma_n$-invariant complement of $sV$. By convention, we define $sV = V$ and $rV=0$ when $n=1$. This is necessary in order that the following result hold. \begin{definition} (cf. \cite[(9.4)]{hain-macp}) A cosimplicial $k$-module $M\supdot$ ($k$ a field of characteristic 0) is called a {\it $\Sigma_\bullet$-module\/} if each $M^n$ has a $\Sigma_{n+1}$-action, and the face maps $$A_j: M^{n-1}\rightarrow M^n$$ satisfy the following relations \begin{equation*} (i-1,i)\circ A_j = \begin{cases} A_j\circ(i-2,i-1), & j<i-1;\\ A_i, & j= i-1;\\ A_{i-1}, & j=i;\\ A_j\circ(i-1,i) & j> i, \end{cases} \end{equation*} for $ j=0,\dots,m$ and $i=1,\dots, m-1$. \end{definition} Natural examples of cosimplicial $\Sigma_\bullet$-modules can be obtained by applying a contravariant $k$-module valued functor to a $\Sigma_\bullet$-space. As usual, let us define $A^\ast: M^{n-1}\rightarrow M^n$ to be $\sum^n_{j=0}(-1)^j A_j$. The sign decomposition generalizes to $\Sigma_\bullet$-cosimplicial modules. The following result is proved in \cite[(9.5)]{hain-macp}. \begin{lemma}\label{signd} The differential $A^\ast$ preserves the sign decomposition $$ M^l = sM^l\oplus rM^l, $$ for $l=0,1,2,\dots$. In particular, $A^\ast sM^n$ is a $\Sigma_{n+2}$-submodule of $M^{n+1}$. \qed \end{lemma} Define the cohomology of a cosimplicial module $M^\bullet$ by $$ H^n(M^\bullet)=\ker(A^\ast: M^n\rightarrow M^{n+1})/\im(A^\ast: M^{n-1}\rightarrow M^n). $$ The following corollary is an immediate consequence of the previous lemma. \begin{corollary} The cohomology groups of a cosimplicial $\Sigma_\bullet$-module $M^\bullet$ have a sign decomposition $$ H^\bullet(M^\bullet)=sH^\bullet(M^\bullet)\oplus rH^\bullet(M^\bullet). $$ The decomposition is natural with respect to $\Sigma_\bullet$-invariant maps between cosimplicial $\Sigma_\bullet$-modules. \end{corollary} Applying the de~Rham complex functor, the Deligne-Beilinson cochain complex functor, or the ${\cal BW}$-cochain complex functor to a $\Sigma_\bullet$-variety, we obtain natural examples of $\Sigma_\bullet$-cosimplicial modules. The following result is an immediate consequence of the previous result. \begin{theorem} If $X_\bullet$ is a smooth complex $\Sigma_\bullet$-variety, then the de~Rham cohomology, Deligne-Beilinson cohomology and the ${\cal BW}$-% cohomology of $X_\bullet$ have sign decompositions $$ H^\bullet(X_\bullet)=sH^\bullet(X_\bullet)\oplus rH^\bullet(X_\bullet), $$ $$ \Hd^\bullet(X_\bullet,\Lambda(m))= s\Hd^\bullet(X_\bullet,\Lambda(m))\oplus r\Hd^\bullet(X_\bullet,\Lambda(m)), $$ $$ H_{\cal BW}^\bullet(X_\bullet,{\Bbb R}(m))= sH_{\cal BW}^\bullet(X_\bullet,{\Bbb R}(m))\oplus rH_{\cal BW}^\bullet(X_\bullet,{\Bbb R}(m)), $$ which are all natural with respect to $\Sigma_\bullet$-invariant map between smooth $\Sigma_\bullet$-varieties. Moreover, the natural maps $$ H^\bullet_{\cal BW}(X_\bullet,{\Bbb R}(m)) \to \Hd^\bullet(X_\bullet,{\Bbb R}(m)) \to H^\bullet(X_\bullet,{\Bbb R}(m)) $$ each preserve the sign decomposition. \qed \end{theorem} \section{Real Grassmann Polylogarithms} \label{polylog} In this section we shall view the truncated simplicial variety $G^m_\bullet$ as a $\Sigma_\bullet$-variety with the $\Sigma_\bullet$ action described in \S \ref{S-variety}. Denote the coordinates of ${\Bbb P}^m$ by $[x_0,x_1,\dots,x_m]$. Denote the hyperplane $x_j = 0$ by $H_j$. There is a natural identification of $G^m_0$ with ${\Bbb P}^m - \cup_{j=0}^m H_j$. This can be identified with $({\Bbb C}^\ast)^m$ by identifying $(x_1,\dots,x_m)\in ({\Bbb C}^\ast)^m$ with $[1,x_1,\dots,x_m]\in {\Bbb P}^m$. Set $$ \mbox{vol}_m = \frac{dx_1}{x_1}\wedge\dots \wedge \frac{dx_m}{x_m}. $$ This is an element of $$ s\Omega^m(G^m_0) \cong sH^m(G^m_0,{\Bbb C}). $$ Recall that $G^m_n$ is placed in dimension $m+n$. There is a canonical homomorphism $$ s\Hd^{2m}(G^m_\bullet,{\Bbb R}(m)) \to sH^m(G^m_0,{\Bbb R}(m-1)) $$ induced by the inclusion $G^m_0 \hookrightarrow G^m\subdot$. \begin{definition} A {\it real Grassmann $m$-logarithm} is an element $L_m$ of $$ sH_{\cal BW}^{2m}(G^m_\bullet,{\Bbb R}(m)) $$ whose image under the composite $$ sH_{\cal BW}^{2m}(G^m_\bullet,{\Bbb R}(m)) \to s\Hd^{2m}(G^m_\bullet,{\Bbb R}(m)) \to sH^m(G^m_0,{\Bbb C}) $$ is $\mbox{vol}_m$. A cocycle in $s{\Bbb R}_{\cal BW}^\bullet(G^m_\bullet)(m)$ that represents $L_m$ will be called a {\it real Grassmann $m$-cocycle.} Finally, the part of a real Grassmann $m$-cocycle that lies in $sW_{2m}{\cal BW}(G^m_{m-1})$ will be called a {\it real Grassmann $m$-logarithm function}. \end{definition} Observe that a real Grassmann $m$-logarithm $D_m$ satisfies the $(2m+1)$-term functional equation $$ A^\ast D_m := \sum_{j=0}^{2m} (-1)^j A_j^\ast D_m = 0, $$ where $A_j : G^m_m \to G^m_{m-1}$, $j=0,\dots,2m$, are the face maps, as well as the skew symmetry property $$ \sigma^\ast D_m = \sgn(\sigma) D_m $$ for all $\sigma \in \Sigma_{2m}$. Now suppose that $U^m_\bullet$ is a Zariski open subset of $G^m_\bullet$. (That is, for each $n$, $U^m_n$ is a Zariski open subset of $G^m_n$ and the inclusion $U^m_\bullet \hookrightarrow G^m_\bullet$ is a simplicial map.) Suppose further that $U^m_0 = G^m_0$, that $U^m_\bullet$ is mapped into itself under the action of the symmetric groups on $G^m_\bullet$, and that the condition \begin{equation}\label{condition} \text{\it Each fiber of each face map $A_j : U^m_l \to U^m_{l-1}$ is non-empty} \end{equation} is satisfied. As above, the inclusion $G^m_0 \hookrightarrow U^m_\bullet$ induces a canonical homomorphism $$ s\Hd^{2m}(U^m_\bullet,{\Bbb R}(m)) \to sH^m(G^m_0,{\Bbb R}(m-1)). $$ \begin{definition}\label{hain:generic} A {\it generic real Grassmann $m$-logarithm} is an element $L_m$ of $$ sH_{\cal BW}^{2m}(U^m_\bullet,{\Bbb R}(m)), $$ where $U^m_\bullet$ is a Zariski open subvariety of $G^p_\bullet$ that satisfies the conditions in the previous paragraph, whose image under the composite $$ sH_{\cal BW}^{2m}(U^m_\bullet,{\Bbb R}(m)) \to s\Hd^{2m}(U^m_\bullet,{\Bbb R}(m)) \to sH^m(G^m_0,{\Bbb C}) $$ is $\mbox{vol}_m$. A cocycle in $s{\Bbb R}_{\cal BW}^\bullet(U^p_\bullet)(m)$ that represents $L_p$ will be called a {\it generic real Grassmann $m$-cocycle.} Finally, the part of a generic real Grassmann $m$-cocycle that lies in $sW_{2m}{\cal BW}(U^m_{m-1})$ will be called a {\it generic real Grassmann $m$ logarithm function}. \end{definition} Observe that every real Grassmann $m$-logarithm (resp.\ cocycle, function) is a generic real Grassmann $m$-logarithm (resp.\ cocycle, function). As in the case of a Grassmann $m$-logarithm, a generic Grassmann $m$-logarithm $D_m$ satisfies the $(2m+1)$-term functional equation $$ A^\ast D_m = 0 $$ and the symmetry relation $$ \sigma^\ast D_m = \sgn (\sigma) D_m $$ for each $\sigma \in \Sigma_{2m}$. In the next section, we will prove that a generic real Grassmann $m$-logarithm (and therefore, every real Grassmann $m$-logarithm) defines a cohomology class $$ d_m \in H^{2m-1}(GL_m({\Bbb C}),{\Bbb R}(m-1)), $$ and in the succeeding section, that $D_m$ defines a cohomology class in the continuous cohomology $$ \delta_m \in \Hcts^{2m-1}(GL_m({\Bbb C}),{\Bbb R}(m-1)) $$ such that the image of $\delta_m$ in $H^{2m-1}(GL_m({\Bbb C}),{\Bbb R}(m-1))$ is $d_m$. \section{Up to the 3-log}\label{up} In this section, we establish the existence of the first 3 real Grassmann logarithms. This results is more or less known from \cite{bloch}, \cite{hain-macp}, \cite{yang:thesis} and \cite{goncharov:trilog}. \begin{proposition} If $m\le 3$, then $sW_{2m}H^{2m}(B_\bullet G^m,{\Bbb Q})$ has dimension one and is spanned by the volume form $\mbox{vol}_m$, while $sW_{2m-1}H^{2m-1}(B_\bullet G^m,{\Bbb Q})$ is trivial. \end{proposition} \begin{pf} When $m\le 3$, the stronger result for all the cohomology (rather than just the alternating part) was proved by direct computation by Hain and MacPherson (cf. \cite[(12.6)]{hain-macp}), although the details were not given. Here we give a complete proof of the weaker statement given in the proposition. The point is that when $m\le 3$, $G^m_n$ is a rational $(m-n)$-$K(\pi,1)$ for all $n$. This is proved in \cite[\S 8]{hain-macp}. This implies that for such $m$ and $n$, the cup product $$ \Lambda^k H^1(G^m_n,{\Bbb Q}) \to H^k(G^m_n,{\Bbb Q}) $$ is surjective, provided that $k\le m-n$. One can easily show that in these cases, $s\Lambda^k H^1(G^m_n,{\Bbb Q})=0$, except when $k=m$ and $n=0$, in which case $$ s\Lambda^m H^1(G^m_0,{\Bbb Q}) \cong sH^m(G^m_0,{\Bbb Q}). $$ The result now follows from the fact that the standard spectral sequence that converges to $H^\bullet(G^m_\bullet,{\Bbb Q})$ is compatible with the $r\oplus s$ decomposition. \end{pf} As a corollary, we obtain the existence and uniqueness of real Grassmann $m$-logarithms for $m=1,2,3,4$. \begin{corollary}\label{one-dim} If $m\le 3$, then the natural map $$ s\Hd^{2m}(B_\bullet G^m,{\Bbb R}(m)) \to \Hd^{2m}(G^m_0,{\Bbb R}(m))\cong {\Bbb R}\mbox{vol}_m $$ is an isomorphism. \end{corollary} \begin{pf} The proposition follows immediately from the following short exact sequence \begin{multline*} 0 \to \Ext^1_{\cal H}({\Bbb Q},H^{2m-1}(X,{\Bbb Q}(m))) \to \\ \Hd^{2m}(X,{\Bbb Q}(m)) \to \Hom_{\cal H}({\Bbb Q},H^{2m}(X,{\Bbb Q}(m)))\to 0 \end{multline*} since by the previous proposition $$ \Ext^1_{\cal H}({\Bbb Q},H^{2m-1}(X,{\Bbb Q}(m)))=0 $$ and $$ \Hom_{\cal H}({\Bbb Q},H^{2m}(X,{\Bbb Q}(m))) $$ is one-dimensional and generated by the volume form $\mbox{vol}_m$. \end{pf} \begin{corollary} If $m \le 3$, there is a canonical $m$-logarithm. \end{corollary} \begin{pf} Since in each case $G^m_n$ is a rational $(m-n)$-$K(\pi,1)$, there is a canonical splitting of the map $$ sH^{2m}_{\cal BW}(G^m_\bullet,{\Bbb R}(m)) \to s\Hd^{2m}(G^m_\bullet,{\Bbb R}(m)) \cong {\Bbb R}\mbox{vol}_m $$ by (\ref{summand}). The $m$-logarithm is the image of $\mbox{vol}_m$ under this splitting. \end{pf} \section{Homology of $GL_m$} \label{homology} Let $k$ be an infinite field. In this section we show that a function $$ f: G^m_n(k) \rightarrow {\Bbb R} $$ satisfying the functional equation $$ A^\ast f = 0 $$ determines an element of $H^{m+n}(GL_m(k);{\Bbb R})$. In fact, with a little bit more work, we will show that the same holds even if $f$ is only defined on a Zariski open subvariety $U^m_n$ of $G^m_n$. Recall that for an abstract group $G$, the group cohomology of $G$ with coefficients in an abelian group $V$ (viewed as a trivial $G$-module) can be computed by taking the homology of the $G$ invariants of the complex $C\supdot(G,V)$, where $C^n(G,V)$ is the group of functions $$ f: \underbrace{G\times\dots\times G}_{n+1}\rightarrow V $$ and the coboundary map is defined by $$ \delta f(g_0,\dots,g_{n+1}) = \sum_{i=0}^{n+1}(-1)^if(g_0,\dots,\hat{g}_i,\dots,g_{n+1}). $$ The group $G$ acts on $C\supdot(G,A)$ on the right via the formula $$ (f\cdot g)(g_0,\dots,g_n) := f(gg_0,\dots,gg_n). $$ The $G$-invariants of the complex $C\supdot(G,A)$ will be denoted by $C\supdot(G,V)^G$. \begin{variant}\label{coho_def} Recall that if $G$ is a topological group, then the continuous group cohomology of $G$ is defined using continuous functions $f: G^{n+1} \to V$ in place of arbitrary functions in the definition above. When $G$ is a Lie group, the locally-$L_p$ cohomology of $G$ is defined using cochains $f:G^{n+1} \to V$ that are locally-$L_p$ with respect to the measure on $G$ given by a left invariant volume form. The cochain complex of continuous cochains of $G$ will be denoted by $C_{\text{cts}}^\bullet(G,V)$, and the locally-$L_p$ cochains by $C_{\text{loc-$L_p$}}^\bullet(G,V)$. The continuous and locally-$L_p$ cohomology groups of a Lie group $G$ will be denoted $H_{\text{cts}}^\bullet(G,V)$ and $H_{\text{loc-$L_p$}}^\bullet(G,V)$, respectively. \end{variant} \medskip Let $k$ be an extension field of ${\Bbb Q}$. Fix a non-zero vector $e$ in $k^m$. Define the subset $X^n_{G(k),e}$ of $E_nGL_m(k)$ to be\footnote{This is the discrete analogue of the variety $B_nGL_m({\Bbb C})^{\text{gen}}$ defined in \S \ref{descent}.} $$ \{(g_0,\dots,g_n)\in GL_m(k)^{n+1} | \; (g_0e,\dots,g_ne)\in E_nG^m(k)\}. $$ Denote the group of functions $X^n_{G(k),e}\rightarrow V$ by $C^n_{G,e}(GL_m(k),V)$. Endowed with the boundary maps induced from those of $C\supdot(GL_m(k),V)$, it is a complex. Denote the subcomplex of $GL_m(k)$-invariants by $C_{G,e}\supdot(GL_m(k),V)^{GL_m(k)}$. \begin{proposition} \label{resoln} The chain map $$ C\supdot(GL_m(k),V)^{GL_m(k)} \rightarrow C\supdot_{G,e}(GL_m(k),V)^{GL_m(k)} $$ induced via restriction from the natural chain map $$ C\supdot(GL_m(k),V)\rightarrow C\supdot_{G,e}(GL_m(k),V) $$ induces an isomorphism on cohomology; i.e., there is a natural isomorphism $$ H^\bullet(C\supdot_{G,e}(GL_m(k),V)^{GL_m(k)})\cong H^\bullet(GL_m(k),V). $$ \end{proposition} \begin{pf} It suffices to prove that $$ 0\rightarrow V \rightarrow C\supdot_{G,e}(GL_m(k),V) $$ is a resolution of $V$ by injective $GL_m(k)$-modules . To prove this, we prove the dual statement. Denote the tensor product of the free abelian group generated by the points of $X^m_{G(k),e}$ with $V$ by $C^{G,e}_n(GL_m(k),V)$. The dual statement is that $$ 0 \leftarrow V\leftarrow C\subdot^{G,e}(GL_m(k),V) $$ is a projective resolution of $V$. Since each $C_n^{G,e}(GL_m(k),V)$ is a free $GL_m(k)$-module, we need only establish exactness. Supposes that \begin{equation*} \partial (\sum_l a_k(g_{l,0},\dots,g_{l,n}))=\sum_k\sum_{i=0}^n(g_{l,0},\dots, \hat{g}_{l,i},\dots,g_{l,n})=0, \end{equation*} where $(g_{l,0}e,\dots, g_{l,n}e)\in E_nG^m(k)$. By elementary linear algebra, there exists $v\in k^m-\{0\}$ such that $(v, g_{l,0}e,\dots, g_{l,n}e)\in E_{n+1}G^m(k)$ for each $l$. Pick $g\in GL_m(k)$ such that $g e = v$. Then $(g,g_{l,0},\dots,g_{l,n})$ lies in $X^{n+1}_{G,e}(k)$ for each $l$. Now it is straightforward to check that \begin{equation} \label{exactness} \partial(\sum_k a_k(g,g_{k,0},\dots,g_{k,n}))=\sum_ka_k(g_{k,0},\dots,g_{k,n}). \end{equation} The exactness follows. \end{pf} Now suppose we are given a function $$ f: G^m_n(k)\rightarrow V $$ that satisfies $A^\ast f = 0$. Since $G^m_n(k)=B_{m+n}G^m(k)$ when $n>0$, $f$ will induce a $GL_m(k)$-invariant map $$ \tilde{f} : E_{m+n}G^m(k) \to V. $$ As before, we fix a non-zero vector $e\in k^m$. Define a map $$ f^e: X^{m+n}_{G,e}(k) \rightarrow V $$ by $$ f^e(g_0,\dots,g_{m+n})= \tilde{f}([g_0e,\dots,g_{m+n}e]). $$ It is obvious that $f^e$ is a $GL_m(k)$-invariant cocycle in $C^{m+n}_{G,e}(GL_m(k),V)$. In fact we have the following result. \begin{proposition}\label{groupclass} The function $f^e$ represents a cohomology class in $$ H^{m+n}(GL_m(k),V). $$ Moreover, the cohomology class it represents is independent of the choice of the base vector $e$ in $k^m-\{0\}$. \end{proposition} \begin{pf} The first statement follows immediately from (\ref{resoln}). To prove the second, suppose that $e'$ is another non-zero vector in $k^m$. There exists a matrix $h\in GL_m(k)$ such that $e'= he$. Define a chain map $$ \phi_h: C\supdot(GL_m(k), V)^{GL_m(k)} \longrightarrow C\supdot(GL_m(k), V)^{GL_m(k)} $$ by $$ \phi_h(f)(g_0,\dots, g_{m+n})= f(g_0h,\dots, g_{m+n}h), $$ where $f\in C\supdot(GL_m(k),V)^GL_m(k)$. It is known (see, e.g., \cite[Chap. IV, Prop.~5.6]{maclane}) that this chain map induces the identity map on cohomology. The map $\phi_h$ induces a chain map $$ \phi^{e,e'}_h: C_{G,e}\supdot(GL_m(k), V)^{GL_m(k)} \longrightarrow C_{G,e'}\supdot(GL_m(k), V)^{GL_m(k)}, $$ which carries $f^e$ to $f^{e'}$. The proposition now follows from (\ref{resoln}) and the commutativity of the following diagram. $$ \begin{CD} C\supdot(GL_m(k),V)^{GL_m(k)} @>\phi_h >> C\supdot(GL_m(k),V)^{GL_m(k)}\\ @VVV @VVV \\ C\supdot_{G,e}(GL_m(k),V)^{GL_m(k)} @>\phi^{e,e'}_h >> C\supdot_{G,e'}(GL_m(k),V)^{GL_m(k)} \end{CD} $$ \end{pf} \begin{corollary} \label{cor-dm} If $D_m$ is a real Grassmann $m$-logarithm function, then $D_m$ defines an element of $H^{2m-1}(GL_m({\Bbb C}),{\Bbb R}(m))$. \qed \end{corollary} \begin{remark}\label{grass_homo} These results can be interpreted in terms of MacPherson's Grassmann homology \cite{b-mcp-s}, as we shall now explain. Further discussion of Grassmann homology and its relation to Suslin's work can be found in \cite{wolf}. Let $k$ be a field. Denote the free abelian group generated by the points of $E_nG^m(k)$ by $C_n(k^m)$. (This is denoted $C_n(GP(k^m))$ in \cite{suslin}.) The face maps of $E_\bullet G^m(k)$ induce a differential $C_n(k^m) \to C_{n-1}(k^m)$. This is a resolution of the trivial module \cite[Lemma 2.2]{suslin}. The group $GL_m(k)$ acts diagonally on $C_n(k^m)$. Applying the functor $\underline{\phantom{X}}\otimes_{GL_m(k)} k$, we obtain the complex $$ 0\leftarrow C_0(k^m)_{GL_m(k)}\stackrel{d}{\leftarrow} C_1(k^m)_{GL_m(k)}\stackrel{d}{\leftarrow} C_2(k^m)_{GL_m(k)} \stackrel{d}{\leftarrow}\cdots $$ where $C_n(k^m)_{GL_m(k)}$ denotes the $GL_m(k)$ coinvariants $C_n(k^m)_{GL_m(k)}\otimes_G k$ --- $C_n(k^m)_{GL_m(k)}$ is simply the free abelian group generated by the points of $B_nG^m(k)$. The homology of this complex is called the {\it (extended) Grassmann homology of $k$}, and is denoted by $GH_\bullet^m(\spec k)$. This differs by a factor of ${\Bbb Z}$ from MacPherson's original definition in dimension $m$ when $m$ is even. The result (\ref{resoln}) simply says that there is a natural map $$ H_n(GL_m(k)) \to GH^m_{m+n}(\spec k) $$ and gives a formula for it. If $f : G^m_n(k) \to V$ satisfies the functional equation $A^\ast f = 0$, then $f$ induces a map $GH^m_{m+n}(\spec k) \to V$. The class $f^e$ of (\ref{groupclass}) is simply the composite of these two maps. \end{remark} \begin{variant}\label{variant} The corollary \ref{cor-dm} can be generalized to generic real Grassmann $m$-logarithm functions. Suppose that $U^m_\bullet(k)$ is a Zariski open subset of $G^m_n(k)$ that satisfies (\ref{condition}) in \S \ref{polylog}. Now suppose that that $f : U^m_n(k) \to V$ satisfies the functional equation $A^\ast f = 0$. We will indicate briefly how to modify the proof of (\ref{cor-dm}) to show that $f$ determines an element $f^e$ of $H_{m+n}(GL_m(k),V)$. Define $$ E_{n+m}U^m = \{(v_0,\dots,v_n)|v_j\in k^m \text{ and } [v_0,\dots,v_n] \in U^m_n(k)\}. $$ Now replace $X^n_{G(k),e}$ by $$ X^n_{U(k),e}:=\{(g_0,\dots,g_n)\in GL_m(k)^{n+1} | \; (g_0e,\dots,g_ne)\in E_nU^m(k)\}. $$ Let $\pi_0: E_{n+1}U^m(k)\rightarrow k^m-\{0\}$ denote the projection map to the first component. Since $k$ is infinite, the condition (\ref{condition}) of \S \ref{polylog} implies that for each finite set of points $x_1, \dots, x_N$ of $E_nU^m(k)$, $$ \bigcap_{l=1}^N \pi_0(A_0^{-1}(x_l))\subset k^m-\{0\} $$ is non-empty. This condition is needed to prove that the complex corresponding to the simplicial set $X^\bullet_{U(k),e}$ is a resolution of ${\Bbb Z}$. Putting all this together, we have: \end{variant} \begin{corollary} If $D_m$ is a generic real Grassmann $m$-logarithm function, then $D_m$ defines an element of $H^{2m-1}(GL_m({\Bbb C}),{\Bbb R}(m))$. \qed \end{corollary} We conclude this section with a useful technical fact. Let $G$ be a discrete group. The standard resolution for computing group cohomology comes from the $\Sigma_\bullet$-variety $BG$. It follows that the group homology $H^\bullet(G,{\Bbb Q})$ has a sign decomposition. \begin{proposition}\label{sign-copy} The homology $H_\bullet(G,{\Bbb Q})$ consists entirely of the alternating part. That is, $rH^\bullet( C^\bullet_{\text{cts}}(B_\bullet G, V))= 0$. \end{proposition} \begin{pf} Let $C_\bullet(G,{\Bbb Q}) \stackrel{\epsilon}{\longrightarrow} {\Bbb Q} \to 0$ be the free resolution of the trivial module that comes from the standard simplicial model of $BG$. Since $BG$ is a $\Sigma_\bullet$-% variety, this resolution has an $r\oplus s$ decomposition. Since $sC_0(G,{\Bbb Q}) = C_0(G,{\Bbb Q})$, and since $r$ and $s$ are exact functors, it follows that $$ sC_\bullet(G,{\Bbb Q}) \stackrel{\epsilon}{\longrightarrow} {\Bbb Q} \to 0 $$ is a projective resolution of the trivial module. The result follows. \end{pf} The analogous results hold for both the continuous cohomology and locally $L_p$ cohomology of a Lie group. The proofs are similar. \section{Bloch-Wigner Functions and Locally $L_p$ Cohomology} \label{loc-l_p} In this section, we show that a Bloch-Wigner function $f:U^m_n({\Bbb C})\rightarrow {\Bbb R}$ that satisfies the functional equation $A^\ast f=0$ represents a {\it continuous} group cohomology class of $GL_m({\Bbb C})$ whose image in $H_{m+n}(GL_m({\Bbb C}),{\Bbb R})$ is the class constructed from $f$ in Section \ref{homology}. Suppose that $M$ is an orientable smooth manifold of dimension $n$ and that $\omega$ is a nowhere vanishing $n$-form on $M$. An almost everywhere defined function $f$ on $M$ is said to be {\it locally $L_p$} if each point $x$ of $M$ has a neighbourhood $U$ such that $$ \int_U |f|^p \omega < \infty. $$ This definition is independent of the choice of volume form $\omega$. \begin{lemma}\label{loc_pres} Suppose that $\pi : M \to N$ is an orientation preserving proper map between orientable manifolds of the same dimension. If $f$ is an almost everywhere defined function on $N$ whose pullback is almost everywhere defined on $M$, then $f$ is locally $L_p$ on $N$ if its pullback $\pi^\ast f$ is locally $L_p$ on $M$. \end{lemma} \noindent{\it Proof.} Choose volume forms $\omega_M$ and $\omega_N$ on $M$ and $N$ respectively. Since $\pi$ is orientation preserving, there is a non-% negative $C^\infty$ function $\phi(x)$ on $M$ such that $$ \pi^\ast \omega_N = \phi(x)\omega_M $$ Now suppose that $x\in N$. Choose a compact neighbourhood $U$ of $x$ in $N$. Since $\pi$ is proper, $\pi^{-1}(U)$ is compact, and there is a real number $C$ such that $\phi$ is bounded by $C$ on $\pi^{-1}(U)$. Since $\pi^{-1}f$ is locally $L_p$ on $M$, and since $\pi^{-1}(U)$ is compact, $$ \int_{\pi^{-1}(U)} |\pi^\ast f|^p \omega_M <\infty. $$ Consequently, $$ \int_U |f|^p \omega_N = \int_{\pi^{-1}(U)} |\pi^\ast f|^p \pi^\ast\omega_N \le C\int_{\pi^{-1}(U)} |\pi^\ast f|^p \omega_M <\infty. \qed $$ \medskip Suppose that $Y$ is a smooth variety which is birational to $X$. Since $X$ and $Y$ differ by sets of measure zero, each almost everywhere defined function on $X$ can be regarded as an almost everywhere function on $Y$. \begin{proposition}\label{loc_Lp} Suppose that $X$ and $Y$ are birational complex algebraic manifolds. If $p> 0$, then each Bloch-Wigner function on $X$ is locally $L_p$ when viewed as a function on $Y$. \end{proposition} \begin{pf} If $U$ is a Zariski open subset of $X$, the restriction map ${\cal BW}(X) \to {\cal BW}(U)$ is injective. By replacing $X$ by a Zariski open subset common to $X$ and $Y$, we may assume that $X$ is a Zariski open subset of $Y$. Let $Z = Y-X$. By Hironaka's resolution of singularities, there is a smooth variety $\widetilde{Y}$, a normal crossings divisor $D$ in $\widetilde{Y}$, and a proper map $$ \pi : (\widetilde{Y},D) \to (Y,Z) $$ which induces an isomorphism $\widetilde{Y} - D \to X$. By (\ref{loc_pres}) a function $f\in {\cal BW}(X)$ is locally $L_p$ on $Y$ if it is locally $L_p$ on $\widetilde{Y}$. It therefore suffices to consider the case where $Z$ is a normal crossings divisor in $Y$. Since $X$ is open in $Y$ and Bloch-Wigner functions are smooth on $X$, elements of ${\cal BW}(X)$ are $L_p$ about points of $X$. Suppose that $x \in Z$. Choose local coordinates $(z_1,\dots,z_n)$ about $x$ defined in a polydisk neighbourhood $\Delta$ of $x$ in $Y$ such that $Z$ is contained in the divisor $z_1 z_2 \dots z_n = 0$. Every multivalued function associated to a relatively closed iterated integral of holomorphic 1-forms on $\Delta$ with logarithmic singularities along $Z$ an be obtained as follows (cf. \cite[\S 3]{hain:geom}): There is a $gl_m({\Bbb C})$ valued 1-form $$ \omega \in \Omega^1(\Delta\log Z)\otimes gl_m({\Bbb C}) $$ with logarithmic singularities along $Z$ which is integrable: $$ d\omega + \omega \wedge \omega = 0 $$ and has strictly upper triangular residue along each component of $Z$. This defines a flat meromorphic connection on the trivial bundle $$ {\Bbb C}^m\times \Delta \to \Delta $$ which is holomorphic on $\Delta - Z$, and has regular singularities along $Z$. The multivalued closed iterated integrals are obtained by taking linear combinations of flat sections of this bundle, and then composing with a linear projection ${\Bbb C}^m \to {\Bbb C}$. By the several variable generalization of \cite[Theorem 5.5]{wasow} (see \cite[5.2]{deligne:de}), every flat section of this bundle is of the form \begin{equation}\label{standard} F(z_1,\dots,z_n) = C P(z_1,\dots,z_n) \prod_{j=1}^n e^{A_j\log z_j}, \end{equation} where $P: \Delta \to GL_m({\Bbb C})$ is a holomorphic function, $C$ is a constant vector, and $A_j$ is the residue of the connection form $\omega$ along $z_j = 0$. In particular, the restriction of an element of $\tilde{\cal O}(X)$ to an angular sector of $\Delta$ about 0 is an entry of (\ref{standard}). Since the matrices $A_j$ are upper triangular, it follows that such functions are polynomials in $\log z_1,\dots, \log z_n$ with coefficients in the ring of holomorphic functions on $\Delta$. Such functions are $L_p$ on each closed angular segment of $\Delta$. Elements of ${\cal BW}(X)$ are linear combinations of real and imaginary parts of such functions. It follows that the restriction of a Bloch-Wigner function to each closed angular sector of $\Delta$ is also $L_p$ for all $p < 0$. The result follows. \end{pf} Suppose that $f: U^m_n({\Bbb C}) \to V$ is a continuous function, where $V$ is a finite dimensional real vector space, such as ${\Bbb R}$ or ${\Bbb R}(m)$. The corresponding function $$ f^e : GL_m({\Bbb C})^{m+n+1} \to V $$ is an almost everywhere defined cocycle. It should be noted, however, that in general this cocycle cannot be made into an everywhere continuous cocycle. If $f$ is $L_p$ for some $p\ge 1$ when viewed as a function on all of the grassmannian, then, by (\ref {loc_Lp}), it will be locally $L_p$ and locally integrable with respect to the measure on $GL_m({\Bbb C})$ associated to a left invariant volume form. Under these conditions, $f$ represents a class in $$ \Hp^{m+n}(GL_m({\Bbb C}),V), $$ where ${\cal H}^\bullet$ denotes the locally $L_p$ group cohomology (cf. \cite{blanc} and (\ref{coho_def}).) The natural chain map $$ j^\ast: C_{\text{cts}}\supdot(GL_m({\Bbb C}),V)\rightarrow C\supdot_{\text{loc-}L_p}(GL_m({\Bbb C}),V) $$ induces isomorphism between continuous group cohomology and locally $L_p$ cohomology \cite[(3.5)]{blanc}. We can write down an explicit inverse of $j^\ast$ as follows. Choose a non-negative continuous function $\chi$ on $GL_m({\Bbb C})$ with compact support and integral $1$ over the group. Define a chain map $r_\chi$ by \begin{multline*} r_\chi: C^n_{\text{loc-}L_p}(GL_m({\Bbb C}),V)\rightarrow C^n_{\text{cts}}(GL_m({\Bbb C}),V)\\ \phantom{r9}(r_\chi f)(g_0,\dots,g_n)=\hfill \\ \int_{GL_m({\Bbb C})^{n+1}} \chi(g_0^{-1}h_0)\dots\chi(g_n^{-1}h_n)f(h_0,\dots,h_n) {\text d}h_0\dots {\text d}h_n. \end{multline*} Here $\text{d}h$ denotes a fixed left invariant volume form on $GL_m({\Bbb C})$. The map $r_\chi$ induces (see \cite[4.11]{blanc}) the inverse $$ (j^\ast)^{-1}: \Hp\supdot(GL_m({\Bbb C}),V) \longrightarrow \Hcts\supdot(GL_m({\Bbb C}),V) $$ of $j^\ast$. In particular, the isomorphism $r_\chi^\ast$ is independent of the choice of the bump function $\chi$. Denote by $\mu^\ast$ the composite $$ \Hp\supdot(GL_m({\Bbb C}),V)\stackrel{(j^\ast)^{-1}}{\rightarrow} \Hcts^\ast(GL_m({\Bbb C}),V) \rightarrow H^\ast(GL_m({\Bbb C}),V). $$ The following result is not unexpected. However, as group cohomology cycles have measure zero, and since locally $L_p$ cochains can be changed with impunity on sets of measure zero, there is something to prove, and the result is not immediately obvious. Recall from (\ref{groupclass}) that a function $f: U^m_n({\Bbb C}) \to V$ that satisfies the functional equation $A^\ast f = 0$ determines a class $f^e$ in $H^{m+n}(GL_m({\Bbb C}),V)$. \begin{proposition} Suppose that $f:U^m_n({\Bbb C}) \to V$ is a continuous function satisfying the functional equation $A^\ast f = 0$. If $f$ is locally $L_p$ on the grassmannian for some $p\ge 1$, then the class $f^e$ determined by $f$ in $H^{m+n}(GL_m({\Bbb C}),V)$ is the image of the class in $\Hp^{m+n}(GL_m({\Bbb C}),V)$ determined by $f$ under the natural map $$ \mu^{m+n}:\Hp^{m+n}(GL_m({\Bbb C});V)\stackrel{\cong}{\leftarrow} \Hcts^{m+n}(GL_m({\Bbb C});V))\rightarrow H^{m+n}(GL_m({\Bbb C});V). $$ In particular, a generic real Grassmann $m$-logarithm function $D_m$ represents a class in $\Hcts^{2m-1}(GL_m({\Bbb C}),{\Bbb R})$ whose image in $H^{2m-1}(GL_m({\Bbb C}),{\Bbb R})$ is the class constructed in (\ref{groupclass}). \end{proposition} \begin{pf} From the preceding discussion, we know that the natural map $\mu^{m+n}$ is induced by the chain map $$ \rho_\chi: C^{m+n}_{{\text loc}-L_p}(GL_m({\Bbb C}), V) \stackrel{r_\chi}\rightarrow C^{m+n}_{\text cts}(GL_m({\Bbb C}),V) \rightarrow C^{m+n}(GL_m({\Bbb C}),V) $$ for each bump function $\chi$ on $GL_m({\Bbb C})$. Fix a non-zero vector $e$ in ${\Bbb C}^m$ as a base point. Then $f^e$ is an $(n+m)-$cocycle in $C^\bullet_{{\text loc}-L_p}(GL_m({\Bbb C}), V)$. To distinguish it from $f^e$ viewed as a discrete cocycle, we shall denote it by $f^e_{L_p}$. By (\ref{resoln}), it suffices to prove that the image of $f^e_{L_p}$ under the chain map $$ C^{m+n}_{{\text loc}-L_p}(GL_m({\Bbb C}), V) \stackrel{\rho_\chi}{\rightarrow} C^{m+n}(GL_m({\Bbb C}), V) \rightarrow C_e^{m+n}(GL_m({\Bbb C}), V) $$ is cohomologous to $f^e$. We prove this by showing that the cocycles $\rho_\chi f^e_{L_p}$ and $f^e$ agree when evaluated on a cycle with support inside $X^{m+n}_{U,e}$. Choose a sequence of bump functions $\chi_i$ on $GL_m({\Bbb C})$ which converge to the 0-current on $GL_m({\Bbb C})$ whose value on a test function is its value at the identity. Since $f^e$ is continuous at $x \in X^{m+n}_{U,e}$, then $$ \lim_{i\rightarrow \infty} \rho_{\chi_i} f^e_{L_p}(x) = f^e(x). $$ Since value of $\rho_{\chi_i}f^e$ evaluated on a cycle supported on $X^{m+n}_{U,e}$ is independent of the choice of $\chi_i$, the proposition follows. \end{pf} \section{Chern Classes in Algebraic $K$-theory} \label{chernclass} We review the construction of the Chern classes from the $K$-groups of an affine complex algebraic variety into its Deligne cohomology. We first fix some notation that will be used throughout this section. Let $X$ be an affine variety over ${\Bbb C}$ and $\eta_X$ be its generic point of $X$. We shall denote the ring of regular functions ${\Bbb C}[X]$ of $X$ by $R$, and the function field ${\Bbb C}(X)$ of $X$ by $F$. The group $GL_m(R)$ will be viewed as the 0-dimensional variety (or, more accurately, as a direct limit of 0-dimensional varieties) whose points are the algebraic functions $f: X \rightarrow GL_m({\Bbb C})$. As usual, $B\subdot GL_m(R)$ will denote the standard simplicial model of the classifying space of $GL_m(R)$. It will be regarded as a simplicial set (or a simplicial variety, each of whose sets of simplices is 0-dimensional). Let us denote the standard simplicial model of the universal complex $m$-bundle by $$ \nu(m): E\subdot GL_m({\Bbb C})\rightarrow B_\bullet GL_m(\C). $$ The pullback of the $l$-th universal Deligne-Beilinson Chern class of $\nu(n)$ $$ c_l\in \Hd^{2l}(B_\bullet GL_N({\Bbb C}),{\Bbb R}(l)) $$ along the evaluation map $$ e_N: X\times B\subdot GL_N(R)\rightarrow B\subdot GL_N({\Bbb C}) $$ gives an element $e_N^\ast (c_m)$ of $$ \Hd^{2m}(X\times B\subdot GL_N(R),{\Bbb R}(m))\cong \bigoplus_{k=0}^{2l} \Hd^{2m-k}(X,{\Bbb R}(m))\otimes H^k(GL_N(R),{\Bbb R}). $$ Evaluating $e_N^\ast(c_m)$ on elements of $H_m(GL_N(R),{\Bbb R})$ gives a natural map $$ H_m(GL_N(R),{\Bbb R})\rightarrow \Hd^{2m-l}(X,{\Bbb R}(m)). $$ The Chern class $$ c_{m,l} : K_l(X) \to \Hd^{2m-l}(X,{\Bbb R}(m)) $$ is obtained by taking $n$ to be sufficiently large and composing the previous map with the Hurewicz homomorphism $$ K_l(X) \to H_l(GL_N(R),{\Bbb R}) $$ Repeating the construction for each Zariski open subset of $X$ and then taking the direct limit over all Zariski open subsets of $X$, we obtain the Chern class maps $$ c_{m,l} : K_l(\eta_X) \rightarrow \Hd^{2m-l}(\eta_X,{\Bbb R}(m)). $$ \section{Descent of the Universal Deligne-Beilinson Chern Classes}\label{descent} In order to prove the existence of the 4-logarithm and all generic Grassmann logarithms, we prove that the alternating part, $\Alt c_m$, of the universal Chern class $$ c_m \in \Hd^{2m}(B_\bullet GL_m(\C), {\Bbb R}(m)) $$ ``descends'' to a class $$ \lambda_m \in s\Hd^{2m}(G^m_\bullet, {\Bbb R}(m)). $$ We begin by making this statement precise. In order to do this, we need to introduce several simplicial varieties. In this section, we will distinguish between genuine simplicial varieties and truncated simplicial varieties. (Cf.\ the conventions in the introduction.) First, let $$ EGL_m({\Bbb C}) \to BGL_m({\Bbb C}) $$ be the standard model of the universal $GL_m({\Bbb C})$ bundle in the category of simplicial varieties. That is, the variety of $n$ simplices, $E_nGL_m({\Bbb C})$, of $EGL_m({\Bbb C})$ is $GL_m({\Bbb C})^{n+1}$, and the $n$ simplices, $B_nGL_m({\Bbb C})$, of $BGL_m({\Bbb C})$ is the quotient of $E_nGL_m({\Bbb C})$ by the diagonal $GL_m({\Bbb C})$ action. The face maps of $EGL_m({\Bbb C})$ are the evident ones. We shall denote this universal bundle by $\nu$. Define the simplicial variety $E_\bullet G^m$ by defining its variety of $n$-simplices $E_nG^m$ to be $$ \left\{(v_0,\dots , v_n) : v_j \in {\Bbb C}^m \text{ and the vectors } v_0, \dots , v_n \text{ are in general position}\right\}. $$ The $j$th face map is the evident one obtained by forgetting the $j$th vector. The simplicial variety $B_\bullet G^m$ is obtained by taking the quotient of $E_\bullet G^m$ by the diagonal $GL_m({\Bbb C})$ action. Observe that its set of $n$ simplices $B_nG^m$ is a point when $n< m$, and that the projection $$ E_\bullet G^m \to B_\bullet G^m $$ is not a principal $GL_m({\Bbb C})$ bundle. Observe also that there is a natural ``map'' of the $[m,2m]$-truncated simplicial variety $G^m_\bullet$ into $B_\bullet G^m$ which is an isomorphism on $n$ simplices when $m\le n \le 2m$.\footnote{It is tempting, though misleading, to think of $G^m_\bullet$ as corresponding to a subspace of $B_\bullet G^m$.} The following result is easily proved by considering the double complexes associated to $G^m_\bullet$ and $B_\bullet G^m$. \begin{proposition}\label{identification} When $m < k \le 2m$, there is a natural isomorphism $$ \Hd^k(G^m_\bullet,{\Bbb R}(m)) \to \Hd^k(B_\bullet G^m,{\Bbb R}(m)) $$ which is compatible with the $r \oplus s$ decomposition. \qed \end{proposition} Next we want to interpolate between $BGL_m({\Bbb C})$ and $B_\bullet G^m$. Fix a non-zero element $e$ of ${\Bbb C}^m$. Define $$ E_nGL_m({\Bbb C})^{\text{gen}} = \left\{(g_0,\dots,g_n)\in GL_m({\Bbb C})^{n+1} : (g_0e,\dots, g_n e) \in E_nG^m\right\} $$ and $B_nGL_m({\Bbb C})^{\text{gen}}$ to be the quotient of this by the diagonal $GL_m({\Bbb C})$ action. We shall denote the principal $GL_m({\Bbb C})$ bundle $$ EGL_m({\Bbb C})^{\text{gen}} \to BGL_m({\Bbb C})^{\text{gen}}. $$ by $\nu^{\text{gen}}$. It is the restriction of the universal $GL_m({\Bbb C})$ bundle. By sending $(g_0,\dots,g_n)$ to $(g_0e,\dots,g_ne)$, we obtain a map $$ \pi : BGL_m({\Bbb C})^{\text{gen}} \to B_\bullet G^m. $$ We have the following diagram of simplicial varieties: $$ B_\bullet G^m \stackrel{\pi}{\leftarrow} BGL_m({\Bbb C})^{\text{gen}} \hookrightarrow BGL_m({\Bbb C}) $$ Denote the component of $$ c_m(\nu) \in \Hd^{2m}(BGL_m({\Bbb C}),{\Bbb R}(m)) $$ in $$ s\Hd^{2m}(BGL_m({\Bbb C}),{\Bbb R}(m)) $$ by $\Alt c_m(\nu)$. Likewise, we denote the alternating part of $$ c_m(\nu^{\text{gen}})\in \Hd^{2m}(BGL_m({\Bbb C})^{\text{gen}},{\Bbb R}(m)) $$ by $\Alt c_m(\nu^{\text{gen}})$. We shall identify $\Hd^{2m}(B_\bullet G^m,{\Bbb R}(m))$ with $\Hd^{2m}(G^m_\bullet,{\Bbb R}(m))$ via the isomorphism given by (\ref{identification}). The precise statement of the descent of the Chern class is: \begin{theorem} There is a unique class $\lambda_m$ in $s\Hd^{2m}(G^m_\bullet,{\Bbb R}(m))$ such that $\pi^\ast \lambda_m = \Alt c_m(\nu^{\text{gen}})$ in $s\Hd^{2m}(BGL_m({\Bbb C})^{\text{gen}},{\Bbb R}(m))$. \end{theorem} \begin{remark} The theorem and our proof are equally valid with ${\Bbb Q}(m)$ coefficients. \end{remark} The remainder of this section is devoted to the proof of this theorem. Because the projection $E_\bullet G^m \to B_\bullet G^m$ is not a principal bundle, it is convenient to introduce a simplicial variety which interpolates between $B_\bullet G^m$ and $BGL_m({\Bbb C})^{\text{gen}}$. Define $\widetilde{B}_\bullet G^m$ to be the homotopy quotient $$ \left(EGL_m({\Bbb C}) \times E_\bullet G^m \right)/GL_m({\Bbb C}) $$ of $E_\bullet G^m$ by $GL_m({\Bbb C})$. Set $$ \widetilde{E}_\bullet G^m = EGL_m({\Bbb C}) \times E_\bullet G^m. $$ Then the natural projection $\widetilde{E}_\bullet G^m \to \widetilde{B}_\bullet G^m$ is a principal $GL_m({\Bbb C})$ bundle which we shall denote by $\mu$. The projection $$ EGL_m({\Bbb C}) \times E_\bullet G^m \to E_\bullet G^m $$ induces a morphism $p : \widetilde{B}_\bullet G^m \to B_\bullet G^m$. Set $$ \widetilde{E} GL_m({\Bbb C})^{\text{gen}} = EGL_m({\Bbb C}) \times EGL_m({\Bbb C})^{\text{gen}} $$ and $\widetilde{B} GL_m({\Bbb C})^{\text{gen}}$ equal to the quotient of this by the diagonal $GL_m({\Bbb C})$ action. We have the diagram $$ \begin{matrix} EGL_m({\Bbb C})^{\text{gen}} & \leftarrow & \widetilde{E} GL_m({\Bbb C})^{\text{gen}} & \to & \widetilde{E}_\bullet G^m \cr \downarrow & & \downarrow & & \downarrow \cr BGL_m({\Bbb C})^{\text{gen}} & \leftarrow & \widetilde{B} GL_m({\Bbb C})^{\text{gen}} & \to & \widetilde{B}_\bullet G^m & \to & B_\bullet G^m \cr \end{matrix} $$ of simplicial varieties where the vertical arrows are principal $GL_m({\Bbb C})$ bundles, and where the right hand map in the top row is induced by evaluation on $e \in {\Bbb C}^m - \{0\}$. The morphism $\widetilde{B} GL_m({\Bbb C})^{\text{gen}} \to BGL_m({\Bbb C})^{\text{gen}}$ is a homotopy equivalence of simplicial varieties, and therefore induces an isomorphism on Deligne cohomology. The class $c_m(\nu^{\text{gen}})$ therefore descends naturally to the class $$ c_m(\mu) \in \Hd^{2m}(\widetilde{B}_\bullet G^m,{\Bbb R}(m)). $$ We will prove the theorem by showing that there is a class $$ \lambda_m \in s\Hd^{2m}(B_\bullet G^m,{\Bbb R}(m))\cong s\Hd^{2m}(G^m_\bullet,{\Bbb R}(m)) $$ such that $$ p^\ast \lambda_m = \Alt c_m(\mu) \in s\Hd^{2m}(\widetilde{B}_\bullet G^m,{\Bbb R}(m)). $$ Observe that each of the varieties defined in this section is a $\Sigma_\bullet$ variety and that all morphisms between them that we have constructed in this section respect the $\Sigma_\bullet$ structures. Denote the $(m-1)$ skeleton of $\widetilde{B}_\bullet G^m$ by $\widetilde{B}_{<m} G^m$. (This is the $[0,m-1]$ truncated simplicial variety whose $n$ simplices are identical with those of $\widetilde{B}_\bullet G^m$ when $n < m$ and empty otherwise.) \begin{proposition}\label{les} There is a long exact sequence \begin{multline*} \dots \to \Hd^{k}(G^m_\bullet,{\Bbb R}(m)) \to \Hd^{k}(\widetilde{B}_\bullet G^m,{\Bbb R}(m)) \to \Hd^{k}(\widetilde{B}_{<m}G^m,{\Bbb R}(m)) \to \\ \dots \to \Hd^{2m}(G^m_\bullet,{\Bbb R}(m)) \to \Hd^{2m}(\widetilde{B}_\bullet G^m,{\Bbb R}(m)) \to \Hd^{2m}(\widetilde{B}_{<m}G^m,{\Bbb R}(m)) \end{multline*} which remains exact when the alternating part functor $s$ is applied. \end{proposition} \begin{pf} Let $B_{\ge m}G^m$ be the $[m,\infty)$-truncated simplicial variety whose $n$ simplices are those of $B_\bullet G^m$ when $n\ge m$ and empty otherwise. Let $\widetilde{B}_{\ge m} G^m$ be the analogous $[m,\infty)$-truncated simplicial variety constructed out of the $n$ simplices of $\widetilde{B}_\bullet G^m$ for $n\ge m$. The natural projection $\widetilde{B}_{\ge m} G^m \to B_{\ge m}G^m$ induces an isomorphism on Deligne cohomology as $\widetilde{B}_n G^m \to B_n G^m$ is a homotopy equivalence whenever $n \ge m$. An easy spectral sequence argument shows that the inclusion $G^m_\bullet \to B_{\ge m}G^m$ induces an isomorphism $$ \Hd^k(B_{\ge m}G^m,{\Bbb R}(m)) \to \Hd^{k}(G^m_\bullet,{\Bbb R}(m)) $$ when $k\le 2m$. This isomorphism is compatible with symmetric group actions. We therefore have an isomorphism $$ \Hd^{k}(G^m_\bullet,{\Bbb R}(m)) \cong \Hd^k(\widetilde{B}_{\ge m}G^m,{\Bbb R}(m)) $$ when $k\le 2m$, also compatible with symmetric group actions. Finally, observe that the sequence $$ 0 \to {\Bbb R}_{\cal D}^\bullet(\widetilde{B}_{\ge m}G^m) \to {\Bbb R}_{\cal D}^\bullet(\widetilde{B}_\bullet G^m) \to {\Bbb R}_{\cal D}^\bullet(\widetilde{B}_{<m}G^m) \to 0 $$ of Deligne cochain complexes is exact and compatible with the symmetric group actions. It induces a long exact sequence on cohomology. The result follows by combining these results. \end{pf} Since the sequence $$ s\Hd^{2m}(G^m_\bullet,{\Bbb R}(m)) \to s\Hd^{2m}(\widetilde{B}_\bullet G^m,{\Bbb R}(m)) \to s\Hd^{2m}(\widetilde{B}_{<m}G^m,{\Bbb R}(m)) $$ is exact, the existence of a lift $\lambda_m$ of $\Alt c_m(\mu^{\text{gen}})$ will be proved if we can show that the image of $\Alt c_m(\mu)$ in $$ s\Hd^{2m}(\widetilde{B}_{<m}G^m,{\Bbb R}(m)) $$ vanishes. \begin{proposition}\label{triv_act} If $0 \le n < m$, then $\Hd^\bullet(\widetilde{B}_nG^m,{\Bbb R}(m))$ is a trivial $\Sigma_{n+1}$-module. \end{proposition} \begin{pf} We begin the proof with an elementary observation. Suppose that $A$ is an $m\times k$ matrix of complex numbers. If $k\le m$ and if the columns of $A$ are linearly independent, then so are the columns of $AB$ for all $B\in GL_k({\Bbb C})$. This is not the case when $k>m$: if each $m$ columns of $A$ are linearly independent, then it is not true that each $m$ of the columns of $AB$ are linearly independent for all $B\in GL_k({\Bbb C})$. In other notation, this says that $E_nG^m$ has a natural right action of $GL_{n+1}({\Bbb C})$ provided that $n<m$. This action commutes with the diagonal left action of $GL_m({\Bbb C})$ on $E_nG^m$. After taking the product with $EGL_m({\Bbb C})$ and taking the quotient by $GL_m({\Bbb C})$, we see that $\widetilde{B}_nG^m$ has a natural right $GL_{n+1}({\Bbb C})$ action, provided $n< m$. Identify $\Sigma_{n+1}$ with the subgroup of $GL_{n+1}({\Bbb C})$ consisting of all permutation matrices. The restriction of the right action of $GL_{n+1}$ on $E_nG^m$ to $\Sigma_{n+1}$ is its standard action. It follows that the action of $\Sigma_{n+1}$ on $\widetilde{B}_nG^m$ is the restriction of the $GL_{n+1}({\Bbb C})$ action. Since $GL_{n+1}({\Bbb C})$ is connected, it follows that the automorphism of $\widetilde{B}_nG^m$ induced by an element of $\Sigma_{n+1}$ is homotopic to the identity. \end{pf} \begin{proposition}\label{equivalence} If $n<m$, then there is a natural map of simplicial varieties $$ BGL_{m-n-1}({\Bbb C}) \to \widetilde{B}_nG^m $$ which is a homotopy equivalence. \end{proposition} \begin{pf} This follows from two facts: First, $GL_m({\Bbb C})$ acts transitively on $E_nG^m$ and the isotropy group of a point is $$ G(n) = {\left ( \begin{array}{cc} I_{n+1} & \ast \\ 0 & {\rm GL}_{m-n-1}({\Bbb C}) \end{array} \right )}. $$ Second, the inclusion of $GL_{m-n-1}({\Bbb C})$ into $G(n)$ is a homotopy equivalence. \end{pf} \begin{lemma}\label{isomorphism} The inclusion $\widetilde{B}_0G^m \hookrightarrow \widetilde{B}_{<m}G^m$ induces an isomorphism $$ s\Hd^\bullet(\widetilde{B}_{<m}G^m, {\Bbb R}(m)) \cong \Hd^\bullet(\widetilde{B}_0G^m,{\Bbb R}(m)). $$ Consequently, there is a natural isomorphism $$ s\Hd^\bullet(\widetilde{B}_{<m}G^m,{\Bbb R}(m))\cong \Hd^\bullet(BGL_{m-1}({\Bbb C}),{\Bbb R}(m)). $$ \end{lemma} \begin{pf} The first isomorphism follows immediately from (\ref{triv_act}) by looking at the spectral sequence associated to $\widetilde{B}_{<m}G^m$. The second assertion follows the previous result. \end{pf} To complete the proof of the theorem, observe that the restriction of $\widetilde{E} G^m \to \widetilde{B} G^m$ to $\widetilde{B}_0 G^m$ has structure group the group $G(0)$ defined in the proof of (\ref{equivalence}). Since this group is homotopy equivalent to $GL_{m-1}({\Bbb C})$, it follows that the image of $c_m(\mu)$ in $$ \Hd^{2m}(\widetilde{B}_0G^m,{\Bbb R}(m)) = s\Hd^{2m}(\widetilde{B}_{<m},{\Bbb R}(m)) $$ vanishes. This establishes the existence of $\lambda_m$. To prove uniqueness, note that it follows from (\ref{les}) that $\lambda_m$ is unique if \begin{equation}\label{silly} s\Hd^{2m-1}(\widetilde{B}_\bullet G^m,{\Bbb R}(m)) \to s\Hd^{2m-1}(\widetilde{B}_{<m}G^m,{\Bbb R}(m)). \end{equation} is surjective. By (\ref{isomorphism}), $$ s\Hd^{2m-1}(\widetilde{B}_{<m}G^m,{\Bbb R}(m)) \cong H^{2m-2}(BGL_{m-1}({\Bbb C}),{\Bbb C}/{\Bbb R}(m)). $$ Thus, to prove that (\ref{silly}) is surjective, it suffices to prove that the restriction mapping $$ H^{2m-2}(\widetilde{B}_\bullet G^m) \to H^{2m-2}(\widetilde{B}_0 G^m)\cong H^{2m-2}(BGL_{m-1}({\Bbb C})). $$ induced by the inclusion $\widetilde{B}_0 G^m \hookrightarrow \widetilde{B}_\bullet G^m$ is surjective. This follows as the restriction of the natural $GL_m({\Bbb C})$ bundle $\mu$ to $\widetilde{B}_0 G^m$ corresponds to the universal $GL_{m-1}({\Bbb C})$ bundle over $BGL_{m-1}({\Bbb C})$. The Chern classes $c_1(\mu),\dots, c_{m-1}(\mu)$ therefore restrict to the generators of the cohomology ring of $\widetilde{B}_0 G^m$. Surjectivity follows and, along with it, the uniqueness of $\lambda_m$. \section{Chern Classes in Algebraic $K$-theory---Addendum} In this section we prove two results needed in the proof of the existence of Grassmann logarithms and in relating them to Chern classes on algebraic $K$-theory. The first result asserts that the class $$ \lambda_m \in s\Hd^{2m}(G^m_\bullet,{\Bbb R}(m)) $$ can be used to represent the restriction $$ c_m : r_mK_p(\eta_X) \to \Hd^{2m-p}(\eta_X,{\Bbb R}(m)) $$ of the Chern class to the rank $m$ part of the algebraic $K$-theory of the generic point $\eta_X$ of each complex algebraic variety $X$. The second result asserts that the restriction of the class $\lambda_m$ to $G^m_0$ is the volume form $\mbox{vol}_m$. Let $U$ be a smooth Zariski open subset of $X$. Denote its coordinate ring by ${\Bbb C}[U]$. Let $G^m_\bullet({\Bbb C}[U])$ denote the simplicial set (i.e., 0-dimensional simplicial variety) whose $n$ simplices consist of all regular maps $U \to G^m_n$. The evaluation map $$ U \times G^m_\bullet({\Bbb C}[U]) \to G^m_\bullet $$ induces a map $$ \Hd^{2m}(B_\bullet G^m,{\Bbb R}(m)) \to \Hd^{2m}(U\times B_\bullet G^m({\Bbb C}[U]),{\Bbb R}(m)) $$ on Deligne cohomology. As in the case of the construction of Chern classes on $K$-theory, by evaluation on $\lambda_m$ we obtain maps $$ l_{p,m} : H_p(GL_m({\Bbb C}[U])) \to \Hd^{2m-p}(U,{\Bbb R}(m)). $$ Denote the function field of $X$ by $F$. \begin{theorem}\label{lambda-chern} The maps $l_{p,m}$ induce the restriction of the $m$-th Chern class $$ c_m : r_mK_p(\eta_X) \to \Hd^{2m-p}(\eta_X,{\Bbb R}(m)) $$ to the rank $m$ part of $K_p(\eta_X)$. That is, if $x\in r_mK_p(\eta_X)$, then $$ c_m(x) = l_{p,m}(\tilde{x}) $$ where $\tilde{x}$ is any element of $H_p(GL_m(F),{\Bbb Q})$ whose image in $$ K_p(\eta_X)_{\Bbb Q} \subseteq H_p(GL(F),{\Bbb Q}) $$ is $x$. \end{theorem} \begin{remark} This construction (and the theorem) are equally valid for the class $\lambda_m|_{U^m_\bullet}$ in $\Hd^{2m}(U^m_\bullet,{\Bbb R}(m))$, where $U^m_\bullet$ is a Zariski open subset of $G^m_\bullet$ that satisfies the condition (\ref{condition}) of \S \ref{polylog} --- see (\ref{variant}). \end{remark} \begin{pf} We begin by showing that elements of $\Hd^{2m}(BGL_m({\Bbb C})^{\text{gen}},{\Bbb R}(m))$ also induce maps $$ H_p(GL_m({\Bbb C}[U])) \to \Hd^{2m-p}(U,{\Bbb R}(m)) $$ for all smooth varieties. The construction is very similar to that of the universal Chern classes and the maps $l_{p,m}$, so we'll be brief. View $BGL_m({\Bbb C}[U])^{\text{gen}}$ as the simplicial set whose $n$ simplices consist of all regular maps $$ U \to B_nGL_m({\Bbb C})^{\text{gen}}. $$ The classes $\Alt c_m(\nu^{\text{gen}})$ and $c_m(\nu_{\text{gen}})$ both induce maps $$ H_p(GL_m({\Bbb C}[U])) \to \Hd^{2m-p}(U,{\Bbb R}(m)). $$ It follows immediately from (\ref{sign-copy}) that these two maps agree. Denote this map by $c_m^{\text{gen}}$. By the naturality of the constructions, the diagram (whose horizontal maps are induced by evaluation) $$ \begin{CD} \Hd^{2m}(B_\bullet G^m,{\Bbb R}(m)) @>>> \Hd^{2m}(U\times B_\bullet G^m({\Bbb C}[U]), {\Bbb R}(m))\\ @VVV @VVV \\ \Hd^{2m}(BGL_m({\Bbb C})^{gen},{\Bbb R}(m)) @>>> \Hd^{2m}(U\times BGL_m({\Bbb C}[U])^{gen},{\Bbb R}(m))\\ @AAA @AAA \\ \Hd^{2m}(BGL_M({\Bbb C}),{\Bbb R}(m)) @>>> \Hd^{2m}(U\times BGL_M({\Bbb C}[U]),{\Bbb R}(m)) \end{CD} $$ commutes for all $M\ge m$. By taking $M$ to be sufficiently large ($M\ge p$ will do by Suslin \cite{suslin}), and taking the limit over all smooth open subsets $U$ of $X$, we see that the diagram $$ \begin{CD} H_p(GL_m(F),{\Bbb Q}) @>l_{p,m}>> \Hd^{2m-p}(\eta_X,{\Bbb R}(m))\\ @VVV @VVV \\ H_p(GL_m(F),{\Bbb Q}) @>c_m^{\text{gen}}>> \Hd^{2m-p}(\eta_X,{\Bbb R}(m))\\ @AAA @AAA \\ H_p(GL(F),{\Bbb Q}) @>c_m>> \Hd^{2m-p}(\eta_X,{\Bbb R}(m))\\ @A\text{Hurewicz}AA @AAA\\ K_p(\eta_X)_{\Bbb Q} @>c_m>> \Hd^{2m-p}(\eta_X,{\Bbb R}(m)) \end{CD} $$ commutes. The result follows. \end{pf} \begin{remark}\label{factorization} The proof actually shows that $\lambda_m$ induces a map $$ \overline{l}_{p,m} : GH^m_p(\eta_X) \to \Hd^{2m-p}(\eta_X,{\Bbb R}(m)) $$ and that $l_{p,m}$ is the composition with $\overline{l}_{p,m}$ with the natural map $$ H_p(GL_m(F)) \to GH^m_p(\eta_X). $$ \end{remark} Next, we determine the restriction of the class $\lambda_m$ to $G^m_0$. \begin{theorem} \label{multiple} The image of $\lambda_m$ under the restriction mapping $$ s\Hd^{2m}(G^m_\bullet,{\Bbb R}(m))\rightarrow sH^m(G^m_0,{\Bbb C}) $$ is $\mbox{vol}_m$. \end{theorem} Suppose that $k$ is a field. In the rest of this section, we shall denote the $K$-theory and Grassmann homology of $\spec k$ by $K_\bullet(k)$ and $GH^m_\bullet(k)$, respectively. Before giving the proof, we review some results of Suslin from \cite{suslin}. For this discussion, $k$ is an infinite field. Define $S_m(k)$ to be $$ \left[\coker\Bigg\{ \bigoplus_{E_{m+2}G^m(k)} {\Bbb Z} \stackrel{A_\ast}{\longrightarrow} \bigoplus_{E_{m+1}G^m(k)}{\Bbb Z}\Bigg\}\right]\otimes_{GL_m(k)}{\Bbb Z}. $$ This is the group of Grassmann $m$-chains mod boundaries. It is generated by the equivalence class of the $(m+1)$ tuples of vectors $$ (e_1,\dots, e_m, \sum a_ie_i), $$ where $e_1,\dots, e_m$ is the standard basis of $k^m$. Following Suslin, we shall denote the corresponding element of $S_m(k)$ by $\langle a_1,\dots, a_m \rangle$. There is a natural inclusion \begin{equation}\label{into_S} GH^m_m(k) \hookrightarrow S_m(k) \end{equation} whose cokernel is trivial or ${\Bbb Z}$ according to whether $m$ is odd or even. By (\ref{grass_homo}), there is a map \begin{equation}\label{into_GH} H_m(GL_m(k)) \to GH^m_m(k). \end{equation} Denote the Milnor $K$-theory of $k$ by $K^M_\bullet(k)$. It is a ring generated by $K^M_1(k) = k^\times$. The symbol $\{a_1,\dots,a_m\}\in K^M_m(k)$ is the product of the $a_i\in k^\times$. Suslin shows that there is a well defined homomorphism \begin{equation}\label{into_milnor} S_m(k) \to K^M_m(k) \end{equation} defined by $$ \langle a_1,\dots, a_m \rangle \mapsto \{a_1,\dots,a_m\}. $$ Suslin also proves that the map $$ H_m(GL_m(k)) \to H_m(GL(k)) $$ is an isomorphism. Consequently, the Hurewicz homomorphism indues a homomorphism $$ K_m(k) \to H_m(GL_m(k)) $$ Composing this with (\ref{into_S}), (\ref{into_GH}), and (\ref{into_milnor}), he obtains a map $$ \phi : K_m(k) \to K^M_m(k). $$ He proves that it has the property that if $a_1,\dots,a_m \in k^\times$, then $$ \phi(a_1\cdot \dots \cdot a_m) = (-1)^{m-1}(m-1)! \{a_1,\dots,a_m\}. $$ \begin{pf*}{Proof of Theorem \ref{multiple}} There is a rational number $K$ such that the image of $\lambda_m$ in $H^m(G^m_0,{\Bbb Q}(m))$ is $K\mbox{vol}_m$. Let $X = {\Bbb C}^m$. We will determine $K$ by evaluating $l_{m,m}$ on a class in $K_m(\eta_X)$ and comparing the answer with the value of the Chern class on it. Deligne cohomology maps to de~Rham cohomology, and it will be sufficient for our needs to replace the Chern class and the map $l_{m,m}$ with their composite with the map to de~Rham cohomology. Denote the function field of $X$ by $F$. The de~Rham cohomology of $\spec F$ is the set of K\"ahler differentials $\Omega_{F/{\Bbb C}}^p$. It is a standard fact that the first Chern class $$ c_1 : K_1(F) \to \Omega^1_{F/{\Bbb C}} $$ takes $f$ to $df/f$. It follows from standard properties of Chern classes on algebraic $K$-theory (see, for example, \cite[p.~28]{schneider}), that \begin{equation}\label{chern_map} c_m : K_m(F) \to \Omega^m_{F/{\Bbb C}} \end{equation} takes $f_1\cdot \dots \cdot f_m$ ($f_i \in F^\times$) to $$ (-1)^{m-1}(m-1)! \frac{df_1}{f_1}\wedge \dots \wedge \frac{d f_m}{f_m} $$ It follows that if we define $$ \psi : K^M_m(F) \to \Omega^m_{F/{\Bbb C}} $$ by $$ \psi : \{f_1,\dots ,f_m\} \to \frac{df_1}{f_1}\wedge \dots \wedge\frac{d f_m}{f_m}, $$ then the restriction $$ K^M_m(F) \to K_m(F) \stackrel{c_m}{\to} \Omega^m_{F/{\Bbb C}} $$ of the Chern class to the Milnor $K$-theory is the composite $$ K^M_m(F) \to K_m(F) \stackrel{\phi}{\to} K^M_m(k) \stackrel{\psi}{\to} \Omega^m_{F/{\Bbb C}}. $$ To compute the constant $K$ that relates $\lambda_m$ and $\mbox{vol}_m$, we use the fact that the map $$ H_m(GL_m(F)) \to H_m(GL(F)) $$ is an isomorphism. From (\ref{lambda-chern}), it follows that we can compute (\ref{chern_map}) using $\lambda_m$. Observe that since $S_m(F)$ is the set of all Grassmann $m$-chains mod boundaries, the class $\lambda_m$ induces a map $$ S_m(F) \to \Omega^m_{F/{\Bbb C}} $$ such that the diagram $$ \begin{CD} K_m(F) @>>> S_m(F) @>\lambda_m>> \Omega^m_{F/{\Bbb C}} \\ @V\text{Hurewicz}VV @AA\text{inclusion}A \\ H_m(GL_m(F)) @>>> GH^m_m(F) \\ \end{CD} $$ commutes. To compute the map $S_m(F) \to \Omega^m_{F/{\Bbb C}}$ induced by $\lambda_m$, it is first necessary to realize that there are two descriptions of $G^m_0$: first, it is the quotient of the set of $(m+1)$-tuples of vectors $(v_0,v_1,\dots, v_m)$ in ${\Bbb C}^m$, in general position, mod the diagonal action $GL_m$. In this case an isomorphism with $(k^\times)^m$ is given by $$ (a_1,\dots,a_m) \mapsto (e_1,\dots,e_m,\sum a_i e_i). $$ The second description is ${\Bbb P}^m$ minus the union of the coordinate hyperplanes. This is identified with $({\Bbb C}^\times)^m$ via the formula $$ (x_1,\dots,x_m) \mapsto [1,x_1,\dots,x_m]. $$ The two descriptions are related by identifying orbit of the vectors $(v_0,v_1,\dots, v_m)$ with the point of ${\Bbb P}^m$ corresponding to the kernel of the linear map $k^{m+1} \to k$ that takes $e_i$ to $v_i$ --- cf.\ \cite[\S 5]{hain-macp}. The volume form $\mbox{vol}_m$ on $G^m_0$ is, by convention (cf. \cite[p.~422]{hain-macp}), $$ \frac{dx_1}{x_1} \wedge \dots \wedge \frac{dx_m}{x_m}. $$ A short computation then shows that this equals the form $$ \frac{da_1}{a_1} \wedge \dots \wedge \frac{da_m}{a_m} $$ with respect to the quotient coordinates. It follows that the map $$ S_m(F) \to \Omega^m_{F/{\Bbb C}} $$ induced by $\lambda_m$ takes $\langle a_1,\dots,a_m \rangle$ to $$ K\frac{da_1}{a_1} \wedge \dots \wedge \frac{da_m}{a_m}. $$ Since $l_{m,m}$ is the composite $$ H_m(GL_m(F)) \to GH_m(F^m) \to H^m(\eta_X,{\Bbb R}(m)), $$ and since $l_{m,m}$ equals $c_m$, we deduce that $K= 1$. \end{pf*} \section{Generic Grassmann Polylogarithms---Existence and Relation to Chern Classes} We first use the results of the preceding sections to prove the existence of generic real Grassmann logarithms and establish their relation to the Beilinson Chern classes. \begin{theorem} For all $m$, there is a canonical choice of a generic real Grassmann $m$-logarithm. Moreover, for all complex algebraic varieties $X$, this $m$-logarithm induces the restriction $$ c_m : r_mK_n(\eta_X) \to \Hd^{2m-n}(\eta_X,{\Bbb R}(m)) $$ of the Beilinson-Chern class to the $m$th part of the rank filtration of $K_\bullet(\eta_X)$. \end{theorem} \begin{pf} By \cite[(7.1)]{hain:generic}, there is a Zariski open subset $V^m_\bullet$ of $G^m_\bullet$ where $U^m_0 = G^m_0$ and where each $V^m_n$ is a rational $K(\pi,1)$. By (\ref{summand}), there is a canonical injection $$ \Hd^{2m}(V^m_\bullet,{\Bbb R}(m)) \hookrightarrow H_{\cal BW}^{2m}(V^m_\bullet,{\Bbb R}(m)). $$ Let $L_m'$ be the restriction of $\lambda_m$ to $V^m_\bullet$, viewed as a class in $H_{\cal BW}^{2m}(V^m_\bullet,{\Bbb R}(m))$. We need to skew symmetrize. Let $U^m_\bullet$ be the Zariski open subset where $U^m_n$ is the intersection of the translates of $V^m_n$ under the action of $\Sigma_{m+n+1}$ on $G^m_n$. Then $U^m_0=G^m_0$ and $U^m_\bullet$ is a $\Sigma_\bullet$ variety. Let $$ L_m \in sH_{\cal BW}^{2m}(U^m_\bullet,{\Bbb R}(m)) $$ be the alternating part of the restriction of $L_m'$ to $U^m_\bullet$. It follows from (\ref{multiple}) that $L_m$ is a generic Grassmann $m$-logarithm. The second assertion is an immediate consequence of the definition of Grassmann logarithms, Theorems \ref{lambda-chern} and \ref{multiple}, and the fact that the open subvarieties $U^m_\bullet$ of $G^m_\bullet$ used above always satisfy the condition (\ref{condition}) of \S \ref{polylog}. \end{pf} \section{Comparison of Cohomologies} \label{comparison} In this section we prove Theorem \ref{summand}. We begin with a brief guide to the proof. For each smooth variety $X$ with $q=0$, we will construct a functorial complex ${\Bbb R}(m)^\bullet_{\cal F}(X)$ which is a formal analogue of the complex ${\Bbb R}(m)^\bullet_{\cal BW}(X)$. There will be a natural chain map $$ {\Bbb R}(m)^\bullet_{\cal F}(X) \to {\Bbb R}(m)^\bullet_{\cal BW}(X). $$ The homology of the formal ${\cal BW}$-complex will be denoted by $H_\F^\bullet(X,{\Bbb R}(m))$. Taking homology, we will have a commutative diagram $$ \begin{matrix} H_\F^\bullet(X,{\Bbb R}(m)) & \to & H_{\cal BW}^\bullet(X,{\Bbb R}(m)) \cr &\searrow & \downarrow \cr & & \Hd^\bullet(X,{\Bbb R}(m)) \cr \end{matrix} $$ We will show, when $X$ is an rational $n$-$K(\pi,1)$, that the composite $$ H_\F^\bullet(X,{\Bbb R}(m)) \to \Hd^\bullet(X,{\Bbb R}(m)) $$ is an isomorphism in degrees $\le n$ and injective in dimension $n+1$. The result in the case when $X_\bullet$ is a single space then follows. The simplicial version will follow using a spectral sequence argument. Our first task is to construct the complex ${\Bbb R}(m)_{\cal F}^\bullet(X)$. To do this, we need to construct a formal analogue $\Omega^\bullet_{\Bbb R}(X)_{\cal F}$ of $\Omega_{\Bbb R}^\bullet(X)$ and a formal analogue ${\cal BW}(X)_{\cal F}$ of the ring of ${\cal BW}(X)$. Since $q(X)=0$, it follows from elementary Hodge theory that there are regular functions $f_j : X \to {\Bbb C}$ such that $\Omega^1(X)$ has basis $df_1/f_1,\dots, df_m/f_m$. Let $A_{\Bbb C}^\bullet(X)$ be the ${\Bbb C}$-subalgebra of $\Omega^\bullet(X)$ generated by the $df_j/f_j$. Let $$ \theta_j = d\Arg f_j \text{ and } \rho_j = d \log|f_j|. $$ Note that $df_j/f_j = \rho_j + i \theta_j$. Observe that $\Lambda^\bullet_{\Bbb C}(df_j/f_j:j = 1,\dots, m)$ is a subalgebra of $\Lambda_{\Bbb C}^\bullet(\theta_j,\rho_j: j=1,\dots,m)$. The latter algebra has the real form $\Lambda_{\Bbb R}^\bullet(\theta_j,\rho_j)$. Each element $u$ of the ideal $$ K := \ker\left\{ \Lambda_{\Bbb C}^\bullet (df_j/f_j:j=1,\dots,m) \to A^\bullet(X)\right\} $$ can be viewed as elements of $\Lambda_{\Bbb C}^\bullet(\theta_j,\rho_j:j=1,\dots,m)$. So we can write each such $u$ in the form $a(u) + i b(u)$, where $a(u), b(u) \in \Lambda_{\Bbb R}^\bullet(\theta_j,\rho_j: j = 1,\dots,m)$. We define $\Omega_{\Bbb R}^\bullet(X)_{\cal F}$ to be the algebra $$ \Lambda_{\Bbb R}^\bullet(\theta_j,\rho_j: j=1,\dots, m)/(a(u), b(u): u\in K). $$ The ring of formal Bloch-Wigner functions ${\cal BW}(X)_{\cal F}$ is defined in terms of the Malcev completion of the fundamental group of $X$. Denote the complex form of the Malcev group associated to $\pi_1(X,x)$ by $G(X,x)$. Denote its real form by $G_{\Bbb R}(X,x)$. Each of these is the inverse limit of its finite dimensional quotients $G(X,x)_s$. The quotient $G_{\Bbb R}(X,x) \backslash G(X,x)$ is a real proalgebraic variety. Its coordinate ring is, by definition, the direct limit of the coordinate rings of its canonical quotients $G_{\Bbb R}(X,x)_s \backslash G(X,x)_s$ Recall that each path $\gamma$ in $X$ from $x$ to $y$ induces a group isomorphism $\mu_\gamma : G(X,x) \to G(X,y)$ which preserves real forms. \begin{proposition}\label{translation} (a) For all $x\in X$, there is a canonical real analytic map $$ \mu_x : X \to G_{\Bbb R}(X,x) \backslash G(X,x) $$ (b) If $\gamma$ is a path in $X$ from $x$ to $y$, then the diagram $$ \begin{matrix} X &\stackrel{\mu_x}{\longrightarrow} & G_{\Bbb R}(X,x) \backslash G(X,x)\cr & \mu_y \searrow & \downarrow \mu_\gamma \cr && G_{\Bbb R}(X,y) \backslash G(X,y)\cr \end{matrix} $$ commutes, where the vertical map is the one induced by $\mu_\gamma$. \newline (c) If $\gamma$ is a loop in $X$ based at $x$, then $$ \mu_\gamma^\ast : \O_{\Bbb R}\left(G_{\Bbb R}(X,x) \backslash G(X,x)\right) \to \O_{\Bbb R}\left(G_{\Bbb R}(X,x) \backslash G(X,x)\right), $$ is the identity. Here $\O_{\Bbb R}(Y)$ denotes the coordinate ring of the real proalgebraic variety $Y$. \end{proposition} \begin{pf} We will use the notation and terminology of \cite[\S 7]{hain-macp}. Observe that $G_{\Bbb R}(X,x)_s$ is the real Zariski closure of $\Gamma_s$ in $G_s$. The map $\mu_x$ is simply the inverse limit of the composites $$ X \stackrel{\theta_x^s}{\longrightarrow} \Alb^s(X,x) \to G_{\Bbb R}(X,x)_s\backslash G_s. $$ If $\gamma$ is a path in $X$ from $x$ to $y$, then we have the element $T_s(\gamma)$ of $G_s$. The sequence $\left\{T_s(\gamma)\right\}$ converges to an element $T(\gamma)$ of $G$. We have $$ G_{\Bbb R}(X,y) = T(\gamma)^{-1}G_{\Bbb R}(X,x)T(\gamma). $$ The map $\mu(\gamma)$ is induced by left multiplication by $T(\gamma)$. The final statement follows as the coordinate ring of $G_{\Bbb R}(X,x) \backslash G(X,x)$ is the ring of functions on $G(X,x)$ that are invariant under left multiplication by elements of $G_{\Bbb R}(X,x)$. \end{pf} Define the ring of {\it formal Bloch-Wigner functions on $X$}, ${\cal BW}(X)_{\cal F}$, to be the coordinate ring of $G_{\Bbb R}(X,x) \backslash G(X,x)$. It follows from (\ref{translation}) that the assignment of ${\cal BW}(X)_{\cal F}$ to $X$ is a well defined contravariant functor. \begin{remark} What is called the ring of Bloch-Wigner functions in \cite[\S 11]{hain-macp} is what we are defining to be the ring of formal Bloch-Wigner functions in this paper. \end{remark}\medskip The {\it formal Bloch-Wigner complex of $X$} is defined by $$ \Omega^\bullet_{\cal BW}(X)_{\cal F} = {\cal BW}(X)_{\cal F} \otimes_{\Bbb R} \Omega^\bullet_{\Bbb R}(X)_{\cal F}. $$ It is a differential graded ${\Bbb R}$-algebra canonically associated to $X$. There are natural weight filtrations on $A^\bullet_{\Bbb C}(X)$, $\Omega_{\Bbb R}^\bullet(X)_{\cal F}$ and ${\cal BW}(X)_{\cal F}$, and therefore on $\Omega^\bullet_{\cal BW}(X)_{\cal F}$. These are defined as follows: The weight filtration on $A^\bullet_{\Bbb C}(X)$ is the one induced by its inclusion into $\Omega^\bullet(X)$. Since each $df_j/f_j$ has weight 2, it follows that all elements of $A^m_{\Bbb C}(X)$ have weight 2m. The weight filtration on $\Omega_{\Bbb R}^\bullet(X)_{\cal F}$ is also defined this way --- all elements of degree $m$ have weight $2m$. The weight filtration on ${\cal BW}(X)_{\cal F}$ is defined in \cite[\S 11]{hain-macp}. It is not difficult to check that the weight filtration of $\Omega^\bullet_{\cal BW}(X)_{\cal F}$ is a filtration by subcomplexes. (Use \cite[(7.7)]{hain-macp}.) Finally, we define $H_\F^\bullet(X,{\Bbb R}(m))$, the {\it formal Bloch-Wigner cohomology of $X$ with coefficients in ${\Bbb R}(m)$}, to be the cohomology of the complex $$ {\Bbb R}(m)_{\cal F}^\bullet(X) = \cone[F^pW_{2p}A_{\Bbb C}\supdot(X) \rightarrow W_{2p}\Omega_{\cal BW}\supdot(X)_{\cal F}\otimes{\Bbb R}(m-1)][-1]. $$ The map $A_{\Bbb C}^\bullet(X) \to \Omega_{\cal BW}^\bullet(X)_{\cal F}\otimes{\Bbb R}(m-1)$ is the composite of $$ A^\bullet_{\Bbb C}(X) \to \Omega^\bullet_{\Bbb R}(X)_{\cal F}\otimes\left({\Bbb R}(m-1) \oplus {\Bbb R}(m)\right) \to \Omega^\bullet_{\Bbb R}(X)_{\cal F}\otimes{\Bbb R}(m-1), $$ where the first map is the algebra homomorphism that takes $df_j/f_j$ to $\rho_j + i \theta_j$, with the natural inclusion $$ \Omega^\bullet_{\Bbb R}(X)_{\cal F}\otimes{\Bbb R}(m-1) \hookrightarrow \Omega_{\cal BW}^\bullet(X)_{\cal F}\otimes{\Bbb R}(m-1). $$ When $X_\bullet$ is a simplicial complex algebraic manifold where each $X_m$ has $q=0$, we define ${\Bbb R}(m)_{\cal F}^\bullet(X_\bullet)$ to be the total complex associated to the cosimplicial chain complex obtained by applying the functor ${\Bbb R}(m)_{\cal F}^\bullet({\phantom{X}})$ to $X_\bullet$. It follows from \cite[pp.~436--7]{hain-macp} that there is a natural $W_\bullet$ filtered algebra homomorphism ${\cal BW}(X)_{\cal F} \to {\cal BW}(X)$. There is an obvious filtered algebra homomorphism $\Omega^\bullet_{\Bbb R}(X)_{\cal F} \to \Omega^\bullet_{\Bbb R}(X)$. These induce chain maps $$ {\Bbb R}(m)_{\cal F}^\bullet(X) \to {\Bbb R}(m)_{\cal BW}^\bullet(X) \to \R(p)_{\cal D}^\bullet(X). $$ Similarly, in the simplicial case, we have chain maps $$ {\Bbb R}(m)_{\cal F}^\bullet(X_\bullet) \to {\Bbb R}(m)_{\cal BW}^\bullet(X_\bullet) \to \R(p)_{\cal D}^\bullet(X_\bullet). $$ Theorem \ref{summand} will follow directly from the following result. \begin{theorem}\label{tech} Suppose that $X_\bullet$ is a simplicial complex algebraic manifold where each $X_m$ has $q=0$. If, for all $m$, $X_m$ is a rational $(n-m)$-$K(\pi,1)$, then the natural map $$ H_\F^t(X_\bullet,{\Bbb R}(m)) \to \Hd^t(X_\bullet,{\Bbb R}(m)) $$ is an isomorphism when $t \le n$. \end{theorem} The proof of this result occupies the rest of this section. The first step is to observe that it follows from the analogue of \cite[(8.2)(iii)]{hain-macp} for rational $n$-$K(\pi,1)$s that if $X$ is a rational $n$-$K(\pi,1)$, then, for all $l$, the natural map $$ W_lA_{\Bbb C}^t(X) \to W_lH^t(X;{\Bbb C}) $$ is an isomorphism when $t\le n$ and injective when $t=n+1$. The second step is more difficult. We will show that if $X$ is a rational $n$-$K(\pi,1)$, then, for all $l$, $$ W_lH^t(\Omega_{\cal BW}^\bullet(X)_{\cal F}) \to W_lH^t(X,{\Bbb R}) $$ is an isomorphism when $t\le n$ and injective when $t=n+1$. First choose a base point $x\in X$. Let ${\frak g}$ be the complex form of the Malcev Lie algebra associated to $(X,x)$. We shall view it as a real Lie algebra with an almost complex structure $J$. Denote its real form by ${\frak g}_{\Bbb R}^{\phantom{X}}$. We shall denote their (real) continuous duals by ${\frak g}^\ast$ and ${\frak g}_{\Bbb R}^\ast$, respectively. In \cite[(7.7)]{hain-macp}, a ${\Bbb C}$-linear map $\theta_x^\ast : \Hom_{\Bbb C}({\frak g}, {\Bbb C}) \to E^\bullet_{\Bbb C}(X)$ is constructed and it is established that the image of $\theta^\ast_x$ is contained in $\Omega^\bullet(X)$. Define an ${\Bbb R}$ linear map $$ \Theta_x : {\frak g}^\ast \to \Omega_{\Bbb R}^\bullet(X)_{\cal F} $$ as follows: each $\phi\in {\frak g}^\ast$ can be extended canonically to a ${\Bbb C}$ linear map $\hat{\phi} : {\frak g} \to {\Bbb C}$. Define $$ \Theta_x(\phi) = \Re \theta_x^\ast(\hat{\phi}). $$ This induces an algebra homomorphism $$ \Lambda^\bullet_{\Bbb R}{\frak g}^\ast \to \Omega_{\Bbb R}^\bullet(X)_{\cal F}. $$ Since $\Theta$ clearly preserves the weight filtration, the induced algebra homomorphism does too. \begin{lemma}\label{first-qism} Suppose that $X$ is a complex algebraic manifold with $q=0$. If $X$ is a rational $n$-$K(\pi,1)$, then, for all $l$, $$ W_lH^t(\Lambda^\bullet_{\Bbb R} {\frak g}^\ast) \to W_lH^t(\Omega^\bullet_{\Bbb R}(X)_{\cal F}) $$ is an isomorphism when $t\le n$ and injective when $t=n+1$. \end{lemma} \begin{pf} Recall that ${\frak g}$ is viewed as a real Lie algebra with almost complex structure $J$. Consequently, $$ {\frak g}\otimes_{\Bbb R} {\Bbb C} = {\frak g}' \oplus {\frak g}'' $$ where ${\frak g}'$ and ${\frak g}''$, the $i$ and $-i$ eigenspaces of $J$, respectively, are commuting complex Lie subalgebras of ${\frak g}\otimes_{\Bbb R}{\Bbb C}$.. It follows that $$ \Lambda^\bullet {\frak g}^\ast\otimes_{\Bbb R} {\Bbb C} \cong \Lambda^\bullet {{\frak g}'}^\ast \otimes \Lambda^\bullet {{\frak g}''}^\ast. $$ This is an isomorphism of $W_\bullet$ filtered cochain complexes. Similarly, there is an almost complex structure on the real vector space $V$ with basis $\theta_j, \rho_j$, where $1\le j \le m$. It is defined by $$ J : \theta_j \to \rho_j \text{ and } J : \rho_j \to -\theta_j. $$ Define $V'$ and $V''$ to be the $i$ and $-i$ eigenspaces of $J$ acting on $V\otimes {\Bbb C}$. Then $V'$ has basis $$ df_j/f_j = \rho_j + i \theta_j, \quad j=1, \dots, m $$ and $V'$ has basis their complex conjugates. It follows that \begin{multline*} \Omega_{\Bbb R}^\bullet(X)_{\cal F} \cong \\ \Lambda_{\Bbb C}^\bullet(df_j/f_j,j=1,\dots,m)/(a(u)+ ib(u)) \otimes \Lambda_{\Bbb C}^\bullet(d\overline{f}_j/\overline{f}_j,j=1,\dots,m)/(a(u)- ib(u)) \\ \cong A^\bullet_{\Bbb C}(X) \otimes \overline{A^\bullet_{\Bbb C}(X)} \end{multline*} Each of these algebras has the property that its degree $m$ part has weight $2m$. Consequently, each of these isomorphisms is a $W_\bullet$ filtered algebra isomorphism. The complexification of the map in the statement of the proposition is the tensor product of the algebra homomorphism $$ \Lambda^\bullet {{\frak g}'}^\ast \to A_{\Bbb C}^\bullet(X) $$ with its complex conjugate. It therefore suffices to prove that this map is a $W_\bullet$ filtered quasi-isomorphism. Note that ${\frak g}'$ is just the complex form of the Malcev Lie algebra associated to $\pi_1(X,x)$ and that the map above is the map induced by the homomorphisms $\theta_x$ of \cite[(7.7)]{hain-macp}. This homomorphism induces a homomorphism \begin{equation}\label{qism} \Lambda^\bullet {{\frak g}'}^\ast \to A^\bullet_{\Bbb C}(X) \subseteq H^\bullet(X;{\Bbb C}) \end{equation} which is the complexification of a morphism of mixed Hodge structures. The result now follows as morphisms of mixed Hodge structures are strict with respect to $W_\bullet$ and since the map on homology induced by (\ref{qism}) is an isomorphism in dimensions $\le n$ and injective in dimension $n+1$ by the definition of a rational $n$-$K(\pi,1)$. \end{pf} View $\Lambda^\bullet_{\Bbb R} {\frak g}^\ast$ as the real left invariant differential forms on $G$, the complex form of the Malcev completion of $\pi_1(X,x)$. Here $G$ is viewed as a real proalgebraic group by restriction of scalars. It follows from the fact that elements of ${\cal BW}(X)_{\cal F}$ are represented by iterated integrals of elements of $\Lambda^\bullet_{\Bbb R} {\frak g}^\ast$ that the exterior derivative of each element of ${\cal BW}(X)_{\cal F}$ is an element of ${\cal BW}(X)_{\cal F} \otimes \Lambda^\bullet_{\Bbb R} {\frak g}^\ast$ (cf. \cite[p.~436]{hain-macp}). Consequently, ${\cal BW}(X)_{\cal F} \otimes \Lambda^\bullet_{\Bbb R} {\frak g}^\ast$ is a subcomplex of $\lim_\to E^\bullet_{\Bbb R}(G_s)$, the de~Rham complex of $G$. \begin{lemma}\label{second-qism} Suppose that $X$ is a complex algebraic manifold with $q=0$. If $X$ is a rational $n$-$K(\pi,1)$, then the natural map $$ \theta^\ast : \Omega_{\cal BW}^\bullet(X)_{\cal F} \to E^\bullet_{\Bbb R}(X) $$ induces maps $$ W_lH^m(\Omega_{\cal BW}^\bullet(X)_{\cal F}) \to W_lH^m(X;{\Bbb R}) $$ which are isomorphisms when $m\le n$ and injective when $m=n+1$. \end{lemma} \begin{pf} By considering the formal analogue of $\Omega_{\cal BW}^\bullet(X)_{\cal F}$ for $\Alb X$, it follows that one can put a differential on $$ {\cal BW}(X)_{\cal F} \otimes \Lambda^\bullet_{\Bbb R} {\frak g}^\ast $$ such that $\Lambda^\bullet_{\Bbb R} {\frak g}^\ast$ is a subcomplex and such that the map to $\Omega_{\cal BW}^\bullet(X)_{\cal F}$ induced by $$ \Lambda^\bullet_{\Bbb R} {\frak g}^\ast \to \Omega_{\Bbb R}^\bullet(X)_{\cal F} $$ is a $W_\bullet$ filtered chain map. It follows from (\ref{first-qism}) that chain map $$ {\cal BW}(X)_{\cal F} \otimes \Lambda^\bullet_{\Bbb R} {\frak g}^\ast \to {\cal BW}(X)_{\cal F} \otimes \Omega_{\Bbb R}^\bullet(X)_{\cal F} = \Omega_{\cal BW}^\bullet(X)_{\cal F} $$ induces an isomorphism on $W_lH^k$ for all $l$ when $k\le n$ and an injection on $W_lH^{n+1}$ for all $l$. (Filter each $Gr^W_l$ by degree.) To prove the result, we need only show that the complexification of the composite $$ {\cal BW}(X)_{\cal F} \otimes \Lambda^\bullet_{\Bbb R} {\frak g}^\ast \otimes {\Bbb C} \to E^\bullet({\overline{X}} \log D) $$ induces an isomorphism on $W_mH^t$ when $t\le n$ and an injection on $W_mH^{n+1}$. As in the proof of (\ref{first-qism}), we have the decomposition $$ {\frak g}\otimes {\Bbb C} = {\frak g}' \oplus {\frak g}'', $$ where ${\frak g}'$ and ${\frak g}''$ are commuting Lie algebras, and the $W_\bullet$ filtered quasi-isomorphism $$ \Lambda^\bullet_{\Bbb C} {\frak g} \cong \Lambda^\bullet {{\frak g}'}^\ast \otimes \Lambda^\bullet {{\frak g}''}^\ast $$ Denote the Malcev group corresponding to ${\frak g}\otimes {\Bbb C}$ by $G_{\Bbb C}$, and the commuting subgroups of $G_{\Bbb C}$ corresponding to ${\frak g}'$ and ${\frak g}''$ by $G'$ and $G''$, respectively. Then $$ G_{\Bbb C} = G' \times G''. $$ Denote the complex points $G_{\Bbb R}({\Bbb C})$ of $G_{\Bbb R}({\Bbb C})$ by $H$. Multiplication induces a continuous map $$ H \times G'' \to G_{\Bbb C}. $$ This is a continuous bijection. To see this, let $H^s$, $G_{\Bbb C}^s$, etc. denote the $s$th terms of the lower central series of $H$, $G_{\Bbb C}$, etc. Note that the $s$th graded quotient of ${\frak g}_{\Bbb C}$ is the direct sum of the $s$th graded quotients of the lower central series of ${\frak h}$ and ${\frak g}''$. It follows that if $g \in G_{\Bbb C}$ is congruent to $hg''$ mod $G_{\Bbb C}^s$, where $h\in H$ and $g''\in G''$, then there exit $v\in H^s$, $u''\in {G'}^s$, unique mod $H^{s+1}$ and ${G''}^{s+1}$, such that $$ g(hg'')^{-1} = vu'' \text{ in } G^s_{\Bbb C}/G^{s+1}_{\Bbb C}. $$ Since $G^s_{\Bbb C}/G^{s+1}_{\Bbb C}$ is central in $G_{\Bbb C}/G^{s+1}_{\Bbb C}$, it follows that $g$ is congruent to $(hv)(g''u'')$ mod $G^{s+1}_{\Bbb C}$. The assertion follows by taking limits. Since ${\cal BW}(X)_{\cal F}$ is the coordinate ring of the real proalgebraic variety $G_{\Bbb R}\backslash G$, it follows that ${\cal BW}(X)_{\cal F}\otimes {\Bbb C}$ is the coordinate ring of the complex proalgebraic variety $H\backslash G_{\Bbb C}$. It follows immediately that the composite $$ G'' \to G_{\Bbb C} \to H\backslash G_{\Bbb C} $$ is an isomorphism of proalgebraic varieties. Consequently, there is a natural algebra isomorphism $$ \O(G'') \cong {\cal BW}(X)_{\cal F} \otimes {\Bbb C}. $$ Since $G''$ is prounipotent, the exponential map induces an isomorphism $$ {\Bbb C}[{{\frak g}''}^\ast] \cong \O(G''). $$ Assembling the pieces, we obtain an algebra isomorphism \begin{equation}\label{filt_alg} {\cal BW}(X)_{\cal F} \otimes \Lambda^\bullet_{\Bbb R} {\frak g}^\ast \otimes {\Bbb C} \cong {\Bbb C}[{{\frak g}''}^\ast] \otimes \Lambda^\bullet {{\frak g}'}^\ast \otimes \Lambda^\bullet {{\frak g}''}^\ast \end{equation} The differential induced on the right hand side can be understood. When the right hand side is quotiented out by the subcomplex $\Lambda^\bullet {{\frak g}'}^\ast$, the resulting complex is isomorphic to the complex $$ {\Bbb C}[{{\frak g}''}^\ast] \otimes \Lambda^\bullet {{\frak g}''}^\ast $$ which is analogous to the complex in the proof of \cite[(7.8)]{hain-macp}. This complex is easily seen to by acyclic by the standard argument given there. It follows that the inclusion \begin{equation}\label{alg_qism} \Lambda^\bullet {{\frak g}'}^\ast \hookrightarrow {\Bbb C}[{{\frak g}''}^\ast] \otimes \Lambda^\bullet {{\frak g}'}^\ast \otimes \Lambda^\bullet {{\frak g}''}^\ast \end{equation} is a quasi-isomorphism. It remains to show that this is a filtered quasi-isomorphism. First observe that since ${\frak g}^\ast$ is the direct limit of complex parts of mixed Hodge structures (albeit, viewed as a real vector spaces), its weight filtration has a canonical splittings by $J$ invariant subspaces. It follows from the definitions that there are canonical splitting of the weight filtrations of ${{\frak g}'}^\ast$ and ${{\frak g}''}^\ast$ such that the isomorphism $$ {\frak g}^\ast \cong {{\frak g}'}^\ast \oplus {{\frak g}''}^\ast $$ is an isomorphism of graded vector spaces. Consequently, there are compatible canonical splittings of the weight filtrations on $$ \Lambda^\bullet {{\frak g}'}^\ast,\quad \Lambda^\bullet {{\frak g}''}^\ast, \quad \Lambda^\bullet_{\Bbb C} {{\frak g}}^\ast, \quad {\Bbb C}[{{\frak g}}^\ast], \quad {\Bbb C}[{{\frak g}'}^\ast], \quad {\Bbb C}[{{\frak g}''}^\ast]. $$ Since ${\cal BW}(X)_{\cal F} = {\Bbb R}[{{\frak g}}^\ast]^{\frak g}$ and since the action of ${\frak g}$ on ${\Bbb R}[{{\frak g}}^\ast]$ comes from a morphism of mixed Hodge structures, this action is compatible with the splittings of the weight filtrations. It follows that the weight filtration of ${\cal BW}(X)_{\cal F}$ has a canonical splitting {\it that depends upon the choice of the base point $x$}. It follows that (\ref{filt_alg}) is an isomorphism of graded algebras and that (\ref{alg_qism}) is a quasi-isomorphism of filtered algebras, and therefore a $W_\bullet$ filtered quasi-isomorphism. To complete the proof, we have to show that the composite $$ \Lambda^\bullet {{\frak g}'}^\ast \hookrightarrow {\cal BW}(X)_{\cal F} \otimes \Lambda^\bullet_{\Bbb R} {\frak g}^\ast \otimes {\Bbb C} \to E^\bullet({\overline{X}} \log D) $$ induces an isomorphism on $W_mH^t$ when $t\le n$ and injection on $W_mH^{n+1}$. Observe that if $\psi \in {{\frak g}'}^\ast$, then $\psi \in {\frak g}^\ast$ and $\psi$ commutes with $J$. That is, $\psi \in \Hom_{\Bbb C}({\frak g},{\Bbb C})$. It follows that the induced map $$ \Lambda^\bullet {{\frak g}'}^\ast\to E^\bullet({\overline{X}}\log D) $$ takes $\psi$ to $\theta_x^\ast(\psi)$. The assertion follows from the fact that $$ \theta^\ast_x : H^\bullet({\frak g}) \to H^\bullet(X;{\Bbb C}) $$ is a morphism of mixed Hodge structures \cite[(7.11)]{hain:cycles} and that $X$ is a rational $n$-$K(\pi,1)$. \end{pf} We are now ready to prove Theorem \ref{tech}. In the case where $X_\bullet$ is a single complex algebraic manifold with $q=0$, we have the morphism $$ \begin{matrix} \to & F^{p}W_{2p}H^{t-1}(A^\bullet_{\Bbb C}(X)) & \to & H_\F^t(X,{\Bbb R}(m)) & \to & W_{2p}H^t(\Omega^\bullet_{\cal BW}(X)_{\cal F}) (m-1) \cr &\downarrow && \downarrow && \downarrow \cr \to & F^{p}W_{2p}H^{t-1}(X,{\Bbb C}) & \to & \Hd^t(X,{\Bbb R}(m)) & \to & W_{2p}H^t(X,{\Bbb R}(m-1)) \cr \end{matrix} $$ of long exact sequences. If $X$ is a rational $n$-$K(\pi,1)$, the result follows directly from Lemma \ref{second-qism}, the very first step in the proof of Theorem \ref{tech}, and the 5-lemma. In the simplicial case, the result follows by a similar argument: Suppose that $X_\bullet$ is a simplicial complex algebraic manifold where each $X_m$ has $q=0$. If each $X_m$ is a rational $(n-m)$-$K(\pi,1)$, then an elementary spectral sequence argument shows that the maps $$ F^{p}W_{l}H^t(A^\bullet_{\Bbb C}(X_\bullet)) \to F^{p}W_{l}H^t(X_\bullet;{\Bbb C}) $$ and $$ W_lH^t(\Omega^\bullet_{\cal BW}(X_\bullet)_{\cal F}) \to W_lH^t(X_\bullet,{\Bbb R}) $$ are isomorphisms for all $l$ and $p$ whenever $t\le n$. Theorem \ref{tech} now follows from the 5-lemma as in the proof of the result for a single space above. \section{The 4-logarithm} In this section, we prove the existence and uniqueness of a real Grassmann 4-logarithm: \begin{theorem} There is a unique Grassmann 4-logarithm. \end{theorem} We will use the formal Bloch-Wigner cohomology defined in Section \ref{comparison} as it behaves better than $H^\bullet_{\cal BW}$. For all $m$, we have the following commutative diagram: \begin{equation*}\label{ladder} \begin{CD} sW_{2m}H^{2m-1}(\Omega_{\cal BW}^\bullet(G^m_\bullet)_{\cal F}) @>\alpha_{2m-1}>> sW_{2m}H^{2m-1}(G^m_\bullet,{\Bbb R}) \\ @VVV @VVV \\ sH_\F^{2m}(G^m_\bullet,{\Bbb R}(m)) @>>> s \Hd^{2m}(G^m_\bullet,{\Bbb R}(m)) \\ @VVV @VVV \\ sF^mW_{2m}H^{2m}(A_{\Bbb C}^\bullet(G^m_\bullet)) @>>> sF^mW_{2m}H^{2m}(\Omega^\bullet(G^m_\bullet))\\ @VVV @VVV\\ sW_{2m}H^{2m-1}(\Omega_{\cal BW}^\bullet(G^m_\bullet)_{\cal F}) @>\alpha_{2m}>> sW_{2m}H^{2m}(G^m_\bullet,{\Bbb R}) \\ \end{CD} \end{equation*} By (\ref{second-qism}), $\alpha_{2m-1}$ is an isomorphism if each $G^m_n$ is a rational $(n-m-1)$-$K(\pi,1)$. If, in addition, $$ sH^{m-n}(G^m_n,{\Bbb R}) $$ vanishes when $n\ge 1$, then $$ sH^{2m}(\Omega_{\cal BW}^\bullet(G^m_\bullet)_{\cal F})) $$ is spanned by the class of $\mbox{vol}_m$. Consequently, $\alpha_{2m}$ is injective. Finally, observe that it follows from \cite[(9.7)]{hain-macp} that the volume form $\mbox{vol}_m$ can be regarded as class in $$ sF^mW_{2m}H^{2m}(A_{\Bbb C}^\bullet(G^m_\bullet)) $$ We now consider the case when $m=4$. It follows from \cite[\S 8]{hain-macp} that $G^4_0$, $G^4_1$ are rational $K(\pi,1)$s, and that $G^4_2$ is a rational 1-$K(\pi,1)$. It follows from \cite[(8.2)]{hain-macp} that the cup products $$ \Lambda^k H^1(G^4_n,{\Bbb R}) \to H^k(G^4_n,{\Bbb R}) $$ are surjective when $k \le 4-n$, except possibly when $n=2$. But in this case, the fibers of the face map $G^4_2 \to G^4_1$ are hyperplane complements with constant combinatorics. It follows that $G^4_2 \to G^4_1$ is a fibration. The surjectivity of the cup product $$ \Lambda^2 H^1(G^4_2) \to H^2(G^4_2) $$ follows as the Leray-Serre spectral sequence of the map collapses at $E^2$ for weight reasons. One can show (e.g., by computer or by hand) that $$ s\Lambda^k H^1(G^4_n,{\Bbb R}) = 0 $$ when $0 \le k < 4-n$ for all $n$, and when $k = 4-n$ for all $n> 0$. By the discussion in the previous paragraph, $\alpha_4$ is injective and $$ sW_8H^7(\Omega_{\cal BW}^\bullet(G^4_\bullet)_{\cal F}) = sW_8H^7(G^4_\bullet,{\Bbb R})=0. $$ Thus, to prove the existence of the 4-logarithm, it suffices to show that $$ \mbox{vol}_4\in sF^4W_8H^8(A_{\Bbb C}^\bullet(G^4_\bullet)) $$ has trivial image in $sW_8H^7(\Omega_{\cal BW}^\bullet(G^4_\bullet)_{\cal F})$. But this is immediate as $\alpha_8$ is injective and $\mbox{vol}_4$ has trivial image in $H^8(G^4_\bullet,{\Bbb R})$. The existence and uniqueness of the 4-logarithm follows. \begin{remark} It is likely that one can prove the existence of a canonical 5-logarithm using a similar argument. \end{remark}

9407

alg-geom/9407008

en

https://arxiv.org/abs/alg-geom/9407008

alg-geom/9407008

Clint McCrory

Gary Kennedy, Clint McCrory, and Shoji Yokura

Natural transformations from constructible functions to homology

5 pages, AMSLaTeX, University of Georgia Math. Preprint Series

20, Volume 2, 1994

For complex projective varieties, all natural transformations from constructible functions to homology (modulo torsion) are linear combinations of the MacPherson-Schwartz-Chern classes. (The authors are willing to mail hard copies of the paper.)

alg-geom

\section{Introduction} \label{intro} \par The MacPherson-Schwartz-Chern class natural transformation $\mpc$ from the constructible functions functor to homology \cite{macp} \cite{bs} satisfies a remarkably stringent normalization requirement: for each nonsingular variety $X$, the element $\mpc(\cf_X)$ of homology assigned to the characteristic function of $X$ is the total Chern class $c(TX) \frown [X]$. Now suppose that we entirely abandon this requirement. Then each individual component $\mpc_i$ of the Chern class natural transformation (assigning to the characteristic function $\cf_X$ of a nonsingular variety the homology class $c_i(TX) \frown [X]$) is likewise a natural transformation, as is any linear combination of these components. \par We will show that, modulo torsion, these linear combinations are the only natural transformations between the two functors. In particular, the MacPherson-Schwartz-Chern class is the only such natural transformation satisfying this weak normalization requirement: for each projective space $\bold{P}$, the top-dimensional component of $\mpc(\cf_{\bold{P}})$ is the fundamental class $[\bold{P}]$. We conjecture that the same statements are valid even for integral homology. We want to note two features of our proofs. First, we never appeal to resolution of singularities. Second, several of our arguments are similar to those in the proof (attributed to A. Landman) of the last Proposition of \cite{fulton}. \par \section{The functors} \label{func} \par Consider the category of complex projective algebraic varieties. If $X$ is such a variety, the {\em characteristic function} of a subvariety $Z$ is the function $\cf_Z$ on $X$ whose value on $Z$ is 1 and whose value elsewhere is 0. If $f: X \to Y$ is a morphism, then $f_* \cf_Z$ is the constructible function on $Y$ whose value at $y$ is the Euler characteristic of $f^{-1}(y) \cap Z$ (a subvariety of $X$). A finite linear combination (over $\bold{Z}$) of characteristic functions of subvarieties of $X$ is called a {\em constructible} function on $X$. We define the pushforward of an arbitrary constructible function by extending the previous definition by linearity, and thus define the {\em constructible functions functor} $\cal C$ to the category of abelian groups. \par We also want to work with an appropriate homology theory. This will be either of the following: \begin{itemize} \item {\bf Ordinary singular or simplicial homology.} The fundamental class of an $n$-dimensional variety $X$ is an element of $H_{2n}(X)$. \item {\bf Algebraic cycles modulo rational equivalence.} The standard reference is \cite[Ch.\ 1]{int}. The fundamental class of an $n$-dimensional variety $X$ is an element of the cycle class group $A_n(X)$. \end{itemize} \par \section{Projective spaces} \label{proj} \par Suppose that $\tau$ is a natural transformation from $\cal C$ to $H_{2i}$ or to $A_i$. Suppose that $n > i$; then $\bold{P}^i$ is naturally a linear subspace of $\bold{P}^n$. \begin{theorem} \label{proj-re} $\tau(\cf_{\bold{P}^n}) = \binom{n+1}{i+1}\tau(\cf_{\bold{P}^i})$. \end{theorem} \begin{pf} Consider the morphism $f:\bold{P}^n \to \bold{P}^n$, \begin{equation*} f: [x_0,x_1,\dots,x_n] \mapsto [x_0^2,x_1^2,\dots,x_n^2]. \end{equation*} Let \begin{equation*} L_k = \{[x_0,x_1,\dots,x_n] \mid \text{ at least } n-k \text{ of the coordinates equal zero}\} \end{equation*} and \begin{equation*} U_k = \{[x_0,x_1,\dots,x_n] \mid \text{ exactly } n-k \text{ of the coordinates equal zero}\}. \end{equation*} Then $L_k$ is a union of $\binom{n+1}{k+1}$ linear subspaces of dimension $k$, and $U_k = L_k \setminus L_{k-1}$. The restriction of $f$ to $U_k$ is a self-covering map of degree $2^k$. Hence the restriction of $f$ to each component of $L_k$ is a branched self-cover of the same degree, and the induced map $f_*:H_{2k}(\bold{P}^n) \to H_{2k}(\bold{P}^n)$, or the induced map $f_*:A_k(\bold{P}^n) \to A_k(\bold{P}^n)$, is multiplication by $2^k$. Consider the constructible function $\cf_{U_k}=\cf_{L_k}-\cf_{L_{k-1}}$. By naturality of $\tau$ we have \begin{equation*} 2^k \tau(\cf_{U_k}) = \tau(2^k \cf_{U_k}) = \tau f_*(\cf_{U_k}) = f_* \tau(\cf_{U_k}) = 2^i \tau(\cf_{U_k}). \end{equation*} If $i \neq k$, then $\tau(\cf_{U_k}) = 0$. Therefore \begin{equation*} \tau(\cf_{\bold{P}^n}) =\tau\left(\cf_{L_i}+\sum_{k=i+1}^n \cf_{U_k}\right) = \tau(\cf_{L_i}). \end{equation*} And \begin{equation*} \tau(\cf_{L_i})=\tau(\sum_K \cf_K), \end{equation*} where the sum is taken over the components of $L_i$, since the two functions in question differ only on a variety of dimension $i-1$. Since $L_i$ is a union of $\binom{n+1}{i+1}$ linear subspaces of dimension $i$, the statement of the theorem follows. \end{pf} \par \section{Galois coverings} \label{gct} \par A morphism $X \to Y$ of (irreducible) varieties is called {\em Galois} if it is finite and surjective, and the corresponding field extension $K(Y) \subset K(X)$ is Galois. \begin{lemma} \label{gct1} Suppose that $X$ is a projective variety of dimension $n$. Then there exists a normal projective variety $Z$, a finite morphism $Z \to X$, and a finite morphism $X \to \bold{P}^n$, such that the composition $Z \to \bold{P}^n$ is Galois. \end{lemma} \begin{pf} To construct a finite morphism $X \to \bold{P}^n$, first embed $X$ in a projective space $\bold{P}^N$ and then project it from a general $\bold{P}^{N-n-1}$. The field extension $K(\bold{P}^n) \subset K(X)$ is finite; hence it can be obtained by adjoining a single element satisfying a minimal polynomial $p$. Let $\cal K$ be the splitting field of $p$. Then $\cal K$ is a normal extension of $K(\bold{P}^n)$ \cite[Thm.\ 8.4, p.\ 82]{stewart}. Let $Z$ be the normalization of $X$ in $\cal K$ \cite[III.8, Thm.\ 3]{mumford}. Then $Z$ is projective \cite[III.8, Thm.\ 4]{mumford}. Since $K(Z) = \cal K$, the morphism $Z \to P^n$ is Galois. \end{pf} \par \begin{lemma} \label{gct2} If $Z$ and $Y$ are normal varieties and $\gamma:Z \to Y$ is Galois, with Galois group $G$, then $Y$ is isomorphic to $Z/G$. \end{lemma} \begin{pf} The finite group $G$ acts by birational maps. Let $g:Z \to Z$ be any one of these maps; let $W \subset Z \times Z$ be the closure of its graph. The projection $W \to Z$ onto the first factor is a birational morphism with finite fibers. By Zariski's Main Theorem \cite[Ch.\ 3, sec.\ 9]{mumford} this projection is an isomorphism. Hence $g$ is in fact a morphism. \par The morphisms $\gamma$ and $\gamma \circ g$ agree on an open dense subset of $Z$; hence they are equal. The quotient $Z/G$ is a projective variety \cite[pp.\ 126--127]{harris}. By the universal property of $Z/G$, there is a morphism to $Y$. Again by Zariski, this morphism is an isomorphism. \end{pf} \par \begin{lemma} \label{gct3} Under the same hypotheses, $\gamma_*$ maps the invariant subgroup $(H_*Z \tq)^G$ isomorphically to $H_*(Z/G)\tq$. Likewise $\gamma_*$ maps $(A_*Z \tq)^G$ isomorphically to $A_*(Z/G)\tq$. \end{lemma} \begin{pf} Triangulate the quotient map \cite{hardt}. The $G$-invariant simplicial homology of $Z$ is isomorphic to the homology of the complex of $G$-invariant simplicial chains of $Z$. (The proof is by averaging over the group $G$.) And the $G$-invariant simplicial chain complex of $Z$ is isomorphic to the simplicial chain complex of $Z/G$. \par For the statement about rational equivalence groups, see \cite[Example 1.7.6]{int}. \end{pf} \par \section{The natural transformations} \label{nt} \par \begin{theorem} \label{main} Suppose that $\sigma$ is a natural transformation from the constructible function functor $\cal C$ to $H_* \tq$, ordinary singular homology with rational coefficients, or to $A_* \tq$, rational equivalence theory with rational coefficients. Suppose that for each projective space $\bold{P}$ the top-dimensional component of $\sigma_{\bold{P}}(\cf_{\bold{P}})$ vanishes. Then $\sigma$ is identically zero. \end{theorem} \begin{corollary} Suppose that $\tau$ is a natural transformation from $\cal C$ to $H_* \tq$ or to $A_* \tq$. Then $\tau$ is a rational linear combination $\sum_{i=0}^{\infty} r_i \mpc_i$ of the components of the MacPherson-Schwartz-Chern class. \end{corollary} \par Note that although the sum appearing in the corollary is nominally infinite, for any particular variety it is a finite sum. \par \begin{pf*}{Proof of corollary} Define $r_i$ to be the rational number for which \begin{equation*} \tau\left(\cf_{\bold{P}^i}\right) = r_i \lbrack\bold{P}^i\rbrack + \text{ terms of lower dimension}. \end{equation*} Then apply the theorem to \begin{equation*} \sigma = \tau - \sum_{i=0}^{\infty} r_i \mpc_i. \end{equation*} \renewcommand{\qed}{} \end{pf*} \begin{pf*}{Proof of theorem} Suppose that $Z$ is a subvariety of $X$. If $\sigma(\cf_Z)$ is zero in the homology group of $Z$, then by naturality it is likewise zero in the homology group of $X$. Hence it will suffice to show that, for each projective variety, \begin{equation} \label{vanish} \sigma(\cf_X) = 0 \end{equation} in the homology group of $X$. \par If $X$ is a projective space, then by hypothesis the top-dimensional component of $\sigma(\cf_X)$ vanishes. From Theorem~\ref{proj-re} we deduce equation~\ref{vanish}. \par In general we use induction on the dimension of $X$, together with the Galois covering techniques of section~\ref{gct}. Let $n$ be the dimension of $X$; suppose that $\sigma(\cf_W) = 0$ for all varieties of smaller dimension. By Lemma~\ref{gct1}, we can construct a normal projective variety $Z$, a finite morphism $\pi: Z \to X$, and a finite morphism $\rho: X \to \bold{P}^n$, such that the composition $\gamma: Z \to \bold{P}^n$ is Galois. By Lemma~\ref{gct2}, $\bold{P}^n$ is the quotient of $Z$ by the Galois group $G$. The characteristic function of $Z$ is fixed by the action of the group; hence by naturality it is an element of the invariant homology group. By naturality and the inductive hypothesis \begin{equation*} \gamma_*\sigma(\cf_Z) = \sigma\left(|G| \cdot \cf_{\bold{P}^n} + \text{ function supported on varieties of smaller dimension} \right) = 0. \end{equation*} By Lemma~\ref{gct3}, $\gamma_*$ maps the invariant homology isomorphically to the homology of $\bold{P}^n$. Hence $\sigma(\cf_Z)$ must be zero. Let $d$ be the degree of $\pi$. Then \begin{equation*} 0 = \pi_* \sigma(\cf_Z) = \sigma\left(d \cdot \cf_X \right) + \sigma(\text{function supported on varieties of smaller dimension}). \end{equation*} By the inductive hypothesis the second term is zero. Hence $\sigma(\cf_X) = 0$. \end{pf*} \par Finally we wish to remark that the theorem and corollary are also valid if we interpret all functors as emanating from the larger category of compact complex algebraic varieties. Indeed, by Chow's lemma \cite[p.\ 282]{shaf} an arbitrary compact variety is the image of a projective variety via a birational morphism. Thus we may prove an extended version of Theorem~\ref{main} by an induction on dimension.

9407

alg-geom/9407012

en

https://arxiv.org/abs/alg-geom/9407012

alg-geom/9407012

Torres Fernando

Fernando Torres

On certain N--sheeted coverings and numerical semigroups which cannot be realized as Weierstrass semigroups

ICTP preprint, 18 pages, Latex v. 2.1. Reason for resubmission: (1) I reformulated the principal result (Theorem A) in order to obtain a better bound on the genus for the results concerning semigroups. Remarks 3.11 contains examples that show the sharpness (or the necessity of the hypothesis) of most of the result stated in the paper

Com. Alg. 23(11) (1995)

A curve $X$ is said to be of type $(N,\gamma)$ if it is an $N$--sheeted covering of a curve of genus $\gamma$ with at least one totally ramified point. A numerical semigroup $H$ is said to be of type $(N,\gamma)$ if it has $\gamma$ positive multiples of $N$ in $[N,2N\gamma]$ such that its $\gamma^{th}$ element is $2N\gamma $ and $(2\gamma+1)N \in H$. If the genus of $X$ is large enough and $N$ is prime, $X$ is of type $(N,\gamma)$ if and only if there is a point $P \in X$ such that the Weierstrass semigroup at $P$ is of type $(N,\gamma)$ (this generalizes the case of double coverings of curves). Using the proof of this result and the Buchweitz's semigroup, we can construct numerical semigroups that cannot be realized as Weierstrass semigroups although they might satisfy Buchweitz's criterion.

alg-geom

\section{Introduction.} In Weierstrass Point Theory one associates a numerical semigroup to any non--singular point $P$ of a projective, irreducible, algebraic curve defined over an algebraically closed field. This semigroup is called the Weierstrass semigroup at $P$ and is the same for all but finitely many points. These finitely many points, where exceptional values of the semigroup occur, are called the Weierstrass points of the curve. They carry a lot of information about the curve. In 1893, Hurwitz asked about the characterization of the numerical semigroups which arise as Weierstrass semigroups. See [E-H] for further historical information. Long after that, in 1980 Buchweitz [B1] showed that not every numerical semigroup can occur as a Weierstrass semigroup, but has to satisfy the following criterion (which can be extended to singular curves by [S1, p. 124]), \medskip {\bf (BC):} ``Let $P$ be a non--singular point of a projective, irreducible, algebraic curve defined over an algebraically closed field. If $n \ge 2$ and $g$ is the arithmetical genus of the curve, then the cardinality of the set of sums of $n$ gaps $G_n$ at $P$ is bounded above by the dimension of the pluri--canonical divisor $nC_{X}$ which is $ (2n-1)(g-1)$ ". \medskip Moreover, Buchweitz showed that for every integer $n \ge 2$ there exist semigroups which do not satisfy the above criterion. However, as was noticed by Oliveira [O, Thm. 1.5] and Oliveira-St\"ohr [O-S, Thm. 1.1], this criterion cannot be applied to symmetric and quasi-symmetric semigroups because, in the first case $\#G_n = (2n-1)(g-1)$ and in the second case $\#G_n = (2n-1)(g-1) - (n-2)$. \medskip Let $X$ be a projective, irreducible, non--singular algebraic curve of genus $g$ defined over an algebraically closed field of characteristic $p\ge 0$. For a point $P$ of $X$, let $H(P)$ denote the Weierstrass semigroup at $P$. A curve is called $\gamma$--hyperelliptic if it is a double covering of a curve of genus $\gamma$. A numerical semigroup $H$ will be called $\gamma$--hyperelliptic if it has $\gamma$ positive even elements in the interval $[2,4\gamma]$ such that its $\gamma^{th}$ element is $4\gamma$, and, $4\gamma+2 \in H$. The reason for this terminology is the following result that has been proved for $p =0$ in [T, Thm. A and Remark 3.10]: \medskip {\bf Theorem 1.} If $g\ge 6\gamma +4$, then the following statements are equivalent: \begin{list} \setlength{\rightmargin 0cm}{\leftmargin 0cm} \itemsep=0.5pt \item[(i)] $X$ is $\gamma$--hyperelliptic. \item[(ii)] There exists $P\in X$ such that $H(P)$ is $\gamma$--hyperelliptic. \item[(iii)] For some integer $2\le i \le \gamma+2$ such that $2\gamma +i \not\equiv 0$ (mod $3$) if $ i < \gamma$, there exists a base--point--free linear system on $X$ of projective dimension $\gamma +i$ and degree $4\gamma+2i$. \end{list} Using the proof of the implication (ii) $\Rightarrow$ (i) above and Buchweitz's example, St\"ohr [T,Scholium 3.5] obtained symmetric numerical semigroups which cannot be realized as Weierstrass semigroups at points of non--singular curves. Since St\"ohr's examples does not depend on the characteristic $p$, we have that these examples cannot even be realized as Weierstrass semigroups for non-singular curves defined in positive characteristic. However, symmetric semigroups can be realized as Weierstrass semigroups of Gorenstein curves [S]. Let $B_{H}$ be the semigroup ring over an algebraically closed field of characteristic zero associated to a numerical semigroup $H$. By the work of Pinkham [P], $B_{H}$ is smoothable if and only if $H$ is a Weierstrass semigroup. On the other hand, $H$ is symmetric if and only if $B_{H}$ is Gorenstein (cf. [Ku]). Hence St\"ohr's examples also show that the Gorenstein condition does not imply smoothability.\medskip The aim of the present paper is to give a generalization of the case $p=0$ of Theorem 1 to certain covers of degree $N$. This will allow us to construct numerical semigroups with a given last gap (the result of Theorem 1 just works for symmetric semigroups) that cannot be realized as Weierstrass semigroups at points of non--singular algebraic curves although they might satisfy Buchweitz's criterion above (cf. Section 4). We introduce the following definitions:\medskip {\bf Definition 1.} A curve $X$ is said to be of type $(N,\gamma )$ if it is an $N$--sheeted covering of a curve of genus $\gamma$ with at least one totally ramified point. \medskip {\bf Definition 2.} Let $N>0$, $\gamma \ge 0$ be integers. A numerical subsemigroup $H$ of $({\rm I}\!{\rm N} ,+)$ is said to be of type $(N,\gamma )$ if the following conditions are satisfied: \begin{list} \setlength{\rightmargin 0cm}{\leftmargin 0cm} \itemsep=0.5pt \item[(a)] $H$ has $\gamma$ positive multiples of $N$ in the interval $[N,2N\gamma]$, \item[(b)] the $\gamma^{th}$ element of $H$ is $2N\gamma$, and, \item[(c)] $(2\gamma +1)N\in H$. \end{list} These definitions are related to each other by Lemma 3.4 below. Let $X$ be a curve of type $(N,\gamma)$ and $P$ be a totally ramified point of $X$. If $p=char(K) \not\!\vert\ N$ or if $N$ is prime and the genus of $X$ is large enough, then the semigroup $H(P)$ satisfies conditions (a), (c) above, and $2N\gamma \in H(P)$. Moreover, $H(P)$ will be of type $(N,\gamma)$ whenever the genus is also large enough (Corollary 3.5) (the appearance of these semigroups in such a context justifies Definition 2). In the case where the covered curve is given as a quotient of $X$ by an automorphism, the above results are included in an implicit way in T. Katos's paper [K, p.395]. From an arithmetical point of view, we can say that every numerical semigroup $H$ is a semigroup of type $(1,\gamma)$ with $\gamma$ equal to the genus of $H$. Moreover, given a positive integer $N$ there exists a natural number $\gamma_N=\gamma_N(H)$ such that $H$ satisfies conditions (a), (c) above (with $N$ and $\gamma_N$) and $2N\gamma_N \in H$ (Lemma 2.3 (ii)). The point is that it does not necessarily fulfil condition (b) (Remark 3.11 (i)). On the other hand, using a result of R\"ohrl [R\"o, Th.3.1] we can construct curves that are not of type $(N,\gamma )$. It follows also that there exist curves $X$ of type $(N,\gamma )$ which do not cover a curve of the form $X/\langle T\rangle$ for $T$ an automorphism of order $N$ defined on $X$. \medskip We now state the main result of this paper. For $A , u, N$ and $\gamma$ integers we define: \begin{equation} \rho_1(A,N,\gamma) = {A(N-1)N\over 2} + N\gamma -N +1, \end{equation} \begin{equation} \rho_2(N,\gamma) = N(2N-1)\gamma - (N-1)(N+2), \end{equation} \begin{equation} \rho_3(N,\gamma) = (2N-1)(N\gamma + N-1), \end{equation} \begin{equation} \rho_4(A,u,N,\gamma) = {1\over 2}(N-u-1)[(A-\gamma-1)(N+u) - 2(N\gamma + N-1)] + \rho_3(N,\gamma). \end{equation} {\bf Theorem A.} Consider the following statements: \begin{list} \setlength{\rightmargin 0cm}{\leftmargin 0cm} \itemsep=0.5pt \item[(i)] $X$ is a curve of type $(N,\gamma )$. \item[(ii)] There exists $P\in X$ such that $H(P)$ is a semigroup of type $(N,\gamma )$. \item[(iii)] There exists $P\in X$ and an integer $A$ such that the linear system $|AP|$ is base--point--free of dimension $A - \gamma$. \end{list} \vspace{-0.5pt} Let $N$ be prime and $A\ge 2\gamma +1$ an integer. \begin{list} \setlenght{\rightmargin 0cm}{\leftmargin 0cm} \itemsep=0.5pt \item[(A1)] If $N \not= p = char(K)$ and $g> \rho_1(2\gamma,N,\gamma) = N^2\gamma - N+1$ or if $g> \rho_2(N,\gamma)$, then (i) $\Rightarrow$ (ii). \item[(A2)] If $g > \rho_3(N,\gamma)$, then (ii)$ \Rightarrow $ (i). \item[(A3)] If $g> \rho_1(A,N,\gamma)$, then (ii) $\Rightarrow$ (iii). \item[(A4)] Let $u= u(A)$ be the biggest integer $\le {N\gamma +N-1 \over A-\gamma -1}$. If the following conditions \begin{equation} A \not\equiv 0\ ({\rm mod}\ t)\ {\rm for}\ t \le {AN\over A-\gamma},\ t\not= N\ {\rm and} \end{equation} \begin{equation} g > \rho_4(A,u,N,\gamma) \end{equation} hold, then (iii) $\Rightarrow$ (i). \end{list} Notice that for $N=2$, statement (iii) of Theorem 1 is equivalent to statement (iii) of Theorem A because the $\gamma$--hyperelliptic involution is unique (if it exists) provided $g> 4\gamma +1$ (see e.g. [A, Lemma 5]). Definitions (3) and (4) are derived from Castelnuovo's number (see Lemma 3.7 below) applied to a linear system of type $g^{A-\gamma}_{AN}$; definition (2) arise from Castelnuovo's bound involving subfields of the field of rational functions of the curve (Lemma 3.2) while definition (1) comes from arithmetical reasons (Lemma 2.1). In Section 2 we study some arithmetical properties of the semigroups of type $(N,\gamma )$. As in the $\gamma$--hyperelliptic case ([T, Lemma 2.6]), we find that the multiples of $N$ contained in the semigroup are determined by properties (a) and (c) (Lemma 2.3 (i)). We also state sufficient conditions for a numerical semigroup to be of type $(N,\gamma)$ (Corollaries 2.5 and 2.6). If $N\ge 2$ is an integer, we have a lower bound for the elements $h$ of the semigroup such that $gcd(h,N) = 1$ (Lemma 2.1). This result generalizes Lemma 2.2 in [T]. Both Lemma 2.1 and Lemma 2.3 are used to obtain linear series on curves having a point with Weiertrass semigroup of type $(N,\gamma)$. In Section 3 we prove Theorem A and some results concerning Weierstrass semigroups at totally ramified points. In Remark 3.11 we discuss the sharpness of the bounds on $g$ used in the results of the paper as well as the necessity of hypothesis (5). In section 4, using the proof of item (A2), we show how to construct numerical semigroups of type $(N,\gamma )$ that are not realized as Weierstrass semigroups of non--singular curves. We also construct numerical semigroups that do not satisfy Buchweitz's criterion for $n =2$. These examples contain Buchweitz's semigroup and they may be well known, but we included them here in order to compare them with the semigroups arising from the proof of item (A2). We remark that an extension of item (A2) to the case of singular curves will provide us with examples of numerical semigroups that cannot be realized as Weierstrass semigroups even for singular curves. \medskip We have employed the methods used in [T], where one of the key tools is Castelnuovo's genus bound for curves in projective space [C], [ACGH, p.116], [R, Corollary 2.8]. Since the coverings considered here can be of degree bigger than two, we will also use as a key tool the other famous genus bound of Castelnuovo which concerns subfields of the field of rational functions of the curve [C1], [St]. \medskip {\bf Conventions.} Throughout this paper, the word curve will mean a projective, irreducible, non--singular algebraic curve defined over an algebraically closed field $K$ of characteristic $p \ge 0$. By a numerical semigroup $H$ we will mean a subsemigroup of $({\rm I}\!{\rm N} ,+)$ whose complement in ${\rm I}\!{\rm N}$ is finite. For $i \in {\rm I}\!{\rm N}$, we denote by $m_i = m_i(H)$ the $i^{th}$ element of the semigroup $H$ and by $G(H)$ the set ${\rm I}\!{\rm N} \setminus H$. When $H$ is the Weierstrass semigroup of some point $P$, we just write $H(P)$, $G(P)$ and $m_i = m_i(P)$. Given a curve $X$, the symbols $g^{r}_{d}$, $K(X)$ and ${\rm div}_\infty (f)$ will denote respectively an $r$-linear system of degree $d$ on $X$, the field of regular functions of $X$ and the polar divisor of $f \in K(X)$. \section{Semigroups of type $(N,\gamma )$.} Let $H$ be a numerical semigroup. The natural number $g=g(H):=\# G(H)$ is called the genus of $H$. The elements of $G(H)$ are called the gaps of $H$ and those of $H$ are called the non--gaps of $H$. Fix a positive non-gap $m \in H$. For $i=1,\dots , m-1$, denote by $s_i=s_i(H,m)$ the smallest element of $H$ such that $s_i\equiv i$ (mod $m$) and then define $e_i=e_i(H,m)$ by the equation \begin{equation} s_i=e_i\ m +i\ . \end{equation} By the semigroup property of $H$, we have that $e_i$ is the number of gaps $\ell$ for which $\ell\equiv i$ (mod $m$). Consequently \begin{equation} g=\sum^{m-1}_{i=1}\ e_i \end{equation} and also \begin{equation} \left\{\matrix{ e_i+e_j\ge e_{i+j},\hfill &{\rm if}\ \ i+j<m.\cr &\cr e_i+e_j\ge e_{i+j-m}-1,&{\rm if}\ \ i+j>m.\cr}\right. \end{equation} Conversely, given numbers $m,e_1,\dots e_{m-1}$ satisfying the above relations one indeed has a semigroup. In particular, $m=m_1$ and the respective $e_i$'s completely determine $H$ (cf. [H]). Let $N$ be a positive integer. We associate to $H$ the number: \begin{equation} \gamma_N := \{\ell \in G(H): \ell \equiv 0\ ({\rm mod}\ N)\}. \end{equation} {\bf Lemma 2.1.} Let $H$ be a numerical semigroup of genus $g$, $N \ge 2$ an integer. If $h\in H$ such that $gcd(h,N) = 1$, then $$ h\ge {{2g-2N\gamma_N}\over {N-1}} +1\ . $$ {\bf Proof.} Set $\gamma = \gamma_N$ and let $m=Nn$ be the least positive non-gap of $H$ which is multiple of $N$. Then, $\gamma = \sum_{i=1}^{n-1} e_{Ni}$ and there exists $i\in \{1,\ldots,m-1\}$ so that $gcd(i,N)=1$ and $h\ge s_i$. {\bf Claim.} For $k=1,\dots ,N-1;\ell =0,\dots ,n-1;ki+N\ell\not\equiv 0$ (mod $N$). Moreover these numbers are pairwise different modulo $Nn$. Indeed, if $ki+N\ell\equiv 0$ (mod $N$), then $ki\equiv 0$ (mod $N$). Since $gcd(i,N)=1$ we have $k\equiv 0$ (mod $N$), a contradiction. On the other hand, $ki+N\ell\equiv k_1i+N\ell_1$ (mod $Nn$) implies $(k-k_1)i\equiv 0$ (mod $N$) and so $k-k_1\equiv 0$ (mod $N$) which gives $k=k_1$. Consequently from $N\ell\equiv N\ell_1$ (mod $Nn$) we obtain that $\ell\equiv\ell_1$ (mod $n$) and so $\ell =\ell_1$. For $k$ and $\ell$ as in the above claim, write $ki+N\ell = a_{k\ell}Nn+r_{k\ell}$ (*) with $0<r_{k\ell}<Nn$. From (9) and induction on $k$ and $\ell$ we have $$ ke_i + e_{N\ell} \ge e_{r_{k\ell}} - a_{k\ell} $$ \noindent where we assume $e_0:= 0$. Adding up these inequalities, from the Claim and (8) we get $$ {(N-1)Nn\ e_i\over 2} +(N-1)\gamma\ge g-\gamma -\sum_{k,\ell}\ a_{k\ell}\ . $$ \noindent Now from (*) and the Claim we have $\sum_{k,\ell}\ a_{k\ell}=(N-1)(i-1)/2$ and hence the proof follows from the above inequality and (7).\qquad $\Box$ \medskip {\bf Remarks 2.2.} (i) The above lemma subsumes the following result due to Jenkins [J]: ``let $H$ be a numerical semigroup of genus $g$ and $0<m<n$ non-gaps of $H$ so that $gcd(m,n) =1$; then $g \le (m-1)(n-1)/2$ ". Indeed, by using the notation of the lemma, take $N=n$; then $\gamma_N =0$ and Jenkins' result follows with $h =m$.\\ (ii) The lower bound of Lemma 2.1 is the number of ramified points minus one of an $N$--sheeted covering of curves of genus $g$ and $\gamma_N$ respectively (defined over a field of characteristic $p\not\!\vert N$), where all the ramified points are totally ramified points.\qquad $\Box$ \medskip The next lemma will help us to understand the structure of the semigroups of type $(N,\gamma)$. \medskip {\bf Lemma 2.3.} Let $H$ be a numerical semigroup. \begin{list} \setlenght{\rightmargin 0cm}{\leftmargin 0cm} \itemsep=0.5pt \item[(i)] Suppose that $H$ fulfils conditions (a) and (c) of Definition 2. Set $F:=\{(2\gamma +i)N:i\in{\rm Z}\!\!{\rm Z}^{+}\}$. Then: \subitem(i.1) $F\subseteq H$, $2N\gamma \in H$, \subitem(i.2) $\gamma = \gamma_N$. \item[(ii)] Conversely, let $N>0$ be an integer. Then, $H$ fulfils condition (a) and (c) of Definition 2 with $N$ and $\gamma_N$, and $2N\gamma_N \in H$. \end{list} {\bf Proof.} (i) If $\gamma =0$ then $N\in H$ and so we have (i.1) and (i.2). Let $\gamma\ge 1$ and denote by $f_1<\ldots <f_\gamma$ the $\gamma$ positive multiples of $N$ non-gaps of $H$ in $[N,2N\gamma]$. So $f_1 > N$. Suppose that $F\not\subseteq H$ and let $(2\gamma + i)N$ be the least element of $F\cap G(H)$. By the semigroup property of $H$ we have $(2\gamma +i)N-f_j\in G(H)$ for $j=1,\dots ,\gamma$. Then, by the selection of $(2\gamma +i)N$ we have that $\{ (2\gamma+i)N-f_j:j=1,\dots ,\gamma\}$ are all the gaps $\ell$ of $H$ such that $\ell \equiv 0$ (mod $N$) and $\ell \le 2N\gamma$. The least of these gaps satisfies $(2\gamma+i)N-f_\gamma \ge iN \ge 2N$ due to condition (c) of Definition 2. Consequently $f_1 =N$ which is a contradiction. The statement (i.2) follows from (i.1) since the gaps $\ell$ for which $\ell\equiv 0$ (mod $N$) belong to the interval $[N,2N\gamma ]$. It remains to proof that $2N\gamma \in H$. Suppose that $f_\gamma <2N\gamma$. Then we get $\gamma+1$ gaps multiples of $N$ namely, $2N\gamma-f_\gamma,\ldots, 2N\gamma-f_1, 2N\gamma$ which is a contradiction due to (i.2).\\ (ii) By the definition of $\gamma=\gamma_N$ (see (10)), there exist at least $\gamma$ positive non-gaps - all of them being multiples of $N$ - in the interval $[N,2N\gamma]$. Denote by $f_1< \ldots < f_\gamma$ such non-gaps. Let $\ell$ be the biggest gap of $H$ so that $\ell \equiv 0$ (mod $N$). We claim that $\ell < f_\gamma$, because on the contrary case we would have - as in the previous proof with $\ell$ instead of $2N\gamma$ - ($\gamma+1$) gaps which is a contradiction with the definition of $\gamma$. This implies that $2N\gamma, (2\gamma+1)N, \ldots $ are non-gaps and we are done. \qquad $\Box$ \medskip {\bf Corollary 2.4.} Let $H$ be a numerical semigroup, $\gamma$ a non--negative integer, $M,N,r$ positive integers so that $2(\gamma +r)M>(2\gamma +r)N$. Then $H$ cannot be both of type $(N,\gamma )$ and of type $(M,\gamma +r)$. {\bf Proof.} Suppose $H$ is both of type $(N,\gamma )$ and of type $(M,\gamma +r)$. From the previous lemma and since $H$ is of type $(M,\gamma +r)$ we have $$ 2(\gamma +r)M=m_{\gamma +r}\le (2\gamma +r)N\ .\qquad{\Box} $$ Using Lemma 2.3 (ii) and Lemma 2.1 we have the following criteria for the type $(N,\gamma_N)$ of numerical semigroups. \medskip {\bf Corollary 2.5.} Let $H$ be a numerical semigroup and $N>0$ an integer. Suppose that every $h \in H$ such that $h\not\equiv 0$ (mod $N$) satisfies $h \ge 2N\gamma_N+1$. Then, $H$ is of type $(N,\gamma_N)$.\qquad $\Box$ \medskip {\bf Corollary 2.6.} Let $H$ be a numerical semigroup of genus $g$ and $N$ prime. If $g>\rho_1(2\gamma_N,N,\gamma) = N^2\gamma_N - N+1$, then $H$ is of type $(N,\gamma_N)$.\qquad $\Box$ \medskip We also have: \medskip {\bf Corollary 2.7.} Let $H$ be a semigroup of type $(N,\gamma)$ with $N$ prime. Let $A \ge \gamma +1$ be an integer and $g$ the genus of $H$. If $g>\rho_1(A,N,\gamma)$ (see (1)), then $$ gcd(m_1,\ldots, m_{A-\gamma}) = N.\qquad \Box $$ \section{Proof of Theorem A.} We study certain $N$--sheeted coverings $$ \pi: X \to \tilde X $$ \noindent of curves. To fix notation, we let $X$ and $\tilde X$ be curves of genus $g$ and $\gamma$, respectively. We assume that there is a point $P\in X$ such that $\pi$ is totally ramified at $P$, i.e., $X$ will be a curve of type $(N,\gamma)$. We are mainly interested in relating the Weierstrass semigroups at $P$ and $\tilde P:= \pi(P)$. Since $P$ is totally ramified, $\tilde m_iN \in H(P)$ for $\tilde m_i \in H(\tilde P)$. Moreover, since $\tilde m_{\gamma + j} = 2\gamma + j$ for $j \in {\rm I}\!{\rm N}$, we have the following statements: \begin{list} \setlength{\rightmargin 0cm}{\leftmargin 0cm} \item[(I)] $\gamma_N = \gamma_N(P) := \#\{\ell \in G(P): \ell \equiv 0\ ({\rm mod}\ N)\} \le \gamma. $ \item[(II)] $ m_\gamma= m_\gamma(P) \le 2\gamma N \in H(P).$ \end{list} \noindent Note that equality in (II) implies equality in (I), and, $H(P)$ is of type $(N,\gamma)$ if and only if equality in (II) holds. Moreover, if $h\in H(P)$ so that $gcd(h,N)=1$, from Lemma 2.1 and (I) we have \begin{equation} h \ge {{2g-2N\gamma_N}\over N-1} +1 \ge {{2g-2N\gamma}\over N-1}+1. \end{equation} \noindent Hence we have the following generalization of [T, Lemma 3.1]. \medskip {\bf Lemma 3.1.} Assume the above notation and suppose $g > \rho_1(2\gamma,N,\gamma)= N^2\gamma-N+1$. Then, every $h \in H(P)$ such that $$ h \le {g+N(N-2)\gamma \over N-1} $$ \noindent satisfies $gcd(h,N) >1$. \qquad $\Box$ \medskip The following result - due to Castelnuovo - will be used, among other things, to prove the implication (ii) $\Rightarrow$ (i) of Theorem A regardless of the characteristic of the base field and to construct examples in order to show that in some cases the bounds of our results are sharp. \medskip {\bf Lemma 3.2 ([C1, St]).} Let $X$ be a curve of genus $g$ and $K_{1}, K_{2}$ be subfields of $K(X)$ with compositum $K(X)$. If $n_{i}$ is the degree of $K(X)$ over $K_{i}$ and $g_{i}$ is the genus of $K_{i}$ for $i=1,2$, then $$ g \le (n_{1}-1)(n_{2}-1) + n_{1}g_{1} + n_{2}g_{2}.\qquad \Box $$ For $N$ prime, we have the uniqueness of $\pi$ above provided $g$ is large enough, and we also have a criterion to decide when a point is totally ramified: \medskip {\bf Corollary 3.3.} Let $X$ be a curve of genus $g$, $N$ prime and $\gamma$ a non--negative integer.\\ (i) If $$ g > \rho_5(N,\gamma):= 2N\gamma +(N-1)^2, $$ \noindent then $X$ admits at most one $N$--sheeted covering of a curve of genus $\gamma$.\\ (ii) Let $P \in X$, $\tilde X$ be a curve of genus $\gamma$ and, $\pi$ an $N$--sheeted covering map from $X$ to $\tilde X$. Then, $P$ is totally ramified for $\pi$ provided there exists $h \in H(P)$ such that \begin{equation} (N-1)h < g -N\gamma +N-1. \end{equation} {\bf Proof.} (i) If K(X) have two differents fields $K_1$ and $K_2$ both of genus $\gamma$, then by Lemma 3.2 we have $g \le \rho_5(N,\gamma)$.\\ (ii) Let $f\in K(X)$ with div$_\infty (f)=hP$ and $K^{'}$ be the compositum of $K(f)$ and $K(\tilde X)$. Using $N$ prime and the hypothesis on $h$, from Lemma 3.2 it follows that $K^{'}=K(\tilde X)$. Then, there exists $\tilde f \in K(\tilde X)$ so that $f = \tilde f\circ \pi$. Consequently, the ramification number of $\pi$ at $P$ is $N$ and so $P$ is totally ramified for $\pi$. \rightline{$\Box$} Next we look for conditions to have equality in (I) or (II). \medskip {\bf Lemma 3.4.} Let $X$, $\tilde X$, $\pi$, $P$, $\tilde P$ and $N$ be as above. If either $p = char(K)$ $\not\!\vert N$ or $N$ prime and $g >\rho_2(N,\gamma)$ (see (2)), then $$ \gamma_N = \gamma. $$ {\bf Proof.} It will be enough to show that: $nN \in H(P) \Rightarrow n \in H(\tilde P)$.\\ {\bf Case 1:} $p \not\!\vert N$. Let $z$ be a local parameter at $P$ so that $z^N$ is also a local parameter at $\tilde P$. Let $\Psi$ (resp. $\tilde \Psi $) denote the inmersion of $K(X)$ (resp. $K(\tilde X)$) into the field of Puiseux series at $P$ (resp. $ \tilde P$) $F_1 = K((z))$ (resp. $F_2 = K((z^N))$). Since ${\Psi \mid} _{K(\tilde X)} = \tilde \Psi$ we have that $Tr_{F_1\mid F_2}\circ \Psi = \tilde \Psi \circ Tr_{K(X)\mid K(\tilde X)}$ (*) (Tr means trace). Let $f \in K(X)$ with ${\rm div}_\infty (f) = nN$. Write $f = \sum_{i=-nN}^{\infty} c_iz^i$. Then, by considering the base $\{1,z,\ldots , z^{N-1} \}$ of $F_1\mid F_2$, we have that $Tr_{F_1\mid F_2} (f) = \sum_{i=-n}^{\infty} {Nc_{iN}z^{iN}}$. Consequently, from (*) it follows that the order of $\tilde f:= Tr_{K(X)\mid K(\tilde X)}(f)$ at $\tilde P$ is $n$ and, since $f$ has no other pole, ${\rm div}_\infty(\tilde f) = n\tilde P$ and we are done.\\ {\bf Case 2:} $g > \rho_2(N,\gamma)$. From the proof of Corollary 3.3 (ii) we have that $f = \tilde f \circ \pi$ for some $\tilde f \in K(\tilde X)$ whenever ${\rm div}_\infty(f) = hP$ with $h$ satisfying (12). Now, from the hypothesis on $g$ we can applied the above statement for $h \in H(P)$ with $h \le 2N\gamma - N$. \qquad $\Box$ \medskip {}From (I), (II), (11) and the lemma above we obtain: \medskip {\bf Corollary 3.5.} Assume the hypothesis of Lemma 3.4.\\ (i) H(P) satisfies conditions (a), (c) of Definition 2 (with $N$ and $\gamma$) and $2N\gamma \in H(P)$.\\ (ii) Suppose $N$ is prime. If either $N \not= p$ and $g > \rho_1(2\gamma,N,\gamma) = N^2\gamma -N+1$ or $g > \rho_2(N,\gamma) $, then $H(P)$ is of type $(N,\gamma)$.\qquad $\Box$ \medskip {\bf Remark 3.6.} Let $\pi: X \to \tilde X$ be an $N$--sheeted covering of curves of genus $g$ and $\gamma$ respectively. Assume $g > \rho_2(N,\gamma)$ and hence, in particular that $\pi $ is ``strongly branched" (cf. [A]). When $\pi$ is a ``maximal strongly branched" (e.g. $N$ prime) we still have the result in Lemma 3.4 [A, Lemma 4].\qquad $\Box$. \medskip To deal with the ``geometry" of Theorem A we need the other Castelnuovo genus bound lemma: \medskip {\bf Lemma 3.7 ([C], [ACGH, p.116], [Ra, Corollary 2.8]).} Let $X$ be a curve of genus $g$ that admits a birational morphism onto a non--degenerate curve of degree $d$ in ${\rm I}\!{\rm P}^r(K)$. Then $$ g\le c(d,r):={m(m-1)\over 2}\ (r-1)+m\varepsilon $$ where $m$ is the biggest integer $\le (d-1)/(r-1)$ and $\varepsilon = d-1 - m(r-1)$. \rightline{$\Box$} {\bf Lemma 3.8.} Let $X$ be a curve of genus $g$, $N$ a prime and $\gamma \ge 0$ an integer. Let $A\ge 2\gamma +1$ be an integer satisfying the hypotheses (5) and (6) of item (A4) ( Theorem A). If $X$ admits a base--point--free linear system $g^{A-\gamma}_{AN}$, then $X$ is an $N$--sheeted covering of a curve of genus $\gamma$. {\bf Proof.} Let $\pi :X\to {\rm I}\!{\rm P}^{A-\gamma}(K)$ be the morphism defined by $g^{A-\gamma}_{AN}$. \medskip {\bf Claim:}\quad $\pi$ is not birational. If by way of contradiction $\pi$ is birational, we can applied the lemma above to obtain $g \le c(AN,A\gamma)=\rho_4(A,u,N,\gamma)$. \smallskip Let $t$ be the degree of $\pi$ and $\tilde X$ the normalization of $\pi (X)$. Then the induced morphism $\pi :X\to\tilde X$ is a covering map of degree $t$ and $\tilde X$ admits a base--point--free linear system $\tilde g^{A-\gamma}_ {AN\over t}$. In particular we have $t \le AN/(A-\gamma)$ and the hypothesis (5) implies $t=N$. Now, by the Clifford's theorem we have that $\tilde g^{A-\gamma}_{A}$ is nonspecial, and consequently by the Riemann--Roch theorem the genus of $\widetilde X$ is $\gamma$ and the proof is complete.\qquad $\Box$ \medskip {\bf Proof of Theorem A.} {\bf (A1):} Corollary 3.5.\\ {\bf (A2):} Since $\rho_3(N,\gamma)> \rho_1(2\gamma+2,N,\gamma)$ from Corollary 2.7 we have $D:= gcd(m_1(P), \ldots, m_{\gamma+2}) = N$. In particular $m_{\gamma + 2} = (2\gamma +2)N$. Now, we can apply the Claim in the proof above with $A=2\gamma+2$ and $\rho_4(2\gamma+2,N-1,N,\gamma)= \rho_3(N,\gamma)$ to conclude that the degree $t$ of the rational map obtained from the liner system $|m_{\gamma +2}P|$ is bigger than 1. Due to the fact that $t\vert D$ and $N$ prime, we conclude that $t=N$, and by a similar argument to the above proof (last lines) we see that the covered curve has genus $\gamma$ and we are done.\\ {\bf (A3):} By Corollary 2.7 we have $m_{A-\gamma}(P) = AN$ and it follows the proof.\\ {\bf (A4):} The above lemma shows that $X$ is an $N$--covering of a curve of genus $\gamma$. Since the covering is given by $|ANP|$, we have that $P$ is a totally ramified point of $\pi$ and the proof is complete. \qquad $\Box$ \medskip {\bf Corollary 3.9.} (i) Let $\pi :X\to\tilde X$ be an $N$--sheeted covering of curves of genus $g$ and $\gamma$ respectively. Suppose $N$ prime and $g> \rho_3(N,\gamma)$. Then $P$ is totally ramified for $\pi$ if and only if $H(P)$ is a semigroup of type $(N,\gamma )$.\\ (ii) Let $H$ be a Weierstrass semigroup of genus $g> \rho_3(N,\gamma)$. Then $H$ is of type $(N,\gamma )$ if and only if there exists an integer $A \in [2\gamma +2, 2\gamma+2 +{\gamma \over N-1}]$ satisfying (5) and such that $m_{A-\gamma} =AN$. {\bf Proof.} (i) Proof of item (A2) of Theorem A.\\ (ii) For the numbers $A$ in that interval we have $\rho_1(A,N,\gamma) \le {N(2N+1)\gamma \over 2} + (N-1)^2 < \rho_3(N,\gamma)$ and hence the ``if" part of the statement is just item (A3). To prove the ``only if" part notice that $u(A) = N-1$ and hence $\rho_4(A,u(A),N,\gamma) = \rho_3(N,\gamma)$. Now the hypotheses on $A$ assure that the degree of the map obtained from $|ANP|$ is $N$, and since this number is the $g.c.d$ of the non--gaps $m_1,\ldots,m_{A-\gamma}$, it follows the proof. \qquad $\Box$ \medskip {\bf Remark 3.10.} Remark 3.11 (ii), (iii) below show that neither the bound on $g$ nor the hypothesis (5) of the above corollary (part (ii)) can be dropped. It would be interesting to have an arithmetical proof of Corollary 3.9 (ii) (i.e. without the assumption that $H$ is a Weierstrass semigroup), because any counter example to the above question would be a numerical semigroup that cannot be realized as Weiertrass semigroup. This numerical semigroup could be see as a ``mid term" between the examples stated in the last section. The most simple case of the above question is for $N=2$. But this case does not provide any counter example [T1]. \medskip {\bf Remarks 3.11.} Let $N$, $\gamma$ be a prime and a non--negative integer respectively and suppose $p \not\!\vert 2N$.\\ {\bf (i)} The following example has respect to Lemma 2.1, Corollary 2.7, Lemma 3.1, Corollary 3.5 (ii) and item (A1) of Theorem A. Let $g>0$ be an integer such that $g-N\gamma \equiv 0$ (mod $(N-1)$) and $L := {{2g-2\gamma N}\over {N-1}}+1$ is coprime with $2N$. Define $i_1:= 2\gamma+1$ and $i_2:= {g-(2N-1)\gamma \over N-1} $ (hence $i_1 +2i_2 =L$). For $j=1,\ldots, i_1, k=1, \ldots i_2$, choose $a_j, b_k$ pairwise distinct elements of $K$. Now consider the curve $X$ defined by the equation \vspace{-0.5pt} $$ y^{2N} = \mathop{\Pi}\limits^{i_1}_{j=1}\ (x-a_j) \mathop{\Pi}\limits^{i_2}_{k=1}\ (x-b_k)^2 $$ \vspace{-0.5pt} Then, by the Riemann--Hurwitz relation we have that the genus of $X$ is $g$. Moreover, $X$ is a $N$--sheeted covering of the hyperelliptic curve $\tilde X$ of genus $\gamma$ whose field of rational functions is $K(x,z)$, where $z = y^{N}/\Pi_{k=1}^{i_2}\ (x-b_k)$. Since $gcd(L,2N)=1$, there exist just one point $P\in X$ over $x = \infty$ and consequently $X$ is a curve of type $(N,\gamma)$ over $\tilde X$. \medskip \noindent {\bf Claim :}\qquad $H(P) = H:= \langle 2N, L, (2\gamma +1)N \rangle$. \medskip \noindent This claim shows that the result in Lemma 2.1 is the best possible. Considering $g = \rho_4(A,N,\gamma)$, we have $L = AN-1$ and hence $m_{A-\gamma}(H) \le AN-1 $ provided $A \ge 2\gamma$. Hence the claim also shows the sharpness of the bound on $g$ of Corollary 2.7. By specializing $A = 2\gamma$ we also see the sharpness of the bound on $g$ of Lemma 3.1 and Corollary 3.5 (ii) (case $p \not\!\vert N$) respectively. With respect to item (A1) is not difficult to see that in the above curve $X$ (with $A = 2\gamma$), all the Weierstrass semigroups at totally ramified points are not of type $(N,\gamma)$ (the case $N=2$ is in [T, Remark 3.9]). However, we cannot say the same about the other Weierstrass semigroups of $X$. \medskip \noindent {\bf Proof of the Claim.} Since the genus of $H$ is at most $g$, it will be enough to show that $H \subseteq H(P)$. This is true because, ${\rm div}_\infty (x) = 2N P$, ${\rm div}_\infty (y) = L P$, and $(2\gamma +1)N \in H(P)$ due to the fact that $P$ is totally ramified over $\tilde X$ which has genus $\gamma$.\qquad $\Box$ \medskip \noindent {\bf (ii)} This example is related to the bound on the genus in Lemma 3.8 and item (A4) of Theorem A. Set $i_1 := 2N\gamma +2N-1$. The curve $X$ defined by the equation \vspace{-0.5pt} $$ y^{2N} = \mathop{\Pi}\limits^{i_1}_{j=1}\ (x-a_j), $$ \vspace{-0.5pt} \noindent where the $a_j's$ are pairwise distinct elements of $K$, has the following properties: \begin{list} \setlength{\rightmargin 0cm}{\leftmargin 0cm} \itemsep=0.5pt \item[(1)] its genus is $g = N(2N-1)\gamma + (N-1)(2N-1) = \rho_3(N,\gamma)$; \item[(2)] the Weierstrass semigroup at the unique point $P$ over $x = \infty$ is generated by $2N$ and $i_1$; \item[(3)] $m_{A-\gamma}(P) = AN$ provided $2\gamma \le A < 4\gamma +4 - {2\over N}$; \item[(4)] it cannot be an $N$--covering of a curve of genus $\gamma$. \end{list} \noindent Consequently, the upper bound $\rho_3(N,\gamma)$ for the genus in both Lemma 3.8 and item (A4) is necessary for $2\gamma +1 \le A \le 4\gamma +4 -{2\over N}$. In particular, if N-1 is the biggest integer $\le {N\gamma +N-1 \over A-\gamma -1}$, $\rho_3(N,\gamma)$ is sharp. In the other cases, we don't know the sharpness of $\rho_4(A,u,N,\gamma)$. \medskip \noindent {\bf Proof of properties (1)--(4).} (1) follows from Riemann--Hurwitz relation. To prove (2), we notice that ${\rm div}(x) = 2NP$ and ${\rm div}(y) = i_1P$ and so $H(P) \supseteq \langle 2N, i_1 \rangle $. Since the last semigroup also has genus $g$, we have (2). To prove (3) we notice that in the interval $[1,AN]$ the number of multiples of $N$ non-gaps of $H(P)$ is $A\over2$ (or $(A+1)/2$); it has ${A\over2} -\gamma$ (or $(A-1)/2 -\gamma $) non--gaps which are congruent to $2N-1$ module $N$, and its other non--gaps are bigger than $AN$ (here we use $A < 4\gamma +4 - {2\over N}$). Finally, if $X$ is an $N$--sheeted covering of a curve of genus $\gamma$, by Castelnuovo's lemma (Lemma 3.2) the genus $g$ would be at most $(2N\gamma +2N-2)(N-1) + N\gamma < \rho_3(N,\gamma)$, a contradiction.\\ {\bf (iii)} Here, we show that the arithmetical conditions (5) cannot be dropped if we suppose \vspace{-0.5pt} $$ g > {\rm max}\ \{{1\over 2}AN[A(N-2) +2\gamma +3], A(N-1)(N-2) +(3N-2)\gamma +3(N-1)\}. $$ Since $A\ge 2\gamma +1$ we have ${AN\over A-\gamma} \le 2N-1$ and then one has to check (5) among the integers of the set $[2,2N-1] \setminus {N}$. Let $t$ be an integer of the above set such that $t\vert A$ and set $r:= {AN\over t} -A +\gamma +1$. Let $g>0$ be an integer satisfying the above bound and such that $i_1:= {2g \over rt -1} +1 $ is also an integer. The curve in the previous remark, with $rt$ instead of $2N$ and the above $i_1$, has genus $g$ and just one point $P$ over $x=\infty$ which satisfies $m_{A-\gamma}(P) = AN$ (here we use the first part of the bound). But $X$ cannot be an $N$--sheeted covering because on the contrary by Castelnuovo's lemma (Lemma 3.2) the genus would be at most the second part of the above bound.\\ {\bf (iv)} Finally, some words about items (A2) and (A3). Since statement (ii) of Theorem A is stronger than (iii), one might expect to sharpen $\rho_3(N,\gamma)$ (this would be relevant to the examples in the next section). In order to do that, one might use Castelnuovo's theory (cf. e.g. [E-H-1, $\oint$ 3], [Ci]) or ``results" extending this theory to Hilbert functions of points in projective spaces [E-G-H]. Specifically, one could use analogous bounds to $c(d,r)$ in order to deal with curves of genus $g \le \rho_3(N,\gamma) = c(AN,A-\gamma)$. The point is that one knows how must look the curves whose genus attain the mentioned bounds. For instance, one can applied the above considerations to double covering of curves of genus one or two and the result is that (A2) is still valid for $g \ge \rho_3(2,\gamma)-2$ ($\gamma \in \{1,2\}$) (see also [G, Lemmas 7 and 9]). In general, we think that item (A2) must be true with a bound of type ``$\rho_3(N,\gamma) - N$". We remark that by applying the arithmetical properties of semigroups of type $(N,\gamma)$ one can find a ``kind of algorithm to compute Hilbert functions". We will intend to describe this in a later paper. With respect to the sharpness of the bound on $g$ of item (A3), we just want to say that it depends on the existence of certain Weierstrass semigroups.\qquad $\Box$ \section{Hurwitz's question.} In this section we construct numerical semigroups with $\ell_g$ given, that cannot be realized as Weierstrass semigroup. These examples will include symmetric and quasi--symmetric semigroups generalizing those in [T, Scholium 3.5] and [O-S, Example 6.5]. \vspace{-0.5pt} \subsection{Corollaries of Buchweitz's criterion; case $n=2$.} Let $X$ be a curve, $H$ a numerical semigroup both of genus $g$. Denote by $\ell_1=\ell_1(H) <,\ldots,\ell_g= \ell_g(H)$ (resp. $G_n=G(H)$) the gaps (resp. the set of sums of $n$ gaps) of $H$. By the definition of $g$ and by the semigroup property of $H$, we have $g\le \ell_g \le 2g-1$ ([B, O]). Semigroups with $\ell_g = g$ are realized for all but finitely many points of $X$ provided it is defined in characteristic zero or characteristic larger than 2g-2 (see e.g. [S-V]). In the remaining cases the situation can be different (see e.g. [Sch], [G-V]). On the other hand, if $\ell_g = 2g-1$, $H$ is called symmetric because between the non--gaps and gaps of $H$ we have the following property: $S(1)$: ``$h \in H \Leftrightarrow \ell_g -h \in G(H)$". These semigroups are important at least for two reasons : 1) they arise in a natural way in the context of Gorenstein rings (cf. [Ku]); 2) for $n \ge 2$, $\#G_n = (2n-1)(g-1)$ ([O, Thm. 1.5]), i.e., they satisfy Buchweitz's criterion (BC) (see Section 1). What can be said if $\ell_g < 2g-1$ ?. If $\ell_g = 2g-2$, $H$ still satisfies property $S(1)$ except for $g-1$ [O, Prop. 1.2] ( consequently $H$ is called quasi--symmetric). Then, it is not surprising that quasi--symmetric semigroups also satisfy (BC). In fact, here we have $\# G_n = (2n-1)(g-1) -(n-2)$ ([O-S, Thm. 1.1]). As in the case of the symmetric semigroups, these semigroups can also be realized as Weierstrass semigroups of Gorestein curve, but in view of the term ``$-(n-2)$" above, here one has to allow reducible curves (cf. [O-S, $\oint$ 3]). In general we have the following properties of type $S(1)$. Suppose $\ell_g \in \{2g-2i+1, 2g-2i\}$ with $i\ge 1$. Considering the pairs $(r, \ell_g -r)$ for $r =1,\ldots, g-i $, $H$ satisfies the property \medskip \noindent $S(i):$\quad If $\ell_g$ is odd, $H$ fulfils property $S(1)$ except $2i-2$ gaps of type: $g-i < h_{i-1}<\ldots < h_1 <\ell_g$, $\ell_g - h_{i-1}, \ldots, \ell_g - h_1$. If $\ell_g$ is even, $H$ fulfils condition $S(1)$ except $2i-2$ gaps of the above type and the gap $(g-i)$. \medskip Since for $i>1$ we have gaps different from those arising in $S(1)$, it seems to be difficult to obtain a closed form for $\#G_n$. However, we think that the following must be true provided $\ell_g \le 2g-2$: \vspace{-0.5pt} $$ G_n = \{n,n+1,\ldots,(n-1)\ell_g \} \mathop{\cup}\limits^{g}_{k=1}\{(n-1)\ell_k +\ell_j : j=1, \ldots g \}. $$ \vspace{-0.5pt} {}From the proof of [O-S, Thm. 1.1] at least the inclusion ``$\supseteq$'' holds. In particular we have \vspace{-0.5pt} $$ \# G_2 = (\ell_g -1) + g + \Lambda, $$ \vspace{-0.5pt} where $\Lambda$ is a non--negative integer. Consequently if $\Lambda \ge 2i-2$ (resp. $2i-1$) for $\ell_g = 2g-2i+1$ (resp. $2g-2i$), by Buchweitz's criterion $H$ is not a Weierstrass semigroup. Now, consider the following sequences of gaps obtained from those of property $S(i)$: \vspace{-0.5pt} $$ 2h_1 >\ldots > h_1 +h_{i-1},\ \ {\rm and} $$ \vspace{-0.5pt} $$ 2h_2>\ldots >2h_{i-1}. $$ \vspace{-0.5pt} With the above notation we have \medskip {\bf Lemma 4.1.1.} Let $H$ be a numerical semigroup with $\ell_g = 2g-2i+1$ and $i \ge 4$. If $h_1 + h_{i-1} > 2h_2$ and $2\ell_g - h_u - h_v \in G(H)$ for $(u,v)= (1,1),\ldots, (1,i-1)$ and $(u,v) = (2,2),\ldots,(2,i-1)$, then $H$ is not a Weierstrass semigroup. {\bf Proof.} The hypothesis involving $G(H)$ means that $h_u + h_v$ is not the sum of $\ell_g$ with some other gap. Consequently from the sequences of gaps above we have $\Lambda \ge 2i-2$ and it follows the proof.\qquad $\Box$ \medskip Using this criterion we can exhibit numerical semigroups with a fixed last gap which cannot be realized as Weierstrass semigroups. The following example with $i=4$, $g=16$ is the well known Buchweitz's semigroup. \medskip {\bf Corollary 4.1.2.} Let $g, i$ be integers so that $g \ge 9i -20$, $i \ge 4$ and $3g + 5i -20 $ even, say equal to $2h_1$. Then the numerical semigroup whose gaps are $$ \{1,2,\ldots,g-i, h_1 - (a+2(i-3)),\ldots, h_1-(a+2), h_1 -a, h_1, 2g-2i+1 \}, $$ \noindent where $a= 2i-5$ is not a Weierstrass semigroup.\qquad $\Box$ \medskip In the above examples one can use $a > 2i-6$ provided that $g \ge 2a -10 +5i$ and $3g+2a +i -10$ even. \subsection{An application of item (A2) of Theorem A.} First we notice that from the proof of item (A2), if $N$ is prime and $H = \{m_0=0,m_1,\ldots \}$ is a Weierstrass semigroup of type $(N,\gamma )$ of genus $g>\rho_3(N,\gamma)$, then the numerical semigroup $$ \pi( H) = \{{m_i\over N}:1\le i\le\gamma\}\cup\{ 2\gamma +i:i\in{\rm I}\!{\rm N}\} $$ \noindent is also a Weierstrass semigroup. We use this remark to prove an analogue of Corollary 4.1.2. The semigroups of this result are also inspired by the properties $S(i)$ of the last subsection. Fix a numerical semigroup $\tilde H$ of genus $\gamma$ such that it is not a Weierstrass semigroup. Let $N$ be a prime and $g$ and integer. Write $g = \lambda N +u$ with $0\le u <N$. Let $f$ be an integer such that $f \le u$ if $u>0$ and $f <N$ otherwise. Set $N\tilde H:= \{hN; h \in \tilde H\}$. We are only going to consider the case $2g-f \not\equiv 0$ (mod $N$) because in the other case we can replace $g$ by $g+1$. \medskip {\bf Corollary 4.2.1.} With the above notation, consider the following sets \begin{list} \setlenght{\rightmargin 0cm}{\leftmargin 0cm} \itemsep=0.5pt \item[(1)] $H_1= N\tilde H\cup\{ 2g-f-r:r\not\in N\tilde H,r\le g-1\}$, if $2u \not\in [N,f+N]$ \item[(2)] $H_2 = H_1 \setminus \{e\}$ otherwise; where $e$ is the biggest integer $\le (2g-f)/2 $. \end{list} If $g > \rho_3(N,\gamma)$, then $H_1$ and $H_2$ are numerical semigroups of type $(N,\gamma )$ of genus $g$ whose last gap is $2g-f$ which are not Weierstrass semigroups. \medskip {\bf Proof.} Set $H$ for $H_1$ or $H_2$. We notice that $\pi(H) = \tilde H$ and so it will be enough to prove the arithmetical statements. By the definition of $H$ and the hypothesis on $g$, it follows that $H$ is a semigroup of type $(N,\gamma )$ such that $H\supseteq\{ 2g,2g+1,\dots\}$ and $\ell_g(H)=2g-f$ (here we use $2g-f\not\equiv 0$ (mod $N$)). Consider $H=H_1$ and let \begin{eqnarray*} U & = & \#\{ h\in H:h\le 2g,h=2g\ \ {\rm or}\ \ h\equiv 0\ ({\rm mod}\ N)\},\\ V & = & \#\{ h\in H:h<2g,h\not\equiv 0\ ({\rm mod}\ N)\}\ . \end{eqnarray*} Then \vspace{-0.5pt} $$ U = \left\{\matrix{ 2\lambda -\gamma,\hfill &{\rm if}\ \ u=0;\hfill\cr 2\lambda +1-\gamma ,\hfill&{\rm if}\ \ 0<2u<N;\cr 2\lambda +2-\gamma ,\hfill &{\rm if}\ \ 2u>N;\hfill\cr}\right. $$ \vspace{-0.5pt} $$ V=g-1-\#\{ 1\leq r\leq g-1:r\in N\widetilde H\}-\#\{ 1\leq r\leq g-1:r\equiv 2u-f ({\rm mod}\ N)\}\ . $$ \noindent Since $2u-f\not= N$, we have $$ V=\left\{\matrix{ g-1-(\lambda -1-\gamma )-\lambda ,\hfill &{\rm if}\ \ u=0;\hfill\cr g-1-(\lambda -\gamma )-\lambda ,\hfill &{\rm if}\ \ 2u-f<N;\cr g-1-(\lambda -\gamma )-(\lambda +1),\hfill &{\rm if}\ \ 2u-f>N.\hfill\cr}\right. $$ Then $U+V=g$ is the number of non--gaps $\le 2g$ of $H_1$ because $2u \not\in [N,n+f]$. In the other case, from the above computations we get $U+V = g+1$ and since we have excluded $e$ we are done.\qquad $\Box$ \medskip The last two corollaries give us numerical semigroups arising from different phenomena. Moreover, notice that the ``intersection " of both families of examples is empty. By considering the distribution of the respective gaps sequences, we can think about of these examples as being the ``extremal" cases of numerical semigroups that cannot be realized as Weierstrass semigroups. Finally, we would like to know if any numerical semigroup which is not a Weierstrass semigroup but satisfies Buchweitz's criterion must be of type $(N,\gamma)$ for some $N$ and $\gamma$. \bigskip \centerline{\bf ACKNOWLEDGMENTS} \medskip Thank you very much to the International Atomic Energy Agency and UNESCO for the hospitality at the International Centre for Theoretical Physics, Trieste. Special thanks to Profs. Oliveira and St\"ohr who showed me how to applied item (A2) of Theorem A to the case of quasi-symmetric semigroups (in fact, their paper [O-S] suggested many ideas to this work); Prof. Ulrich for pointing out to me the remark on semigroups rings stated in the introduction and the referee for suggesting to improve the earlier version of the paper. \bigskip \centerline{\bf REFERENCES} \small \begin{description} \itemsep=-0.5pt \item{[A]} Accola, R.D.M.: Strongly branched coverings of closed Riemann Surfaces, Proc. Amer. Math. Soc. {\bf 26}, 315--322 (1970). \item{[ACHG]} Arbarello, E., Cornalba, M., Griffiths, P.A. and Harris, J.: Geometry of algebraic curves, Vol.I, Springer--Verlag, New York (1985). \item{[B]} Buchweitz, R.O.: \"Uber deformationem monomialer kurvensingularit\"aten und Weierstrasspunkte auf Riemannschen fl\"achen, Thesis, Hannover (1976). \item{[B1]} Buchweitz, R.O.: On Zariski's criterion for equisingularity and non--smoothable monomial curves, Th\'ese, Paris VII (1981). \item{[C]} Castelnuovo, G.: Ricerche di geometria sulle curve algebriche, Atti. R. Acad. Sci. Torino {\bf 24}, 196--223 (1889). \item{[C1]} Castelnuovo, G.: Sulle serie albegriche di gruppi di punti appartenenti ad una curvea algebrica, Rendiconti della Reale Accademia dei Lincei (5) {\bf 15}, 337--344 (1906). \item{[Ci]} Ciliberto, C.: Hilbert functions of finite sets of points and the genus of a curve in a projective space, Space Curves, Rocca di Papa 1985, ed. F. Ghione, C. Peskine and E. Sernesi (1987). \item{[E-G-H]} Eisenbud, D., Green, M., Harris, J.: Some Conjectures Extending Castelnuovo Theory, preprint (1993). \item{[E-H]} Eisenbud, D., Harris, J.: Existence, decomposition and limits of certain Weierstrass points, Invent. math. {\bf 87}, 499--515 (1987). \item{[E-H-1]} Eisenbud, D., Harris, J.: Curves in projective space, Les Preses de l'Universit\'e de Montr\'eal, Montr\'eal, (1982). \item{[G]} Garcia, A.: Weights of Weierstrass points in double coverings of curves of genus one or two, Manuscripta Math. {\bf 55}, 419--432 (1986). \item{[G-V]} Garcia, A., Viana, P.: Weierstras points on certain non--classical curves, Arch. Math. {\bf 46}, 315--322 (1986). \item{[H]} Hurwitz, A.: \"Uber algebraische gebilde nit eindeutigen transformationen in sich, Math. Ann. {\bf 41}, 403--441 (1893). \item{[J]} Jenkins, J.A.: Some remarks on Weierstrass points, Proc. Amer. Math. {\bf 44}, 121--122 (1974). \item{[K]} Kato, T.: On the order of a zero of the theta function, Kodai Math. Sem. Rep. {\bf 28}, 390--407 (1977). \item{[Ku]} Kunz, E.: The value-semigroup of a one-dimensional Gorenstein ring, Proc. Amer. Math. Soc. {\bf 25}, 748--751 (1970). \item{[O]} Oliveira, G.: Weierstrass semigroups and the canonical ideal of non--trigonal curves, Manuscripta Math. {\bf 71}, 431--450 (1991). \item{[O-S]} Oliveira, G., St\"ohr, K.O.: Gorenstein curves with quasi-symmetric Weierstrass semigroups, Preprint (1994). \item{[P]} Pinkham, H.: Deformations of Algebraic Varieties with ${\rm G}\!{\rm G}_m$ Action, Asterisque {\bf 20}, Soc. Math. France, (1974). \item{[R]} Rathmann, J.: The uniform position principle for curves in characteristic $p$, Math. Ann. {\bf 276}, 565--579 (1987). \item{[R\"o]} R\"ohrl, H.: Unbounded coverings of Riemann surfaces and extensions of rings of meromorphic functions, Trans. Amer. Math. Soc. {\bf 107}, 320--346 (1963). \item{[Sch]} Schmidt, F.K.: Zur arithmetischen Theorie der algebraischen Funktionen, II, Allgemaire Theorie der Weierstrasspunke, Math. Z. {\bf 45}, 75--96 (1939). \item{[S]} St\"ohr, K.O.: On the modulli spaces of Gorenstein curves with symmetric Weierstrass semigroups, J. reine angew. Math. {\bf 441}, 189--213 (1993). \item{[S1]} St\"ohr, K.O.: On the poles of regular differentials of singular curves, Bol. Soc. Bras. Mat. {\bf 24}, 105--136 (1993). \item{[S-V]} St\"ohr, K.O., Voloch, J.F.: Weierstrass points and curves over finite fields, Proc. London Math. Soc. (3), {\bf 52}, 1--19, (1986). \item{[St]} Stichtenoth, H.: Die ungleichung von Castelnuovo, J. Reine Angew. Math. {\bf 348}, 197--202 (1984). \item{[T]} Torres, F.: Weierstrass points and double coverings of curves with application: Symmetric numerical semigroups which cannot be realized as Weierstrass semigroups, Manuscripta Math. {\bf 83}, 39--58 (1994). \item{[T1]} Torres, F.: Remarks on Weierstrass semigroups at totally ramified points (provisory title), in preparation. \end{description} \end{document}

9407

alg-geom/9407003

en

https://arxiv.org/abs/alg-geom/9407003

alg-geom/9407003

Raghu Nyshadham

I.Biswas and N.Raghavendra

Canonical Generators for the Cohomology of Moduli of Parabolic Bundles on Curves

22 pages, Latex Version 2.09

We determine generators of the rational cohomology algebras of moduli spaces of parabolic vector bundles on a curve, under some `primality' conditions on the parabolic datum. These generators are canonical in a precise sense. Our results are new even for usual vector bundles (i.e., vector bundles without parabolic structure) whose rank is greater than 2 and is coprime to the degree; in this case, they are generalizations of a theorem of Newstead on the moduli of vector bundles of rank 2 and odd degree.

alg-geom

\section{Introduction} The aim of this paper is to determine generators of the rational cohomology algebras of moduli spaces of parabolic vector bundles on a curve, under some `primality' conditions (see Assumptions 1.1 and 1.2) on the parabolic datum. These generators are canonical in a sense which will be made precise below. Our results are new even for usual vector bundles (i.e., vector bundles without parabolic structure) whose rank is greater than $2$ and is coprime to the degree; in this case, they are generalizations of a theorem of Newstead \cite{N}, where the case of vector bundles of rank $2$ and odd degree is studied. Let $X$ be a compact Riemann surface, fix integers $n$ and $d$ with $n$ positive, and let $\Delta $ be a parabolic datum of rank $n$ on $X$ (see Section 3 below). Denote by $\mbox{${\cal U}_{X}(n,d,\Delta )$} $ the moduli space of parabolic vector bundles of rank $n$ and degree $d$, which are parabolic semistable with respect to $\Delta $. Fix a holomorphic line bundle $L$ of degree $d$ on $X$, and let $\mbox{${\cal S}{\cal U}_{X}(n,L,\Delta )$} $ be the subvariety of $\mbox{${\cal U}_{X}(n,d,\Delta )$} $ consisting of vector bundles with determinant isomorphic to $L$. We make the following two hypotheses on the parameters $n$, $d$ and $\Delta$. \begin{assume} \begin{em} Every parabolic vector bundle of rank $n$ and degree $d$ on $X$ which is parabolic semistable with respect to the parabolic datum $\Delta $ is in fact parabolic stable. \end{em} \end{assume} \begin{assume} \begin{em} There exists a universal parabolic bundle (or briefly, a universal bundle) on $\mbox{${\cal U}_{X}(n,d,\Delta )$} \times X$. \end{em} \end{assume} Recall that a universal bundle on $\mbox{${\cal U}_{X}(n,d,\Delta )$} \times X$ is a vector bundle $U$ on $\mbox{${\cal U}_{X}(n,d,\Delta )$} \times X$ together with a flag of subbundles $\mbox{$j_{x}^{\ast}$} U = \uxi{1} \supset \uxi{2} \supset \cdots \supset \uxi{k_{x}} \supset \uxi{k_{x}+1} = 0$ in $\mbox{$j_{x}^{\ast}$} U$ for each $x \in J$ ($J$ being the set of parabolic points), where $\mbox{$j_{x}$} : \mbox{${\cal U}_{X}(n,d,\Delta )$} \mbox{$\rightarrow$} \mbox{${\cal U}_{X}(n,d,\Delta )$} \times X$ is the map $E \mapsto (E,x)$, such that for each $E \in \mbox{${\cal U}_{X}(n,d,\Delta )$}$, the restriction of $U$ to $\{ E \} \times X$ is parabolically isomorphic to $E$. We use the same symbols to denote the restrictions of $U$ and $\uxi{i}$ etc. to $\mbox{${\cal S}{\cal U}_{X}(n,L,\Delta )$} $. \begin{nota} \begin{em} \begin{itemize} \item[ \item If $S$ and $T$ are topological spaces, and if $\alpha \in H^{\ast }(S\times T,\mbox{${\bf Q}$})$, then $\sigma (\alpha ):H_{\ast}(T,\mbox{${\bf Q}$}) \rightarrow H^{\ast} (S,\mbox{${\bf Q}$})$ is the map $\sigma (\alpha )z=\alpha /z$, where $/$ denotes the slant product.\\ \item If $V$ is a vector bundle, then $\mbox{${\bf P}$}(V)$ denotes its projectivization.\\ \item All cohomologies in the paper have $\mbox{${\bf Q}$}$-coefficients. \end{itemize} \end{em} \end{nota} \label{1.3} Having settled on the notation, we now have the following two theorems which are the main results of this paper. \begin{thm} Suppose that Assumptions 1.1 and 1.2 hold, and let $U$ be any universal bundle on $\mbox{${\cal U}_{X}(n,d,\Delta )$} \times X$. Then, the rational cohomology algebra of $\mbox{${\cal U}_{X}(n,d,\Delta )$} $ is generated by the Chern classes $c_{j}(\mbox{${\cal H}{\it om}$} (U^{x,i} ,U^{x,i-1} ))~(x\in J)$ and the images of $$\sigma (c_{1}(U)): H_{1}(X) \rightarrow H^{1}(\mbox{${\cal U}_{X}(n,d,\Delta )$} )~~~{\it and}$$ $$\sigma (a_{i}(\mbox{${\bf P}$} (U))): H_{r}(X) \rightarrow H^{2i-r}(\mbox{${\cal U}_{X}(n,d,\Delta )$} )~~(2\leq i \leq n,~0\leq r \leq 2)$$ where $a_{i}(.)$ denote the characteristic classes of projective bundles introduced in Definition 2.4 below. \end{thm} \begin{thm} Suppose that Assumptions 1.1 and 1.2 hold, and let $U$ be any universal bundle on $\mbox{${\cal S}{\cal U}_{X}(n,L,\Delta )$} \times X$. Then the rational cohomology algebra of $\mbox{${\cal S}{\cal U}_{X}(n,L,\Delta )$} $ is generated by the Chern classes $c_{j}(\mbox{${\cal H}{\it om}$} (U^{x,i},U^{x,i-1}))~(x\in J)$ and the images of $$\sigma (a_{i}(\mbox{${\bf P}$} (U))): H_{r}(X) \rightarrow H^{2i-r}(\mbox{${\cal S}{\cal U}_{X}(n,L,\Delta )$} )~~(2\leq i\leq n,~0\leq r \leq 2).$$ \end{thm} Note that the generators given in Theorems 1.4 and 1.5 are {\em canonical}, i.e., independent of the choice of a universal bundle (which is easily seen to be non-unique). Indeed, if $U'$ is another universal bundle, then there exists a line bundle $\xi$ on $\mbox{${\cal U}_{X}(n,d,\Delta )$}$ such that $U' \cong U\otimes p^{*}\xi$, where $p : \mbox{${\cal U}_{X}(n,d,\Delta )$} \times X \mbox{$\rightarrow$} \mbox{${\cal U}_{X}(n,d,\Delta )$}$ is the canonical projection. Now, for every $z\in H_{1}(X)$, we have $\sigma (c_{1}(U'))z = \sigma (c_{1}(U))z$, since $\sigma (c_{1}(p^{*}\xi ))z = 0$; on the other hand, it is obvious that $\mbox{${\bf P}$} (U') \cong \mbox{${\bf P}$} (U)$ and $\mbox{${\cal H}{\it om}$} ((U')^{x,i},(U')^{x,i-1}) \cong \mbox{${\cal H}{\it om}$} (\uxi{i}, \uxi{i-1})$. We now relate the above theorems to certain results of Atiyah and Bott \cite{AB}. Let $\mbox{${\cal U}_{X}(n,d)$}$ be the moduli space of stable vector bundles of rank $n$ and degree $d$, where $n$ and $d$ are coprime. In this case, Atiyah and Bott \cite{AB} proved that the Kunneth components (with respect to any basis of $H^{*}(X,\mbox{${\bf Q}$} )$) of the Chern classes of any universal bundle on $\mbox{${\cal U}_{X}(n,d)$} \times X$ generate the rational cohomology algebra of $\mbox{${\cal U}_{X}(n,d)$}$. Theorem 1.4 above differs from this result in the following respects. Firstly, we work throughout in the setup of parabolic bundles, whereas Atiyah and Bott were working with usual vector bundles. Secondly, as we have observed above, the generators we obtain are canonical, i.e., they are independent of the choice of a universal bundle, whereas the Kunneth components of the Chern classes of a universal bundle $U$ do depend on the choice of $U$. Finally, by specializing to the case where the parabolic set is empty, and applying Lemma 2.6 below, we obtain the above result of Atiyah and Bott from Theorem 1.4; whereas it does not seem possible to deduce Theorem 1.4 from the result of Atiyah and Bott: the difficulty is due to the fact that the slant product does not behave well with the cup product. We should remark that Beauville \cite{Beau} has given another proof of the above result of Atiyah and Bott. In the parabolic setup, and under Assumptions 1.1 and 1.2, the method of Beauville can be used to deduce that the Kunneth components of the Chern classes of any universal bundle $U$ and the Chern classes $c_{j}(\mbox{${\cal H}{\it om}$} (\uxi{i},\uxi{i-1}))$ generate the cohomology of $\mbox{${\cal U}_{X}(n,d,\Delta )$}$, a statement which, as we have seen above, is weaker than Theorem 1.4. The following is a consequence of the above theorems. \begin{sloppy} \begin{cor} Let n=2, suppose Assumptions 1.1 and 1.2 are true, and let $U$ be any universal bundle on $\mbox{${\cal SU}_{X}(2,L,\Delta)$} \times X$. Then, the rational cohomology algebra of $\mbox{${\cal SU}_{X}(2,L,\Delta)$} $ is generated by the Chern classes $c_{j}(\mbox{${\cal H}{\it om}$} (S^{x}, Q^{x}))$ ($x \in J$), and the image of $$\sigma(c_{2}(\mbox{${\cal E}{\it nd}$} U)) : H_{r}(X) \mbox{$\longrightarrow$} H^{4-r}(\mbox{${\cal S}{\cal U}_{X}(n,L,\Delta )$} ) {}~~~(0 \leq r \leq 2),$$ where $\mbox{$j_{x}^{\ast}$} U = \uxi{1} \supset \uxi{2} = S^{x} $ is the flag in $\mbox{$j_{x}^{\ast}$} U$ ($x\in J$), and $Q^{x} = \mbox{$j_{x}^{\ast}$} U / S^{x}$. \end{cor} \end{sloppy} The above corollary is a generalization to parabolic bundles of a theorem of Newstead \cite{N}. In our approach, Assumption 1.1 is natural and indispensable; it is a technical necessity which guarantees that the action of a certain gauge group on a certain space of holomorphic structures is free. Granted this, Assumption 1.2 is not too stringent a restriction, as the following observation shows. \begin{propn} Suppose the parameters $n$, $d$ and $\Delta$ satisfy Assumption 1.1. Then, they satisfy Assumption 1.2 if any one of the following three conditions is satisfied: \begin{itemize} \item The rank $n$ and the degree $d$ are coprime. \item There exists a parabolic point $x \in J$ such that $\sum_{i=j}^{k_{x}}n_{x,i}$ is coprime to $n$ for some $j$ ($1 \leq j \leq k_{x}$), where $n_{x,1},\ldots ,n_{x,k_{x}}$ denote the parabolic multiplicities at $x$. \item There exists a parabolic point $x \in J$ such that $\sum_{i=j}^{k_{x}}n_{x,i}$ and $n+d$ are coprime for some $j$. \end{itemize} \end{propn} Here is a brief outline of the contents of the paper. In Section 2, we define the characteristic classes of projective bundles which occur in the statements of the theorems. The next section contains a description of generators of the rational cohomology of the classifying spaces of certain gauge groups. The final section contains proofs of the above results. \section{Projective Bundles} This preliminary section deals with some universal aspects of projective bundles, the aim being to define explicit characteristic classes for these bundles. If $G$ is a topological group, then $EG \rightarrow BG$ will denote a universal principal $G$-bundle. Cohomology groups will have rational coefficients throughout, unless otherwise indicated. Fix a positive integer $n$. The natural epimorphism $\pi :U(n) \rightarrow PU(n)$ induces a fibration $B\pi :BU(n) \rightarrow BPU(n)$ with fibre $BU(1)$. Let $x_{1},\ldots ,x_{n}$ be the Chern roots of $EU(n)$, so $H^{\ast}(BU(n))$ is the algebra $S[x_{1},\ldots ,x_{n}]$ of symmetric polynomials in the $x_{i}$ with rational coefficients (or, equivalently, $H^{\ast }(BU(n))$ is the polynomial algebra $\mbox{${\bf Q}$} [c_{1},\ldots ,c_{n}]$, where $c_{i}=c_{i}(EU(n))$). The Leray-Hirsch theorem implies that the map $(B\pi )^{\ast}:H^{\ast}(BPU(n))\rightarrow H^{\ast}(BU(n))$ is injective, and (see \cite{BH}, Section 15.2) its image equals the subalgebra $I[x_{1},\ldots ,x_{n}]$ of $S[x_{1},\ldots ,x_{n}]$ consisting of symmetric polynomials invariant under the affine change of variables $x_{i} \mapsto x_{i}+d$, where $d$ is an indeterminate. \begin{rem} \begin{em} The above fact means that if $E\rightarrow M$ is a vector bundle, the characteristic classes of its projectivization $\mbox{${\bf P}$} (E)$ are precisely the characteristic classes of $E$ which are invariant under tensoring by a line bundle. \end{em} \end{rem} \begin{sloppy} \begin{lemma} The above al\-gebra $I[x_{1}\ldots ,x_{n}]$ is a poly\-no\-mial al\-gebra $\mbox{${\bf Q}$} [z_{2},\ldots ,z_{n}]$, where $$ z_{k} = \sum _{1\leq j_{1}<\ldots ,j_{k}\leq n}y_{j_{1}}\ldots y_{j_{k}},~~~~2\leq k \leq n,$$ are the elementary symmetric functions in $y_{1},\ldots ,y_{n}$, where $$y_{i}=nx_{i}-\sum _{j=1}^{n}x_{j},~~~~1\leq i \leq n.$$ \end{lemma} \end{sloppy} The proof of the lemma is quite easy, and we omit it. Note that the first elementary symmetric polynomial in the $y_{i}$, namely their sum, is zero. The following assertion follows immediately from Lemma 2.2. \begin{cor} For $k=2,\ldots ,n$, define $a_{k} \in H^{2k}(BPU(n))$ by $(B\pi )^{\ast }a_{k}=z_{k}$. Then $H^{\ast }(BPU(n))$ is the polynomial algebra $\mbox{${\bf Q}$} [a_{2},\ldots ,a_{n}]$. \end{cor} \begin{defn} \begin{em} Let $P$ be a principal $PU(n)$-bundle on a CW-complex $M$. Then, for $k=2,\ldots ,n$, the {\em $k$-th characteristic class} $a_{k}(P)$ of $P$ is, by definition, the element $f^{\ast }a_{k} \in H^{2k}(M)$, where $f:M \rightarrow BPU(n)$ is some classifying map for $P$ and $a_{k}$ is as in Corollary 1.3. (As usual, $a_{k}(P)$ is independent of the choice of $f$.) \end{em} \end{defn} \begin{exam} \begin{em} \begin{enumerate} \item[] ~~~ \item[1.] If $E\rightarrow M$ is a vector bundle of rank 2, with Chern roots $x_{1},x_{2}$, then $$ a_{2}(\mbox{${\bf P}$} (E))= -(x_{1}-x_{2})^{2}=c_{2}(\mbox{${\cal E}{\it nd}$} ~E),$$ where $\mbox{${\bf P}$}(E)$ is the projectivization and $\mbox{${\cal E}{\it nd}$} ~ E$ is the endomorphism bundle of $E$.\\ \item[2.] Let $E\rightarrow M$ be a vector bundle of rank 3 such that $c_{i}(E)=0~(i=1,2),~c_{3}\not= 0$, and let $x_{1},x_{2},x_{3}$ be the Chern roots of $E$. Then $$a_{3}(\mbox{${\bf P}$}(E))=27x_{1}x_{2}x_{3} =27c_{3}(E) \not= 0,$$ while $c_{1}(\mbox{${\cal E}{\it nd}$} ~E)=c_{3}(\mbox{${\cal E}{\it nd}$}~E)=0$, $\mbox{${\cal E}{\it nd}$}~ E$ being the complexification of a real vector bundle. \end{enumerate} \end{em} \end{exam} The above examples illustrate the fact that any characteristic class of $\mbox{${\cal E}{\it nd}$}~ E$ can be written as a polynomial in $a_{i}(\mbox{${\bf P}$}(E))$, but the converse is not true, for $\mbox{${\bf P}$}(E)$ has more characteristic classes than $\mbox{${\cal E}{\it nd}$} ~E$. \begin{lemma} If $n\geq k\geq 2$, there exist a polynomial $P_{n,k}(T_{1},\ldots ,T_{k-1})$ with rational coefficients, and a non-zero constant $\lambda _{n,k} \in \mbox{${\bf Q}$}$ such that for every vector bundle $E$ of rank $n$ on a CW-complex $M$, we have \begin{eqnarray} a_{k}(\mbox{${\bf P}$}(E)) = P_{n,k}(c_{1}(E),a_{2}(\mbox{${\bf P}$}(E)),\ldots ,a_{k-1}(\mbox{${\bf P}$}(E))) + \lambda _{n,k}c_{k}(E). \end{eqnarray} \end{lemma} {\em Proof}.~~It suffices to find $P_{n,k}$ and non-zero $\lambda _{n,k}$ such that (1) holds for $EU(n)$. Let $u_{i}=a_{i}(\mbox{${\bf P}$}(EU(n)))$ and $c_{i}=c_{i}(EU(n))$; then $u_{i}=(BU)^{\ast }a_{i}$. We prove the result only for $k=2$, the general case following easily by induction on $k$. Since $c_{1}^{2}$ and $c_{2}$ generate $H^{4}(BU(n))$, we can find $\alpha ,\lambda \in \mbox{${\bf Q}$}$ such that $u_{2}=\alpha c_{1}^{2}+\alpha c_{2}$. If $\lambda =0$, then $u_{2}-\alpha c_{1}^{2}=0$; since $\{1,c_{1},c_{1}^{2},\ldots \}$ is an $H^{\ast }(BPU(n))$-basis of $H^{\ast}(BU(n))$, this implies that $u_{2}=0$, contradicting the injectivity of $(B\pi )^{\ast}$.~~$\Box$ \begin{lemma} Let $1\leq r \leq 2$, and fix a $C^{\infty }$ vector bundle $E$ of rank $n$ on the $r$-sphere $S^{r}$. Then, for each $k=2,\ldots ,n$, there exists a $C^{\infty }$ vector bundle $V$ on $S^{2k-r} \times S^{r}$ such that: \begin{enumerate} \item For each $t \in S^{2k -r}$, we have $V_{t} \simeq E$, where $V_{t} = \mbox{$i_{t}$} ^{\ast}V,~\mbox{$i_{t}$} :S^{r}\mbox{$\rightarrow$} S^{2k-r}\times S^{r}$ being the map $x\mapsto (t,x).$\\ \item For each $x\in S^{r},~\mbox{$j_{x}$} ^{\ast}V$ is trivial, where $\mbox{$j_{x}$} :S^{2k-r} \mbox{$\rightarrow$} S^{2k-r}\times S^{r}$ is the map $t\mapsto (t,x).$\\ \item $a_{k}(\mbox{${\bf P}$} (V)) \not= 0$. \end{enumerate} \end{lemma} {\em Proof}.~~ The existence of $V$ satisfying (1) and (2) with $c_{k}(V) \not= 0$ follows by standard arguments of K-theory. If $a_{k}(V)=0$, then Lemma 2.6 implies that $c_{k}(V)$ is a multiple of $c_{1}(V)^{k}$, which is zero since $c_{1}(V)$ is the pull-back of $c_{1}(E)$ by the second projection, a contradiction.~~$\Box$ \section{Cohomology of Some Classifying Spaces} In this section we describe, for later use, generators of the rational cohomology algebras of certain mapping spaces. The computations here are motivated by Section 2 of \cite{AB}, \cite{Dre}, and Section 5.1 of \cite{DK}. All CW-complexes here are assumed to be finite, and cohomologies are over $\mbox{${\bf Q}$}$. Let $M$ be a pointed CW-complex, and let $E$ be a complex vector bundle of rank $n$ over $M$. Fix a base point $b_{0} \in BU(n)$. Let $\mbox{${\cal G}$}(M,E)$ denote the complex gauge group of $E$, with the compact-open topology. Then as shown in \cite{AB} and \cite{Dre}, the space ${\rm Map}_{E}(M,BU(n))$ of all maps $f:M\mbox{$\rightarrow$} BU(n))$ such that $f^{\ast }EU(n) \simeq E$ is a classifying space for $\mbox{${\cal G}$} (M,E)$, so let us denote ${\rm Map}_{E}(M,BU(n))$ by $B\mbox{${\cal G}$}(M,E)$. The subspace ${\rm Map}_{E}^{\ast} (M,BU(n))$ consisting of all pointed maps is closed in $B\mbox{${\cal G}$} (M,E)$. We denote ${\rm Map}_{E}^{\ast}(M,BU(n))$ by $\mbox{${\cal B}$} (M,E)$. When there is no scope for confusion, we shall write just $\mbox{${\cal G}$} $ for $\mbox{${\cal G}$} (M,E)$, $\mbox{${\cal B}$} $ for $\mbox{${\cal B}$} (M,E)$, etc. Let $\varepsilon :B\mbox{${\cal G}$} \times M \mbox{$\rightarrow$} BU(n)$ denote the evaluation map, $(f,x) \mapsto f(x)$. Then the bundle $\varepsilon ^{\ast}EU(n)$ is called the {\it universal bundle} on $B\mbox{${\cal G}$} \times M$, and is denoted $\mbox{${\cal E}$} (M,E)$. We denote the restriction of $\mbox{${\cal E}$} (M,E)$ to $\mbox{${\cal B}$} \times M$ also by $\mbox{${\cal E}$} (M,E)$. If $\phi :M'\mbox{$\rightarrow$} M$ is a pointed map of CW-complexes, and if $E$ is a vector bundle over $M$, then, denoting $E'=\phi ^{\ast}E$, there is a natural map $\phi ^{\#}:B\mbox{${\cal G}$} (M,E)\mbox{$\rightarrow$} B\mbox{${\cal G}$} (M',E')$, $f\mapsto f\circ \phi$, carrying $\mbox{${\cal B}$} (M,E)$ into $\mbox{${\cal B}$} (M',E')$. These constructs have the following functorial property. \begin{propn} Suppose $\phi :M\mbox{$\rightarrow$} M'$ is a pointed map of CW- complexes, let $E$ be a complex vector bundle over $M$, and let $E'=\phi ^{\ast }E$. Write $\mbox{${\cal G}$} $ for $\mbox{${\cal G}$}(M,E)$, $\mbox{${\cal G}$} '$ for $\mbox{${\cal G}$} (M',E')$, etc. If $d(.)$ denotes $c_{k}(.)$ or $a_{k}(\mbox{${\bf P}$} (.))$ for some $k$, then the diagram \begin{center} \setlength{\unitlength}{0.5pt} \begin{picture}(288,210)(0,-20) \put(20,2){\makebox(0,0){$H^{*}(B\mbox{${\cal G}$} ')$}} \put(256,2){\makebox(0,0){$H^{*}(B\mbox{${\cal G}$} )$}} \put(256,162){\makebox(0,0){$H_{*}(M)$}} \put(20,162){\makebox(0,0){$H_{*}(M')$}} \put(74,2){\vector(1,0){128}} \put(256,144){\vector(0,-1){124}} \put(74,162){\vector(1,0){128}} \put(20,144){\vector(0,-1){124}} \put(138,-15){\makebox(0,0){$H^{*}(\phi ^{\# })$}} \put(304,82){\makebox(0,0){$\sigma (d(\mbox{${\cal E}$} ))$}} \put(138,179){\makebox(0,0){$\phi _{*}$}} \put(-28,82){\makebox(0,0){$\sigma (d(\mbox{${\cal E}$} '))$}} \end{picture} \end{center} \noindent commutes, where $\sigma (\alpha )z=\alpha /z$ is the slant product with $\alpha $. \end{propn} {\em Proof.}~~The commutativity of the diagram \begin{center} \setlength{\unitlength}{0.5pt} \begin{picture}(288,210)(0,-20) \put(20,2){\makebox(0,0){$B\mbox{${\cal G}$} \times M$}} \put(256,2){\makebox(0,0){$BU(n)$}} \put(256,162){\makebox(0,0){$B\mbox{${\cal G}$} ' \times M'$}} \put(20,162){\makebox(0,0){$B\mbox{${\cal G}$} \times M'$}} \put(74,2){\vector(1,0){128}} \put(256,144){\vector(0,-1){124}} \put(74,162){\vector(1,0){128}} \put(20,144){\vector(0,-1){124}} \put(138,-15){\makebox(0,0){$\varepsilon$}} \put(286,82){\makebox(0,0){$\varepsilon '$}} \put(138,179){\makebox(0,0){$\phi ^{\# } \times 1$}} \put(-18,82){\makebox(0,0){$1 \times \phi$}} \end{picture} \end{center} \noindent taken with the functorial properties of the slant product, leads directly to the desired conclusion. {}~~$\Box $ The space $\mbox{${\cal B}$} (M,E)$ has the following semi-universal property. \begin{propn} Suppose $E$ is a complex vector bundle on a pointed CW-complex, and $V$ a vector bundle on $T\times M$ such that: \begin{itemize} \item For each $t \in T$, we have $V_{t}\cong E$, where $V_{t}=\mbox{$i_{t}^{\ast}$} V$, $\mbox{$i_{t}$} :M\mbox{$\rightarrow$} T\times M$ being the map $x\mapsto (t,x)$.\\ \item If $x_{0}$ is the base point of $M$, and if $j_{x_{0}}:T\mbox{$\rightarrow$} T\times M$ is the map $t\mapsto (t,x_{0})$, then $j_{x_{0}}^{*}V$ is trivial. \end{itemize} \noindent Then, there exists a map $\psi :T\mbox{$\rightarrow$} \mbox{${\cal B}$} (M,E)$ such that $(\psi \times 1)^{\ast}\mbox{${\cal E}$} (M,E) \cong V$. \end{propn} {\em Proof}.~~General properties of classifying spaces give a map $\theta :(T\times M, T\times \{x_{0}\})\mbox{$\rightarrow$} (BU(n),b_{0})$ such that $\theta ^{*}EU(n)\cong V$, where $b_{0}$ is the base point in $BU(n)$. If we define $\psi :T\mbox{$\rightarrow$} \mbox{${\cal B}$} $ by $\psi (t)x=\theta (t,x)$, then $\varepsilon \circ (\psi \times 1)=\theta $, proving that $(\psi \times 1)^{*}\mbox{${\cal E}$} \cong V$. {}~~$\Box $ Let us study the space $\mbox{${\cal B}$} (M,E)$ a little more when $M=S^{r}$, $r=1,2$. Take $M=S^{1}$ first. \begin{propn} Let $E$ be a (necessarily trivial) complex vector bundle of rank $n$ on $S^{1}$. Then the cohomology algebra of $\mbox{${\cal B}$} (S^{1},E)$ is generated by $c_{1}(\mbox{${\cal E}$})/[S^{1}]$ and $a_{i}(\mbox{${\bf P}$}(\mbox{${\cal E}$}))/[S^{1}]$, $i=2,\ldots ,n$, where $[S^{1}]$ denotes the fundamental class of $S^{1}$. \end{propn} {\em Proof}.~~ Note first that $\mbox{${\cal B}$} (S^{1},E)$ equals $\Omega BU(n)$, which is homotopically equivalent to $U(n)$. Thus $H^{*}(\mbox{${\cal B}$} )$ is an exterior algebra on $n$ generators $\theta _{i} \in H^{2i-1}(\mbox{${\cal B}$} )$, $i=1,\ldots ,n$. Introduce the ad-hoc notation $\omega _{1}=c_{1}(\mbox{${\cal E}$})/[S^{1}]$ and $\omega _{i}=a_{i}(\mbox{${\bf P}$} (\mbox{${\cal E}$} ))/[S^{1}]$, $i=2,\ldots ,n$. Since $H^{1}(\mbox{${\cal B}$} )=\mbox{${\bf Q}$}.\theta _{1}$, we can write $\omega _{1}=\lambda \theta _{1}$ for some $\lambda \in \mbox{${\bf Q}$}$. Choose a vector bundle $V$ on $S^{1}\times S^{1}$ such that $c_{1}(V)\not= 0$. By Proposition 3.2, there exists a map $\psi :S^{1}\mbox{$\rightarrow$} \mbox{${\cal B}$} $ such that $(\psi \times 1)^{*}\mbox{${\cal E}$} = V$, hence $c_{1}(V)/[S^{1}] = \psi ^{*}\omega _{1}= \lambda \psi ^{*}\theta _{1}$. Since $c_{1}(V)\not= 0$, this implies that $\lambda \not=0$, and $\theta _{1}=\lambda ^{-1}\omega _{1}$. Now let $2\leq k \leq n$ and assume that for each $i=1,\ldots ,k-1$, $\theta _{i}$ is a polynomial in $\omega _{1},\ldots ,\omega _{i}$. Thus $H^{*}(\mbox{${\cal B}$})$ is generated by $\omega _{1},\ldots ,\omega _{k-1},\theta _{k},\ldots ,\theta _{n}$. Write \begin{eqnarray} \omega _{k} = P(\omega _{1},\ldots ,\omega _{k-1}) + \mu \theta _{k}, \end{eqnarray} where $P$ is some polynomial and $\mu \in \mbox{${\bf Q}$}$. By Lemma 2.7, there exists a vector bundle $V$ on $S^{2k-1}\times S^{1}$ satisfying the conditions of Proposition 3.2, such that $a_{k}(\mbox{${\bf P}$}(V))\not= 0$. Choose a map $\psi :S^{2k-1} \mbox{$\rightarrow$} \mbox{${\cal B}$} $ such that $(\psi \times 1)^{*}\mbox{${\cal E}$} =V$. We see then that $\psi ^{*}\omega _{k}=a_{k}(\mbox{${\bf P}$}(V))/[S^{1}]$; moreover, for $1\leq i\leq k-1$, $\psi ^{*}\omega _{i}\in H^{2i-1}(S^{2k-1})=0$. Therefore, pulling back equation (1) by $\psi $, we get $a_{k}(\mbox{${\bf P}$}(V))/[S^{1}]=\mu \psi ^{*}\theta _{k}$. Since $a_{k}(\mbox{${\bf P}$}(V))\not= 0$, we conclude that $\mu \not= 0$, and dividing by $\mu $, we express $\theta _{k}$ as a polynomial in $\omega _{1},\ldots ,\omega _{k}$. {}~~$\Box $ \begin{propn} If $E$ is a complex vector bundle of rank $n$ on $S^{2}$, then the cohomology algebra of $\mbox{${\cal B}$} (S^{2},E)$ is generated by $a_{i}(\mbox{${\bf P}$}(\mbox{${\cal E}$}))/[S^{2}]$, $2\leq i \leq n$. \end{propn} {\em Proof}.~~By definition, $\mbox{${\cal B}$} (S^{2},E)$ is a connected component of $\Omega ^{2}BU(n)$, which is homotopically equivalent to $\Omega U(n)$. Therefore (see \cite{PS}, p.68), $H^{*}(\mbox{${\cal B}$} )$ is generated by $n-1$ elements $\theta _{i} \in H^{2i}(\mbox{${\cal B}$} )$, $1\leq i\leq n-1$. If $\omega _{i}=a_{i+1}(\mbox{${\bf P}$}(\mbox{${\cal E}$}))/[S^{2}]$, $1\leq i \leq n-1$, then using Lemma 2.7 and Proposition 3.2, we see, as in the proof of Proposition 3.3, that each $\theta _{k}$ is a polynomial in $\omega _{1},\ldots ,\omega _{k}$. {}~~$\Box $ We now use these results to obtain generators for $\mbox{${\cal B}$} (X,E)$ when $X$ is a 2-manifold. \begin{propn} Let $X$ be a pointed, compact, connected and oriented surface, and let $E$ be a complex vector bundle of rank $n$ on $X$. Then the cohomology algebra of $\mbox{${\cal B}$} (X,E)$ is generated by the images of $$\sigma (c_{1}(\mbox{${\cal E}$})):H_{1}(X)\mbox{$\longrightarrow$} H^{1}(\mbox{${\cal B}$} )~~~~{\it and}$$ $$\sigma (a_{i}(\mbox{${\bf P}$}(\mbox{${\cal E}$}))):H_{r}(X)\mbox{$\rightarrow$} H^{2i-r}(\mbox{${\cal B}$} ) {}~~~(2\leq i\leq n,1\leq r\leq 2),$$ where $\sigma $ denotes, as usual, the slant product. \end{propn} {\em Proof}.~~Write $X$ as a cofibration $B\stackrel{i}{\hookrightarrow}X \stackrel{\pi}{\mbox{$\rightarrow$}} S^{2}$, where $B$ is a wedge of $2g$ circles, and assume, without loss of generality, that the centre of the wedge is the base point $x_{0}$ of $X$. Denote $G=i^{*}E$. Let $z_{0}=\pi (x_{0})$, and let $F$ be a vector bundle over $S^{2}$ such that $\pi ^{*}F\cong E$. Since the mapping functor transforms cofibrations into fibrations, we get a fibration $$\mbox{${\cal B}$} (S^{2},F) \stackrel{\pi ^{\#}}{\hookrightarrow}\mbox{${\cal B}$} (X,E) \stackrel{i^{\#}}{\mbox{$\rightarrow$}}\mbox{${\cal B}$} (B,G).$$ Proposition 3.1 implies that the pull-back of $a_{i}(\mbox{${\bf P}$}\mbox{${\cal E}$}(X,E))/[X]$ by $\pi ^{\#}$ equals $a_{i}(\mbox{${\bf P}$}\mbox{${\cal E}$}(S^{2},F))/\pi _{*}[X]$. Since $H_{2}(\pi )$ is an isomorphism, Proposition 3.4 now tells us that the Leray-Hirsch theorem applies. Thus the cohomology algebra of $\mbox{${\cal B}$} (X,E)$ is generated by the $a_{i}(\mbox{${\bf P}$}\mbox{${\cal E}$}(X,E))/[X]$ together with the image of $H^{*}(i^{\#})$. Let $B=\vee_{\alpha =1}^{2g}S_{\alpha }$, where each $S_{\alpha }$ is a circle; then $\gamma _{\alpha }=[S_{\alpha }]$ form a basis of $H_{1}(B)$. Since $\mbox{${\cal B}$} (B,G)=\prod _{\alpha =1}^{2g}\mbox{${\cal B}$} (S_{\alpha },G)$, and since $H_{1}(i)$ is an isomorphism, Propositions 3.1 and 3.3, applied as above, lead us to the finish. {}~~$\Box $ \begin{thm} Let $X$ be a 2-manifold as in Proposition 3.5. Then, the cohomology algebra of $\mbox{${\cal B}$}\mbox{${\cal G}$}$ is generated by the images of $$\sigma (c_{1}(\mbox{${\cal E}$})):H_{r}(X)\mbox{$\longrightarrow$} H^{2-r}(\b\mbox{${\cal G}$})~~(0\leq r\leq 1)~~{\it and}$$ $$\sigma (a_{i}(\mbox{${\bf P}$}(\mbox{${\cal E}$}))): H_{r}(X) \mbox{$\longrightarrow$} H^{2i-r}(\b\mbox{${\cal G}$})~~(0\leq r\leq 2,2\leq i \leq n).$$ \end{thm} {\em Proof}.~~ Let $x_{0}$ be the base point of $X$, and consider the fibration $\mbox{${\cal B}$} \hookrightarrow B\mbox{${\cal G}$} \stackrel{\varepsilon _{x_{0}}}{\mbox{$\rightarrow$}} BU(n)$, where $\varepsilon _{x_{0}}(f)=f(x_{0})$. Since the images of $H_{1}(X)$ and $H_{2}(X)$ under the various slant products restrict, by Proposition 3.5, to generators of $H^{*}(\mbox{${\cal B}$} )$, the Leray-Hirsch theorem applies. By Lemma 2.6, $c_{1}(EU(n))$ and $a_{i}(\mbox{${\bf P}$} EU(n))$, $2\leq i\leq n$, generate $H^{*}(BU(n))$. Since $\varepsilon _{x_{0}}^{*}d(EU(n))=d(\mbox{${\cal E}$})/[x_{0}]$ for $d=c_{1}(.)$ or $d=a_{i}(\mbox{${\bf P}$}(.))$, the result follows. {}~~$\Box $ \begin{rem} \begin{em} In view of Lemma 2.6, Theorem 2.6 implies the assertion concerning rational cohomology in Proposition 2.20 of \cite{AB}. Actually Lemma 2.6 may give one the impression that the above theorem can be deduced from Proposition 2.20 of \cite{AB}, but this impression is hard to substantiate; the difficulty is due to the fact that the slant product does not behave well with the cup product. \end{em} \end{rem} We now apply the above results in the context of parabolic bundles over a curve. The standard reference for parabolic bundles is Mehta and Seshadri \cite{MS}, and we refer to Nitsure \cite{Nit} for the gauge theoretic aspects of parabolic bundles. Let $X$ be a compact, connected and oriented surface, fix a positive integer $n$, and let $\Delta $ be a parabolic datum of rank $n$ on $X$. Thus, $\Delta $ consists of: \begin{itemize} \item a finite subset $J$ of $X$; and \\ \item for each $x\in J$, a sequence $(n_{x,1},\ldots ,n_{x,k_{x}})$ of positive integers such that $\sum _{i=1}^{k_{x}} n_{x,i}=n$, and a sequence $0\leq \alpha _{x,1}<\ldots <\alpha _{x,k_{x}}<1$ of real numbers. \end{itemize} \noindent Fix a quasi-parabolic vector bundle of rank $n$ and type $\Delta $ on $X$, and let $\mbox{${\cal G}_{\rm par}$} $ denote the subgroup of the gauge group $\mbox{${\cal G}$} $ of $E$, consisting of parabolic gauge transformations. Let $\mbox{${\cal E}$} $ denote the universal bundle on $B\mbox{${\cal G}$} \times X$, and for each $x\in J$, let $\mbox{${\cal F}$} _{x}$ denote the bundle of flags of type $\Delta $ in $\mbox{$j_{x}^{\ast}$} \mbox{${\cal E}$}$, where $\mbox{$j_{x}$} :B\mbox{${\cal G}$} \mbox{$\rightarrow$} B\mbox{${\cal G}$} \times X$ is the map $f\mapsto (f,x)$. Define $\phi :\mbox{${\cal F}$} \mbox{$\rightarrow$} B\mbox{${\cal G}$} $ to be the fibre product of $\mbox{${\cal F}$} _{x}~~(x\in J)$ over $B\mbox{${\cal G}$}$. With these preparations out of the way, we can identify $B\mbox{${\cal G}_{\rm par}$} $. \begin{lemma} The space $\mbox{${\cal F}$}$ is a classifying space for $\mbox{${\cal G}_{\rm par}$}$. \end{lemma} {\em Proof}.~~General considerations give $B\mbox{${\cal G}_{\rm par}$} $ as $E\mbox{${\cal G}$} /\mbox{${\cal G}_{\rm par}$}$; further, by \cite{AB}, $E\mbox{${\cal G}$}$ is the space of all maps $\tilde{f}:E\mbox{$\rightarrow$} EU(n)$ which carry each fibre of $E$ isomorphically to some fibre of $EU(n).$ Note that such an $\tilde{f}$ defines an element $f\in B\mbox{${\cal G}$}$ such that $\tilde{f}$ is an isomorphism of $E$ with $f^{*}EU(n)$. On the other hand, the fibre of $\mbox{${\cal F}$}$ at $f\in B\mbox{${\cal G}$}$ is the product of certain flag manifolds of $EU(n)_{f(x)}~(x\in J)$. Define $\alpha :E\mbox{${\cal G}$} \mbox{$\rightarrow$} \mbox{${\cal F}$}$ by $\alpha (\tilde{f})=(\tilde{f} (F^{i}E_{x}) \subset EU(n)_{f(x)})$, where $f \in B\mbox{${\cal G}$}$ is induced by $\tilde{f}$, and $F^{i}E_{x}$ are given by the quasi-parabolic structure of $E$. By the definition of $\mbox{${\cal G}_{\rm par}$}$, $\alpha $ factors through a map $\tilde{\alpha }:E\mbox{${\cal G}$} /\mbox{${\cal G}_{\rm par}$} \mbox{$\rightarrow$} \mbox{${\cal F}$}$, which is easily seen to be a homeomorphism.~~$\Box$ So, denote $\mbox{${\cal F}$}$ by $B\mbox{${\cal G}_{\rm par}$}$. The pull-back $\mbox{${\cal E}_{\rm par}$}$ of $\mbox{${\cal E}$}$ by $\phi \times 1 :B\mbox{${\cal G}_{\rm par}$} \times X \mbox{$\rightarrow$} B\mbox{${\cal G}$} \times X$ is a family of quasi-parabolic bundles, i.e., for each $x\in J$, there is a decreasing flag $\exi{1}\supset \exi{2}\supset \ldots $ of type $\Delta $ in $\mbox{$j_{x}^{\ast}$} \mbox{${\cal E}_{\rm par}$}$, where, as usual, $\mbox{$j_{x}$} (t)=(t,x)$ for $t\in B\mbox{${\cal G}_{\rm par}$}$. We call $\mbox{${\cal E}_{\rm par}$} $ the universal bundle on $B\mbox{${\cal G}_{\rm par}$} \times X$. \begin{thm} With notation as above, the cohomology algebra of $B\mbox{${\cal G}_{\rm par}$} $ is generated by $c_{j}(\mbox{${\cal H}{\it om}$} (\exi{i},\exi{i-1}))~(x\in J)$ and the images of $$\sigma (c_{1}(\mbox{${\cal E}_{\rm par}$} )):H_{r}(X)\mbox{$\rightarrow$} H^{2-r}(B\mbox{${\cal G}_{\rm par}$})~~(0\leq r\leq 1)~~{\it and}$$ $$\sigma (a_{i}(\mbox{${\bf P}$}\mbox{${\cal E}_{\rm par}$})):H_{r}(X)\mbox{$\rightarrow$} H^{2i-r}(B\mbox{${\cal G}_{\rm par}$} )~~(0\leq r\leq 2,~2\leq i\leq n).$$ \end{thm} {\em Proof}.~~As already remarked, the fibre of $\phi :B\mbox{${\cal G}_{\rm par}$} \mbox{$\rightarrow$} B\mbox{${\cal G}$}$ over $f$ is a product of flag manifolds $M_{f}^{x}$ of the vector spaces $\mbox{${\cal E}$} _{(f,x)}~(x\in J)$. Each of these flag manifolds carries a tautological flag $F_{f}^{x,i}$ of vector bundles, and the flag $\exi{i}$ on $B\mbox{${\cal G}_{\rm par}$}$, in fact, restricts to the tautological flag on each factor $M_{f}^{x}$ of the fibre. Now, in general, if $M$ is a flag manifold, and if $F^{1}\supset F^{2}\supset \ldots $ is its tautological flag of vector bundles, then $c_{j}(\mbox{${\cal H}{\it om}$} (F^{i},F^{i-1}))$ generate the cohomology algebra of $M$. In our context, this fact implies that the Leray-Hirsch theorem holds for the fibration $\phi $, and the result follows from Theorem 3.6. {}~~$\Box $ \section{Proofs} This section brings together the results of the previous sections to prove Theorems 1.4 and 1.5. The notation is the same as before. Let $X$ be a compact Riemann surface, and let $n$ and $d$ be integers with $n$ positive, and let $\Delta $ be a parabolic datum of rank $n$ on $X$. {\em Suppose that Assumptions 1.1 and 1.2 are satisfied.} Let $E$ be a $C^{\infty}$ quasi-parabolic bundle on $X$ of rank $n$, degree $d$ and parabolic type $\Delta $. Let ${\cal A}$ be the space of holomorphic structures in $E$, and $\mbox{${\cal A}^{s}_{\rm par}$} $ the open subset of ${\cal A} $ consisting of holomorphic structures which are parabolic stable with respect to the datum $\Delta $. Let $\mbox{${\cal G}$}$ denote the gauge group of $E$, and denote $\mbox{$\bar{\cal G}$} = \mbox{${\cal G}$} /{\bf C}^{*}$ and $\mbox{$\bar{\cal G}_{\rm par}$} = \mbox{${\cal G}_{\rm par}$} /{\bf C}^{*}$, where ${\bf C}^{*}$ is the constant scalar subgroup of $\mbox{${\cal G}$}$. There is a natural action of $\mbox{$\bar{\cal G}$} $ on ${\cal A}$, which induces a free action of $\mbox{$\bar{\cal G}_{\rm par}$}$ on $\mbox{${\cal A}^{s}_{\rm par}$}$, and there is a canonical homeomorphism of $\mbox{${\cal A}^{s}_{\rm par}$} /\mbox{$\bar{\cal G}_{\rm par}$}$ with $\mbox{${\cal U}_{X}(n,d,\Delta )$}$, hence we will identify them with each other from now on. \begin{nota} \begin{em} If $G$ is a topological group, and $T$ is a $G$-space, then $T(G)$ denotes the homotopy quotient $EG\times _{G}T$. (We write $T(G)$ instead of the standard notation $T_{G}$ for reasons of convenience.) \end{em} \end{nota} \begin{rem} \begin{em} Note that for any $G$-space $T$, there are two canonical maps $T(G) \mbox{$\rightarrow$} BG$ and $T(G)\mbox{$\rightarrow$} T/G$. The first map is a fibre bundle over $BG$ with fibre $T$, and is a homotopy equivalence if $T$ is contractible. The second map is a homotopy equivalence of $T(G)$ with $T/G$ if $T$ is a free $G$-space. \end{em} \end{rem} Consider the diagram \begin{center} \def\normalbaselines{\baselineskip17pt \lineskip3pt \lineskiplimit3pt } \def\mapright#1{\smash{ \mathop{\longrightarrow}\limits^{#1}}} \def\mapdown#1{\Big\downarrow \rlap{$\vcenter{\hbox{$\scriptstyle#1$}}$}} $$\matrix{1&\mapright{}&{\bf C}^{*}&\mapright{}&{\mbox{${\cal G}_{\rm par}$}}&\mapright{f}&{\mbox{$\bar{\cal G}_{\rm par}$}}&\mapright{}&1\cr &&\mapdown{}&& \mapdown{}&&\mapdown{}&&\cr 1&\mapright{}&{\bf C}^{*}&\mapright{}& {\mbox{${\cal G}$}}&\mapright{f'}&\mbox{$\bar{\cal G}$} &\mapright{}& 1\cr} $$ \end{center} \noindent where the vertical maps are the canonical inclusions, and $f$ and $f'$ denote the canonical projections. This induces a diagram \begin{center} \setlength{\unitlength}{0.5pt} \begin{picture}(288,210)(0,-20) \put(20,2){\makebox(0,0){$B\mbox{${\cal G}$} $}} \put(256,2){\makebox(0,0){$B\mbox{$\bar{\cal G}$} $}} \put(256,162){\makebox(0,0){$B\mbox{$\bar{\cal G}_{\rm par}$} $}} \put(20,162){\makebox(0,0){$B\mbox{${\cal G}_{\rm par}$} $}} \put(74,2){\vector(1,0){146}} \put(256,144){\vector(0,-1){124}} \put(74,162){\vector(1,0){146}} \put(20,144){\vector(0,-1){124}} \put(138,-15){\makebox(0,0){$\pi '$}} \put(138,179){\makebox(0,0){$\pi $}} \end{picture} \end{center} \noindent of fibrations; the fibres of $\pi $ and $\pi '$ are homeomorphic to $BU(1)$. Since ${\cal A} $ is contractible, by Remark 4.2, the natural map\\ $\phi : \mbox{${\cal A}$} (\mbox{${\cal G}_{\rm par}$})\mbox{$\rightarrow$} B\mbox{${\cal G}_{\rm par}$}$ is a homotopy equivalence. Let $\mbox{${\cal E}_{\rm par}$} $ denote the universal bundle on $B\mbox{${\cal G}_{\rm par}$} \times X$, and let $V$ denote the bundle on $\mbox{${\cal A}^{s}_{\rm par}$}(\mbox{${\cal G}_{\rm par}$}) \times X$ obtained by pulling back $\mbox{${\cal E}_{\rm par}$}$ via the composition $$\mbox{${\cal A}^{s}_{\rm par}$} (\mbox{${\cal G}_{\rm par}$}) \times X \stackrel{\lambda \times 1} {\hookrightarrow}\mbox{${\cal A}$}(\mbox{${\cal G}_{\rm par}$})\times X \stackrel{\phi \times 1}{\mbox{$\rightarrow$}}B\mbox{${\cal G}_{\rm par}$} \times X,$$ where $\lambda :\mbox{${\cal A}^{s}_{\rm par}$} (\mbox{${\cal G}_{\rm par}$})\hookrightarrow \mbox{${\cal A}$}(\mbox{${\cal G}_{\rm par}$})$ denotes the inclusion map. \begin{rem} \begin{em} The $\mbox{${\cal G}_{\rm par}$}$-equivariant perfectness of a certain stratification (see Nitsure \cite{Nit}) implies that the inclusion $\lambda $ induces a surjection in rational cohomology. Thus, by Theorem 3.9, the Chern classes $c_{j}(\mbox{${\cal H}{\it om}$} (V^{x,i}, V^{x,i-1}))$ and the slant products $c_{1}(V)/z~(z\in H_{1}(X))$, $c_{1}(V)/[x_{0}]$ ($x_{0}$ a fixed base point in $X$) and $a_{i}(\mbox{${\bf P}$}(V))/[y]~~(y\in H_{r}(X), ~0\leq r\leq 2,~2\leq i \leq n)$ generate the algebra $H^{\ast }(\mbox{${\cal A}^{s}_{\rm par}$} (\mbox{${\cal G}_{\rm par}$} ))$. \end{em} \end{rem} \noindent {\bf Proof of Theorem 1.4:} ~~Let notation be as above, and as in Theorem 1.4. Since the action of $\mbox{$\bar{\cal G}_{\rm par}$}$ on $\mbox{${\cal A}^{s}_{\rm par}$} $ is free, the canonical map $\psi :\mbox{${\cal A}^{s}_{\rm par}$} (\mbox{$\bar{\cal G}_{\rm par}$})\mbox{$\rightarrow$} \mbox{${\cal U}_{X}(n,d,\Delta )$} $ is a homotopy equivalence. Let $V'$ denote the bundle on $\mbox{${\cal A}^{s}_{\rm par}$}(\mbox{${\cal G}_{\rm par}$})\times X$ obtained by pulling back the universal bundle\\ $U\mbox{$\rightarrow$} \mbox{${\cal U}_{X}(n,d,\Delta )$} \times X$ by the composition $$ \mbox{${\cal A}^{s}_{\rm par}$} (\mbox{${\cal G}_{\rm par}$}) \times X \stackrel{\pi \times 1}{\mbox{$\longrightarrow$}} \mbox{${\cal A}^{s}_{\rm par}$} (\mbox{$\bar{\cal G}_{\rm par}$})\times X \stackrel{\psi \times 1}{\mbox{$\longrightarrow$}}\mbox{${\cal U}_{X}(n,d,\Delta )$} \times X$$ where $\pi $ is induced by $\pi :B\mbox{${\cal G}_{\rm par}$} \mbox{$\rightarrow$} B\mbox{$\bar{\cal G}_{\rm par}$}$ above. Recall now that we have constructed above another family $V$ on $\mbox{${\cal A}^{s}_{\rm par}$} (\mbox{${\cal G}_{\rm par}$} )\times X$ using $B\mbox{${\cal G}_{\rm par}$}$. Now $V$ and $V'$ are families of parabolic stable bundles parametrized by $\mbox{${\cal A}^{s}_{\rm par}$} (\mbox{${\cal G}_{\rm par}$} )$ such that for each $t \in \mbox{${\cal A}^{s}_{\rm par}$} (\mbox{${\cal G}_{\rm par}$})$, $V_{t}\cong V'_{t}$. Therefore, there exists a line bundle $\xi $ on $\mbox{${\cal A}^{s}_{\rm par}$} (\mbox{${\cal G}_{\rm par}$})$ such that $V'\cong V\otimes p^{*}\xi $, where $p:\mbox{${\cal A}^{s}_{\rm par}$} (\mbox{${\cal G}_{\rm par}$})\times X \mbox{$\rightarrow$} \mbox{${\cal A}^{s}_{\rm par}$} (\mbox{${\cal G}_{\rm par}$})$ is the canonical projection. This implies that $\mbox{${\bf P}$}(V)\cong \mbox{${\bf P}$}(V')$, $\mbox{${\cal H}{\it om}$} (V^{x,i},V^{x,i-1}) \cong \mbox{${\cal H}{\it om}$} ((V')^{x,i},(V')^{x,i-1})$, and $c_{1}(V)/z = c_{1}(V')/z$ for all $z\in H_{1}(X)$. Thus, if $W=(\psi \times 1)^{*}U$, then $\mbox{${\bf P}$}(V)=(\pi \times 1)^{*}\mbox{${\bf P}$}(W)$, $\mbox{${\cal H}{\it om}$} (V^{x,i},V^{x,i-1})=\pi ^{*}(\mbox{${\cal H}{\it om}$} (W^{x,i}, W^{x,i-1}))$, and $c_{1}(V)/z = \pi ^{*}(c_{1}(W)/z)$ for all $z\in H_{1}(X)$. Further, we easily see that under the inclusion $BU(1) \hookrightarrow \mbox{${\cal A}^{s}_{\rm par}$} (\mbox{${\cal G}$})$ as a fibre of $\pi ,~c_{1}(V)/[x_{0}]$ restricts to a generator of $H^{2}(BU(1))$. Now in general, if $\pi :E \mbox{$\rightarrow$} B$ is a fibration with fibre $F$ such that: (a) the algebra $H^{*}(E)$ is generated by certain classes $\alpha ,\beta _{1},\ldots ,\beta _{k}$; (b) the classes $\beta _{i}$ are pull-backs of certain classes $\theta _{i} \in H^{\ast}(B)$ by $\pi $; and (c) $H^{\*}(F)$ is a polynomial algebra on $\alpha _{F}$, where $\alpha _{F}$ denotes the restriction of $\alpha $ to $F$; then, $H^{*}(B)$ is generated by $\theta _{1},\ldots ,\theta _{k}$. This fact applies in our situation because of the above observations and because of Remark 4.3, and implies that the Chern classes $c_{j}(\mbox{${\cal H}{\it om}$} (W^{x,i},W^{x,i-1}))$ and the slant products $c_{1}(W)/z~(z\in H_{1}(X))$ and $a_{i}(\mbox{${\bf P}$}(W))/y~(y\in H_{r}(X),~0\leq r\leq 2,~2\leq i\leq n)$ generate $H^{*}(\mbox{${\cal A}^{s}_{\rm par}$}(\mbox{$\bar{\cal G}_{\rm par}$}))$. Since $\psi :\mbox{${\cal A}^{s}_{\rm par}$}(\mbox{$\bar{\cal G}_{\rm par}$})\mbox{$\rightarrow$} \mbox{${\cal U}_{X}(n,d,\Delta )$}$ is a homotopy equivalence, we are done. ~~$\Box $ \noindent {\bf Proof of Theorem 1.5:}~~Suppose $U$ is a universal bundle on $\mbox{${\cal S}{\cal U}_{X}(n,L,\Delta )$} \times X$. Consider the right action of the $n$-torsion subgroup $\Gamma _{X}(n)$ of the Jacobian $J_{X}$ on $\mbox{${\cal S}{\cal U}_{X}(n,L,\Delta )$} \times X$, defined by $(E,\alpha ).\zeta = (E\otimes \zeta ,\zeta ^{-1}\otimes \alpha )$, where $E\in \mbox{${\cal S}{\cal U}_{X}(n,L,\Delta )$} ,~\alpha \in J_{X}$ and $\zeta \in \Gamma _{X}(n)$. Then the map $$\pi :\mbox{${\cal S}{\cal U}_{X}(n,L,\Delta )$} \times J_{X} \mbox{$\longrightarrow$} \mbox{${\cal U}_{X}(n,d,\Delta )$} , ~~~(E,\alpha) \mapsto E \otimes \alpha $$ is a principal $\Gamma _{X}(n)$-bundle, i.e., a Galois covering with Galois group $\Gamma _{X}(n)$. On the other hand, the Poincar\'{e} polynomials of $\mbox{${\cal U}_{X}(n,d,\Delta )$}$ and\\ $\mbox{${\cal S}{\cal U}_{X}(n,L,\Delta )$} \times J_{X}$ are equal (see Nitsure \cite{Nit}, Remark 3.11). Since $\Gamma _{X}(n)$ is a finite group, this means that the action of $\Gamma _{X}(n)$ on $\mbox{${\cal S}{\cal U}_{X}(n,L,\Delta )$} \times J_{X}$ induces a trivial action on the cohomology of $\mbox{${\cal S}{\cal U}_{X}(n,L,\Delta )$} \times J_{X}$, or equivalently that the map $\pi $ induces an isomorphism in rational cohomology. (In the case of usual vector bundles, the triviality of the action of $\mbox{$\Gamma _{X}(n)$}$ on the rational cohomology of ${\cal SU}_{X}(n,L)$, where $n$ and the degree of $L$ are coprime, is a theorem of Harder and Narasimhan \cite{HN}, who proved it using arithmetic techniques. It was reproved by Atiyah and Bott \cite{AB} using gauge theory. The methods of Nitsure \cite{Nit} generalize the approach of Atiyah and Bott \cite{AB} to parabolic bundles.) Now, if $$i:\mbox{${\cal S}{\cal U}_{X}(n,L,\Delta )$} \mbox{$\longrightarrow$} \mbox{${\cal S}{\cal U}_{X}(n,L,\Delta )$} \times J_{X}$$ denotes the map $E\mapsto (E,{\cal O}_{X})$, and if $ j :\mbox{${\cal S}{\cal U}_{X}(n,L,\Delta )$} \mbox{$\rightarrow$} \mbox{${\cal U}_{X}(n,d,\Delta )$} $ denotes the inclusion, then $j=\pi \circ i$. Since $\pi ^{*}$ is an isomorphism and $i^{*}$ is surjective, we see that $$j^{*}:H^{*}(\mbox{${\cal U}_{X}(n,d,\Delta )$})\mbox{$\longrightarrow$} H^{*}(\mbox{${\cal S}{\cal U}_{X}(n,L,\Delta )$} )$$ is surjective. Now, let $\tilde{V}$ be an arbitrary universal bundle on $\mbox{${\cal U}_{X}(n,d,\Delta )$} \times X$, and denote the restriction of $\tilde{V}$ to $\mbox{${\cal S}{\cal U}_{X}(n,L,\Delta )$} \times X$ by $V$. Then, Theorem 1.4 applied to $\tilde{V}$, and the surjectivity of $j^{*}$ imply that the Chern classes $c_{j}(\mbox{${\cal H}{\it om}$} (V^{x,i},V^{x,i-1}))$ and the slant products $c_{1}(V)/z~(z\in H_{1}(X))$ and $a_{i}(\mbox{${\bf P}$}(V))/y~(y\in H_{r}(X),~0\leq r\leq 2,~2\leq i\leq n)$ generate $H^{*}(\mbox{${\cal S}{\cal U}_{X}(n,L,\Delta )$})$. But $\mbox{${\cal S}{\cal U}_{X}(n,L,\Delta )$} $ is simply connected, so the classes $c_{1}(V)/z$ ($z \in H_{1}(X)$) are all zero. Finally, since $U$ and $V$ are both universal bundles on \\ $\mbox{${\cal S}{\cal U}_{X}(n,L,\Delta )$} \times X$, they differ by a line bundle coming from $\mbox{${\cal S}{\cal U}_{X}(n,L,\Delta )$}$, hence\\ $\mbox{${\cal H}{\it om}$} (V^{x,i},V^{x,i-1}) \cong \mbox{${\cal H}{\it om}$} (U^{x,i},U^{x,i-1})$ and $\mbox{${\bf P}$}(V)\cong \mbox{${\bf P}$}(U)$.~~~~~$\Box $ \noindent {\bf Proof of Corollary 1.6:}~~If $U^{x}=\mbox{$j_{x}^{\ast}$} U~(x\in J)$, then the exact sequence $$0\longrightarrow S^{x} \longrightarrow U^{x} \longrightarrow Q^{x} \longrightarrow 0 $$ implies that $U^{x}\cong S^{x} \oplus Q^{x}$ topologically. Since $S^{x}$ is either zero or a line bundle, $S_{x}^{*} \otimes S_{x}$ is either zero or a trivial line bundle, and hence \\ $c_{j}(\mbox{${\cal H}{\it om}$} (S^{x},S^{x}))=c_{j}(\mbox{${\cal H}{\it om}$} (S^{x},Q^{x}))$. Finally, Example 2.5 (1) implies that \\ $a_{2}(\mbox{${\bf P}$}(U))=c_{2}(\mbox{${\cal E}{\it nd}$} U)$.~~~$\Box $ \noindent {\bf Proof of Proposition 1.7:}~~As in Atiyah and Bott (see \cite{AB}, Section 9), the crux of the proof consists in finding a holomorphic $\mbox{${\cal G}_{\rm par}$} $-line bundle $\xi $ on $\mbox{${\cal A}^{s}_{\rm par}$} $ on which ${\bf C}^{*} \subset \mbox{${\cal G}_{\rm par}$} $ acts via the identity homomorphism ${\bf C}^{*}\mbox{$\rightarrow$} {\bf C}^{*},~t \mapsto t$. Let $U =\mbox{${\cal A}^{s}_{\rm par}$} \times E$ and $U^{x,i}=\mbox{${\cal A}^{s}_{\rm par}$} \times F^{i}E_{x}~(x\in J)$, where $E$ is the fixed $C^{\infty }$ quasi-parabolic bundle under consideration. If we let $\mbox{${\cal G}_{\rm par}$} $ act trivially on $X$, then $U$ and $U^{x,i}$ are naturally $\mbox{${\cal G}_{\rm par}$}$-vector bundles on which ${\bf C}^{*}$ acts by the identity homomorphism. Fix a line bundle ${\cal O}_{X}(1)$ of degree 1 on $X$, and for each $k\in {\bf Z}$, let $U(k)=U\otimes q^{*}{\cal O}_{X}(k)$, where $q:\mbox{${\cal A}^{s}_{\rm par}$} \times X \mbox{$\rightarrow$} X$ denotes the canonical projection. Denote by Det $U(k)$ the determinant line bundle of $U(k)$ in the sense of Quillen \cite{Q}. Then Det $U(k)$ is a holomorphic $\mbox{${\cal G}_{\rm par}$}$-line bundle over $\mbox{${\cal A}^{s}_{\rm par}$} $ on which ${\bf C}^{*}$ acts by the homomorphism $t\mapsto t^{N+kn}$, where $N=d+n(1-g),~g$ being the genus of $X$. If $(n,d)=1$, let $a,b \in {\bf Z}$ be such that $an+bN =1$, and take $$\xi = ({\rm Det}~U(1))^{a}\otimes ({\rm Det}~U)^{b-a};$$ then ${\bf C}^{*}$ acts by the identity homomorphism on $\xi $. If $\sum _{i=j}^{k_{x}} n_{x,j}$ and $n$ are coprime for some $x$ and $j$, then the rank $m$ of $U^{x,j}$ and $n$ are coprime; let $a,b \in {\bf Z}$ be such that $am+bn =1$, and take $$\xi = ({\rm det}~U^{x,j})^{a} \otimes ({\rm Det}~U)^{-b}\otimes ({\rm Det}~U(1))^{b};$$ then $\xi $ has the required property. Lastly, if $\sum _{i=j}^{k_{x}} n_{x,i}$ and $n+d$ are coprime for some $x$ and $j$, let $a,b \in {\bf Z}$ be such that $am +b(d+n)=1$, where $m$ is the rank of $U^{x,i}$; then $$\xi =({\rm det}~U^{x,j})^{a}\otimes ({\rm Det}~U(1))^{b} \otimes ({\rm det}~U^{x})^{b(g-1)}$$ will do, where $U^{x}=\mbox{$j_{x}^{\ast}$} U$.~~~~~$\Box $ \noindent {\em Acknowledgement}.~~ This paper is the outcome of a suggestion of Professor M.S.Narasimhan. We would like to thank him for his interest and encouragement.

9208

alg-geom/9208004

en

https://arxiv.org/abs/alg-geom/9208004

alg-geom/9208004

Rick Miranda

Rick Miranda

Torsion Sections of Semistable Elliptic Surfaces

16 pages, AmsLatex 1.0

Let S be a torsion section of an elliptic surface with only I_n fibers. This article addresses the question: which components of singular fibers can S pass through? We give necessary criteria for the "component numbers", and show an equidistribution result for torsion sections of prime order.

alg-geom

\section{Introduction} In this article we would like to address the following situation. Let $f:X \to C$ be an elliptic surface with section $S_0$. We further assume that $X$ is relatively minimal and smooth, and that all singular fibers are semistable, that is, are cycles of ${\Bbb P}^1$'s (type ``$I_m$'' in Kodaira's notation, see \cite{kodaira}). Given a section $S$ of $f$, it will meet one and only one component in every singular fiber. The fundamental aim of this paper is to try to say something specific about which components are being hit, in the case that $S$ is a torsion section (i.e., of finite order in the Mordell-Weil group of sections of $f$). Most work on the Mordell-Weil group for elliptic surfaces focuses on properties of the group itself, (for example its rank, etc.) and not on properties of the elements of the group. See for example \cite{shioda} and \cite{cox-zucker} for general properties, and papers such as \cite{schwartz} and \cite{stiller} for computations in specific cases. Although the methods used in this article are elementary, they are suprisingly powerful, enabling us to obtain rather detailed information about the possibilities for component intersection for a torsion section. To be specific, suppose that the $j^{th}$ singular fiber is a cycle of length $m_j$, and that the components are numbered cyclically around the cycle from $0$ to $m_j-1$. Suppose that in this $j^{th}$ fiber, a section $S$ of order exactly $n$ hits the $k_j^{th}$ component. The main restrictions on these ``component'' numbers $k_j$ are the following: \medskip \noindent {\bf The Quadratic Relation} (Proposition \ref{formula1}): \newline \[ \sum_j k_j(1 - k_j/m_j) = 2 \chi({\cal O}_X). \] \noindent {\bf The Component Number Sum Relation} (Corollaries \ref{formula2} and \ref{formula4}): \newline If the component numbers satisfy $k_j \leq m_j/2$, (which can always be arranged by the suitable ``reorienting'' of the cycle), then \[ \sum_{j=1}^s k_j(S) = \begin{cases} 4\chi({\cal O}_X) & \text{ if } n=2 \\ 3\chi({\cal O}_X) & \text{ if } n \geq 3. \end{cases} \] These relations hold for the general torsion section, but actually we have more to say in the case of a torsion section $S$ of prime order $p$. In this case we are able to calculate exactly the distribution of the ``distances'' between the components hit by $S$ and the components hit by the zero-section $S_0$, in the following sense. If a fiber has $m$ components, indexed from $0$ to $m-1$ cyclically, then $S$, having order $p$, meets either the component indexed by $0$ (which we take to be the component meeting $S_0$) or one of the components indexed by $im/p$ for some $i$. In other words, a section of order $p$ must meet either $1/p$ of the way around the cycle, or $2/p$ of the way around, etc. Denote by $M_{i,p}(S)$ the fraction of the total Euler number $e=\sum_j m_j$ contributed by those fibers in which $S$ meets $i/p$ of the way around the cycle, in either direction. Our main result is a computation of these numbers, which turn out to be independent of $S$, and surprisingly, independent of $i$ (as long as $i\neq 0$): \medskip \noindent {\bf The Distribution Numbers} (Theorem \ref{Minvalues}): \newline Let $S$ be a torsion section of odd prime order $p$. Then \[ M_{i,p}(S) = 2p/(p^2-1) \; \text{ if }\; 1 \leq i \leq (p-1)/2, \] and \[ M_{0,p}(S) = 1/(p+1). \] Thus we obtain an ``equidistribution'' of these distances: the total sum of those $m_j$'s contributed from fibers of type $I_{m_j}$ in which $S$ meets $i/p$ of the way around the cycle is constant, independent of $i$. Thus a torsion section of prime order, as it travels across the elliptic surface, is forced to be completely ``fair'' in its choit may hit, in the above sense. The above results generalize various statements concerning the component numbers $\{k_j\}$ for a torsion section which appeared in \cite{miranda-persson1}, \cite{miranda-persson2}, \cite{miranda-persson3}, and \cite{miranda}. The results in this paper could be proved by appealing to the appropriate elliptic modular surface (for those primes where such exists) and its universal properties. The equidistribution property is invariant under base change, so if it is true for the (universal) modular surface, it will be true in general. This is a computation involving the modular group, which could be made once and for all. However it is my intention to show that these properties are somehow more elementary than the theory of modular surfaces, and follow from straightforward considerations involving only basic facts concerning the Mordell-Weil group and a little intersection theory of surfaces. In a further paper we hope to discuss the case of torsion of non-prime order more fully. I would like to thank Igor Dolgachev, Jeanne Duflot, Ulf Persson, Miles Reid, and Peter Stiller for some useful conversations. I would also like to thank the organizers of the Conference in L'Aquila for their kind invitation to allow me to speak on this subject. I would especially like to thank Prof. E. Laura Livorni for her wonderful efforts to make the conference such a success, and Prof. Ciro Ciliberto for his excellent hospitality in Rome while this work was completed. \section{Basic Facts} We work over the field of complex numbers. Let $C$ be a smooth curve, and let $f:X \to C$ be a smooth semistable elliptic surface over $C$. We suppose that $f$ has a chosen section $S_0$ which we treat as the zero for the group law on the sections of $f$. Let $g$ be the genus of $C$, and let $\chi = \chi({\cal O}_X)$ be the holomorphic Euler characteristic of $X$. If $e$ is the topological Euler number of $X$, then $e = 12 \chi$. We suppose that the map $f$ has $s$ singular fibers $F_1,\dots,F_s$, each semistable, i.e., each a cycle of rational curves (of type ``$I_m$'' in Kodaira's notation \cite{kodaira}). Indeed, let us say that the fiber $F_j$ is of type $I_{m_j}$. Therefore \[ e = 12 \chi = \sum_{j=1}^s m_j. \] Choose an ``orientation'' of each fiber $F_j$ and write the $m_j$ components of $F_j$ as \[ C^{(j)}_0,C^{(j)}_1,\dots,C^{(j)}_{m_j-1}, \] where the zero section $S_0$ meets only $C^{(j)}_0$ and for each $k$, $C^{(j)}_k$ meets only $C^{(j)}_{k\pm 1} \mod m_j$. If $m_j = 1$, then $F_j = C^{(j)}_0$ is a nodal rational curve, of self-intersection $0$. If $m_j \geq 2$, then each $C^{(j)}_k$ is a smooth rational curve with self-intersection $-2$. If we denote by $\operatorname{NS}(X)$ the Neron-Severi group of the elliptic surface $X$, we can consider the sublattice $L \subseteq \operatorname{NS}(X)$ generated by the zero-section $S_0$ and the components $C^{(j)}_k$ of the singular fibers. The class of the fiber $F$ is of course in this sublattice $L$; moreover, the only relations among these classes are that \begin{equation} \label{Frelation} F \equiv \sum_{k=0}^{m_j-1} C^{(j)}_k \text{ for each } j = 1,\dots,s. \end{equation} If we let $U$ denote the sublattice of $L$ spanned by $F$ and $S_0$, then $U$ is a unimodular lattice of rank $2$, isomorphic to a hyperbolic plane, and so splits off $L$. Its orthogonal complement $R$ is freely generated by the components $C^{(j)}_k$ for $j = 1 \dots s$ and $k \neq 0$. $R$ is therefore a direct sum of $s$ root lattices of type $A$, with the $j^{th}$ summand isomorphic to $A_{m_j-1}$, generated by $C^{(j)}_k$ for $k \neq 0$. The Shioda-Tate formula for the Mordell-Weil group $\operatorname{MW}(X)$ of sections of $X$ is derived from the exact sequence (see \cite{shioda}) \[ 0 \to L \to \operatorname{NS}(X) \to \operatorname{MW}(X) \to 0 \] where the first map is the inclusion of the sublattice $L$ into $\operatorname{NS}(X)$. The second map is the fiber-by-fiber summation map, sending a divisor class $D \in \operatorname{NS}(X)$ to the closure of the sum of the points of $D$ on the generic fiber of $X$. We therefore obtain information about both the rank and the torsion of the Mordell-Weil group $\operatorname{MW}(X)$. If we denote by $\rho$ the Picard number of $X$, which is the rank of the Neron-Severi group, we see that since $L$ has rank equal to $2 + \sum_{j=1}^s (m_j-1)$, \[ \operatorname{rank}\operatorname{MW}(X) = \rho - 2 - \sum_{j=1}^s (m_j-1) = s + \rho - 2 - e. \] The torsion $\operatorname{MW}_{tor}(X)$ in the Mordell-Weil group corresponds to those classes in $\operatorname{NS}(X)$ for which some multiple lies in the sublattice $L$. These classes form an intermediate lattice $L^{sat}$ containing $L$ as a sublattice of finite index, and we see that \[ \operatorname{MW}_{tor}(X) \cong L^{sat}/L. \] For any lattice $N$, denote by $N^\#$ the dual lattice $\operatorname{Hom}_{\Bbb Z}(N,\Bbb Z)$. The inclusion of $L$ into $L^{sat}$ gives the sequence of inclusions \[ L \subseteq L^{sat} \subseteq {(L^{sat})}^\# \subseteq L^\# \] which shows that the quotient $L^{sat}/L$ is isomorphic to a subgroup of the discriminant-form group $G_L = L^\#/L$. Note that $L^\# = U^\#\oplus R^\#$ since $L = U \oplus R$ and since $U^\# = U$ ($U$ is unimodular), we have $G_L \cong R^\#/R = \bigoplus_{j=1}^s A_{m_j-1}^\#/A_{m_j-1}$. A computation shows that for the root lattices of type $A$, $A_m^\#/A_m \cong \Bbb Z/m\Bbb Z$; therefore we have that the torsion part $\operatorname{MW}_{tor}(X)$ of the Mordell-Weil group of sections of $X$ is isomorphic to a subgroup of $\bigoplus_{j=1}^s \Bbb Z/m_j\Bbb Z$. (See for example \cite{miranda-persson1} or \cite{miranda}.) A choice of orientation for each singular fiber $F_j$ gives an identification of the cyclic group of components $C^{(j)}_k$ for $k =0,\dots,m_j-1$ with $\Bbb Z/m_j\Bbb Z$. For any section $S$ of $f$, and any singular fiber $F_j$, denote by $k_j(S)$ the index of the component of $F_j$ which $S$ meets. Thus \begin{equation} \label{Sdot} S \cdot C^{(j)}_l = \delta_{l k_j(S)} \end{equation} where $\delta$ above is the Kronecker delta. Note that with our notation above, $k_j(S_0) = 0$ for every $j$. The numbers $\{k_j(S)\}$ will be called the {\em component numbers} of the section $S$. The upshot of the remarks above is that, for torsion sections, this assignment of integers $\{k_j(S)\}$ to $S$ is $1$-$1$. Thus a torsion section is determined by its component numbers (but not all sets of components numbers can occur). It is the purpose of this article to study the properties of the component numbers of torsion sections for semistable elliptic surfaces. Although it is clear from the above discussion that the component numbers $k_j(S)$ are well-defined modulo $m_j$, it is more useful for our purposes to take them to be integers in the range $0,\dots, m_j-1$. Two facts are necessary for what follows; they can be found in \cite{miranda} as well as the other basic references. \begin{lemma} \label{2facts} \begin{num} \item If $S_1$ and $S_2$ are two different sections in $\operatorname{MW}(X)$ with $S_1-S_2$ torsion, then $S_1\cdot S_2 = 0$, i.e., they are disjoint. \item If $S$ is any section in $\operatorname{MW}(X)$, then $S\cdot S = -\chi$. \end{num} \end{lemma} \section{The Divisor Class of a Torsion Section} Let $S$ be a torsion section of the semistable elliptic surface $f:X \to C$. In this section we wish to write down the class of $S$ in $\operatorname{NS}(X)$. Since the class of $S$ lies in $L^{sat}$, we have that $S$ is a $\Bbb Q$-linear combination of the zero section $S_0$, and the fiber components $C^{(j)}_k$. Because (\ref{Frelation}) is the only relation among these classes, the classes $S_0$, $F$, and $C^{(j)}_k$ for $j = 1,\dots,s$ and $k \neq 0$ form a basis for the module spanned by all these classes. For fixed $j=1,\dots,s$ and $k = 1,\dots,m_j-1$, set \begin{eqnarray*} D^{(j)}_k &=& (m_j - k) \sum_{i=1}^k i C^{(j)}_i + k \sum_{i=k+1}^{m_j-1} (m_j - i) C^{(j)}_i \\ &=& (m_j-k)[ C^{(j)}_1 + 2C^{(j)}_2 + \dots + kC^{(j)}_k ] + \\ & & k [ (m_j-k-1)C^{(j)}_{k+1} + \dots + 2C^{(j)}_{m_j-2} + C^{(j)}_{m_j-1} ]. \end{eqnarray*} We set $D^{(j)}_0 = 0$. Note that \begin{equation} \label{S0dot} D^{(j)}_k \cdot S_0 = 0 \end{equation} for every $k$. \begin{lemma} \label{Dlemma} Fix $j = 1,\dots,s$, and indices $k$ and $l$ with $0 \leq k,l \leq m_j-1$. Then \[ D^{(j)}_k \cdot C^{(j)}_l = \begin{cases} m_j & \text{ if } l = 0 \text{ and } k \neq 0 \\ -m_j & \text{ if } l = k \text{ and } k \neq 0 \\ 0 & \text{ otherwise.} \end{cases} \] \end{lemma} \begin{pf} This is a straightforward check; let us first notice that if $k = 0$, then $D^{(j)}_k = 0$, so certainly the intersection product is $0$. Assume then that $k \neq 0$. If $l=0$, then since $C^{(j)}_0$ meets only $C^{(j)}_1$ and $C^{(j)}_{m_j-1}$, each once, we have \[ D^{(j)}_k \cdot C^{(j)}_0 = (m_j-k)[1] + (k)[1] = m_j, \] which proves the first statement. If $l = k$, then $C^{(j)}_k$ meets only $C^{(j)}_{k\pm 1}$ (each once) and itself (-2 times), so that \[ D^{(j)}_k \cdot C^{(j)}_k = (m_j-k)[(k-1)(1) + (k)(-2)] + (k)[(m_j-k-1)(1)] = -m_j, \] proving this case. We leave to the reader to check that all other intersection products are $0$, as stated. \end{pf} It is useful to note that the above lemma can be re-expressed using the Kronecker delta function as \begin{equation} \label{Ddot} D^{(j_1)}_k \cdot C^{(j_2)}_l = m_j \delta_{j_1j_2} (1-\delta_{k0})(\delta_{l0} - \delta_{lk}). \end{equation} \begin{theorem} With the above notation, if $S$ is a torsion section of order $n$, then \[ S \equiv S_0 + (S - S_0 \cdot S_0) F - \sum_{j=1}^s {1 \over m_j} D^{(j)}_{k_j(S)}. \] \end{theorem} \begin{pf} Since the intersection form on $L$ is nondegenerate, we need only check that both sides of the equation intersect the generators $S_0$, $F$, and the $C^{(j)}_l$ in the same number. Let us begin with $S_0$. Using (\ref{S0dot}) and the equations \[ S \cdot F = S_0 \cdot F = 1, \] we have that the right hand side intersects $S_0$ to $S\cdot S_0$, agreeing with the left hand side. Similarly, $F$ meets only the $S_0$ term on the right hand side, so the intersections with $F$ are also equal. Now fix indices $j = 1,\dots,s$ and $l = 0,\dots, m_j-1$, and let us check the intersection with $C^{(j)}_l$; Lemma \ref{Dlemma} is the critical part of the computation. Intersecting with the right hand side gives $S_0 \cdot C^{(j)}_l - {1 \over m_j} D^{(j)}_{k_j(S)} \cdot C^{(j)}_l$, which reduces to $\delta_{l 0} - (1-\delta_{k_j(S) 0})(\delta_{l 0} - \delta_{l k_j(S)})$ using (\ref{Ddot}). This is after some simplification equal to $\delta_{l k_j(S)}$, which is also $S \cdot C^{(j)}_l$ by (\ref{Sdot}). \end{pf} The reader familiar with \cite{cox-zucker} will recognize the above expression as the ``correction'' term in their pairing on the Mordell-Weil group. They were more interested in questions of rank in that paper, and less so in the torsion sections. The formula for the linear equivalence class of $S$ given above can be simplified somewhat in the case that $S$ is not the zero section $S_0$. Then $(S\cdot S_0)=0$ and $(S_0\cdot S_0) = -\chi$ by Lemma \ref{2facts}, so we have \begin{equation} \label{nonzeroS} S \equiv S_0 + \chi F - \sum_{j=1}^s {1 \over m_j} D^{(j)}_{k_j(S)}. \end{equation} \section{The Quadratic Relation for the Component Numbers} Let us now take the above formula for the torsion section $S$ and intersect it with $S$ itself. If $S = S_0$, then of course all $k_j(S) = 0$, both sides of the equation are $S_0$, so we recover no information. However if $S \neq S_0$, the formula (\ref{nonzeroS}) is nontrivial. Dotting $S$ with the right-hand side gives \[ \chi - \sum_{j=1}^s {1 \over m_j} D^{(j)}_{k_j(S)} \cdot S. \] Since $S$ meets only the curve $C^{(j)}_{k_j(S)}$ in the $j^{th}$ singular fiber, and it meets it exactly once, we have \[ D^{(j)}_{k_j(S)} \cdot S = (m_j - k_j(S))k_j(S). \] Therefore the above reduces to \[ \chi - \sum_{j=1}^s k_j(S) (1-{k_j(S) \over m_j}). \] Finally, using Lemma \ref{2facts}.2, dotting the left-hand side with $S$ gives $-\chi$. Hence we obtain the following formula, which we call the {\em quadratic relation} for the component numbers. \begin{proposition} \label{formula1} Let $S$ be a torsion section of $f$, not equal to the zero section $S_0$. Then \[ \sum_{j=1}^s k_j(S) (1-{k_j(S) \over m_j}) = 2\chi({\cal O}_X). \] \end{proposition} Note that the quadratic relation is independent of the choice of orientation of each singular fiber, as it should be. (If one reverses the orientation of $F_j$, then $k_j$ is replaced by $m_j-k_j$ if $k_j \neq 0$, so that the two terms of each summand are simply switched.) Miles Reid has pointed out to me that the quadratic relation given above can be derived directly via the use of the Riemann-Roch theorem for Weil divisors, as expounded in \cite[section 9]{reid}. One should contract all components of fibers except $C_0^{(j)}$, obtaining a surface with only $A_{m_j-1}$ singularities, and consider on that surface the Weil divisor $S$. The Riemann-Roch formula for $S$ has as corrections to the usual Riemann-Roch the terms $k_j(S) (1-{k_j(S) \over m_j})$. (See also \cite[section 2]{giraud}.) This approach is cleaner and perhaps conceptually simpler; however in the spirit of keeping things as elementary as possible, I have decided to present this ``low road'' version. The interested reader will have no problem making the computation as Reid suggests and re-deriving the quadratic relation in this manner. \section{The Component Number Sums} In this section we will develop formulas for the component number sums $\sum_j k_j(S)$ for a torsion section $S$. Such formulas follow rather directly from the quadratic relation for the component numbers. Suppose first that $S$ has order $2$. Then for each $j$ with $k_j(S) \neq 0$, we must have $m_j$ even and $k_j(S) = m_j/2$. Therefore the quadratic relation reduces to the following. \begin{corollary} \label{formula2} Let $S$ be an order $2$ section of $f$. Then \[ \sum_{j=1}^s k_j(S) = 4\chi({\cal O}_X). \] \end{corollary} Note that in the order $2$ case, each $k_j$ is either $0$ or $m_j/2$, and hence the component numbers for a torsion section of order $2$ are independent of the orientation of the fibers. This is clearly not true for other values of the component numbers. In general, reversing the orientation of the fiber $F_j$ changes the component number from $k_j$ to $m_j-k_j$. Hence the analogue of Corollary (\ref{formula2}) for torsion sections of order at least $3$ must take this into account. To this end define a function $d_m:\{0,\dots,m-1\} \to \{0,\dots,[m/2]\}$ by setting \[ d_m(k) = \min\{k, m-k\}. \] Note that for any given torsion section $S$, it is possible to choose the orientations of the fibers so that the component numbers $k_j(S)$ are minimal, that is, $k_j(S) = d_{m_j}(k_j(S))$ for each $j$. If this condition holds, we say that the section $S$ has {\em minimal component numbers}. Next suppose that a torsion section $S$ has order $n \geq 3$. Then both $S$ and $2S$ are nonzero torsion sections of $X$. If for each index $j$, we choose the orientation of the components so that $0 \leq k_j(S) \leq m_j/2$, then we have the formulas \begin{equation} \label{kjfor2S} k_j(2S) = \begin{cases} 0 \text{ if }k_j(S) = m_j/2, \text{ and } \\ 2k_j(S) \text{ if }k_j(S) < m_j/2 \end{cases} \end{equation} for every $j$. Therefore applying Proposition \ref{formula1} to $2S$, after dividing by $2$ we obtain \[ \sum\begin{Sb} j \text{ with }\\ k_j(S)<m_j/2 \end{Sb} k_j(S) (1-2{k_j(S) \over m_j}) = \chi({\cal O}_X). \] Multiplying the formula of Proposition \ref{formula1} by $2$ gives \[ \sum\begin{Sb} j \text{ with }\\ k_j(S)=m_j/2 \end{Sb} k_j(S) + \sum\begin{Sb} j \text{ with }\\ k_j(S)<m_j/2 \end{Sb} k_j(S) (2-2{k_j(S) \over m_j}) = 4\chi({\cal O}_X), \] and subtracting the above two equations yields the following. \begin{corollary} \label{formula3} Let $S$ be a section of order $n \geq 3$ with minimal component numbers $\{k_j(S)\}$. Then \[ \sum_{j=1}^s k_j(S) = 3\chi({\cal O}_X). \] \end{corollary} In case the orientations of the fibers are not chosen so as to give $S$ minimal component numbers, a similar formula holds, expressed with the $d_m$ function. \begin{corollary} \label{formula4} Let $S$ be a section of order $n \geq 3$. Then \[ \sum_{j=1}^s d_{m_j}(k_j(S)) = 3\chi({\cal O}_X). \] \end{corollary} Using these component number sum relations, one can re-express the quadratic relation in a simpler form. \begin{corollary} \label{formula5} Let $S$ be a section of order $n \geq 2$, with minimal component numbers $\{k_j=k_j(S)\}$. Then \[ \sum_{j=1}^s k_j^2/m_j = \begin{cases} \chi({\cal O}_X) & \text{ if } n \geq 3 \\ 2\chi({\cal O}_X) & \text{ if } n = 2 \end{cases} \] \end{corollary} \begin{pf} The quadratic relation can be written as \[ \sum_j k_j^2/m_j = \sum_j k_j - 2\chi({\cal O}_X), \] and so the result follows from Corollaries \ref{formula2} and \ref{formula4} \end{pf} These results generalize a certain divisibility statement concerning the component numbers obtained in \cite[Proposition 4.6]{miranda-persson1} (see also \cite[Lemma 3(a)]{miranda-persson2} and \cite[X.3.1]{miranda}). Although the above cited works deal with elliptic K3 surfaces, the statement holds in general. It is: \[ \sum_{j=1}^s {m_j-1 \over 2m_j} k_j^2 \; \text{ is an integer.} \] This was proved using lattice-theoretic arguments involving the discriminant-form group $L^{\#}/L$. We can recover this result as follows. First note that in the above sum, if any $k_j$ is replaced by $m_j-k_j$, the sum changes by an integer; hence it suffices for us to derive the divisibility result using minimal component numbers. Now the above sum in question can be rewritten as \[ \sum_j k_j^2/2 - \sum_j k_j^2/2m_j. \] Note that $\sum_j k_j^2$ has the same parity as $\sum_j k_j$, so that the first term $\sum_j k_j^2/2$ is equal to $\sum_j k_j/2$ modulo ${\Bbb Z}$. Now modulo ${\Bbb Z}$, we have \begin{eqnarray*} \sum_j k_j^2/2 - \sum_j k_j^2/2m_j &=& \sum_j k_j/2 - \sum_j k_j^2/2m_j \\ &=& \begin{cases} 4\chi/2 - 2\chi/2 & \text{ if } n=2 \\ 3\chi/2 - \chi/2 & \text{ if } n \geq 3 \end{cases} \\ &=& \chi \end{eqnarray*} which is an integer. \section{The Distribution Numbers for a Torsion Section} In this section we will apply the relations developed in the previous sections for the component numbers of a non-zero torsion section $S$ to its multiples $\alpha S$. To this end let us express the quadratic relation by using the following notation. For each singular fiber $F_j$, denote by $f_j(S)$ the fraction $k_j(S)/m_j$. Let $P(x) = x(1-x)$ for real numbers $x$. Then the quadratic relation can be expressed as \begin{equation} \label{qrforS} \sum_{j=1}^s P(f_j(S))\, m_j = 2\chi({\cal O}_X). \end{equation} Note again that this formula is independent of the choice of orientation of the fibers, since changing the orientation of the components of $F_j$ simply switches the two factors $f_j$ and $1-f_j$ in each $P$ term of the above sum. Now note that if one fixes orientations for all of the singular fibers, and if a torsion section $S$ has component numbers $\{k_j(S)\}$ using these orientations, then for any $\alpha$, the component numbers for the multiple $\alpha S$ of $S$ in these orientations will be \[ k_j(\alpha S) \equiv \alpha k_j(S) \mod m_j. \] Let $\langle x \rangle$ denote the smallest non-negative real number in the residue class modulo $\Bbb Z$ of a real number $x$; we have $\langle x \rangle \in [0,1)$ always. The above formula can be expressed using the fractions $f_j$ by \[ f_j(\alpha S) = \langle \alpha f_j(S) \rangle. \] Hence the quadratic relation (\ref{qrforS}) applied to the section $\alpha S$ is \begin{equation} \label{qrforalphaS} \sum_{j=1}^s P( \langle \alpha f_j(S) \rangle ) m_j = 2\chi({\cal O}_X). \end{equation} We want to reorganize this sum to collect terms with the same $f_j$. Let $S$ be any torsion section whose order divides $n$. Then $nS$ is the zero-section $S_0$, all of whose component numbers are zero; hence for each $j$, $n k_j(S)$ is divisible by $m_j$. In terms of the fractions $f_j$, this is equivalent to having $f_j(S)$ being a multiple of $1/n$ for every $j$. Define rational numbers ${M'}_{i,n}(S)$ for $0 \leq i \leq n-1$ by the formula \[ {M'}_{i,n}(S) = (\sum\begin{Sb} j\text{ with }\\ f_j(S) = i /n \end{Sb} m_j)/12\chi. \] Roughly speaking, ${M'}_{i,n}(S)$ is the fraction of the total sum $\sum_j m_j = 12\chi$ contributed by fibers where $S$ meets the component which is ``distance $i$'' from the component meeting the zero-section $S_0$. (This distance is measured in units of $m_j/n$, and is in the direction of the orientation of the fiber.) We will call these fractions ${M'}_{i,n}(S)$ the {\em oriented distribution numbers} for the section $S$. Obviously \[ \sum_{i=0}^{n-1} {M'}_{i,n}(S) = 1. \] Reorganizing the quadratic relation (\ref{qrforalphaS}) for $\alpha S$ to sum over the $i$'s gives \[ \sum_{i=0}^{n-1} \sum\begin{Sb} j\text{ with }\\ f_j(S) = i /n \end{Sb} P( \langle \alpha i / n \rangle ) m_j = 2\chi({\cal O}_X), \] which, after dividing through by $12\chi({\cal O}_X)$, we rewrite using the oriented distribution numbers as \[ \sum_{i=0}^{n-1} P( \langle \alpha i / n \rangle ) {M'}_{i,n}(S) = 1/6. \] If we disregard orientation, we are led to defining rational numbers $M_{i,n}(S)$ for $0 \leq i \leq [n/2]$ by \[ M_{i,n}(S) = \begin{cases} {M'}_{0,n}(S) & \text{ if } i=0,\\ {M'}_{i,n}(S) + {M'}_{n-i,n}(S) & \text{ if } 1 \leq i < n/2 ,\text{ and } \\ {M'}_{n/2,n}(S) & \text{ if } i = n/2. \end{cases} \] Note that $M_{i,n}(S)$ is the fraction of the total sum $\sum_j m_j = 12\chi$ contributed by fibers where $S$ meets the component which is ``distance $i$'' from the component meeting the zero-section $S_0$, in either of the two orientations. We therefore refer to the $M_{i,n}(S)$ as the {\em unoriented distribution numbers} for the section $S$. As with the oriented distribution numbers, we have \begin{equation} \label{sumMin1} \sum_{i=0}^{[n/2]} M_{i,n}(S) = 1. \end{equation} Since $\langle -x \rangle = 1 - \langle x \rangle$, and $\langle \ell + x \rangle = \langle x \rangle$ for integers $\ell$, we see that $\langle \alpha (n-i)/n \rangle = 1 - \langle \alpha i/n \rangle$. Therefore since the polynomial $P(x)$ has the same value at $x$ as at $1-x$, the quadratic relation written in terms of the oriented distribution numbers has the coefficient of ${M'}_{i,n}(S)$ equal to the coefficient of ${M'}_{n-i,n}(S)$. Hence we may combine these terms, obtaining the following relation for the unoriented distribution numbers. (Note that the $i=0$ term has been dropped, since it is zero.) \begin{lemma} \label{qrforM} If $S$ is a torsion section of order exactly $n$, and $\alpha$ is not divisible by $n$, then \[ \sum_{i=1}^{[n/2]} P( \langle \alpha i / n \rangle ) M_{i,n}(S) = 1/6. \] \end{lemma} Note that one gets the same equation relating the unoriented distribution numbers $M_{i,n}(S)$ using either $\alpha$ or $n-\alpha$. Hence the above set of equations are, a priori, only $[n/2]$ different equations. One can express the above set of equations in matrix form. Let $ {\bold{P}} _n$ be the square matrix of size $[n/2]$ whose ${\alpha i}^{th}$ entry is $P( \langle \alpha i / n \rangle )$. Let $ {\bold{1}} _n$ be a vector of length $[n/2]$, which has every coordinate equal to $1$. Finally let $ {\bold{M}} _n(S)$ be the column vector of $M_{i,n}(S)$'s, for $1 \leq i \leq [n/2]$. The equations of \ref{qrforM} can then be expressed in matrix form as follows. \begin{lemma} \label{PMnequation} If $S$ is a nonzero torsion section of order exactly $n$, then \[ {\bold{P}} _n {\bold{M}} _n(S) = (1/6) {\bold{1}} _n. \] \end{lemma} For example, suppose that $n=2$, so that we are discussing a $2$-torsion section $S$ of $X$. The linear system above reduces to the single equation $(1/2) M_{1,2}(S) = 1/6$, so that $M_{1,2}(S) = 1/3$ and therefore $M_{0,2}(S) = 2/3$ by (\ref{sumMin1}). In case $n=3$, the linear system again reduces to the single equation $(2/9) M_{1,3}(S) = 1/6$, so that $M_{1,3}(S) = 3/4$ and therefore $M_{0,3}(S) = 1/4$ by (\ref{sumMin1}). Note that these distribution numbers are independent of $S$! This is our result in general for $p$-torsion sections, which will be discussed in the next section. As a last example, suppose that $n=5$. Then the linear system above is the two equations $(4/25)M_{1,5}(S) + (6/25)M_{2,5} = 1/6$ and $(6/25)M_{1,5}(S) + (4/25)M_{2,5} = 1/6$ which leads to $M_{1,5}(S) = M_{2,5}(S) = 5/12$, and $M_{0,5}(S)=1/6$. Note again that the distribution numbers are independent of $S$, as we have seen in the previous examples. However here we see more: the distribution numbers $M_{i,5}$ for $i\neq 0$ are {\em equal}. This is the ``equidistribution'' property mentioned in the Introduction, and is our main result for general primes $p$. \section{Equidistribution for Torsion Sections of Prime Order} In this section we will use the linear system describing the unoriented distribution numbers $M_{i,n}(S)$ to compute these numbers for a torsion section $S$ of prime order $p$. Our first task is to show that the matrix $ {\bold{P}} _p$ is invertible, which therefore shows that the distribution numbers are determined by the linear system of Lemma \ref{PMnequation}. \begin{proposition} \label{P_pinvertible} Fix an odd prime $p$. Let $ {\bold{P}} _p$ be the square matrix of size $(p-1)/2$ whose $\alpha\,i$ entry is $P(\langle \alpha i / p \rangle )$. Then $ {\bold{P}} _p$ is invertible. \end{proposition} \begin{pf} Let $V$ be ${\Bbb C}^p$, with coordinates indexed by residue classes modulo $p$. A vector $v \in V$ will have as its $k^{th}$ coordinate the number $v_k$. The group of units $G = {({\Bbb Z}/p)}^\times$ acts on $V$, by the formula \[ {(a \cdot v)}_k = v_{ak \mod p} \] for $a \in G$. Define vectors $S_\alpha \in V$ by setting ${(S_\alpha)}_k = P(\langle \alpha k /p \rangle)$. We obviously have $S_\alpha = S_{p-\alpha}$. Also note that ${(S_\alpha)}_0 = 0$ for every $\alpha$, and ${(S_\alpha)}_{p-k} = {(S_\alpha)}_k$ for every $\alpha$ and $k$. Therefore $S_\alpha$ lies in the subspace $V^+$ defined by \[ V^+ = \{v \in V \,|\, v_0=0 \text{ and } v_{p-k} = v_k \text{ for all }k \}. \] The dimension of $V^+$ is $(p-1)/2$. Moreover $V^+$ is stable under the $G$-action, so that we can consider $G$ to be acting on $V^+$ also, with the same formula. Further note that $a\cdot S_1 = S_a$ for every $a \in G$. Note that the $\alpha^{th}$ row of $ {\bold{P}} _p$ consists of the coordinates ${(S_\alpha)}_k$ for $k=1,\dots,(p-1)/2$; hence to prove that $ {\bold{P}} _p$ is invertible, it suffices to show that the vectors $S_\alpha$, for $\alpha = 1,\dots,(p-1)/2$, are independent. Equivalently, we can show that the vectors $\{S_\alpha\}$ span $V^+$. Let $\chi:G \to {\Bbb C}^\times$, be a character of $G$. For $v \in V^+$, define \[ \omega_\chi(v) = \sum_{a \in G} \chi(a) \, a \cdot v. \] Note that $\omega_\chi(v)$ is an eigenvector of the action of $G$ on $V^+$, with eigenvalue $\chi^{-1}$, that is, $a \cdot \omega_\chi(v) = \chi^{-1}(a) \omega_\chi(v)$. Let us focus in on the special element \[ S_\chi = \omega_\chi(S_1) = \sum_{a \in G} \chi(a) S_a. \] Since this vector is a linear combination of the $S_a$'s, to show that the $S_a$'s span $V^+$, it suffices to show that the $S_\chi$'s do. We focus attention on the {\em even} characters $\chi$, that is, those characters with $\chi(-1)=1$. (If $\chi(-1)=-1$, $\chi$ is called {\em odd}.) There are exactly $(p-1)/2$ even characters for $G$, and we will finish the proof if we can show that for the $(p-1)/2$ even characters $\chi$, the vectors $S_\chi$ are non-zero; if so, they will be linearly independent, since they are eigenvectors for distinct characters. Hence they will span $V^+$. In fact we need only show that the first coordinate of each $S_\chi$ (for $\chi$ even) is non-zero. This is the number \[ s_\chi = {(S_\chi)}_1 = \sum_{a \in G} \chi(a) P(\langle a/p \rangle ). \] If $\chi$ is the trivial character which is identically $1$, then $s_\chi$ is obviously non-zero, so we may assume $\chi$ is one of the $(p-3)/2$ nontrivial even characters of $G$. Let $ {\bold{B}} _k(X)$ denote the $k^{th}$ Bernoulli polynomial. In particular, we have \[ {\bold{B}} _2(X) = X^2 - X + 1/6 = 1/6 - P(X), \text{ or } P(X) = 1/6 - {\bold{B}} _2(X) = {\bold{B}} _2(0) - {\bold{B}} _2(X). \] Therefore the number $s_\chi$ can be written as \[ s_\chi = \sum_{a \in G} \chi(a) [ {\bold{B}} _2(0) - {\bold{B}} _2(\langle a/p \rangle ) ], \] and since $\chi$ is nontrivial, $\sum_{a \in G} \chi(a) = 0$, so that \[ s_\chi = - \sum_{a \in G} \chi(a) {\bold{B}} _2(\langle a/p \rangle ). \] Now the definition of the generalized Bernoulli numbers $B_{k,\chi}$ according to Leopoldt (see \cite[page 37]{lang}, or \cite[Proposition 4.1]{washington}), gives \[ B_{k,\chi} = p^{k-1} \sum_{a=0}^{p-1} \chi(a) {\bold{B}} _k(\langle a/p \rangle); \] hence we have \[ s_\chi = {-1 \over p} B_{2,\chi} \] for an even nontrivial character $\chi$ on $G$. Now the classical theorem that $B_{k,\chi}$ is nonzero when $k$ and $\chi$ have the same parity shows that for every even character $\chi$, $s_\chi \neq 0$, finishing the proof. This theorem is proved by using the associated $L$-series \[ L(s,\chi) = \sum_{n=1}^\infty {\chi(n) \over n^s} \] and noting first that for $n \geq 1$, $B_{n,\chi} = -n L(1-n,\chi)$ (see \cite[Proposition 16.6.2]{ireland-rosen} or \cite[Theorem 4.2]{washington}). Hence it suffices to show that for even $\chi$, $L(1-n,\chi)$ is nonzero. This follows from considering the functional equation for $L$, which for even $\chi$ can be written as (see \cite[page 29]{washington}) \[ L(1-s,\chi) = \Gamma(s) {2 \over \tau(\chi)}{({p \over 2\pi})}^s L(s,\overline{\chi}) \] where $\tau(\chi) = \sum_{a=1}^p \overline{\chi}(a) e^{2\pi i a/p}$ is a Gauss sum, and is known to be non-zero (it has absolute value $\sqrt(p)$, see \cite[Lemma 4.8]{washington}). Therefore to finish we must check that for even $n \geq 2$, $L(n,\overline{\chi}) \neq 0$, and this follows since for any $\chi$, $L(s,\chi)$ is nonzero whenever $ {\bold{Re}} (s) >1$, by considering the Euler product expansion for $L$, which is \[ L(s,\chi) = \prod_p {(1 - \chi(p)p^{-s})}^{-1}. \] \end{pf} The techniques of the proof given above was inspired by those used in a similar situation in \cite[appendix to section 5]{reid}, following \cite{morrison-stevens}. I am indebted to Miles Reid for leading me in this direction. The first corollary of the above Proposition is that the unoriented distribution numbers $M_{i,n}(S)$ for a torsion section $S$ of prime order are determined by the linear system of Lemma \ref{PMnequation}. This proves that they are independent of $S$! Moreover, we can easily calculate them from the linear system. We require one lemma concerning the coefficient matrix $ {\bold{P}} _p$: \begin{lemma} \label{Pprowsums} For an odd prime $p$, and for any fixed $\alpha = 1,\dots,(p-1)/2$, \[ \sum_{i=1}^{(p-1)/2} P(\langle \alpha i /p \rangle) = { p^2-1 \over 12p}. \] \end{lemma} \begin{pf} For fixed $\alpha$, the integers $\alpha i$ as $i$ ranges from $1$ to $p-1$ themselves range over a complete set of representatives for the nonzero residue classes modulo $p$. Therefore, since $P(\langle \alpha i /p \rangle)$ is an even function of $i$, the set of values of this function for $i=1,\dots, (p-1)/2$ are the same as the set of values for the function when $\alpha = 1$. In this range we have $\langle i /p \rangle = i/p$, so that for any $\alpha$, the sum above is simply \[ \sum_{i=1}^{(p-1)/2} (i/p)(1-i/p) = { p^2-1 \over 12p} \] as claimed. \end{pf} Our main result is the following. \begin{theorem} \label{Minvalues} Let $S$ be a torsion section of odd prime order $p$. Then \[ M_{i,p}(S) = 2p/(p^2-1) \; \text{ if }\; 1 \leq i \leq (p-1)/2, \] and \[ M_{0,p}(S) = 1/(p+1). \] If $S$ is a torsion section of order $2$, then \[ M_{1,2}(S) = 1/3 \;\text{ and }\; M_{0,2}(S) = 2/3. \] \end{theorem} \begin{pf} If $p$ is odd, Lemma \ref{Pprowsums} shows that the matrix $ {\bold{P}} _p$ has constant row sums. In other words, $ {\bold{P}} _p {\bold{1}} = ((p^2-1)/ 12p) {\bold{1}} $, where $ {\bold{1}} $ is the constant vector with every coordinate $1$. Therefore a solution to the linear system $ {\bold{P}} _p {\bold X} = (1/6) {\bold{1}} $ (of which the vector $ {\bold{M}} _p$ of $M_{i,p}(S)$'s is a solution by Lemma \ref{PMnequation}) is given by a constant vector $(2p / (p^2-1)) {\bold{1}} $. Since $ {\bold{P}} _p$ is invertible, such a solution is unique, showing that for $i=1,\dots,(p-1)/2$, we must have $M_{i,p}(S) = 2p/(p^2-1)$. The value of $M_{0,p}$ is then gotten from (\ref{sumMin1}). For $p=2$, the computations were made at the end of the last section. \end{pf} We will use the notation $M_{i,p}$ for $M_{i,p}(S)$ from now on. The computation of $M_{0,p}$ was done previously in \cite{miranda-persson3} using a quotient argument. We view the above theorem as an ``equidistribution'' result: for a $p$-torsion section $S$, the number of times $S$ meets a component which is ``distance $i$'' away from the zero component $C_0$, (measured in units of $m_j/p$), counted with multiplicity ($m_j$ for the fiber $F_j$), is independent of $i$, for $i \neq 0$. As a concrete example, take the elliptic $K3$ surface with $6$ singular fibers, having $[m_j] = [1,1,1,7,7,7]$. Such a $K3$ surface exists, (see \cite{miranda-persson1}) and has a torsion section $S$ of order $7$ (see \cite{miranda-persson2}). The component numbers $\{k_j(S)\}$ must be $(0,0,0,a,b,c)$, where we can take the orientations of the $I_7$ fibers so that $0 \leq a,b,c \leq 3$. The equidistribution property above says that $M_{i,7} = 7/24$ for $i=1,2,3$; hence we are forced to have $(k_j) = (0,0,0,1,2,3)$ after possibly reordering the fibers. Since $\chi({\cal O}_X) = 2$, the sum of the component numbers $k_j(S)$ must be $6$ by Corollary \ref{formula3}, so this is consistent. The quadratic relation for $S$ in this case says that $1\cdot 6/7 +2\cdot 5/7+3\cdot 4/7 = 2\chi({\cal O}_X) = 4$, which also checks out. Finally note that any multiple of $S$ (except $S_0$ of course) has component numbers $\{0,0,0,1\text{ or }6,2\text{ or }5,3\text{ or }4\}$ so that all of the relations above are satisfied for all multiples. As another example, take the Beauville surface $X_{6321}$, which is a rational elliptic surface ($\chi({\cal O}_X)=1$) with four singular fibers of types $I_6$, $I_3$, $I_2$, and $I_1$ (see \cite{beauville}). Such a surface has a finite Mordell-Weil group of order $6$ (see \cite{miranda-persson4}). Using the relations given here for the component numbers, we can easily deduce them. Let $S_2$ be a torsion section of order $2$ and $S_3$ be a torsion section of order $3$. The component numbers for $S_2$ must be $(0\text{ or }3,0,0\text{ or }1,0)$, and since their sum must be $4$ by Corollary \ref{formula2}, we must have $(k_j(S_2)) = (3,0,1,0)$. Note that these numbers do not change if we change orientations, since $S_2$ has order $2$. The component numbers for $S_3$ must be $(0\text{ or }2\text{ or }4,0\text{ or }1\text{ or }2,0,0)$, and we may choose orientations so that they are $(0\text{ or }2,0\text{ or }1,0,0)$. By Corollary \ref{formula3}, these numbers must sum to $3$, so that we must have $(k_j(S_3)) = (2,1,0,0)$. In particular we have with these orientations that the sum of these sections, which is a section $S_6$ of order $6$ generating the Mordell-Weil group of $X$, has component numbers $(5,1,1,0)$. (We may re-orient the fibers to achieve $(k_j(S_6)) = (1,1,1,0)$ if we desire.) As a final example let us consider the Beauville surface $X=X_{3333}$, which is the Hessian pencil surface obtained by taking a smooth cubic curve $C$ in the plane and considering the pencil formed by $C$ and its Hessian $H$. One obtains after blowing up the $9$ base points of this pencil (which are the $9$ flexes of $C$) a rational elliptic surface ($\chi=1$) with four $I_3$ fibers. (One gets the same surface starting from any smooth $C$!). The Mordell-Weil group of $X$ is of order $9$, with the zero-section $S_0$ and $8$ other sections of order $3$. The component numbers for a section are all $0$, $1$, or $2$, and if $S$ is not the zero section, then the component number sum formula Corollary \ref{formula4} says that exactly one $k_j(S)$ is $0$, all others are non-zero. We may choose an ordering of the fibers and orientations so that one of the order $3$ sections $S_1$ has component numbers $(0,1,1,1)$. Let $S_2$ be another order $3$ section, not equal to $2S_1$, so that $S_1$ and $S_2$ generate the Mordell-Weil group. By reordering the fibers we may assume that the component numbers for $S_2$ are $(a,0,b,c)$ where $a,b,c$ are not $0$; after reorienting the first fiber we may further assume that $a=1$. Since the sum $S_3=S_1+S_2$ will then have component numbers $(1,1,b+1,c+1)$ and must have three non-zero component numbers, we may (after possibly switching the last two fibers) assume that $b=2$ and $c=1$, so that $(k_j(S_2)) = (1,0,2,1)$. This then determines the component numbers for all $8$ nonzero sections of $X$. As an application of these computations of the unoriented distribution numbers $M_{i,p}$ for prime $p$, we obtain a divisibility result. Since each of the numbers $12\chi M_{i,p}(S)$ is an integer, and since $p$ and $p^2-1$ are always relatively prime, we obtain the following. \begin{corollary} \label{pdiv} Let $p$ be an odd prime and suppose that $f$ admits a section of order exactly $p$. Then $(p^2-1)/2$ divides $e = 12\chi$. \end{corollary} This improves a divisibility result stated in \cite[page 255]{miranda-persson3} that $p+1$ divides $e=12\chi$, which was obtained by using $M_{0,p}$ only. Moreover this is sharp; for the elliptic modular surface (associated to the group $\Gamma_1(p)$) which has a section of order $p$, the Euler number is exactly $e = (p^2-1)/2$ (see \cite{shimura} or \cite{cox-parry}). In some sense we can view this result as enabling us to ``divine'' the Euler number of such a universal surface, without constructing it: if it existed, it surely would have minimal $e$. We also have recovered what Persson and I referred to as the ``Fixed Point Rule'' in \cite[Corollary 5.5]{miranda-persson1} (see also \cite[Lemma 3(c)]{miranda-persson2} and \cite[Corollaries X.3.3 and X.3.4]{miranda}). Again these works concentrated on the K3 surface case, which has $\chi({\cal O}_X) = 2$; the statement in this case was that if $S$ is a $p$-torsion section on a K3 elliptic surface, with component numbers $\{k_j\}$, then \[ \sum \{m_j \; : \; k_j \neq 0\} = 24p/(p+1). \] In general of course we have (using the results of this article) that \begin{eqnarray*} \sum \{m_j \; : \; k_j \neq 0\} &=& 12\chi - \sum \{m_j \; : \; k_j = 0\} \\ &=& 12\chi - 12\chi\cdot M_{0,p}(S) \\ &=& 12\chi(1 - 1/(p+1)) \\ &=& 12\chi p / p+1, \end{eqnarray*} which generalizes the Fixed Point Rule as stated in the above articles. Finally, I would like to mention a ``dual'' equidistribution property which I have not been able to prove, but which should be true. It concerns not the indices of the components other than the zero component which a $p$-torsion section $S$ hits, but rather the position in $C_0^{(j)}$ which $S$ hits, when it does hit this zero component. Specifically, each $C_0^{(j)}$ can be identified with ${\Bbb C}^*$ by sending the nodes to $0$ and $\infty$, and sending the point $S_0 \cap C_0^{(j)}$ to $1$. Any torsion section $S$ of order $p$ will then hit $C_0^{(j)}$ in a $p^{th}$ root of unity. The equidistribution property in this context should be that each $p^{th}$ root of unity occurs equally often, (counted properly, of course). I do not know how to detect this root of unity using the elementary types of techniques used in this paper. Of course this property, properly formulated, would be invariant under base change, and so it would suffice to check it for the modular surfaces. Peter Stiller has informed me that it is true for $p \equiv 3 \mod 4$, by checking the modular surface.

9210

alg-geom/9210003

en

https://arxiv.org/abs/alg-geom/9210003

alg-geom/9210003

Tyurin

Andrej Tyurin

The simple method of distinguishing the underlying differentiable structures of algebraic surfaces

24 pages, Latex

The simplest version of the Spin-polynomial invariants of the underlying differentiable structures of algebraic surfaces were considered and the simplest arguments were used in order to distinguish the underlying smooth structures of certain algebraic surfaces.

alg-geom

\section{ Introduction} The purpose of this preprint is to construct a new invariant of the smooth structure of a simply connected 4-manifold M so called Spin-polynomials \[ \gamma^{g,C}_1 (2,c_1,c_2) \in S^{d_1} H^2(M, {\mathchoice {\hbox{$\sf\textstyle Z\kern-0.4em Z$}) {~~~~~} (0.1) \] and to show how to use it to compare the smooth structures of rational surfaces and surfaces of general type. This Spin-polynomial (0.1) is the analogue of the original Donaldson polynomial $$ \gamma^g(2,c_1,c_2) \in S^d H^2(M, {\mathchoice {\hbox{$\sf\textstyle Z\kern-0.4em Z$}) {~~~~~} (0.1') $$ and depends on one extra index C given by a so called $ Spin^{ \bbbc} $-structure on M . It specifies a lift of the Stiefel-Whitney class $w_2(M) \in H^2(M, {\mathchoice {\hbox{$\sf\textstyle Z\kern-0.4em Z$} _2) $ to some integer class $C \in H^2(M, {\mathchoice {\hbox{$\sf\textstyle Z\kern-0.4em Z$} ) $ . The technical basis of the construction of the Spin-polynomial is the same as of the ordinary polynomials (0.1') .It is our aim to compare the properties of polynomials (0.1) and (0.1') Here our aim is to consider only the simplest version of invariants of such type and to use in applications the simplest arguments of proofs (now standart for Donaldson's stuff ). Much more sophisticated constructions and a vague discussion of the properties of these invariants are contained in forthcoming article [T 3]. A special case of our construction has been used in the article [P-T], where some important basic theorems were proved. By this reason we will follow the english translation of this article in terminology and notation (see [P-T]). As applications of our techniques we will prove the non existence of algebraic fake planes,Hirzebruch surfaces or quadrics.(Here "fake" means "diffeomorphic to ...,but algebraically non equivalent"). I should like to thanks the SONDERFORSCHUNGSBEREICH 170 in G\"ottingen for its kind hospitality, support and the use of its facilities. \section {$ Spin ^ \bbbc $-structure } The Stiefel-Whitney class $ w_2 (M) \in H^2(M, {\mathchoice {\hbox{$\sf\textstyle Z\kern-0.4em Z$}_2 ) $ of a smooth,compact 4-manifold M is the characteristic class of the lattice $ H^2(M, {\mathchoice {\hbox{$\sf\textstyle Z\kern-0.4em Z$}) $ with its intersection form $ q_M $ . This means that for every $ \sigma \in H^2(M, {\mathchoice {\hbox{$\sf\textstyle Z\kern-0.4em Z$}) $ \[ \sigma^2 \equiv \sigma.w_2 (M){~} mod {~}2 {~~~~~~~} (1.1) \] {\bf Definition 1.1. } Let M be a smooth, simply connected,compact 4-manifold ,then a class $ C \in H^2(M, {\mathchoice {\hbox{$\sf\textstyle Z\kern-0.4em Z$}) $ such that $ C \equiv w_2(M) mod 2 $ is called a $Spin^ \bbbc $-structure of M.Thus to equip M with a $Spin^ \bbbc $-structure is the same as to lift up $w_2 $ to an integer class. The set of all $Spin^ \bbbc $-structures on M is the affine sublattice \[ H_w(M) = \{ \sigma \in H^2(M, {\mathchoice {\hbox{$\sf\textstyle Z\kern-0.4em Z$}) \mid \sigma \equiv w_2(M) {~}mod {~}2 \} {~~~~}(1.2) \] {\bf Remarks} 1) For every $C \in H_w(M) $ \[ C^2 \equiv b^+_2- b^- _2 = I {~} mod {~} 8 {~~~~~~~~} (1.3) \] (here I is the index or signature of M) 2) The diffeomorphism group Diff M of M acts on $H_w(M)$ by the affine transformations. 3) the lattice $ H^2(M, {\mathchoice {\hbox{$\sf\textstyle Z\kern-0.4em Z$}) $ 1-transitive acts on $H_w(M) $ by the formula: for $ C \in H_w(M) $ \[ \sigma(C)=C+2 \sigma {~~~~~} (1.4) \] Hence, the choice of a $ C_0 $ of $ H_w(M)$ gives an identification : $H_w(M)=H^2(M).$ For the rest of this paper the symbol $ L_ \sigma $ for $\sigma \in H^2(M, {\mathchoice {\hbox{$\sf\textstyle Z\kern-0.4em Z$}) $ will denote a complex line bundle with the first Chern class \[ c_1( L_ \sigma )= \sigma {~~~~~~} (1.5)\] If M is equipped with a Riemannian metric g the $Spin ^\bbbc $ -structure C on M defines a pair of rank 2 Hermitian vector bundles $ W^+ $ and $ W^- $ such that the complexification of the tangent bundle can be decomposed as a tensor product \[ TM_ \bbbc=(W ^-)^* \otimes W^+ {~~~~~} (1.6) \] with \[ \Lambda^2 W^ {+,-} = L_C {~~~~~~~}(1.7) \] Moreover,if we equip $ L_C $ with any Hermitian connection $ \bigtriangledown_0 ,$ then for any U(2)-bundle E on M and any Hermitian connection \[ a \in {A \cal } _h (E) {~~~~~~} (1.8) \] on E we have a coupled Dirac operator \[ D ^{C, \bigtriangledown _0 }_a : \Gamma ^ {\infty }(E \otimes W^+ ) \rightarrow \Gamma^ { \infty }(E \otimes W^-) {~~~~~~~~~} (1. 9) \] and an integer number \[ \chi _C (E) = ind D ^{C,\bigtriangledown _0 } _a {~~~~~~~} (1.10 ) \] which is the index of the Fredholm operator (1.9). It is easy to see that this integer number doesn't depend on the continuous parameters g, $ \bigtriangledown _0 $ and $ a $ and depends on the $ Spin ^ \bbbc $-structure $ C \in H _w (M)$ only. If we change the $ Spin ^ \bbbc $ -structure from C to C' (we can consider it as a result of the action (1.4) of $ \delta =(C'- C) / 2 $ on C ) we have \[ \chi _ {C+ 2 \delta } (E) = \chi _C (E) + c_1 . \delta - \delta ( \delta + C ) {~~~~~~} (1.11) \] because of \[ \chi _{C + 2 \delta } (E) = \chi _C ( E \otimes L _ \delta ) {~~~~~~~} (1.12) \] and by the Atiyah - Singer formula \[ \chi _C (E) = c_1 (c_1 + C)/2 + 2 \chi _C (L_o) -c_2 {~~~~~~~~} (1.13) \] .Here $ L_0 $ denotes the trivial line bundle on M and $ c_1, c_2 $ are the Chern- classes of E. Finally we shoud point out that special structures on M can define a canonical $Spin ^ \bbbc $-structure: for example if M is the underlying structure of a complex surface S, then there is the natural $Spin ^ \bbbc $-structure $ C = c_1 (S) = - K_S $ given by the anticanonical class or if M admits a symplectic structure $ \omega $,then there exists the canonical set of complex structures on the tangent bundle TM=E with the same first Chern class $c_1 (M, \omega ) $ which is the natural $Spin ^ \bbbc $-structure for $(M, \omega )$. \section { The definition of Spin-polynomials } If a $Spin ^ \bbbc $ -4-manifold (M,C) is equipped with a Riemannian metric g then for every $U(2)$-- bundle E the gauge - orbit space \[{ \cal B } (E) = {\cal A^*}_h (E) /{ \cal G} \] of irreducible connections contains the subspace \[ {\cal M}^g (E) \subset {\cal B}(E) {~~~~~~~} (2.1) \] of anti self dual connections with respect to the Riemannian metric g. We can consider the subspace of $\cal M$ $^g (E)$: \[ {\cal M}^{g,C} _1 (E) = \{ (a) \in {\cal M}^g (E) | rk {~} ker D ^ {C,\bigtriangledown _0} _a \geq 1 \} {~~~~~~} (2.2) \] Analogously, \[ {\cal M}^{g,C}_2 (E) = \{ (a) \in {\cal M}^g (E) | rk {~} ker D ^ {C,\bigtriangledown _0} _a \geq 2 \} {~~~~~~} \] and so on. If $ (a) \in {\cal M}^{g,C}_1 - {\cal M}^{g,C}_2 $ and the family of Dirac operators is in "general position" near (a), then the fibre of the normal bundle to ${\cal M}^{g,C}_1 (E)$ at (a) is given by \[ (N_{{\cal M} _1 \subset {\cal M }} ) _{{~} (a)} = Hom ( ker D_a, coker D_a ) {~~~} (2.3) \] with $ker D_a = \bbbc $, $coker D_a = \bbbc^{1 - \chi _C (E)} $ (if the index of the Dirac operators is not positive ). Thus the virtual (expected) codimension of $ {\cal M}^{g,C}_1 (E) $ \[ v.codim {\cal M}^{g,C} _1 (E) = 2 - 2 \chi _C (E) {~~~~~~~~~~~~~} (2.4) \] On the analogy of the Freed - Uhlenbeck theorem,which says that for generic metric g the moduli space$ \cal M$$^g (E) $ (2.1) is a smooth manifold of the expected dimension with regular ends (see Theorem 3.13 of [F-U]) the following fact was proved in section 3 of Ch.2 of [P-T] . \[ \] {\bf Transversality Theorem}.For generic pair (g, $ \bigtriangledown _0) \in {\cal S} \times \Omega^1 $ of metric and connection on $L_C$, the moduli space ${\cal M}^{g,C}_1 (E)$ is smooth outside $\cal M$$^{g,C} _2 (E)$ of expected codimension (2.4). \[ \] Moreover, $\cal M$$^g (E)$ admits a natural orientation (see [D 1] and [K]).But $\cal M$$^{g,C} _1 (E) $ admits the special orientation becouse its normal bundle (2.4) has a natural complex structure. This orientation is described in details in section 5 of Ch.2 of [P-T]. Now, we need the usual restrictions on the topology of M.We will suppose \[ b^+ _2 (M)= 2 p_g (M) +1 \ \ \ \ (2.5) \] to be odd.Then both v.dim $\cal M$$^g (E) = 2 d $ and v.dim $\cal M$$^ {g,C} _1(E) = 2 d_1 $ must be even . To compute the value of $ \gamma ^{g,C}_1(E) $ evaluated at an argument $ (\sigma_1,...,\sigma_{d_1}) $ we need to consider Donaldson's realisation of this collection of 2-cycles as a collection of smoothly embedded Riemannian surfaces $ ( \Sigma_1,...,\Sigma_ {d_1}) $ which are in general position in the following sense : 1) Any two surfaces $ \Sigma _i $ and $ \Sigma _j $ meet transversally. And let \[ \{ m_1,...,m_N \} {~~~~} (2.6) \] be the set of all points wich are intersection points for some i and j. 2) exactly two surfaces pass through any point of intersection $m_i $ so that in the flag diagram \[ \begin{array}{rcl} & \{ m_ i \in \Sigma _j \} & {~~~~~~~~} \\ & & {~~~~~~~~~~~~~~~~~~~} (2.7) \\ \swarrow & & \searrow \\ \{ m_i,...,m_ N \} & & {~} \{ \Sigma _1,...,\Sigma _{d_1} \} \end {array} \] the projection to the set of all intersection points is an "unramified double cover"; For every 2-cocycle $\sigma $ Donaldson constructed a so called fundamental cycle $ D_{\sigma}$ in the space $ \cal B $$(E) $ of gauge-orbits,which is a closed subspace of codimension 2 in $\cal B $$(E) $ (see [D 1] or the formulas (4.18) - (4.20) in the survey article [T 1]). Then the third condition is 3) the collection of fundamental cycles $ D_{\sigma_1},...,D_{\sigma _{d_1}}$ is in general position with respect to the strata of ends of $\cal M$$^g _1 (E)$. Then we can define the value of the $Spin ^\bbbc$-polynomial as the algebraic number of points of intersection \[ \gamma ^{g,C}_1(E) (\sigma_1,...,\sigma_{d_l})=D_{\sigma_1}\cap...\cap D_{\sigma_{d_1}}\cap {\cal M} ^{g,C}_1 (E) {~~~} (2.8) \] This definition makes sense because of \[ \] {\bf Analogue of Donaldson's Lemma }. 1) If $\Sigma _1,...,\Sigma _{d_1}$ are chosen in general position and \[ c_2 (E) \geq \frac{3}{2} (b^+_2 + 1 ) - \frac{1}{2} c_1. C - 2\chi_C(L_0) {~~~~~} (2.9) \] then the intersection (2.8) is compact. 2) If $ g_t$ is a generall path in the space of metrics (which doesn't intersect walls (see (2.19)), then the union of all intersections (2.8) is smooth and compact. 3) The intersection number (2.8) depends only on the homology classes of the $\sigma_i$'s. {\bf Proof}. Since we use the same ideas and constructions as in [D 1] we will prove only first statement where our constants are a little bit different from Donaldson's. The proof of other statements is left to the reader. {\bf Remark}. In our applications we will use the $SO(3)$ - bundles with $w_2(E) \neq 0$, so one doesn't even need the estimate (2.9). If the intersection (2.8) is not compact, then there exists a sequence of connections \[ \{ a_i \} \in \bigcap_{i=1}^{d_1} D_{\sigma _i} \cap {\cal M}^{g,C}_1 (E) \] which after suitable gauge transformations will converge uniformly (with bounded norm of the curvature) on $ M - \{m_1,...,m_l \} $, where $\{m_i\}$ is a finite set of points of M, which can be regularized by an $ L^p$ - gauge transformation. The limit connection can be regularized to an anti self dual connection \[ a_ {\infty } \in {\cal M}^{g,C}_1 (E') with \] \[ c_1 (E') = c_1 (E), c_2 (E')= c_2 (E) - l \ \] \[ \chi _C (E') = \chi _C (E) + l \] (because of the regularisation theorem of Uhlenbeck [ U ]). Moreover $l \leq c_2 -\frac{1}{4} c_1^2 $ (otherwise ${\cal M}^g(E')$ is empty by the Bogomolov inequality). Since every $ D_{\sigma_i}$ is closed, there are two posibilities: \[ either \ \ a _{\infty} \in D _{\sigma _i} {~~~~~} (2.10) \] \[ or {~~~}\exists {~~~}j{~~~} such{~~} that \ \ m_j \in {\Sigma }_i {~~~~~~~~} \] Consider first the extremal case $l = c_2 - \frac{1}{4}.c_1^2 + 1$. Then the flag diagram (2.7) gives the inequality \[ 2 l =\#\{ m_i \in \Sigma_j \} \geq \# \{\Sigma_1,...,\Sigma_{d_1} \} {~~~} (2.11) \] But \[ d = \frac{1}{2} dim_{\bbbr}{\cal M}^g(E) = 4c_2 - c_1^2 -\frac{3}{2} (b^+_2 + 1) {~~~} (2.12)\] \[d_1 = \frac{1}{2} dim_{\bbbr}{\cal M}^{g,C}_1(E) = 3c_2 - 1 - \frac{1}{2} c_1 (c_1 - C) - \frac{3}{2}(b^+_2 + 1) - 2\chi_C(L_0) \] {}From this the inequality (2.11) is equivalent to \[ c_2 < \frac{3}{2} (b^+_2) -\frac{1}{2} c_1.C - 2\chi_C(L_0) {~~~} (2.13) \] contradicting (2.9). In the general case \[a_{\infty } \in \bigcap_{i=1}^{l'} D_{\sigma _i} {~~~~~~} (2.14) \] and each surface $ \Sigma _{l' + 1},...,\Sigma _{d_1} $ contains at least one point in $ \{ m_1,...,m_ l\} $. By the general position condition 3) \[ \bigcap_{i=1}^{l'} D_{\sigma _i} \cap {\cal M}^{g,C}_1 (E') \neq \phi \Rightarrow \frac{1}{2} dim_{\bbbr} {\cal M}^{g,C}_1 (E') \geq l' {~~~~~~~}(2.15) \] On the other hand the flag diagram (2.7) gives the inequality \[ 2 l \geq d_1 - l' \Longleftrightarrow l' \geq d_1 - 2 l {~~~~~~~} (2.16) \] From (2.15) and (2.16) we have \[ \frac{1}{2} dim_{\bbbr} {\cal M}^{g,C}_1 (E') \geq \frac{1}{2} dim_{\bbbr} {\cal M}^{g,C}_1 (E) - 2 l \] that is \[ 4. c_2 - 4 l - c^2_1 - 3 (b^+_2 + 1 ) - 1 + \chi_C (E) + l \geq 4. c_2 - c^2_1 - 3 (b^+_2 + 1 ) - 1 + \chi_C (E)) - 2 l \] and this is a contradiction if l is positive. This proves the lemma and completes - with the usual additions (see 3.1-3.3 from [D 1]) the construction of the Spin - polynomials \[ \gamma^{g,C}_1 (2,c_1,c_2) \in S^{d_1} H^2(M, {\mathchoice {\hbox{$\sf\textstyle Z\kern-0.4em Z$}) {~~~~~} (2.17) \] for a regular (in the sense the Transversality Theorem ) metric g avoiding reducible connections. Recall that if $ b^+_2 = 1$,then associating to the metric g the ray of harmonic selfdual forms on M defines the so called period map of the space of Riemannian metrics to the Lobachevski space \[ K^+ \subset H^2 (M,\bbbr) / \bbbr^+ {~~~~} (2.18) \] The Lobachevski space $ K^+$ is divided by the collection of walls $ \{W_e = e^{\perp} \}$ \[ e \in H^2 (M,{\mathchoice {\hbox{$\sf\textstyle Z\kern-0.4em Z$}), e \equiv c_1 mod 2, c^2_1 - 4 c_2 \leq e^2 \leq 0 {~~} (2.19) \] into chambers of type $(c_1,c_2)$, which form a set $\Delta$ . Actually we can lift $ SO(3) $ - connections up to $U (2) $- connections, then the reducibility conditions give the decomposition of our vector bundle as $ E = L_e \oplus L_{c_1 - e} $.The wall e will be important for us if $ \chi_C (L_e) $ or $ \chi_C (L_{c_1 - e}) $ will be positive. Now we can compute the link of the singularity of the bordism of the moduli spaces given by one dimensional path in the parameter space ${\cal S} \times \Omega^1 $ and the number of points of the intersection (2.8) which disappeared (appeared) in (from) this singularity. This number is given by the pure topological formula actually by Porteus formula for the virtual index vector bundle of the family of Dirac operators (the details you can see in the forthcoming preprint of Victor Pidstrigach) .From this by the same bordism arguments as in [D 1] and [K] we obtain the description of the dependence of the Spin-polynomials on the parameters: \[ \] {\bf Theorem 2.1. } If $ b^+_2 = 1$ then for every pair of regular metrics $g_1,g_2$ from the same chamber $ C\in \Delta $ \[ \gamma^{g_1,C}_1 (2,c_1,c_2) = \gamma^{g_2,C}_1 (2,c_1,c_2) {~~~~~~} (2.20) \] (if $ b^+_2 \geq 3 $ this is true without any chamber condition). \[ \] On the other hand the dependence of the Spin-polynomials on changing the $ Spin ^ \bbbc $-structure $ C \in H_w(M) $ was given in section 1: \[ \gamma^{g_1,C+ 2 \delta}_1 (2,c_1,c_2) = \gamma^{g_1,C}_1 (2,c_1- 2 \delta,c_2 -c_1. \delta + \delta ^2 ) {~~~~}(2.21) \] by the formula (1.12). \section {Algebraic surfaces } If M is the underlying manifold of an algebraic surface S, then there exist the canonical $ Spin ^ \bbbc $-structure given by the anticanonical class $ -K_S$ (we will drop the index as long as there is no danger of confusion). In this case for Hodge metric $ g_ H $ given by a polarization \[ H \in Pic S \subset H^2 (S,{\mathchoice {\hbox{$\sf\textstyle Z\kern-0.4em Z$}) {~~~~~~} (3.1) \] the Donaldson-Uhlenbeck identification theorem gives \[ {\cal M}^{g_H} (E) = M^H (2,c_1,c_2 ) {~~~~~~~} (3.2) \] where the right side is the moduli space of H - slope stable bundles on S with Chern classes $ c_1,c_2.$ Under the identification (3.2) $(a) = E $ we have an identification \[ ker D^{g_H,-K}_a = H^0 (E) \oplus H^2 (E) {~~~~~~~~} (3.3) \] \[ coker D^{g_H,-K}_a = H^1 (E) \] where the $H^i(E)$ denote the coherent cohomology groups and \[ ind D^{g_H,-K}_a = \chi (E) {~~~~~~~~~} (3.4) \] \[\chi _{-K} (L_0) = \chi ( {\cal O}_S ) = p_g + 1 \] The subspace (2.2) then is the Brill - Noether locus \[ {\cal M}^{g_H,-K}_1 (2,c_1,c_2) = \{ E \in M^H (2,c_1,c_2 ) | h^1 (E) \geq -\chi (E) + 1 \} {~~~~} (3.5) \] But in the situation of surfaces the last inequality \[ h^1 (E) \geq - \chi (E) + 1 \Longleftrightarrow h^0 (E) + h^2 (E) \geq 1 {~~~~~~}(3.6) \] Hence we have a decomposition \[ { \cal M}^{g_H,-K}_1 (2,c_1,c_2) = M_{1,0}^H (2,c_1,c_2) \cup M_{0,1}^H (2,c_1,c_2) \] where the components are algebraic subvarities \[ M_{1,0}^H (2,c_1,c_2) = \{ E \in M^H (2,c_1,c_2 ) | h^0 (E) \geq 1 \} {~~~~~~~~~} (3.7) \] \[ M_{0,1} ^H(2,c_1,c_2 ) = \{ E \in M^H (2,c_1,c_2 ) | h^2 (E) \geq 1 \} \] On the other hand the transformation \[ E \leadsto E^* (K) = E^* \otimes {\cal O}_S (K) {~~~~~~~~~~} (3.8) \] gives the identification \[ M^H (2,c_1,c_2 ) = M^H (2,2 K - c_1,c_2 -c_1 .K + K^2 ) {~~~~~~~~~~~~}(3.9) \] and by Serre-duality \[ M_ {0,1}^H (2,c_1,c_2 ) = M_{1,0}^H (2,2 K - c_1,c_2 -c_1 .K + K^2 ) {~~~~~~~} (3.10) \] By this reason we have in the algebraic geometric situation two polynomials \[ \gamma^{g_H,-K}_{1,0} (2,c_1,c_2) ,{~} and {~} \gamma^{g_H,-K}_{0,1} (2,c_1,c_2) {~~~~~~~~} (3.11) \] given by the construction in section 2 with the subspaces (3.7) ( of course, if our Hodge metric $ g_H $ is regular, then the spaces (3.7) have the expected dimension ). Now to compute the Spin-polynomial (2.17) we must sum the individual polynomials (3.11) but here we must be careful because the natural orientations of the components (3.7) can be different .Actually in section 5 of Ch.1 of [P-T] following the orientation law was proved: \[ \] {\bf Orientation Rules }.1) If the number \[ 1 - \chi ( E ) {~~~~} (3.12) \] is even,then the natural orientations of $ M_ {1,0}^H (2,c_1,c_2 ) $ and $ M_ {0,1}^H (2,c_1,c_2 ) $ coincide (compared with the complex orientation). 2) otherwise they have different orientations. \[ \] It means that \[ c_2 = \frac{1}{2} ( c^2_1 - c_1.K) + 1 mod 2 \Rightarrow \gamma^{g_H,-K}_1 = \gamma^{g_H,-K}_{1,0} + \gamma^{g_H,-K}_{0,1} {~~~} (3.13) \] and \[ c_2 = \frac{1}{2} ( c^2_1 - c_1.K) + mod 2 \Rightarrow \gamma^{g_H,-K}_1 = \gamma^{g_H,-K}_{1,0} - \gamma^{g_H,-K}_{0,1} \] On the analogy of the Non-degeneracy Theorem for the original Donaldson polynomial we can prove \[ \] {\bf Theorem 3.1}.Assume our Hodge metric $g_H$ avoids reducible connections and \[ 1) c_2 \geq 5 (p_g + 1) + \frac{1}{2}c_1 .K {~~~} (see (2.9);\] \[ 2) c_2 = \frac{1}{2} ( c^2_1 - c_1.K) + 1 mod 2 \] \[ 3) M_ {1,0}^H (2,c_1,c_2 ) {~~~} and {~~~} M_ {0,1}^H (2,c_1,c_2 ) \] have an expected positive dimension,then \[ \gamma^{g_H,-K}_1 (2,c_1,c_2) \neq 0{~~~~~~~~~} \] \[ \] {\bf Proof.} Because of condition 1) the polynomial exists. We can choose some smooth curve $ C $ in the complete linear system $| N H |, N \gg 0 $, such that the restriction map \[ res_C : M_ {1,0} ^H(2,c_1,c_2 ) \cup M_ {0,1}^H (2,c_1,c_2 ) \rightarrow M_C (2,c_1.C ){~~~}(3.14) \] is an embedding (see,for example, [T 3]).On the other hand, there is an ample divisor $ \Theta \in Pic M_C (2,c_1.C ) $ and the value of Spin-polynomial on the class C is the sum of degrees of the image subvarieties $( res_C ( M_ {1,0}^H (2,c_1,c_2 ) \cup M_ {0,1}^H (2,c_1,c_2 ) ) $ with respect to this $\Theta $. It must be a sum, not a difference because of condition 2) (see (3.13)). As in the case of the original Donaldson polynomials we are done. The condition 3) is very important because the properties of Spin-polynomials depend on degrees \[ deg_H c_1 = H.c_1 ,{~~} deg_H K_S = H.K_S {~~~~~~~~~~~} (3.15) \] with respect to the polarisation H. Namely, in contrast to the behaviour of the Donaldson polynomials the Spin-polynomials may vanish for all values of $c_2$: {\bf Lemma 3.1 }If \[ 2 K_S .H \leq c_1.H \leq 0 {~~~~~~~} (3.16) \] then \[ M_ {1,0} ^H(2,c_1,c_2 ) \cup M_ {0,1}^H (2,c_1,c_2 ) = \phi \] and ,hence \[ \gamma^{g_H,-K}_1 (2,c_1,c_2) = 0 \] for every $c_2$. {\bf Proof.} Indeed, \[ h^0 (E) > 0 \Longleftrightarrow \exists s: {\cal O}_S \rightarrow (E), s \neq 0 \] On the other hand \[ h^2 (E) > 0 \Longleftrightarrow \exists j: E \rightarrow {\cal O}_S (K_S),j\neq 0 \] but $ s \neq 0, c_1 .H \leq 0 $ contradicts the stability condition for E and $ c_1 .H \geq 2 H.K_S $ is a contradiction to the stability condition for E , too. We are done. {\bf Remark.} Of course, the inequalities (3.16) are possible for rational surfaces only. The vanising condition (3.16) is crucial (it is actually due to Donaldson [D 2]). Of course the original Donaldson polynomials don't vanish under this conditions as for example in the case $ S = {\bbbc}{{\rm I\!P}}^2 $ for the sequence \[ (2,-2,c_2) , c_2 \in {{\mathchoice {\hbox{$\sf\textstyle Z\kern-0.4em Z$}}^+ \] \section { Asymptotic regularity } Let S be a algebraic surface, H a polarisation on S and $ c_1 \in Pic S $ a divisor class. {\bf Definition 4.1}A class $ c_1 \in Pic S $ is called H-semisimple, if for any effective curve $ C \subset S $ \[ c_1.H > 2 C.H \Longleftrightarrow C.K_S + C^2 \leq c_1.C {~~~~~~~} (4.1) \] (I would like to emphasize that the left side of the last inequality is the degree of the canonical class on C). On the analogy of Donaldson's Non-degeneracy Theorem we prove \[ \] {\bf Theorem 4.1.} For every H-semisimple $ c_1 \in Pic S $ with $ c_1.H > 0 $ there exists a constant $ N(H,c_1) $ such that for $c_2 \geq N (H,c_1)$ \[v.dim M_ {1,0} ^H(2,c_1,c_2 ) = dim M_ {1,0} ^H(2,c_1,c_2 ) > 0 {~~~} (4.2) \] and general point of $ M_ {1,0} ^H(2,c_1,c_2 ) $ is smooth. \[ \] {\bf Proof} 7 Each $ E \in M_ {1,0} ^H(2,c_1,c_2 ) $ has a section,that is a non zero homomorphism \[ s: {\cal O}_S \rightarrow (E) {~~~~}(4.3) \] The subscheme of zeroes of this homomorphism contains a priori subschemes of different dimensions: \[ (s )_0 = C \cup \xi {~} with {~~~} dim C = 1, dim \xi = 0 {~~~} (4.4) \] Because E is H-stable , we have \[ 2 C.H < c_1.H {~~~~~} ( 4.5)\] There exists a finite set of non empty complete linear systems \[ | 0 |, | C_1 |,...,| C_N | {~~~~~}(4.6) \] satisfying the inequality (4.5) ( $|0|$ is the complete linear system of the class $0 \in Pic S $ ). For every $ i = 0,1,...,N $ consider the variety \[ GAM _ {C_i} (2,c_1,c_2 ) = \{ 0 \rightarrow {\cal O}_S (C_i) \rightarrow E \rightarrow J_{\xi} (c_1 - C_i ) \rightarrow 0 \} / {\bbbc }^* {~~~} (4.7) \] of all non trivial extensions up to homotheties, where $J_{\xi} $ is the ideal sheaf of a 0-dimensional subscheme $ \xi $ (of a cluster $\xi$ for short). (GAM alias GAMBURGER ). We need to prove that \[ dim \bigcup_{i=0}^{N} GAM _ {C_i} (2,c_1,c_2 ) \leq v. dim M_ {1,0} ^H(2,c_1,c_2 ) {~~~~}(4.8) \] and that $ M_ {1,0} ^H(2,c_1,c_2 ) \neq \phi $. But under the operation $E \leadsto E(-C_i)$ \[ GAM _ {C_i} (2,c_1,c_2 ) = GAM _ 0 (2,c_1 -2 C_i,c_2 - c_1.C_i + C^2_i) {~~~~~}(4.9) \] The constants $ \{C_i^2 - c_1.C_i \} $ are bounded, hence we are done if we prove the following \[ \] {\bf Lemma 4.1}For $c_2 \gg 0$ \[ dim GAM _ 0 (2,c_1,c_2 ) \leq v. dim M_ {1,0} ^H(2,c_1,c_2 ) = \] \[ = 3 c_2 - 1 - \frac {c_1 (c_1 + K ) }{2} - (p_g + 1) {~~~} ( 4.10) \] \[ \] (see (2.12) with $C = -K_S $ ) Note that \[ 3 (c_2 - c_1.C_i + C^2_i ) - 1 - \frac {(c_1 - 2 C_i)( K + c_1 - 2 C_i)}{2} - (p_g + 1) = \] \[ = 3 c_2 - 1 - \frac {c_1 (c_1 + K )}{2} - (p_g + 1) + (C_i.K + C_i^2 - c_1.C_i ) \] and the tail is non positive due to the inequality (4.1). \[ \] {\bf Proof of Lemma 4.1}The natural projection \[ \pi : GAM _ 0 (2,c_1,c_2 ) \rightarrow Hilb^{c_2} S {~~~~}(4.11) \] given by sending the extension (4.7) to the cluster $\xi $ as element of the Hilbert scheme is surjective for big $ c_2$ . A fibre \[ \pi ^{-1}(\xi ) = {{\rm I\!P}} Ext^1( J_{\xi} (c_1),{\cal O}_S ) = {{\rm I\!P}} H^1 ( J_{\xi} (c_1 + K))^* (4.12) \] by Serre-duality. For every $ \xi \in Hilb^{c_2} S $ we have a short exact sequence \[ 0 \rightarrow J_{\xi} (c_1 + K) \rightarrow {\cal }O_S(c_1 + K) \rightarrow {\cal O}_{\xi} (c_1 + K) \rightarrow 0 \] giving rise to a cohomology exact sequence \[H^0 ( J_{\xi} (c_1 + K) )\rightarrow H^0({\cal }O_S (c_1 + K)) \rightarrow {\bbbc}^{c_2} \rightarrow H^1 ( J_{\xi} (c_1 + K) )\rightarrow H^1({\cal }O_S (c_1 + K)) \rightarrow 0 {~}(4.13) \] Moreover, \[ h^0 ( J_{\xi} (c_1 + K) ) = 0 \Rightarrow dim {{\rm I\!P}} H^1 ( J_{\xi} (c_1 + K)) = c_2 - \chi ( {\cal }O_S(c_1 + K) ) - 1 = \] \[ = c_2 - 1 - \frac {c_1 (c_1 + K )}{2} - (p_g + 1) {~~~}(4.14) \] Consider the subvariety \[ \Delta = \{\xi \in Hilb^{c_2} S | h^0 ( J_{\xi} (c_1 + K) ) > 0 \} {~~~} (4.15) \] It is easy to see that \[ dim \Delta \leq c_2 + dim | c_1 + K | = c_2 + h^0({\cal }O_S(c_1 + K) ) - 1 \] On the other hand from (4.13) we have \[ dim \pi^{-1} (\xi ) = {{\rm I\!P}} H^1 ( J_{\xi} (c_1 + K)) \leq h^1({\cal }O_S(c_1 + K) ) +c _2 \] Hence \[ dim \pi^{-1} (\Delta ) \leq 2 c_2 - 2 + h^0({\cal O} _S(c_1 + K) ) + h^1 ({\cal }O_S(c_1 + K) ) \] and \[ c_2 > 2 h^0({\cal }O_S(c_1 + K) ) \Rightarrow dim \pi^{-1} (\Delta ) < 3 c_2 - 2 - \frac {c_1 (c_1 + K )}{2} - (p_g + 1) . \] This proves Lemma 4.1.To finish the proof of Theorem 4.1 we prove {\bf Lemma 4.2 }.If $ c_1.H >0 $,then for $c_2 \gg 0$ \[ M_ {1,0} ^H(2,c_1,c_2 ) \neq \phi \] and hence by Theorem 4.1 it has the expected dimension. {\bf Proof }.We need to prove that for generic $ \xi \in Hilb^{c_2} $ and $ c_2 \gg 0 $ any non trivial extension \[ 0 \rightarrow {\cal O}_S \rightarrow E \rightarrow J_{\xi} (c_1 ) \rightarrow 0 \] is H-stable. Twisting E by $ (-c_1) $ we have \[ 0 \rightarrow {\cal O}_S (-c_1) \rightarrow E (-c_1)\rightarrow J_{\xi} \rightarrow 0 \] The hypothetical destabilizing line bundle must be of type $ {\cal }O_S(- C)$,where C is an effective curve subject to the inequality (4.1) and the cluster $\xi $ must be supported on this effective curve. But the collection (4.6) of such curves is finite and for $ c_2 \gg 0 $ (as in Theorem 4.1) a generic $ \xi $ is not contained in any curve in this collection of complete linear systems (see (4.15)). {\bf Definition 4.2 }. A class $c_1 \in Pic S $ is called H-simple, if it is semisimple and the class $ 2K_S - c_1 $ is semisimple too. As a corollary of Theorems 3.1 and 4.1 we provide {\bf Theorem 4.2}.Assume that our Hodge metric g avoids reducible connections . Then for every H-simple $c_1 \in Pic S $ with $ c_1.H >0 $ there exists a constant $ N(H,c_1)$ such that for \[ c_2 \geq N(H,c_1) , c_2 = \frac{1}{2} ( c^2_1 - c_1.K) + p_g mod 2 {~~} (4.16)\] \[ \gamma^{g_H,-K}_1 (2,c_1,c_2) \neq 0 \] At last (but not at least ) we need to explain what we have to do if our Hodge metric does not avoid reducible connections.Certainly in case when $rk Pic S > 1 $ we may use the following extremely useful trick: {\bf Definition 4.3}.A polarization $ H^{\varepsilon} $ is called close to H if the ray $ {\bbbr}^+.H^{\varepsilon} $ in $K^+ $ (2.18) is close to the ray $ {\bbbr}^+.H $ in Lobachevski metric. {\bf Lemma 4.3}. If a class $ c_1 \in Pic S $ is a H-simple ,then for a polarisation $ H^{\varepsilon} $ sufficiently close to H \[ 1) {~~~} c_1 {~~~} is {~~~} H^{\varepsilon } -simple {~}too.\] \[ 2) {~} 2 K_S .H < c_1.H < 0 \Longrightarrow 2 K_S .H ^{\varepsilon}< c_1.H ^{\varepsilon} < 0 \ {~~~~} (4.17) \] 3) for every polarisation H there exists a sufficiently close to H polarisation $ H^{\varepsilon}$ such that the Hodge metric $ g_{H^{\varepsilon}} $ avoids the reducible connections. {\bf Proof }.For a sufficiently close polarisation $ H^{\varepsilon} $ \[ 2 C.H < c_1. H \Longrightarrow 2 C. H^{\varepsilon} < c_1. H^{\varepsilon} {~~~~~} (4.18) \] Hence the collection of linear systems (4.6) for $ H^{\varepsilon} $ is the same as for H and we have 1) and 2). To prove 3) it is enough to remark that the set of rays of polarisations is dense in the projectivisation of the K\"ahler cone in $ K^+ $ and the set of walls is discrete and locally finite. In the last section we consider three very simple examples to show how we can use Theorem 4.2 and Lemma 3.1 to distinguish the underlying smooth structures of rational surfaces and surfaces of general type. \section { Applications }. For the beginning we prove {\bf Theorem 5.1 }.If an algebraic surface S is diffeomorphic to $ {\bbbc}{{\rm I\!P}}^2 $ then $ S = {\bbbc}{{\rm I\!P}}^2 $ (as algebraic surface). {\bf Proof}.Let \[ f: {\bbbc}{{\rm I\!P}}^2 \rightarrow S {~~~~~~~~~~~} (5.1) \] be a diffeomorphism, $ h \in Pic S $ be the positive generator of $Pic S$ ($h^2 =1$).Then (using the real anti involution of $ {\bbbc}{{\rm I\!P}}^2 $ if necessary) we may consider the case when \[ f^* \ (h) = l \] is the class of the line on $ {\bbbc}{{\rm I\!P}}^2 $. For the canonical class we have $K_S = 3 h$ (otherwize $K_S =-3h $ and S is rational). Then the increment of the canonical class with respect to f is \[ \delta _f (K) = \frac { f^*(K_S) -K_{ {\bbbc}{{\rm I\!P}}^2 } }{2} = -K_{ {\bbbc}{{\rm I\!P}} }^2 {~~~~} (5.2) \] Consider the following topological type of vector bundles on S \[ (2,h,c_2), c_2 \gg 0 {~~~~~~~~~~} (5.3) \] {\bf Lemma 5.1 }.For all $c_2$ the Spin-polynomial \[ \gamma^{g_h,-K}_1 (2,h,c_2) = 0 . \] {\bf Proof}.The operation $f^*$ gives the equality \[ \gamma^{g_h,-K}_1 (2,h,c_2) = \gamma^{f^*(g_h),K_{\bbbc {\rm I\!P} ^2}}_1 (2,l,c_2) \] By the equality (2.20) \[ \gamma^{f^*(g_h),K_{\bbbc {\rm I\!P} ^2}}_1 (2,l,c_2) = \gamma^{g_{F-S},K_{\bbbc {\rm I\!P} ^2}}_1 (2,l,c_2) \] where $g _{F-S} $ is the Fubini-Study metric on $ \bbbc {\rm I\!P} ^2 $. By the equality (2.21) \[ \gamma^{g_{F-S},K_{\bbbc {\rm I\!P} ^2}}_1 (2,l,c_2) = \gamma^{g_{F-S},-K_{\bbbc {\rm I\!P} ^2}}_1 (2,2 K_{\bbbc {\rm I\!P}^2}+l,c_2+ 6)= \gamma^{g_{F-S},-K_{\bbbc {\rm I\!P} ^2}}_1 (2,- 5l,c_2 + 6) \] The first Chern class $c_1 = -5l $ satisfies the inequality (3.16) and by Lemma 3.1 we are done. To provide a contradiction to the existence of an f (5.1) we prove {\bf Lemma 5.1' }.On S the class h is h-simple. Hence by Theorem 4.2 if $ c_2$ is odd then \[ \gamma^{g_h,-K}_1 (2,h,c_2) \neq 0 \] {\bf Proof} For h the set of linear systems (4.6) is $|0|$.Hence h is h-semisimple. For the class $2K_S - h = 5h $ the set (4.6) is \[ |0|, |h|, |2h| \] For these classes we have \[ h^2 + h.K_S =4 <5 , (2h)^2 + 2h.K_S = 10 \leq 10 \] and thus $2K_S -h$ is semisimple too . This implies that h is simple.We are done by Theorem 4.2. Let $ F_1$ be the projective plane blown up in one point (Hirzebruch surface of number 1). {\bf Theorem 5.2}.If an algebraic surface S is diffeomorphic to $F_1$ then $ S = F_{2n+1} $ that is the odd Hirzebruch surface. {\bf Proof}.Let \[ f: F_1 \rightarrow S {~~~~~~} (5.4) \] be a diffeomorphism.We can find a basis h,e in $Pic S$ such that \[ h^2 = 1, e^2 = -1, h.e = 0, K_S = 3h - e {~~~~} (5.5)\] Again it is sufficient to consider the case when \[ f^* (h) = l, f^* (e) = E {~~~~~~~~~} (5.6) \] where l is the class of line and E is the exceptional divisor on $F_1$.Hence \[ K_{F_1} = -3l + E = f^*(-K_S) {~~~~~} (5.7) \] Then the increment of the canonical class \[ \delta _f K = -K_{F_1} \] We only need to investigate the case when S is a surface of the general type and minimal. Then $ K_S $ is a polarisation on S and $f^* (K_S)=-K_{F_1}$ is a polarisation on $F_1$. Let $ H $ be a polarisation on S sufficiently close to $K_S$ such that \[ f^*(H) = H_1 \] is a polarisation on $F_1$ sufficiently close to $(-K_{F_1})$. {\bf Lemma 5.2}.For all $c_2$ the Spin-polynomial \[ \gamma^{g_H,-K_S} _1 (2,h,c_2) = 0 . \] {\bf Proof}.The operation $f^*$ gives the equality \[ \gamma^{g_H,-K_S}_1 (2,h,c_2) = \gamma^{f^*(g_{H_1}),K_{F_1}}(2,l,c_2) . \] But the metrics $f^*(g_H) $ and $ g_{ H_1} $ on $F_1$ are contained in the same chamber. More precisely they have the same image of the period map .Then by (2.20) \[ \gamma^{f^*(g_H),K_{F_1}}(2,l,c_2) = \gamma^{g_{H_1},K_{F_1}}(2,l,c_2) \] Moreover \[ \gamma^{g_{H_1},K_{F_1}}(2,l,c_2) = \gamma^{g_{H_1},-K_{F_1}}(2,-5h+2l,c_2+5) \] The first Chern class $c_1 = -5h +2l$ satisfies the inequality (3.16). Thus by Lemma3.1 we are done. To provide a contradiction to existence of f (5.4) we prove {\bf Lemma 5.2'}.On S the class h is $K_S$-simple. {\bf Proof} Let C be an effective curve C of the form $ C=xh -ye $. We will check whether the inequality (4.1) does hold for C : First \[2 \leq 2.C.K_S \leq h.K_S =3 \Rightarrow C.K_S = 1 \Rightarrow y=3x-1 \] Then \[C^2 = x^2 -y^2 =-8x^2 + 6x -1,C^2 + C.K_S = -8x^2 +6x <C.h = x \] for every $x \in {\mathchoice {\hbox{$\sf\textstyle Z\kern-0.4em Z$} $.Hence h is $K_S$-semisimple. Now for $c_1 = 2 K_S - h = 5h - 2l $ we have \[ 2 \leq 2C.K_S \leq 13 \Rightarrow C.K \in \{1,...,6\} \Rightarrow y = 3x - \{ 1,...,6\} \] Then \[ C^2 + C.K_S = -8x^2 + 6x .\{1,...,6\} - \{0,2,6,12,20,30\} \] \[ C. (5h-2e) = - x + 2.\{1,...,6\} \] It is easy to check that for any of the six cases the inequality (4.1) holds. Hence $5h - 2e $ is $K_S$-semisimple ,too, and we are done. At last let us go to the Hirzebruch Problem. {\bf Theorem 5.3}.On $ S^2 \times S^2 $ there exists the unique algebraic structure $ Q = \bbbc {\rm I\!P}^1 \times \bbbc {\rm I\!P}^1 $ up to the elementary transformations to the even Hirzebruch surface $F_{2n}$. {\bf Proof}.In this case for any topological type $(2,c_1,c_2) $ of vector bundle the virtual dimension of $ M^H(2,c_1,c_2) $ is odd.We will use a simple trick: Let \[ f:Q \rightarrow S {~~~~} (5.8) \] be a diffeomorphism. We can find basis $ h_+,h_- $ in $Pic S$ such that \[ h_+^2 = h_-^2 = 0, h_+.h_- = 1,K_S = 2h_+ + 2h_- {~~~} (5.9) \] such that $f^* (h_+)= h'_+$ , $f^*(h_-)=h'_- $ is the standard basis of $ Pic \bbbc {\rm I\!P}^1 \times \bbbc {\rm I\!P}^1 $ and $f^*(K_S) = -K_Q $. We only need to consider the case when $K_S$ is nef. Let's blow up a point p on Q and $ f(p) $ on S. Then the diffeomorphism f (5.8) can be extended to a diffeomorphism \[ \tilde{f} : \tilde{Q} \rightarrow \tilde {S} {~~~~} (5.8') \] and \[ K_{\tilde{Q}} = K_Q + E' , K_{\tilde{S}} = K_S + E {~~~} (5.10) \] where E and E' are the respective exceptional curves. The increment of the canonical class with respect to $\tilde{f}$ is \[ \delta _{\tilde{f}} K = -K_Q \] The divisor class $ (h_+ + h_-) \in Pic \tilde{S} $ is nef and we consider a polarisation H on $\tilde{S} $ sufficiently close to $ (h_+ + h_-) $ such that $ \tilde{f}^*(H) = H'$ is a polarisation on $\tilde{Q} $ sufficiently close to $ (h'_+ + h'_-) $ . {\bf Lemma 5.3}.For all $c_2$ the Spin-polynomial \[\gamma ^{g_H,-K_S -E}_1 (2,h_+ +h_- +E,c_2) = 0 \] {\bf Proof} As usual by (2.20) and (2.21) \[\gamma ^{g_H,-K_S - E}_1 (2,h_+ +h_- +E,c_2) = \gamma ^{g_H',-K_Q - E'}_1 (2,-3 (h'_+ + h'_- + -E') ,c_2 + 2) \] The first Chern class $ c_1 = -3 (h'_+ + h'_- -E') $ satisfies the inequality (3.16) so by Lemma 3.1 we are done. To provide a contradiction to the existence of $\tilde{f} $ (and hence of f (5.8)) as before we prove {\bf Lemma 5.3'}. On $\tilde{S} $ the divisor class $ h_+ +h_- +E $ is H - simple. {\bf Proof}.For an effective curve C \[ 0 \leq C (h_+ +h_-) <1 \Rightarrow C = m(h_+ -h_-) + nE \] But then \[ C.K_{\tilde{S}} + C^2 = -2m^2 - n^2 -n \leq C.(h_+ +h_- +E) = -n \] and $ h_+ +h_- +E $ is H - semisimple. Now for $ 2 K_{\tilde{S}} - c_1 =3 (h_+ + h_-)+E $ let \[ C = xh_+ + m(h_+ -h_-) + nE . \] Then \[0 \leq C (h_+ +h_-) <3 \Rightarrow x =\{0,1,2\} \] and \[ C.K_{\tilde{S}} + C^2 = \{0,2,4\}m - 2m^2 - n^2 +\{0,2,4\} -n ,{~} C.(3h_+ + 3h_- +E) = \{0,3,6\} -n \] It is easy to check that the right side of the inequality (4.1) holds for all n and m .Hence $h_+ +h_- +E$ is H-simple. The reader may continue these purely arithmetical investigations himself. Good luck!. \[ \] {\bf Reference } [D 1] S.Donaldson "Polynomial invariants for smooth 4-manifolds" ,Topology 29 (1990),257-315. \\ [D 2] S.Donaldson "Differential topology and complex varieties",Arbeitstagung Proceedings 1990,MPI (1990). \\ [F-U] D.Freed,K.Uhlenbeck "Instantons and four-manifolds" M.S.R.I.Publ. Springer, New York, 1984. \\ [K] D.Kotschick "SO(3)-invariants for 4-manifolds with $b^+_2 =1$", Proc.London Math.Soc.3:63(1991),426-448. \\ [P-T] V.Pidstrigach A.Tyurin"Invariants of the smooth structures of an \ \ \ algebraic surface arising from Dirac operator."Izv.AN SSSR Ser.Math.,56:2(1992),279-371.(english translation:Warwick preprint 22 (1992). \\ [T 1] A.Tyurin "Algebraic geometric aspects of smooth structure.1.The Donaldson polynomials." Russian Math.Surveys 44:3 (1989),113-178. \\ [T 2] A.Tyurin "A slight generalization of the theorem of Mehta-Ramanathan."Springer LNM 1479 (1989),258-272. \\ [T-3] A.Tyurin " Spin-polynomial invariants of smooth structures of algebraic surfaces."forthcoming Izv.AN Russia Ser.Math.,57:2(1993) \\ [U] K.Uhlenbeck " Connections with $L^p$ bounds on curvature" Comm.Math . Phys.83 (1982),31-42. \end{document}

9210

alg-geom/9210007

en

https://arxiv.org/abs/alg-geom/9210007

alg-geom/9210007

Michael Thaddeus

Michael Thaddeus

Stable pairs, linear systems and the Verlinde formula

40 pages, LaTeX

We study the moduli problem of pairs consisting of a rank 2 vector bundle and a nonzero section over a fixed smooth curve. The stability condition involves a parameter; as it varies, we show that the moduli space undergoes a sequence of flips in the sense of Mori. As applications, we prove several results about moduli spaces of rank 2 bundles, including the Harder-Narasimhan formula and the SU(2) Verlinde formula. Indeed, we prove a general result on the space of sections of powers of the ideal sheaf of a curve in projective space, which includes the Verlinde formula.

alg-geom

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\newcommand{\mbox{$\longrightarrow$}}{\mbox{$\longrightarrow$}} \newcommand{\mbox{$\Lambda_{0}$}}{\mbox{$\Lambda_{0}$}} \newcommand{\mbox{$\tilde{L}$}}{\mbox{$\tilde{L}$}} \newcommand{\tilde{M}_b}{\tilde{M}_b} \newcommand{\tilde{M}^-_{i-1}}{\tilde{M}^-_{i-1}} \newcommand{\tilde{M}_i}{\tilde{M}_i} \newcommand{\tilde{M}^+_i}{\tilde{M}^+_i} \newcommand{n_{j}}{n_{j}} \newcommand{\mbox{$M^{\Lambda}$}}{\mbox{$M^{\Lambda}$}} \newcommand{n}{n} \newcommand{\mbox{$\tilde{M}$}}{\mbox{$\tilde{M}$}} \newcommand{\mbox{$N^{\Lambda}$}}{\mbox{$N^{\Lambda}$}} \newcommand{\mbox{$\tilde{N}$}}{\mbox{$\tilde{N}$}} \newcommand{{\em Proof}}{{\em Proof}} \newcommand{p_{2}}{p_{2}} \newcommand{\mbox{$\tilde{Q}$}}{\mbox{$\tilde{Q}$}} \newcommand{\mathop{{\rm rank}}\nolimits}{\mathop{{\rm rank}}\nolimits} \newcommand{\mbox{$\rightarrow$}}{\mbox{$\rightarrow$}} \newcommand{\sigma}{\sigma} \newcommand{\mbox{$S_{3}$}}{\mbox{$S_{3}$}} \newcommand{\mbox{$S_{4}$}}{\mbox{$S_{4}$}} \newcommand{\mbox{$\overline{\rm SL}$}}{\mbox{$\overline{\rm SL}$}} \newcommand{\mbox{$\overline{\rm SL}_{2}$}}{\mbox{$\overline{\rm SL}_{2}$}} \newcommand{\mbox{${\rm SL}_{2}{\rm(\C)}$}}{\mbox{${\rm SL}_{2}{\rm(\mbox{\bbb C})}$}} \newcommand{\mbox{${\rm SL}_{2}$}}{\mbox{${\rm SL}_{2}$}} \newcommand{\mbox{SL($\chi$)}}{\mbox{SL($\chi$)}} \newcommand{\mbox{$S_{0}$}}{\mbox{$S_{0}$}} \newcommand{\mbox{${\rm SU(2)}$}}{\mbox{${\rm SU(2)}$}} \newcommand{\mathop{{\rm td}}\nolimits}{\mathop{{\rm td}}\nolimits} \newcommand{\otimes}{\otimes} \newcommand{\mathop{{\rm tr}}\nolimits}{\mathop{{\rm tr}}\nolimits} \newcommand{\mbox{$\tilde{\cal U}$}}{\mbox{$\tilde{\cal U}$}} \newcommand{\mbox{${\cal W}^{\bla}_{0}$}}{\mbox{${\cal W}^{{\bf \Lambda}}_{0}$}} \newcommand{\mathop{{\rm vol}}\nolimits}{\mathop{{\rm vol}}\nolimits} \newcommand{\mbox{$\tilde{\cal W}^{\bla}_{0}$}}{\mbox{$\tilde{\cal W}^{{\bf \Lambda}}_{0}$}} \newcommand{\mbox{${\cal W}^{\bla}$}}{\mbox{${\cal W}^{{\bf \Lambda}}$}} \newcommand{\mbox{$\tilde{\cal W}_{0}$}}{\mbox{$\tilde{\cal W}_{0}$}} \newcommand{\mbox{${\cal W}_{0}$}}{\mbox{${\cal W}_{0}$}} \newcommand{\mbox{${\cal W}_{z}$}}{\mbox{${\cal W}_{z}$}} \newcommand{\chi}{\chi} \newcommand{\mbox{$X_{2}$}}{\mbox{$X_{2}$}} \newcommand{\mbox{$X_{n}$}}{\mbox{$X_{n}$}} \newcommand{\mbox{$X_{0}$}}{\mbox{$X_{0}$}} \newcommand{\mbox{$\tilde{X}$}}{\mbox{$\tilde{X}$}} \newcommand{\mbox{$X_{z}$}}{\mbox{$X_{z}$}} \newcommand{\mbox{$Z_{k}$}}{\mbox{$Z_{k}$}} \newcommand{\mbox{$Z_{k}^{\Lambda}$}}{\mbox{$Z_{k}^{\Lambda}$}} \newcommand{\mbox{$Z_{k}^{\tilde{\Lambda}}$}}{\mbox{$Z_{k}^{\tilde{\Lambda}}$}} \newcommand{\mbox{$\tilde{Z}$}}{\mbox{$\tilde{Z}$}} \renewcommand{\det}{\mathop{{\rm det}}\nolimits} \renewcommand{\bullet}{-} \catcode`\@=12 \noindent {\LARGE \bf Stable pairs, linear systems} \\ {\LARGE \bf and the Verlinde formula} \\ $\phantom{.}$ \\ {\bf Michael Thaddeus }\\ \smallskip Mathematical Sciences Research Institute, 1000 Centennial Drive, Berkeley, Cal. 94720 \\ \setcounter{section}{-1} \bit{Introduction} Let $X$ be a smooth projective complex curve of genus $g \geq 2$, let $\Lambda \mbox{$\rightarrow$} X$ be a line bundle of degree $d > 0$, and let \mbox{$(E, \phi)$}\ be a pair consisting of a vector bundle $E \mbox{$\rightarrow$} X$ such that $\Lambda^{2} E = \Lambda$ and a section $\phi \in H^0(E) - 0$. This paper will study the moduli theory of such pairs. However, it is by no means a routine generalization of the well-known theory of stable bundles. Rather, it will discuss at least three remarkable features of the moduli spaces of pairs: 1. Unlike bundles on curves, pairs admit many possible stability conditions. In fact, stability of a pair depends on an auxiliary parameter $\sigma$ analogous to the weights of a parabolic bundle. This parameter was first detected by Bradlow-Daskalopoulos \cite{bd} in the study of vortices on Riemann surfaces, and indeed the spaces we shall construct can also be interpreted as moduli spaces of rank 2 vortices. As $\sigma$ varies, we will see that the moduli space undergoes a sequence of flips in the sense of Mori theory, whose locations can be specified quite precisely. 2. For some values of $\sigma$ the moduli space $M(\sigma, \Lambda)$ is the blow-up of $\mbox{\bbb P} H^1(\Lambda^{-1})$ along $X$, embedded as a complete linear system. Thus we can use $M(\sigma, \Lambda)$ to study the projective embeddings of $X$. In particular, we obtain a very general formula \re{6m} for the dimension of the space of hypersurfaces of degree $m+n$ in $\mbox{\bbb P} H^1(\Lambda^{-1})$ with a singularity at $X$ of order $n-1$. This formula does not depend on the precise choice of $X$ and $\Lambda$, only on $g$ and $d$, which is rather surprising. 3. For other values of $\sigma$, stability of the pair implies semistability of the bundle, so $M(\sigma, \Lambda)$ plays the role in rank 2 Brill-Noether theory of the symmetric product in the usual case, and there is an Abel-Jacobi map from $M(\sigma, \Lambda)$ to the moduli space of semistable bundles. For large $d$ this is generically a fibration, so we can use moduli spaces of pairs to study moduli spaces of bundles. In particular, we recover the known formulas for Poincar\'e polynomials \cite{ab,hn} and Picard groups \cite{dn}; more spectacularly, we prove, and generalize, the rank 2 Verlinde formula \re{6n} for both odd and even degrees. We will not fully discuss the many other fascinating aspects of the subject, but we will briefly touch on one of them---the relation with Cremona transformations and Bertram's work on secant varieties---in an appendix, \S8. We hope to treat the relation with vortices and Yang-Mills-Higgs theory in a later paper. An outline of the other sections is as follows. In \S1 we prove some basic facts about pairs, in analogy with bundles. Following Gieseker \cite{g1}, we then use geometric invariant theory to construct the moduli space $M(\sigma, \Lambda)$ of $\sigma$-semistable pairs, and a universal family over the stable points of $M(\sigma, \Lambda)$. The choice of $\sigma$ corresponds to a choice of linearization for our group action. In \S2 we discuss the deformation theory of the moduli problem. In \S3 we show that the $M(\sigma, \Lambda)$ are reduced, rational, and smooth at the stable points. We then show that as $\sigma$ varies, $M(\sigma, \Lambda)$ undergoes a sequence of flips whose centres are symmetric products of $X$. We also define the rank 2 Abel-Jacobi map mentioned above. In \S4 we calculate the Poincar\'e polynomial of $M(\sigma, \Lambda)$, and extract from it the Harder-Narasimhan formula for the Poincar\'e polynomial of the moduli space of rank 2 bundles of odd degree. Thereafter we concentrate on studying the line bundles over $M(\sigma, \Lambda)$, and their spaces of sections. In \S5 we compute the Picard group of $M(\sigma, \Lambda)$, and its ample cone. We explain how any section of a line bundle on $M(\sigma, \Lambda)$ can be interpreted as a hypersurface in projective space, singular to some order on an embedded $X$. We also make the connection with the Verlinde vector spaces. Finally in \S\S6 and 7 we use the Riemann-Roch theorem to calculate Euler characteristics of the line bundles in $M(\sigma, \Lambda)$. Combined with the information from \S5, Kodaira vanishing, and some residue calculations which were carried out by Don Zagier, this gives a formula for the dimensions of the spaces of sections of line bundles on $M(\sigma, \Lambda)$, under some mild hypotheses. We conclude by extracting the Verlinde formula from this. For convenience we work over the complex numbers, but much of the paper should be valid over any algebraically closed field: certainly \S\S1--3 and 5. Kodaira vanishing is of course crucial in \S6, but the computation of the Euler characteristics ought to make sense in general, if integral cohomology is replaced with intersection theory. A few notational habits should be mentioned: $X_i$ refers to the $i$th symmetric product of $X$; $\pi$ denotes any obvious projection, such as projection on one factor, or down from a blow-up; tensor products of vector bundles are frequently indicated simply by juxtaposition; and likewise a pullback such as $f^*L$ is often called just $L$. Also, in \S3 and thereafter, $M(\sigma, \Lambda)$ is referred to simply as $M_i$, where $i$ depends on $\sigma$ in a manner explained in \S3. These conventions are not meant to be elliptical, but to clean up what would otherwise be some very messy formulas. We also make the following assumptions, which are explained in the text but are repeated here for emphasis. We always assume $g \geq 2$. In the geometric invariant theory construction of \S1, we assume $d$ is large, an assumption which is justified by \re{3o} and the discussion following it. From \S3 to the end we assume $d \geq 3$. However, this assumption is implicit in other inequalities---so for example our main formula \re{6m} is valid as it stands. {\em Acknowledgments.} My principal debt of gratitude is of course to Don Zagier, whose exquisite computations are indispensable to the paper. The proof of \re{4u}, and the entire \S7, are due to him. I am also very grateful to Simon Donaldson for his advice, encouragement and patience, and to Arnaud Beauville, Aaron Bertram, Steven Bradlow, Jack Evans, Oscar Garcia-Prada, Rob Lazarsfeld, David Reed, Miles Reid, and Eve Simms for helpful conversations. Finally, I thank Krzysztof Gaw\c edzki and the Institut des Hautes Etudes Scientifiques for their hospitality while much of the research for this paper was carried out. \bit{Constructing moduli spaces of $\sigma$-semistable pairs} Our main objects of study, which we refer to simply as {\em pairs}, will be pairs \mbox{$(E, \phi)$}\ consisting of a rank 2 algebraic vector bundle $E$ over our curve $X$, and a nonzero section $\phi \in H^{0}(E)$. A careful study of such pairs was made by Steven Bradlow \cite{brad}. He defined a stability condition for pairs and proved a Narasimhan-Seshadri-type theorem relating stable pairs to vortices on a Riemann surface. The vortex equations depend on a positive real parameter $\tau$, and so the stability condition also depends on $\tau$. Bradlow and Georgios Daskalopoulos went on \cite{bd} to give a gauge-theoretic construction of the moduli space of $\tau$-stable pairs, under certain conditions on $\tau$ and $\deg E$. Oscar Garcia-Prada later showed \cite{gp} that there always exists a projective moduli space, by realizing it as a subvariety of a moduli space of stable bundles on $X \times \mbox{\bbb P}^1$. In this section we will give a geometric invariant theory construction of the moduli space of $\tau$-stable pairs for arbitrary $\tau$ and $\deg E$ (though for convenience we assume $\mathop{{\rm rank}}\nolimits E = 2$). Aaron Bertram has informed me that he has done something similar \cite{bert2}, and I apologize to him for any overlap. The Bradlow-Daskalopoulos stability condition is in general rather complicated, but in the rank 2 case it simplifies to the following. Let $\sigma$ be a positive rational number. It is related to $\tau$ by $\sigma = \tau \mathop{{\rm vol}}\nolimits X/4 \pi - \deg E /2$, where $\mathop{{\rm vol}}\nolimits X$ is the volume of $X$ with respect to the metric chosen in \cite{bd}. \begin{defn} \label{3l} The pair \mbox{$(E, \phi)$}\ is $\sigma$-{\em semistable} if for all line bundles $L \subset E$, $$\begin{array}{cl} \deg L \leq {\scriptstyle{\frac{1}{2}}} \deg E - \sigma & \mbox{if $\phi \in H^{0}(L)$ and} \\ \deg L \leq {\scriptstyle{\frac{1}{2}}} \deg E + \sigma & \mbox{if $\phi \not\in H^{0}(L)$.} \end{array}$$ It is $\sigma$-{\em stable} if both inequalities are strict. \end{defn} The main result of this section is then the following. \begin{thm} \label{3b} Let $\Lambda \mbox{$\rightarrow$} X$ be a line bundle of degree $d$. There is a projective moduli space $M(\sigma, \Lambda)$ of $\sigma$-semistable pairs \mbox{$(E, \phi)$}\ such that $\Lambda^{2} E = \Lambda$, nonempty if and only if $\sigma \leq d/2$. \end{thm} Our construction will be modelled on that of Gieseker \cite{g1}. We begin with a few basic facts about $\sigma$-stable and semistable pairs, parallel to those for bundles. We write $\Lambda$ for $\Lambda^{2} E$, and $d$ for $\deg E = \deg \Lambda$. \begin{lemma} For $\sigma > 0$, there exists a $\sigma$-semistable pair of determinant $\Lambda$ if and only if $\sigma \leq d/2$. \end{lemma} {\em Proof}. If $\sigma > d/2$, then $\sigma$-semistability implies $\deg L < 0$ if $\phi \in H^0(L)$, which is absurd. If $\sigma \leq d/2$, let $L \mbox{$\rightarrow$} X$ be a line bundle of degree $[d/2-\sigma]$ having a nonzero section $\phi$. Let $E$ be a nonsplit extension $$0 \mbox{$\longrightarrow$} L \mbox{$\longrightarrow$} E \mbox{$\longrightarrow$} \Lambda L^{-1} \mbox{$\longrightarrow$} 0.$$ Then the first inequality in the definition \re{3l} is obvious. As for the second, if $M \subset E$ and $\deg M > d/2 + \sigma$, then there is a nonzero map $M \mbox{$\rightarrow$} \Lambda L^{-1}$. Since $\deg \Lambda L^{-1} < d/2 + \sigma + 1$, this is an isomorphism, so the extension is split, which is a contradiction. \mbox{ $\Box$} \begin{lemma} \label{3a} Let \mbox{$(E, \phi)$}\ be a pair. There is at most one $\sigma$-destabilizing bundle $L \subset E$ such that $\phi \in H^{0}(L)$, and at most one $\sigma$-destabilizing $M \subset E$ such that $\phi \not\in H^{0}(M)$. If both $L$ and $M$ exist, then $E = L \oplus M$. \end{lemma} {\em Proof}. The first statement is obvious, and the second follows from the uniqueness of ordinary destabilizing bundles, since $\deg M \geq {\scriptstyle{\frac{1}{2}}} \deg E + \sigma > {\scriptstyle{\frac{1}{2}}} \deg E$. If both $L$ and $M$ exist, then the map $M \mbox{$\rightarrow$} E \mbox{$\rightarrow$} \Lambda L^{-1}$ is nonzero since $\phi \in H^{0}(L)$ but $\not\in H^{0}(M)$. But $\deg M \geq d/2 + \sigma \geq \deg \Lambda L^{-1}$, so $M = \Lambda L^{-1}$ and $E$ is split. \mbox{ $\Box$} \begin{lemma} Let $(E_{1}, \phi_{1})$ and $(E_{2}, \phi_{2})$ be $\sigma$-stable pairs of degree $d$, and let $\psi: E_{1} \mbox{$\rightarrow$} E_{2}$ be a map such that $\psi \phi_{1} = \phi_{2}$. Then $\psi$ is an isomorphism. \end{lemma} {\em Proof}. The kernel of $\psi$ is a subsheaf of a locally free sheaf on a smooth curve, so it is locally free. If $\mathop{{\rm rank}}\nolimits\ker \psi = 2$, then $\psi$ is generically zero, so $\psi = 0$ and $\psi \phi_{1} \neq \phi_{2}$. If $\mathop{{\rm rank}}\nolimits\ker \psi = 1$, then $\ker \psi$ is a line subbundle $L$ of $E_{1}$, since $E_{1}/\ker \psi$ is contained in the torsion-free sheaf $E_{2}$. Hence $\psi$ descends to a map $\Lambda L^{-1} \mbox{$\rightarrow$} E_{2}$ (possibly with zeroes) such that $\phi_{2} \in H^{0}(\Lambda L^{-1})$. Since $(E_{2}, \phi_{2})$ is $\sigma$-stable, $\deg \Lambda L^{-1} < d/2 - \sigma$, so $\deg L > d/2 + \sigma$, contradicting the $\sigma$-stability of $(E_{1}, \phi_{1})$. Finally, if $\mathop{{\rm rank}}\nolimits\ker \psi = 0$, then $\ker \psi = 0$ and $\psi$ is injective. Moreover, $\mathop{{\rm coker}}\nolimits\psi$ is a coherent sheaf on a curve with rank and degree 0, so $\mathop{{\rm coker}}\nolimits\psi = 0$ and $\psi$ is an isomorphism. \mbox{ $\Box$} \begin{lemma} \label{3k} Let \mbox{$(E, \phi)$}\ be a $\sigma$-stable pair. Then there are no endomorphisms of $E$ annihilating $\phi$ except 0, and no endomorphisms preserving $\phi$ except the identity. \end{lemma} {\em Proof}. Subtracting from the identity interchanges the two statements, so they are equivalent. We prove the second. Any endomorphism annihilating $\phi$ annihilates the subbundle $L$ generated by $\phi$, so descends to a map $E/L \mbox{$\rightarrow$} E$. But by $\sigma$-stability $E/L$ is a line bundle of degree $\geq d/2 + \sigma$, so the image of this map, if it were nonzero, would generate a line bundle of degree $\geq d/2 + \sigma$, which would be destabilizing. \mbox{ $\Box$} \begin{lemma} \label{3n} Let $(\mbox{$\bf E$},{\bf \Phi}), (\mbox{$\bf E$}',{\bf \Phi}') \mbox{$\rightarrow$} T \times X$ be two families over $T$ parametrizing the same pairs. Then $(\mbox{$\bf E$},{\bf \Phi}) = (\mbox{$\bf E$}',{\bf \Phi}')$. \end{lemma} {\em Proof}. For any $t \in T$, the subspace of $H^0(X;\mathop{{\rm Hom}}\nolimits(\mbox{$\bf E$}_t,\mbox{$\bf E$}'_t))$ consisting of homomorphisms $\psi$ such that $\psi {\bf \Phi}_t = \lambda {\bf \Phi}'_t$ for some $\lambda \in \mbox{\bbb C}$ is one-dimensional by \re{3k}. This determines an invertible subsheaf of the direct image $(R^0\pi)\mathop{{\rm Hom}}\nolimits(\mbox{$\bf E$}_t,\mbox{$\bf E$}'_t)$. But this subsheaf is trivialized by the section $\lambda = 1$, which produces the required isomorphism. \mbox{ $\Box$} The notion of a Harder-Narasimhan filtration for rank 2 pairs is quite a simple one. For \mbox{$(E, \phi)$}\ stable, define $\mathop{{\rm Gr}}\nolimits\mbox{$(E, \phi)$} = \mbox{$(E, \phi)$}$. Otherwise, define $\mathop{{\rm Gr}}\nolimits\mbox{$(E, \phi)$}$ to be a direct sum of line bundles, one of them containing the section $\phi$, as follows. If $L$ is the destabilizing bundle and $\phi \in H^{0}(L)$, define $\mathop{{\rm Gr}}\nolimits\mbox{$(E, \phi)$} = (L \oplus \Lambda L^{-1}, \phi)$. If $M$ is the destabilizing bundle and $\phi \not\in H^{0}(M)$, project $\phi$ to a nonzero section $\phi' \in H^{0}(\Lambda M^{-1})$ and define $\mathop{{\rm Gr}}\nolimits\mbox{$(E, \phi)$} = (M \oplus \Lambda M^{-1}, \phi')$. Note that if there are destabilizing bundles of both sorts, then by \re{3a} $E = L \oplus \Lambda L^{-1}$ and the two definitions agree. \begin{lemma} \label{3j} There exists a degeneration of \mbox{$(E, \phi)$}\ to $\mathop{{\rm Gr}}\nolimits\mbox{$(E, \phi)$}$, but $\mathop{{\rm Gr}}\nolimits\mbox{$(E, \phi)$}$ degenerates to no semistable bundle. \end{lemma} {\em Proof}. The first statement is vacuous when \mbox{$(E, \phi)$}\ is stable. If it is unstable, say with destabilizing bundle $M$, we can construct a pair $(\mbox{$\bf E$}, {\bf \Phi}) \mbox{$\rightarrow$} X \times \mbox{\bbb C}$ such that $(\mbox{$\bf E$}_{z}, {\bf \Phi}_{z}) \cong \mbox{$(E, \phi)$}$ for $z \neq 0$, but $(\mbox{$\bf E$}_{0}, {\bf \Phi}_{0}) \cong \mathop{{\rm Gr}}\nolimits\mbox{$(E, \phi)$}$, as follows. Pull back \mbox{$(E, \phi)$}\ to $X \times \mbox{\bbb C}$, and tensor by ${\cal O}(0)$ when $\phi \not\in H^0(M)$. This gives a pair $(\mbox{$\bf E$}', {\bf \Phi}') \mbox{$\rightarrow$} X \times \mbox{\bbb C}$ such that ${\bf \Phi}'$ is annihilated by the natural map $\mbox{$\bf E$}' \mbox{$\rightarrow$} \Lambda M^{-1}|_{X \times \{ 0 \} }$. Let $\mbox{$\bf E$}$ be the kernel of this map; then ${\bf \Phi}'$ descends to ${\bf \Phi} \in H^0(\mbox{$\bf E$})$, and it is straightforward to check that $(\mbox{$\bf E$}, {\bf \Phi})$ has the desired properties. As for the second statement, suppose first that \mbox{$(E, \phi)$}\ is stable. If $C$ is a curve, $p \in C$, and $(\mbox{$\bf E$}, {\bf \Phi}) \mbox{$\rightarrow$} X \times C$ is a flat family of pairs such that $(\mbox{$\bf E$}_{z}, {\bf \Phi}_{z}) \cong \mbox{$(E, \phi)$}$ for $z \neq p$, then ${\bf \Phi}_{p}$ has the same zero-set $D$ as $\phi$, so $E$ and $\mbox{$\bf E$}_{p}$ are both extensions of $L = {\cal O}(D)$ by $\Lambda(-D)$; indeed, $\mbox{$\bf E$}$ is a family of such extensions. The extension class varies continuously, so the extension class of $\mbox{$\bf E$}_{p}$ is in the same ray as that of $E$. If it is nonzero, $\mbox{$(E, \phi)$} \cong (\mbox{$\bf E$}_{p}, {\bf \Phi}_{p})$, and if it is zero, $(\mbox{$\bf E$}_{p}, {\bf \Phi}_{p})$ is destabilized by $\Lambda L^{-1}$. Now suppose that \mbox{$(E, \phi)$}\ is not stable, so that for some $L$, $\mathop{{\rm Gr}}\nolimits\mbox{$(E, \phi)$} = L \oplus \Lambda L^{-1}$ and $\phi \in H^{0}(L)$. Then as above $\mbox{$\bf E$}_{p}$ is an extension of $L$ by $\Lambda L^{-1}$, but now by continuity the extension class must be zero, so $\mathop{{\rm Gr}}\nolimits\mbox{$(E, \phi)$} = (\mbox{$\bf E$}_{p}, {\bf \Phi}_{p})$. \mbox{ $\Box$} \begin{lemma} \label{3o} If \mbox{$(E, \phi)$}\ is $\sigma$-(semi)stable, then so is $(E(D), \phi(D))$ for any effective divisor $D$. Likewise, if $\phi$ vanishes on an effective divisor $D$ and \mbox{$(E, \phi)$}\ is $\sigma$-(semi)stable, then so is $(E(-D), \phi(-D))$. \end{lemma} {\em Proof}. If $L \subset E$ is any line bundle, $\phi(D) \in H^{0}(L(D))$ if and only if $\phi \in H^{0}(L)$, and $\deg L(D) = \deg L + \deg D$. But ${\scriptstyle{\frac{1}{2}}} \deg E(D) = {\scriptstyle{\frac{1}{2}}} \deg E + \deg D$ also, so both inequalities are preserved by tensoring with $D$. The second statement is proved similarly. \mbox{ $\Box$} Hence if the moduli spaces $M(\sigma, \Lambda)$ exist for large enough $d$, then the moduli spaces for smaller $d$ will be contained inside them as the locus of pairs \mbox{$(E, \phi)$}\ such that $\phi$ vanishes on some effective $D$. So to prove our existence theorem \re{3b} it suffices to construct $M(\sigma, \Lambda)$ for $d$ large relative to $g$ and $\sigma$, and {\em we will assume for the remainder of\/ {\rm \S\thesection\ }that $d$ is large in this sense}. For such a large $d$, we then have the following useful fact. \begin{lemma} \label{3e} For fixed $g$ and $\sigma$ and large $d$, \mbox{$(E, \phi)$}\ $\sigma$-semistable implies that $H^{1}(E) = 0$ and $E$ is globally generated. \end{lemma} {\em Proof}. Suppose that $H^{1}(E) \neq 0$. Then $H^{0}(KE^{*}) \neq 0$, so there is an injection $0 \mbox{$\rightarrow$} K^{-1}(D) \mbox{$\rightarrow$} E^{*}$ for some effective $D$. Hence there is an injection $0 \mbox{$\rightarrow$} K^{-1}\Lambda(D) \mbox{$\rightarrow$} E$. Since $\deg K^{-1}\Lambda(D) \geq 2 - 2g + d$, the $\sigma$-semistability condition implies that $2 - 2g + d \leq d/2 + \sigma$, so that $d \leq 4g-4+2\sigma$. So for $d$ larger than this, $H^{1}(E) = 0$. Similarly, if $d > 4g-2+2\sigma$, then $H^{1}(E(-x)) = 0$ for all $x \in X$, so $E$ is globally generated. \mbox{ $\Box$} Since we are assuming that $d$ is large, the above lemma implies that for \mbox{$(E, \phi)$}\ $\sigma$-stable, $\dim H^{0}(E) = \chi(E) = d+2-2g$. Call this number $\chi$. If we fix an isomorphism $s: \mbox{\bbb C}^{\chi} \mbox{$\rightarrow$} H^{0}(E)$, we obtain a map $\Lambda^{2} \mbox{\bbb C}^{\chi} \stackrel{s}{\mbox{$\rightarrow$}} \Lambda^{2} H^{0}(E) \stackrel{\wedge}{\mbox{$\rightarrow$}} H^{0}(\Lambda)$, which is nonzero because $E$ is globally generated. Thus to any bundle $E$ appearing in a $\sigma$-semistable pair, and any isomorphism $s$, we associate a point $T(E,s) \in \mbox{\bbb P} \mathop{{\rm Hom}}\nolimits(\Lambda^{2} \mbox{\bbb C}^{\chi}, H^{0}(\Lambda))$. We will consider the pair $(T(E,s), s^{-1}\phi) \in \mbox{\bbb P} \mathop{{\rm Hom}}\nolimits \times \mbox{\bbb P}\mbox{\bbb C}^{\chi}$, where $\mbox{\bbb P} \mathop{{\rm Hom}}\nolimits$ is short for $\mbox{\bbb P} \mathop{{\rm Hom}}\nolimits(\Lambda^{2} \mbox{\bbb C}^{\chi}, H^{0}(\Lambda))$. Roughly speaking, $M(\sigma, \Lambda)$ will be a geometric invariant theory quotient of the set of such pairs. The quotient is necessary to remove the dependence on the choice of $s$. Since two such isomorphisms are related by an element of $\mbox{SL($\chi$)}$, the group action will be the obvious diagonal action of $\mbox{SL($\chi$)}$ on $\mbox{\bbb P} \mathop{{\rm Hom}}\nolimits \times \mbox{\bbb P}\mbox{\bbb C}^{\chi}$. As usual in geometric invariant theory, we must {\em linearize} the action by choosing an ample line bundle and lifting the action of $\mbox{SL($\chi$)}$ to its dual. So let the ample bundle be any power of ${\cal O}(\chi + 2\sigma, 4\sigma)$, with the obvious lifting. (Of course $\chi + 2\sigma$ and $4\sigma$ may not be integers, but by abuse of notation we will refrain from clearing denominators, since the choice of power does not matter.) We can then define stable and semistable points in the sense of geometric invariant theory with respect to this linearization. \begin{propn} \label{3f} If \mbox{$(E, \phi)$}\ is $\sigma$-(semi)stable, then $(T(E,s), s^{-1}\phi)$ is a (semi)stable point with respect to the linearization above. \end{propn} {\em Proof}. Suppose $T = (T(E,s), s^{-1}\phi)$ is not semistable. Then by Mumford's numerical criterion \cite{mf,new} there exists a nontrivial 1-parameter subgroup $\lambda: \mbox{\bbb C}^{\times} \mbox{$\rightarrow$} \mbox{SL($\chi$)}$ such that for any $\tilde{T}$ in the fibre of the dual of our ample bundle over $T$, $\lim_{t \mbox{$\rightarrow$} 0}\lambda(t) \cdot \tilde{T} = 0$. We interpret this limit concretely as follows. Any 1-parameter subgroup of $\mbox{SL($\chi$)}$ can be diagonalized, so there exists a basis $e_{i}$ of $\mbox{\bbb C}^{\chi}$ such that $\lambda(t) \cdot e_{i} = t^{r_{i}}e_{i}$, where $r_{i} \in \mbox{\bbb Z}$ are not all zero and satisfy $\sum_{i} r_{i} = 0$ and $r_{i} \leq r_{j}$ for $i \leq j$. Then $\lim_{t \mbox{$\rightarrow$} 0}\lambda(t) \cdot \tilde{T} = 0$ means that any basis element $(e_i^* \wedge e_j^* \otimes v, e_k) \in \mathop{{\rm Hom}}\nolimits(\Lambda^{2} \mbox{\bbb C}^{\chi}, H^0(\Lambda)) \oplus \mbox{\bbb C}^{\chi}$ which is acted on with weight $\leq 0$ has coefficient zero in the basis expansion of $\tilde{T}$. Because of our choice of linearization, this means that $T(E,s)(e_{i},e_{j}) = 0$ whenever \begin{equation} \label{3c} r_{i} + r_{j} \leq \frac{2\sigma}{\raisebox{.5ex}{$\chi$}\!/2 + \sigma}\,\, r_{\ell}, \end{equation} where $\ell = \max \{ i: \mbox{coefficient of $e_{i}$ in $s^{-1}\phi$ is $\neq 0$} \}$. Let $L \subset E$ be the line bundle generated by $s(e_{1})$. We distinguish between two cases, according to whether $\phi \in H^{0}(L)$. First case: $\phi \in H^{0}(L)$. For $i \leq \raisebox{.5ex}{$\chi$}\!/2 - \sigma + 1$, note that $$ (\raisebox{.5ex}{$\chi$}\!/2 - \sigma)\, r_{1} + (\raisebox{.5ex}{$\chi$}\!/2 + \sigma)\, r_{i} \leq \sum_{i} r_{i} = 0,$$ since the left-hand side can be regarded as the integral over $[0,\chi)$ of a (two-step) step function whose value on $[j-1, j)$ is $\leq r_{j}$. Hence for $i \leq \raisebox{.5ex}{$\chi$}\!/2 - \sigma + 1$, $$ r_{1} + r_{i} \leq \frac{2\sigma}{\raisebox{.5ex}{$\chi$}\!/2 + \sigma}\,\, r_{1} \leq \frac{2\sigma}{\raisebox{.5ex}{$\chi$}\!/2 + \sigma}\,\, r_{\ell},$$ so $T(E,s)(e_{1}, e_{i}) = s(e_{1}) \wedge s(e_{i}) = 0$. Hence $s(e_{i})$ is a section of the same line bundle as $s(e_{1})$, namely $L$. So $\dim H^{0}(L) > \raisebox{.5ex}{$\chi$}\!/2 - \sigma$; since $d$ is large relative to $g$ and $\sigma$, this implies that $\deg L > d/2 - \sigma$, so \mbox{$(E, \phi)$}\ is not $\sigma$-semistable. Second case: $\phi \not\in H^{0}(L)$. For $i \leq \raisebox{.5ex}{$\chi$}\!/2 + \sigma + 1$, $$ (\raisebox{.5ex}{$\chi$}\!/2 + \sigma)\, r_{1} + (\raisebox{.5ex}{$\chi$}\!/2 - \sigma)\, r_{i} \leq 0,$$ for the same reason as above. Hence $$ r_{1} + r_{i} \leq \frac{2\sigma}{\raisebox{.5ex}{$\chi$}\!/2 + \sigma}\,\, r_{i}.$$ We claim that $\ell > \raisebox{.5ex}{$\chi$}\!/2 + \sigma + 1$. If not, then for all $i \leq \ell$, $$r_{1} + r_{i} \leq \frac{2\sigma}{\raisebox{.5ex}{$\chi$}\!/2 + \sigma}\,\, r_{\ell},$$ so that $s(e_{i})$ would be in the same line bundle as $s(e_{1})$. Since $\phi$ is a linear combination of $e_{i}$ for $i \leq \ell$, we would conclude $\phi \in H^{0}(L)$, a contradiction. This proves the claim. So for $i \leq \raisebox{.5ex}{$\chi$}\!/2 + \sigma + 1$, actually $$ r_{1} + r_{i} \leq \frac{2\sigma}{\raisebox{.5ex}{$\chi$}\!/2 + \sigma}\,\, r_{\ell};$$ hence $s(e_{i}) \in H^{0}(L)$ as in the first case. So $\dim H^{0}(L) > \raisebox{.5ex}{$\chi$}\!/2 + \sigma$, and again \mbox{$(E, \phi)$}\ is not $\sigma$-semistable. The proof for stability is similar: the numerical criterion now just says $\lim_{t \mbox{$\rightarrow$} 0}\lambda(t) \cdot \tilde{T} \neq \infty$, so we replace the $\leq$ in \re{3c} by $<$. We just need to note that if $i < \raisebox{.5ex}{$\chi$}\!/2 - \sigma + 1$, then $$ (\raisebox{.5ex}{$\chi$}\!/2 - \sigma)\, r_{1} + (\raisebox{.5ex}{$\chi$}\!/2 + \sigma)\, r_{i} < 0$$ strictly, because either the two step functions are different just to the left of $\raisebox{.5ex}{$\chi$}\!/2 - \sigma$, or the smaller one is identically $r_{1} < 0$. \mbox{ $\Box$} \begin{propn} \label{3g} Let \mbox{$(E, \phi)$}\ be a pair, let $s: \mbox{\bbb C}^{\chi} \mbox{$\rightarrow$} H^{0}(E)$ be a linear map, and let $v \in \mbox{\bbb C}^{\chi}$ satisfy $s(v) = \phi$. Write $T_{s}$ for the composition $\Lambda^{2} \mbox{\bbb C}^{\chi} \stackrel{s}{\mbox{$\rightarrow$}} \Lambda^{2} H^{0}(E) \stackrel{\wedge}{\mbox{$\rightarrow$}} H^{0}(\Lambda)$. If $(T_{s}, v)$ is semistable, then $s$ is an isomorphism and \mbox{$(E, \phi)$}\ is $\sigma$-semistable. \end{propn} {\em Proof}. First of all, note that if $s$ is not injective, then $(T_{s}, v)$ is certainly not semistable. Indeed, if $s(w) = 0$ for some $w$, put $e_{1} = w$, $e_{2} = v$, extend to a basis $\{ e_{i} \}$ of $\mbox{\bbb C}^{\chi}$, and then take the 1-parameter subgroup defined by $r_{1} = -\chi + 2$, $r_{2} = 0$, $r_{3} = \cdots = r_{\chi} = 1$. Then $\ell = 2$, so $$r_{i} + r_{j} \leq \frac{2\sigma}{\raisebox{.5ex}{$\chi$}\!/2 + \sigma}\,\, r_{l}$$ means just $r_{i} + r_{j} \leq 0$. Hence either $i=1$, or $j=1$, or $i=j=2$; in any case, clearly $T_{s}(e_{i},e_{j}) = 0$. Suppose then that $s$ is injective and \mbox{$(E, \phi)$}\ is $\sigma$-unstable. We will prove $(T_{s}, v)$ is unstable. Let $L \subset E$ be the destabilizing bundle. We distinguish two cases, depending on the sign of $d - \deg L - 2g + 2$. First case: $d - \deg L > 2g-2$. Then $H^{1}(\Lambda L^{-1}) = 0$, but $H^{1}(L) = 0$ also since $\deg L > d/2 - \sigma$ which is large relative to $g$. Hence from the long exact sequence of \begin{equation} \label{3d} 0 \mbox{$\longrightarrow$} L \mbox{$\longrightarrow$} E \mbox{$\longrightarrow$} \Lambda L^{-1} \mbox{$\longrightarrow$} 0 \end{equation} we find that $H^{1}(E) = 0$, so $\dim H^{0}(E) = \chi$ and $s$ is an isomorphism. Choose a basis $ e_{1}, \dots , e_{p}$ for $s^{-1}(H^{0}(L))$ and extend to a basis $ e_{1}, \dots , e_{\chi}$ for $\mbox{\bbb C}^{\chi}$. Take the 1-parameter subgroup defined by $r_{i} = p-\chi$ for $i \leq p$, $p$ for $i > p$. Then $r_{\ell} = p - \chi$ if $\phi \in H^{0}(L)$, $p$ if $\phi \not\in H^{0}(L)$. Since $L$ is destabilizing, $p > \raisebox{.5ex}{$\chi$}\!/2 - \sigma$ if $\phi \in H^{0}(L)$, $p > \raisebox{.5ex}{$\chi$}\!/2 + \sigma$ if $\phi \not\in H^{0}(L)$. Either way, $$r_{i} + r_{j} \leq \frac{2\sigma}{\raisebox{.5ex}{$\chi$}\!/2 + \sigma}\,\, r_{l}$$ implies $i,j \leq p$; if $\phi \in H^{0}(L)$, and say $i > p$, then $$r_{i} + r_{j} - \frac{2\sigma}{\raisebox{.5ex}{$\chi$}\!/2 + \sigma}\,\, r_{l} \geq p + (p-\chi)(1 - \frac{2\sigma}{\raisebox{.5ex}{$\chi$}\!/2 + \sigma}) = p\,\,\frac{\chi}{\raisebox{.5ex}{$\chi$}\!/2+\sigma} - \chi\,\frac{\raisebox{.5ex}{$\chi$}\!/2-\sigma}{\raisebox{.5ex}{$\chi$}\!/2+\sigma}$$ $$> (\raisebox{.5ex}{$\chi$}\!/2 - \sigma)\frac{\chi}{\raisebox{.5ex}{$\chi$}\!/2+\sigma} - \chi\,\frac{\raisebox{.5ex}{$\chi$}\!/2-\sigma}{\raisebox{.5ex}{$\chi$}\!/2+\sigma} = 0,$$ whereas if $\phi \not\in H^{0}(L)$, and say $j > p$, then $$r_{i} + r_{j} - \frac{2\sigma}{\raisebox{.5ex}{$\chi$}\!/2 + \sigma}\,\, r_{l} \geq p - \chi + p\,(1 - \frac{2\sigma}{\raisebox{.5ex}{$\chi$}\!/2 + \sigma}) = p\,\,\frac{\chi}{\raisebox{.5ex}{$\chi$}\!/2+\sigma} - \chi > \chi - \chi = 0.$$ But if $i, j \leq p$, then $s(e_{i}), s(e_{j}) \in H^{0}(L)$, so $T_{s}(e_{i}, e_{j}) = 0$. Hence $(T_{s}, v)$ is unstable. Second case: $d - \deg L \leq 2g-2$. Then $\dim H^{0}(\Lambda L^{-1}) \leq g$, so from the long exact sequence of \re{3d} we deduce that the codimension of $H^{0}(L)$ in $H^{0}(E)$ is $\leq g$. Hence the codimension of $s^{-1}(H^{0}(L))$ in $\mbox{\bbb C}^{\chi}$ is $\leq g$. Choose a basis $ e_{1}, \dots , e_{p}$ for $s^{-1}(H^{0}(L))$ and extend to a basis $ e_{1}, \dots , e_{\chi}$ for $\mbox{\bbb C}^{\chi}$. Take the 1-parameter subgroup defined by $r_{i} = p-\chi$ for $i \leq p$, $p$ for $i > p$. Since $p \geq \chi - g$ and $\chi = d + 2 - 2g$ is large relative to $\sigma$ and $g$, certainly $p > \raisebox{.5ex}{$\chi$}\!/2 + \sigma$. The remainder of the proof proceeds as in the first case. So far we have proved that if $(T_{s}, v)$ is semistable, then $s$ is injective and \mbox{$(E, \phi)$}\ is $\sigma$-semistable. But then by \re{3e}, $\dim H^{0}(E) = \chi$, so $s$ is an isomorphism. \mbox{ $\Box$} \begin{propn} \label{3h} Suppose $(E_{1}, \phi_{1})$ and $(E_{2}, \phi_{2})$ are $\sigma$-semistable, and there exist $s_{1}, s_{2}$ such that $(T(E_{1}, s_{1}), s_{1}^{-1}\phi_{1}) = (T(E_{2}, s_{2}), s_{2}^{-1}\phi_{2})$. Then there is an isomorphism $(E_{1}, \phi_{1}) \cong (E_{2}, \phi_{2})$ under which $s_{1} \cong s_{2}$. \end{propn} {\em Proof}. By \re{3e} each $E_{i}$ is globally generated, so the components $s_{i}(e_{j}) \wedge s_{i}(e_{k})$ of $T(E_{i}, s_{i})$ give a map from $X$ to the Grassmannian of $(\chi - 2)$-planes in $\mbox{\bbb C}^{\chi}$ such that $E_{i}$ is the pullback of the tautological rank 2 bundle, $\phi_{i}$ is the pullback of the section defined by $s_{i}^{-1}(\phi_{i})$, and $s_{i}$ is the natural map from $\mbox{\bbb C}^{\chi}$ to the space of sections of the tautological bundle. So we can recover $(E_{i}, \phi_{i})$ and $s_{i}$, up to isomorphism, from $(T(E_{i}, s_{i}), s_{i}^{-1}\phi_{i})$. \mbox{ $\Box$} \begin{propn} \label{3i} Let $C$ be a smooth affine curve and $p \in C$. Let $(\mbox{$\bf E$}, {\bf \Phi})$ be a locally free family of pairs on $X \times C - \{ p \}$, and suppose \mbox{$\bf E$}\ is generated by finitely many sections $s_{i}$. Then after possibly rescaling ${\bf \Phi}$ by a function on $C - \{ p \}$, $(\mbox{$\bf E$}, {\bf \Phi})$ and the $s_{i}$ extend over $p$ so that \mbox{$\bf E$}\ is still locally free, ${\bf \Phi}_{p} \neq 0$, and the $s_{i}$ generate $\mbox{$\bf E$}_{p}$ at the generic point. \end{propn} The reason for proving the last fact is to ensure that $T(E,s)$ is nonzero at $p$, so defines an element of $\mbox{\bbb P} \mathop{{\rm Hom}}\nolimits$. {\em Proof}. Choose an ample line bundle $L$ on $X \times C - \{ p \}$ such that $\mbox{$\bf E$}^{*} \otimes L$ is globally generated. Then \mbox{$\bf E$}\ embeds in a direct sum of copies of $L$, and $\oplus_{j} L$ can be extended over $p$ as a sum of line bundles in such a way that the $s_{i}$ extend too. Consider the subsheaf of the extended $\oplus_{j} L$ generated by the $s_{i}$. This is a subsheaf of a locally free sheaf, so it is torsion-free, and hence \cite{oss} has singular set $S$ of codimension $\geq 2$. Furthermore, it injects into its double dual, whose singular set has codimension $\geq 3$ \cite{oss}, hence is empty. Hence the double dual is a locally free extension of \mbox{$\bf E$}\ over $p$, and is generated by $s_{i}$ away from $S$. As for ${\bf \Phi}$, it certainly extends with a possible pole at $p$, so it is just necessary to multiply it by a function on $C$ vanishing to some order at $p$. \mbox{ $\Box$} We can finally proceed to construct the geometric invariant theory quotient. Consider the Grothendieck Quot scheme \cite{grot} parametrizing flat quotients of ${\cal O}^{\chi}_{X}$ with degree $d$, let $\mathop{{\rm Quot}}\nolimits(\Lambda) \subset \mathop{{\rm Quot}}\nolimits$ be the locally closed subset consisting of locally free quotients $E$ with $\Lambda^{2} E = \Lambda$, and let $U \subset \mathop{{\rm Quot}}\nolimits(\Lambda)$ be the open set where the quotient induces an isomorphism $s: \mbox{\bbb C}^{\chi} \mbox{$\rightarrow$} H^{0}(E)$. Then the pair $E,s$ specifies a point in $U$. By \re{3e}, if $(E_{p}, \phi)$ is $\sigma$-semistable for any section $\phi$, then $p \in U$. Now $U$ is acted upon by \mbox{SL($\chi$)}\ in the obvious way, and the map $$T \times 1: U \times \mbox{\bbb P} \mbox{\bbb C}^{\chi} \mbox{$\rightarrow$} \mbox{\bbb P} \mathop{{\rm Hom}}\nolimits \times \mbox{\bbb P} \mbox{\bbb C}^{\chi}$$ intertwines the group actions on the two sets. By \re{3f} and \re{3g}, the $\sigma$-semistable set $V(\sigma) \subset U \times \mbox{\bbb P} \mbox{\bbb C}^{\chi}$ is the inverse image of the semistable set $V'(\sigma) \subset \mbox{\bbb P} \mathop{{\rm Hom}}\nolimits \times \mbox{\bbb P} \mbox{\bbb C}^{\chi}$ with respect to the linearization ${\cal O}(\chi+2\sigma, 4\sigma)$. In future, we restrict $T \times 1$ to a map $V(\sigma) \mbox{$\rightarrow$} V'(\sigma)$. Now Gieseker proves the following. \begin{lemma} Let $G$ be a reductive group and $M_{1}$ and $M_{2}$ be two $G$-spaces. Suppose that $f: M_{1} \mbox{$\rightarrow$} M_{2}$ is a finite $G$-morphism and that a good quotient $M_{2} / / G$ exists. Then a good quotient $M_{1} / / G$ exists, and the induced morphism $M_{1} / / G \mbox{$\rightarrow$} M_{2} / / G$ is finite. \mbox{ $\Box$} \end{lemma} So to show that $V(\sigma)$ has a good quotient it suffices to prove: \begin{lemma} On $V(\sigma)$, $T \times 1$ is finite. \end{lemma} {\em Proof}. By \re{3h}, $T \times 1$ is injective. We use the valuative criterion to check that $T \times 1$ is proper. Let $C$ be a smooth curve, $p \in C$, and let $\Psi: C - \{ p \} \mbox{$\rightarrow$} V(\sigma)$ be a map such that $(T \times 1) \Psi$ extends to a map $C \mbox{$\rightarrow$} V'(\sigma)$. On $C - \{ p \}$, we then have a family $(\mbox{$\bf E$}, {\bf \Phi})$ of pairs such that \mbox{$\bf E$}\ is generated by the sections $s(e_{1}), \dots , s(e_{\chi})$. By \re{3i}, on an open affine of $C$ containing $p$, $(\mbox{$\bf E$}, {\bf \Phi})$ extends over $p$ in such a way that ${\bf \Phi}_{p} \neq 0$ and the $s(e_{i})$ generically generate $\mbox{$\bf E$}_{p}$. Thus $T(\mbox{$\bf E$}_{p}, s)$ is defined, and so by continuity $(T(\mbox{$\bf E$}_{p}, s), s^{-1}{\bf \Phi}_{p}) = ((T \times 1) \Psi)(p)$. Hence by \re{3g} $s: \mbox{\bbb C}^{\chi} \mbox{$\rightarrow$} H^{0}(\mbox{$\bf E$}_{p})$ is an isomorphism and $(\mbox{$\bf E$}_{p}, {\bf \Phi}_{p})$ is $\sigma$-semistable. So $(\mbox{$\bf E$}_{p}, s^{-1}{\bf \Phi}_{p}) \in V(\sigma)$ and $\Psi$ extends to a map $C \mbox{$\rightarrow$} V(\sigma)$. \mbox{ $\Box$} Hence $V(\sigma)$ has a good projective quotient. By \re{3j}, the closure of the orbit of $\mbox{$(E, \phi)$}$ contains the orbit of $\mathop{{\rm Gr}}\nolimits \mbox{$(E, \phi)$}$, which is closed in the $\sigma$-semistable set. But the closure of any orbit in the $\chi$-semistable set contains only one closed orbit \cite[3.14 (iii)]{new}. Hence if two pairs are $\sigma$-semistable, then the closures of their orbits intersect if and only if they have the same $\mathop{{\rm Gr}}\nolimits$. This completes the proof of our main theorem \re{3b}. \mbox{ $\Box$} \begin{remk} \label{3p} If $D$ is any effective divisor, by \re{3o} there is an inclusion $\iota_D: M(\sigma, \Lambda) \hookrightarrow M(\sigma, \Lambda(2D))$. Indeed, if $(\mbox{$\bf E$}^{\Lambda}, {\bf \Phi}^{\Lambda})$ and $(\mbox{$\bf E$}^{\Lambda(2D)}, {\bf \Phi}^{\Lambda(2D)})$ are the corresponding universal pairs, there is a sequence $$0 \mbox{$\longrightarrow$} \mbox{$\bf E$}^{\Lambda} \stackrel{\iota_D}{\mbox{$\longrightarrow$}} \iota_D^* \mbox{$\bf E$}^{\Lambda(2D)} \mbox{$\longrightarrow$} {\cal O}_D (\iota_D^* \mbox{$\bf E$}^{\Lambda(2D)}) \mbox{$\longrightarrow$} 0$$ such that $\iota_D({\bf \Phi}^{\Lambda}) = {\bf \Phi}^{\Lambda(2D)}$. \end{remk} One other pleasant fact should be mentioned: that the stable subsets of these moduli spaces are fine. \begin{propn} \label{3m} There exists a universal pair over the $\sigma$-stable set $M_s(\sigma,\Lambda)$. \end{propn} {\em Proof}. There is a universal bundle $\mbox{$\bf E$} \mbox{$\rightarrow$} \mathop{{\rm Quot}}\nolimits(\Lambda) \times X$ and a surjective map ${\cal O}^{\chi} \mbox{$\rightarrow$} \mbox{$\bf E$}$. Hence there is a natural $\mbox{SL($\chi$)}$-invariant section ${\bf \Phi} \in H^0(\mathop{{\rm Quot}}\nolimits(\Lambda) \times \mbox{\bbb P}\mbox{\bbb C}^{\chi} \times X; \mbox{$\bf E$}(1))$, and $(\mbox{$\bf E$}(1), {\bf \Phi})$ is a universal pair. By \re{3k} the only stabilizers of elements of the $\sigma$-stable subset of $V(\sigma)$ are the $\chi$th roots of unity. These act oppositely on $\mbox{$\bf E$}$ and on ${\cal O}(1)$, hence trivially on $\mbox{$\bf E$}(1)$, so on the $\sigma$-stable set $\mbox{$\bf E$}(1)$ is invariant under stabilizers. Hence by Kempf's descent lemma \cite{dn} $\mbox{$\bf E$}(1)$ descends to a bundle on $M_s(\sigma, \Lambda) \times X$, and the section ${\bf \Phi}$, being invariant, also descends. This pair over $M_s(\sigma, \Lambda) \times X$ then has the desired universal property. \mbox{ $\Box$} \bit{Their tangent spaces} We now turn to the deformation theory of our spaces. By semicontinuity $\sigma$-stability is an open condition, so the Zariski tangent spaces to our moduli spaces at the $\sigma$-stable points will just be deformation spaces. Hence we may refer to $T_{(E, \phi)} M(\sigma, \Lambda)$ simply as $T_{(E, \phi)}$. \begin{propn} \label{4c} If $\mbox{$(E, \phi)$} \in M(\sigma, \Lambda)$ is $\sigma$-stable, then {\rm (i) \,\,}(cf.\ {\rm \cite{bd}}) \,$T_{(E, \phi)}$ is canonically isomorphic to $H^{1}$ of the complex $$ C^{0}(\mathop{{\rm End}}\nolimits_{0} E) \oplus \mbox{\bbb C} \stackrel{p}{\mbox{$\longrightarrow$}} C^{1}(\mathop{{\rm End}}\nolimits_{0} E) \oplus C^{0}(E) \stackrel{q}{\mbox{$\longrightarrow$}} C^{1}(E), $$ where $p(g,c) = (dg, (g+c)\phi)$ and $q(f, \psi) = f \phi - d\psi$; {\rm (ii) \,\,}$H^0$ and $H^2$ of this complex vanish; {\rm (iii) \,\,}there is a natural exact sequence $$0 \mbox{$\longrightarrow$} H^{0}(\mathop{{\rm End}}\nolimits E) \stackrel{\phi}{\mbox{$\longrightarrow$}} H^{0}(E) \mbox{$\longrightarrow$} T_{(E, \phi)} \mbox{$\longrightarrow$} H^{1}(\mathop{{\rm End}}\nolimits_{0} E) \stackrel{\phi}{\mbox{$\longrightarrow$}} H^{1}(E) \mbox{$\longrightarrow$} 0. $$ \end{propn} {\em Proof}. Let $R = \mbox{\bbb C}[\varepsilon]/(\varepsilon^{2})$. By a well-known result \cite[II Ex.\ 2.8]{h} $T_{(E, \phi)}$ is the set of isomorphism classes of maps $\mathop{{\rm Spec}}\nolimits R \mbox{$\rightarrow$} M(\sigma, \Lambda)$ such that $(\varepsilon) \mapsto \mbox{$(E, \phi)$}$. Since $\sigma$-stability is an open condition, $T_{(E, \phi)}$ is just the set of isomorphism classes of families $(\mbox{$\bf E$}, {\bf \Phi})$ of pairs on $X$ with base $\mathop{{\rm Spec}}\nolimits R$, such that $(\mbox{$\bf E$}, {\bf \Phi})_{(\varepsilon)} = \mbox{$(E, \phi)$}$ and $\Lambda^{2} \mbox{$\bf E$}$ is the pullback of $\Lambda$. We will explain how to construct any such family. The only open set in $\mathop{{\rm Spec}}\nolimits R$ containing $(\varepsilon)$ is $\mathop{{\rm Spec}}\nolimits R$ itself, so any bundle $\mbox{$\bf E$}$ over $\mathop{{\rm Spec}}\nolimits R \times X$ can be trivialized on $\mathop{{\rm Spec}}\nolimits R \times U_{\alpha}$ for some open cover $\{ U_{\alpha} \}$ of $X$. Thus if $\mbox{$\bf E$}_{(\varepsilon)} = E$, the transition functions give a \v{C}ech cochain of the form $1 + \varepsilon f_{\alpha \beta}$ where $f \in C^{1}(\mathop{{\rm End}}\nolimits E)$. In order for $\Lambda^{2} \mbox{$\bf E$}$ to be isomorphic to the pullback of $\Lambda$, the transition functions of $\Lambda^{2} \mbox{$\bf E$}$ must be conjugate to $1 \in C^{0}({\cal O})$. But the transition functions are $\det (1 + \varepsilon f_{\alpha \beta}) = 1 + \varepsilon \mathop{{\rm tr}}\nolimits f_{\alpha \beta}$, so we are asking that $$(1 + \varepsilon g_{\alpha}) (1 + \varepsilon \mathop{{\rm tr}}\nolimits f_{\alpha \beta}) (1 - \varepsilon g_{\beta}) = 1$$ for some $g \in C^{0}({\cal O})$, that is, $\mathop{{\rm tr}}\nolimits f = -d g$. But if such a $g$ exists, then $\tilde{f} = f + dg/2$ is trace-free, and $1 + \varepsilon \tilde{f}$ is obviously conjugate to $1 + \varepsilon f$, so determines the same bundle \mbox{$\bf E$}. Hence up to isomorphism we can obtain any \mbox{$\bf E$}\ even if we consider only trace-free $f \in C^{1}(\mathop{{\rm End}}\nolimits_{0} E)$. Now if there is a section ${\bf \Phi} \in H^{0}(\mbox{$\bf E$})$ such that ${\bf \Phi}_{(\varepsilon)} = \phi$, then with respect to the local trivializations of \mbox{$\bf E$}\ described above, ${\bf \Phi} = \phi + \varepsilon \psi_{\alpha}$ for some \v{C}ech cochain $\psi \in C^{0}(E)$. Of course, $\psi$ must be compatible with the transition functions; this means that $$(1 + \varepsilon f_{\alpha \beta}) (\phi + \varepsilon \psi_{\beta}) = (\phi + \varepsilon \psi_{\alpha}),$$ that is, $f \phi = d \psi$. Hence any pair $(\mbox{$\bf E$}, {\bf \Phi})$ having the desired properties can be obtained from some $(f, \psi) \in C^{1}(\mathop{{\rm End}}\nolimits_{0} E) \oplus C^{0}(E)$ satisfying $f \phi - d\psi = 0 \in C^{1}(E)$. We now need only check which $(f, \psi)$ give us isomorphic $(\mbox{$\bf E$}, {\bf \Phi})$. Of course the two choices will be related by a change of trivialization on $\mathop{{\rm Spec}}\nolimits R \times U_{\alpha}$, but we may assume that the change of trivialization is of the form $1 + \varepsilon g_{\alpha}$ on $U_{\alpha}$, since \mbox{$(E, \phi)$}\ itself has no automorphisms \re{3k}. Furthermore, $g$ must belong to $C^{0}(\mathop{{\rm End}}\nolimits_{0} E) \oplus \mbox{\bbb C}$ in order to keep $f$ trace-free, since the action of $g$ is given by $$1 + \varepsilon f_{\alpha \beta} \mapsto (1 + \varepsilon g_{\alpha}) (1 + \varepsilon f_{\alpha \beta}) (1 - \varepsilon g_{\beta}),$$ that is, $f \mapsto f + dg$, and $dg$ is trace-free if and only if $g \in C^{0}(\mathop{{\rm End}}\nolimits E)$ is the sum of a trace-free cocycle and a constant. Similarly the action of $g$ on $\psi$ is $$\phi + \varepsilon \psi_{\alpha} \mapsto (1 + \varepsilon g_{\alpha}) (\phi + \varepsilon \psi_{\alpha}), $$ that is, $\psi \mapsto \psi + g \phi$. Hence two pairs $(f, \psi)$ and $(\tilde{f}, \tilde{\psi})$ determine isomorphic pairs $(\mbox{$\bf E$}, {\bf \Phi})$ if and only if they are in the same coset of the image of the map $C^{0}(\mathop{{\rm End}}\nolimits_{0} E) \oplus \mbox{\bbb C} \mbox{$\rightarrow$} C^{1}(\mathop{{\rm End}}\nolimits_{0} E) \oplus C^{0}(E)$ given by $g + c \mapsto (dg, (g+c)\phi)$. This completes the proof of (i). As for (ii) and (iii), substituting $H^{0}(\mathop{{\rm End}}\nolimits_{0} E) \oplus \mbox{\bbb C} = H^{0}(\mathop{{\rm End}}\nolimits E)$ into the long exact sequence of the double complex with exact rows $$ \begin{array}{ccccccccc} 0 & \mbox{$\longrightarrow$} & 0 & \mbox{$\longrightarrow$} & C^{0}(\mathop{{\rm End}}\nolimits_{0} E) \oplus \mbox{\bbb C} & \mbox{$\longrightarrow$} & C^{0}(\mathop{{\rm End}}\nolimits_{0} E) \oplus \mbox{\bbb C} & \mbox{$\longrightarrow$} & 0 \\ & & \Bdal{} & & \Bdal{} & & \Bdal{} & & \\ 0 & \mbox{$\longrightarrow$} & C^{0}(E) & \mbox{$\longrightarrow$} & C^{1}(\mathop{{\rm End}}\nolimits_{0} E) \oplus C^{0}(E) & \mbox{$\longrightarrow$} & C^{1}(\mathop{{\rm End}}\nolimits_{0} E) & \mbox{$\longrightarrow$} & 0 \\ & & \Bdal{} & & \Bdal{} & & \Bdal{} & & \\ 0 & \mbox{$\longrightarrow$} & C^{1}(E) & \mbox{$\longrightarrow$} & C^{1}(E) & \mbox{$\longrightarrow$} & 0 & \mbox{$\longrightarrow$} & 0 \end{array} $$ gives $$0 \mbox{$\longrightarrow$} H^{0} \mbox{$\longrightarrow$} H^{0}(\mathop{{\rm End}}\nolimits E) \mbox{$\longrightarrow$} H^{0}(E) \mbox{$\longrightarrow$} H^{1} \mbox{$\longrightarrow$} H^{1}(\mathop{{\rm End}}\nolimits_{0} E) \mbox{$\longrightarrow$} H^{1}(E) \mbox{$\longrightarrow$} H^{2} \mbox{$\longrightarrow$} 0,$$ where $H^{i}$ is the cohomology of the complex from (i). But the map $H^{0}(\mathop{{\rm End}}\nolimits E) \stackrel{\phi}{\mbox{$\longrightarrow$}} H^{0}(E)$ is injective for \mbox{$(E, \phi)$}\ $\sigma$-stable by \re{3k}, and the map $H^{1}(\mathop{{\rm End}}\nolimits_{0} E) \stackrel{\phi}{\mbox{$\longrightarrow$}} H^{1}(E)$ is always surjective: indeed this is equivalent to the Serre dual map $H^{0}(K E^* ) \stackrel{\phi}{\mbox{$\longrightarrow$}} H^{0}(K \mathop{{\rm End}}\nolimits_{0} E^{*})$ being injective, which is obvious since the map $K E^{*} \stackrel{\phi}{\mbox{$\longrightarrow$}} K \mathop{{\rm End}}\nolimits_{0} E^{*}$ is an injection of sheaves. Hence $H^{0}$ and $H^{2}$ vanish, and we get the exact sequence in (iii). \mbox{ $\Box$} As a corollary, we obtain the following. \begin{cor} \label{4w} If $\mbox{$(E, \phi)$} \in M(\sigma, \Lambda)$ is $\sigma$-stable, then $\dim T_{(E, \phi)} = d+g-2$. \end{cor} {\em Proof}. By \re{4c}(iii) $$\dim T_{(E, \phi)} = \chi(E) - \chi(\mathop{{\rm End}}\nolimits_{0} E) - 1 = (d+2-2g) - (3-3g) -1 = d+g-2. \mbox{ $\Box$} $$ We will see in the next section that $\dim M(\sigma, \Lambda) = d+g-2$; hence $M(\sigma, \Lambda)$ will be smooth at the stable points. \bit{How they vary with $\sigma$} For obvious numerical reasons the $\sigma$-semistability condition remains the same, and implies $\sigma$-stability, for any $\sigma \in (\max (0, d/2 - i - 1), d/2 - i)$, where $i$ is an integer between 0 and $(d-1)/2$. Hence for $\sigma$ in that interval we get a fixed smooth projective moduli space $M(\sigma, \Lambda)$, which we will henceforth denote $M_{i}(\Lambda)$ or just $M_{i}$. The remainder of this paper will concentrate on these smooth moduli spaces $M_i$, ignoring the special values of $\sigma$ for which there exist $\sigma$-semistable pairs which are not $\sigma$-stable. In the extreme case $i=0$, it is then easy to construct the moduli space: \begin{equation} \label{4n} M_{0}(\Lambda) = \mbox{\bbb P} H^{1}(\Lambda^{-1}). \end{equation} {\em Proof}. The first inequality in the $\sigma$-stability condition \re{3l} says that $\phi \in H^{0}(L)$ implies $\deg L \leq 0$. Hence $L = {\cal O}$, $E$ is an extension of ${\cal O}$ by $\Lambda$, and $\phi \in H^{0}({\cal O})$ is a constant section. The second inequality says that $E$ has no subbundles of degree $\geq d$: this is equivalent to not being split, since $M \mbox{$\rightarrow$} E \mbox{$\rightarrow$} \Lambda$ nonzero and $\deg M \geq d = \deg \Lambda$ implies $M = \Lambda$. Hence $M_{0}(\Lambda)$ is simply the moduli space of nonsplit extensions of ${\cal O}$ by $\Lambda$, which is of course just $\mbox{\bbb P} H^{1}(\Lambda^{-1})$. \mbox{ $\Box$} We will not attempt such a direct construction of $M_{i}(\Lambda)$ for $i > 0$. Rather, we will carefully study the relationship between $M_{i-1}$ and $M_{i}$. Of course, this will only be of interest if there exists an $M_i$ for $i>0$, so {\em we will assume for the remainder of the paper that $[(d-1)/2] > 0$, that is, $d \geq 3$}. Anyhow, the first step is to construct families parametrizing those pairs which appear in $M_{i}$ but not $M_{i-1}$, or $M_{i-1}$ but not $M_{i}$. To do this, we first define two vector bundles over the $i$th symmetric product $X_i$. Let $\pi: X_i \times X \mbox{$\rightarrow$} X_i$ be the projection and let $\Delta \subset X_i \times X$ be the universal divisor. Then define $W^-_i = (R^0 \pi) {\cal O}_{\Delta}\Lambda(-\Delta)$ and $W^+_i = (R^{1}\pi)\Lambda^{-1} (2\Delta)$. These are locally free sheaves of rank $i$ and $d+g-1-2i$, respectively. \begin{propn} \label{4g} For $i \leq (d-1)/2$, there is a family over $\mbox{\bbb P} W^+_i$ parametrizing exactly those pairs which are represented in $M_i$ but not $M_{i-1}$. \end{propn} {\em Proof}. As we pass from $i$ to $i-1$, the first inequality in the stability condition \re{3l} gets stronger and the second gets weaker. So we look for pairs which almost violate the first inequality. That is, $E$ must be an extension $$0 \mbox{$\longrightarrow$} {\cal O}(D) \mbox{$\longrightarrow$} E \mbox{$\longrightarrow$} \Lambda(-D) \mbox{$\longrightarrow$} 0,$$ where $\deg D = i$, and $\phi$ is the section of ${\cal O}(D)$ vanishing on $D$. Conversely, any such pair is stable unless it splits $E = {\cal O}(D) \oplus \Lambda(-D)$. Indeed, if $L \subset E$ and $\phi \not\in H^0(L)$, then the map $L \mbox{$\rightarrow$} \Lambda(-D)$ is nonzero, so $\deg L \leq \deg \Lambda(-D) = d-i$, with equality only if $L = \Lambda(-D)$. But $\mbox{\bbb P} W^+_i$ is the base of a family parametrizing all such nonsplit pairs: indeed \mbox{$\bf E$}\ is the tautological extension $$0 \mbox{$\longrightarrow$} {\cal O}(\Delta) \mbox{$\longrightarrow$} \mbox{$\bf E$} \mbox{$\longrightarrow$} \Lambda(-\Delta)(-1) \mbox{$\longrightarrow$} 0,$$ and ${\bf \Phi}$ is the section of ${\cal O}(\Delta)$ vanishing on $\Delta$. \mbox{ $\Box$} \begin{propn} \label{4h} For $i \leq (d-1)/2$, there is a family over $\mbox{\bbb P} W^-_i$ parametrizing exactly those pairs which are represented in $M_{i-i}$ but not $M_{i}$. \end{propn} {\em Proof}. This time the first inequality in \re{3l} gets weaker and the second gets stronger. So we look for pairs which almost violate the second inequality. That is, $E$ is an extension $$ 0 \mbox{$\longrightarrow$} M \mbox{$\longrightarrow$} E \mbox{$\longrightarrow$} \Lambda M^{-1} \mbox{$\longrightarrow$} 0 $$ where $\deg M = d-i$, and $\phi \not\in H^0(M)$. Hence projecting $\phi$ in the exact sequence, we get a nonzero $\gamma \in H^0(\Lambda M^{-1})$ vanishing on a divisor $D$ of degree $i$ such that $\Lambda M^{-1} = {\cal O}(D)$. Then at $D$, $\phi$ lifts to $M = \Lambda(-D)$, so we get an element $p\mbox{$(E, \phi)$} \in H^0({\cal O}_D\Lambda(-D))$, defined up to a scalar as usual. On the other hand, we can recover \mbox{$(E, \phi)$}\ from $D$ and $p$. Indeed, choose a \v Cech cochain $\psi \in C^0 (\Lambda(-D))$ such that $\psi|_D = p$. Then $d\psi|_D = dp = 0$, so $d\psi$ vanishes on $D$ and descends to a closed cochain $f = d\psi/\gamma \in C^1((\Lambda(-2D))$. This determines an extension $$ 0 \mbox{$\longrightarrow$} \Lambda(-D) \mbox{$\longrightarrow$} E' \mbox{$\longrightarrow$} {\cal O}(D) \mbox{$\longrightarrow$} 0.$$ The compatibility condition for $\gamma + \psi$ to define a section $\phi' \in H^0(E')$ is $\gamma f = d \psi$, which is automatic. Thus we get a new pair $(E', \phi')$ satisfying $p(E', \phi') = p$. Up to isomorphism, $(E', \phi')$ is independent of the choice of $\psi$, since adding $\xi \in C^0(\Lambda(-2D))$ to $\psi$ is simply equivalent to acting by $\left( \begin{array}{cc} 1 & \xi_{\alpha} \\ 0 & 1 \end{array} \right)$ on the local splittings of $E'$ with which the extension is defined. In particular, we can choose local splittings of the old $E$ and let $\psi$ be the projection of the old $\phi$ on $M = \Lambda(-D)$ with respect to these splittings. Then the construction of the previous paragraph recovers \mbox{$(E, \phi)$}, so $(E', \phi') = \mbox{$(E, \phi)$}$. The construction above can be generalized to produce a family $(\mbox{$\bf E$}, {\bf \Phi}) \mbox{$\rightarrow$} \mbox{\bbb P} W^-_i \times X$, as follows. Let $p: \mbox{\bbb P} W^-_i \mbox{$\rightarrow$} X_i$ be the projection, and choose a cochain $\Psi \in C^0(\Lambda(-\Delta)(1))$ such that $\Psi|_{p^{-1}\Delta}$ is the tautological section. Then $d\Psi$ vanishes on $p^{-1}\Delta$, so descends to $C^1(\Lambda(-2\Delta)(1))$. This determines an extension $$0 \mbox{$\longrightarrow$} \Lambda(-\Delta)(1) \mbox{$\longrightarrow$} \mbox{$\bf E$} \mbox{$\longrightarrow$} {\cal O}(\Delta) \mbox{$\longrightarrow$} 0,$$ and if $\gamma \in H^0({\cal O}(\Delta))$ is the section vanishing on $\Delta$, then $\gamma + \Psi$ defines the desired section ${\bf \Phi} \in H^0(\mbox{$\bf E$})$. \mbox{ $\Box$} By the universal properties of $M_{i-1}$ and $M_i$, we thus get injections $\mbox{\bbb P} W^+_i \hookrightarrow M_{i}$ and $\mbox{\bbb P} W^-_i \hookrightarrow M_{i-1}$. As an example, consider the case $i = 1$. By \re{4n}, $M_0 = \mbox{\bbb P} H^1(\Lambda^{-1})$. Moreover, $W^-_1$ is a line bundle and hence $\mbox{\bbb P} W^-_1 = X_1 = X$. Hence the inclusion of \re{4h} is a map $X \hookrightarrow \mbox{\bbb P} H^1(\Lambda^{-1})$; it can be identified explicitly as follows. \begin{propn} The inclusion $X \hookrightarrow \mbox{\bbb P} H^1(\Lambda^{-1})$ is given by the complete linear system $|K_X\Lambda|$. \end{propn} {\em Proof}. There is an alternative way to see what pairs are represented in $M_0$ but not $M_1$. Any pair $\mbox{$(E, \phi)$} \in M_0$ is an extension \begin{equation} \label{4o} 0 \mbox{$\longrightarrow$} {\cal O} \mbox{$\longrightarrow$} E \mbox{$\longrightarrow$} \Lambda \mbox{$\longrightarrow$} 0, \end{equation} say with extension class $s \in H^1(\Lambda^{-1})$, and with $\phi \in H^0({\cal O})$. Such a pair is the image of $x \in X$ under the injection of \re{4h} if there is an inclusion $0 \mbox{$\rightarrow$} \Lambda(-x) \mbox{$\rightarrow$} E$ such that the composition $\gamma_x: \Lambda(-x) \mbox{$\rightarrow$} E \mbox{$\rightarrow$} \Lambda$ vanishes at $x$. Hence we ask for what extension classes $s \in H^1(\Lambda^{-1})$ the map $\gamma_x: \Lambda(-x) \mbox{$\rightarrow$} \Lambda$ lifts to $E$. Twisting \re{4o} by $\Lambda^{-1}(x)$ and taking the long exact sequence yields $$H^0(E \otimes \Lambda^{-1}(x)) \mbox{$\longrightarrow$} H^0({\cal O}(x)) \stackrel{s}{\mbox{$\longrightarrow$}} H^1(\Lambda^{-1}(x)),$$ where the second map is the cup product with $s$. Hence $\gamma_x \in H^0({\cal O}(x))$ lifts to $H^0(E \otimes \Lambda^{-1}(x))$ as desired if and only if $\gamma_x s = 0$. That is, $s$ must be in the kernel of the map $\gamma_x: H^1(\Lambda^{-1}) \mbox{$\rightarrow$} H^1(\Lambda^{-1}(x))$, or Serre dually, $\gamma_x: H^0(K_X\Lambda)^* \mbox{$\rightarrow$} H^0(K_X\Lambda(-x))^*$. Since $\gamma_x$ is dual to the injection $H^0(K_X\Lambda) \mbox{$\rightarrow$} H^0(K_X\Lambda(-x))$, it is surjective, so $$\dim \ker \gamma_x = \dim H^0(K_X\Lambda(-x)) - \dim H^0(K_X\Lambda).$$ But since $\deg K_X\Lambda(-x) > 2g-2$, this is 1. Hence for each $x \in X$, there is a unique $s \in \mbox{\bbb P} H^1(\Lambda^{-1})$ such that $\gamma_x s = 0$. What is this $s$? Regarded as a linear functional on $H^0(K_X\Lambda)$, $s \in \ker \gamma_x$ if it annihilates all sections vanishing at $x$. Certainly evaluation at $x$ does this, so this is the $s$ generating $\ker \gamma_x$. But it is also the image of $x$ in the map $X \hookrightarrow \mbox{\bbb P} H^1(\Lambda^{-1})$ given by $|K_X\Lambda|$. Hence the two maps are identical. \mbox{ $\Box$} \begin{propn} \label{4t} The $M_i$ are all smooth rational integral projective varieties of dimension $d +g-2$, and for $i>0$, there is a birational map $M_i \leftrightarrow M_1$, which is an isomorphism except on sets of codimension $\geq 2$. \end{propn} {\em Proof}. By \re{4n} and Riemann-Roch, the first statement is certainly true of $M_0$. For $i>0$, suppose by induction on $i$ that it is true of $M_{i-1}$. By \re{4g} and \re{4h} there is an isomorphism $M_{i-1} - \mbox{\bbb P} W^-_i \leftrightarrow M_i - \mbox{\bbb P} W^+_i$. But $\dim \mbox{\bbb P} W^-_i = 2i-1 < d-1 < d+g-2$, and $\dim \mbox{\bbb P} W^+_i = d+g-2-i < d+g-2$, so $\dim M_i = \dim M_{i-1} = d+g-2$ and $M_i$ is birational to $M_{i-1}$, hence to $M_0$. Moreover by \re{4w}, the Zariski tangent space to $M_i$ has constant dimension $d+g-2$, so $M_i$ is a smooth reduced variety. The second statement is also proved by induction: just note that for $i > 1$, $\mathop{{\rm codim}}\nolimits \mbox{\bbb P} W^-_i/M_{i-1} = d+g-2i-1 \geq 2$ and $\mathop{{\rm codim}}\nolimits \mbox{\bbb P} W^+_i/M_i = i \geq 2$. \mbox{ $\Box$} \begin{propn} \label{4e} Let $\mbox{$(E, \phi)$} \in \mbox{\bbb P} W^+_i$, let $D$ be the zero-set of $\phi$, and let $\gamma$ be the map $$E \otimes \Lambda^{-1}(D) \mbox{$\rightarrow$} \Lambda(-D) \otimes \Lambda^{-1}(D) = {\cal O}.$$ Then $T_{(E, \phi)}\mbox{\bbb P} W^+_i$ is canonically isomorphic to $H^{1}$ of the complex $$ C^{0}(E \otimes \Lambda^{-1}(D)) \oplus \mbox{\bbb C} \stackrel{p}{\mbox{$\longrightarrow$}} C^{1}(E \otimes \Lambda^{-1}(D)) \oplus C^{0}({\cal O}(D)) \stackrel{q}{\mbox{$\longrightarrow$}} C^{1}({\cal O}(D)), $$ where $p(g,c) = (dg, (\gamma g + c) \phi)$ and $q(f, \psi) = \gamma f \phi - d\psi$. Moreover, $H^0$ and $H^2$ of this complex vanish. \end{propn} {\em Proof}. The proof is modelled on that of \re{4c}. We regard $\mbox{\bbb P} W^+_i$ as a moduli space of triples $(L, E ,\phi)$, where $L$ is a line bundle of degree $i$, $E$ is an extension of $L$ by $\Lambda L^{-1}$, and $\phi \in H^0(L)$, and consider the deformation theory of this moduli problem. Let $R = \mbox{\bbb C}[\varepsilon ]/(\varepsilon ^2)$ as before. Then $T_{(L, E ,\phi)} \mbox{\bbb P} W^+_i$ is the set of isomorphism classes of families $({\bf L}, \mbox{$\bf E$}, {\bf \Phi})$ of triples on $X$ with base $\mathop{{\rm Spec}}\nolimits R$, such that $({\bf L}, \mbox{$\bf E$}, {\bf \Phi})_{(\varepsilon )} = (L, E ,\phi)$. We will explain how to construct any such family. Any bundle over $\mathop{{\rm Spec}}\nolimits R \times X$ can be trivialized on $\mathop{{\rm Spec}}\nolimits R \times U_{\alpha}$ for some open cover $\{ U_{\alpha} \}$ of $X$. Thus if ${\bf L}_{(\varepsilon )} = {\cal O}(D)$ and $\mbox{$\bf E$}_{(\varepsilon )} = E$, then the transition functions for \mbox{$\bf E$}\ give a \v{C}ech cochain of the form $1 + \varepsilon f_{\alpha \beta}$ where $f \in C^{1}(\mathop{{\rm End}}\nolimits E)$. Since \mbox{$\bf E$}\ is to be a family of extensions of $L$ by $\Lambda L^{-1}$, it must have $\Lambda^{2} \mbox{$\bf E$} = \Lambda$, so as explained in the proof of \re{4c} we may take $f \in C^1(\mathop{{\rm End}}\nolimits_0 E)$. Furthermore, the transition functions must preserve $\bf L$, so if $f'$ is the projection of $f$ to $C^1(\Lambda(-2D))$ in the natural exact sequence $$0 \mbox{$\longrightarrow$} E \otimes \Lambda^{-1}(D) \mbox{$\longrightarrow$} \mathop{{\rm End}}\nolimits_0 E \mbox{$\longrightarrow$} \Lambda(-2D) \mbox{$\longrightarrow$} 0,$$ then $1 + \varepsilon f'_{\alpha \beta}$ must be conjugate to 1. Hence $$(1 - \varepsilon g_{\alpha}) (1 + \varepsilon f'_{\alpha \beta}) (1 - \varepsilon g_{\beta}) = 1$$ for some $g \in C^0(\Lambda(-2D))$, that is, $f' = dg$. But if such a $g$ exists, then for any lifting $\tilde{g}$ of $g$ to $C^0(\mathop{{\rm End}}\nolimits_0 E)$, $\tilde{f} = f - d\tilde{g}$ projects to $0 \in C^1(\Lambda(-2D))$, and $1 + \varepsilon \tilde{f}$ is obviously conjugate to $1 + \varepsilon f$, so determines the same bundle \mbox{$\bf E$}. Hence up to isomorphism we can obtain any \mbox{$\bf E$}\ that is an extension of some $\bf L$ by $\Lambda {\bf L}^{-1}$ even if we consider only those $f$ in the kernel of $C^1(\mathop{{\rm End}}\nolimits_0 E) \mbox{$\rightarrow$} C^1(\Lambda(-2D))$, that is, in $C^1(E \otimes \Lambda^{-1}(D))$. The transition functions for $\bf L$ are then just $1 + \varepsilon \gamma f_{\alpha \beta}$. Now if there is a section ${\bf \Phi} \in H^0({\bf L})$ such that ${\bf \Phi}_{(\varepsilon )} = \phi$, then with respect to the local trivializations of \mbox{$\bf E$}, ${\bf \Phi} = \phi + \varepsilon \psi_{\alpha}$ for some \v{C}ech cochain $\psi \in C^0({\cal O}(D))$. Of course, $\psi$ must be compatible with the transition functions; this means that $$(1 + \varepsilon \gamma f_{\alpha \beta}) (\phi + \varepsilon \psi_{\beta}) = (\phi + \varepsilon \psi_{\alpha}),$$ that is, $\gamma f \phi = d \psi$. Hence any triple $({\bf L}, \mbox{$\bf E$}, {\bf \Phi})$ having the desired properties can be obtained from some $(f, \psi) \in C^{1}(E \otimes \Lambda^{-1}(D)) \oplus C^{0}({\cal O}(D))$ satisfying $\gamma f \phi - d\psi = 0 \in C^{1}({\cal O}(D))$. We now need only check which $(f, \psi)$ give us isomorphic $({\bf L},\mbox{$\bf E$}, {\bf \Phi})$. This part of the argument follows that of \re{4c} exactly, except that $g$ ends up being in $C^1(E \otimes \Lambda^{-1}(D)) \oplus \mbox{\bbb C}$, and acts on $\psi$ by $\psi \mapsto \psi + \gamma g \phi$. This completes the proof of the first statement. As for the second, taking the long exact sequence of the double complex $$ \begin{array}{ccccccccc} 0 & \mbox{$\longrightarrow$} & 0 & \mbox{$\longrightarrow$} & C^{0}(E \otimes \Lambda^{-1}(D)) \oplus \mbox{\bbb C} & \mbox{$\longrightarrow$} & C^{0}(E \otimes \Lambda^{-1}(D)) \oplus \mbox{\bbb C} & \mbox{$\longrightarrow$} & 0 \\[.2em] & & \Bdal{} & & \Bdal{} & & \Bdal{} & & \\[.2em] 0 & \mbox{$\longrightarrow$} & C^{0}({\cal O}(D)) & \mbox{$\longrightarrow$} & \def1{.6} \begin{array}{c} C^{1}(E \otimes \Lambda^{-1}(D)) \\ \oplus C^{0}({\cal O}(D)) \end{array} & \mbox{$\longrightarrow$} & C^{1}(E \otimes \Lambda^{-1}(D)) & \mbox{$\longrightarrow$} & 0 \\[.6em] & & \Bdal{} & & \Bdal{} & & \Bdal{} & & \\[.2em] 0 & \mbox{$\longrightarrow$} & C^{1}({\cal O}(D)) & \mbox{$\longrightarrow$} & C^{1}({\cal O}(D)) & \mbox{$\longrightarrow$} & 0 & \mbox{$\longrightarrow$} & 0 \end{array} $$ gives \begin{eqnarray*} \lefteqn{0 \mbox{$\longrightarrow$} H^0 \mbox{$\longrightarrow$} H^0(E \otimes \Lambda^{-1}(D)) \oplus \mbox{\bbb C} \mbox{$\longrightarrow$} H^0({\cal O}(D)) \mbox{$\longrightarrow$} H^1 } \\ & & \mbox{$\longrightarrow$} H^1(E \otimes \Lambda^{-1}(D)) \mbox{$\longrightarrow$} H^1({\cal O}(D)) \mbox{$\longrightarrow$} H^2 \mbox{$\longrightarrow$} 0, \end{eqnarray*} where $H^{i}$ is the cohomology of the complex in the statement. Now $H^0(\Lambda^{-1}(2D)) = 0$ since $\deg \Lambda^{-1}(2D) < 0$, and $E$ is a nonsplit extension of ${\cal O}(D)$ by $\Lambda(-D)$, so $$H^0(E \otimes \Lambda^{-1}(D)) = H^0(\mathop{{\rm Hom}}\nolimits(\Lambda(-D), E)) = 0.$$ But the map $\mbox{\bbb C} \mbox{$\rightarrow$} H^0({\cal O}(D))$ is injective: indeed, it is multiplication by $\phi$. Hence $H^0 = 0$. Likewise, the map $H^1(E \otimes \Lambda^{-1}(D)) \mbox{$\rightarrow$} H^1({\cal O}(D))$ is surjective: indeed this is equivalent to the Serre dual map $H^0(K(-D)) \mbox{$\rightarrow$} H^0(E^* \otimes K\Lambda(-D))$ being injective, which is obvious since the map $K(-D) \mbox{$\rightarrow$} K \mbox{$\rightarrow$} E^* \otimes K\Lambda(-D)$ is an injection of sheaves. Hence $H^2 = 0$. \mbox{ $\Box$} The following proposition is proved similarly. \begin{propn} \label{4v} Let $\mbox{$(E, \phi)$} \in \mbox{\bbb P} W^-_i$, and let $D = p\mbox{$(E, \phi)$}$. Then $T_{(E, \phi)}\mbox{\bbb P} W^-_i$ is canonically isomorphic to $H^{1}$ of the complex $$ C^0 (E(-D)) \oplus \mbox{\bbb C} \mbox{$\longrightarrow$} C^1 (E(-D)) \oplus C^0 (E) \mbox{$\longrightarrow$} C^1 (E). $$ Moreover, $H^0$ and $H^2$ of this complex vanish. \mbox{ $\Box$} \end{propn} \begin{propn} \label{4f} The injection $\mbox{\bbb P} W^+_i \hookrightarrow M_i$ induces an exact sequence on $\mbox{\bbb P} W^+_i$ $$0 \mbox{$\longrightarrow$} T\mbox{\bbb P} W^+_i \mbox{$\longrightarrow$} TM_i |_{\mbox{\blb P} W^+_i} \mbox{$\longrightarrow$} W^-_i(-1) \mbox{$\longrightarrow$} 0.$$ \end{propn} {\em Proof}. The complex $$ C^0 (\Lambda(-2\Delta)) \mbox{$\longrightarrow$} C^1 (\Lambda(-2\Delta)) \oplus C^0 (\Lambda(-\Delta)) \mbox{$\longrightarrow$} C^1 (\Lambda(-\Delta)) $$ with the obvious maps has $R^0 \pi = 0$, $R^1 \pi = W^-_i$ from the long exact sequence of the double complex $$ \begin{array}{ccccccccc} 0 & \mbox{$\longrightarrow$} & C^0 (\Lambda(-2\Delta)) & \mbox{$\longrightarrow$} & C^0(\Lambda(-2\Delta)) & \mbox{$\longrightarrow$} & 0 & \mbox{$\longrightarrow$} & 0 \\[.2em] & & \Bdar{(1,0)} & & \Bdar{} & & \Bdal{} & & \\[.2em] 0 & \mbox{$\longrightarrow$} & \def1{.6} \begin{array}{c} C^0(\Lambda(-2\Delta)) \\ \oplus C^1(\Lambda(-2\Delta)) \end{array} & \mbox{$\longrightarrow$} & \def1{.6} \begin{array}{c} C^0(\Lambda(-\Delta)) \\ \oplus C^1(\Lambda(-2\Delta)) \end{array} & \mbox{$\longrightarrow$} & C^0({\cal O}_{\Delta} \Lambda(-\Delta)) & \mbox{$\longrightarrow$} & 0 \\[.6em] & & \Bdar{(0,1)} & & \Bdar{} & & \Bdal{} & & \\[.2em] 0 & \mbox{$\longrightarrow$} & C^1(\Lambda(-2\Delta)) & \mbox{$\longrightarrow$} & C^1(\Lambda(-\Delta)) & \mbox{$\longrightarrow$} & C^1({\cal O}_{\Delta} \Lambda(-\Delta)) & \mbox{$\longrightarrow$} & 0. \end{array} $$ Hence the result follows from the long exact sequence of the double complex $$ \begin{array}{ccccccccc} 0 & \mbox{$\longrightarrow$} & C^0 (\mbox{$\bf E$} \Lambda^{-1}(\Delta)) \oplus \mbox{\bbb C} & \mbox{$\longrightarrow$} & C^0(\mathop{{\rm End}}\nolimits_0 \mbox{$\bf E$}) \oplus \mbox{\bbb C} & \mbox{$\longrightarrow$} & C^0 (\Lambda(-2\Delta))(-1) & \mbox{$\longrightarrow$} & 0 \\[.2em] & & \Bdar{(1,0)} & & \Bdar{p} & & \Bdal{} & & \\[.2em] 0 & \mbox{$\longrightarrow$} & \def1{.6} \begin{array}{c} C^1(\mbox{$\bf E$} \Lambda^{-1}(\Delta)) \\ \oplus C^0({\cal O}(\Delta)) \end{array} & \mbox{$\longrightarrow$} & \def1{.6} \begin{array}{c} C^1(\mathop{{\rm End}}\nolimits_0 \mbox{$\bf E$}) \\ \oplus C^0(\mbox{$\bf E$}) \end{array} & \mbox{$\longrightarrow$} & \def1{.6} \begin{array}{c} C^1(\Lambda(-2\Delta))(-1) \\ \oplus C^0(\Lambda(-\Delta))(-1) \end{array} & \mbox{$\longrightarrow$} & 0 \\[.6em] \def1{1} & & \Bdar{(0,1)} & & \Bdar{q} & & \Bdal{} & & \\[.2em] 0 & \mbox{$\longrightarrow$} & C^1({\cal O}(\Delta)) & \mbox{$\longrightarrow$} & C^1(\mbox{$\bf E$}) & \mbox{$\longrightarrow$} & C^1(\Lambda(-\Delta))(-1) & \mbox{$\longrightarrow$} & 0, \end{array} $$ together with \re{4c} and \re{4e}. \mbox{ $\Box$} \begin{cor} The map $\mbox{\bbb P} W^+_i \hookrightarrow M_i$ is an embedding. \end{cor} {\em Proof}. By \re{4e}, it is an injection, and by \re{4f}, so is its derivative. \mbox{ $\Box$} The following proposition and corollary are proved similarly, using \re{4c} and \re{4v}. \begin{propn} \label{4q} The injection $\mbox{\bbb P} W^-_i \hookrightarrow M_{i-1}$ induces an exact sequence on $\mbox{\bbb P} W^-_i$ $$0 \mbox{$\longrightarrow$} T\mbox{\bbb P} W^-_i \mbox{$\longrightarrow$} TM_{i-1} |_{\mbox{\blb P} W^-_i} \mbox{$\longrightarrow$} W^+_i(-1) \mbox{$\longrightarrow$} 0. \mbox{ $\Box$} $$ \end{propn} \begin{cor} The map $\mbox{\bbb P} W^-_i \hookrightarrow M_{i-1}$ is an embedding. \mbox{ $\Box$} \end{cor} By \re{4g} and \re{4h} every pair in $M_i - \mbox{\bbb P} W^+_i$ is also in $M_{i-1} - \mbox{\bbb P} W^-_i$, and vice-versa. Hence there is a natural isomorphism $M_i - \mbox{\bbb P} W^+_i \mbox{$\rightarrow$} M_{i-1} - \mbox{\bbb P} W^-_i$. Our next task is to extend this to a proper map. Let $\tilde{M}^+_i$ be the blow-up of $M_i$ at $\mbox{\bbb P} W^+_i$. Then by \re{4f} the exceptional divisor is $E^+_i = \mbox{\bbb P} W^-_i \oplus \mbox{\bbb P} W^+_i$, and ${\cal O}_{E^+_i}(E^+_i) = {\cal O}(-1,-1)$. \begin{propn} \label{4l} There is a map $\tilde{M}^+_i \mbox{$\rightarrow$} M_{i-1}$ such that the following diagram commutes: $$\begin{array}{ccccc}M_i - \mbox{\bbb P} W^+_i & \mbox{$\longrightarrow$} & \tilde{M}^+_i & \longleftarrow & E^+_i \\ \Big\updownarrow & & \Big\downarrow & & \Big\downarrow \\ M_{i-1} - \mbox{\bbb P} W^-_i & \mbox{$\longrightarrow$} & M_{i-1} & \longleftarrow & \mbox{\bbb P} W^-_i.\end{array}$$ \end{propn} {\em Proof}. Let $(\mbox{$\bf E$}, {\bf \Phi}) \mbox{$\rightarrow$} \tilde{M}^+_i \times X$ be the pullback of the universal family. We will construct a new family $(\mbox{$\bf E$}', {\bf \Phi}')$ of pairs all of which are in $M_{i-1}$. By uniqueness of families \re{3n}, $(\mbox{$\bf E$}, {\bf \Phi})|_{E^+_i \times X}$ is the pullback of the family over $\mbox{\bbb P} W^+_i$ constructed in \re{4g}. Thus there is a surjective sheaf map $\mbox{$\bf E$} \mbox{$\rightarrow$} {\cal O}_{E^+_i \times X}\Lambda(-\Delta)(0,-1)$ annihilating ${\bf \Phi}$. Define $\mbox{$\bf E$}'$ to be the kernel of this map, so that \begin{equation} \label{4p} 0 \mbox{$\longrightarrow$} \mbox{$\bf E$}' \mbox{$\longrightarrow$} \mbox{$\bf E$} \mbox{$\longrightarrow$} {\cal O}_{E^+_i \times X}\Lambda(-\Delta)(0,-1) \mbox{$\longrightarrow$} 0. \end{equation} Then $\mbox{$\bf E$}$ is locally free, and ${\bf \Phi}$ descends to ${\bf \Phi}' \in H^0(\mbox{$\bf E$}')$. For $z \in M_i - \mbox{\bbb P} W^+_i$, clearly $(\mbox{$\bf E$}', {\bf \Phi}')_z = (\mbox{$\bf E$}, {\bf \Phi})_z$. So to prove the proposition it suffices to show that $(\mbox{$\bf E$}', {\bf \Phi}')_{E^+_i}$ is the pullback of the family over $\mbox{\bbb P} W^-_i$ constructed in \re{4h}. The first promising thing to note is that there certainly a surjection $\mbox{$\bf E$}' \mbox{$\rightarrow$} {\cal O}_{E^+_i \times X}(\Delta) \mbox{$\rightarrow$} 0$, and $\Lambda^{2} \mbox{$\bf E$}' = \Lambda^{2} \mbox{$\bf E$}(-E^+_i \times X)$, so we get an extension $$0 \mbox{$\longrightarrow$} \Lambda(-\Delta)(1,0) \mbox{$\longrightarrow$} \mbox{$\bf E$}'_{E^+_i \times X} \mbox{$\longrightarrow$} {\cal O}(\Delta) \mbox{$\longrightarrow$} 0,$$ just as in the family of \re{4h}. Now fix $s \in E^+_i$ over $\mbox{$(E, \phi)$} \in M_i$, and let $D$ be the zero-set of $\phi$. Let $R = \mbox{\bbb C}[\varepsilon]/(\varepsilon^2)$ as before, and choose a map $\mathop{{\rm Spec}}\nolimits R \mbox{$\rightarrow$} \tilde{M}^+_i$ representing an element of $T_s \tilde{M}^+_i - T_s E^+_i$. Then \re{4p} restricts to an exact sequence $$0 \mbox{$\longrightarrow$} {\cal O}_{\mathop{{\rm Spec}}\nolimits R \times X}(\mbox{$\bf E$}') \mbox{$\longrightarrow$} {\cal O}_{\mathop{{\rm Spec}}\nolimits R \times X}(\mbox{$\bf E$}) \mbox{$\longrightarrow$} {\cal O}_{(\varepsilon) \times X}\Lambda(-D) \mbox{$\longrightarrow$} 0.$$ On some open cover $\{ U_{\alpha} \}$ of $X$, $E$ splits as \begin{equation} \label{4i} E|_{U_{\alpha}} = {\cal O}(D)|_{U_{\alpha}} \oplus \Lambda(-D)|_{U_{\alpha}}, \end{equation} and this splitting can be extended to a splitting of $\mbox{$\bf E$}|_{\mathop{{\rm Spec}}\nolimits R \times U_{\alpha}}$. Then \begin{equation} \label{4j} \mbox{$\bf E$}'|_{U_{\alpha}} = {\cal O}(D)|_{U_{\alpha}} \oplus \Lambda(D)|_{U_{\alpha}} \otimes {\cal I}_{(\varepsilon)}. \end{equation} The section ${\bf \Phi}$ is then of the form $\phi + \varepsilon\psi_{\alpha}$ for some $\psi \in C^0(E)$, and the transition functions are $1 + \varepsilon f_{\alpha\beta}$ for some $f \in C^1(\mathop{{\rm End}}\nolimits_0 E)$. The latter hence act as 1 on the second factor of \re{4j}. Now decompose $\psi_{\beta} = \psi^{{\cal O}(D)}_{\beta} + \psi^{\Lambda(-D)}_{\beta}$ and $f_{\alpha\beta} = f^{{\cal O}(D)}_{\alpha\beta} + f^{\Lambda(-D)}_{\alpha\beta}$ corresponding to the splitting on $U_{\beta}$. If $\mbox{$\bf E$}'$ is restricted to $(\varepsilon ) \times U_{\alpha}$, then $\varepsilon \psi^{{\cal O}(D)}_{\beta} = 0$ and $\varepsilon f^{{\cal O}(D)}_{\alpha\beta} = 0$, since everything divisible by $\varepsilon $ is now set to zero. However, $\varepsilon \psi^{\Lambda(-D)}_{\beta}$ and $\varepsilon f^{\Lambda(-D)}_{\alpha\beta}$ are not necessarily zero, since not everything in their images is divisible by $\varepsilon $ {\em in the module} $\Lambda(-D) \otimes {\cal I}_{(\varepsilon )}$. Hence ${\bf \Phi}_{(\varepsilon )} = \phi + \varepsilon \psi^{{\cal O}(D)}_{\beta}$ on $U_{\beta}$, and $\mbox{$\bf E$}_{(\varepsilon )}$ has transition functions $\left( \begin{array}{cc} 1 & f^{\Lambda(-D)}_{\alpha\beta} \\ 0 & 1 \end{array} \right)$ with respect to the splitting \re{4j}. In other words, the extension class of $E' = \mbox{$\bf E$}_{(\varepsilon )}$ is the projection of $f \in C^1(\mathop{{\rm End}}\nolimits_0 E)$ to $C^1(\Lambda(2D))$, and the lifting of $\phi'$ is the projection of $\psi \in C^1(E)$ to $C^1(\Lambda(-D))$. Hence $(E', \phi')$ is the bundle over the image of \mbox{$(E, \phi)$}\ in $\mbox{\bbb P}^-_i$ in the family of \re{4h}. By uniqueness of families \re{3n} this means that $(\mbox{$\bf E$}', {\bf \Phi}')|_{E^+_i \times X}$ is the pullback of the family of \re{4h}. \mbox{ $\Box$} There is a result similar to \re{4l} for the inverse map $M_{i-1} - \mbox{\bbb P} W^-_i \mbox{$\rightarrow$} M_i - \mbox{\bbb P} W^+_i$. Let $\tilde{M}^-_{i-1}$ be the blow-up of $M_{i-1}$ at $\mbox{\bbb P} W^-_i$. Hence by \re{4q} the exceptional divisor is $E^-_i = \mbox{\bbb P} W^-_i \oplus \mbox{\bbb P} W^+_i$, and ${\cal O}_{E^-_i}(E^-_i) = {\cal O}(-1,-1)$. Note that there is an isomorphism $E^-_i \leftrightarrow E^+_i$. \begin{propn} \label{4m} There is a map $\tilde{M}^-_{i-1} \mbox{$\rightarrow$} M_i$ such that the following diagram commutes: $$\begin{array}{ccccc} M_{i-1} - \mbox{\bbb P} W^-_i & \mbox{$\longrightarrow$} & \tilde{M}^-_{i-1} & \longleftarrow & E^-_i \\ \Big\updownarrow & & \Big\downarrow & & \Big\downarrow \\ M_i - \mbox{\bbb P} W^+_i & \mbox{$\longrightarrow$} & M_i & \longleftarrow & \mbox{\bbb P} W^+_i. \end{array}$$ \end{propn} {\em Proof}. Let $(\mbox{$\bf E$}, {\bf \Phi}) \mbox{$\rightarrow$} \tilde{M}^-_{i-1} \times X$ be the pullback of the universal family. We will construct a new family $(\mbox{$\bf E$}', {\bf \Phi}')$ of pairs all of which are in $M_i$. By uniqueness of families \re{3n}, $(\mbox{$\bf E$}, {\bf \Phi})|_{E^-_i \times X}$ is the pullback of the family over $\mbox{\bbb P} W^-_i$ constructed in \re{4h}. Thus there is a surjective sheaf map $\mbox{$\bf E$} \mbox{$\rightarrow$} {\cal O}_{E^-_i \times X}(-\Delta)$. This time the map does not necessarily annihilate ${\bf \Phi}$. However, if we tensor by ${\cal O}(E^-_i)$, then the twisted map $\mbox{$\bf E$}(E^-_i) \mbox{$\rightarrow$} {\cal O}_{E^-_i \times X}(\Delta)(-1,-1)$ of course annihilates ${\bf \Phi}(E^-_i)$. If we define $\mbox{$\bf E$}'$ to be the kernel of this twisted map, so that $$0 \mbox{$\longrightarrow$} \mbox{$\bf E$}' \mbox{$\longrightarrow$} \mbox{$\bf E$}(E^-_i) \mbox{$\longrightarrow$} {\cal O}_{E^+_i \times X}(\Delta)(-1,-1) \mbox{$\longrightarrow$} 0,$$ then $\mbox{$\bf E$}'$ is locally free, and ${\bf \Phi}(E^-_i)$ descends to ${\bf \Phi}' \in H^0(\mbox{$\bf E$}')$. The remainder of the proof is analogous to that of \re{4l}. \mbox{ $\Box$} At last we come to the goal of all the above work. \begin{propn} There is a natural isomorphism $\tilde{M}^+_i \leftrightarrow \tilde{M}^-_{i-1}$ such that the following diagram commutes: $$\begin{array}{ccccc}M_i - \mbox{\bbb P} W^+_i & \mbox{$\longrightarrow$} & \tilde{M}^+_i & \longleftarrow & E^+_i \\ \Big\updownarrow & & \Big\updownarrow & & \Big\updownarrow \\ M_{i-1} - \mbox{\bbb P} W^-_i & \mbox{$\longrightarrow$} & \tilde{M}^-_{i-1} & \longleftarrow & E^-_i.\end{array}$$ \end{propn} {\em Proof}. Both $\tilde{M}^+_i$ and $\tilde{M}^-_{i-1}$ are smooth, and by \re{4l} and \re{4m} they both inject into $M_{i-1} \times M_i$. Indeed, both injections are embeddings, since as is easily checked they annihilate no tangent vectors, and both have the same image. This image is precisely the closure of the graph of the isomorphism $M_i - \mbox{\bbb P} W^+_i \leftrightarrow M_{i-1} - \mbox{\bbb P} W^-_i$, which proves the left-hand square; for both $E^-_i$ and $E^+_i$ it is the map $\mbox{\bbb P} W^-_i \oplus \mbox{\bbb P} W^+_i \mbox{$\rightarrow$} \mbox{\bbb P} W^-_i \times \mbox{\bbb P} W^+_i$, which proves the right-hand square. \mbox{ $\Box$} {\em Note.} In light of this result, we will henceforth refer to $\tilde{M}^+_i = \tilde{M}^-_{i-1}$ simply as $\tilde{M}_i$, and $E^+_i = E^-_i$ as $E_i$. Thus $M_i$ is obtained from $M_{i-1}$ by blowing up $\mbox{\bbb P} W^-_i$, and then blowing down the same exceptional divisor in another direction. Such a blow-up and blow-down is an example of what is called a {\em flip} in Mori theory. This paper will not use any of the deep results of Mori theory, but we will see some of its basic principles in action. In one case the flip degenerates to an ordinary blow-up. \begin{propn} \label{4r} The moduli space $M_1$ is the blow-up of $M_0 = \mbox{\bbb P} H^1(\Lambda^{-1})$ along $X$ embedded via $|K_X\Lambda|$. \end{propn} {\em Proof}. Since $W^-_1$ is a line bundle, there is nothing to blow down. \mbox{ $\Box$} The other extreme case is also of interest. Let $w = [(d-1)/2]$, so that $M_w$ is the last moduli space in our sequence. Let $N$ be the moduli space of ordinary rank 2 semistable bundles of determinant $\Lambda$. \begin{propn} \label{4s} There is a natural ``Abel-Jacobi'' map $M_w \mbox{$\rightarrow$} N$ with fibre $\mbox{\bbb P} H^0(E)$ over a stable bundle $E$. It is surjective if $d > 2g-2$. \end{propn} {\em Proof}. If $i = w$, then $\sigma \in (0, [d/2] + 1 - d/2)$, so $\sigma$-stability of \mbox{$(E, \phi)$}\ implies ordinary semistability of $E$. Thus there is a map $M_w \mbox{$\rightarrow$} N$. Moreover, ordinary stability of $E$ implies $\sigma$-stability of \mbox{$(E, \phi)$}, so the fibre over a stable $E$ is just $\mbox{\bbb P} H^0(E)$. For $d > 2g-2$, any bundle $E$ has a nonzero section $\phi$ by Riemann-Roch. Hence every stable bundle in $N$ is certainly in the image of $M_w$. But $M_w$ is complete, so its image is a complete variety containing the stable set, which must be $N$ itself. \mbox{ $\Box$} We may sum up our findings in the following diagram. \def5pt{2pt} $$\begin{array}{ccccccccccccccccc} & & \tilde{M}_2 & & & & \tilde{M}_3 & & & & \tilde{M}_4 & & & & \tilde{M}_w & & \\ & \swarrow & & \searrow & & \swarrow & & \searrow & & \swarrow & & \searrow & & \swarrow & & \searrow & \\ M_1 & & & & M_2 & & & & M_3 & & & & \, \cdots \, & & & & M_w \\ \downarrow & & & & & & & & & & & & & & & & \downarrow \\ M_0 & & & & & & & & & & & & & & & & N \end{array}$$ \def5pt{5pt} All the arrows are birational morphisms except sometimes the one to $N$. \bit{Their Poincar\'e polynomials} Before going on to our main application in the next section, let us pause to see how the flips described above can be used to compute the Poincar\'e polynomials of our moduli spaces. \begin{equation} P_t(M_i) = \frac{1}{1-t^2} \mathop{{\rm Coeff}}_{x^i} \left(\frac{t^{2d+2g-2-4i}}{xt^4-1} - \frac{t^{2i+2}}{x-t^2} \right) \left( \frac{(1+xt)^{2g}}{(1-x)(1-xt^2)} \right). \end{equation} {\em Proof}. Since $\tilde{M}_j$ is the blow-up of $M_{j-1}$ at $\mbox{\bbb P} W^-_j$, by the formula for the Poincar\'e polynomial of a blow-up \cite[p.\ 605]{gh}, $$P_t(\tilde{M}_j) = P_t(M_{j-1}) + P_t(E_j) - P_t(\mbox{\bbb P} W^-_j).$$ But $\tilde{M}_j$ is also the blow-up of $M_j$ at $\mbox{\bbb P} W^+_j$, so $$P_t(\tilde{M}_j) = P_t(M_j) + P_t(E_j) - P_t(\mbox{\bbb P} W^+_j)$$ as well. Hence $$P_t(M_j) - P_t(M_{j-1}) = P_t(\mbox{\bbb P} W^+_j) - P_t(\mbox{\bbb P} W^-_j).$$ But the Poincar\'e polynomial of any projective bundle splits, so \begin{eqnarray*} P_t(\mbox{\bbb P} W^+_j) - P_t(\mbox{\bbb P} W^-_j) & = & P_t(\mbox{\bbb P}^{d+g-2-2j})P_t(X_j) - P_t(\mbox{\bbb P}^{i-1})P_t(X_j) \\ & = & \frac{t^{2j}-t^{2d+2g-2-4j}}{1-t^2} P_t(X_j). \end{eqnarray*} A formula for $P_t(X_j)$ was given by Macdonald \cite{mac}: $$P_t(X_j) = \mathop{{\rm Coeff}}_{x^j} \frac{(1+xt)^{2g}}{(1-x)(1-xt^2)}.$$ Hence $$P_t(M_j) - P_t(M_{j-1}) = \mathop{{\rm Coeff}}_{x^j} \frac{(t^{2j}-t^{2d+2g-2-4j})(1+xt)^{2g}}{(1-t^2)(1-x)(1-xt^2)}.$$ Notice that this formula also produces $P_t(M_0)$ when $j=0$. So to sum up, $$\begin{array}{ccl} P_t(M_i) & = & \displaystyle\frac{1}{1-t^2} \mathop{{\rm Coeff}}_{x^i} \sum_{j=0}^{i} \frac{x^{i-j}(t^{2j}-t^{2d+2g-2-4j})(1+xt)^{2g}}{(1-x)(1-xt^2)} \\[15pt] & = & \displaystyle\frac{1}{1-t^2} \mathop{{\rm Coeff}}_{x^i} \left( \frac{x^{i+1}-t^{2i+2}}{x-t^2} + \frac{t^{2d+2g-2-4i}(1-t^{4i-4}x^{i+1})}{xt^4-1} \right) \left( \frac{(1+xt)^{2g}}{(1-x)(1-xt^2)} \right), \end{array} $$ which agrees with the formula stated after the terms containing $x^{i+1}$ are removed. \mbox{ $\Box$} We can use this formula to recover the formula of Harder-Narasimhan \cite{hn} for the Poincar\'e polynomial of the moduli space $N$ of stable bundles of rank 2, determinant $\Lambda$, and odd degree $d$: \begin{equation} \label{4u} P_t(N) = \frac{(1+t^3)^{2g}-t^{2g}(1+t)^{2g}}{(1-t^2)(1-t^4)}. \end{equation} {\em Proof}. When $d > 2g-2$ is odd and $i = w$, then by \re{4s} there is a surjective map $M_w \mbox{$\rightarrow$} N$ with fibre $\mbox{\bbb P} H^0(E)$ over a bundle $E$. If moreover $d > 4g-4$, then $H^1(E) = 0$ for all stable $E$ (see for example the proof of \re{3e}), so $M_w$ is then just the $\mbox{\bbb P}^{d-2g+1}$-bundle $\mbox{\bbb P} (R^0\pi) \mbox{$\bf E$}$, where \mbox{$\bf E$}\ is a universal bundle over $N$, and $$P_t(N) = \frac{1-t^2}{1-t^{2d-4g+4}} P_t(M_w).$$ For simplicity we may as well assume that $d = 4g-3$. Then $w = 2g-2$ and $$P_t(N) = \frac{1}{1-t^{4g-2}} \mathop{{\rm Coeff}}_{x^{2g-2}}\left(\frac{t^{2g}}{xt^4-1} - \frac{t^{4g-2}}{x-t^2}\right) \left( \frac{(1+xt)^{2g}}{(1-x)(1-xt^2)} \right).$$ The following argument, due to Don Zagier, then shows that this equals the Harder-Nara\-sim\-han formula. Let $$F(a,b,c,t) = \mathop{{\rm Coeff}}_{x^{2g-2}} \frac{(1+xt)^{2g}}{(1-ax)(1-bx)(1-cx)}.$$ Then $$P_t(N) = \frac{t^{4g-4}F(1,t^2,t^{-2},t) - t^{2g}F(1,t^2,t^4,t)}{1-t^{4g-2}}.$$ On the other hand, $$F(a,b,c,t) = \mathop{{\rm Res}}_{x=0}\left\{ \frac{x^{1-2g}(1+xt)^{2g}dx}{(1-ax)(1-bx)(1-cx)}\right\};$$ since this has no pole at infinity, by the residue theorem \begin{eqnarray*} F(a,b,c,t) & = & (-\mathop{{\rm Res}}_{x = 1/a} -\mathop{{\rm Res}}_{x = 1/b} -\mathop{{\rm Res}}_{x = 1/c}) \left\{ \frac{x^{1-2g}(1+xt)^{2g}dx}{(1-ax)(1-bx)(1-cx)}\right\} \\ & = & \frac{(a+t)^{2g}}{(a-b)(a-c)} + \frac{(b+t)^{2g}}{(b-a)(b-c)} + \frac{(c+t)^{2g}}{(c-a)(c-b)}. \end{eqnarray*} After this substitution, it is a matter of high-school algebra to verify \re{4u}. \mbox{ $\Box$} \bit{Their ample cones} We now turn to a study of the line bundles over the $M_i$. Indeed, our goal is a formula for the dimension of the space of sections of any line bundle over any $M_i$. Since $M_0$ is just a projective space, the first interesting case is $M_1$; so we first of all ask what line bundles there are on $M_1$. \begin{propn} $\mathop{{\rm Pic}}\nolimits M_1 = \mbox{\bbb Z} \oplus \mbox{\bbb Z}$, generated by the hyperplane $H$ and the exceptional divisor $E_1$. \end{propn} {\em Proof}. Obvious from \re{4r}. \mbox{ $\Box$} The case of $M_1$ will be crucial for us, so we introduce the notation $${\cal O}_1(m,n) = {\cal O}((m+n)H - nE_1),$$ $$V_{m,n} = H^0(M_1; {\cal O}_1(m,n)).$$ Pushing down to $M_0 = \mbox{\bbb P} H^1(\Lambda^{-1})$ then yields $V_{m,n} = H^0(M_0; {\cal O}(m+n) \otimes {\cal I}_X^n)$. That is, an element of $\mbox{\bbb P} V_{m,n}$ is a hypersurface of degree $m+n$ with a singularity of order $n-1$ at $X$. The dimension of $V_{m,n}$, which we shall attempt to calculate, is thus a number canonically associated to $X$, $\Lambda$, $m$, and $n$. Of course, in many cases this number is easy to compute. If $m < 0$, for example, then $V_{m,n} = 0$, since no hypersurface can have a singularity of order greater than its degree. If $n < 0$, then $V_{m,n} = H^0(M_0; {\cal O}(m+n) \otimes {\cal I}_X^n) = H^0(M_0; {\cal O}(m+n))$, because $\mathop{{\rm codim}}\nolimits X/M_0 = d+g-3 > 1$ by our assumptions on $d$ and $g$, and a section cannot have a pole on a set of codimension $>1$. So in this case $\dim V_{m,n} = {m+n+d+g-2 \choose m+n}$. However, for $m,n \geq 0$, it is quite an interesting problem to calculate $\dim V_{m,n}$. When $n=1$, these are of course precisely the spaces whose syzygies are studied by Green and Lazarsfeld \cite{gl}, but for $n > 1$ very little appears to be known. What about $M_i$ for $i > 1$? These give exactly the same information as $M_1$, for the following simple reason. \begin{propn} \label{5a} For $i > 0$ there is a natural isomorphism $\mathop{{\rm Pic}}\nolimits M_1 = \mathop{{\rm Pic}}\nolimits M_i$. Moreover, if by abuse of notation we denote by ${\cal O}_i(m,n)$ the image of ${\cal O}_1(m,n)$ in $\mathop{{\rm Pic}}\nolimits M_i$, then for any $m,n$ there is a natural isomorphism $V_{m,n} = H^0(M_i; {\cal O}_i(m,n))$. \end{propn} {\em Proof}. By \re{4t}, $M_1$ is isomorphic to $M_i$ except on sets of codimension $\geq 2$. Hence divisors, functions, line bundles, and sections can be pulled back from one to the other and extended over the bad sets in a unique way. \mbox{ $\Box$} However, we will certainly not ignore the higher $M_i$ for the rest of the paper. Instead, they will be indispensable tools in the study of the cohomology of $M_1$, to be used as follows. A naive approach to calculating $\dim V_{m,n}$ would be to calculate $\chi(M_1; {\cal O}_1(m,n))$, which is easy using Riemann-Roch, and then to apply Kodaira vanishing to show that the higher cohomology all vanished. This will not work: the hypothesis of Kodaira vanishing, which is that $K^{-1}_{M_1} {\cal O}_1(m,n)$ must be ample, will not typically be satisfied, and the higher cohomology will not vanish. But this problem can be cured by shifting attention to some other $M_i$. Indeed, under some mild hypotheses on $m$ and $n$, there will be some $i$ such that $K^{-1}_{M_i} {\cal O}_i(m,n)$ will be ample on $M_i$. Hence $\dim V_{m,n} = \chi(M_i; {\cal O}_i(m,n))$, which will be calculated by an inductive procedure on $i$. To carry out this programme, of course, we need to know the ample cone of each $M_i$. So our goal in this section will be to prove the following theorem. \begin{thm} \label{5b} For $0 < i < w$, the ample cone of $M_i$ is bounded by ${\cal O}_i(1,i-1)$ and ${\cal O}_i(1,i)$. For $d > 2g-2$, the ample cone of $M_w$ is bounded by ${\cal O}_w(1,w-1)$ and ${\cal O}_w(2,d-2)$; for $d \leq 2g-2$, it is bounded on one side by ${\cal O}_w(1,w-1)$, and contains the cone bounded on the other side by ${\cal O}_w(2,d-2)$. \end{thm} So as we pass from $i-1$ to $i$, the ample cone flips across the ray of slope $i-1$, as illustrated for $d = 7$ in the figure. This is exactly the behaviour which is predicted by Mori theory; indeed, flips are so named for precisely this reason. \begin{figure}[hbt] \begin{center} \font\thinlinefont=cmr5 \mbox{\beginpicture \setcoordinatesystem units < 0.500cm, 0.500cm> \unitlength= 0.500cm \linethickness=1pt \setplotsymbol ({\makebox(0,0)[l]{\tencirc\symbol{'160}}}) \setshadesymbol ({\thinlinefont .}) \setlinear \linethickness= 0.500pt \setplotsymbol ({\thinlinefont .}) \putrule from 6.462 17.638 to 6.938 17.638 \plot 6.811 17.607 6.938 17.638 6.811 17.670 / \linethickness= 0.500pt \setplotsymbol ({\thinlinefont .}) \plot 4.398 9.542 6.938 18.749 / \plot 6.932 18.488 6.938 18.749 6.809 18.521 / \linethickness= 0.500pt \setplotsymbol ({\thinlinefont .}) \plot 4.398 9.542 7.415 18.590 / \plot 7.394 18.329 7.415 18.590 7.274 18.370 / \linethickness= 0.500pt \setplotsymbol ({\thinlinefont .}) \plot 4.398 9.542 8.684 18.114 / \plot 8.628 17.859 8.684 18.114 8.514 17.915 / \linethickness= 0.500pt \setplotsymbol ({\thinlinefont .}) \plot 4.398 9.542 11.066 16.209 / \plot 10.931 15.985 11.066 16.209 10.841 16.075 / \linethickness= 0.500pt \setplotsymbol ({\thinlinefont .}) \plot 4.335 7.891 4.398 7.637 4.462 7.891 / \putrule from 4.398 7.637 to 4.398 19.067 \plot 4.462 18.813 4.398 19.067 4.335 18.813 / \linethickness= 0.500pt \setplotsymbol ({\thinlinefont .}) \plot 2.747 9.605 2.493 9.542 2.747 9.478 / \putrule from 2.493 9.542 to 13.923 9.542 \plot 13.669 9.478 13.923 9.542 13.669 9.605 / \put {$\scriptstyle m$} [lB] at 13.982 9.442 \put {$\scriptstyle n$} [lB] at 4.239 19.226 \put {$\scriptstyle i=3$} [lB] at 5.357 17.519 \put {$\scriptstyle i=2$} [lB] at 6.535 15.892 \put {$\scriptstyle i=1$} [lB] at 7.473 14.304 \put {$\scriptstyle i=0$} [lB] at 9.378 12.082 \linethickness=0pt \putrectangle corners at 1.873 19.416 and 14.082 7.017 \endpicture} \end{center} \end{figure} The first thing to notice is that, since all the $M_i$ have unique universal pairs $(\mbox{$\bf E$}, {\bf \Phi}) \mbox{$\rightarrow$} M_i \times X$, an expression such as $\det \pi_! \mbox{$\bf E$}$, or $\Lambda^{2} \mbox{$\bf E$}_x$ for some $x \in X$, defines line bundles on all the $M_i$, which agree with one another on the open sets where the maps between different $M_i$ are defined, and which consequently correspond under the natural isomorphism of \re{5a}. Since $\Lambda^{2} \mbox{$\bf E$}_x$ and $\det \pi_! \mbox{$\bf E$}$ are the canonical (indeed, essentially the only) examples, we work out what they are on $M_1$. \begin{propn} \label{5d} On $M_1$, $\Lambda^{2} \mbox{$\bf E$}_x = {\cal O}_1(0,-1)$ and $\det \pi_! \mbox{$\bf E$} = {\cal O}_1(-1,g-d)$; that is, ${\cal O}_1(m,n) = \det^{-m} \pi_! \mbox{$\bf E$} \otimes (\Lambda^{2} \mbox{$\bf E$}_x)^{(d-g)m - n}$. \end{propn} {\em Proof}. The universal pair on $M_0 \times X$ is easy to construct directly: it is the tautological extension $$0 \mbox{$\longrightarrow$} {\cal O} \mbox{$\longrightarrow$} \mbox{$\bf E$}_0 \mbox{$\longrightarrow$} \Lambda(-1) \mbox{$\longrightarrow$} 0$$ determined by the class $id \in \mathop{{\rm End}}\nolimits H^{1}(X;\Lambda^{-1}) = H^{0}(\mbox{\bbb P} H^{1}(\Lambda^{-1}); {\cal O}(1)) \otimes H^{1}(X;\Lambda^{-1}) = H^{1}((\mbox{\bbb P} H^{1}(\Lambda^{-1}) \times X; \Lambda^{-1}(1))$, together with the constant section ${\bf \Phi}_0 \in H^0({\cal O})$. Recall from \re{4m} that the universal pair $(\mbox{$\bf E$}_1, {\bf \Phi}_1) \mbox{$\rightarrow$} M_1 \times X$ is constructed by pulling back $(\mbox{$\bf E$}_0, {\bf \Phi}_0)$, twisting by ${\cal O}(E^+_1)$, and modifying at $E^+_1$: $$0 \mbox{$\longrightarrow$} \mbox{$\bf E$}_1 \mbox{$\longrightarrow$} \mbox{$\bf E$}_0(E^+_1) \mbox{$\longrightarrow$} {\cal O}_{E^+_1 \times X}(\Delta)(-1) \mbox{$\longrightarrow$} 0.$$ Hence $\Lambda^{2} (\mbox{$\bf E$}_1)_x = \Lambda^{2} (\mbox{$\bf E$}_0(E^+_1))_x \otimes {\cal O}(-E^+_1) = \Lambda^{2} \mbox{$\bf E$}_0 \otimes {\cal O}(E^+_1) = {\cal O}_1(0,-1)$, and \begin{eqnarray*} \det \pi_! \mbox{$\bf E$}_1 & = & \det \pi_! \mbox{$\bf E$}_0(E^+_1) \otimes {\cal O}((g-2)(E^+_1)) \\ & = & \det \pi_! {\cal O}(E^+_1) \otimes \det \pi_! \Lambda(-1)(E) \otimes {\cal O}_1(g-2,2-g) \\ & = & {\cal O}_1(1-g,g-1) \otimes {\cal O}_1(0,-d-1+g) \otimes {\cal O}_1(g-2,2-g) \\ & = & {\cal O}_1(-1,g-d). \mbox{ $\Box$} \end{eqnarray*} The next three results collect some basic information about pullbacks of ${\cal O}_i(m,n)$. \begin{propn} \label{5h} The restriction of ${\cal O}_i(m,n)$ to \begin{tabbing} {\rm (iii)} \= \kill {\rm (i)} \= a fibre of $\mbox{\bbb P} W^+_i$ is ${\cal O}(n-(i-1)m)$; \\ {\rm (ii)} \= a fibre of $\mbox{\bbb P} W^-_i$ is ${\cal O}((i-1)m-n)$; \\ {\rm (iii)} \= $f^{-1}(E) \subset M_w$, where $E$ is a stable bundle and $f$ is the Abel-Jacobi map \\\phantom{xxxxxxxx} of \re{4s}, is ${\cal O}(m(d-2)-2n)$. \end{tabbing} \end{propn} {\em Proof}. By \re{4g}, the bundle \mbox{$\bf E$}\ in the universal pair restricts to an extension $$0 \mbox{$\longrightarrow$} {\cal O}(D) \mbox{$\longrightarrow$} \mbox{$\bf E$} \mbox{$\longrightarrow$} \Lambda(-D)(-1) \mbox{$\longrightarrow$} 0$$ on the fibre of $\mbox{\bbb P} W^+_i$ over $D \in X_i$. Hence on this fibre $\Lambda^{2} \mbox{$\bf E$}_x = {\cal O}(-1)$ and $$\det \pi_! \mbox{$\bf E$} = \det \pi_! {\cal O}(D) \otimes \det \pi_! \Lambda(-D)(-1) = {\cal O}(-\chi(\Lambda(-D))) = {\cal O}(-d+g-1+i). $$ So by \re{5d} ${\cal O}_i(m,n)$ restricts to ${\cal O}((d-g+1-i)m -(d-g)m + n) = {\cal O}((1-i)m +n)$, which proves (i). Similarly by \re{4h}, \mbox{$\bf E$}\ restricts to an extension $$0 \mbox{$\longrightarrow$} \Lambda(-D)(1) \mbox{$\longrightarrow$} \mbox{$\bf E$} \mbox{$\longrightarrow$} {\cal O}(D) \mbox{$\longrightarrow$} 0$$ on the fibre of $\mbox{\bbb P} W^-_i$ over $D \in X_i$. Hence $\Lambda^{2} \mbox{$\bf E$}_x = {\cal O}(1)$ and $$\det \pi_! \mbox{$\bf E$} = \det \pi_! \Lambda(-D)(1) \otimes \det \pi_! {\cal O}(D) = {\cal O}(\chi(\Lambda(-D))) = {\cal O}(d-g+1-i). $$ So the previous situation is reversed, and ${\cal O}_i(m,n)$ restricts to ${\cal O}((i-1)m-n)$, which proves (ii). Finally, on a fibre $\mbox{\bbb P} H^0(E)$ of the Abel-Jacobi map, the universal pair restricts to $E(1)$ with the tautological section. Hence on this fibre $\Lambda^{2} \mbox{$\bf E$}_x = {\cal O}(2)$ and $\det \pi_! \mbox{$\bf E$} = {\cal O}(d+2-2g)$. So by \re{5d} ${\cal O}_i(m,n)$ restricts to ${\cal O}((2g-2-d)m +2((d-g)m-n)) = {\cal O}(m(d-2)-2n)$, which proves (iii). \mbox{ $\Box$} \begin{cor} \label{5j} On $\tilde{M}_i$, ${\cal O}_i(m,n) = {\cal O}_{i-1}(m,n)(((i-1)m-n)E_i)$. \end{cor} {\em Proof}. Certainly ${\cal O}_i(m,n)$ and ${\cal O}_{i-1}(m,n)$ are isomorphic away from $E_i$, so ${\cal O}_i(m,n)$ $= {\cal O}_{i-1}(m,n)(q E_i)$ for some $q$. But ${\cal O}_i(m,n)$ must be trivial on the fibres of $\mbox{\bbb P} W^-_i$, and ${\cal O}_{E_i}(qE_i) = {\cal O}(-q,-q)$, so by \re{5h}(ii) $q = (i-1)m-n$. \mbox{ $\Box$} \begin{cor} \label{5k} For an effective divisor $D$, let $\iota_D$ be the inclusion of moduli spaces defined in \re{3p}. Then $\iota_D^*{\cal O}_i(m,n) = {\cal O}_i(m,n - m |D|)$. \end{cor} {\em Proof}. Choose $x \in X-D$. Then from \re{5d} and the long exact sequence in \re{3p}, ${\cal O}_i(0,-1) = \Lambda^{2} \mbox{$\bf E$}^{\Lambda}_x = \Lambda^{2} (\iota^* \mbox{$\bf E$}^{\Lambda(2D)}_x) = \iota^* {\cal O}_i(0,-1)$. Likewise, \begin{eqnarray*} {\cal O}_i(-1,g-d) & = & \det \pi_! \mbox{$\bf E$}^{\Lambda} \\ & = & \det \pi_! \iota^* \mbox{$\bf E$}^{\Lambda(2D)} \otimes \det^{-1} \pi_! {\cal O}_D(\mbox{$\bf E$}^{\Lambda(2D)}) \\ & = & \det \pi_! \iota^* \mbox{$\bf E$}^{\Lambda(2D)} \otimes \bigotimes_{x \in D} (\Lambda^{2} \mbox{$\bf E$}^{\Lambda(2D)}_x)^{-1} \\ & = & \iota^* {\cal O}_i(-1, g-d-2|D|) \otimes \iota^* {\cal O}_i(0, |D|) \\ & = & \iota^* {\cal O}_i(-1, g-d-|D|). \mbox{ $\Box$} \end{eqnarray*} We now pause to apply these ideas to compute the Picard group of the moduli space $N$ of ordinary semistable bundles of determinant $\Lambda$: \begin{equation} \mathop{{\rm Pic}}\nolimits N = \mbox{\bbb Z}. \end{equation} {\em Proof}. If $g=2$ and $d$ is even, then $N = \mbox{\bbb P}^3$ \cite{nr}, so the result is obvious. Otherwise, the complement of the stable set $N_s \subset N$ has codimension $\geq 2$; since $N$ is normal \cite{dn}, this implies $\mathop{{\rm Pic}}\nolimits N_s = \mathop{{\rm Pic}}\nolimits N$. By \re{4s} the Abel-Jacobi map $f: M_w \mbox{$\rightarrow$} N$ has fibre $\mbox{\bbb P} H^0(E)$ over a stable bundle $E$. Tensoring by a line bundle, we may of course assume $d > 4g-4$. But then $H^1(E) = 0$ (see for example the proof of \re{3e}), so $\dim \mbox{\bbb P} H^0(E) = d+2g-1$ always and $f$ is locally trivial over $N_s$. Hence $\mathop{{\rm Pic}}\nolimits N_s$ is the subgroup of $\mathop{{\rm Pic}}\nolimits M_w$ whose restriction to each projective fibre of $f$ is trivial. By \re{5h}(iii) this consists of the bundles ${\cal O}_w(k,k(d/2-1))$ for $k \in \mbox{\bbb Z}$ (where $k$ is even if $d$ is odd). \mbox{ $\Box$} Denote by ${\cal O}(\Theta)$ the $\mbox{\bbb Q}$-Cartier divisor class such that $f^*{\cal O}(\Theta) = {\cal O}_w(1,d/2-1)$. Note that this differs slightly from the normalization in \cite{dn}. The following is then true for any $d$, not just $d > 4g-4$: \begin{equation} \label{5e} f^*{\cal O}(\Theta) = {\cal O}_w(1,d/2-1). \end{equation} {\em Proof}. True by definition if $d > 4g-4$; follows otherwise from \re{5k}, since $$\iota_D^* {\cal O}_w(1,d/2 + |D| -1) = {\cal O}_w(1, d/2-1). \mbox{ $\Box$} $$ Now that we know $\mathop{{\rm Pic}}\nolimits N$, we can make the following definition. \begin{defn} The {\em Verlinde vector spaces} are $$Z_k(\Lambda) = H^0(N; {\cal O}(k \Theta)),$$ with the convention that $Z_k(\Lambda) = 0$ if $d$ and $k$ are both odd. \end{defn} Verlinde's original papers \cite{dv,v} conjectured a striking formula for the dimensions of these vector spaces, which has since been proved by several authors. We will give our own proof in \S7; the first step, however, is the following result, originally due to Bertram \cite{bert}. \begin{propn} \label{5f} For $d > 2g-2$, there is a natural isomorphism $Z_k(\Lambda) = V_{k,k(d/2-1)}$. \end{propn} The proof requires the following lemma. \begin{lemma} \label{5i} Let $M$, $N$ be varieties with $N$ normal, and let $f:M \mbox{$\rightarrow$} N$ be a morphism which is generically a projective bundle. Then $f_* {\cal O}_M = {\cal O}_N$. \end{lemma} {\em Proof}. This is essentially Stein factorization. Let $U \subset N$ be the open set such that $f: f^{-1}(U) \mbox{$\rightarrow$} U$ is a projective bundle. Then certainly $f_* {\cal O}_{f^{-1}(U)} = {\cal O}_U$, so $N' = \mathop{{{\bf Spec}}}\nolimits f_* {\cal O}_M$ is birational to $N$. By construction there is a map $f': M \mbox{$\rightarrow$} N'$ such that $f'_* {\cal O}_M = {\cal O}_{N'}$. On the other hand, since $f_* {\cal O}_M$ is a coherent sheaf of ${\cal O}_N$-algebras, the birational morphism $N' \mbox{$\rightarrow$} N$ is finite. But a birational finite morphism to a normal variety is an isomorphism---this is essentially Zariski's main theorem; the proof in \cite[III 11.4]{h} goes through, or see \cite[III.9]{red}. Hence $N' = N$ and $f_* {\cal O}_M = {\cal O}_N$. \mbox{ $\Box$} {\em Proof}\ of \re{5f}. Recall again from \re{4s} that for $d > 2g-2$, the Abel-Jacobi map $f: M_w \mbox{$\rightarrow$} N$ is surjective with fibre $\mbox{\bbb P} H^0(E)$ over a stable bundle $E$. If $U \subset N$ is the set of bundles $E$ such that $E$ is stable and $\dim H^0(E)$ is minimal, then certainly $f: f^{-1}(U) \mbox{$\rightarrow$} U$ is a projective bundle; for example it is the descent of a trivial projective bundle over the $\mathop{{\rm Quot}}\nolimits$ scheme. Moreover, $N$ is always normal \cite{dn}. So by \re{5i}, $f_* {\cal O}_{M_w} = {\cal O}_{N}$. Hence $f_*f^* {\cal O}(k\Theta) = {\cal O}(k\Theta)$, so that $$f^*: H^0(N;{\cal O}(k \Theta)) \mbox{$\rightarrow$} H^0(M_w; {\cal O}_w(k, k(d/2-1)))$$ has inverse $f_*$. \mbox{ $\Box$} It is worth mentioning, if not proving, a generalization of this result. Over the stable set $N_s \subset N$, let $\mbox{$\bf E$} \mbox{$\rightarrow$} N_s \times X$ be a universal bundle, normalized so that $\Lambda^{2} \mbox{$\bf E$} |_{N_s \times \{ x \} } = {\cal O}$. (Actually, such a normalization is impossible for $d$ odd, and $\mbox{$\bf E$}$ will not even exist for $d$ even! However, the obstructions are all in $\mbox{\bbb Z}/2$, and will cancel in the cases we are considering; for details see \cite{glue}.) Then let $U = (R^0 \pi) \mbox{$\bf E$} \mbox{$\rightarrow$} N_s$. \begin{propn} For $d > 2g-2$, there is a natural isomorphism $H^0(N_s; S^{m(d-2)-2n}U(m\Theta)) = V_{m,n}$ unless $g=2$ and $d$ is even. \end{propn} {\em Sketch of proof.} The complement of $f^{-1}(N_s) \subset M_w$ has codimension $\geq 2$ unless $g=2$ and $d$ is even (in which case $N = \mbox{\bbb P}^3$ \cite{nr}), so $V_{m,n} = H^0(f^{-1}(N_s); {\cal O}(m,n))$. Also $(R^0\pi){\cal O}(m,n)|_{N_s} = S^{m(d-2)-2n}U(m\Theta)$, so $$ H^0(f^{-1}(N_s); {\cal O}(m,n)) = H^0(N_s; S^{m(d-2)-2n}U(m\Theta)) $$ as in the proof of \re{5f}. \mbox{ $\Box$} Hence seeking a formula for $\dim V_{m,n}$ can be regarded as seeking a generalization of the Verlinde formula. At last we return to the determination of the ample cone of $M_i$. It can of course be quite difficult to decide whether a given line bundle on a projective variety is ample. However, a geometric invariant theory quotient is naturally endowed with an ample bundle, which is the descent of the ample bundle used in the linearization. So we shall work out how the line bundles used in the linearizations of \S1 descend to $M_i$. Recall that the linearization was some power of ${\cal O}(\chi + 2\sigma, 4\sigma) \mbox{$\rightarrow$} \mbox{\bbb P} \mathop{{\rm Hom}}\nolimits \times \mbox{\bbb P} \mbox{\bbb C}^{\chi}$, or more precisely, its pullback to $\mathop{{\rm Quot}}\nolimits(\Lambda) \times \mbox{\bbb P} \mbox{\bbb C}^{\chi}$, which by abuse of notation we still denote ${\cal O}(\chi + 2\sigma, 4\sigma)$. By further abuse of notation we refrain from worrying about whether $\chi + 2 \sigma$ and $4\sigma$ are actually integers. \begin{propn} \label{5c} The bundle ${\cal O}(\chi + 2\sigma, 4\sigma) \mbox{$\rightarrow$} \mathop{{\rm Quot}}\nolimits(\Lambda) \times \mbox{\bbb P} \mbox{\bbb C}^{\chi}$ descends to ${\cal O}_i(1,d-1-2\sigma) \mbox{$\rightarrow$} M_i$. \end{propn} {\em Proof}. As in \S1, let $U \subset \mathop{{\rm Quot}}\nolimits(\Lambda)$ be the set of quotients ${\cal O}^{\chi} \mbox{$\rightarrow$} E \mbox{$\rightarrow$} 0$ of determinant $\Lambda$ such that the induced map $\mbox{\bbb C}^{\chi} \mbox{$\rightarrow$} H^0(E)$ is an isomorphism. If ${\cal O}^{\chi} \mbox{$\rightarrow$} \mbox{$\bf E$} \mbox{$\rightarrow$} 0$ is the universal quotient over $U \times X$, then as in \re{3m} there is a universal pair $(\mbox{$\bf E$}(1), {\bf \Phi}) \mbox{$\rightarrow$} U \times \mbox{\bbb P} \mbox{\bbb C}^{\chi} \times X$ descending to the universal pair $(\mbox{$\bf E$}, {\bf \Phi})$ on each $M_i$. Hence $\det \pi_! \mbox{$\bf E$}(1) \mbox{$\rightarrow$} U \times \mbox{\bbb P} \mbox{\bbb C}^{\chi}$ descends to $\det \pi_! \mbox{$\bf E$} = {\cal O}_i(-1,g-d) \mbox{$\rightarrow$} M_i$, and for any $x \in X$, $\Lambda^{2} \mbox{$\bf E$}(1)_x \mbox{$\rightarrow$} U \times \mbox{\bbb P} \mbox{\bbb C}^{\chi}$ descends to $\Lambda^{2} \mbox{$\bf E$}_x = {\cal O}_i(0,-1) \mbox{$\rightarrow$} M_i$. By \cite[III Ex.\ 12.6(b)]{h} $\mathop{{\rm Pic}}\nolimits (U \times \mbox{\bbb P}\mbox{\bbb C}^{\chi}) = \mathop{{\rm Pic}}\nolimits U \oplus \mathop{{\rm Pic}}\nolimits \mbox{\bbb P} \mbox{\bbb C}^{\chi}$. So to determine a bundle on $U \times \mbox{\bbb P}\mbox{\bbb C}^{\chi}$, it suffices to determine it on $\{ E \} \times \mbox{\bbb P} \mbox{\bbb C}^{\chi}$ and $U \times \{ \phi \}$ for some $E \in U$, $\phi \in \mbox{\bbb P} \mbox{\bbb C}^{\chi}$. On $\{ E \} \times \mbox{\bbb P} \mbox{\bbb C}^{\chi}$, $\mbox{$\bf E$}(1) = E(1)$, so $\det \pi_! \mbox{$\bf E$}(1) = {\cal O}(\chi)$ and $\Lambda^{2} \mbox{$\bf E$}_x = {\cal O}(2)$. On $U \times \{ \phi \}$, $\mbox{$\bf E$}(1) = \mbox{$\bf E$}$, so $\det \pi_! \mbox{$\bf E$}(1) = \det \pi_! \mbox{$\bf E$}$. But for all $E \in U$, $H^0(E) = H^0({\cal O}^{\chi})$ and $H^1(E) = 0$. Consequently $\det \pi_! \mbox{$\bf E$} = {\cal O}$. Moreover, there is a canonical map $$\Lambda^{2} \mbox{\bbb C}^{\chi} = \Lambda^{2} H^0({\cal O}(\chi)) \mbox{$\longrightarrow$} \Lambda^{2} H^0(\mbox{$\bf E$}) \mbox{$\longrightarrow$} H^0(\Lambda^{2} \mbox{$\bf E$}), $$ so the pullback of ${\cal O}(1) \mbox{$\rightarrow$} \mbox{\bbb P} \mathop{{\rm Hom}}\nolimits (\Lambda^{2} \mbox{\bbb C}^{\chi}, H^0(\Lambda))$ to $U$, also denoted by ${\cal O}(1)$, is precisely $(R^0\pi)\mathop{{\rm Hom}}\nolimits(\Lambda, \Lambda^{2} \mbox{$\bf E$})$. This is clearly isomorphic to $\Lambda^{2} \mbox{$\bf E$}_x = \mathop{{\rm Hom}}\nolimits(\Lambda, \Lambda^{2} \mbox{$\bf E$})_x$, since $\mathop{{\rm Hom}}\nolimits(\Lambda, \Lambda^{2} \mbox{$\bf E$})$ is trivial on every fibre of $\pi$. Putting it all together, we find that ${\cal O}(0, \chi)$ descends to ${\cal O}_i(-1,g-d)$ and ${\cal O}(1,2)$ descends to ${\cal O}_i(0,-1)$. The result follows after a little arithmetic. \mbox{ $\Box$} {\em Proof} of \re{5b}. For any $\sigma \in (\max (0, d/2 - i - 1), d/2 - i)$, the quotient of $U \times \mbox{\bbb P} \mbox{\bbb C}^{\chi}$ by the action of $\mbox{SL($\chi$)}$, linearized by ${\cal O}(\chi + 2\sigma, 4\sigma)$, gives the same quotient $M_i$. Hence the descent of ${\cal O}(\chi + 2\sigma, 4\sigma)$ to $M_i$ is ample for any $\sigma$ in that interval. By \re{5c} and a little arithmetic these bundles span exactly the cones in the statement of \re{5b}. Hence those cones are contained in the ample cones of the $M_i$. It remains to show that no bundles over $M_i$ outside those cones are ample, except possibly on one side for $i=w$ and $d \leq 2g-2$. By \re{5h}(i), the restriction of ${\cal O}_i(m,n)$ to a fibre of $\mbox{\bbb P} W^+_i$ is ${\cal O}(n-(i-1)m)$. So ${\cal O}_i(m,n)$ can only be ample over $M_i$ if this is positive, that is, if $(i-1)m < n$. Thus one side of the ample cone of $M_i$ is where it should be. Likewise by \re{5h}(ii) the restriction of ${\cal O}_{i-1}(m,n) \mbox{$\rightarrow$} M_{i-1}$ to a fibre of $\mbox{\bbb P} W^-_i$ is ${\cal O}((i-1)m-n)$. So for $1 < i \leq w$, that is, when the dimension of this fibre is positive, ${\cal O}_{i-1}(m,n)$ can only be ample over $M_{i-1}$ if $(i-1)m > n$. Thus the other side of the ample cone of $M_{i-1}$ is where it should be. The only case we have not yet treated is the other side of the ample cone of $M_w$ for $d > 2g-2$. In that case there is by \re{4s} a surjective map $M_w \mbox{$\rightarrow$} N$ onto the moduli space of stable bundles of determinant $\Lambda$. It is not an isomorphism, since for example $\mathop{{\rm Pic}}\nolimits M_w = \mbox{\bbb Z} \oplus \mbox{\bbb Z}$ while $\mathop{{\rm Pic}}\nolimits N = \mbox{\bbb Z}$. Hence the pullback of the ample bundle ${\cal O}(2 \Theta) \mbox{$\rightarrow$} N$ is nef but not ample, that is, it is in the boundary of the ample cone. But by \re{5e} this is precisely ${\cal O}(2,d-2)$. \mbox{ $\Box$} \bit{Their Euler characteristics} Now that we know the ample cones of the $M_i$, we can calculate $\dim V_{m,n}$ following the programme outlined in the last section. We first need a formula for the canonical bundle of $M_i$: \begin{equation} \label{6a} K_{M_i} = {\cal O}_i(-3,4-d-g). \end{equation} {\em Proof}. Clearly the canonical bundle is preserved by the isomorphism of \re{5a}, so it suffices to work it out on $M_1$. But this is easy using \re{4r} and the standard formulas for the canonical bundle of projective space and of a blow-up. \mbox{ $\Box$} \begin{propn} \label{6b} Suppose that $m,n \geq 0$ and that $m(d-2) - 2n > -d+2g-2$. Let $b = \left[\frac{n+d+g-4}{m+3}\right] + 1$. Then $\dim V_{m,n} = \chi(M_b; {\cal O}_b(m,n))$. \end{propn} The idea of the proof is that $\dim V_{m,n}$ will be an Euler characteristic by Kodaira vanishing provided that ${\cal O}(m,n)$ lies inside some cone in the translate of the ample fan by $K$. This is illustrated in the figure for the case $d=7$. \begin{figure}[htb] \begin{center} \font\thinlinefont=cmr5 \mbox{\beginpicture \setcoordinatesystem units < 0.500cm, 0.500cm> \unitlength= 0.500cm \linethickness=1pt \setplotsymbol ({\makebox(0,0)[l]{\tencirc\symbol{'160}}}) \setshadesymbol ({\thinlinefont .}) \setlinear \linethickness= 0.500pt \setplotsymbol ({\thinlinefont .}) \putrule from 6.462 17.638 to 6.938 17.638 \plot 6.811 17.607 6.938 17.638 6.811 17.670 / \linethickness= 0.500pt \setplotsymbol ({\thinlinefont .}) \plot 4.398 9.542 6.938 18.749 / \plot 6.932 18.488 6.938 18.749 6.809 18.521 / \linethickness= 0.500pt \setplotsymbol ({\thinlinefont .}) \plot 4.398 9.542 7.415 18.590 / \plot 7.394 18.329 7.415 18.590 7.274 18.370 / \linethickness= 0.500pt \setplotsymbol ({\thinlinefont .}) \plot 4.398 9.542 8.684 18.114 / \plot 8.628 17.859 8.684 18.114 8.514 17.915 / \linethickness= 0.500pt \setplotsymbol ({\thinlinefont .}) \plot 4.398 9.542 11.066 16.209 / \plot 10.931 15.985 11.066 16.209 10.841 16.075 / \linethickness= 0.500pt \setplotsymbol ({\thinlinefont .}) \plot 5.287 7.891 5.351 7.637 5.414 7.891 / \putrule from 5.351 7.637 to 5.351 19.067 \plot 5.414 18.813 5.351 19.067 5.287 18.813 / \linethickness= 0.500pt \setplotsymbol ({\thinlinefont .}) \plot 2.747 11.827 2.493 11.764 2.747 11.700 / \putrule from 2.493 11.764 to 13.923 11.764 \plot 13.669 11.700 13.923 11.764 13.669 11.827 / \putrule from 4.398 9.542 to 13.923 9.542 \plot 13.669 9.478 13.923 9.542 13.669 9.605 / \put {$\scriptstyle m$} [lB] at 13.982 11.715 \put {$\scriptstyle n$} [lB] at 5.142 19.226 \put {$\scriptstyle i=3$} [lB] at 5.357 17.519 \put {$\scriptstyle i=2$} [lB] at 6.535 15.892 \put {$\scriptstyle i=1$} [lB] at 7.473 14.304 \put {$\scriptstyle i=0$} [lB] at 9.378 12.082 \put {$\scriptstyle K$} [lB] at 3.810 9.366 \linethickness=0pt \putrectangle corners at 1.873 19.416 and 14.082 7.017 \endpicture} \end{center} \end{figure} {\em Proof}\ of \re{6b}. Note first that the inequality can be rewritten $$(d/2-1)(m+3) > n+d+g-4,$$ which guarantees that $b \leq [(d-1)/2]$ and hence that $M_b$ exists. Suppose that $\frac{n+d+g-4}{m+3}$ is not an integer. Then $b(m+3) > n+d+g-4 > (b-1)(m+3)$, so ${\cal O}_b(m+3, n+d+g-4)$, which by \re{6a} equals $K_{M_b}^{-1}{\cal O}_b(m,n)$, is in the ample cone of $M_b$ by \re{5b}. The result then follows from \re{5a} and Kodaira vanishing. If $\frac{n+d+g-4}{m+3}$ is an integer, then ${\cal O}_{b-1}(m+3, n+d+g-4)$ and ${\cal O}_b(m+3, n+d+g-4)$ are merely nef, so Kodaira vanishing does not apply. Instead, we move up to $\tilde{M}_b$. By \re{5i} the 0th direct image of ${\cal O}_{\tilde{M}_b}$ in the projection $\tilde{M}_b \mbox{$\rightarrow$} M_b$ is ${\cal O}_{M_b}$, and by the theorem on cohomology and base change \cite[III 12.11]{h} the higher direct images vanish, so for all $j$, $H^j(\tilde{M}_b; {\cal O}_b(m,n)) = H^j(M_b; {\cal O}_b(m,n))$. By \re{6a} and the standard formula for the canonical bundle of a blow-up, $K_{\tilde{M}_b} = {\cal O}_b(-3, 4-d-g)((b-1)E_b)$. Unfortunately $K_{\tilde{M}_b}^{-1} {\cal O}_b(m,n)$ may not be ample, so Kodaira vanishing still does not apply. Instead, we make the following two claims: first, that $H^j(\tilde{M}_b; {\cal O}_b(m,n)) = H^j(\tilde{M}_b; {\cal O}_b(m,n)((b-2)E_b))$ for all $j$, and second, that ${\cal O}_b(m+3, n+d+g-4)(-E_b)$ is ample on $\tilde{M}_b$. The desired result follows immediately from these claims, since at last Kodaira vanishing applies to ${\cal O}_b(m,n)((b-2)E_b)$. To prove the first claim, note that for $0 < k < b$, $H^j(E_b; {\cal O}_b(m,n)(kE_b)) = 0$ for all $j$, since ${\cal O}_b(m,n)(kE_b)$ is ${\cal O}(-k)$ on each fibre of $\mbox{\bbb P}^{b-1} \mbox{$\rightarrow$} E_b \mbox{$\rightarrow$} \mbox{\bbb P} W^+_b$, so that every term in the Leray spectral sequence vanishes. Hence from the long exact sequence on $\tilde{M}_b$ of $$0 \mbox{$\longrightarrow$} {\cal O}_b(m,n)((k-1)E_b) \mbox{$\longrightarrow$} {\cal O}_b(m,n)(kE_b) \mbox{$\longrightarrow$} {\cal O}_b(m,n) {\cal O}_{E_b}(kE_b) \mbox{$\longrightarrow$} 0,$$ we get isomorphisms $H^j(\tilde{M}_b; {\cal O}_b(m,n)((k-1)E_b)) = H^j(\tilde{M}_b; {\cal O}_b(m,n)(kE_b))$. The first claim follows by induction. As for the second claim, note that on $\tilde{M}_b$, the line bundles ${\cal O}_{b-1}(1,b-2)$, ${\cal O}_b(1,b-1)$, and ${\cal O}_b(1,b)$ (or ${\cal O}_b(2,2b-1)$ if $b = w$) are all nef, since they are pulled back from nef bundles on $M_{b-1}$ or $M_b$. It is easy using \re{5j}, the constraints on $m$ and $n$, and a little arithmetic to check that ${\cal O}_b(m+3, n+d+g-4)(-E_b)$ is in the interior of the cone generated by these three bundles. \mbox{ $\Box$} We will have to assume in future that \begin{equation} \label{6o} m(d-2) - 2n > -d+2g-2, \end{equation} since otherwise there is no analogue of the last result and $K^{-1}_{M_i}{\cal O}_i(m,n)$ may not be ample for any $i$. However, for $d \geq 2g$, we still get a complete answer to our problem, for the following reason. \begin{propn} \label{6p} For $d \geq 2g$ and $m(d-2) - 2n < 0$, $V_{m,n} = 0$. \end{propn} {\em Proof}. By Riemann-Roch $\deg E \geq 2g$ implies $\dim H^0(E) \geq 2$, so for any stable bundle $E$, by \re{4s} the fibre $f^{-1}(E)$ of the Abel-Jacobi map is a projective space of positive dimension. By \re{5h}(iii), the restriction of ${\cal O}_w(m,n)$ to this is ${\cal O}(m(d-2)-2n)$, so any section of ${\cal O}_w(m,n)$ must vanish on $f^{-1}(E)$. Hence it must vanish on the inverse image $f^{-1}(N_s)$ of the stable subset of $N$. But this is open, so it must vanish everywhere. \mbox{ $\Box$} Let $L_i \mbox{$\rightarrow$} X_i$ be the line bundle defined by $L_i = \det^{-1} \pi_! \Lambda(-\Delta) \otimes \det^{-1} \pi_! {\cal O}(\Delta)$. Also put $q_i = n-(i-1)m$. \begin{propn} \label{6c} The restriction of ${\cal O}_{i-1}(m,n)$ to $\mbox{\bbb P} W^-_i$ is $L_i^m(-q_i)$. \end{propn} {\em Proof}. Easy from \re{5d} and the description of the universal pair over $\mbox{\bbb P} W^-_i$ in \re{4h}. \mbox{ $\Box$} Now let $U_i \mbox{$\rightarrow$} X_i$ be the vector bundle $(W^-_i) \oplus (W^+_i)^*$, and define numbers $$N_i = \chi(X_i; L_i^m \otimes \Lambda^i W^-_i \otimes S^{q_i-i}U_i), $$ with of course the convention that this is zero when $q_i-i < 0$. On $M_0$, which is just projective space, make the additional convention that ${\cal O}_0(m,n) = {\cal O}(m+n)$. \begin{equation} \label{6d} N_0 = \chi(M_0; {\cal O}_0(m,n)) = {m+n+d+g-2\choose m+n}. \end{equation} {\em Proof}. Since $X_0$ is just a point and $W^-_0 = 0$, $U_0 = (W^+_0)^*$ is just the vector space $H^1(\Lambda)^*$. Hence $S^{m+n}U_0 = H^0(M_0;{\cal O}_0(m,n))$ with our conventions and the result follows. \mbox{ $\Box$} \begin{propn} \label{6e} Let $0 < i \leq b$, and suppose that $m,n \geq 0$ satisfy \re{6o}. Then $$\chi(M_i;{\cal O}_i(m,n)) - \chi(M_{i-1};{\cal O}_{i-1}(m,n)) = (-1)^i N_i.$$ \end{propn} {\em Proof}. By \re{5i} the 0th direct image of ${\cal O}_{\tilde{M}_i}$ in the projection $\tilde{M}_i \mbox{$\rightarrow$} M_i$ is ${\cal O}_{M_i}$, and by the theorem on cohomology and base change \cite[III 12.11]{h} the higher direct images vanish, so $\chi(\tilde{M}_i; {\cal O}_i(m,n)) = \chi(M_i; {\cal O}_i(m,n))$. Likewise $\chi(\tilde{M}_i; {\cal O}_{i-1}(m,n)) = \chi(M_{i-1}; {\cal O}_{i-1}(m,n))$, so it suffices to work on $\tilde{M}_i$. Suppose first that $q_i \leq 0$, so that $N_i = 0$. For $0 < j \leq -q_i$, consider the exact sequence $$0 \mbox{$\longrightarrow$} {\cal O}_{i-1}(m,n)((j-1)E_i) \mbox{$\longrightarrow$} {\cal O}_{i-1}(m,n)(jE_i) \mbox{$\longrightarrow$} {\cal O}_{i-1}(m,n) \otimes {\cal O}_{E_i}(jE_i) \mbox{$\longrightarrow$} 0.$$ By \re{6c} the restriction of ${\cal O}_{i-1}(m,n)$ to $E_i = \mbox{\bbb P} W^-_i \oplus \mbox{\bbb P} W^+_i$ is $L_i^m(-q_i,0)$, and ${\cal O}_{E_i}(E_i) = {\cal O}(-1,-1)$, so the third term of the exact sequence becomes ${\cal O}(-q_i-j,-j)$ and we get $$\chi(\tilde{M}_i;{\cal O}_{i-1}(m,n)(jE_i)) - \chi(\tilde{M}_i;{\cal O}_{i-1}(m,n)((j-1)E_i)) = \chi(E_i; L_i^m(-q_i-j,-j).$$ Summing over $j$ and using \re{5j} yields $$\chi(\tilde{M}_i;{\cal O}_i(m,n)) - \chi(\tilde{M}_i;{\cal O}_{i-1}(m,n)) = \sum_{j=1}^{q_i} \chi(E_i; L_i^m(-q_i-j,-j).$$ However, for $0 < i \leq b$ and $m,n,d,g \geq 0$, a little high-school algebra shows $-q_i < d+g-1-2i$. Hence for all $j$ in the sum above, $0 < j < d+g-1-2i$, so every term in the Leray sequence of the fibration $\mbox{\bbb P}^{d+g-2-2i} \mbox{$\rightarrow$} E_i \mbox{$\rightarrow$} \mbox{\bbb P} W^-_i$ vanishes. Hence all terms are zero, as desired. Now suppose $q_i > 0$. By an argument similar to the one above, $$\chi(\tilde{M}_i;{\cal O}_i(m,n)) - \chi(\tilde{M}_i;{\cal O}_{i-1}(m,n)) = \sum_{j=0}^{q_{i-1}} \chi(E_i; L_i^m(-q_i+j,j).$$ Each term of the right-hand side can be evaluated using the Leray sequence of the fibration $\mbox{\bbb P}^{i-1} \times \mbox{\bbb P}^{d+g-2-2i} \mbox{$\rightarrow$} E_i \mbox{$\rightarrow$} X_i$. Because $-q_i+j < 0 \leq j$, the only nonzero direct image of $L_i^m(-q_i+j,j)$ is the $i$th, which is just $L_i^m \otimes \Lambda^i W^-_i \otimes S^{-q_i+j-i}(W^-_i) \otimes S^j(W^+_i)^*$. Here the factor of $\Lambda^i W^-_i$ comes from Serre duality, since the isomorphism ${\cal O}(-i) = K_{\mbox{\blb P}^{i-1}}$ is not canonical unless the right-hand side is tensored by such a factor. Hence $$\chi(E_i; L_i^m(-q_i+j,j)) = (-1)^i \chi(X_i; L_i^m \otimes \Lambda^i W^-_i \otimes S^{-q_i+j-i}(W^-_i) \otimes S^j(W^+_i)^*).$$ Of course the right-hand side is zero if $q_i-j-i < 0$, so the sum need only run up to $q_i-i$. The result follows because certainly $$S^{q_i-i}U_i = \bigoplus_{j+0}^{q_i-i}S^{q_i-j}(W^-_i) \otimes S^j(W^+_i)^*. \mbox{ $\Box$} $$ \begin{propn} \label{6f} For $i > b$, $N_i = 0$. \end{propn} {\em Proof}. It suffices to show that if $i > b$, then $q_i-i < 0$, that is, $(m+n)/(m+1) < i$. But using $m,n \geq 0$, the definition of $b$, and the inequality \re{6o}, it is a matter of high-school algebra to check $(m+n)/(m+1) \leq b$. \mbox{ $\Box$} \begin{equation} \dim V_{m,n} = \sum_{i = 0}^{\infty}(-1)^i N_i. \end{equation} {\em Proof}. Put together \re{6b}, \re{6d}, \re{6e}, and \re{6f}. \mbox{ $\Box$} Since each $N_i$ can be evaluated using Riemann-Roch on $X_i$, the right-hand side depends only on $g$, $d$, $m$, and $n$, not on the precise geometry of $X$ and $\Lambda$. So even before doing the hard work of the next section, we have found that $\dim V_{m,n}$ depends only on $g$, $d$, $m$, and $n$, which is rather surprising. \bit{Don Zagier to the rescue} All of the results in this section (except \re{6h} and \re{6l}) are due to Don Zagier and were communicated by him to the author. In this section we will compute the $N_i$, using the Riemann-Roch theorem and Macdonald's description \cite{mac} of the cohomology ring of $X_i$. So we begin with a review of Macdonald's results. Let $e_i, \dots , e_g, e'_1, \dots e'_g \in H^i(X;\mbox{\bbb Z})$ be generators such that the intersection form is $\sum_j e_j \otimes e'_j$. Define classes $\xi, \xi' \in H^1(X_i: \mbox{\bbb Z})$ and $\eta \in H^2(X_i; \mbox{\bbb Z})$ as the K\"unneth components of the divisor $\Delta \subset X_i \times X$, regarded as belonging to $H^2(X_i \times X; \mbox{\bbb Z})$: $$\Delta = \eta + \sum_j (\xi'_j e_j - \xi_j e'_j) + iX. $$ These generate the ring $H^*(X_i;\mbox{\bbb Z})$. Moreover, if we put $\sigma_j = \xi_j \xi'_j$, then for any multiindex $I$ without repeats, \begin{equation} \langle \eta^{i-|I|} \sigma_I , X_i \rangle = 1. \end{equation} This implies that for any two power series $A(x)$, $B(x)$, \begin{eqnarray} \langle A(\eta) \exp (B(\eta)\sigma), X_i \rangle & = & \sum_{k=0}^{\infty} \langle A(\eta) B(\eta)^k \sigma^k/k!, X_i \rangle \nonumber \\ & = & \sum_{k=0}^{g} {g \choose k} \mathop{{\rm Res}}_{\eta = 0} \left\{ \frac{A(\eta) B(\eta)^k}{\eta^{i-k+1}} d\eta \right\} \nonumber \\ & = & \mathop{{\rm Res}}_{\eta = 0} \left\{ \frac{A(\eta)(1+\eta B(\eta))^g}{\eta^{i+1}} d\eta \right\} , \label{6j} \end{eqnarray} where $\sigma = \sum_j \sigma_j$. Note that since $\sigma_j^2 = 0$, $\sigma^k/k!$ is just the $k$th symmetric polynomial in the $\sigma_j$. Since we will be doing Riemann-Roch, we need to know the Todd class of $X_i$; luckily this can be worked out in a useful form. \begin{equation} \label{6g} \mathop{{\rm td}}\nolimits X_i = \left( \frac{\eta}{1-e^{-\eta}} \right)^{i-g+1} \exp \left( \frac{\sigma}{e^{\eta}-1} - \frac{\sigma}{\eta} \right). \end{equation} {\em Proof}. Macdonald \cite{mac} shows that the total Chern class of the tangent bundle of $X_i$ is $$ c(X_i) = (1+ \eta)^{i-2g+1}\prod_{j=1}^{g}(1+n-\sigma_i). $$ Let $h(x) = x/(1-e^{-x})$, so that $$\mathop{{\rm td}}\nolimits X_i = h(\eta)^{i-2g+1} \prod_{j+1}^{g} h(\eta-\sigma_j).$$ Expanding $h(\eta-\sigma_j)$ in a power series around $\eta$ and using $\sigma_j^2 = 0$, \begin{eqnarray*} \mathop{{\rm td}}\nolimits X_i & = & h(\eta)^{i-g+1} \prod_{j+1}^{g} \left( 1-\sigma_j \frac{h'(\eta)}{h(\eta)} \right) \\ & = & h(\eta)^{i-g+1} \sum_{k=0}^{\infty} (-1)^k \frac{\sigma^k}{k!} \left( \frac{h'(\eta)}{h(\eta)} \right) ^k \\ & = & h(\eta)^{i-g+1} \exp\left( -\sigma \frac{h'(\eta)}{h(\eta)}\right) , \end{eqnarray*} which yields the desired formula. \mbox{ $\Box$} \begin{lemma} \label{6h} For any line bundle $M \mbox{$\rightarrow$} X$ and any $k \in \mbox{\bbb Z}$, $$\mathop{{\rm ch}}\nolimits \pi_! M(k\Delta) = ((\deg M + ki + 1 - g) - k^2 \sigma) e^{k\eta}.$$ \end{lemma} {\em Proof}. By Grothendieck-Riemann-Roch \begin{eqnarray*} \mathop{{\rm ch}}\nolimits \pi_! M(k\Delta) & = & \pi_* \mathop{{\rm ch}}\nolimits M(k\Delta) \mathop{{\rm td}}\nolimits X \\ & = & \pi_* \exp((\deg M + ki)X +k \Xi +k \eta) (1+(1-g)X) \\ & = & \pi_* (1 + (\deg M + ki)X) (1+k \Xi -k^2 \sigma X) e^{k\eta} (1+(1-g)X) \\ & = & ((\deg M + ki + 1 - g) - k^2 \sigma) e^{k\eta}, \end{eqnarray*} where $\Xi = \sum_j (\xi'_j e_j - \xi_j e'_j)$, so that $\Xi^2 = -2\sigma X$. \mbox{ $\Box$} \begin{manynotop} \label{6l} \begin{tabbing} {\bf (\theequation )} \= (iii) \= $\mathop{{\rm ch}}\nolimits(\Lambda^i W^-_i)$ \= = \= \kill {\bf (\theequation )} \>(i) \> \> $\mathop{{\rm ch}}\nolimits(L_i)$ \' = \> $\exp((d-2i)\eta+2\sigma)$;\+ \\ (ii) \> \> $\mathop{{\rm ch}}\nolimits(\Lambda^i W^-_i)$ \' = \> $\exp((d-3i+1-g)\eta + 3\sigma)$; \\ (iii) \> \> $\mathop{{\rm ch}}\nolimits(U_i)$ \' = \> $(d-i+1-2g) e^{-\eta} + (2g-2) e^{-2\eta} + \sum_{j=1}^{g}e^{-\eta-\sigma_i}$. \end{tabbing} \end{manynotop} {\em Proof}. Since $L_i = \det^{-1} \pi_! \Lambda(-\Delta) \otimes \det^{-1} \pi_! {\cal O}(\Delta)$, by \re{6h} $$c_1(L_i) = -c_1(\pi_! \Lambda(-\Delta)) -c_1(\pi_! {\cal O}(\Delta)) = (d-i+1-g)\eta + \sigma + (-i-1+g)\eta + \sigma = (d-2i)\eta+2\sigma,$$ which implies (i). {}From the exact sequence $$0 \mbox{$\longrightarrow$} \Lambda(-2\Delta) \mbox{$\longrightarrow$} \Lambda(-\Delta) \mbox{$\longrightarrow$} {\cal O}_{\Delta}\Lambda(-\Delta) \mbox{$\longrightarrow$} 0,$$ it follows that $W^-_i = \pi_! {\cal O}_{\Delta}\Lambda(-\Delta) = \pi_! \Lambda(-\Delta) - \pi_! \Lambda(-2\Delta)$ in $K$-theory. Hence by \re{6h} $$ \mathop{{\rm ch}}\nolimits W^-_i = ((d-i+1-g)-\sigma)e^{-\eta} - ((d-2i+1-g)-4\sigma)e^{-2\eta}. $$ In particular $$c_1(\Lambda^i W^-_i) = c_1(W^-_i) = -(d-i+1-g)\eta -\sigma +2(d-2i+1-g)\eta +4\sigma = (d-3i+1-g)\eta + 3\sigma,$$ which implies (ii). Again by \re{6h}, $$\mathop{{\rm ch}}\nolimits (W^+_i)^* = \mathop{{\rm ch}}\nolimits \pi_! \Lambda^{-1}(2\Delta) = ((d-2i+g-1)-4\sigma) e^{-2\eta}.$$ Hence \begin{eqnarray*} \mathop{{\rm ch}}\nolimits U_i & = & \mathop{{\rm ch}}\nolimits \: (W^-_i) \oplus (W^+_i)^* \\ & = & ((d-i+1-g)-\sigma)e^{-\eta} + (2g-2)e^{-2\eta} \\ & = & (d-i+1-2g)e^{-\eta} + (2g-2)e^{-2\eta} + \sum_{j=1}^{g} e^{-\eta-\sigma_i}, \end{eqnarray*} which is (iii). \mbox{ $\Box$} \begin{propn} \label{6i} $\mathop{{\rm ch}}\nolimits (L_i^m \otimes \Lambda^iW^-_i \otimes S^{q_i-i} U_i)$ $$= \mathop{{\rm Coeff}}_{t^{q_i-i}} \left[ e^{ (m(d-2)-2n)\eta } \exp\left((2m+3)\sigma-\frac{t\sigma}{e^{-\eta}-t}\right) \frac{(e^{-\eta}-t)^{-d+i-1+g}}{(1-t)^{2g-2}} \right]. $$ \end{propn} {\em Proof}. The Chern roots of $S^k U_i$ are the sums of $k$ (not necessarily distinct) Chern roots of $U_i$, so by \re{6l}(iii) \begin{eqnarray*} \sum_{k=0}^{\infty} \mathop{{\rm ch}}\nolimits (S^k U_i) t^k & = & \prod_{ \def1{.5}\begin{array}{c}\mbox{\rm \small Chern roots} \\ \mbox{ \rm \small $\alpha$ of $U_i$}\end{array}\def1{1} } \frac{1}{1-te^{\alpha}} \\ & = & \left( \frac{1}{1-te^{-\eta}} \right)^{d-i+1-2g} \left( \frac{1}{1-te^{-2\eta}} \right)^{2g-2} \prod_{j=1}^{g} \left( \frac{1}{1-te^{-\eta-\sigma_j}} \right) \\ & = & \frac{(1-te^{-\eta})^{-d+i-1+g}}{(1-te^{-2\eta})^{2g-2}} \exp \left( \frac{-t\sigma}{e^{\eta}-t} \right). \end{eqnarray*} Replacing $t$ by $te^{2\eta}$ and taking coefficients of $t^{q_i-i}$ yields $$\mathop{{\rm ch}}\nolimits (S^{q_i-i} U_i) = \mathop{{\rm Coeff}}_{t^{q_i-i}} \left[ e^{-2(q_i-i)\eta}\frac{(1-t e^{\eta})^{-d+i-1+g}}{(1-t)^{2g-2}} \exp\left(\frac{-t\sigma}{e^{-\eta} - t} \right) \right] .$$ The result then follows using \re{6l}(i) and (ii) and the pleasing identity $$m(d-2i) + (d-3i+1-g) - 2(q_i-i) = m(d-2)-2n + (d-i+1-g). \mbox{ $\Box$} $$ We are now ready to perform our Riemann-Roch calculation: \begin{manynotop} \label{6k} \def1{2} \begin{tabbing} {\bf (\theequation )} \= = \= xxx \= \kill \> $N_i$ \' = \> $\langle \mathop{{\rm ch}}\nolimits(L_i^m \otimes \Lambda^i W^-_i \otimes S^{q_i-i}U_i) \mathop{{\rm td}}\nolimits(X_i), X_i \rangle$ \\[7pt] \> = \> $\displaystyle \mathop{{\rm Coeff}}_{t^{q_i-i}} \Bigg\langle e^{ (m(d-2)-2n)\eta } \exp\left((2m+3)\sigma-\frac{t\sigma}{e^{-\eta}-t}\right)$ \\[7pt] \> \> \> $\displaystyle \frac{(e^{-\eta}-t)^{-d+i-1+g}}{(1-t)^{2g-2}} \left(\frac{\eta}{1-e^{-\eta}} \right)^{i-g+1} \exp \left( \frac{\sigma}{e^{\eta}-1} - \frac{\sigma}{\eta}\right), X_i \Bigg\rangle $ \\[7pt] \> = \> $\displaystyle \mathop{{\rm Coeff}}_{t^{q_i-i}} \mathop{{\rm Res}}_{\eta = 0} \Bigg\{ \frac{e^{((d-2)m-2n)\eta}(e^{-\eta}-t)^{-d+i-1+g}} {(1+t)^{2g-2}(1-e^{-\eta})^{i+1}}$ \\[7pt] {\bf (\theequation )} \> \> \> $\displaystyle \left(e^{-\eta}+\left(2m+3-\frac{t}{e^{-\eta}-t}\right)(1-e^{-\eta})\right)^g d\eta \Bigg\}$; \end{tabbing} \end{manynotop} \noindent the first equality by Riemann-Roch, the second by \re{6g} and \re{6i}, and the third by taking $$A(x) = \left(\frac{x}{1-e^{-x}}\right)^{i-g+1} e^{((d-2)m-2n)x} \frac{(e^{-x}-t)^{-d+i-1+g}}{(1+t)^{2g-2}} $$ and $$B(x) = 1/(e^x-1) - 1/x + 2m + 3 - t/(e^{-x}-t)$$ in \re{6j}, then combining $g$th powers. The term in braces is the product of $\left(\frac{e^{-\eta}-t}{1-e^{-\eta}}\right)^i$ with something independent of $i$, so make the substitution $$y = \frac{e^{-\eta}-t}{1-e^{-\eta}},\,\,\,\,\,\,\,\,\,\, e^{-\eta} = \frac{1+ty}{1+y},\,\,\,\,\,\,\,\,\,\, 1 - e^{-\eta} = \frac{(1-t)y}{1+y},$$ $$ e^{-\eta} - t = \frac{1-t}{1+y}, \,\,\,\,\,\,\,\,\,\,d\eta = \frac{(1-t)dy}{(1+y)(1+ty)}.$$ Then the residue in \re{6k} becomes $$\mathop{{\rm Res}}_{y=0}\left\{ \frac{a(y)dy}{y^{i+1}} \right\} = \mathop{{\rm Coeff}}_{y^i} a(y) $$ for $$a(y) = \frac{(1+ty)^{q_{d/2}-1}(1+y)^{-q_{d/2}+d-2g+1}}{(1-t)^{d+g-1}} \Big(1+(2m+3)(1-t)y - ty^2 \Big)^g.$$ Then since $q_i - i = (m+n) - (m+1)i$, \begin{eqnarray*} \dim V_{m,n} & = & \sum_{i = 0}^{\infty}(-1)^i N_i \\ & = & \sum_{i = 0}^{\infty}(-1)^i \mathop{{\rm Coeff}}_{t^{q_i-i}} \mathop{{\rm Coeff}}_{y^{i}} a(y) \\ & = & \mathop{{\rm Coeff}}_{t^{m+n}} \left( \sum_{i = 0}^{\infty} (-t^{m+1})^i \mathop{{\rm Coeff}}_{y^i} a(y) \right) \\ & = & \mathop{{\rm Coeff}}_{t^{m+n}} a(-t^{m+1}). \end{eqnarray*} Thus we obtain the following theorem. We repeat the definition of $V_{m,n}$ for convenience. \begin{thm} \label{6m} Let $X$ be embedded in $\mbox{\bbb P} H^1(\Lambda^{-1})$ via the linear system $|K_X\Lambda|$. For any $m,n \geq 0$, let $V_{m,n} = H^0(\mbox{\bbb P} H^1(\Lambda^{-1}); {\cal O}(m+n) \otimes {\cal I}_X^n)$. Define $$F(t) = \frac{(1-t^{m+2})^{-h-1}(1-t^{m+1})^{-h'-1}}{(1-t)^{d+g-1}t^{m+n}} \Big(1-(2m+3)(1-t)t^{m+1}-t^{2m+3} \Big)^g,$$ where $h = (d-2)m-2n$ and $h' = -h -d +2g -2$. Then if $m(d-2) - 2n > -d+2g-2$, $$\dim V_{m,n} = \mathop{{\rm Res}}_{t=0} \left\{ \frac{F(t) dt}{t} \right\} ,$$ that is, the constant term in the Laurent expansion of $F(t)$ at $t=0$. Moreover, if $d \geq 2g$ and $m(d-2) - 2n < 0$, then $V_{m,n} = 0$. \mbox{ $\Box$} \end{thm} This is the most explicit formula for $\dim V_{m,n}$ we will obtain in general. However, in some cases we could obtain completely explicit formulas. If $m+n$ is small, for example, we could calculate directly, since we would then be looking at the residue of a function with a pole of low order; for fixed $m+n$, we would get an explicit polynomial in $g$, $d$, $m$, and $n$. Otherwise, we can still use the residue theorem, which says that the sum of the residues at all the poles of $F(t) dt/t$ is zero. These poles are of five possible kinds: $t=0$, $t=\infty$, $t=1$, $t^{m+1}= 1$ but $t \neq 1$, and $t^{m+2}= 1$ but $t \neq 1$ (note that the last two cases are disjoint). But in fact $t=1$ is never a pole, since at that point $1-(2m+3)(1-t)t^{m+1} -t^{2m+3}$ has a triple zero, and hence the order of $F(t)$ is $$(-h-1)+(-h'-1)-(d+g-1)+3g = 1 \geq 0.$$ Also, it is straightforward to check that $F(1/t) = -F(t)$, which implies that $$\mathop{{\rm Res}}_{t = \infty}\left\{\frac{F(t)dt}{t}\right\} = \mathop{{\rm Res}}_{t = 0}\left\{\frac{F(t)dt}{t}\right\}.$$ Hence \begin{equation} \label{6q} -2\dim V_{m,n} = \Bigg( \sum_{\stackrel{\scriptstyle \zeta^{m+1} = 1}{\zeta \neq 1}} \mathop{{\rm Res}}_{t=\zeta} + \sum_{\stackrel{\scriptstyle \zeta^{m+2} = 1}{\zeta \neq 1}} \mathop{{\rm Res}}_{t=\zeta}\Bigg)\left\{\frac{F(t)dt}{t}\right\}. \end{equation} There are poles at the $(m+2)$th roots of unity if and only if $h \geq 0$, and at the $(m+1)$th roots of unity if and only if $h' \geq 0$. Thus $\dim V_{m,n}$ is a sum over the residues at the $(m+2)$th roots if $h' < 0 \leq h$, a sum over the residues at the $(m+1)$th roots if $h < 0 \leq h'$, and is 0 if $h, h' < 0$. (Note that this last case agrees with \re{6p}.) For $h \geq 0$ it is necessary to calculate the residue of a function with a pole of order $1+h$, which gets more and more difficult as $h$ grows. However, when $h=0$, the calculation is easy, and we can prove the celebrated Verlinde formula. \begin{equation} \label{6n} \dim Z_k(\Lambda) = \left( \frac{k+2}{2} \right)^{g-1} \sum_{j=1}^{k+1}\frac{(-1)^{d(j+1)}}{( \sin \frac{j\pi}{k+2})^{2g-2}}. \end{equation} {\em Proof}. If $d$ and $k$ are both odd, then on symmetry grounds the right-hand side is zero as desired. So assume $d$ and $k$ are not both odd. By \re{5f} $\dim Z_k(\Lambda) = \dim V_{k,k(d/2-1)}$ for any $d > 2g-2$. Then $h=0$ and $h' < 0$, so by \re{6q} \begin{eqnarray*} \lefteqn{-2 \dim V_{k,k(d/2-1)}} \\ & = & \sum_{\stackrel{\scriptstyle \zeta^{k+2} = 1}{\zeta \neq 1}} \mathop{{\rm Res}}_{t=\zeta} \left( \frac{-dt/t}{t^{k+2}-1} \right) \frac{(1-\zeta^{-1})^{d-2g+1}}{(1-\zeta)^{d+g-1}\zeta^{kd/2}} \Big(1-(2k+3)(\zeta^{-1}-1) -\zeta^{-1}\Big)^g. \end{eqnarray*} But $(1-(2k+3)(\zeta^{-1}-1) -\zeta^{-1}) = (2k+4)(1-\zeta^{-1})$, the residue is $-1/(k+2)$, and $$\frac{(1-\zeta^{-1})^d}{(1-\zeta)^d\zeta^{kd/2}} = \frac{(1-\zeta^{-1})^d}{(1-\zeta)^d\zeta^{-d}\zeta^{(k+2)d/2}} = (-1)^d \zeta^{(k+2)d/2},$$ so \begin{eqnarray*} \dim V_{k,k(d/2-1)} & = & (2k+4)^{g-1} \sum_{\stackrel{\scriptstyle \zeta^{k+2} = 1}{\zeta \neq 1}} (-1)^{d} \zeta^{(k+2)d/2} \left(\frac{-\zeta}{(1-\zeta)^2}\right)^{g-1} \\ & = & {\scriptstyle{\frac{1}{2}}} (2k+4)^{g-1} \sum_{\stackrel{\scriptstyle{\xi^{2k+4}=1}} {\scriptstyle{\xi \neq \pm 1}}} \frac{(-1)^{d+g-1}\xi^{(k+2)d}}{(\xi^{-1} - \xi)^{2g-2}}, \end{eqnarray*} which is equivalent to the Verlinde formula. \mbox{ $\Box$} \bit{Relation with Bertram's work} In this appendix we explain briefly, without proving anything, how this paper is related to Bertram's work on secant varieties. In \cite{bert}, Bertram considers how to resolve the rational map $\mbox{\bbb P} H^1(\Lambda^{-1}) \mbox{$\rightarrow$} N$. He shows that blowing up first $X \subset \mbox{\bbb P} H^1(\Lambda^{-1})$, then the proper transform of each of its secant varieties in turn, produces after $[(d-1)/2]$ steps a smooth variety $\tilde{\mbox{\bbb P}}$ having a morphism to $N$ that agrees with the rational map away from the blow-ups. The existence of the morphism is proved by constructing a sequence of families of bundles, each obtained by an elementary transformation of the last, starting with the pullback of the tautological family on $\mbox{\bbb P} H^1(\Lambda^{-1}) \times X$, and ending with a family of bundles that are all semistable. Bertram's families of bundles can be interpreted, after some twisting, as families of pairs in our sense, and it follows that his $\tilde{\mbox{\bbb P}}$ dominates all of the $M_i$. In other words, he performs all of our blow-ups but none of our blow-downs. In particular, our blow-up loci are birational to his, that is, our $\mbox{\bbb P} W^-_i$ in $M_{i-1}$ is the proper transform of the $i$th secant variety in $\mbox{\bbb P} H^1(\Lambda^{-1}) = M_0$. This makes sense, since both are essentially $\mbox{\bbb P}^{i-1}$-bundles over $X_i$. However, this correspondence is a little more delicate than it seems, because the $\mbox{\bbb P}^{i-1}$-bundles are different: ours is $\mbox{\bbb P} W^-_i = \mbox{\bbb P} (R^0 \pi) {\cal O}_{\Delta}\Lambda(-\Delta)$, but as Bertram explains, the secant variety is the image in $\mbox{\bbb P} H^1(\Lambda^{-1})$ of $\mbox{\bbb P} (R^0 \pi) {\cal O}_{\Delta} K\Lambda$. How is one projective bundle transformed into another? If we pull back the lower secant varieties to $\mbox{\bbb P} (R^0 \pi) {\cal O}_{\Delta} K\Lambda$ we find that blowing them up and down induces a {\em Cremona transformation} on each fibre of the projective bundle. For example, consider the $\mbox{\bbb P}^2$ fibre over $x_1 + x_2 + x_3 \in X_3$ of the 3rd secant variety. This of course meets $X \subset \mbox{\bbb P} H^1(\Lambda^{-1})$ in the 3 points $x_1, x_2, x_3$, so if $X$ is blown up, then $\mbox{\bbb P}^2$ gets blown up at those 3 points. The proper transform of the 2nd secant variety meets this blown-up $\mbox{\bbb P}^2$ in the proper transforms of the 3 lines between the points, so blowing it up does nothing, and blowing it down blows down the 3 lines. All in all we have blown up the vertices of a triangle in the plane, then blown down the proper transforms of the edges. This is well-known to recover $\mbox{\bbb P}^2$ \cite[V 4.2.3]{h}; indeed it is given in coordinates by $[z_0, z_1, z_2] \mapsto [z_1 z_2, z_0 z_2, z_0 z_1]$. If we do the same thing to $\mbox{\bbb P}^3$, we find ourselves blowing up the vertices of a tetrahedron, then blowing up and down---that is to say, flipping---the proper transforms of the edges, and finally blowing down the proper transforms of the faces. Notice that by the time we get to the faces, they have already undergone Cremona transformations themselves. More generally, starting with a simplex in $\mbox{\bbb P}^n$, we may flip all of the subsimplices, starting with the vertices and working our way up. The varieties we obtain thus fit into a diagram shaped exactly like that at the end of \S3. It is not so well-known that this recovers $\mbox{\bbb P}^n$, or that it is given in coordinates by $[z_i] \mapsto [z_0 \cdots z_{i-1} z_{i+1} \cdots z_n]$, but these facts can be proved using the theory of toric varieties. Even that is not quite the end of the story, since over divisors in $X_i$ with multiple points the transformations are somewhat different. Over $2x_1 + x_2 \in X_3$, for example, we want to blow up one reduced point and one doubled point, then blow down one reduced line and one doubled line. In coordinates, this is $[z_0,z_1,z_2] \mapsto [z_0^2, z_0 z_1, z_1 z_2]$. It is an amusing exercise to work out coordinate expressions for the Cremona transformations over other divisors with multiple points.

9703

alg-geom/9703023

en

https://arxiv.org/abs/alg-geom/9703023

alg-geom/9703023

Lev A. Borisov

On Betti numbers and Chern classes of varieties with trivial odd cohomology groups

5 pages, LaTeX. It turned out that most of the results of the paper are already known. The appropriate reference is added

It was noticed in a very recent preprint of T. Eguchi, K. Hori, and Ch.-Sh. Xiong (hep-th/9703086) that a curious identity between Betti numbers and Chern classes holds for many examples of Fano varieties. The goal of this paper is to prove that for varieties with trivial odd cohomology groups this identity is equivalent to having zero Hodge numbers $h^{p,q}$ for $p\neq q$.

alg-geom

\section{Introduction} \noindent Let $X$ be a smooth complex projective variety of dimension $n$ whose odd cohomology groups $H^{2k+1}({\bf C})$ are zero. It was noticed in \cite{tekhcx} that a curious identity $$\frac 14 \sum_k h^{2k} (k-\frac{n-1}2)(1-k+\frac{n-1}2)= \frac1{24}\left(\frac{3-n}2\chi(X)-\int_Xc_1(X)\wedge c_{n-1}(X)\right)$$ holds in many examples. The authors of \cite{tekhcx} were primarily concerned with Fano varieties $X$ and the above identity was a prerequisite for the conjectural construction of Virasoro operators that control quantum cohomology of $X$. The above condition could be rewritten as $$\sum_{k=0}^n h^{2k} (k-\frac n2)^2=\frac16 c_1c_{n-1}+\frac n{12} c_n$$ where $c_l$ is the $l$-th Chern class of $X$. The goal of this paper is to prove the following result. \begin{prop} If $X$ is a smooth complex projective variety of dimension $n$ with $H^{odd}(X,{\bf C})=0$ then $$\sum_{k=0}^n h^{2k} (k-\frac n2)^2\leq\frac16 c_1c_{n-1}+\frac n{12} c_n$$ and equality holds if and only if $$h^{p,q}=0 ~{\rm for~all}~p\neq q.$$ \end{prop} \label{main} In addition, we give an application of this result to the combinatorics of reflexive polytopes that describe smooth toric Fano varieties. After this preprint was submitted to the archive, the author was informed by Anatoly Libgober that the proof of the crucial Proposition 2.2 is contained in \cite{liwo}. In addition, Victor Batyrev presented to the author a combinatorial proof of Corollary 2.3 for arbitrary smooth toric varieties. \section{Proof of the main result} \noindent Let $X$ be a smooth complex projective variety of dimension $n$. We introduce the E-polynomial $$E(u,v)=\sum_{p,q}(-1)^{p+q}h^{p,q}u^pv^q$$ where $h^{p,q}={\rm dim}H^q(X,\Lambda^p T^*X)$ are Hodge numbers of $X$. We also introduce $$\chi_p=(-1)^p\chi(\Lambda^p T^*X)=\sum_q(-1)^{p+q}h^{p,q}$$ and the polynomial $$\hat E(t)=\sum_p\chi_p t^p=E(t,1).$$ \begin{rem} If $h^{p,q}=0$ for all $p\neq q$ then $\chi_p=h^{p,p}$. \end{rem} \label{hpp} \begin{prop} In the above notations $$\sum_{p=0}^n\chi_p(p-\frac n2)^2=\frac16c_1c_{n-1}+\frac n{12}c_n.$$ \end{prop} \label{chiequal} {\em Proof.} This result comes as an easy application of Hirzebruch-Riemann-Roch theorem \cite{hirz}. We start by rewriting the left hand side in terms of the polynomial $\hat E$. $$\sum_p\chi_p(p-\frac n2)^2=\sum_p\chi_p p(p-1) + \sum_p \chi_p(1-n)(p-\frac n2)+(\sum_p \chi_p)(\frac n2-\frac {n^2}4)$$ $$=\frac {d^2}{dt^2}\hat E(t)|_{t=1}+\hat E(1)(\frac n2-\frac {n^2}4).$$ We have used $\chi_p=\chi_{n-p}$ to get rid of the second sum. By Hirzebruch-Riemann-Roch theorem, $$\hat E(t)=\sum_p t^p(-1)^p\chi(\Lambda^pT^*X)= \int_X Td(X)\sum_p(-t)^p ch(\Lambda^pT^*X).$$ As usual, we introduce Chern roots $\alpha_i$ such that $c(TX)(w)=\prod_i(1+\alpha_iw)$, see for example \cite{fult}. Then $$\hat E(t)=\int_X (\prod_i\frac{\alpha_i}{1-{\rm e}^{-\alpha_i}}) \sum_p(-t)^p\sum_{i_1<...<i_p}{\rm e}^{-\alpha_{i_1}-...-\alpha_{i_p}}$$ $$=\int_X \prod_i \alpha_i(1+(1-t)\frac{e^{-\alpha_i}}{1-{\rm e}^{-\alpha_i}}).$$ This shows, of course, that $\hat E(1)=\chi(X)=c_n$. Besides we can calculate $$ \frac {d^2}{dt^2}\hat E(t)|_{t=1}= 2\sum_{i<j}\int_X (\prod_{k\neq i,j} \alpha_k) (1-\frac12\alpha_i+\frac 1{12}\alpha_i^2)(1-\frac12\alpha_j+\frac 1{12}\alpha_j^2) $$ $$=\frac16 c_1c_{n-1}+(\frac {n^2}4-\frac {5n}{12})c_n$$ and the rest is straightforward. \hfill $\Box$ We combine Proposition 2.2 with Remark 2.1 to get the following corollary. \begin{coro} If $h^{p,q}=0$ for all $p\neq q$ then $$\sum_{p=0}^n h^{2p}(p-\frac n2)^2=\frac16c_1c_{n-1}+\frac n{12}c_n.$$ \end{coro} This corollary gives a sufficient condition for the identity of \cite{tekhcx}. Now we will see that this condition is also necessary, that is we will prove Proposition 1.1. {\em Proof of Proposition 1.1.} Because of $h^{odd}=0$, we have $\chi_p=\sum_q h^{p,q}$ and $$\sum_{p=0}^n h^{2p}(p-\frac n2)^2= \sum_{p,q}h^{p,q}(\frac{p+q}2-\frac n2)^2 $$ $$=\sum_p (\sum_q h^{p,q})(p-\frac n2)^2+ \sum_{p,q} h^{p,q}(\frac {q-p}2)(\frac {3p+q}2-n)$$ $$ =\sum_p \chi_p (p-\frac n2)^2- \sum_{p,q}h^{p,q}(\frac {q-p}2)^2+ \sum_{p,q}h^{p,q}(\frac {q-p}2)(p+q-n).$$ Because of $h^{p,q}=h^{q,p}$, the last sum is zero. Together with the result of Proposition 2.2, we get $$\sum_{p=0}^n h^{2p}(p-\frac n2)^2= \frac 16 c_1c_{n-1}+\frac 1{12} c_n-\sum_{p,q}h^{p,q}(\frac {q-p}2)^2$$ which proves the proposition. \hfill ${\Box}$ \section{Application to toric Fano varieties} The goal of this section is to give a combinatorial equivalent of Corollary 2.3 for the case of smooth toric Fano varieties. Recall (see for example \cite{baty}) that a smooth toric Fano variety can be defined in terms of the polytope $\Delta \in M$ that supports the sections of the anticanonical line bundle. To calculate the left hand side, notice that $X$ is a disjoint union of algebraic tori $T_\theta$. Using the additivity of the cohomology with compact support, we get $$\hat E(t)=\sum_{\theta\subseteq \Delta}(t-1)^{{\rm dim}(\theta)}$$ and $$\frac {d^2}{dt^2}\hat E(t)|_{t=1}=2\#\{\theta\subseteq \Delta, {\rm dim}(\theta)=2\}.$$ Analogously, $$\hat E(1)=\#\{\theta\subseteq \Delta, {\rm dim}(\theta)=0\}.$$ To calculate the right hand side of the identity, notice that $$c(TX)(w)=\prod_{\theta, {\rm dim}(\theta)=n-1} (1+D_\theta w),$$ where $D_\theta$ is the closure of the strata that corresponds to $\theta$, and that $\sum D_\theta$ is a divisor with normal crossings. This gives $$c_1=\sum_{\theta, {\rm dim}(\theta)=n-1} D_\theta$$ $$c_{n-1}=\sum_{\theta, {\rm dim}(\theta)=1} l_\theta$$ where in the second identity $l_\theta$ is the closure of $T_\theta$. One can show that $c_1 l_\theta$ equals the number of interior points of $\theta$ plus one. Now an easy calculation shows that the identity of Corollary 2.3 becomes $$ \#\{\theta\subseteq \Delta, {\rm dim}(\theta)=2\}= \frac1{12}\sum_{\theta,{\rm dim}(\theta)=1} \#\{P\in M,~P\in {\rm interior}(\theta)\}$$ $$+ (\frac {n^2}8-\frac n6)\#\{\theta\subseteq \Delta, {\rm dim}(\theta)=0\}. $$ \bigskip

9703

alg-geom/9703011

en

https://arxiv.org/abs/alg-geom/9703011

alg-geom/9703011

Philip A. Foth

Jean-Luc Brylinski and Philip Foth

Moduli of flat bundles on open Kaehler manifolds

LaTeX 21p, revised

We consider the moduli space MN of flat unitary connections on an open Kaehler manifold U (complement of a divisor with normal crossings) with restrictions on their monodromy transformations. Using intersection and L2 cohomologies with degenerating coefficients we construct a natural symplectic form F on MN. When U is quasi-projective we prove that F is actually a Kaehler form.

alg-geom

\section{Introduction} \setcounter{equation}{0} Let $X$ be a compact K\"{a}hler manifold and $D$ a divisor on $X$ with normal crossings. There exists a moduli space ${{\cal M}_{\cal N}}$ of flat irreducible unitary bundles on $U=X\setminus D$ such that the monodromy transformation around each smooth irreducible component of $D$ lies in a prescribed conjugacy class in $U(N)$. We develop a theory of deformations of representations of a discrete group relative to our situation. We obtain a condition (similar to Goldman-Millson \cite{GM}) when ${{\cal M}_{\cal N}}$ has a manifold structure and further we work under this assumption. If we pick a representation $\rho$ of $\pi_1(U)$ satisfying these conditions, then we get a local system ${\tilde{\frak{g}}}$ on $U$ associated with the representation $\rho$. Since ${\tilde{\frak{g}}}$ has singularities on $X$ (along $D$) it is quite natural to express the tangent space $T_{\rho}{{\cal M}_{\cal N}}$ in terms of the intersection cohomology groups of $X$ with coefficients in ${\tilde{\frak{g}}}$. Lemma \ref{lem:l32} shows that this tangent space identifies with the group $IH^1(X, {\tilde{\frak{g}}})$. We introduce a natural $2$-form on ${{\cal M}_{\cal N}}$ as a pairing $IH^1(X, {\tilde{\frak{g}}})\times IH^1(X, {\tilde{\frak{g}}})\to{\Bbb R}$. To see that our form is actually closed, we map our manifold ${{\cal M}_{\cal N}}$ to an infinite-dimensional affine space such that its tangent space at any point consists of $L_2$ forms on $X$ having certain additional properties. This affine space admits a constant coefficient $2$-form such that its pull-back to ${{\cal M}_{\cal N}}$ is given by a pairing in $L_2$ cohomology. We further use the isomorphism between intersection cohomology and $L_2$ cohomology constructed by Cattani-Kaplan-Schmid \cite{CKS}, and Kashiwara-Kawai \cite{KK}. As an auxiliary tool we introduce the notion of $L_2$ vector bundle, which seems to be of independent interest. We prove \begin{Th} The moduli space ${{\cal M}_{\cal N}}$ is symplectic. \label{Th:t11} \end{Th} Intersection cohomology enjoys such properties as Poincar\'{e} duality and the Hard Lefschetz theorem. This follows from an unpublished work of Deligne and is partially explained by Zucker in \cite{Zu}. The point is that the Hard Lefschetz theorem holds true for the $L_2$ cohomology groups, because they are finite-dimensional, which again is a consequence of the isomorphism between $L_2$ and intersection cohomologies. The Hard Lefschetz theorem provides us with an isomorphism $IH^{d-j}(X, {\tilde{\frak{g}}})\simeq IH^{d+j}(X, {\tilde{\frak{g}}})$, where $d=\dim_{{\Bbb C}}X$. This allows us to see that our $2$-form is non-degenerate. We also prove \begin{Th} When $X$ is projective, the moduli space ${{\cal M}_{\cal N}}$ is {K\"{a}hler\ }. \label{Th:t12} \end{Th} When $X$ is a curve we use the identification of ${{\cal M}_{\cal N}}$ with the moduli space of stable parabolic vector bundles given by Mehta-Seshadri \cite{MS} to get a {K\"{a}hler\ } structure on the space ${{\cal M}_{\cal N}}$. Then we reduce the case of general projective manifold $X$ to the case of a curve by taking appropriate number of hyperplane sections of $X$. The result was known in some special cases. For example, in the compact case (when $D$ is empty), the symplectic structure on the moduli space appears in the works of Atiyah-Bott \cite{AB}, Goldman \cite{G}, and Karshon \cite{Kar}. In the case of a Riemann surface with punctures the symplectic structure on the moduli space was described by Atiyah \cite{A3}, Biquard \cite{BiqTh} (see also \cite{Biq2}), and Witten \cite{W}. It was also the subject of a paper by Biswas-Guruprasad \cite{BG}; a proof in this case using group cohomology is due to Guruprasad-Huebschmann-Jeffrey-Weinstein \cite{GHJW}. We would like to thank P. Deligne for his useful comments. \section{Description of the moduli space} \setcounter{equation}{0} Let $X$ be a compact K\"ahler manifold of complex dimension $d$ endowed with a K\"{a}hler form ${\lambda}$, and let $D$ be a divisor on $X$ with normal crossings such that $D=\cup_{i=1}^rD_i$ is a decomposition of $D$ into the union of smooth irreducible complex analytic subvarieties. Let $G$ be a compact connected Lie group and ${\frak{g}}$ its Lie algebra with fixed non-degenerate invariant bilinear form $B\langle\ ,\ \rangle$. From now on we fix the set ${\cal N}=({\cal C}_1, {\cal C}_2, ..., {\cal C}_r)$ of $r$ conjugacy classes in $G$. On the complement of $D$ we also have a K\"{a}hler form - the restriction of the form ${\lambda}$ so that one has an open K\"{a}hler manifold $U:=X\setminus D$. We notice that any quasi-projective smooth algebraic variety can be obtained this way. Take a base point $b\in U$ and denote by $\pi_1(U)=\pi_1(U,b)$ the fundamental group of $U$ with base point $b$ and by $\pi_1(X)=\pi_1(X,b)$ the fundamental group of $X$ with the same base point. Let us define ${\bar{\cal M}}_{{\cal N}}$ - the moduli space of flat $G$-bundles on $U$ such that the monodromy transformation around $D_i$ lies in ${\cal C}_i$. Later on we will provide the reader with some examples when ${\bar{\cal M}}_{{\cal N}}$ is actually smooth. The moduli space ${\cal M}$ of flat $G$-bundles over $X$ identifies with the space $Hom(\pi_1(X), G)/G$, where the group $G$ acts by conjugation. As it is well-known \cite{G} \cite{Kar} the space ${\cal M}$ admits a natural symplectic structure. Let $\rho$ be a smooth point on ${\cal M}$. Then the Zariski tangent space at the class of $\rho$ is $T_{[\rho]}{\cal M}=H^1(\pi_1(X), {\frak{g}})$, where ${\frak{g}}$ is considered as a $\pi_1(X)$-module via the adjoint representation followed by $\rho$. Let ${\tilde{\frak{g}}}$ stand for the local system associated with the representation $\rho$. Due to the isomorphism $H^1(\pi_1(X), {\frak{g}})\simeq H^1(X, {\tilde{\frak{g}}})$ and the natural map $H^2(\pi_1(X), {\frak{g}})\to H^2(X, {\tilde{\frak{g}}})$ we get the operation $$ \begin{array}{ccccc} H^1(\pi_1(X), {\frak{g}}) & \cup & H^1(\pi_1(X), {\frak{g}}) & \to & H^2(\pi_1(X), {\frak{g}}) \\ \parallel & {} & \parallel & {} & \downarrow \\ H^1(X, {\tilde{\frak{g}}}) & \cup & H^1(X, {\tilde{\frak{g}}}) & \to & H^2(X, {\tilde{\frak{g}}}) \end{array} $$ Then we compose $$ B\langle\cdot,\cdot\cup\ [{\lambda}]^{d-1}\rangle: H^1(X, {\tilde{\frak{g}}})\times H^1(X, {\tilde{\frak{g}}}) \to{\Bbb R} $$ to complete the pairing. Our purpose is to construct a symplectic structure (and, moreover a K\"{a}hler structure) on the moduli space ${\bar{\cal M}}_{{\cal N}}$ (at least on its smooth locus) satisfying the following condition: if $X$ is smooth and ${\cal C}_i=1$ for each $i$ (under this assumption of course ${\cal M}={\bar{\cal M}}_{{\cal N}}$) then this form is the one described above. The moduli space ${\bar{\cal M}}_{{\cal N}}$ is the same as $Hom_{{\cal N}}(\pi_1(U), G)/G$, where the subscript ${\cal N}$ means that the prescribed generator of $\pi_1(U)$ corresponding to a given loop ${\gamma}_i$ around $D_i$ goes to ${\cal C}_i\in G$. We will denote by ${{\cal M}_{\cal N}}$ the locus of ${\bar{\cal M}}_{{\cal N}}$ corresponding to the {\it irreducible} representations of $\pi_1(U)$. \ \noindent{\it Remark.} The conjugacy classes ${\cal C}_i$ may be chosen in such a fashion that the moduli space is empty (on ${\Bbb C}{\Bbb P}^1$ with one puncture one can take ${\cal C}\ne Id$, see also \cite{SPM}). Also we notice that the obvious map $$\pi_1(U)\to\pi_1(X) $$ is surjective. \ \begin{prop} The Zariski tangent space to the moduli space ${{\cal M}_{\cal N}}$ at the point $\rho$ is $$ T_{[\rho]}{{\cal M}_{\cal N}}=Ker(H^1(\pi_1(U), {\frak{g}})\to\prod_i H^1(\Gamma_i, {\frak{g}})), $$ where $\Gamma_i\simeq{\Bbb Z}$ is generated by the class of a loop encircling $D_i$. \label{prop:p21} \end{prop} {\noindent{\it Proof.\ \ }} We shall repeatedly use the fact that for every connected manifold $Z$ the two groups $H^1(Z, {\tilde{\frak{g}}})$ and $H^1(\pi_1(Z), {\frak{g}})$ are canonically isomorphic. To understand the tangent space $T_{[\rho]}{{\cal M}_{\cal N}}$ we recall the well-known fact that it is a subspace of the tangent space $T_{[\rho]}(Hom(\pi_1(U),G)/G)$. Also we note that the tangent space to the conjugacy class ${\cal C}_g$ of $g\in G$ is the subspace of ${\frak{g}}$ given as the range of $Ad(g)-1$ if we identify as usual the tangent space $T_gG$ with ${\frak{g}}$ via the action of the left translation by $g$. Now if $\gamma_i$ is the class of a loop encircling $D_i$ and if $\Gamma_i\subset\pi_1(U)$ is the cyclic subgroup generated by $\gamma_i$ then the cohomology group $H^1(\Gamma_i, {\frak{g}})$ is the cokernel of the map $Ad(g)-1:\ {\frak{g}}\to{\frak{g}}$, so the equality follows. $\bigcirc$ \ The space $Ker(H^1(\pi_1(U), {\frak{g}})\to\prod_i H^1(\Gamma_i, {\frak{g}}))$ is called {\it the parabolic cohomology} of $X$ with coefficients in the local system ${\tilde{\frak{g}}}$ on $U$ associated to the action of $\pi_1(U)$ on ${\frak{g}}$. We will briefly discuss some of the known results for punctured Riemann surfaces. Here $D_1, ...,D_r$ are just distinct points on ${\Sigma}$. Let $G_{{\Bbb C}}=SL(n, {\Bbb C})$, $SO(n, {\Bbb C})$, or $Sp(2n, {\Bbb C})$ and let $G$ be its standard maximal compact subgroup. Let ${\Bbb C}^r$ be the standard representation space of $G_{{\Bbb C}}$ (so that $r=n$ or $2n$) and consider $V=\oplus_{i=1}^r\wedge^i{\Bbb C}^r$ which is naturally a representation space of $G_{{\Bbb C}}$ too. We say that an element $A\in G_{{\Bbb C}}$ (or its conjugacy class) satisfies {\it property P} if two stabilizers have the same dimension: $\dim(V^{A_s})=\dim(V^T)$, where $T$ is a maximal torus in $G_{{\Bbb C}}$ (or in $G$) and $A=A_sA_n$ is a Jordan decomposition of $A$ into the product of commuting unipotent and semisimple elements. In terms of the eigenvalues ${\lambda}_1, ..., {\lambda}_n$ of $A$ property P means that the product ${\lambda}_{i_1}\cdots{\lambda}_{i_k}\ne 1$ for any $i_1<\cdots <i_k$ and $k<n$. For more details we refer to a paper of the second author \cite{F} where it was shown that in the case of a Riemann surface with one puncture the moduli space of flat $SL(n,{\Bbb C})$- (or $SU(n)$-) bundles is smooth if and only if the monodromy transformation around the puncture has property P. It was also shown that if $G_{{\Bbb C}}=SO(n, {\Bbb C})$, or $Sp(2n, {\Bbb C})$, then the moduli space has at worst quotient singularities (i.e. it is a quotient of a smooth manifold by an action of a finite group). \section{Deformations of representations of a discrete group} \setcounter{equation}{0} We have to make an additional assumption in order to guarantee that each point $[\rho]\in{{\cal M}_{\cal N}}$ (corresponding to an irreducible representation $\rho$) is a smooth point and the tangent space $T_{[\rho]}{{\cal M}_{\cal N}}$ to the {\it manifold} is given by Proposition 2.1. For this it is enough to show that every infinitesimal deformation (an element of the Zariski tangent space) is tangent to an analytic path in ${{\cal M}_{\cal N}}$. We recall a theorem of M. Artin \cite{Art}, which asserts that an infinitesimal deformation is tangent to an analytic path if and only if there exists a formal power series deformation with the infinitesimal deformation as its leading term. In the compact {K\"{a}hler\ } case, the criterion is established by Goldman-Millson \cite{GM}. For example, it is enough to have $H^2(\pi_1(X), {\frak{g}})=0$. Analogously to their methods we will show that in our situation a sufficient condition is the vanishing of the second relative cohomology group $H^2(\pi_1(U), (\Gamma_i), {\frak{g}})$. Therefore, for the rest of the paper we will assume that we are dealing with the situation when $H^2(\pi_1(U), (\Gamma_i), {\frak{g}})=0$ and thus we have no obstruction for representing the actual tangent space as the kernel of the map $H^1(\pi_1(U), {\frak{g}})\to\prod_iH^1(\Gamma_i, {\frak{g}})$. In all other situation we only get "formal symplectic structure" and it is a separate problem which will be treated elsewhere to understand its actual meaning. In the present section we will establish all the results in the following generality. Let $\pi$ be a discrete finitely generated group and let $(\Gamma_i)_{i=1}^r$ be a system of its subgroups such that $\Gamma_i\simeq{\Bbb Z}$ is generated by an element ${\gamma}_i$. We consider the representation variety ${{\cal M}_{\cal N}}=Hom(\pi, G)_{{\cal N}}/G$ ($G=U(N)$), consisting of classes of such group homomorphisms that $Im(\pi)$ is not contained in any proper parabolic subgroup of $G$ (irreducibility condition) and $Im({\gamma}_i)\in{\cal C}_i$. We saw that the Zariski tangent space to a point $[\rho_0]$ is given by $$T_{[\rho_0]}{{\cal M}_{\cal N}}=Ker[H^1(\pi, {\frak{g}})\to\oplus_iH^1(\Gamma_i, {\frak{g}})].$$ Our task is to prove \begin{prop} If $H^2(\pi, (\Gamma_i), {\frak{g}})=0$ then for any $\eta\in T_{[\rho_0]}{{\cal M}_{\cal N}}$ there exists a formal power series $\rho_t$ of $\rho_0$ representing $\eta$. \label{prop:pg} \end{prop} We devote the rest of the section to the proof of this statement. Let us have $$\rho_t(g)=\rho_0(g)\exp(\sum_{i=1}^{\infty}f_i(g)t^i),$$ where $f_i: \pi\to{\frak{g}}$ is a group $1$-cochain and $t$ is a formal parameter. We recall that ${\frak{g}}$ is a $\pi$ - module via the adjoint representation followed by $\rho_0$. We have the following two conditions to satisfy. First, $\rho_t$ is a group homomorphism, therefore $\rho_t(g_1g_2)=\rho_t(g_1)\rho_t(g_2)$. Secondly, the condition of mapping ${\gamma}_i$ to fixed conjugacy classes is written as \begin{equation}\rho_t({\gamma}_j)=\exp(\sum_{i=1}^{\infty}c_i^jt^i) \rho_0({\gamma}_j) \exp(-\sum_{i=1}^{\infty}c_i^jt^i),\label{eq:e30}\end{equation} where $c_i^j\in{\frak{g}}$. Next we make a change of notation $h_i(g)=f_i(g^{-1})$ and using the Campbell-Hausdorff formula spell out these conditions. To the first order of $t$ we have $${\partial} h_1 (g_1, g_2)={\rm Ad}\rho_0(g_1)h_1(g_2)-h_1(g_1g_2)+h_1(g_1)=0, $$ $$h_1({\gamma}_j)={\rm Ad}\rho_0({\gamma}_j)c_1^j-c_1^j={\partial} c_1^j({\gamma}_j),$$ where $c_1^j$ is considered as a zero-cochain. As one sees, these equations are equivalent to the fact that $h_1\in Ker[H^1(\pi, {\frak{g}})\to\oplus_iH^1(\Gamma_i, {\frak{g}})]$. Our purpose is to find such $h_2, h_3, ...$ and $c_2^j, c_3^j, ...$ which satisfy those two conditions. We will do this by the induction process. Let us first show explicitly the existence of such $h_2$ and $c_2^j$. To the second order of $t$ we have \begin{equation} \begin{array}{c} {\partial} h_2(g_1,g_2)=-{1\over 2}[{\rm Ad}\rho_0(g_1)h_1(g_2),h_1(g_1)], \\ h_2({\gamma}_j)-{\partial} c_2^j=-{1\over 2}[{\rm Ad}\rho_0({\gamma}_j)c_1^j, c_1^j]. \end{array} \label{eq:e1} \end{equation} We recall that given a group $\pi$ and a system of its subgroups $\Gamma_j$ together with the restriction maps $Map(\pi, {\frak{g}})\to Map(\Gamma_j, {\frak{g}})$ the relative cochain complex is defined as the cone of the system of maps of complexes $R_j: C^{{\bullet}}(\pi, {\frak{g}})\to C^{{\bullet}}(\Gamma_j, {\frak{g}})$. By definition, $$Cone_{{\frak{g}}}^k(R_1, ..., R_r)=C^k(\pi, {\frak{g}})\oplus \bigoplus_jC^{k-1}(\Gamma_j, {\frak{g}})$$ with the differential $(-{\partial}, R_j + {\partial})$. Therefore, we would be able to find such $h_2$ and $c_2^j$ if the relative group $H^2(\pi, ({\cal G}_i), {\frak{g}})$ vanishes and the right hand side of (\ref{eq:e1}) is a relative $2$-cocycle. One easily checks that the cocycle condition is satisfied: $${\partial}[{\rm Ad}\rho_0(x)h_1(y), h_1(x)]\ (g_1, g_2, g_3)=0, \ \ g_1, g_2, g_3\in\pi, \ \ {\rm and}$$ $$[{\rm Ad}\rho_0(\tau_1)h_1(\tau_2), h_1(\tau_1)]={\partial}[{\rm Ad}\rho_0(x)c_1^j, c_1^j]\ (\tau_1, \tau_2), \ \ \tau_1, \tau_2\in\Gamma_j.$$ (To verify the equalities we use that ${\partial} h_1(g_1, g_2)=0$ and $h_1({\gamma}_j)={\partial} c_1^j({\gamma}_j)$.) This procedure serves several purposes. First, we get an idea that the obstruction for an element of Zariski tangent space to be tangent to an analytic path lies in the second relative group cohomology. Besides, we notice that in all successive steps we will deal with the system of the form \begin{equation} \begin{array}{c}{\partial} h_{k+1} (g_1, g_2, g_3)=F(h_1, ..., h_k) \\ h_{k+1}({\gamma}_j)-{\partial} c_{k+1}^j({\gamma}_j)=H(c_1^j, ..., c_k^j) \end{array} \label{eq:e2} \end{equation} Also, we see that in order to show that the right hand side of these equations is a relative cocycle, we can restrict ourselves to the case of just one subgroup $\Gamma\subset\pi$ generated by an element ${\gamma}$. We need the following simple \begin{lem} Let $\pi$ be a finitely generated discrete group and let ${\gamma}\in\pi$. There exists a free group ${\cal F} $ with $g\in{\cal F}$ and a surjective homomorphism $\phi: {\cal F}\to\pi$ such that $\phi(g)={\gamma}$ and $g$ is an element of a basis of ${\cal F} $. \end{lem} {\noindent{\it Proof.\ \ }} Let ${\cal F}'$ be any free group such that there is a surjective map $\phi': {\cal F}'\to\pi$. If $(g_2, ..., g_l)$ is a basis for ${\cal F}'$ then we consider a free group ${\cal F} $ obtained from ${\cal F}'$ by adding one generator $g$. Then $(g, g_2, ..., g_l)$ is a basis of ${\cal F} $ and we let $\phi(g)={\gamma}$ and $\phi(g_i)=\phi'(g_i)$ for $2\le i\le l$. $\bigcirc$ \ Now we make the inductive step. Let ${\cal F}$ be such a free group that satisfies the condition of the above lemma for our group $\pi$ and its subgroup $\Gamma$ generated by ${\gamma}$. Let $\Gamma'\simeq{\Bbb Z}$ be the subgroup of ${\cal F}$ generated by the element $g$ from the above lemma. (We notice that $H^2({\cal F}, \Gamma', {\frak{g}})=0$.) Let us lift the system of equations (\ref{eq:e2}) to the free group ${\cal F}$. We were able to find a solution of this system up to order $k$. The character variety of a free group in $l$ generators $(g, g_2, ..., g_l)$ such that the image of $g$ goes to ${\cal C}\subset G$ is just $${\cal C}\times\underbrace{G\times G\times \cdots \times G}_{l-1}$$ and it is non-singular. Thus there is no obstruction to finding $\rho_t:{\cal F}\to G$ up to order $k+1$, inducing the given $k$-th order formal homomorphism, as well as $C_{k+1}^j$ such that \ref{eq:e30} helds up to $T^{k+1}$. We conclude that in the free group ${\cal F}$ it is possible to find a solution of the lift of (\ref{eq:e2}). This means that the right hand side not only of the lift of (\ref{eq:e2}) but also of (\ref{eq:e2}) itself is a relative cocycle. It is a relative coboundary as well, because we assumed the vanishing of the group $H^2(\pi, (\Gamma_i), {\frak{g}})$. Therefore there exist $h_{k+1}$ and $c_{k+1}^j$ satisfying the equations (\ref{eq:e2}), which completes the inductive step. This finishes the proof of Proposition \ref{prop:pg}. We refer to results \cite{KM} of Kapovich-Millson for another approach to the relative deformation theory, where they work with differential graded algebras of differential forms on a manifold. \section{Intersection cohomology and the construction of the $2$-form} \setcounter{equation}{0} We will construct a non-degenerate $2$-form on the space ${{\cal M}_{\cal N}}$. We first consider the dimension $1$ case. The case of a Riemann surface ${\Sigma}$ with punctures is quite simple because we know the explicit structure of the fundamental group and ${\Sigma}$ is a $K(\pi, 1)$. The latter property allows us to conclude that $H^i(U, {\tilde{\frak{g}}})=H^i(\pi_1(U), {\frak{g}})$. Consider the exact sequence $$ \cdots\to H^1(Cone,{\frak{g}})\to H^1(\pi_1(U), {\frak{g}})\to \oplus_iH^1(\Gamma_i, {\frak{g}})\to H^2(Cone_{{\frak{g}}})\to 0, $$ where $Cone_{{\frak{g}}}$ is the mapping cone for the morphism of complexes $$ C^{{\bullet}}(\pi_1(U), {\frak{g}})\to \bigoplus_iC^{{\bullet}} (\Gamma_i, {\frak{g}}). $$ Similarly one can define $Cone_{{\Bbb Z}}$, $Cone_{{\Bbb R}}$, etc. If we apply the bilinear form $B\langle ,\rangle$ together with the pairing in cohomology then we get a map $$ H^i(Cone_{{\frak{g}}})\times H^j(Cone_{{\frak{g}}})\to H^{i+j}(Cone_{{\Bbb R}}). $$ \begin{lem} $H^2(Cone_{{\Bbb Z}})\simeq{\Bbb Z}$. \end{lem} {\noindent{\it Proof.\ \ }} Let $\Delta_i$ be a small disk centered at $i$-th marked point. Then $\Delta_i^*$ be obtained from $\Delta_i$ by removing the marked point. We have the class $\gamma_i'\in{\Bbb H}^1 (\Gamma_i,{\Bbb Z})\simeq H^1(\Delta_i^*, {\Bbb Z})$, which can be defined as follows. In terms of group cohomology it corresponds to the map $\phi: \Gamma_i\to{\Bbb Z}$ sending $\gamma_i$ to $1$. Under the above isomorphism this class goes to the class defined in $H^1(\Delta_i^*, {\Bbb Z})$ by the loop $\gamma_i$. Our point is that $$ H^2(Cone_{{\Bbb Z}})=Coker(H^1(\pi_1(U), {\Bbb Z}) \stackrel{\alpha}{\to}\oplus_iH^1(\Gamma_i, {\Bbb Z})); $$ the cokernel of the map $\alpha$ is ${\Bbb Z}$ and is generated by the class of the element $$ ({\gamma}_1', ..., {\gamma}_r')\in\oplus_iH^1(\Delta_i^*, {\Bbb Z}) \simeq\oplus_iH^1(\Gamma_i, {\Bbb Z}). \ \bigcirc $$ This lemma gives us the idea how to construct a symplectic form on ${{\cal M}_{\cal N}}$: $T_{[\rho]}{{\cal M}_{\cal N}}\times T_{[\rho]}{{\cal M}_{\cal N}}\to{\Bbb R}$, because $$ T_{[\rho]}{{\cal M}_{\cal N}}=Ker(H^1(\pi_1(U), {\frak{g}})\to\prod_iH^1(\Gamma_i, {\frak{g}}))= $$ $$ Im(H^1(Cone_{{\frak{g}}})\to H^1(\pi_1(U), {\frak{g}})) $$ and we have a natural pairing $$ H^1(Cone_{{\frak{g}}})\times H^1(Cone_{{\frak{g}}}) \stackrel{B\langle, \rangle}{\longrightarrow}H^2(Cone_{{\Bbb R}})\to{\Bbb R}. $$ This pairing is clearly non-degenerate. Now we move on to arbitrary dimensions. Unfortunately, the usual cohomology does not allow us to construct a non-degenerate pairing on the tangent space of ${{\cal M}_{\cal N}}$ when the local system ${\tilde{\frak{g}}}$ cannot be extended from $U$ to $X$, i.e. when at least one of the conjugacy classes ${\cal C}_i$ is not the class of the identity of $G$. That is the main reason why we need to use intersection cohomology instead. We introduce the following (canonical) filtration on $X$: $X_0\subset X_1\subset\cdots\subset X_{n-1}\subset X_n=X$, where $X_j$ consists of all points that belong to at least $n-j$ smooth irreducible components $D_i$ of $D$. We notice that $X\setminus X_{n-1}=U$. We will always work with intersection cohomology for the middle perversity. We define the truncated complex $\tau_jC^{{\bullet}}$ for a complex of sheaves $C^{{\bullet}}$. It is the complex which in degree $i$ is $$ C^i \ \ \ \ \ \ \ {\rm if} \ \ i < j, $$ $$ Ker(C^j\to C^{j+1})\ \ {\rm if} \ \ i=j, $$ $$ 0 \ \ \ \ \ \ \ {\rm if} \ \ i>j. $$ \begin{lem} $Ker(H^1(\pi_1(U), {\frak{g}})\to\prod_iH^1(\Gamma_i, {\frak{g}}))=IH^1(X, {\tilde{\frak{g}}})$. \label{lem:l32} \end{lem} {\noindent{\it Proof.\ \ }} An important thing is that the codimension of $D$ in $X$ is $1$. That is why in the complex $i_!i^!IC^{{\bullet}}_X({\tilde{\frak{g}}})$ has its cohomology sheaves only in degree $\ge 2$. If $j: U\hookrightarrow X$ and $i: D\hookrightarrow X$ are the inclusions then one has an exact (distinguished) triangle of complexes (see \cite{B}, p.109) \begin{equation} i_!i^!IC^{{\bullet}}_X({\tilde{\frak{g}}})\to IC^{{\bullet}}_X({\tilde{\frak{g}}})\to Rj_*j^*IC^{{\bullet}}_X({\tilde{\frak{g}}})\to i_!i^!IC^{{\bullet}}_X({\tilde{\frak{g}}})[1], \label{eq:e31} \end{equation} which gives rise to the long exact sequence in cohomology (we need only a small part of it): $$ \cdots\to H^1(X, i_!i^!IC^{{\bullet}}_X({\tilde{\frak{g}}}))\to H^1(X, IC^{{\bullet}}_X({\tilde{\frak{g}}}))\to $$ $$ H^1(U, IC^{{\bullet}}_U({\tilde{\frak{g}}}))\to H^2(X, i_!i^!IC^{{\bullet}}_X({\tilde{\frak{g}}}))\to H^2(X,IC^{{\bullet}}_X({\tilde{\frak{g}}}))\to\cdots $$ The vanishing mentioned above implies that $H^1(X, i_!i^!IC^{{\bullet}}_X({\tilde{\frak{g}}}))=0$. Besides, from definition it follows that $H^1(X, IC^{{\bullet}}_X({\tilde{\frak{g}}}))=IH^1(X, {\tilde{\frak{g}}})$, and since $U$ is non-singular and the local system ${\tilde{\frak{g}}}$ is defined on $U$ one has $H^1(U, IC^{{\bullet}}_U({\tilde{\frak{g}}}))=H^1(U,{\tilde{\frak{g}}})$. All this means that $$ IH^1(X, {\tilde{\frak{g}}})=Ker(H^1(U, {\tilde{\frak{g}}})\to H^2(X, i_!i^!IC^{{\bullet}}_X({\tilde{\frak{g}}}))). $$ We will show that there is a natural injection $$ H^2(X, i_!i^!IC^{{\bullet}}_X({\tilde{\frak{g}}}))\hookrightarrow\prod_iH^1(\Gamma_i, {\frak{g}}), $$ and in view of Proposition \ref{prop:p21} it is enough to prove the lemma. First, we consider the case $r=1$, when the divisor $D$ is smooth and irreducible. We will use Deligne's construction of the intersection sheaf complex as in \cite{GMII}. Let us consider a point $x\in D\subset X$ and a small neighbourhood $V$ of $x$. The group $\Gamma=\pi_1(V\setminus(V\cap D))$ is isomorphic to ${\Bbb Z}$ and is generated by a loop $\gamma$ encircling $D$. If $k: V\hookrightarrow X$ is the inclusion of this neighbourhood, then the sheaf $k_*{\tilde{\frak{g}}}$ has a stalk $(k_*{\tilde{\frak{g}}})_x=\underbar H^0(\pi_1(V\setminus(V\cap D), G)$. Passing to the derived functor and taking the sheaf cohomology we see that a stalk $\underbar H^1(Rk_*{\tilde{\frak{g}}})_x=H^1(\Gamma, {\frak{g}})$, which is the cokernel of $\rho(\gamma)-Id$ in ${\frak{g}}$. (We recall that ${\frak{g}}$ is a $\pi_1(U)$-module via the adjoint representation followed by $\rho: \pi_1(U)\to G$.) For $i>1$ the sheaf cohomology groups vanish: $\underbar H^i(Rk_*{\tilde{\frak{g}}})=0$. We also notice that $H^1(V\setminus(V\cap D), {\tilde{\frak{g}}})=H^1(\pi_1(V\setminus(V\cap D)), {\frak{g}})$. Let ${\cal S}$ stand for a local system on $D$ with a stalk ${\cal S}_x=H^1(\Gamma, {\frak{g}})$. Now we have $IC^{{\bullet}}_X({\tilde{\frak{g}}})=\tau_0Rk_*{\tilde{\frak{g}}}$, since the canonical filtration simply amounts to $D\subset X$. Thus we get an exact triangle $$ \cdots\to IC^{{\bullet}}_X\to Rk_*{\tilde{\frak{g}}}\to i_*{\cal S}[-1]\to IC^{{\bullet}}_X[1]\to\cdots, $$ because the mapping cone for $IC^{{\bullet}}_X\to Rk_*{\tilde{\frak{g}}}$ is homotopic to the complex having in degree $1$ the sheaf $i_*{\cal S}$ and nothing else. Comparing this triangle with (\ref{eq:e31}) we see that $i_!IC^{{\bullet}}_X({\tilde{\frak{g}}})\simeq {\cal S}[-2]$ and it means that $$ H^2(D, i^!IC^{{\bullet}}_X({\tilde{\frak{g}}}))=H^0(D, {\cal S}), $$ and it injects into $H^1(\Gamma, {\frak{g}})$ by definition of ${\cal S}$. Similarly we can consider the case $r>1$. Here the canonical filtration is $\cdots\subset M\subset D\subset X$, where the subvariety $M$ consists of all points that belong to at least $2$ irreducible components of $D$. For any $x\in M$ we take its small enough neighbourhood $V$; the fundamental group $\pi_1(V\setminus(D\cap V))\simeq{\Bbb Z}^m$, where $m$ is the number of intersecting components of $D$ at $x$. If $l:M\hookrightarrow X$ is the inclusion of $M$ into $X$ then the complex $l_!l^!IC^{{\bullet}}_X({\tilde{\frak{g}}})$ has its cohomology sheaves only in degree $\ge 3$, since the codimension of $M$ in $X$ is at least $2$. Thus from the exact sequence $$ \cdots\to H^2(X, l_!l^!IC^{{\bullet}}_X({\tilde{\frak{g}}}))\to IH^2(X,{\tilde{\frak{g}}})\to IH^2(X\setminus M,{\tilde{\frak{g}}})\to\cdots $$ one concludes that we have an injection $$ IH^2(X,{\tilde{\frak{g}}})\hookrightarrow IH^2(X\setminus M,{\tilde{\frak{g}}}). $$ Now the same arguments as above show that the group $H^2(X, i_!i^!IC^{{\bullet}}_X({\tilde{\frak{g}}}))$ naturally injects into the product $\prod_iH^1(\Gamma_i, {\frak{g}})$. $\bigcirc$ \ The above lemma proves that $T_{\rho}{{\cal M}_{\cal N}}\simeq IH^1(X, {\tilde{\frak{g}}})$, and now we are ready to construct a $2$-form $F$ on the space ${{\cal M}_{\cal N}}$. The idea is as follows. We have the class $[{\lambda}]\in H^2(X, {\Bbb R})$ of the K\"{a}hler form (when $X$ is a projective variety the class $[{\lambda}]$ is just the class of a hyperplane section). Now the form $F$ is given by the pairing \begin{equation} IH^1(X, {\tilde{\frak{g}}})\times IH^1(X, {\tilde{\frak{g}}})\to{\Bbb R}:\ \ \ \langle x,y\rangle=B\langle x, y\cup{\lambda}^{d-1}\rangle. \label{eq:e32} \end{equation} Here we used the intersection pairing described in \cite{GMII}. Now we shall make a choice of Riemannian metric on $U$ with singularities along $D$. Locally the description of our metric is as follows. Let $\Delta$ stand for the standard open disc in ${\Bbb C}$ given by $|z|<\varepsilon$ for some positive number $\varepsilon$ and let $\Delta^*=\Delta\setminus 0$. The intersection of a small neighbourhood of $x\in X$ with $U$ looks like $(\Delta^*)^r\times\Delta^{d-r}$ if $r$ components of $D$ meet in $x$. In local coordinates $z_1, ..., z_d$ these components are defined by equations $z_1=0$, $z_2=0$, ..., $z_r=0$. On $\Delta$ we take the metric $dzd\bar{z}$ and on $\Delta^*$ there is the Poincar\'{e} metric given in polar coordinates $r, \theta$ by $$ {dr^2+(rd\theta)^2\over (r\ln r)^2}. $$ So we assume that on $U$ we have a metric that is quasi-isometric to this one over any such open set $(\Delta^*)^r\times\Delta^{d-r}$. It is important to notice that the local system ${\tilde{\frak{g}}}$ is unitary, since the representation ${\frak{g}}$ of $G$ is unitary. By a well-known theorem (\cite{CKS}, \cite{KK}) one has the isomorphism $IH^i(X, {\tilde{\frak{g}}})\simeq H^i_{(2)}(U,{\tilde{\frak{g}}})$ between the intersection cohomology and the $L_2$-cohomology with coefficients in ${\tilde{\frak{g}}}$. Therefore when $i=1$ this space carries a pure Hodge structure of weight $1$, with only Hodge types $(0,1)$ and $(1, 0)$. This induces a complex structure on the tangent space to ${{\cal M}_{\cal N}}$. We refer the reader to \cite{BZ} for a survey of Hodge theory and its relation with $L_2$ cohomology. \section{The universal bundle} \setcounter{equation}{0} From now on we shall concentrate on the case when $G=U(N)$ and $G_{{\Bbb C}}=GL(N, {\Bbb C})$. Let us consider a principal flat $G$-bundle $P$ over $U$; then there exists canonically a holomorphic vector bundle $\bar V$ over $X$ such that $$ \bar{V}_{|U}=V:=P\times_{G}{\Bbb C}^N, $$ where ${\Bbb C}^N$ is the standard $G$-module. The holomorphic bundle $\bar V$ is called the {\it Deligne extension} of $V$ (see \cite{De1}). It has the property that the corresponding connection $\nabla_0$ has at worst logarithmic singularities along $D$. The canonical Deligne extension relies on fixing a unit interval: it is assumed that if $\mu$ is an eigenvalue of $Res_{D_i}(\nabla_0)$ then $0\le Re(\mu)< 1$. Over $U$ we have the Lie algebra bundle $End(V)$ and it extends to the holomorphic bundle $End({\bar V})$ over $X$. Let $Z$ be the subvariety of $Hom(\pi_1(U), G)$ consisting of all irreducible representations with the monodromy transformation around $D_i$ lying in the fixed conjugacy class ${\cal C}_i$, so that ${{\cal M}_{\cal N}}=Z/G$. Let us consider the vector bundle ${\Bbb U}$ over $Z\times U$ given by $${\Bbb U}=({\Bbb C}^N\times\tilde{U}\times Z)/\pi_1(U), $$ where $\tilde{U}$ is the universal covering of $U$ and the action of an element $a\in\pi_1(U)$ is given by $$ a(x, \tilde{y}, \rho)=(\rho(a)(x), a(\tilde{y}), \rho), \ \ x\in{\Bbb C}^N, \ {\tilde y}\in \tilde{U}. $$ Also one can construct the universal bundle $E$ over the product ${{\cal M}_{\cal N}}\times U$: $$ E=({\Bbb C}^N\times{\tilde U}\times Z)/\pi_1(U)\times G $$ $$ (a,g)(x, {\tilde y}, \rho)= (g\rho(a)g^{-1}(x), a({\tilde y}), g\rho g^{-1}),\ \ \ g\in G. $$ We will need the following \begin{lem} For any $x\in D$ there exists a neighbourhood $V$ of $x$ such that as $\rho$ varies in a connected component of $Z$ the local monodromy representation ${\Bbb Z}^k\simeq\pi_1(V\setminus(V\cap D))\to G$ does not change (up to conjugacy). \end{lem} {\noindent{\it Proof.\ \ }} Clearly one only has to show that the map $\xi$: $$\begin{array}{c} \{\ k\ -{\rm\ tuples\ of\ commuting\ elements\ of\ } G\} /{\rm conjugacy} \\ \xi\downarrow \\ (G {\rm\ mod\ conjugacy})^k \end{array}$$ is a finite map. It is also important to notice that the source of this map is Hausdorff. Let us exhibit this for $k=2$, because for $k>2$ the arguments are the same. Assume that for $a,b,c,d\in G$ such that $[a,b]=[c,d]=1$ one has $\xi(a,b)=\xi(c,d)$. Due to the fact that we consider pairs up to conjugacy we may assume that $a=c$. Moreover, $b$ is conjugate to $d$. We identify pairs $(a,b)$ and $(a,d)$ if $b$ is conjugate to $d$ by means of an element from $Z(a)$ - the centralizer of $a$. Next, we observe that each $G$-conjugacy class intersects $Z(a)$ by only finitely many $Z(a)$-conjugacy classes. In fact, the number of these classes is bounded from above by the cardinality of the Weyl group of $G$. $\bigcirc$ \ This lemma allows us to prove the next important result. \begin{prop} There exists a vector bundle $\tilde {\Bbb U}$ over $X\times Z$ extending ${\Bbb U}\to U\times Z$ such that for every smooth point $\rho\in Z$ the restriction $\tilde{{\Bbb U}}_{|X\times\rho}$ is the holomorphic Deligne extension. \end{prop} {\noindent{\it Proof.\ \ }} The above lemma allows one to construct the bundle $\tilde{{\Bbb U}}$ locally near $D\subset X$, because if one picks a point $x\in D$ then in a neighbourhood $V\ni x$ the local representation of $\pi_1(V\setminus(V\cap D))$ does not vary. So one can take the bundle over $V\times Z$ as a pullback of the bundle over $V$ under the projection onto the first coordinate. Now one notices that the Deligne construction is compatible with restrictions to smaller open sets. It means that for neighbourhoods $V_1$ and $V_2$ of two points $x_1, x_2\in D$ respectively the restrictions of ${\tilde {\Bbb U}}$ onto $V_1$ and $V_2$ agree on the intersection $V_1\cap V_2$. $\bigcirc$ \ Repeating the above arguments verbatim one can prove the existence of the extension $\tilde{E}\to{{\cal M}_{\cal N}}\times X$ of the universal bundle $E$. In fact, the bundle ${\tilde E}$ is the one we use in the rest of the paper. The bundle $\tilde E$ can also be obtained using the quotient of ${\tilde{\Bbb U}}$ by the action of $G$. We also notice that if the monodromy representation of $\pi_1(U)$ is irreducible then the corresponding logarithmic connection is stable in the sense of \cite{Nits}. \section{Gauge group and $L_2$ bundles} \setcounter{equation}{0} In this section we introduce an "$L_2$ gauge group" ${\cal G}$, later we will use it to prove Theorem \ref{Th:t11}. As before we consider a (holomorphic) vector bundle $\bar V$ on $X$ which is the Deligne extension of the vector bundle $V$ on $U$ with a fixed flat connection $\nabla_0$. The connection $\nabla_0$ has logarithmic poles along the divisor $D$ and it corresponds to an irreducible unitary representation $\rho_0$ of $\pi_1(U)$. This representation sends monodromy transformations around irreducible components of $D$ to the prescribed set ${\cal N}$ of conjugacy classes in $U(N)$. Now we are ready to introduce the gauge group ${\cal G}$ that acts on the space of connections on $\bar V$. It is the set of smooth unitary automorphisms of the bundle $V$ over $U$ such that if $g\in{\cal G}$ then $g^{-1}\nabla_0g$ is a unitary $L_2$ $1$-form with coefficients in $End(V)$. The topology on ${\cal G}$ is the coarsest topology satisfying the following three conditions. First, for any compact subset $K$ of $U$, the map from ${\cal G}$ to the Fr\'echet space of $C^{\infty}$ sections of $End(V)$ over $K$ should be continuous. Second, our topology is such that the distance function $G\times G\to{\Bbb R}$ defined by $${\rm dist}(g_1,g_2)=\sup_{x\in U, v\in V_x, ||v||=1}||g_1^{-1}g_2(v)-v||,$$ is continuous. The above supremum is well-defined since every $g\in{\cal G}$ is bounded. Finally, we require that in our topology the function ${\cal G}\to{\Bbb R}$ given by the $L_2$-norm of the covariant derivative of $g\in G$ is continuous. To see explicitly the manifold structure, we will give a description of the algebra $Lie({\cal G})$ and construct a map from a small neighbourhood of zero in $Lie({\cal G})$ to a small neighbourhood of $1\in{\cal G}$. The algebra $Lie({\cal G})$ consists of bounded sections $u$ of $u(V)$ - the unitary Lie algebra bundle corresponding to $V$ such that $\nabla_0u$ is $L_2$. Now the map $Lie({\cal G})\to{\cal G}$ in the neighbourhood of $0$ is just the Cayley transform $({\sqrt{-1}} I-A)({\sqrt{-1}} I+A)^{-1}$. Let $\{ Y_i\}$ be a finite open cover of $D$ in $X$ by contractible open polycylinders $\Delta^d$ so that $Y_i\simeq (Y_i\cap D)\times \Delta$. Next we need to define {\it a hermitian $L_2$ vector bundle} over $X$. It is a triple $(V, h, \{ B_i\})$, where $V$ is a vector bundle over $U$ with hermitian metric $h$, and $B_i$ is a class modulo ${\cal G}$ of frames of $V$ over $Y_i\cap U$ such that $B_i$ and $B_j$ agree over the intersection $Y_i\cap Y_j\cap U$. If we have a refinement $\{ \tilde{Y}_j\}$ of $\{ Y_i\}$ with similar properties then we naturally obtain a triple $(V, h, \{ \tilde{B_j}\})$, defining the same hermitian $L_2$ vector bundle. Two hermitian $L_2$ vector bundles are isomorphic if they are isomorphic over a common refinement. We call elements of $B_i$ {\it local $L_2$-frames}. We define a section of a hermitian $L_2$ bundle to be a section of $V$ which is square integrable near $D$. Let $x\in D$ be an arbitrary point of the divisor $D$ and let $Y$ be a small open polycylinder with Poincar\'{e} metric containing $x$. Let $f$ be a smooth compactly-supported function on $Y$. Then it is easy to see that $||df||$ in Poincar\'{e} metric is bounded from above. This allows us to use partitions of unity in the $L_2$-context. Let $Z$ be a smooth manifold (possibly with boundary). We introduce the $L_2$ gauge group ${\cal G}'$ , which consists of those smooth maps $g\in Map(Z\times(X\setminus D), G)$, which satisfy $$\sup_{z\in C}||\nabla_Xg(\cdot,z)||<\infty,$$ where $C$ is a compact subset of $Z$, $\nabla_X$ is the covariant derivative in $X$-direction, and $||\cdot ||$ is the global $L_2$-norm on $X$. Let $p_2$ be the projection $Z\times X\to X$. Then $p_2^{-1}(D)$ has real codimension $2$ in $Z\times X$ (and still has normal crossings), and the notion of hermitian $L_2$ bundle over $Z\times X$ makes perfect sense. It is a triple $(E,h,\{ p_2^{-1}B_i\})$, where $E$ is a vector bundle over $Z\times(X\setminus D)$ equipped with a hermitian metric $h$. One can think of a hermitian $L_2$ bundle over $Z\times X$ as of a family of hermitian $L_2$ bundles over $X$ varying smoothly with $z\in Z$. The important step in our construction is the following Lemma which has a well-known prototype in topological K-theory. \begin{lem} Let $Z_1, Z_2$ be smooth contractible manifolds and let $\Phi$ be a hermitian $L_2$ bundle over $Z_2\times X$. Let $a_0$ and $a_1$ be two smooth homotopic maps $Z_1\to Z_2$. Then $(a_0\times Id)^*\Phi$ and $(a_1\times Id)^*\Phi$ are isomorphic as hermitian $L_2$ vector bundles over $Z_1\times X$. \end{lem} {\noindent{\it Proof.\ \ }} Let $I$ denote the unit interval and let ${\cal L}$ be a hermitian $L_2$ bundle over $X\times I\times Z$. Let $K$ be a closed subset of $X\times I$ (actually, we use $K=X\times\{ t\}$). The restriction ${\cal L}_{K\times Z}$ is a hermitian $L_2$ bundle over $K\times Z$. Locally, a section $s$ of this bundle is a vector-valued function on $K\times Z$ which by definition satisfies the following estimates on its $L_2$-norms: $$||s||<\infty, \ \ ||\nabla_X s||<\infty.$$ Therefore the Tietze extension theorem can be applied and for each $x\in X\times I\times Z$ we may find an open set $Y\subset X\times I$ satisfying $x\in Y\times Z$ and a section $s'$ of ${\cal L}$ such that $s$ and $s'$ agree over $(Y\cap K)\times Z$. Since $X\times I$ is compact, we can find a finite cover by such open sets and apply $L_2$ partition of unity to see that the section $s$ can be extended to $X\times I\times Z$. To finish the proof, we apply the classical argument as given on page 17 of \cite{AK}. $\bigcirc$ Now we pick a small contractible neighbourhood $W\subset{{\cal M}_{\cal N}}$ of $\nabla_0$. Let $a_0, a_1: W\times X\to W\times X$ be two homotopic maps, where the map $a_0$ is just the identity map and $a_1$ is the map $W\times X\to \{\nabla_0\}\times X$ which induces identity on the second coordinate. The bundle we are interested in is $\tilde E$ from Section 5. We see that the bundle $a_1^*({\tilde E})$ is naturally isomorphic to the bundle $p_2^*{\bar V}$ over $W\times X$, where $p_2: W\times X\to X$ be the projection onto the second coordinate. Now we apply the above Lemma to the maps $a_0$ and $a_1$ to get an isomorphism $$\psi: {\tilde E}\simeq p_2^*{\bar V}$$ of hermitian $L_2$ bundles over $W\times X$. For any $\rho\in W$ we let $\psi_{\rho}$ stand for the restriction of this isomorphism to $\rho\times X$. \section{The $2$-form is symplectic.} \setcounter{equation}{0} Our next goal is to show that the form $F$ constructed earlier is actually closed. For this we shall construct another $2$-form $H$ on a bigger space such that its closeness is apparent and then we shall see that our $2$-form $F$ is equal to a pullback of the form $H$. Let $Af$ be the affine space of all unitary connections $\nabla$ on $\bar V$ such that the difference $\nabla-\nabla_0$ is an $L_2$ $1$-form with coefficients in $End({\bar V})$. For each $\rho\in W$ we have a pull-back $\nabla_{\rho}=\psi_{\rho}^*\nabla_0$ of the connection $\nabla_0$ in $V$. The corresponding map $\beta: W\to Af$, $\beta(\rho) = \nabla_{\rho}$ is a smooth embedding. We define the following $2$-form on $Af$ \begin{equation} H(v_1, v_2)=\int_{U}Tr(v_1\cup v_2)\cup {\lambda}^{d-1}, \ \ \ v_1, v_2\in T_a{Af}. \label{eq:e61} \end{equation} (The integral in question is defined since ${\lambda}$ is a bounded form on $U$, and the $v_j$'s are $L_2$.) The form $H(.,.)$ is naturally closed, because it is a constant coefficient $1$-form over an affine space. The isomorphism mentioned in Section 4 between $L_2$ and intersection cohomologies (with coefficients in ${\tilde{\frak{g}}}$) has the property that corresponding pairings on degree $1$ cohomology given by (\ref{eq:e32}) and (\ref{eq:e61}) correspond to one another. Thus we get \begin{lem} The pull-back $\beta^*H(.,.)$ is equal to $F$. \end{lem} The pull-back $\beta^*H(.,.)$ is closed and the point $\nabla_0$ can be chosen to be an arbitrary representation with trivial stabilizer, hence $F$ is closed as well. \ \noindent{\bf Remark.} Let ${\cal A}$ be the space consisting of flat irreducible connections $\nabla$ on $V$ which are compatible with the unitary structure and have the property that the difference $\nabla-\nabla_0\in A^1(U, End({\bar V}))$ is an $L_2$ $1$-form. We also require that $\nabla$ and $\nabla_0$ have the same monodromies (up to conjugation) around the irreducible components of $D$. (It is a mater of simple computations to see that this condition is redundant and follows from the others.) The tangent space $T_{\nabla_0}{\cal A}$ is the space of closed unitary $L_2$-forms on $U$ such that their restriction to a small enough punctured polycylinder $\Delta^r\times(\Delta^*)^{d-r}$ is $\nabla_0$-exact. The smooth structure on ${\cal A}$ naturally comes from the fibering $$ \begin{array}{ccc} {\cal G} & \to & {\cal A} \\ {} & {} & \downarrow\pi \\ {} & {} & {{\cal M}_{\cal N}} \end{array}. $$ It comes down to considering the natural smooth map $s: {\cal A}\to Af$. The form $H(.,.)$ on $Af$ is invariant under the action of ${\cal G}$ and the pull-back $s^*H$ of the form $H(.,.)$ to ${\cal A}$ (which we denote by $H_{\cal A}$) is closed too. The form $H_{\cal A}$ is ${\cal G}$-invariant and vertical, meaning that $H_{\cal A}(v_1, v_2)=0$, where $v_1, v_2\in T_{\nabla_0}{\cal A}$ and $v_1$ is $\nabla_0$-exact. This shows that the form $H_{\cal A}$ descends to a $2$-form on ${{\cal M}_{\cal N}}$ and it can be seen directly from the definitions that this form coincides with $F$. \ To finish the proof of Theorem \ref{Th:t11} we need \begin{lem} The form $F$ is non-degenerate. \end{lem} {\noindent{\it Proof.\ \ }} Let $\rho\in{{\cal M}_{\cal N}}$. The $2$-form $F$ on $T_{\rho}{{\cal M}_{\cal N}}=IH^1(X, {\tilde{\frak{g}}})$ is given by $$ IH^1(X, {\tilde{\frak{g}}})\times IH^1(X, {\tilde{\frak{g}}})\to{\Bbb R}:\ \ \ F(x,y)=B\langle x, y\cup{\lambda}^{d-1}\rangle. $$ It is clearly skew-symmetric; it is also non-degenerate, because the Poincar\'{e} duality pairing is non-degenerate and the Hard Lefschetz theorem says that iterated cap-product by ${\lambda}$ induces an isomorphism $$ IH^{d-k}(X ,{\tilde{\frak{g}}})\simeq IH^{d+k}(X, {\tilde{\frak{g}}}).$$ Here we use the fact that the Hard Lefschetz theorem holds for $L_2$ cohomology \cite{Zu} and we recall (Section 4) the isomorphism between $L_2$ and intersection cohomologies. $\bigcirc$ \ Combining this result with our above observation that $F$ is closed, we conclude that $F$ gives a natural symplectic structure on ${{\cal M}_{\cal N}}$. This proves Theorem \ref{Th:t11}. \section{Projective case.} In this section we assume that $X$ is a projective manifold and we intend to show that the form $F$ defined in Section 4 is actually a {K\"{a}hler\ } form. (One can conjecture that this is true under the assumption that $X$ is just a {K\"{a}hler\ } manifold, but we do not know the proof of this more general fact.) Let $X\subset{\Bbb C}{\Bbb P}^M$ be an embedding of $X$. Let us take a curve $C$ in $X$ obtained by intersection with $d-1$ generic hyperplanes in ${\Bbb C}{\Bbb P}^M$. This curve satisfies the following properties. First, $C$ is smooth and, secondly, if $D=\cup_{i=1}^rD_i$ is the decomposition of $D$ into the union of irreducible complex analytic subvarieties then $C$ intersects $D_i$ transversally and $C$ does not meet $D_i\cap D_j$ for $i\ne j$. Thus $C\cap D$ is a finite set of points on $C$. Let $S:=C\setminus(C\cap D)$ be the open Riemann surface. There is a natural set of conjugacy classes ${\cal N}'$ in the group $G$ assigned to $S$, which comes from the set ${\cal N}$. There is also the moduli space ${{\cal M}_{\cal N}}'$ of flat irreducible $G$-bundles on $S$ with monodromies around the punctures defined by the set ${\cal N}'$. The following result, which is a direct consequence of Lemma 1.4 in \cite{D3} comes handy. \begin{lem} Let $z\in S$. The natural morphism $$\pi_1(S, z)\to \pi_1(U, z)$$ is surjective. \end{lem} This result allows us to get a smooth inclusion: $$T: {{\cal M}_{\cal N}} \hookrightarrow {{\cal M}_{\cal N}}'.$$ If we pick an irreducible representation $\rho$ of $\pi_1(U)$, it gives rise to an irreducible representation $\rho'$ of $\pi_1(S)$. There is an obvious compatibility of images by $\rho$ and $\rho'$ of loops encircling irreducible components of the divisor $D$ and the punctures on $S$ respectively. The tangent space to ${{\cal M}_{\cal N}}'$ is identified (by Proposition \ref{prop:p21}) with the group $Ker[H^1(\pi_1(S), {\frak{g}})\to\prod_j H^1({\Gamma_j}', {\frak{g}})]$, where the groups ${\Gamma_j}'\simeq{\Bbb Z}$ are generated by the classes of loops encircling the punctures. The above Lemma also gives us the existence of injective homomorphisms: $$H^1(\pi_1(U), {\frak{g}})\hookrightarrow H^1(\pi_1(S), {\frak{g}}), $$ $$\prod_i H^1(\Gamma_i, {\frak{g}}))\hookrightarrow \prod_j H^1({\Gamma_j}', {\frak{g}})).$$ (Recall that $\Gamma_i$ are generated by the classes of loops encircling irreducible components of the divisor $D$.) Thus we get the following injection $$Ker[H^1(\pi_1(U), {\frak{g}})\to\prod_i H^1(\Gamma_i, {\frak{g}})]\hookrightarrow Ker[H^1(\pi_1(S), {\frak{g}})\to\prod_j H^1({\Gamma_j}', {\frak{g}})],$$ which is equal to $dT:\ T_{\rho}{{\cal M}_{\cal N}}\hookrightarrow T_{\rho'}{{\cal M}_{\cal N}}'.$ (The Lefschetz hyperplane theorem for intersection cohomology \cite{GMII} gives the same result.) Thus we proved \begin{prop} The map $T: {{\cal M}_{\cal N}} \to{{\cal M}_{\cal N}}'$ defined above is an embedding. \end{prop} Let $F'$ be the $2$-form on the moduli space ${{\cal M}_{\cal N}}'$ defined by Equation (\ref{eq:e32}) with $d=1$. Theorem \ref{Th:t11} shows that $F'$ is a symplectic form. We have the following relation between those two $2$-forms: \begin{prop} $$F'_{|{{\cal M}_{\cal N}}}=F.$$ \label{prop:p73} \end{prop} {\noindent{\it Proof.\ \ }} Let $Y\stackrel{\alpha}{\hookrightarrow} X$ be a generic hyperplane section of $X$. Then we have a well-defined map $$\alpha^*:\ IH^j(X, {\tilde{\frak{g}}})\to IH^j(Y, {\tilde{\frak{g}}}),$$ (which is the transpose to the map $\alpha_*$ of 7.1 \cite{GMII}). Let $B_X\langle, \rangle$ and $B_Y\langle, \rangle$ be the pairings on $IH^*(X, {\tilde{\frak{g}}})$ and $IH^*(Y, {\tilde{\frak{g}}})$ respectively defined as in (\ref{eq:e32}). We have \begin{lem} Let ${\lambda}\in H^2(X)$ be the class of a hyperplane section and let $a\in IH^j(X, {\tilde{\frak{g}}})$, $b\in IH^{d-j-1}(X, {\tilde{\frak{g}}})$. Then $$B_X\langle a, b\cup{\lambda}\rangle =B_H\langle\alpha^*a, \alpha^* b\rangle.$$ \end{lem} Let $i:C\hookrightarrow X$ be the inclusion and let $[C]={\lambda}^{d-1}\in H^{2d-2}(X)$ be the class of $C$. We apply the above result $(d-1)$ times to see that if $a\in IH^k(X, {\tilde{\frak{g}}})$, $b\in IH^j(X, {\tilde{\frak{g}}})$, and $k+j=2$, then $$B_C\langle i^*a, i^*b\rangle= B_X\langle a,b\cup [C]\rangle.$$ We notice that when $k=j=1$ the pairings in the left and right hand sides are by definition $F'$ and $F$. $\bigcirc$ \ By a well-known theorem of Mehta and Seshadri \cite{MS} the moduli space ${{\cal M}_{\cal N}}'$ identifies with the moduli space of parabolic stable vector bundles and hence it has a natural structure of complex manifold. In view of the isomorphism between intersection and $L_2$-cohomologies, the complex structure on ${{\cal M}_{\cal N}}'$ coming from moduli space of stable parabolic bundles coincides with the almost complex structure $$J:\ IH^1(C, {\tilde{\frak{g}}})\to IH^1(C, {\tilde{\frak{g}}})$$ we have introduced in Section 4. To conclude that ${{\cal M}_{\cal N}}'$ is a {K\"{a}hler\ } manifold, it remains to notice that if $v,w\in IH^1(X, {\tilde{\frak{g}}})$ then $F'(Jv,Jw)=F'(v,w)$. Moreover, since $C$ is a general curve, the holomorphic $L_2$ $1$-forms on $U$ restrict to holomorphic $L_2$ $1$-forms on $S$ (both with coefficients in ${\tilde{\frak{g}}}$). Thus $IH^1(X, {\tilde{\frak{g}}})$ is a complex subspace of $IH^1(C, {\tilde{\frak{g}}})$ and Proposition \ref{prop:p73} follows that ${{\cal M}_{\cal N}}$ is a {K\"{a}hler\ } submanifold of ${{\cal M}_{\cal N}}'$. This ends the proof of Theorem \ref{Th:t12}. \ \noindent{\bf Remark.} Let us assume that the monodromy around each $D_i$ is of finite order $k$. Then the same is true for the corresponding monodromy transformations around the punctures in $S$. It is well-known (see e.g. \cite{DW}, \cite{W}) that in this case the form $${\omega}={kF'\over 4\pi^2}$$ defines an integral cohomology class $[{\omega}]$ in $H^2({{\cal M}_{\cal N}}', {\Bbb R})$. Moreover, there exists a natural holomorphic line bundle ${\cal L}'$ on ${{\cal M}_{\cal N}}'$ with curvature equal to $-2\pi{\sqrt{-1}} [{\omega}]$. Explicit constructions of ${\cal L}'$ can be found, for instance, in \cite{DW} and \cite{Kon}. Using the fact that ${{\cal M}_{\cal N}}$ is a {K\"{a}hler\ } submanifold of ${{\cal M}_{\cal N}}'$, we obtain \begin{prop} Let the monodromy transformation around each $D_i$ be of finite order $k$. Then there exists a natural line bundle ${\cal L}$ on ${{\cal M}_{\cal N}}$ with the curvature equal to $k F / 2\pi{\sqrt{-1}}$. \end{prop} The line bundle ${\cal L}$ is simply the restriction of ${\cal L}'$ to ${{\cal M}_{\cal N}}$. The space of $L_2$-sections of ${\cal L}$ can serve as a (partial) goal in the geometric quantization program. \thebibliography{123} \bibitem{Art}{M. Artin, On the solution of analytic equations, {\it Inv. Math.}, {\bf 5}, 1968, 277-291} \bibitem{AB}{M. Atiyah and R. Bott. The Yang-Mills Equations over a Riemann Surface. {\it Phil. Trans. Roy. Soc}, {\bf A308}, 523 (1982)} \bibitem{A3}{M. Atiyah, {\it The geometry and physics of knots}, Cambridge U. Press, 1990} \bibitem{AK}{M. Atiyah, {\it K-theory,} Benjamin, New York, 1967} \bibitem{BiqTh}{O. Biquard, These, Ecole Polytechnique, 1991} \bibitem{Biq2}{O. Biquard, Fibr\'es paraboliques stables et connexions singuli\`eres plates, {\it Bull. Soc. math. France}, {\bf 119}, 1991, 231-257} \bibitem{BG}{I. Biswas and K. Guruprasad, Principal bundles on open surfaces and invariant functions on Lie groups, {\it Int. J. Math.}, {\bf 4}, 1993, 535-544} \bibitem{B}{A. Borel {\it at al}. Intersection Cohomology, Birkh\"{a}user, 1984} \bibitem{BZ}{J.-L. Brylinski and S. Zucker, An Overview of Recent Advances in Hodge Theory, {\it Encyclopaedia of Mathematical Sciences}, {\bf 69}, Several Complex Variables VI, Springer-Verlag, 1990, 39-142} \bibitem{CKS}{E. Cattani, A. Kaplan and W. Schmid, $L^2$ and intersection cohomologies for a polarizable variation of Hodge structure, {\it Invent. Math.}, {\bf 87}, 1987, 217-252} \bibitem{DW}{G. Daskalopoulos and R. Wentworth, Geometric Quantization for the Moduli Space of Vector Bundles with Parabolic Structures, preprint, 1992} \bibitem{De1}{P. Deligne, Equations Differentielles a Points Singuliers Reguliers, {\it Lect. Not. in Math.} {\bf 163}, Springer-Verlag, 1970} \bibitem{D3}{P. Deligne, Le Groupe Fundamental du Compl\'{e}ment d'une Courbe Plane n'ayant que des Points Doubles Ordinaires Est Ab\'{e}lien (d'apr\`{e}s W. Fulton), S\'{e}m. Bourbaki 543, Nov. 1979, {\it Lect. Not. Math.}, {\bf 524}} \bibitem{F}{P. A. Foth, Geometry of Moduli Spaces of Flat Connections on Punctured Surfaces, preprint, {\bf alg-geom/9703004}, 1996} \bibitem{GM}{W. Goldman and J. Millson, Deformations of flat bundles over {K\"{a}hler\ } manifolds, {\it Lect. Not. in Pure and Appl. Math.}, {\bf 105}, Dekker, NY, 1987, 129-145} \bibitem{G}{W. Goldman, The Symplectic Nature of Fundamental Groups of Surfaces, {\it Adv. in Math.}, {\bf 54}, 1984, 200-225} \bibitem{GMII}{M. Goresky and R. MacPherson, Intersection Homology II, {\it Inventiones Mathematicae} {\bf 71}, 1983, 77-129} \bibitem{GHJW}{K. Guruprasad, J. Huebschmann, L. Jeffrey and A. Weinstein, Group Systems, Groupoids, and Moduli Spaces of Parabolic Bundles, preprint, {\bf dg-ga/9510006}, 1995} \bibitem{KM}{M. Kapovich and J. Millson, The relative deformation theory of representations and flat connections and deformations of linkages in constant curvature spaces, {\it Composito Math.}, {\bf 103}, 1996, 287-317} \bibitem{Kar}{Y. Karshon, An algebraic proof for the symplectic structure of moduli space, {\it Proc. AMS}, {\bf 116}, 3, 1992, 591-605} \bibitem{KK}{M. Kashiwara and T. Kawai, The Poincar\'{e} lemma for variations of polarized Hodge structures, {\it Publ. R.I.M.S. Kyoto Univ.}, {\bf 23}, 1987, 345-407} \bibitem{Kon}{H. Konno, On the Natural Line Bundle on the Moduli Space of Stable Parabolic Bundles, {\it Comm. Math. Phys.}, {\bf 155}, 1993, 311-324} \bibitem{MS}{V. Mehta and C. Seshadri, Moduli of Vector Bundles on Curves with Parabolic Structures, {\it Math. Ann.}, {\bf 248}, 1980, 205-239} \bibitem{Nits}{N. Nitsure, Moduli of semistable logarithmic connections, {\it J. Amer. Math. Soc.}, {\bf 6}, 1993, 597-609} \bibitem{Simgar}{C. Simpson, Harmonic Bundles on Noncompact Curves, {\it J. Amer. Math. Soc.}, {\bf 3}, 1990, 713-770} \bibitem{SPM}{C. Simpson, Product of Matrices, {\it Diff. Geom., Global Anal. \& Top.}, CMS Conf. Proc., {\bf 12}, 1992, 157-185} \bibitem{W}{E. Witten, On Quantum Gauge Theories in Two Dimensions, {\it Comm. Math. Phys.}, {\bf 141}, 1991, 153-209} \bibitem{Zu}{S. Zucker, Hodge theory with degenerating coefficients: $L_2$-cohomology in the Poincar\'e metric, {\it Ann. Math.}, {\bf 109}, 1979, 415-476} \vskip 0.3in Department of Mathematics \\ Penn State University \\ University Park, PA 16802 \\ [email protected], [email protected] \ \noindent{\it AMS subj. class.} \ \ primary 32G13 \end{document}

9703

alg-geom/9703038

en

https://arxiv.org/abs/alg-geom/9703038

alg-geom/9703038

Vladimir Baranovsky

Vladimir Baranovsky

On Punctual Quot Schemes for Algebraic Surfaces

Latex2e, amssymb, amsmath packages; 4 pages. Proofs simplified

The punctual Quot scheme parametrizes all length d quotients of a (locally) trivial rank r sheaf which are supported at a fixed point. The author shows that this scheme is irreducible and (rd-1)-dimensional. The same result was proved independently by Ellingsrud and Lehn.

alg-geom

\section*{Introduction.} Let $S$ be a smooth projective surface over the field of complex numbers $\mathbb{C}$. Fix a closed point $s \in S$ and a pair of positive integers $r, d$. By results of Grothendieck (cf. \cite{G}, \cite{S}) there exists a projective scheme $\mathrm{Quot}_{[s]}(r, d)$ parametrizing all quotient sheaves $\mathcal{O}^{\oplus r}_S \to A $ of length $d$ supported at $s$. We consider this scheme with its \emph{reduced} scheme structure and call it the \emph{punctual Quot scheme}. Note that $\mathrm{Quot}_{[s]}(1,d)$ is nothing but the punctual Hilbert scheme $\mathrm{Hilb}_{[s]}^d$ studied by Brian\c con and Iarrobino in \cite{B}, \cite{I}. The main result of this paper is the following extension of their results to the case $r > 1$: \medskip \noindent\textbf{Main Theorem.} $\mathrm{Quot}_{[s]}(r, d)$ is an irreducible scheme of dimension $(rd-1)$. \medskip We prove this by exibiting a dense open subset in $\mathrm{Quot}_{[s]}(r, d)$ isomorphic to a rank $(r-1)d$ vector bundle over $\mathrm{Quot}_{[s]}(1, d) = \mathrm{Hilb}_{[s]}^d$. One can show that, for a quotient $\mathcal{O}^{\oplus r} \to A$ as above, the $d$-th power of the maximal ideal $\mathfrak{m}_{S, s}$ acts trivially on $A$. Hence the punctual Quot scheme $\mathrm{Quot}_{[s]}(r, d)$ does not depend on $S$ and in our proof we can assume that $S =\mathbb{C}^2$. In this case a straghtforward generalization of Nakajima's construction for Hilbert schemes allows to prove the result. \medskip \noindent \textbf{Remark.} The original results of Brian\c con and Iarrobino were used by G\"ottsche and Soergel in \cite{GS} to show that the natural map $\pi: Hilb^d(S) \to Sym^d(S)$ is strictly semismall with respect to the natural stratifications. This leads to a simple proof of G\"ottsche's formula for the Poincar\'e polynomials of $Hilb^d(S)$. Similarly, the Main Theorem above can be used to show that the natural map $ \pi: M^G(r, d) \to M^U(r, d)$ from the Gieseker moduli space of stable rank $r$ sheaves to the Uhlenbeck compactification of the instanton moduli space, is also strictly semismall (at least in the coprime and unobstructed case). This allows one to find a connection between some homological invariants of these spaces. A systematic treatment of this questions will appear in the author's forthcoming paper. \bigskip \noindent\textbf{Acknowledgments.} This work was originally motivated by a conjecture due to V. Ginzburg on semismallness of the map $\pi: M^G(r, n) \to M^U(r, n)$. The author thanks him for providing the motivation and also for his helpful discussions and support. The author also thanks J.Li who explained the role of the punctual Quot scheme (and also suggested an alternative proof of the irreducibility statement). \bigskip After the first draft of this paper was finished, the author learned about a preprint by G.Ellingsrud and M.Lehn \cite{EL} who prove the same result (among others) by a different method. \section{Punctual Hilbert scheme.} The result of this section is well known (cf. \cite{B}, \cite{BI}, \cite{I}). The outline of the proof is given here for convenience of the reader. It is a slight modification of Corollary 1.2 in \cite{ES}. \begin{theorem} $\mathrm{Hilb}_{[s]}^d$ is irreducible of dimension $(d-1)$. \end{theorem} \begin{proof} First of all, we can assume that $S=\mathbb{P}^2$. There exists a $\mathbb{C}^*$-action on $\mathbb{P}^2$ such that our point $s$ is a zero-dimensional cell of the corresponding Bialynicki-Birula decomposition. It follows that $\mathrm{Hilb}_{[s]}^d$ is stable under the induced $\mathbb{C}^*$-action on the global Hilbert scheme $\mathrm{Hilb}^d(\mathbb{P}^2)$ of points on the projective plane. Recall that $\mathrm{Hilb}^d(\mathbb{P}^2)$ is \emph{smooth}, (cf. \cite{N}) and hence one has the Bialynicki-Birula decomposition for the torus action. Then $\mathrm{Hilb}_{[s]}^d$ is a union of cells of this decomposition. One can prove that $\mathrm{Hilb}_{[s]}^d$ has a unique $(d-1)$-dimensional cell and no cells of higher dimension (cf. \cite{ES}). Hence $\mathrm{dim}(\mathrm{Hilb}_{[s]}^d)= d-1.$ To prove the irreducibility of $\mathrm{Hilb}_{[s]}^d$ consider the universal subscheme $Z \subset \mathrm{Hilb}^d(\mathbb{P}^2)\times \mathbb{P}^2$. By definition of the Hilbert scheme $Z$ is finite and flat over $\mathrm{Hilb}^d(\mathbb{P}^2)$. Denote by $Z_{d-1}$ the subscheme of all points in $Z$ where $d$ sheets of the map $f: Z \to \mathrm{Hilb}^d(\mathbb{P}^2)$ come together (i.e. $Z_{d-1}$ is the $(d-1)$-st ramification locus of $f$, cf. \cite{GL} for a rigorous definition). Then $(Z_{d-1})_{red}$ is a locally trivial bundle over $\mathbb{P}^2$ with fibers isomorphic to $\mathrm{Hilb}_{[s]}^d$. Since $Z$ is normal (cf. \cite{F}) and $\mathrm{Hilb}^d(\mathbb{P}^2)$ is smooth, we can apply the following result due to Lazarsfeld (cf. \cite{GL} for the statement of the result, the proof of it is contained in Lazarsfeld's PhD thesis): \emph{Let $f: Z \to H$ be a finite surjective morphism of irreducible varieties, with $Z$ normal and $H$ non-singular. If $Z_{d-1}$ is not empty , then every irreducible component of $Z_{d-1}$ has codimension $\leq(d-1)$ in $Z$.} It follows that any irreducible component of $\mathrm{Hilb}_{[s]}^d$ should be at least $(d-1)$-dimensional. Since $\mathrm{Hilb}_{[s]}^d$ has only one $(d-1)$-dimensional cell, it can have only one irreducible component. \end{proof} \section{Proof of the Main Theorem.} Our strategy is to find a dense irreducible open subset $W \subset \mathrm{Quot}_{[s]}(r, d)$ of dimension $(rd-1)$. We define $W$ as the set of all quotients $\mathcal{O}^{\oplus r} \stackrel{\phi}\longrightarrow A$, $\phi=(\phi_1+ \phi_2 + \ldots + \phi_r)$ such that the first component $\phi_1: \mathcal{O} \to A$ is surjective (this is clearly an open condition). Such a $\phi_1$ corresponds to a point in $\mathrm{Hilb}_{[s]}^d$. Once $\phi_1$ is chosen, the other components $(\phi_2, \ldots, \phi_r)$ are given by an arbitrary element of $Hom(\mathcal{O}^{\oplus(r-1)}, A) = \mathbb{C}^{(r-1)d}$. Therefore $W$ is a rank $(r-1)d$ vector bundle over $\mathrm{Hilb}_{[s]}^d$. By results of Brian\c con and Iarrobino, $W$ is irreducible of dimension $(rd-1)$. Now we want to show that $W$ is dense in $\mathrm{Quot}_{[s]}(r, d)$. In fact, for any point $x \in \mathrm{Quot}_{[s]}(r, d)$ we will find an irreducible rational curve $C \subset \mathrm{Quot}_{[s]}(r, d)$ connecting it with some point in $W$. To that end, we generalize Nakajima's construction (cf. \cite{N}) of the global Hilbert scheme $\mathrm{Hilb}^d(\mathbb{C}^2)$ to the Quot scheme. Once we do that, the existence of the irreducible curve will amount to an exercise in linear algebra (cf. Lemma 2.3). \bigskip Fix a complex vector space $V$ of dimension $d$, and $N_d$ let be the space of pairs of commuting nilpotent operators on $V$. The space $N_d$ is naturally a closed affine subvariety of $\mathrm{End}(V) \oplus \mathrm{End}(V)$. Consider a subspace $U_r$ of $N_d \times V^{\oplus r}$ formed by all elements $(B_1, B_2, v_1, \ldots, v_r)$ such that there is no proper subspace of $V$ which is invariant under $B_1, B_2$ and contains $v_1, \ldots, v_r$. Then $U_1 \times V^{\oplus (r-1)} \subset U_2 \times V^{\oplus (r-2)} \subset \ldots \subset U_r$ is a chain of open subsets in $N_d \times V^{\oplus r}$ (each of them is given by a condition saying that some system of vectors in $V$ has maximal rank). \medskip Note thate the general linear group $GL(V)$ acts naturally on $V_r$ and it is easy to prove that $U_r$ is $GL(V)$-stable. \begin{lemma} $GL(V)$ acts freely on $U_r$. \end{lemma} \begin{proof} Suppose $g \in GL(V)$ stabilizes $(B_1, B_2, v_1, \ldots, v_r) \in U_r$. Then $\mathrm{Ker}(1-g)$ contains $ v_1, \ldots, v_r$. Since it is also preserved by $B_1, B_2$ , we have $Ker(1-g)=V$ and therefore $g=1$. \end{proof} The following lemma gives an explicit construction of the punctual Quot scheme: \begin{lemma} There exists a morphism $\pi: U_r \to \mathrm{Quot}_{[s]}(r, d)$ such that (i) $\pi$ is surjective; (ii) the fibers of $\pi$ are precisely the orbits of $GL(V)$ action on $U_r$; (iii) $\pi^{-1}(W) = U_1 \times V^{\oplus (r-1)}$. \end{lemma} \begin{proof} We can assume that $S= \mathbb{C}^2 = Spec \;\mathbb{C}[x_1, x_2]$ and $s=0 \in \mathbb{C}^2$. To construct $\pi$ suppose that $(B_1, B_2, v_1, \ldots, v_r)$ is a point in $U_r$ and consider a $\mathbb{C}[x_1, x_2]$-module structure on $V$ in which $x_1$ acts by $B_1$ and $x_2$ acts by $B_2$. We can view $V$ as a quotient of a free $\mathbb{C}[x_1, x_2]$-module with generators $v_1, \ldots, v_r$. Since $B_1$ and $B_2$ are nilpotent $\sqrt{\mathrm{Ann}(V)} = (x_1, x_2)$. Therefore a coherent sheaf $A$ on $\mathbb{C}^2$ associated with $V$ is a quotient of $\mathcal{O}^{\oplus r}$ supported at $s$. Moreover, $V \simeq \mathrm{H}^0(S, A)$ as vector spaces. A different point in the same $GL(V)$-orbit defines an isomorphic quotient, hence the fibers of $\pi: U_r/ \mathrm{GL(V)} \to \mathrm{Quot}_{[s]}(r,d)$ are $GL(V)$-invariant. Moreover, suppose that two points $u_1, u_2$ of $U_r$ give rise to isomorphic quotients $A_1$, $A_2$. Then the induced isomorphism between $\mathrm{H}^0(S, A_1)$ and $\mathrm{H}^0(S, A_2)$ defines an element of $GL(V)$ taking $u_1$ to $u_2$. Therefore, each fiber of $\pi$ is precisely one $GL(V)$-orbit. This proves $(ii)$. To prove $(i)$, suppose we have a quotient $ \mathcal{O}^{\oplus r} \to A \to 0 $ of length $d$ supported at zero. Multiplication by $x_1$ and $x_2$ induces a pair of commuting nilpotent operators on $\mathrm{H}^0(S, A)$. Choose a $\mathbb{C}$-linear isomorphism $\mathrm{H}^0(S, A) \simeq V$. The generators of the free $\mathbb{C}[x_1, x_2]$-module $\mathrm{H}^0(S, \mathcal{O}^{\oplus r})$ project to some vectors $v_1, \ldots, v_r$ in $V$. Since $v_1, \ldots, v_r$ generate $V$ as a $\mathbb{C}[x_1, x_2]$-module, $(x_1, x_2, v_1, \ldots v_r)$ is a point of $U_r$. Thus $(i)$ is proved. Finally, $(iii)$ follows from definitions of $W$ and $U_1$. \end{proof} Now we want to show that any point in $U_r$ can be deformed to a point in the preimage of $W$. The above construction will allow us to construct this deformation using the following lemma \begin{lemma} Let $B_1$, $B_2$ be two commuting nilpotent operators on a vector space V. There exists a third nilpotent operator $B_2'$ and a vector $w \in V$ such that (i) $B_2'$ commutes with $B_1$; (ii) any linear combination $\alpha B_2 + \beta B_2'$ is nilpotent; (iii) $(B_1, B_2', w) \in U_1$, \;i.e. $w$ is a cyclic vector for the pair of operators $(B_1, B_2')$. \end{lemma} This lemma will be proved later. Now we will show how it can be used to give a \bigskip \noindent\emph{Proof of the Main Theorem:} Let $x$ be a point of $\mathrm{Quot}_{[s]}(r, d)$ and $u_1 = (B_1, B_2, v_1, \ldots, v_r)$ be any point of $\pi^{-1}(x) \subset U_r$. Choose a nilpotent operator $B_2'$ and a vector $w \in V$ as in Lemma 2.3. Connect the points $u_1$ and $u_2 = (B_1, B_2', w, v_2 , \ldots, v_r)$ with a straight line $\Phi(t)$, $t \in \mathbb{C}$ \; such that $\Phi(1) = u_1$ and $\Phi(0) = u_2$. This $\Phi(t)$ is given by equation: $$ \Phi(t)= (B_1, tB_2' + (1-t)B_2, tw +(1-t)v_1, v_2, \ldots, v_r) $$ Note that for all $t \in C$,\; $B_2(t) = tB_2' + (1-t)B_2 $ is nilpotent and commutes with $B_1$. Therefore the image of $\Phi(t)$ is a subset of $N_d \times V^{\oplus r}$. Since $U_r$ is open in $N_d \times V^{\oplus r}$, there is a dense open subset $C \subset \mathbb{C}$ such that $\Phi(C) \subset U_r$. Similarly, there exists a dense open subset $C_1 \subset C$ such that $\Phi(C_1) \subset U_1 \times V^{\oplus (r-1)}$. Hence the image $\pi(\Phi(C)) \subset \mathrm{Quot}_{[s]}(r, d)$ is an irreducible rational curve connecting $x = \pi(u_1)$ with $\pi(u_2) \in W$. Note that $\pi(\Phi(C_1)) \subset W$. Therefore $x$ belongs to the closure of $W$. Since by Theorem 1.1 $W$ is irreducible of dimension $(rd-1)$, the scheme $\mathrm{Quot}_{[s]}(r, d)$ is also irreducible of dimension $(rd-1)$. The Main Theorem is proved. \bigskip \noindent\emph{Proof of Lemma 2.3:} \medskip \emph{Step 1.} We will find a basis $e_{i,j}$ of $V$, where $1 \leq i \leq k$ and $1 \leq j \leq \mu_i$ such that (a) $B_1^{j-1}(e_{i, 1}) = e_{i, j}$ for $j \leq \mu_i$ and $B_1^{\mu_j}(e_{i, 1}) = 0$ (i.e. $B_1$ has Jordan canonical form in the basis $e_{i, j}$); (b) $B_2(e_{i, 1}) \in \Big(\bigoplus_{k \geq i +1} \mathbb{C}\; e_{k, 1}) \oplus B_1 \cdot V$. \bigskip To that end, recall one way to construct a Jordan basis for $B_1$. Let $d = \dim V$ and $V_i = Ker(B_1^{d-i})$. The subspaces $V_i$ form a decreasing filtration $V = V_0 \supset V_1 \supset V_2 \ldots $. Moreover, $B_1 \cdot V_i \subset V_{i+1}$. Firstly, we choose a basis $(w_1, \ldots, w_{a_1})$ of $W_1: = V_0/V_1$. Lift this basis to some vectors $e_{1, 1}, e_{2, 1}, \ldots, e_{a_1, 1}$ in $V_0$ and set all $\mu_1, \ldots, \mu_{a_1}$ equal to $d$. Secondly, choose a basis $(w_{a_1 + 1}, \ldots, w_{a_2})$ of $W_2: = V_1/ (B_1\cdot V_0 + V_2)$. Lift this basis to some vectors $e_{a_1 + 1, 1}, e_{2, 1}, \ldots, e_{a_2, 1}$ in $V_1$ and set all $\mu_{a_1+1}, \ldots, \mu_{a_2}$ equal to $d-1$. Continue in this manner by choosing bases of the spaces $W_{i+1} = V_i/ (B_1 \cdot V_{i-1} + V_{i+1})$ and lifting them to $V_i$. This procedure gives us vectors $e_{1, 1}, e_{2, 1}, \ldots, e_{k, 1}$ and the formula (a) tells us how to define $e_{i, j}$ for $j \geq 2$. It is easy to check that the system of vectors $\{e_{i, j}\}$ is in fact a basis of $V$. \bigskip If we want to have the property (b) we should be more careful with the choice of $w_i$. Note that all the subspaces $V_i$ and $B_1 \cdot V_i$ are $B_2$-invariant. Therefore we have an induced action of $B_2$ on each $W_i$. We can choose our basis $(w_{a_{i-1} + 1}, \ldots, w_{a_i})$ of $W_i$ in such a way that $B_2(w_i) \in \bigoplus_{s=i+1}^{a_i} \mathbb{C} \; w_s$ for all $i \in \{a_{i-1} + 1, \ldots, a_i \}$. This ensures that $(b)$ holds as well. \bigskip \emph{Step 2.} Define $B_2'$ by $B_2'(e_{i, j}) = e_{i+1, j}$ if $j \leq \mu_{i+1}$ and 0 otherwise. It is immediate that $B_2'$ is nilpotent and that $[B_1, B_2']=0$. Let $w = e_{1, 1}$. Then $e_{i, j} = B_1^{j-1} (B_2^{i-1}(w))$ hence $(B_1, B_2', w) \in U_1$. \bigskip \emph{Step 3.} Note that both $B_2$ and $B_2$ are lower-triangular with zeros on the diagonal in the basis of $V$ given by $$ e_{1, 1}, e_{2, 1}, \ldots, e_{k, 1}, e_{1, 2}, e_{2, 2}, \ldots e_{k, 2}, \ldots $$ Hence any linear combination of $B_2$ and $B_2'$ as also lower-triangular and has zeros on the diagonal. Therefore $\alpha B_2 + \beta B_2'$ is nilpotent for any complex $\alpha$ and $\beta$. This completes the proof of Lemma 2.3. \bibliographystyle{amsplain}

9703

alg-geom/9703002

en

https://arxiv.org/abs/alg-geom/9703002

alg-geom/9703002

Ludmil Katzarkov

Fedor Bogomolov and Ludmil Katzarkov

Complex projective surfaces and infinite groups

29 pages, some comments and examples added LaTeX 2.09

The paper contains a general construction which produces new examples of non simply-connected smooth projective surfaces. We analyze the resulting surfaces and their fundamental groups. Many of these fundamental groups are expected to be non-residually finite. Using the construction we also suggest a series of potential counterexamples to the Shafarevich conjecture which claims that the universal covering of smooth projective variety is holomorphically convex. The examples are only potential since they depend on group theoretic questions, which we formulate, but we do not know how to answer. At the end we formulate an arithmetic version of the Shafarevich conjecture.

alg-geom

\section{Introduction} It is well known that complex projective surfaces can have highly nontrivial fundamental groups. It is also known that not all finitely presented groups can occur as fundamental groups of projective surfaces. The fundamental problem in the theory then, is to determine which groups can occur as the fundamental groups of the complex projective surfaces and to describe complex manifolds which occur as nonramified coverings of the surface. The only interesting case is when the group is infinite since finite coverings of projective surfaces are projective and every finite group occurs as a fundamental group of some projective surface. In this paper we mainly consider a surface with a representation as a family of projective curves over a curve. It does not put much restriction on the choice of a surface since any surface has such a representation after blowing up a finite number of points. We use a base change construction combined with a finite ramified covering to associate to a given surface a collection of surfaces with infinite fundamental groups. Most of these fundamental groups were not previously known to be fundamental groups of smooth projective surfaces. Every surface we construct comes equipped with a regular map to a curve of high genus. The kernel of the corresponding map of fundamental groups is obtained from the fundamental group of a generic fiber by imposing torsion relations on some elements and a very big class of infinite groups occur in this way. We also analyze the universal coverings of these surfaces in the context of the Shafarevich's conjecture which states that the universal covering of the smooth complex projective variety must be holomorphically convex. Partial affirmative results \cite{LM} regarding Shafarevich's conjecture can be applied to the fundamental groups we obtain. It leads to nontrivial purely algebraic results on the structure of this groups. On the other hand the generality of our construction indicates that not all fundamental groups we construct will satisfy the algebraic restrictions imposed by Shafarevich's conjecture. It would appear then that the conjecture may be false in general. We describe a series of potential counterexamples in section 4. We begin with a local version of the general construction. Namely we have a local fibration without multiple components and with only double singular points in the central fiber. As a first step we make a local base change which produces a new surface with singular points corresponding to the singular points of the central fiber. This move changes the image of the fundamental group of a generic fiber in the fundamental group of the Zariski open subset of nonsingular points. Namely the kernel of this map is generated by the $N$-th powers of the initial vanishing cycle where $N$ is the degree of the local base change. As a second step we desingularize the surface by taking a finite fiberwise covering of the singular surface which is ramified at singular points only. The global construction follows the same pattern, but as a result we obtain a surface with a highly nontrivial fundamental group coming from the fiber even if the surface at the beginning was simply connected. First we construct a singular projective surface with a big fundamental group of the compliment to the set of singular points. Smooth projective surface is obtained at the second step as a finite covering of the singular surface ramified at singular points only. We prove a general theorem (Theorem 2.4) which establishes a close similarity between the fundamental groups and universal coverings for two classes of surfaces: normal projective surfaces and the surfaces obtained from them by deleting a finite number of points. The above construction enlarges the image of the fundamental group of the fiber in the fundamental group of the whole surface. The resulting group can be described in purely algebraic terms. Let $\pi_g$ be a fundamental group of a projective curve of genus $g$ (generic fiber of the fibration). Consider a finite set of pairs of $(s_i \in \pi_g, N_i \in \Bbb{N})$ and a subgroup $M$ of the automorphisms of $\pi_g$. We assume that all $s_i$ are vanishing cycles. These are special conjugacy classes in the fundamental group of the curve which constitute a finite number of orbits under the action of the mapping class group $Map(g)$( see section 2). The orbits $Ms_i^{N_i}$ generate the normal subgroup $\Xi(M,s_i,N_i)$ of $\pi_g$. Now we can give an algebraic version of the description of the corresponding group. \begin{defi} Define a Burnside type quotient of $\pi_g$ to be the group $\pi_g/\Xi(M,s_i,N_i)$. \end{defi} In the geometric situation $s_i$ are the vanishing cycles of the initial fibration and $M$ is its monodromy group. Geometric Burnside type groups constitute a proper subset among all Burnside type groups. In particular not every data $ s_i,N_i,M$ can be geometrically realized. The problem which data can appear in geometry is the most substantial problem of the above construction. It is clear that we can change $M$ into its own subgroup of finite index but at the expense of changing the set of elements $s_i$. The following lemma shows that we are free to vary $N_i$. \begin{lemma} Let $s_i,N_i = 1 ,M$ describe the data of the smooth projective family of curves of a given genus $g$. Then $s_i,N_i ,M$ corresponds to the geometric Burnside group for any choice $N_i$ satisfying the condition: $N_i = N_j$ if the singular points corresponding to the cycles $s_i$ belong to the same singular fiber of the fibration. \end{lemma} The topological structure of the algebraic family of curves can be rather arbitrary if we consider its restriction on a disc inside the base curve. We summarize relevant results of th article in the following proposition. \begin{prop} Let $\Gamma$ be any finitely presented group and $f_g:\pi_g \to \Gamma$ be any surjective map with $Ker f_g$ generated by a finite subset $VS$ of the conjugacy classes in $\pi_g$. There exists a surjective map $p_h:\pi_h\to \pi_g$ which corresponds to the contraction of a set of nonintersecting handles $H_i$ inside the curve of genus $h$ with the following properties : 1) Every element $s_j\in VS$ has a preimage $s'_j$ in $\pi_h$ which is realized by a smooth cycle. 2) There is a smooth holomorphic family $X$ of curves of genus $h$ with simple singularities over an algebraic curve which has all cycles $s_j'$ and the generating cycles $a_i,b_i$ of the handles $H_i$ as vanishing cycles. 3) The family $X$ above can have as a monodromy group any subgroup of finite index in $Map(h)$ containing the elements corresponding to positive Dehn twists for all $s_j',a_i,b_i$. \end{prop} Thus any topological data $ s_i, M $ can be realized as part of some geometric data $ s_i, s_i' M'$ where $M'$ is any subgroup of finite index in $Map(g)$ containing $M$. The previous lemma allows to transfer all additional relations $s_i'$ into their $N$-powers. Hence we have two parameter ``approximation''of any topological data by geometric data . One of the parameters corresponds to $N$ and converges to infinity . Another parameter runs through subgroups of finite index in $Map(h)$ which containing the monodromy group $M$ ot the fibration over the disc. Since $Map(h)$ is residually finite the corresponding sequence of groups converges to $M$. \begin{rem} These results provide with a tool to produce a big variety of fundamental groups and universal coverings using our construction. It is worth noticing that though the initial group with $N_i = 1$ can be small (even trivial) the groups which appear for other choices of $N_i$ are quite diverse. \end{rem} \begin{defi} Let us take a free group $\Bbb{F}^{l}$ on $l$ generators, the normal subgroup $\Xi$ generated by $N$-th powers of all the primitive elements can be described as in definition 1.1 . Namely $\Xi$ is generated by $Ms^N$ where $s$ is a primitive element of $\Bbb{F}^{l}$ and $M$ is the group of all automorphisms of $\Bbb{F}^{l}$. We will denote $\Bbb{F}^l/ \Xi $ by $BT(l,N) $. \end{defi} \begin{defi} Denote by $\pi_{g}/(x^{N}=1)$ the quotient of the fundamental group of a Riemann surface of genus $g$ by the group generated by the $N$-th powers of all primitive elements $x$ in $\pi_{g}$. \end{defi} \begin{con} (Zelmanov) For big $l ,g,N$ the groups $BT(l,N) $ and $\pi_{g}/(x^{N}=1)$ are nonresidually finite groups. \end{con} If the above conjecture is true then the base change construction provides us with many new simple examples of nonresidually finite fundamental groups of smooth projective surfaces. The first example of such a group was constructed by Toledo ( \cite{TOL}, see also \cite{FL}). In section four we discuss the holomorphic convexity of the universal coverings of the surfaces we have constructed. Recently many new powerful methods have been developed to investigate the structure of the fundamental groups and universal coverings of complex projective surfaces. These methods lead to many new remarkable results. In particular quite a few positive results on Shafarevich's conjecture were proved e.g. the results of F. Campana, H. Grauert, R. Gurjar, J. Koll\'ar, B. Lasell, R. Narasimhan, T. Napier, M. Nori, M. Ramachandran, C. Simpson, S. Shasrty, K. Zuo, S.T. Yau (see e.g. \cite{CM}, \cite{LM}, \cite{LN}, \cite{KR}, \cite{K1}, \cite{K2}, \cite{N1}, \cite{SIM}, \cite{YAU}). In particular holomorphic convexity is established (see \cite{LM}) for coverings of normal projective surface $X$ corresponding to the homomorphisms of $\pi_1(X)$ to $GL(n,\Bbb{C})$ such that the image of these homomorphisms is virtually not equal to $\Bbb{Z}$ (see also \cite{FT}). We suggest possible applications of the construction related to the Shafarevich's conjecture. Holomorphic convexity of a variety implies the absence of infinite chains of compact curves. This can be expressed in our case as a restriction on the images on the fundamental groups of the components of a singular fiber ( see lemma 4.1). We provide with a scheme how to control the behavior of the monodromy and vanishing cycles to construct a counterexample to the Shafarevich conjecture. A free group with a given number of generators can be identified with a fundamental group of a Riemann surface with one or two ends. The number of generators in the free group defines the genus of the surface and the number of ends (two if the number of generators is odd and one if the latter is even). \begin{defi} Denote by $\Bbb{F}^{g}_k$ a free group with $k$ generators realized as a fundamental group of a Riemann surface $B$ with one or two ends. Denote by $P^g(k,N)$ the quotient of $\Bbb{F}^{g}_k$ by a normal subgroup generated in $\Bbb{F}^{g}_k$ by all primitive elements in $\Bbb{F}^{g}_k$ which map into primitive elements (embedded curves) in $B$. \end{defi} The following group theoretic question is closely related to the Shafarevich conjecture: {\bf Question} Are there such a $k>1$ and $N$ that that $P^g( 2k,N)$ is a finite group and $\pi_{ 2k }/(x^{N}=1)$ is an infinite group? If the answer to this question is affirmative then one gets a counterexample to the Shafarevich conjecture (see section 4). We also suggest potential counterexmaples by considering simplest nontrivial case $N = 3$ in which we establish the finiteness of all the groups $P^g(k,3)$( Appendix B). In this case we suggest a family of potential counterexamples to the conjecture which depends on a non-finiteness of some groups in the family ``approximating'' some infinite group. The corresponding family of groups can be described as follows. Let us take a chain or ring of curves $X_0$ containing more than two curves of genus greater than zero. A natural contraction of a generic curve of genus $g$ onto this special fiber defines a subgroup $M_D$ inside the mapping class group $Map(g)$ which commutes with the contraction. If every component $C_i$ of $X_0$ contains at least one vanishing cycle $s_i$ then the group $\pi_g/ ( M_D s_i^3 )$ is infinite though the images of the groups corresponding to different components $C_i\subset X_0$ are finite( Appendix B ). It is true even if we consider smaller subgroups $M^f \subset M_D$. Now by our ``approximation'' results we can realize $X_0$ as a fiber in an algebraic fibration with the corresponding fiber group $\pi_g / (M^f_j s_i^3 , M^f_j s_{k(N})^{3^N}$. Here $M^f_j$ is any open subgroup of finite index in $Map(g)$ containing $M^f$, $N$ is any integer and $s_{k(N)}$ runs through the set of additional vanishing classes. We conjecture that this set of groups contains many infinite groups which will imply that there are many counterexamples to the Shafarevich's conjecture obtained from the surfaces above. \begin{rem} Since the group we ``approximate'' surjects onto a nontrivial free product of finite groups we expect that the proof of the above conjecture can be found within the modern theory of infinite groups \end{rem} In this section we briefly discuss the symplectic version of our construction. We also formulate an arithmetic variant of the Shafarevich conjecture which is presumably easier to prove though the conjecture is formally stronger then a direct analogue of the complex case. \bigskip \noindent {\bf Acknowledgments:} The authors would like to thank M.Gromov, J.Koll\'ar, T.Pantev, G.Tian, D.Toledo for useful conversations and comments. The second author would like to thank A.Beilinson, J.Carlson, H.Clemens, K.Corlette, P.Deligne, R.Donagi, S.Gersten, S.Ivanov, M.Kapovich, M.Newman, \newline M.Nori, A.Olshanskii, M.Ramachandran, C.Simpson, Y.T.Siu, S.Weinberger, E.Zelmanov and S.T.Yau for useful conversations and constant attention to work. We would like also to thank the referee for pointing out an erroneous statement in the initial version and for helpful suggestions on the organization of the paper. We thank M.Fried for looking through the arithmetic part of the paper. \section{The general construction} \subsection{Vanishing cycles - a local construction} In this subsection we explain the local computation with the vanishing cycles on which the whole construction is based. We begin with some classical results on degeneration of curves that can be found in \cite{DK}. Let $X_D$ be a smooth complex surface fibered over a disc $D$. We assume that fibers over a punctured disc $D^* = D - 0$ are smooth curves of genus $g$ and the projection $ t: X\to D$ is a complex Morse function. In particular the fiber $X_0$ over $0\in D$ has only quadratic singular points and it has no multiple components. Denote by $P$ the set of singular points of $X_0$ and by $T : X_t \to X_t$ the monodromy transformation acting on the fundamental group of the general fiber $ X_t $. This action can be described in terms of Dehn twists. Obviously this action defines an action on the first homology group of the general fiber. The following proposition describes completely the topology of $X_D$ and the projection $t: X_D \to D$. \begin{prop} 1) There is a natural topological contraction $ cr: X_D \to X_0$. 2) The restriction of $cr$ to $X_t$ is an isomorphism outside singular points $P_i \in X_0$. It contracts the circle $S_i \in X_t$ into $P_i$. The monodromy transformation $T$ is the identity outside small band $B_i$ around $S_i$ and in $B_i$ the transformation $T$ coincides with a standard Dehn twist. \end{prop} {\bf Proof} See \cite{DK}. The above contraction is an isomorphism from $X_t $ minus preimage of $P$ on $X_{0} - P$. The preimage of any singular point $P_i$ is a smooth, homotopically nontrivial curve $S_i\subset X_t$. \begin{defi} We will call the free homotopy class of $S_{i}$ in the fundamental group of $X_t$ a geometric vanishing cycle. It defines a conjugacy class in $\pi_g$(vanishing cycle). \end{defi} \begin{rem} The direction of the standard Dehn twist is defined by the orientation of $S_i$ which in turn is defined by the complex structure of the neighborhood of the corresponding singular point of the singular fiber. \end{rem} Let us denote by $De_i$ the topological Dehn transformation of $X_t$ and by $De_{i,H} $ its action on the homology of $X_{t}$. In a neighborhood of $X_{0}$ the monodromy transformation $T$ is a product of Dehn transformations $De_i$ with non-intersecting support. \begin{lemma} 1)The monodromy transformation $T$ acts via unipotent transformation $T_H$ on the homology group $H_1( X_t, \Bbb{Z})$. 2) $( 1 - T_H)^2 = 0$. 3) $(1 - T_H^N ) = 0(mod N)$ for any $N$. \end{lemma} {\bf Proof} It is enough to prove (2) for $De_{i,H}$. The topological description of $De_i$ implies that $(1 -De_{i,H})x = (x,s_i)s_i$, where $s_i$ a homology class of the vanishing cycle $s_i$ and $(x,s_i)$ is the intersection number. Since $(s_i,s_i) = 0$ we obtain $(1 - De_{i,H})^2 = 0 $. The image of $(1 - De_{i,H})$ consists of the elements proportional to $s_i$. We also have $(1 - De_{i,H}^N ) = (1 - NDe_{i,H}) = 0 (mod N)$. The topological transformations $De_i$ commute for different $i$ since they have disjoint supports. Therefore the same holds for $De_{i,H}$. We also have $ De_{i,H} s_j = 0$ for any vanishing cycle $s_j$ since the corresponding circles $S_i, S_j$ don't intersect in $X_t$. We now see that \[(1 - T_H)^2 = (1 - \prod De_{i,H})^2 = \prod ( 1 - De_{i,H})^2 = 0 \] where the last equality follows from the above formulas. Similarly we obtain 1) and 3). $\Box$ The geometric vanishing cycles consist of two different types: 1) The first type includes homologically nontrivial vanishing classes. They are all equivalent under the mapping class group $Map(g)$. The latter can be described either as group of connected components of the orientation preserving homeomorphisms of the Riemann surface of genus $g$ or as the group of exterior automorphisms of the group $\pi_g $. The vanishing cycle from this class is a primitive element of $\pi_g$ which means it can be included into a set of generators of $\pi_g$ satisfying standard relation defining the fundamental group of the curve. We shall denote the vanishing cycles of the first type type as $NZ$-cycles. 2) The second type consists of elements in $\pi_g$ which are homologous to zero. Any vanishing cycle of this type cuts the Riemann surface $X_t$ into two pieces and the number of handles in these pieces is the only invariant which distinguishes the type of a cycle under the action of $Map(g)$. We shall denote the vanishing cycles of the second type by $Z$-cycles. Vanishing $Z$-cycles correspond to the singular points of the singular fiber which divide this fiber into two components and $NZ$-cycles to the ones which do not. Each $Z$-cycle defines a primitive element in the center of the quotient $\pi_g/[[\pi_g,\pi_g],\pi_g] $, while each $NZ$-cycle defines a primitive element in the abelian quotient $\pi_g/[\pi_g,\pi_g] = \Bbb{Z}^{2g}$. Assume now that we made a change of the variable $ t = u^N$ and consider the induced family of curves over $D$ with a coordinate $u$. Denote the resulting family as $X_U$. It is a singular surface and the singular set can be identified with the set $P$ o f the singular points of the fiber $X_0$. For the new monodromy transformation we have $T^U = T^N$. Therefore by lemma 2.3 it acts trivially on $H_1(X_t, \Bbb{Z}_N), t\neq 0$. \begin{lemma} The surface $X_U$ contracts to the central fiber $X_0$. \end{lemma} Indeed the fiberwise contraction of $X_D$ to $X_0$ can be lifted into a contraction of $X_U$. \begin{rem} The fundamental group $\pi_1(X_D - P) = \pi_1(X_D) =\pi_1(X_0)$ since the singular points of the fiber $X_0$ are nonsingular points of $X_D$. The analogous statement is not true however for $X_U$. \end{rem} \begin{theo} The fundamental group $\pi_1(X_U - P)$ is equal to the quotient of $\pi_1(X_t) = \pi_g$ by a normal subgroup generated by the elements $s_i^N$. \end{theo} {\bf Proof} The fundamental group of a complex coincides with the fundamental group of any two-skeleton of the complex. There is a natural two-dimensional complex with a fundamental group as in the theorem. Namely let us take a curve $X_t$ and attach two-dimensional disks $D_i$ via the boundary maps $f_i : dD_i\to S_i$ of degree $N$. The resulting two dimensional complex $X_t^a$ evidently has a fundamental group isomorphic to the group described in the theorem. We are going to show that $X_t^a$ can be realized as a two skeleton of $X_U - P$. We prove first the following statement: \begin{lemma} The surface $X_U - P$ retracts onto a three dimensional complex $X^3$ which is a union of generic curve $X_t$ and a set of three-dimensional lens spaces $L_N^i$. Each lens space corresponds to the singular point $P_i$ of the singular fiber and $L_N^i$ intersects $X_t$ along the band $B_i$. \end{lemma} {\bf Proof }Locally near $P_i$ the surface $X_U$ is described by the equation $ t^N = z_1z_2 $. Hence a neighborhood $U_i$ of $P_i$ is a cone over the three-dimensional lens space $L_N^i= S^3/ \Bbb{Z}_N$. Here $S^3$ is a three dimensional sphere and $\Bbb{Z}_N$ is generated by the matrix with eigenvalues $\chi, \chi^{-1}$, where $\chi$ is a primitive root of unity of order $N$. We can assume that the surface $X_t$ intersects $L_N^i$ along a two-dimensional band $B_i$ with a central circle $S_i$ defining a generator of $\pi_1(L_N^i)$. Though the band $B_i$ is a direct product of $S_i$ by the interval its embedding into $L_N^i$ is nontrivial : the boundary circles have nonzero linking number. We use a fiberwise contraction to contract $X_U - P$ to the union to $X^3$. It coincides with a standard contraction outside of the cones over $L_N^i$. Therefore we obtain a contraction of $X_U -P$ onto $X^3$. $\Box$ The two-skeleton of $X^3$ can be obtained as a union of $X_t$ and a two-skeleton of $L_N^i$. The latter can be seen as a retract of a complimentary set to the point in $L_N^i$. Here is the topological picture we are looking at. \begin{lemma} Let $L_N^i$ be a lens space as above. Then the complimentary set to a point retracts on complex $L^2_i$ obtained by attaching a disc to the circle $S_i$ via the boundary map of degree $N$. \end{lemma} {\bf Proof} The sphere $S^3$ can be represented as a joint of two circles $S_i$ and $S'$. In other words it consists of intervals connecting different points of $S_i, S'$. The action of $\Bbb{Z}_N$ with $ L_N^3 = S^3/ \Bbb{Z}_N$ rotates both circle. Define the disc $D_x$ to be a cone over $S_i$. Different discs $D_x$, $x \in S'$, don't intersect and the image of $D_x$ in $L_N^3$ is $L^2_i$. The fundamental domain of $ \Bbb{Z}_N$-action lies between discs $D_x ,D_{gx}$, where $g$ is a generator of $\Bbb{Z}_N$. Hence this domain is isomorphic to $D^3$ and coincides with complimentary of $L^2_i$ in $L_N^i$. The last proves the lemma. $\Box$ \begin{corr} There is an embedding of $X_t^a $ into $X_U - P$ which induces an isomorphism of the fundamental groups. \end{corr} Indeed we obtain the two skeleton of $X^3$ gluing $X_t$ and $L^2_i$ along $S_i$, but the resulting two-complex coincides with $X_t^a$. Since $X^3$ is a retract of $X_U - P$ we obtain the corollary and finish the proof of the theorem. $\Box$ Let us denote by $G_X$ the fundamental group of $X_U - P$ and by $\widetilde X$ its universal covering. The description of the group $G_X$ can be obtained in pure geometric terms. Namely the vanishing cycles $S_i$ don't intersect and therefore we can first contract $Z$-cycles to obtain the union of smooth Riemann surfaces $X_i$ with normal intersections. The graph corresponding to this system of surfaces is tree since every point corresponding to $Z$-cycle splits it into two components. Remaining $NZ$- cycles lie on different surfaces $X_j$ and constitute a finite isotropic subset of primitive elements in $\pi_1(X_j)$. The vanishing cycle $S_i$ which contracts to the point of $X_j$ defines a map of a free group onto $\pi_1(X_j)$ with $S_i$ corresponding to standard relations in $\pi_1(X_j)$. We also have the following lemma. \begin{lemma} If $N$ is odd or divisible by four then the group $G_X$ has a natural surjective projection on the quotient group $\pi_g/[[\pi_g,\pi_g],\pi_g]$ with additional relation $x^N = 1$. If $N = 2$ then $G_X$ maps surjectively on a central extension of $\Bbb{Z}_2^{2g}$ by $\Bbb{Z}_2$ and the images of all $s_i$ have order $2$. \end{lemma} {\bf Proof} The case when $N$ is odd or divisible by four is clear since all the elements $s_i$ have order $N$. In the case $N = 2$ we use the fact that $NZ$ cycles $s_i$ contained in the component $X_j$ lie in the isotropic subspace. Hence there exists a standard $\Bbb{Z}_2$ central extension of $\Bbb{Z}_2^{2g_j}$ where the images of all $s_i\in X_j$ are exactly of order $2$. Denote this group by $G_j$ and the generator of its center by $c_j$. Now consider the product of $G_j$ for all $j$ and factor it by a central subgroup generated by $ c_k - c_j$ if $X_k$ and $X_j$ intersect. Since the graph is a tree we obtain that the quotient of the center by the above group is equal to $\Bbb{Z}_2$ which is identified with the zero homology with coefficients $\Bbb{Z}_2$. The image of $\Bbb{Z}$-cycle $s_i$ coincide with $c_j$ if $s_i\in X_j$ and hence is never zero. Vanishing $NZ$-cycles project into nonzero elements of the abelian quotient of the group. $\Box$ \begin{rem} If $N$ is odd or divisible by four we obtain a canonical quotient of $G_X$ which is equivariant under $Aut(\pi_g)$. We denote this group by $UC_g^N$. For $N = 2$ our construction is less canonical, it depends on the choice of isotropic subspace in $H_1(X_t, \Bbb{Z}_2)$ \end{rem} Here we start to develop an idea that will be constantly used through out the paper. Namely we show that we can work with open surfaces and get results concerning the universal coverings of the closed surfaces. \begin{theo} There is a natural $G_X$-invariant embedding of $\widetilde X$ into a smooth surface $\widetilde X_U $ with $\widetilde{X}_U/G_X = X_U$. The complimentary of $\widetilde X$ in $\widetilde X_U$ consists of a discrete subset of points. \end{theo} {\bf Proof} Let $P_i $ is a point that corresponds to a $NZ$ vanishing cycle. The preimage of a local neighborhood $U_i $ of $P_i$ in $\widetilde X$ consists of a number of nonramified coverings of $U_i - P_i$. Since $\pi_1(U_i - P_i) = \pi_1(L_N^i) = \Bbb{Z}_N$ we obtain that nonramified covering we get is finite. Therefore by local pointwise completion we obtain $\widetilde X_U$ with the action of $G_X$ extending on it. Now the universal nonramified covering of $U_i - P_i$ is a punctured unit disk in $\Bbb{C}^2$. Therefore if the map $\pi_1(U_i - P_i)\to G_X$ is injective then the surface $\widetilde X_U$ is nonsingular. Since the group $\pi_1(U_i - P_i) = \Bbb{Z}_N$ and is generated ed by the vanishing cycle $s_i$ we can easily get the result. Indeed the groups $H_1(G_X, \Bbb{Z}_N)$ and $H^1( X_t, \Bbb{Z}_N)$ are isomorphic. The general result follows from theorem 2.1 since we have constructed finite quotients for $G_X$ which map every $s_i$ into an element of order $N$. $\Box$ We also have the following: \begin{lemma} Let $G_X^0$ be the kernel of projection of $G_X$ into one of the finite groups defined in lemma 1.13. Then $G_X^0$ acts freely on $\widetilde X_U$ and the quotient is a family of compact curves without multiple fibers and with a fundamental group $G_X^0$. \end{lemma} {\bf Proof} Indeed the surface $\widetilde X_U$ was obtained from $\widetilde X$ by adding points and since $\widetilde X$ was simplyconnected the former is symplyconnected either. Any element of $G_X$ which has invariant point in $\widetilde X_U$ is conjugated to the power of $s_i$, but the latter are not contained in $G_X^0$. $\Box$ We are done with our description of the local computations. By taking finite coverings and taking away the singularities of the covering we were able to make all local geometric vanishing cycles to be torsion elements. \subsection{ The global construction} In this subsection we globalize the construction of section 2.1 to a compact surface. Let $X$ be smooth surface with a proper map to a smooth projective curve $C$. We assume that the map $f:X\to C$ is described locally by a set of holomorphic Morse functions and hence satisfies the conditions of the previous section in the neighborhood of any fiber. The generic fiber is a smooth curve $X_t, t\in C$ of genus $g$. We denote by $P$ the set of all singular points of the fibers and by $P_C$ the set of points in $C$ corresponding to the singular fibers , $ f(P) = P_C$. The main difference of the global situation lies in the presence of the global monodromy group which is the image of $\pi_1( C - P_C)$ in the mapping class group $Map(g)$. We denote this group by $M_X$. Let us choose an integer $N$ and consider a base change $h: R\to C$ where $R$ such that the map $h$ is $N$-ramified at all the preimages of the points from $P_C$ in $R$. Consider a surface $S$ obtained via a base change $h: R\to C$. We have the finite map $h' : S \to X$ defined via $h$ and the projection $g: S\to R$ with a generic fiber $S_t = X_{h(t)} $. The surface $S$ is singular with the set of singular points equal to $h^{-1}(P) = Q$ and the set of singular fibers over the points of $h^{-1}P_C = P_R $.. The monodromy group $M_S$ of the family $S$ is subgroup of finite index of the group $M_X$. \begin{theo} The fundamental group $\pi_1(S - Q)$ surjects onto $\pi_1(R)$. The kernel of this surjection is a quotient of $\pi_g$ by a normal subgroup generated by the orbits of $M_S (s_i^N)$. \end{theo} {\bf Proof} Indeed we have a natural surjection of $\pi_g$ on the kernel of projection onto $\pi_1(R)$ since there are no multiple fibers in the projection $g$. A standard argument reduces all relations to the local ones and the local relations where described in the previous section (see theorem 2.1). $\Box$ Now we move to the second step of our construction getting out of $S -Q$ a smooth compact surface $S^{N}$ with almost the same fundamental group. In the next subsection we will develop some partial theory of this second step summarizing and generalizing some known results. Let us assume that $N$ is either odd or divisible by $4$. \begin{lemma} There exists a smooth projective surface $S^{N}$ with a finite map $f : S^{N}\to S$ such that the image of the homomorphism $f_{*}:\pi_1(S^{N})\to \pi_1(S -Q)$ is a subgroup of finite index in $\pi_1(S -Q)$. \end{lemma} As it was already shown in lemma 2.5 there exists a projection of $\pi_1(X_g)$ to a finite group $UC_g^N$ which is invariant under $Aut(\pi_g)$ and factors through the fundamental group of a small neighborhood of a singular fiber in $S-Q$. Therefore, if the map $h: S \to R$ has a topological section we obtain a finite fiberwise covering of $S$ which is ramified only over the singular set $Q$. The resulting surface coincides locally with the smooth surface described in theorem 2. It is also smooth and the map $f$ is finite. If there is not a topological section we first make a base change nonramified at $P_R$ in order to obtain such a section ( see theorem 5.2 Appendix A) and then apply the previous argument. Since the map $f$ is finite the preimage $f^{-1}(Q)$ consists of a finite number of smooth points and therefore $ \pi_1(S^{N}) = \pi_1( S^{N} - f^{-1}(Q)) $ (see theorem 5.1 Appendix A). This also proves that the image of the homomorphism $f_{*}: \pi_1(S^{N})\to \pi_1(S -Q)$ is a subgroup of finite index in $\pi_1(S -Q)$. $\Box$ As we have said in the introduction to find a counterexample to the Shafarevich's conjecture we try to control the existence of an infinite connected chain of compact curves in the universal covering of $S^{N}$. The above construction shows that we can do it by controlling the image of the fundamental groups of the open irreducible components of the reducible singular fibers in the fundamental group of the open surface $S-Q$. \subsection{Comparison theorem} The construction discussed in the previous section can be applied to both projective and quasiprojective fibered surfaces. This indicates that the universal coverings and the fundamental groups for these two classes of surfaces have a similar structure. In this section we illustrate another flavor of the same principle by describing a procedure comparing the fundamental groups of surfaces with quotient singularities to the fundamental groups of certain smooth surfaces. For future reference we will set up this transition in a slightly bigger generality. Let $V$ be a normal projective complex surface and $Q \subset V$ be the finite set of its singular points. The fundamental group $\pi_1(V)$ is the quotient of $\pi_1(V - Q)$ by the normal subgroup generated by the images of the local fundamental groups of the points $q\in Q$. Recall that the local fundamental group $L_{q}$ of a point $q \in Q$ is defined as the fundamental group of a deleted neighborhood of $q$, i.e. as the group $\pi_1( U_q - \{ q \})$ where $U_q$ is a small analytic neighborhood of the point $q\in V$. The topology of the neighborhood $U_q$ is completely determined by $L_q$ as it was shown by D. Mumford \cite{MUM}. The following theorem is a generalization of a theorem by J. Koll\'ar (see \cite{K2}). \begin{theo} Let $V$ be a normal projective surface and let $Q$ be the finite set of its singular points. Consider for any $q\in Q$ a normal subgroup of finite index $K_q\triangleleft L_q$ which contains the kernel of the natural map $ L_q\to \pi_1(V - Q)$. Then there exists a smooth projective surface $F$ and a surjective finite map $r : F \to V$ which induces an isomorphism between $\pi_1(F)$ and the quotient of $\pi_1(V - Q) $ by the normal subgroup generated by the images of $K_q \in \pi_1(V - Q) , q\in Q$. \end{theo} {\bf Proof.} Denote by $K_Q$ the normal subgroup of $\pi_1(V - Q)$ generated by the subgroups $K_q\subset L_q$. We obtain the surface $N$ as a generic hyperplane section of a singular projective variety $W$ with the following property: $W$ contains a subvariety $S$ of codimension $\geq 3$ with $\pi_1(W-S) = \pi_{1}(V-Q)/K_{Q}$. We may assume that $F$ does not intersect $S$ since the latter has codimension at least $3$ in $W$. The fundamental group $\pi_1(F) = \pi_1(W-S)$ since $F$ is a generic complete intersection in $W$. We are going to construct $W$ as a union of two quasiprojective subvarieties. Denote by $G_q$ the finite quotient $L_q/K_q$ and by $G$ the direct product of all the groups $G_q$. Denote by $g_q$ the coordinate projection of $G$ onto $G_q$ and by $i_q$ the coordinate embedding of $G_q$ into $G$. For any $q$ there is a natural finite covering $ M_q$ of $U_q$ corresponding to the projection $L_q\to G_q$. The preimage of $q$ in $M_{q}$ consists of a single point and the projection $M_q\to U_q$ is nonramified outside $q$. In the next lemma we prove the existence of an algebraic extension of this local covering. \begin{lemma} There exist an open affine subvariety $V_q\subset V$ containing $q$ and an affine variety $B_q$ which is a $G_{q}$-Galois covering of $V_q$ ramified only at $q$ so that $B_{q}\times_{V_{q}} U_{q}$ and $M_{q}$ are isomorphic. \end{lemma} {\bf Proof} Let $\hat A_q$ be the completed local ring of $q\in V$. A local $G_q$-covering defines a finite algebraic extension $\hat B_q$ of $\hat A_q$. By Artin's approximation theorem \cite{ART} there exists an affine ring $A\subset {\Bbb C}(V)$ and a finite algebraic extension $B$ of $A$ which locally at $q$ corresponds to the extension $\hat B_q$ over $\hat A_q$. Explicitly the extension $\hat B_q$ is described by a monic polynomial $f(x)$ with coefficients in $\hat A_q$. If we now consider any monic polynomial $g(x)$ over the ring $A$ with $g(x) = f(x) \; {\rm mod} \; {\frak m}_q^N $ for a big enough $N$ then the resulting algebraic extension $B$ will be the one we need. The ring $A$ defines an open algebraic subvariety ${\rm Spec}(A) \subset V$ containing $q$. Similarly $B$ defines an affine variety ${\rm Spec}(B)$ with a finite projection $p_q : {\rm Spec}(B) \to {\rm Spec}(A)$. This projection is unramified outside $q$ in the formal neighborhood of $q$. Since we have the freedom to impose any finite number of extra conditions on $g(x)$ we can choose $p_q $ to be unramified at any finite number of points. In particular we may assume that the projection $p_q$ is nonramified over $Q-\{ q\}$. That means that the divisorial part $D\subset {\rm Spec}(A)$ of the ramification of $p_q$ does not intersect $Q$. Now we can take ${\rm Spec}(A)/D$ as $V_q$. Let $B_q $ denote ${\rm Spec}(B) \times_{{\rm Spec}(A)} V_{q}$. It is an affine variety with affine $G_q$-action since it extends the local nonramified Galois covering $U_q-\{ q\}$ and has the same degree. \hfill $\Box$ \medskip Let $B_{0}$ be the product of all $B_{q}$'s over $V$. This is an affine variety with the action of $G$. The quotient $B_0/G = V_0'$ is an open affine subvariety of $V$ which contains $Q$. The action of $G$ on $B_0$ is free outside of the preimage of $Q$. Let $G \to GL(E)$ be a (not-necessarily irreducible) faithful linear representation of $G$ of dimension $e$ with the property that only $1\in G$ is represented by scalar matrix. Consider the diagonal action of $G$ on the product $B_0 \times E$. There exists a natural $\Bbb{C}^*$ action on $E$ - multiplication by scalars. It extends to a $\Bbb{C}^*$-action on the product which commutes with the $G$-action. Let $F_0' = (B_0\times E)/G$ be the quotient variety. It is an affine variety with induced $\Bbb{C}^*$ action which has a natural projection $\pi_0 : F_0' \to V_0'$ and a zero section $i(V_0') = (B_0\times 0)/G$. For any $s\in V_0'-Q$ the preimage $\pi_0^{-1}(s)$ is a vector space isomorphic to $E$. Moreover $F_0'$ contains a natural vector bundle $I$ over $V_0'-Q$. Its sheaf of sections coincides with the sheaf of $G$-equivariant sections of the constant sheaf ${\cal O}\otimes E$ over $B_0$. Let us choose a smaller affine variety $V_0\subset V_0', Q\subset V_0$ with the property that $I$ is constant on $V_0-Q$. We define $F_0$ as the preimage of $V_0$ in $F_0'$. Let $V_1$ be an open subvariety of $V$ which does not contain $Q$ and such that the union of $V_1$ and $V_0$ is equal to $V$. Let $J$ be the trivial bundle of rank $e$ on $V_1$. Choose a linear algebraic isomorphism of $F_0$ and $J$ over the intersection of $V_0$ and $V_1$. Use this isomorphism to glue the projectivization ${\Bbb P}(J)$ with the singular variety $X = (N_0 -i(V_0')/{\Bbb C}^*$. The resulting proper variety $W$ has a natural projection $p: W\to V$ with all the fibers outside $Q$ isomorphic to projective space $\Bbb{P}^{e-1}$. Moreover the preimage of $V-Q$ in $W$ coincides with the projectivization of a vector bundle according to the construction of $W$. The fiber $W_q$ over $q$ coincides with $\Bbb{P}^{e-1}/i_q(G_q)$. The action of $i(G_q)$ on $\Bbb{P}^{e-1}$ is effective because of our assumption on the representation of $G \to GL(E)$. We denote by $S_q$ the singular subset of $W_q$.. It lies in the image of the fixed sets ${\rm Fix}_{g}(\Bbb{P}^{e-1})\subset \Bbb{P}^{e-1}$ for different elements $g\in i_q(G_q), g\neq 1$. Define a subvariety $S$ as the union of the varieties $S_q , q\in Q$. The set $S$ has codimension $\ge 3$ in $W$ since the codimension of $S_q$ in $W_q$ is at least $1$. \begin{lemma} The variety $W-S$ has a fundamental group isomorphic to $\pi_1(V-Q)/ K_Q$. \end{lemma} {\bf Proof} The fundamental group of $W-S$ is the quotient of the fundamental group of $\pi_1(V-Q)$ since $W$ contains an open subvariety which is $\Bbb{P}^{e-1}$ fibration over $V-Q$ and therefore has the same fundamental group. The group $K_q$ maps into zero under the surjective map $\pi_1(V-Q)\to \pi_1(W-S)$ since the image of a neighborhood of $q$ via the zero section $i$ has $K_q$ as a local fundamental group. All the relations are local and concentrated near special fibers. A formal neighborhood of $W_q - S_q$ in $W - S$ is topologically isomorphic to a fibration over $W_q - S_q$ with $M_q$ as a fiber. Therefore all local relations follow from $K_q =1$. It finishes the proof that $\pi_1( W - S) = \pi_1(V - Q)/K_Q$. \hfill $\Box$ Finally we prove the projectivity of $W$ by constructing an ample line bundle on it. Start with a line bundle $L$ on $W$ whose sections give an embedding of $X$ into a projective space. To see that such an $L$ exists consider first the $G$-invariant and $\Bbb{C}^*$ homogeneous sections of the trivial bundle ${\cal O}\otimes E$ over $B_0$. Assume that the degree of homogeneity is big enough and divisible by the order of $G$. It is known that such sections separate the points in the quotient variety $X$ and we obtain and embedding of $X$ into a projective space. Thus the induced bundle ${\cal O}(1)$ is defined on $X$. Denote by $L$ some extension of ${\cal O}(1)$ to $W$. Such extension exists since the complement of $X$ in $W$ is smooth. Next by choosing a polarization $H$ on $V$ appropriately we may assume that the global sections of $L\otimes p^*H$ on $W$ separate all the points of $X$. The restriction of $L$ on ${\Bbb P}(J)$ coincides with ${\cal O}_{{\Bbb P}(J)}(m)$ for some positive integer $m$. By replacing $H$ with a high power of $H$ if necessary we can produce enough sections of $L \otimes p^*H$ to separate the points of ${\Bbb P}(J)$. Therefore $L \otimes p^*H$ gives an embedding of $W$ into a projective space. \hfill $\Box$ \begin{rem} J.Koll\`{a}r (\cite{K1}) obtained similar result under additional assumption of existence of a surjective map of $\pi_1(V - Q)$ onto a finite group $H$ with $K_q$ as a kernel of the induced map on $L_q$ for every $q\in Q$. \end{rem} The universal covering $\tilde F$ of the smooth projective surface $F$ is very similar to the ramified covering $\tilde V$ of $V$ corresponding to the quotient group $\pi_1(V - Q)/K_Q$. Namely there is a natural finite map $p : F \to V$ which induces a finite map from $\tilde F$ to $\tilde V$. Hence both $\tilde F$ and $\tilde V$ are simultaneously either holomorphically convex or not. \subsection{Fiber groups} The global construction described in section 2.2 treats separately the part of the fundamental group of the fibered surface which lies in the image of the fundamental group of the fiber. Let $V$ be normal projective complex surface and $Q$ be a set of its singular points. Suppose that there is a projection of $V$ on a smooth curve $C$ which has no multiple fibers and the generic fiber of the projection is a curve of genus $g > 1$. \begin{defi} Denote by $\pi_{1,f} (V - Q) $ the image of the fundamental group $\pi_g$ of the general fiber in $\pi_1(V - Q)$. We will call this group a general fiber group. \end{defi} In this article we mostly consider the case when the set of singular points in $V$ includes only singularities with finite local fundamental groups. It is well known that these are exactly the quotient singularities. \begin{defi} We will call the group $\pi_{1,f}(V - Q)$ above a fiber group if $Q$ consists of the quotient singularities only. \end{defi} We shall also give a special notation for the case when $Q$ is empty. \begin{defi} We will call the group $\pi_{1,f}(V)$ projective fiber group if $V$ is a projective fibered surface over $C$ without multiple fibers. \end{defi} \begin{rem} Though we don't allow multiple fibers in the above definition we allow some multiple components in the singular fibers. We need that at least one component of each singular fiber has multiplicity one. \end{rem} Thus we have defined three classes of groups. These groups are equipped with a surjective map from the group $\pi_g$. The principal difference between these three classes of groups lies in the nontriviality of the local fundamental groups of normal surface singularity. \begin{rem} It follows that the general fiber group occurs also as a fiber group of a projective surface if the images of all local fundamental groups are finite. In particular this is true if all singular points have finite local fundamental groups. \end{rem} As a consequences of theorem 2.4 we get: \begin{corr} The classes of fundamental groups and fiber groups are the same for projective smooth surfaces and projective surfaces minus quotient singularities. \end{corr} The above results suggest that finding examples of smooth projective surfaces with pathological behavior of fundamental groups and universal coverings can be reduced to a similar problem for normal projective surfaces minus singular points. The latter seems to be an easier task. \section {Nonresidually finite groups} This section contains some material that shows opportunities to make our construction applicable to a big variety of examples. The second subsection shows how one can use our construction and a conjecture by Zelmanov to obtain a variety of potential ex amples of surfaces with nonresidually finite fundamental groups. It is a pleasure for the second author to thank M. Nori for many illuminating discussions concerning the related ideas. \subsection{ Variety of constructions} In this subsection we analyze the groups that can be obtained as fiber groups. Recall that the fiber group depends on the genus $g$ of the generic fiber, the monodromy group and the set of geometric vanishing cycles. Thus we can define an abstract algebraic data which gives us an abstract analogue of the fiber group. Let $M$ be a subgroup of the mapping class group $Map(g)$ and $g_1,...,g_N$ be any finite collection of elements in $\pi_g$ which are powers of vanishing cycles. Let $Mg_i$ be the orbit of $g_i$ under the action of $M$. \begin{defi} An abstract fiber invariant is a set of the form $(g,M, Mg_i)$ for some $M\subset Map(g)$ and some finite set of $g_i\in \pi_g$ as above. \end{defi} An abstract fiber invariant defines an abstract fiber group as the quotient of $\pi_g$ by a normal subgroup generated by $Mg_i$. We are interested in determining conditions under which this abstract fiber group is the actual fiber group of some geometric fibration which means a complex quasiprojective surface with a fibration over a curve. \begin{rem} The answer is rather simple in smooth or symplectic categories because all the elements of $Map(g)$ can be realized by the automorphisms of the Riemann surface of genus $g$ which preserve a given volume form. However the question about geometric fiber invariants (smooth projective case) is substantially more delicate. \end{rem} The following theorem shows that we still have a significant freedom to vary geometric fiber invariants. \begin{theo} Assume that $(g, M, Ms_i)$ is a fiber invariant of a projection $p: X \to C$ where $X$ is a smooth compact surface and $p$ has only Morse singularities and $s_i$ are the vanishing cycles corresponding to the singular points $P_i$ of the fibers of $p$. Then $(g,M,Ms_i^{N_i})$ is also a fiber invariant associated with some other fibration provided $N_i = N_j$ if $P_i, P_j$ are contained in the same singular fiber. \end{theo} {\bf Proof} We shall construct a new fibration with the desired properties by applying the base change construction to the fibration $p : X \to C$. Let $P$ be the set of points of $C$ corresponding to the singular fibers of $p$ and $N(p)$ be a function on $P$ with positive integer values such that $N(s) = N_i = N(P_i)$ if a singular point $P_i\in X_s$ for some $s\in P$. Let us take a base change $ h : R\to C$ where $R$ is a cyclic covering of $C$ of degree $N$ - the minimal integer divisible by all $N_i$, and with ramification indices $N_i$ at $P_i$ and $N$ at some $c\notin P$. This covering satisfies the following properties: 1) The preimage of $s\in P$ in $R$ is $N(s)$ ramified. 2) The map $\pi_1(R - h^{-1}(P)) \to \pi_1(C - M) $ is surjective. Indeed the first property is obvious from the construction of $h$. The point $c$ has exactly one preimage in $R$. Hence any closed loop in $C - P$ containing $c$ lifts into a closed loop in $R - h^{-1}(P)$ which proves surjectivity of the corresponding map of fundamental groups. Now we can induce the family of curves on $R$ by the map $h$. The resulting surface $Y$ is a singular surface with a finite map $f : Y\to X$. The singular points of $Y$ are the preimages of the points $P_i\in X$ with $N_i > 1$. All the singularities of $Y$ are quotient singularities. If we denote the set of singular points by $Q$ then the fiber invariant of the projection $ p_h : ( Y - Q) \to R$ is described by the data $ g, M, Mg_i$. Indeed the monodromy of the new family coincides with the image of the group $\pi_1 ( R - h^{-1}( P)$ in $Map(g)$ but the latter is equal to the image of $\pi_1(C - P)$ in $Map(g)$ as it was proved above. Hence the monodromy of the newly obtained family is $M$. All relations in the fiber group of $Y - Q$ are generated by local relations . The latter correspond to the singular points of the fibers of $p_h$. If $f(Q_j) = P_i$ then the corresponding relation is described by $s_i^{N_i}$ as it was shown in section 2.1. It finishes the proof of the theorem. $\Box$ The above theorem suggests that we can construct a big class of Burnside type groups as fiber groups. We can consider the set of geometric vanishing cycles $s_i$ as a set of simple curves on one copy of the fiber. The following construction allows to approximate any topological data by the geometric ones. Let $M_{g}^{L}$ be the compactified moduli space of curves of genus $g$ corresponding to a subgroup of finite index $M_L$ in the group $Map(g)$. It is an algebraic variety with quotient singularities only which contains a family of similar type irreducible divisors $S_I$ with normal crossings corresponding to different type of stable degenerations. Singular points of $M_{g}^L$ correspond to stable curves with automorphisms and constitute a subset of codimension more than $1$ if genus $g > 2$. Let $M_g^{L_0}$ be an open nonsingular subvariety in $M_g^L$ which corresponds to smooth curves of genus $g$ without extra automorphisms. Then $\pi_1(M_g^{L_0}) = M_L$ . The space $M_g^0 , g > 3$ is far from being affine. If fact there is a natural map of $M_g^L$ into the Satake compactification of the moduli of abelian varieties of dimension $g$ with a principal polarization. It maps each stable curve to a point corresponding to the Jacobian of its normalization. Thus all divisors corresponding to degenerate curve have images of codimension at least $2$ if $g > 2$ under this map. In particular generic hyperplane sections of Satake compactification produce complete curves which lie in $M_g^0$. We are interested in constructing holomorphic families of curves with a given set of singularities. The following construction shows that there almost no restrictions in constructing such families over a disc. \begin{lemma} Let $X_0$ be curve of genus $g > 2$ with a given set of smooth noncontractible cycles $s_i^k$ on it. Assume that for a given $k$ all cycles $s_i^k, s_j^k$ don't intersect and correspond to different conjugated classes in the fundamental group of $X_0$. Then there is a holomorphic family of curves over a disc $D$ which contains $X_0$ as nonsingular fiber, the cycles $s_i^k$ correspond to the vanishing cycles for degenerate fibers and monodromy group is generated by the products of Dehn twists over $s_i^k$ for each $k$. \end{lemma} {\bf Proof} The condition on $s_i^k$ means that each set $s_i^k$ corresponds to some type of stable degeneration $I(k)$ modulo the action of $Map(g)$. Consider a small complex disc $D_k$ around a generic point of $S_{I(k)}$ in $M_g$. There is a local family of stable curves over $D_k$ which consists of smooth curves outside the point of intersection of $D_k$ and $S_{I(k)}$. Let $p_k$ be a point on the boundary circle $dD_k$. We can find a path $t_0^k$ connecting $0$ and $D_k$ inside $M_g^0$ which provides with a diffeomorpisms $X_0 \to X_{p_k}$ which maps $s_i^k$ into a family of vanishing cycles on $X_{p_k}$. Indeed different paths provide with maps which differ by the elements of $Map(g) = \pi_1(M_g^0)$. Let us take an extension of $t_0^k$ into smooth real analytic curves which ends up transversally at the intersection point of $D_k$ and $S_{I(k)}$. We can assume that all the curves $t_k^0$ meet at $0$ being tangent to some one-dimensional complex subspace. Thus we constructed a one dimensional ``octopus'' $W$ consisting of the extended curves $t_k^0$. After a small variation we can complexify the resulting one-dimensional real set into a complex disc $D$ which contains $0$ and intersects a given set of divisors $S_{I(k)}$ with a prescribed monodromy corresponding to the connecting path $t_k^0$. It follows from the fact that a small neighborhood of $W$ is Stein and holomorphic functions on it approximate continuous functions on $W$. The family of stable curves induced on $D$ satisfies the lemma. $\Box$ \begin{lemma} For any finitely presented group $\Gamma$ we can construct a relatively projective family of curves $X_t$ over a holomorphic disc which has $\Gamma$ as a fundamental group. \end{lemma} {\bf Proof} For any group $\Gamma$ above we can find a surjective homomorphism $r : \pi_g \to \Gamma$ for some $g$. Let $N$ be the kernel of $r$. It has a finite number of generators $k_i$ as normal subgroup of $\pi_g$. The elements $k_i$ can be realized as cycles with normal intersections (including selfintersections ) only on the curve $X_g$. Let us add a handle at each intersection. We can lift $k_i$ in a new Riemann surface $X_h$ into a family of cycles $\tilde k_i$ without selfintersections. Let us add to this family of cycles the generating cycles $a_j,b_j$ of the additional small handles. We obtain the set of conjugated classes which generates the kernel of the projection $\pi_h \to \Gamma$ and each of the elements in this set is realized by a smooth cycle. Introduce a complex structure on $X_h$ and consider positive Dehn twists corresponding to $\tilde k_i, a_j, b_j$. All these cycles correspond to the conjugation classes which belong to the kernel of the projection $r_h : \pi_h \to \Gamma$. Note that the Dehn twist along the cycle $s$ acts trivially on the quotient of $\pi_h$ by a normal subgroup generated by $s$. Therefore we can apply lemma 3.1 to $ X_h , \tilde k_i,a_j,b_j$ and obtain a relatively projective family $X$ of curves of genus $h$ over a disc with $\pi_1 (X) =\Gamma $. $\Box$ Though the holomorphic families of curves look very different from algebraic ones we can rather easily embed a small deformation of such a family into an algebraic one. Denote by $M$ the monodromy group of the family of curves over disc described in the lemma. Note that $D$ is a Stein subvariety in $M_g$. We can lift it into any covering of $M_g$ which is unramified along $D$. In particular we can lift $D$ into any variety $M_g^L$ for a subgroup of finite index $M^L\subset Map(g)$ containing $M$. Since $D$ lies in affine subset of $M_g$ we can find an algebraic curve $C\subset M_g$( respectively $ M_g^L$) which contains a small variation of $D$. The family over $C$ induced from $M_g$ or $M_g^L)$ extends globally a small variation of the family over $D$ without changing its topological data. By taking generic $C$ we can assume that the resulting family $Y$ of curves over a normalization of $C$ is smooth and the projection $p: Y\to C$ is a Morse type map. Thus having a family over disc with arbitrary data $ g, Ms_i$ where $M$ is generated by the Dehn twists over $s_i$ we obtain the following geometric data $ (g, M_Ls_i, M_L s_j)$ for any subgroup of finite index in $M_g$ containing $M$. Accordingly considering any data $ g,Ms_i $ we can obtain a Burnside type approximation $ (g, M_L s_i^{N_1}, M_L s_j^{N_2})$ for any $N_1, N_2$. By taking $N_1 = N$ and $N_2 = N^B$, increasing the integer $B$ and decreasing $M_L$ we obtain series which approximate the group given by $(g, M s_i^N)$. \begin{rem} It seems plausible that most of the groups in such a series are nonresidually finite and violate any other good properties of linear groups if the group $ (g, Ms_i)$ violates them. \end{rem} \begin{rem} The above construction can be applied also to algebraic manifolds parametrizing special curves instead of moduli spaces. It results in different series of groups as monodromy groups. \end{rem} \subsection {Potential examples} Now we suggest a construction that can lead to rather simple new examples of surfaces with nonresidually finite groups. We thank V. Alexeev , S. Keel and M.Nori , for the fruitful discussions of the construction. Consider the map $f$ of the moduli space $M_g^L$ into Satake compactification $S_g$ of the moduli space of abelian varieties of dimension $g$ with a principal polarization. The latter is a projective variety which has a representation as a union of $A_g, A_{g-1}..... $ where $A_g$ is a quotient of the space of positively defined hermitian matrices of rank $g$ by the action of $Sp( 2g,\Bbb{Z})$. If $x\in M_g^L$ corresponds to a stable curve $X$ then $f(x)\in S_g$ is a point corresponding to the Jacobian of the normalization of the curve $X$. Denote by $SM_g$ the closure of the image $f(M_g)$ in $S_g$. If $g > 3$ the map $f$ contracts analytic subvarieties corresponding to the degenerate curves and curves with nontrivial automorphisms into proper analytic subvarieties in $SM_g$ of codimension at least two. Denote by $\Delta_0$ the divisor in $M_g^L$ corresponding to irreducible stable curves with one node. The image of $f(\Delta_0)$ consists of all the points of $S_{g-1}\subset S_g$ which correspond to the Jacobian varieties of dimension $g-1$. Generic jacobian is known to be a simple abelian variety. On the other hand the images of divisors corresponding to different type of stable degeneration intersect $S_{g-1}$ in proper subvarieties corresponding to nonsimple abelian subvarieties (they decompose into a product after isogeny). Consider the surface $V$ in $SM_g$ obtained by a set of hyperplane sections. We can assume that : 1) The fundamental group of an open part of $V$ surjects onto $Map(g)$. 2) The surface $V$ intersects the images of different divisors $S_{I}$ at a point only. 3) The intersection of $V$ and $f(\Delta_0)$ does not include points from the images of other divisors $S_I$ or subsets corresponding to curves with automorphisms. Denote the latter as $R \subset V$. By resolving $V$ at $R$ only we again obtain a projective surface $V'$. General hyperplane section $C$ of $V'$ will lift into a curve $C'$ in $M_g^L$ which intersects only $\Delta_0$ and $C' - \Delta_0$ is contained in $M_g^{L,0}$. Thus we have a family of curves $X_g$ over $C'$ which has singular fibers of one type only and the monodromy group of it coincides with $M^L$. Now we apply our construction to the above surface $V'$ and get a surface $V^{N}$ whose fundamental group $\pi_{1}(V^{N})$ is a of a finite index in an extension of the fundamental group of a Riemann surface by $\pi^{g}/(x^N = 1)$. Therefore Zelmanov's conjecture implies that the group $\pi_{1}(V^{N})$ is nonresidually finite. Hence we obtain a series of simple potential examples of surfaces with nonresidually finite fundamental groups. The considerations from the previous subsection allow us to get even bigger variety of examples. Let us make a: \begin{defi} Let $x$ be a primitive element in the fundamental group $\pi_{1}(g)$ of a Riemann surface of genus $g>1$ and $M^{L}$ be a subgroup of finite index in $Map(g)$. Consider the orbit of $x$ under $M^{L}$, $(M^{L} x)$ and take the $N$-th powers of all this elements. Consider the normal closure of this powers in $\pi_{1}(g)$. Let us denote this normal closure by $(M^{L} x)^{N}=1$. We will denote the quotient group by $\pi_{1}(g)/(M^{L} x)^{N}=1$. ( Observe that definition depends on the choice of $x$.) \end{defi} Now the following generalization of the conjecture of Zelmanov's gives us a way of constructing more examples of surfaces with nonresidually finite fundamental groups. {\bf Question} (Zelmanov) For big $g$ and $N$ the groups $\pi_{1}(g)/(M^{L} x)^{N}=1$ are nonresidually finite for any primitive $x$ and $M^{L}$ a subgroup of finite index in $Map(g)$. \section{ Some remarks on Shafarevich's conjecture for fibered surfaces} \subsection{ The case of projective surfaces} In this section we consider potential counterexamples to the Shafarevich's conjecture based on our construction. We begin with the general setting. Let $f:X\to R$ be a Morse type fibration with $X_t$ as generic fiber. Suppose that the fiber $X_0$ is singular and has more than one component $X_0 = \cup C_i$. We also assume that all components $C_i$ are smooth and without selfintersection. Denote the intersection graph of $X_0$ by $\Gamma_0$. Consider the retraction $cr : X_t\to X_0$ of generic fiber on special fiber( see section 1). The preimage $cr^{-1}(C_i) $ in $X_t$ is an open Riemann surface with a boundary consisting of geometric vanishing cycles corresponding to the intersection points of $C_i$ with other components of $X_0$. The fundamental group $\pi_1(cr^{-1}(C_i)$ is free. The natural embedding $cr^{-1}(C_i)$ into generic fiber $X_g$ defines an embedding of the fundamental groups $\pi_1(cr^{-1}(C_i)) = \Bbb{F}_i\to \pi_g$. Similar construction holds for any proper subgraph of curves in $X_0$. \begin{defi} For any proper subgraph $ K\subset \Gamma_0$ define a subgroup $ \Bbb{F}_K \subset \pi_g$ as a fundamental group of the preimage $cr^{-1}(\cup C_i) ,i\in K$. \end{defi} \begin{rem} If the graph $K$ is connected then its preimage in $X_t$ has only one component and vise versa. \end{rem} Let $\pi_{1,f}$ be a fiber group obtained from $\pi_g$ by our base change construction for some $N$. \begin{lemma} Suppose that there is a decomposition of a connected subgraph $K\subset \Gamma_0$ into a union $K_1 \cup K_2$ so that the image of $\Bbb{F}_K$ in $\pi_{1,f}$ is infinite, but the image of both $\Bbb{F}_{K_1} , \Bbb{F}_{K_2}$ is finite then the Shafarevich conjecture is not true. \end{lemma} {\bf Proof} Indeed under the conditions of the lemma we obtain an infinite connected graph of compact curves in the universal covering of the surface $S^{N}$. $\Box$ Now we choose $N = 3$. In this case we can apply the above lemma due to the group theoretic result which concerns the quotients of the free groups. \begin{defi} Let $\Bbb{F}^g_k$ be a free group with $k$ generators with a realization as a fundamental group of curve minus one or two points ( depending on $k$). Define $P^g( k,3)$ as the quotient of $\Bbb{F}^g_k$ by the set of relations $x^3 = 1$ for all primitive elements of $\Bbb{F}_k$ which can be realized by smooth nonintersecting curves in the above geometric realization $ \Bbb{F}^g_k$ of $\Bbb{F}_k$. \end{defi} \begin{theo} The group $P^g (k ,3)$ is equal to the Burnside group $B(k ,3)$ and hence finite. \end{theo} {\bf Proof} See Appendix B. $\Box$ \begin{corr} The group $\pi_g/(x^3 = 1)$ is a quotient of $B(2g,3)$ by one additional relation. \end{corr} Let $X_0$ be a graph of smooth curves $C_i$ with each curve intersecting at most two others ( chain or ring). Suppose that there exists a Morse family of curves $X \to R$ with a set of vanishing cycles $VS$ and a monodromy group $M$ which has $X_0$ as fiber. Assume that cycles from $VS$ correspond to different singular points of $R$ unless they correspond to $X_0$. Assume that $VS$ is decomposed into a union $S_0\bigcup S_1\bigcup S_2$. Assume that cycles from different subsets $S_i,i=0,1,2$ correspond to different singular points of $R$. \begin{lemma} Suppose that the monodromy group $M$ and the sets of vanishing cycles $S_0,S_1,S_2$ satisfy the following properties: 1. The image of $\pi_1(cr^{-1}(C_i)) $ in $\pi_g/ Ms_j^3 $ is finite for any $i, s_j \in S_0\bigcup S_1$. 2. The quotient group $\pi_g / ((Ms_j^3), Ms_k^{3^S}) $ is infinite for some integer $S$ and $s_j \in S_0\bigcup S_1 , s_k\in S_2 $. Then the universal covering $\tilde X$ is holomorphically nonconvex. \end{lemma} {\bf Proof} Indeed the universal covering $\tilde X$ contains an infinite covering of $X_0$ which is connected and consists of compact curves. Hence $\tilde X$ is not holomorphically convex. $\Box$ We are going to construct a family of surfaces which presumably contain an infinite number of surfaces with above property. We would like to produce such families from a standard family over an interval $I$. Let $g > 3$ and consider a family over an interval $I = [0,1]$ which has a fiber $X_0$ over $0$ and a singular fiber $X_1$, The generic fiber $X_t$ surjects on $X_0$ and $X_1$ and vanishing cycles for both singular fibers are realized as smooth curves on $X_t$. Denote the corresponding set of cycles as $S_0, S_1$ respectively. We assume that they don't intersect and for any component $C_i\subset X_0$ there is a corresponding cycle $s_i\in S_1$ which projects into a smooth homologically nontrivial cycle in $X_i$. This family over interval can be complexified into an algebraic family over a complete curve $R$. We can assume that the monodromy of the resulting family is any subgroup of finite index in the group $Map(g)$ which contains commuting monodromy transformations $T_0,T_1$ defined by the fibers $X_0, X_1$. Let $M_D$ be a subgroup of $Map(g)$ which commutes a contraction map $\pi_g\to \pi_1(X_0)$. Since the group $M_D$ contains $Map(g)$ for any curve $C_i$ we have obtain that the image $\pi_1(cr^{-1}(C_i))/(M_D s_i^3) $ is a finite group of exponent $3$ for any component $C_i$. \begin{lemma} The quotient $\pi_g/ (M_D s_i^3)$ is infinite if the number of components $C_i, g(C_i) > 0$ is more than $1$. \end{lemma} {\bf Proof} Indeed the group above maps surjectively onto a free product of nontrivial Burnside groups of exponent $3$ corresponding to different components of $X_0$ with nonzero genus. The latter is infinite which implies the lemma. $\Box$ Let $M^f$ be any subgroup of $M_D$ which contains $T_0,T_1$ and has the property that the image of $\pi_1(cr^{-1}C_i) $ in $\pi_g /(M^f s_i^3)$ is finite. The group $M^f$ defines a set of subgroups $M_j^f$ of finite index in $Map(g)$ containing $M^f$. The intersection of this set of subgroups coincides with $M^f$ since $Map(g)$ is residually finite. For each $M_j^f$ we can find a curve $R_j$ with a Morse family of curves $X_j$ which contains a topological family over interval constructed above and with a monodromy $M_j^f$. Let $S_2^j$ be a complementary set of vanishing cycles in the family $X_j$. We can now consider any $S$ and construct a new family $X_j^S$ using the theorem 3.1 with $N_0 =N_1 = 3 ,N_2 = 3^A$. \begin{corr} For $M_j^f $ and integer $A > 1$ we obtain a group $\pi_g /((M_j^f)s_i^3 , M_j^f s_k^{3^A})$ as a fiber group. Here $s_i \in S_0\subset S_1 , s_k\subset S_2$. \end{corr} \begin{rem} The subset $S_2$ depends on the actual curve $R_j$. The dependence of the fiber group on $S_2$ weakens with $A$ converging to infinity. The resulting family of groups approximates the infinite group $\pi_g/ (M^f s_i^3)$ as $M_j^f$ converges to $M^f$ and $A$ converges to infinity. \end{rem} \begin{con} Let $X_0$ be a nontrivial chain or ring of curves of genus greater than $0$. Assume that $g$ is an arithmetic genus of $X_0$ and $X_I$ is a family of curves over an interval described above. For any small enough subgroup $M^f\subset M_D$ defined above there exists a subgroup $M^f_j\subset Map(g)_f$ of finite index and an integer $A > 1$ such that $\pi_g/ (M^f_j s_i^3, M_j^f s_k^{3^A})$ is infinite for $s_i\subset S_0\bigcup S_1$ and any finite subset $s_k\in \pi_g$ \end{con} If the answer to the above conjecture is positive then the Shafarevich conjecture is not true. The fact that the family of groups parameterized by $M_j^f$ and $A$ approximates a group which has a nontrivial free product of groups as a quotient provides with a strong evidence supporting the above conjecture. On the other and the resulting fiber group does not have infinite linear representations which are equivariant with respect to the action of the monodromy group ( \cite{LN}). There is also another possibility to satisfy condition 2 . We can easily construct a family of curves of genus $g=2k, k>1$ such that there is a fiber in this family which consists of two components each of genus $k$ that give us a tree of components. Applying the base change construction for a given $N$ we obtain a surface $S - Q$ with the image of the fundamental group of every component in $\pi_1(S - Q)$ being equal to $P^g(2k,N) $. The fiber group of $S-Q$ is equal to $\pi_{2k }/(x^{N}=1)$. Now as we have shown we have the same behavior on $S^{N}$ for the closed curves and surfaces. We can formulate the following question: {\bf Question} Are there such a $k$ and $N$ such that $P^g(2k,N)$ is a finite group and $\pi_{ 2k }/(x^{N}=1)$ is an infinite group? If the answer of the above question is affirmative for some $ N, k$ we get a counterexample to the Shafarevich conjecture. We should point out that if the groups obtained from the components are finite then $\pi_{2g}/(x^{N}=1)$ does not have infinite linear representation. If the Shafarevich conjecture is correct the answer of the above question is negative. It also implies the answer to many similar group theoretic questions. The most basic question seems to be the following: {\bf Question} Is there such an $N$ and such $2\le m_{1}< m_{2}$ for which $B(m_{1},N)$ is finite and $B(m_{2},N)$ is infinite? Recently we were informed by Zelmanov that he can show that there exists an integer $d(0)$ so that for every prime number $p$ and an integer $d$ the group $B(d_{0},p)$ is finite if and only if the group $B(d,p)$ is finite. This result suggests the existence of an abstract group theoretic version of holomorphic convexity. The above considerations indicate a possibility for analysis of infinite groups by analytic methods. \subsection{Other applications} In this subsection we discuss symplectic and arithmetic versions of our construction. As Gompf has shown \cite{GOMPF}, every finitely presented group can be realized as a fundamental group of a symplectic manifold. It is reasonable to ask the if being symplectic puts any restrictions on the structure of universal coverings. Our construction easily extends to the symplectic category. It leads to interesting examples of symplectic four dimensional manifolds ( see \cite{BKS}). Using this construction we have defined in \cite{BKS} an obstruction to a symplectic Lefschetz pencil being a K\"{a}hler Lefschetz pencil. We begin with a symplectic fourfold $X$. Consider the corresponding Lefschetz pencil with reducible fiber and apply to it the our construction. So we get for a fix integer $N$ a symplectic fourfold $S^{N}$. Let $\rho$ be a generic representation $\rho :\pi_{1}(S^{N}) \to GL(n,\Bbb{C})$ whose image is not virtually equal to $\Bbb{Z}$. Denote by $Y_{i}$ the components of the preimage of the reducible fiber of $S$ in $S^{N}$ and denote by $F$ the general fiber of $S^{N}$. Denote by $\Gamma$ the image of $\pi_{1}(F)$ in $\pi_{1}(S^{N})$ and by $\Gamma_{i}$ the images of the fundamental groups of $Y_{i}$ in $\pi_{1}(S^{N})$. If the restrictions of $\rho$ on $\Gamma$ and $\Gamma_{i}$ for all $i$ are both finite or infinite we will say that the obstruction $O(X)^{N,n}$ is equal to zero and to one otherwise. \begin{prop} If $S$ is K\"{a}hler $O(X)^{N,n}$ is trivial for every pair $N,n$. \end{prop} Indeed otherwise we contradict the holomorphic convexity of the covering corresponding to infinite linear representation of the fundamental group. In \cite{BKS} we have constructed examples of symplectic fourfolds with nontrivial $O(X)^{N,n}$. In this article we consider only surfaces with given projection on a curve. Algebraically this means that the field of rational functions on the surface is provided with a structure of a one-dimensional field over a field of rational functions of the base curve. This picture is parallel to that of a curve defined over number field. It suggests that there should be a natural arithmetic version of the Shafarevich's conjecture. The notion of holomorphic convexity is not well defined in the arithmetic case. Instead we can describe the analogue of the absence of infinite chains of compact curves in the universal covering in the arithmetic case. The absence of infinite chains of compact curves seems to be the only obstruction to the holomorphic convexity in the case of compact complex surfaces. Let $C$ be a projective semi-stable curve over $K$. Since $K$ is not algebraically closed we obtain a nontrivial map $C\to Spec (O_K)$ where $O(_K)$ is the ring of integers. Extending $K$ if necessary we can assume that $C$ is a semistable curve. That means $C$ is a normal variety with semi-stable fibers consisting of normally intersecting divisors. The function field $K(C)$ is regular and has dimension one over $K$. Any maximal ideal $\nu$ in $O_K$ defines a subring $O_{\nu}$ of $K(C)$ which consists of the elements of $K(C)$ which are regular at the generic point of any component of the preimage of $\rho$ in the scheme of $C$. The ring $O_{\nu}$ contains the ideal of elements which are trivial on the preimage of $\nu$. Denote this ideal by $M_{\nu}$. The quotient ring $O_{\nu} / M_{\nu}$ is a finite sum of fields of rational functions on the components of the fiber of $C$ over $\nu$. Consider the maximal nonramified extension $K(C)^{nr}$ of $K(C)$. It is a Galois extension with a profinite Galois group $Gal^{nr}(K(C)$. The field $K(C)^{nr}$ contains the maximal nonramified extension $K^{nr}$ of the field $K$ as a subfield. (Observe that the field $\Bbb{Q}^{nr} = \Bbb{Q}$ but for many other fields $K$ the field $K^{nr}$ is an infinite extension.) The group $Gal^{nr}(K(C)$ maps surjectively onto $Gal^{nr}(K)$. Denote the kernel of the corresponding projection as $Gal_g^{nr}K(C)$. Any maximal ideal $\rho$ in the ring of integers $O_{K^{nr}}$ contains the unique maximal ideal $\nu$ of $O_K$. We can now define the subring $A_{\rho}$ as an integral algebraic closure of the subring $O_{\nu}$ in $K^{nr}(C)$. Let $Res_{\rho} = A_{\rho}/I(\rho)$ be the quotient ring by the ideal generated by $\rho$ in $A_{\rho}$. It is a semisimple ring of finite characteristics. We formulate a strong arithmetic analogue of the Shafarevich's conjecture. \begin{con} Let $K(C)$ be a field as above. For some finite extension $F$ of $K$ we can find a semistable model $C'$ of the field $F(C)$ such that the ring $Res_{\rho}$ is a direct sum of a finite number of fields for any ideal $\rho$ of the field $F^{nr}$. \end{con} We can also formulate this conjecture in a more geometric language. Namely if $C'$ is a semistable curve over $F$ then for any finite nonramified extension $L$ of $F(C')$ we have a model $C_L$ with a finite map onto $C'$. The fibers of this new model $C_L$ are uniquely defined by $C'$. \begin{con} There is a number $J(F(C'))$ such that the number of components of any fiber of $C_L$ is bounded by $J(F(C'))$ for any finite extension $L$ of $F(C')$ containing in $F(C')^{nr}$. \end{con} \begin{rem}The result of the conjecture depends ( at least formally) on the chosen model $C'$. If we blow up a generic point on a fiber the conjecture becomes false if the corresponding covering induces an infinite covering of the fiber. Thus the arithmetic version is sensitive to the change of the semistable model whereas the geometric conjecture is not. \end{rem} \begin{rem} The fields $F(C')^{nr}$ correspond to the factor of $\pi_{1}^{fin}(C')$ that is acted trivially by all inertia groups. ( Here we denote by $\pi_{1}^{fin}(C')$ the profinite completion of the geometric fundamental group of $C'$ ). Geometrically this means that we consider coverings of $C_L$ that are unramified over a generic point of every irreducible divisor in $C_L$. \end{rem} There exists some evidence for the above arithmetic conjectures. Partial results in this direction were obtained by the first author several years ago (1982). Namely he proved that the torsion group of an abelian variety $A$ is finite for any infinite algebraic extension of $K$ which contains only finite abelian extensions of $K$. In particular it is true for infinite nonramified extension of $K$ where $K$ is a finite extension of $\Bbb{Q}$. (This result was announced at Delange-Puiso seminar in Paris, May 1982 and later appeared in \cite{Co}). Yu. Zarhin proved that the same is true if the infinite extension of $K$ contains only finite number of roots of unity under some conditions on the algebra of endomorphisms $A$. The above result states that the group $Gal_g^{nr}K(C)$ has a finite abelian quotient and hence the conjecture 4.2 is evidently true for the quotient of $Gal^{nr}K(C)$ by the commutant of $Gal_g^{nr}K(C)$. For the same reason it is true for the quotient of $Gal_g^{nr}K(C)$ by any iterated commutant. \begin{rem} Any curve over arithmetic field can be obtained as a covering of $\Bbb{P}^1$ ramified at $ (0,1,\infty) $ according to the famous Bely's theorem. As it was pointed out by Yu.Manin any arithmetic curve has a ramified covering which is a nonramified covering of a modular curve. Therefore one might reduce the above conjecture to modular curves being considered over different number fields. In the complex case we don't have similar simple class of dominant manifolds( see the discussion in \cite{BH}. \end{rem} If Conjecture 4.2 is correct for every curve over every finite extension of $\Bbb{Q}$ then we get that there isn't any infinite chains of compact curves on the universal covering of a projective surface with a residually finite fundamental group. According to a conjecture of M. Ramachandran this is the only obstruction to holomorphic convexity for the universal coverings of a projective surfaces. \section{ Appendix A - Fiber groups and monodromy} This appendix includes several technical results from the theory of surfaces which we use in our article. First theorem generalizes the result we used in order to pass from quasiprojective surface to the projective smooth surface in section 2. \begin{theo} Let $V$ be a normal projective surface with $Q$ being the set of singular points in $V$ and $f: V' \to V$ be a finite surjective map from another normal projective surface $V'$. Assume that for any $q\in f^{-1}(Q)$ the map of the local fundamental groups $ f_* :\pi_1( U_q - q) \to \pi_1(V - Q)$ is zero, where $U_q$ is small topological neighborhood of $q$. Then there is a natural map $ f_* : \pi_1(V') \to \pi_1(V - Q)$ which is a surjection on a subgroup of finite index bounded from above by the degree of $f$. \end{theo} {\bf Proof} Indeed $Q$ is a set of isolated singular points. Following the proof of the lemma 2.5 we obtain a map $f_* : \pi_1 (V' - f^{-1}(Q)) \to \pi_1(V - Q)$. In order to prove the theorem it is sufficient to show that the kernel of natural surjection $i_* : \pi_1(V' - f^{-1} (Q))\to \pi_1(V')$ lies in the kernel of $f_* $. The preimage $f^{-1}(Q)$ also consists of a finite number of points and the kernel of $i_*$ is generated as a normal subgroup in $\pi_1(V' - f^{-1}(Q)$ by the local subgroups $\pi_1(U_q - q), q \in f^{-1}(Q)$. Due to the condition of the theorem the images of these groups are trivial in $\pi_1(V - Q)$ and hence we obtain the map $f_*$. $\Box$ We shall also need the following general result which allows to compare the properties of fiber groups and fundamental groups of the smooth quasiprojective surfaces. \begin{theo} Let $ f : X \to R $ be a quasiprojective surface which has a structure of a family of curves over a smooth curve $R$. Assume that generic fiber is an irreducible smooth projective curve and any fiber contains a component of multiplicity one. Let $h: \pi_{1,f} (X) \to G$ be $M_X$ invariant homomorphism into a finite group $G$ with $ M_X$ action. Let $K$ be the kernel of $h$. Then there exists a subgroup of finite index $ H \in \pi_1(X)$ that the intersection of $H$ with $\pi_{1,f}(X)$ is equal to $K $. \end{theo} \begin{rem} This is true if there exists a section $s:\pi_1(R) \to \pi_1(X)$, since we can define $H$ as a subgroup of $\pi_1(X)$ generated by products of the elements of $ s(\pi_1(R))$ and $ K$. If the curve $R$ is open then the group $\pi_1(R)$ is free and hence a section always exists. \end{rem} {\bf Proof} The group $K$ is a normal subgroup of $\pi_{1}(X)$ since it is invariant under the conjugations from both $\pi_{1}(R)$ and $\pi_{1,f}(X)$. Denote by $Q$ the quotient $\pi_{1}(X) / K$. It is an extension of $\pi_{1}(R)$ by $G$. Hence there is an action of $\pi_{1}(R)$ over $G$. Since $G$ is a finite group $\pi_{1}(R)$ contains a subgroup of finite index which acts trivially on $G$ via interior endomorphisms. This subgroup corresponds to a finite nonramified covering $ \phi : C \to R$ and will be denoted by $\pi_1(C)$. Its preimage $Q'$ in $Q$ is a subgroup of finite index. Consider a subgroup $K_G$ of $Q'$ consisting of the elements commuting with all the elements of $G$. The group $K_G$ projects onto $\pi_1(C)$ by the definition of $\pi_1(C)$ and therefore its intersection with $G$ is a cyclic subgroup of the center of $G$. The group $\pi_1(C)$ contains a subgroup of finite index $K'$ where the corresponding central extension splits. Namely if the order of the corresponding cyclic extension is $n$ then it splits over any cyclic covering of order $n$. The preimage of $K'$ in $Q'$ splits into a direct product of $G$ and $K'$. Hence on the preimage of a subgroup $ K'\subset \pi_1(C)\subset \pi_1(R)$ we have a natural extension of the projection from $\pi_{1,f}(X) \to G$. $\Box$ \section{ Appendix B - Some group theoretic results} This appendix contains several group theoretic results. We present proofs though most of the results are presumably known to the experts in the group theory.. Recall that we denote by $BT(n,m)$ the quotient of a free group $\Bbb{F}^n$ by a normal subgroup generated by the elements $x^m = 1$ where $x$ runs through all primitive elements of $\Bbb{F}^n$. \begin{prop} If $m$ is divisible by 4 and $n \ge 2$ then $BT(2,m)$ is infinite. \end{prop} {\bf Proof} The group $BT(2,4)$ has an infinite representation. Namely let $\Bbb{Q}_8 $ be the group of the unit quaternions of order $8$. It acts on $ H =\Bbb{R}^4$ by multiplications. Consider a group of the affine transformations of $\Bbb{R}^4$ generated by two generating rotations $g_1, g_2$ in $\Bbb{Q}_8$ but with different invariant points. The resulting group $G$ will be infinite. It has two generators and $G$ has $\Bbb{Q}_8$ as its quotient group. Any element of $g \in G$ which projects nontrivially into $G$ has order $4$. Indeed $g$ is an affine transformation of $\Bbb{R}^4$ with a linear part of order $4$ and without $1$ as eigenvalue. Hence $g$ in the conjugacy class of its linear part and has order $4$. $\Box$ \begin{rem} Similar argument can be applied to any finite subgroup of a skewfield instead of $\Bbb{Q}_8$ (see the description of such finite groups in J.Amitsur, Ann. of Math., 1955, vol. 62, p. 8). The above result makes it plausible that the groups $P(n,m)$ are infinite if $ m > 3, n > 1$. \end{rem} The case $BT(n,3)$ is different. The following result gives a hint on the effects which occur with the exponent $3$. \begin{lemma} Let $G_i, i=1,2$ be the groups generated by $ a,b$ with relations 1) $G_1 : {a^3 = b^3 = (ab)^3 = 1}$ 2) $G_2 : {a^3 = b^3 = (ab)^3 = (ab^2)^3 = 1}$. Then $G_1$ is infinite and $G_2$ coincides with the Burnside group $B(2,3)$ and hence it is finite. \end{lemma} {\bf Proof} The group $G_1$ has a natural geometric realization. Namely let us take $\Bbb{P}^1$ minus three points $p_i , i = 1,2,3$. The fundamental group $\pi_1(\Bbb{P}^1 - p_i) =\Bbb{F}_2$. If we impose relations $x^3$ on the elements which can be realized by smooth curves in $\Bbb{P}^1 - p_i$ then we obtain the set 1). Let us take a $\Bbb{Z}_3$ character $\chi$ of $\Bbb{ F}_2$ which is nontrivial on $a,b,ab$. We obtain a covering of $\Bbb{P}^1 $ ramified over three points. Imposing the above relations corresponds to the completion of the curve. Hence the subgroup of $G_1$ which is the kernel of $\chi$ coincides with the fundamental group of the corresponding complete curve. Since the above curve is a torus the group $G_1$ is an extension of $ \Bbb{Z}_3 $ by a free abelian group $ \Bbb{Z}+ \Bbb{Z}$. The description of 2) immediately follows since it is the quotient group of the group in 1). The element $ab^2$ generates $\Bbb{Z}+ \Bbb{Z} $ as a $\Bbb{Z}_3$-module. Hence $(ab^2)^3$ generates $3(\Bbb{Z}+ \Bbb{Z})$ and the resulting group is an extension of $\Bbb{Z}_3$ by the group $\Bbb{Z}_3 + \Bbb{Z}_3$. $\Box$ We shall use the following standard notations. Denote by $(a,b)$ the commutator of the elements $a,b$ and we put a sequence of brackets to denote an element obtained by iteration of the procedure. The group $B(n,3)$ has a rather simple description. It is a metabelian group with a central series of length three. The elements $(a,x)$ where $x\in [B(n,3),B(n,3)]$ are in the center of $B(n,3)$. In fact $((a,b),c)$ varies under permutation according to standard $\Bbb{Z}_2$ character of the group $S_3$ for any $((a,b),c)$. In particular $((x,r),x) = 1$ for any $x$. \begin{lemma} Let $G$ be a finitely generated group with a given set $S$ of generators. Assume that $((a,b), f)$ is invariant under any even permutation of $a,b,f$ for any $a,b\in S, f\in S\bigcup (S,S)$ where the latter denotes the set of pairwise commutators of the elements from $S$. Then $G$ is a metabelian group and it has a central series of length at most $3$. \end{lemma} {\bf Proof} The proof closely follows the proof from \cite{MS}. Indeed we can write $((a, b),(c, d))= (a,(c,d)), b)$. We deduce next that the above expression is invariant under even permutations. The latter implies that it is equal to 1 and hence $((a,b), c)$ commutes with any element from $S$ and hence lies in the center of $G$. The quotient of $G$ by the center $Z\in G$ is also generated by $S$ with equality $((a ,b),c) = 1$ for any $a,b,c\in S$ which means that $(a,b)$ is in the center of $G/Z$ for any $a,b\in S$. That means $G/Z$ is a central extension of abelian group which finishes the proof. $\Box$ \begin{corr} Under the above conditions the commutant $[G,G]$ is additively generated by the elements $(a,b),((a,b)c),a,b,c \in S $. \end{corr} \begin{lemma} Assume that $S$ in the above lemma consists of $n$ elements , $[S] , [S,S]$ consists of elements of order $3$ and there is a surjective map $p: G\to B(n,3)$. Then $p$ is an isomorphism. \end{lemma} {\bf Proof} The commutant $[G,G]$ is additively generated as $G^{ab}$-module by the elements $(a,b)$ under the above conditions. Hence it is of exponent $3$ and the number of elements in $G$ is not greater than the number of elements in the commutant of $B(n,3)$. The abelian quotient $G^{ab}$ is isomorphic to $B(n,3)^{ab}$. Therefore the number of elements in $G$ is not greater than in $B(n,3)$ and a surjective map $p : G\to B(n,3)$ is an isomorphism. $\Box$ \begin{prop} The group $BT(n,3)$ is finite and coincides with the Burnside group $B(n,3)$ if $BT(3,3) = B(3,3)$. \end{prop} {\bf Proof} Indeed this is true for $n= 2$ as it was shown above ( see e.g. \cite{MS}). The group $BT(n,3)$ has a natural surjective map onto $B(n,3)$ and hence we have to check the condition on $(a,b),c)$. The latter is enough to check for the group with three generators. $\Box$ \begin{lemma} $BT(3,3) = B(3,3)$. \end{lemma} {\bf Proof} The group $BT(n,3)$ is obtained as an extension of $B(3,2)$ by $c$ and since $cx$ is a generator for any $x \in B(3,2)$ we obtain that $ (cx)^3 = 1$. Therefore $x^{-1}c x , c$ commute for any $x \in B(3,2)$ and the kernel $K$ of the projection $p_c :BT(3,3) \to B(3,2)$ is an abelian group of exponent $3$. The sum $x^{-1}c x + c + x c x^{-1} = 0$ and we can easily deduce that $K$ has $4$ generators as a $\Bbb{Z}_3$-space. Therefore the group $BT(3,3)$ has the same number of elements as $B(3,3)$ and since there is a natural surjection $ BT(3,3)\to B(3,3)$ they are isomorphic. $\Box$ \begin{lemma} Let $S$ be a finite set of generators of the group $G$. Assume that any subset of $4$ elements in $S$ generates a subgroup of exponent $3$. Then $G$ is of exponent $3$. \end{lemma} {\bf Proof} The assumption implies that any three-commutator of $a,b,c\in S$ lies in the center $C$ of the group $G$ and the group satisfies lemmas 6.2, 6.3. $\Box$ \begin{lemma} Assume that $G$ has four generators $a,b,c,d$ that $G$ contains a set of subgroups of exponent $3$ which includes groups generated by triples of generators $ a,b,c, d$ and also the groups $a,b,(cd)$, $ c,d, ( a,b)$. Then $G$ is of exponent $3$. \end{lemma} {\bf Proof} It follows from the above results that $((a,b)c)$ and similar combinations are invariant under cyclic permutations. We also have $((a,b)(c,d)) = (((c,d)a)b) = (((d,c)b)a) = (((a,b)c)d) = (((a,b)d)c)$. Thus the value of the above commutator does not depend on any even permutation of the symbols. On the other hand it transforms into opposite if we permute $(c,d)(a,b)$. Hence all the above elements are equal to 1. This implies the lemma. $\Box$ We are interested in the groups which occur as the quotients of the fundamental groups of curves. Namely we will study the groups $P^{g}(n,m)$ defined in section four. We are going to use a representation of the fundamental group of a Riemann surface as a subgroup of index $2$ in the group generated by involutions which was successfully used by J.Birman, M.Nori, W.Thurston , B.Wainrieb and many others. Let $X$ be a curve of genus $g$. It can be represented as a double covering of $\Bbb{P}^1$ ramified over $2g+ 2$ points and $\pi_g$ is a subgroup of index two in the group generated by $2g + 2$ involutions $x_i$ with additional relation that the product of all involutions is an involution again. Similarly the group $\pi_1(X_g - pt)$ is realized as subgroup of index two in the group generated by $2g + 1$ involutions. The fundamental group of a curve minus two points is realized as a subgroup of index two in the group generated by $2g + 2$ involutions without any additional relations. The groups above are free groups, but they are provided with a special realization as the fundamental groups of open curves. \begin{defi} We shall denote by $\Bbb{F}^{g}_{n} $ a free group of $n$ generators provided with a realization as a fundamental group of a curve of genus $g$ minus one or two points. In case $n$ is odd then $\Bbb{F}^{g}_{n} $ is realized as a fundamenta l group of a curve of genus $g$ minus one point. In case $n$ is even $\Bbb{F}^{g}_{n} $ is realized as a fundamental group of a curve of genus $g$ minus two points. \end{defi} Recall that $P^{g}(n,m)$ is a quotient of $\Bbb{F}^{g}_{n} $ by the relations $x^{m}=1$ where $x$ runs through the primitive elements of $\Bbb{F}^{g}_{n} $ which can be realized as a smooth loops in the above geometric realization. The following lemma reduces a general case to $ n \leq 4$ \begin{lemma} If $P^g(4,3) = B(4,3)$ then $P^g(n,3) = B(n,3)$ for any $n\geq 4$. \end{lemma} {\bf Proof} The group $\Bbb{F}_n^g$ is represented as subgroup of index two generated by involutions $ x_1,..,x_{n+1}$. The set of standard generators of $\Bbb{F}_n^g$ can be taken as $ x_1x_i, i\neq 1$. Any four elements $x_1 x_j $ generate a subgroup of $P^g(n,3)$ which is a quotient of $P^g(4,3)$ and by assumption of the lemma the latter is of exponent $3$. Hence by lemma 6.5 $P^g(n,3)$ is of exponent $3$. Since the set of generators includes only primitive elements $\Bbb{F}_n$ the group $P^g(n,3)$ coincides with $B(n,3)$. $\Box$ \begin{lemma} If $P^g(3,3) = B(3,3)$ then $P^g(4,3) = B(4,3)$. \end{lemma} {\bf Proof} The group $P^g(4,3)$ is obtained from the curve of genus 2 minus a point. Consider a standard decomposition of $X^2$ into a union of two handles corresponding to pairs of generators $ a,b$ and $c,d$ of $\Bbb{F}^g_4$. There are topological embeddings of tori with two discs removed corresponding to the subgroups generated by any three symbols from the set $(a,b,c, d)$. The group $P^g(3,3)$ is realized as the quotient of the fundamental group of torus minus two discs. The above tori are obtained from one of the handles by adjoining a neighborhood of a generator in another handle. By assumption of the lemma $((a,b),c)$ is transformed into itself or the opposite element under the permutation of symbols. Similar groups correspond to the triples $( a,b (cd)) ,( c,d, (ab))$. Namely we can consider a corresponding handle minus a point. Thus we can apply lemma 6.6 and obtain that $P^g(4,3)$ is of exponent $3$ if the group $P^g(3,3)$ is. $\Box$ \begin{theo} $P^g(3,3) = B(3,3)$. \end{theo} Let us first describe the geometric picture. Consider the torus $T^2$ with a small embedded interval $I$. Let $p_1,p_2$ be two different points in the interval $I$ and $I_1$ be the interval between $p_1,p_2$ inside $I$. We assume that $ p_0$ is a point in $I - I_1$ and identify $p_0$ as an initial point for the fundamental group $\pi_1(T^2 - p_1- p_2)= \Bbb{F}^g_3$. Assume that $T^2,I$ are given with orientation. We consider smooth oriented loops through $p_0$ which are transversal to $I$. \begin{lemma} Any element of $P^g(2,3)$ with $p_0$ as an initial point is represented by an oriented curve $A$ without selfintersection which does not intersect $I$ and such that $A,I$ defines a standard orientation of $T$ at $p_0$. \end{lemma} {\bf Proof} Let $a,b$ be standard generators of $\Bbb{F}_2^g$ with a given orientation. The group $Out(\Bbb{F}_2^g) = SL(2,\Bbb{Z})$ and can be realized by linear periodic map of torus. In particular we can represent topologically the elements of $SL(2,\Bbb{Z})$ by maps which stabilize the points of $I$. In this way we obtain any map of $\Bbb{F}_2^g$ into itself which transforms $aba^{-1}b^{-1} = C$ into itself. Any homomorphism of the free group with above property is induced by this action of $SL(2,\Bbb{Z})$. Thus $g(a), g\in SL(2,\Bbb{Z}), a \in B(3,2)$ can be any element such that there exists $b'\in B(3,2)$ with $g(a)b'g(a)^{-1}b'^{-1}= C$ where $C$ is a given generator of the center. But we can find such $b'$ for any $g(a)$ which is not in the center. The curve $g(a)= A$ will be the image of a map which is linear outside a neighborhood of $I$ and keeps orientation intact. $\Box$ \begin{rem} Since $A^{-1} I$ represents the opposite orientation the lemma actually shows that the element $A^{-1}$ can be represented by another simple closed curve $B$ with orientation $B I$ opposite to $A^{-1} I$. \end{rem} The group $P^g(3,3)$ is generated by $B(2,3)$ realized as above and an element $r$ realized by a curve $R$ with one selfintersection. We have a natural representation of $r$ as $x_1 x_2^{-1}$ where $x_1, x_2$ are simple curves with the same orientation which move around $p_1, p_2$ respectively. The element $c = x_1 x_2$ is a natural central loop in $P^g(2,3) = B(2,3)$. \begin{lemma} Any element $bx_i $ is realized by a simple loop if $b$ is not in the center of $B(2,3)$. \end{lemma} {\bf Proof} We have to find a simple representative of $b$ through $p_0$ with an appropriate orientation. The latter exists due to the previous lemma. $\Box$ \begin{lemma} Any element $br\in P^g(3,3), b\in B(2,3)$ can be represented by a simple curve in its conjugation class unless $b$ is in the center of $B(2,3)$. \end{lemma} We have $ br = bx_1x_2$. The element $bx_1$ is represented by a simple curve $B$. Let $S$ be curve which contains $I_1$ and intersects $B$ transversally at exactly one point inside $I_1$. The complimentary $ T - ( S- I_1)$ defines another group $B(3,2)$. We assume that $p_0$ is not in $S$ and hence $bx_1$ is equivalent to a simple curve with a desired orientation with respect to $x_2$. That means $ (bx_1)x_2$ can be realized by a class of simple curve in $ P^g(3,3)$. The classes $x_i$ are also realized by simple curves . $\Box$ \begin{corr} The elements $(br)^3 = 1$ for any $b\neq c = x_1x_2$. \end{corr} Indeed if $b$ is not in the center then $br$ is realized by a simple curve and we get the result. The element $c^{-1}r = x_2^{-1}x^{-1}x_1 x_2^{-1} = x_2^{-2} = x_2$ in $P^g(3,3)$. In dealing with the extension of $B(2,3)$ by an element $r$ we will be using the following general argument. Let $G$ be a finite group of exponent $3$ and $G'$ is obtained from $G$ by adding $r$ and some relations of type $(br)^3 = 1$. Then the kernel of a natural projection $p: G'\to G$ is generated by the elements $r^a = ara^{-1}, a \in G$. The group $G$ acts on this set of elements by left translation $ g : r^a \to r^{ga}$. Any relation $(br)^3 = 1$ implies the relation : $1 = brbr br = brb^{-1} b^{-1}r b r = r^b r^{b^{-1}}r =1$ and similar relation for left translations of the orbits of the cyclic group $B = (1,b,b^{-1})$. If in addition $(b^{-1}r)^3 = 1$ all the elements $r^b,r^{b^{-1}}, r$ commute and any pair of them generate the same abelian group. \begin{lemma} The group $P^g(3,3)$ is an abelian extension of $B(2,3)$. \end{lemma} {\bf Proof} The kernel of the projection $p_r :P^g(3,3)\to B(2,3)$ which maps $r$ to $1$ is generated by the elements $r^b = brb^{-1}$. All these elements commute with $r$ unless $b = c$. We also have $r^b r r^{b^{-1}} = 1$ and they all commute if $b$ is not in the center. Therefore $ r^a$ commutes with $r$ if it commutes with both $r^b,r^{b^{-1}}$. On the other hand $ r^{ab},r^{a}, r^{ab^{-1}}$ commute if $r^b,r, r^{b^{-1}}$ commute. Let $a$ be a generator of $B(2,3)$. Then $ r^{ca } r^{c} r^{ca^{-1}}= 1$ and all these elements commute, but $r^{ca}, r^{ca^{-1}}$ commute with $r$. This implies that all the elements $r^b, b\in B(2,3)$ commute with $r$. After translation by $B(2,3)$ we obtain that all the elements $r^g$ commute. $\Box$ \begin{lemma}Let $T$ be the group generated by $r^b, b\in B(2,3)$. Then $T$ is an abelian group with $4$ generators. \end{lemma} {\bf Proof} Indeed the set of cyclic subgroups which don't lie in the center generate a family of relations. Since we have established that $T$ is an abelian group we shall write them in the additive form $ r^{x}+r^{ax}+ r^{a^2 x} = 0$, $x \in B(2,3) a$ but not in the center $C$. Denote by $T_S$ a subgroup of $T$ generated by a subset $S\subset B(2,3)$. Let $A$ be an abelian subgroup generated by $a$ and $c$ which generates the center of $B(2,3)$. The summation over orbits of cyclic noncentral subgroups gives zero. Thus if we consider $T_A$ modulo a subgroup $T_C$ corresponding to the center we obtain $r^g + r^{gh} = 0, g\notin C$. Hence the elements $r^g = -r^{g^{-1}}$ and $r^{gc}= r^{g}$ modulo subgroup $r^c, r$. We obtain that $r^{a}$ generates $T_A$ modulo the subgroup $(r_c,r) $ and a sum over any orbit of $C$ is also zero. Thus we have $ r(x^{-1}) = - r - r^{x}$ and $ r^{x} + r^{y} + r^{y^{-1}x^{-1}} = 0$ for the elements $x,y$ which lie in one abelian subgroup of $B(3,2)$. The same is true modulo $r^c$ since we can apply the same argument to the quotient of $B(3,2)$ by the center. Hence $r^{x} + r^{y} - r = r^{xy}$ (modulo($r^c$)) for any $x,y\in B(2,3) $. In particular $ r^{a}, r^{b}, r^{c},r$ generate the group $T$. $\Box$ \begin{lemma} $T$ is an elementary abelian group. \end{lemma} {\bf Proof} We have $r^{x^2} = 2 r^x - r$ and $r = 3 r^x- 2r$. Hence $3r = 3r^x$ for any $x$. Hence $ 3(r - r^x) = 0$ for any $x$. Thus the elements of zero degree in $T$ constitute an elementary $3$-group $T_0$ which is a normal subgroup of $P^g(3,3)$. The quotient $T/T_0$ is a cyclic group. The group $P^g(3,3)/T_0$ is a central extension of $B(2,3)$. Since $P^g(3,3)$ is generated by elements of order $3$ we obtain that $T/T_0 =\Bbb{Z}_3$. $\Box$ \begin{corr} The number of elements in $T$ is $3^4$. \end{corr} Hence the number of elements in $P^g(3,3)$ is equal to $ 3^7$ and coincides with the number of elements in $B(3,3)$. Since there exists a surjective map $p: P^g(3,3)\to B(3,3)$ the groups coincide. Thus we have proved that group $P^g(n,3)$ coincide with $B(n,3)$ for all $n$.

9703

alg-geom/9703009

en

https://arxiv.org/abs/alg-geom/9703009

alg-geom/9703009

Klaus Hulek

C. Ciliberto, K. Hulek

A Remark on the Geometry of Elliptic Scrolls and Bielliptic Surfaces

LaTeX2e with theorem, amstex, amssymb, amscd packages; 11 pages

The union of two quintic elliptic scrolls in P^4 intersecting transversally along an elliptic normal quintic curve is a singular surface Z which behaves numerically like a bielliptic surface. In the appendix to the paper [W. Decker et al.: Syzygies of abelian and bielliptic surfaces in P^4, alg-geom/9606013] where the equations of this singular surface were computed, we proved that Z defines a smooth point in the appropriate Hilbert scheme and that Z cannot be smoothed in P^4. Here we consider the analogous situation in higher dimensional projective spaces P^{n-1}, where, to our surprise, the answer depends on the dimension n-1. If n is odd the union of two scrolls cannot be smoothed, whereas it can be smoothed if n is even. We construct an explicit smoothing.

alg-geom

\section{Introduction} The union of two quintic elliptic scrolls in ${\Bbb{P}}^4$ intersecting transversally along an elliptic normal quintic curve is a singular surface $Z$ which behaves numerically like a bielliptic surface. In the appendix to the paper \cite{ADHPR} where the equations of this singular surface were computed, we proved that $Z$ defines a smooth point in the appropriate Hilbert scheme and that $Z$ cannot be smoothed in ${\Bbb{P}}^4$. Here we consider the analogous situation in higher dimensional projective spaces ${\Bbb{P}}^{n-1}$, where, to our surprise, the answer depends on the dimension $n-1$. If $n$ is odd the union of two scrolls cannot be smoothed, whereas it can be smoothed if $n$ is even. We construct an explicit smoothing. \section{Elliptic scrolls} To every elliptic normal curve $E\subset {\Bbb{P}}^{n-1}$ of degree $n$ and every point $P\in E$ one can associate a {\em translation scroll} $S=S(E,P)$ by defining $S$ as the union of all secants of $E$ joining the points $x$ and $x+P,\ (x\in E)$. If $n\ge 5$ and $P$ is not a 2-torsion point, then $S$ is a singular surface of degree $2n$ with $E$ as its singular locus. If, however, $P$ is a non-zero 2-torsion point, then $S$ becomes a smooth scroll of degree $n$ (the secants spanned by $x, x + P$ and $x-P, x$ coincide). Varying the point $P$ does, of course, not define a flat degeneration: as a scheme the general translation scrolls degenerates to a multiplicity-$2$ scheme whose support is the smooth scroll of degree $n$ (cf. \cite {HVdV}). In this paper we are interested in the degree $n$ scrolls defined by non-zero 2-torsion points. We will simply call these scrolls {\em degree} $n$ {\em elliptic scrolls}. (If $n=5$ this is the unique irregular smooth scroll in ${\Bbb{P}}^4$.) Our first aim is to determine these scrolls as abstract surfaces. Here, as in the sequel, we shall notice a difference between the cases $n$ even and $n$ odd.\\ We shall first treat the $n$ odd case. For this purpose we fix an elliptic curve~$F$. Recall that there is a unique ${\Bbb{P}}^1$-- bundle over $F$ with invariant $e=-1$ (in the sense of \cite[Chapter V]{Ha}). It was already observed in \cite[p. 451]{A} that this is the symmetric product $S^2F$ where the ${\Bbb{P}}^1-$bundle structure is given by summation $$ \begin{array}{rll} \pi: & S^2 F & \rightarrow F\\ & \{x,y\} & \mapsto x+y. \end{array} $$ {}Fix an origin $0$ of $F$, and let $p:F\times F\rightarrow S^2F$ be the natural projection. The curve $$ {}F_0=p(F\times\{0\}) $$ is a section of the ${\Bbb{P}}^1$ -- bundle $S^2F$. We choose the point $p(0,0)$ as its origin and, by abuse of notation, shall denote it again by $0$. If $\{P_i; i=1,2,3\}$ are the non-zero 2-torsion points of $F$, then the curves $$ \Delta_i=\{(x,x+P_i); x\in F\} (\cong F)\quad (i=1,2,3) $$ are mapped 2:1 under $p$ to 2-sections $F_i\subset S^2F$. As abstract curves $F_i=F/\langle P_i\rangle$. We shall choose the point $0_i=p(0,P_i)=p(P_i,0)$ as the origin of $F_i$. The group of 2-torsion points of $\Delta_i$ is mapped to two points $(0_i,Q_i)$. Note that $F_i/\langle Q_i\rangle\cong F$. Every fibre $f$ of $\pi$ intersects $F_i$ in two points which differ by $Q_i$. We shall denote the fibre of $\pi$ over $P\in F$ by $f_P$ and put $S=S^2F$. The following formulae follow immediately from the above description: $$ \begin{array}{ll} (1) & {\cal O}_S(F_0)|_{F_0}={\cal O}_{F_0}(0)\\[1mm] (2) & {\cal O}_S(F_i)={\cal O}_S (2F_0-f_{P_i})\\[1mm] (3) & {\cal O}_S(F_i)|_{F_i}={\cal O}_{F_i}(0_i-Q_i)\\[1mm] (4) & K_S= {\cal O}(-2F_0+f_0). \end{array} $$ \begin{proposition}\label{pro1.1} Assume $n\ge 5$ odd. The line bundle ${\cal O}_S(H)={\cal O}_S(F_0+\left(\frac{n-1}2\right)f_0)$ is very ample and embeds $S$ as smooth surface of degree $n$ in ${\Bbb{P}}^{n-1}$. This surface is the translation scroll of the elliptic normal curves $F_i, i=1,2,3$ defined by the 2-torsion points $Q_i$. Conversely every translation scroll of an elliptic normal curve of degree $n$ by a 2-torsion point arises in this way. \end{proposition} \begin{Proof} Very ampleness of ${\cal O}_S(H)$ follows e.g.\ from \cite[Exercise V.2.12]{Ha}. A straightforward calculation using Riemann-Roch shows $h^0({\cal O}_S(H))=n$. Since $H.F_i=n$ and $h^1({\cal O}_S(H-F_i))=0$ (the latter can be seen e.g. by Kodaira vanishing), the 2-sections $F_i$ are mapped to elliptic normal curves of degree $n$. By construction $S$ is then the translation scroll defined by the pair $(F_i, Q_i)$. Conversely given any pair $(F_i, Q_i)$ consisting of an elliptic curve and a 2-torsion point, then $F_i$ is a 2-section of $S^2F$ with $F=F_i/\langle Q_i\rangle$ such that the rulings of $S^2F$ cut $F_i$ in two points differing by $Q_i$. \end{Proof} We now turn to the case $n$ even where we assume $n\ge 6$. Let $Y\subset {\Bbb{P}}^{n-1}$ be a degree $n$ scroll given by a pair $(E,Q_i)$ where $E$ is an elliptic normal curve of degree $n$ and $P_i$ a non-zero 2-torsion point of $E$. Then $Y$ can also be constructed as follows: Embed $E$ as a normal curve of degree $n+1$ in ${\Bbb{P}}^n$ and let $X$ be the degree $(n+1)$-scroll given by $E\subset {\Bbb{P}}^n$ and the point $Q_i$. Projection from say the origin $0\in E$ then maps $X$ to $Y$. More precisely, the surface ${\tilde X}$ which is the blow-up of $X$ in $0$ is mapped to $Y$. Under this projection map the fibre of $X$ over the origin $0$ is contracted, whereas the map is bijective otherwise. In fact this is the geometric realization of an elementary transformation of $X$. \begin{proposition}\label{pro1.2} The degree $n$ scroll $Y$ is smooth. It is isomorphic to the ${\Bbb{P}}^1$ -- bundle ${\Bbb{P}}(\cal E)$ over the elliptic curve $F=E/\langle Q_i\rangle$ where ${\cal E}={\cal O}_F\oplus{\cal O}_F(0-P_i)$ and $P_i$ is the image of the 2-torsion points $Q_j, j\neq i$ under the projection to $F$. The embedding is given by the complete linear system defined by the line bundle ${\cal O}_Y(H)={\cal O}_Y(F_0+\frac n2 f_0)$ where $F_0$ (by abuse of notation) is the image of the section $F_0$ of $X$. \end{proposition} \begin{Proof} Recall the situation on $X=S^2F$ where $E$ is the 2-section $F_i$. There are two sections of $X$ which intersect $F_i$ (transversally) in the point $p(0,P_i)$, namely $F_0$ and $F_{Pi}=p(F\times \{P_i\})$. Since $F^2_0=F^2_{P_i}=F_0 F_{P_i}=1$ these sections become disjoint sections of the ${\Bbb{P}}^1$-bundle ${\Bbb{P}}({\cal E})$ which is defined by the elementary transformation with centre $p(0,P_i)$. We shall denote these sections again by $F_0$ resp. $F_{P_i}$. Since after the elementary transformation $F^2_0=F^2_{P_i}=0$ it follows that we can assume that ${\cal E}={\cal O}\oplus {\cal M}$ where ${\cal M}$ has degree $0$. It follows from the elementary transformation that the normal bundle of $F_0$ in ${\Bbb{P}}({\cal E})$ is isomorphic to ${\cal O}_F(0-P_i)$. This shows the claim about ${\cal E}$. The above description of the elementary transformation immediately gives ${\cal O}_{{\Bbb{P}}({\cal E})}(H)={\cal O}_{{\Bbb{P}}({\cal E})}(F_0+\frac n2 f_0)$. Clearly $H^2=n$. Since $h^0({\cal O}_{{\Bbb{P}}({\cal E})}(H))=n$ and since $H$ is very ample \cite[Exercise V.2.12]{Ha} the proposition follows. \end{Proof} \begin{uremark} Using the adjunction formula we immediately obtain the following results: $$(5)\quad K_Y={\cal O}_Y(-2F_0)\otimes{\cal O}_Y(f_0-f_{P_i})$$ \noindent The curve $E$ is again a 2-section of $Y$ with self-intersection number $E^2=0$. Since $E$ and $F_0$ do not intersect we obtain $$(6)\quad {\cal O}_Y(E)={\cal O}_Y(2F_0)$$ \noindent and combining (5) and (6) gives $$(7)\quad {\cal O}_Y(E)={\cal O}_Y(-K)\otimes {\cal O}_Y(f_0-f_{P_i}).$$ \noindent Note that an analogous formula holds for $E=F_i$ on $S=S^2F$. \end{uremark} \section{Rigidity for $n$ odd} We fix an elliptic normal curve $E$ in ${\Bbb{P}}^{n-1}$ of odd degree $n$ and two non-zero 2-torsion points $P_i\neq P_j$ on $E$. These define degree $n$ elliptic scrolls $X_i$ and $X_j$. The union $Z=X_i\cup X_j$ of these scrolls is a singular surface of degree $2n$ whose singular locus is the curve $E$, which is a double curve of $Z$. Numerically $Z$ is a bielliptic surface, its dualizing sheaf is a line bundle $\omega_Z$ with $\omega^2_Z={\cal O}_Z$. In the case $n=5$ those surfaces were considered in connection with abelian and bielliptic surfaces in \cite{ADHPR}, where their equations were determined. It was also shown \cite[appendix]{ADHPR} that they define smooth points in their Hilbert scheme and that they are rigid, in the sense that every small deformation of $Z$ is again of the same type. In particular, these surfaces cannot be smoothed. In this section we shall see that this is the same for all odd degrees, whereas, surprisingly, the situation is very different for $n$ even.\\ Our first aim is to study the normal bundle of the degree $n$ scroll in ${\Bbb{P}}^{n-1}$. Let $X_i$ be one of these scrolls. Then we have the following exact sequence for the rulings $f$ of this scroll: $$ \begin{array}{rccccl} 0\rightarrow & N_{f/X_i} & \rightarrow & N_{f/{\Bbb{P}}^{n-1}} & \rightarrow & N_{X_i/{\Bbb{P}}^{n-1}}|_f\quad \rightarrow 0\\ & || & & || & &\\ & {\cal O}_f & & (n-2){\cal O}_f(1) & & \end{array} $$ It follows that $$ N_{X_i/{\Bbb{P}}^{n-1}}|_f = (n-4){\cal O}_f(1)\oplus {\cal O}_f(2). $$ The degree 2 subbundle is uniquely determined and varying $f$ we obtain a line subbundle $$ {\cal L} = \pi^*(\pi_*N_{X_i/{\Bbb{P}}^{n-1}}(-2))(2)\subset N_{X_i/{\Bbb{P}}^{n-1}} $$ and as in \cite{HVdV} one proves that ${\cal L}=K^{-1}_{X_i}$. Thus we have an exact sequence $$(8)\quad 0\rightarrow K^{-1}_{X_i} \rightarrow N_{X_i/{\Bbb{P}}^{n-1}}\rightarrow Q\rightarrow 0$$ \noindent where $Q$ is a vector bundle of rank $(n-4)$ with $Q|_f=(n-4){\cal O}_f(1)$. \begin{lemma}\label{lem2.1} \begin{enumerate} \item[${\rm(i)}$] $h^0(N_{X_i/{\Bbb{P}}^{n-1}})=n^2$ \item[${\rm(ii)}$] $h^j(N_{X_i/{\Bbb{P}}^{n-1}})=0 \mbox{ for } j\ge1.$ \end{enumerate} \end{lemma} \begin{Proof} Since $h^j(K^{-1}_{X_i})=0$ for all $j$, it follows from sequence (8) that the claim is equivalent to $h^0(Q)=n^2$ and $h^j(Q)=0$ for $j\ge 1$. The defining sequences for $Q$ and the normal bundle $N_{X_i/{\Bbb{P}}^{n-1}}$ together with Riemann-Roch give $$ \chi(Q)=\frac 12 (c^2_1(Q)-2c_2(Q))+\frac 12 c_1(Q)(-K_{X_i})+(n-4)\chi({\cal O}_X)=n^2 $$ Hence it is enough to prove that $h^j(Q)=0$ for $j\ge 1$. By Serre duality $h^2(Q)=h^0(Q^{\vee} \otimes K_{X_i})=0$ since $Q^{\vee}\otimes K_{X_i|f}=(n-4){\cal O}_f(-3)$. To prove vanishing of $h^1(Q)$ we first remark that $Q(-1)$ is trivial on the fibres $f$ and hence $Q(-1)=\pi^*{\cal F}$ where ${\cal F}$ is a rank $n-4$ bundle on $F$. Since $T_{{\Bbb{P}}^{n-1}}(-1)$ is generated by global sections the same is true for $N_{X_i/{\Bbb{P}}^{n-1}}(-1)$ and hence also for $Q(-1)$ and ${\cal F}$. But now, using the classification of vector bundles on elliptic curves \cite{A} it follows that $h^1({\cal F}(D))=0$ for every divisor $D$ on $F$ of positive degree. Recall that $H=F_0+\left( \frac{n-1}2\right) f_0$. Let $Q'=Q(-1)\otimes \left( \frac{n-1}2\right) f_0$. Then $h^1(Q')=h^1({\cal F}(\frac{n-1}2 0))=0$. Finally $h^1(Q)=0$ follows from the exact sequence $$ 0\rightarrow Q'\rightarrow Q\rightarrow Q|_{F_0}\rightarrow 0 $$ since $Q|_{F_0}={\cal F}(\frac{n+1}2 f_0)$ and $h^1({\cal F}(\frac{n+1}2 f_0))=0.$ \end{Proof} \begin{lemma}\label{lem2.2} $h^j(N_{X_i/{\Bbb{P}}^{n-1}}(-F_i))=0$ for all $j$. \end{lemma} \begin{Proof} Twisting (8) with ${\cal O}_{X_i}(-F_i)$ we obtain the exact sequence $$ 0\rightarrow K^{-1}_{X_i}(-F_i)\rightarrow N_{X_i/{\Bbb{P}}^{n-1}}(-F_i)\rightarrow Q(-F_i)\rightarrow 0. $$ The line bundle $K^{-1}_{X_i}(-F_i)={\cal O}_{X_i}(f_{P_i}-f_0)$ has no cohomology. Since for the restriction to a ruling $Q(-F_i)|_f=(n-4){\cal O}_f(-1)$ it follows that $h^0(Q(-F_i))=h^2(Q(-F_i))=0$. Finally we obtain $-h^1(Q(-F_i))=\chi(Q(-F_i))=0.$ \end{Proof} We now turn to the normal bundle $N_{Z/{\Bbb{P}}^{n-1}}$ of $Z$. \begin{proposition}\label{pro2.3} \begin{enumerate} \item[${\rm(i)}$] $h^0(N_{Z/{\Bbb{P}}^{n-1}})=n^2$ \item[${\rm(ii)}$] $h^j(N_{Z/{\Bbb{P}}^{n-1}})=0$ for $j\ge 1$. \end {enumerate} \end{proposition} \begin{Proof} The line bundle $$ T=N_{E/X_i}\otimes N_{E/X_j}={\cal O}_E(20-Q_i-Q_j) $$ is a non trivial 2-torsion bundle. As in \cite{CLM} we have the following exact sequences $$(9)\quad 0\rightarrow N_{X_i/{\Bbb{P}}^{n-1}}\rightarrow N_{Z/{\Bbb{P}}^{n-1}}|_{X_i}\rightarrow T\rightarrow 0$$ $$(10)\quad 0\rightarrow N_{X_i/{\Bbb{P}}^{n-1}}(-E)\rightarrow N_{Z/{\Bbb{P}}^{n-1}}|_{X_i}\otimes {\cal O}_{X_i}(-E)\rightarrow T\otimes {\cal O}_{X_i}(-E)\rightarrow 0$$ $$(11)\quad 0\rightarrow N_{Z/{\Bbb{P}}^{ n-1}}|_{X_j}\otimes {\cal O}_{X_j}(-E)\rightarrow N_{Z/{\Bbb{P}}^{n-1}}\rightarrow N_{Z/{\Bbb{P}}^{n-1}}|_{X_i}\rightarrow 0.$$ By formula (3) $T\otimes{\cal O}_{X_i}(-E)={\cal O}_E(0-Q_j)$. Together with Lemma \ref{lem2.2} it follows from (10) that $$ h^j (N_{Z/{\Bbb{P}}^{n-1}}|_{X_i}\otimes {\cal O}_{X_i}(-E))=0 \mbox{ for all } j. $$ {}From sequence (9) and Lemma \ref{lem2.1} we obtain $$ h^0(N_{Z/{\Bbb{P}}^{n-1}}|_{X_i})=n^2,\quad h^j(N_{Z/{\Bbb{P}}^{n-1}}|_{X_i})=0 \mbox { for } j\ge 1. $$ The result now follows from sequence (11). \end{Proof} \begin{theorem}\label{theo2.4} (Rigidity) The component of the Hilbert scheme of surfaces containing $Z$ is smooth of dimension $n^2$ at [Z]. Every small deformation of $Z$ is again a union of two degree $n$ elliptic scrolls intersecting transversally along an elliptic normal curve. \end{theorem} \begin{Proof} The statement about smoothness and the dimension of the Hilbert scheme follows immediately from Proposition \ref{pro2.3}. Let $X(2,n)$ be the modular curve parametrizing elliptic curves with a level $n$ structure and a non-zero 2-torsion point. Every point of $X^0(2,n)=X(2,n)-\mbox{\{cusps\}}$ gives rise to a union $Z$ of two degree $n$ elliptic scrolls. Indeed the elliptic curves with level $n$ structure are in 1:1 correspondence with Heisenberg invariant elliptic normal curves in ${\Bbb{P}}^{n-1}$. Given a non-zero 2 torsion point we have exactly two other such points. We can use these two points to construct~$Z$. Conversely every point $Z$ arises in this way up to a change of coordinates. Let ${\cal H}$ be the component of the Hilbert scheme containing a given surface $Z$. Then we have a natural map $$ \Phi: X^0(2,n)\times \operatorname{PGL } (n,{\Bbb{C}})\rightarrow {\cal H}. $$ Since every elliptic normal curve has a finite automorphism group this map is finite and hence surjective in a neighbourhood of [Z]. \end{Proof} \begin{uremark} Of course there exist global deformations of $Z$ which are not a union of two degree $n$ elliptic scrolls. E.g. $Z$ can degenerate into non-reduced union of $n$-planes. For a discussion of possible degenerations in the case of ${\Bbb{P}}^4$, i.e. $n=5$ see \cite[section 9]{ADHPR}. \end{uremark} \section{Smoothing for $n$ even} The case $n$ even is subtly different from the case $n$ odd, as can already be seen in the computation of the cohomology of the normal bundle of the union of two scrolls. Again we fix an elliptic normal curve $E$ in ${\Bbb{P}}^{n-1}$ of degree $n$ and two non-zero 2-torsion points $P_i\neq P_j$ defining degree $n$ elliptic scrolls $X_i$ and $X_j$. Let $Z=X_i\cup X_j$. As before we find for the normal bundle of $X_i$ an exact sequence $$ 0\rightarrow K^{-1}_{X_i}\rightarrow N_{X_i/{\Bbb{P}}^{n-1}}\rightarrow Q\rightarrow 0 $$ with $Q|_f=(n-4){\cal O}_f(1)$. In this case, however, $h^0(K_{X_i}^{-1})=h^1(K_{X_i}^{-1})=1$. Nevertheless the arguments of Lemma \ref{lem2.2} still go through and give $$(12)\quad h^j(N_{X_i/{\Bbb{P}}^{n-1}}(-E))=0 \mbox{ for all } j.$$ We also have an exact sequence $$(13)\quad 0\rightarrow N_{X_i/{\Bbb{P}}^{n-1}}(-E)\rightarrow N_{X_i/{\Bbb{P}}^{n-1}}\rightarrow N_{X_i/{\Bbb{P}}^{n-1}}|_E\rightarrow 0.$$ Since $N_{X_i/{\Bbb{P}}^{n-1}}(-1)$ is globally generated we can conclude as in the proof of Lemma \ref{lem2.1} that $h^j(N_{X_i/{\Bbb{P}}^{n-1}}|_E)=0$ for $j\ge 1$. But then sequence (13) together with Riemann-Roch (numerically the cases $n$ odd and $n$ even behave in exactly the same way) gives $$ (14)\quad h^0( N_{X_i/{\Bbb{P}}^{n-1}})=n^2,\quad h^j (N_{X_i/{\Bbb{P}}^{n-1}})=0\mbox { for } j\ge 1 $$ which is as in the degree $n$ odd case. The main difference between the two cases lies in the fact that $N_{E/X_i}=N_{E/X_j}={\cal O}_E$ and hence $$(15)\quad T=N_{E/X_i} \otimes N_{E/X_j}={\cal O}_E.$$ It now follows from sequences (9) and (11) together with formula (14) that $h^1(N_{Z/{\Bbb{P}}^{n-1}}|_{X_i})=1$. Since it follows by sequence (10) and by (12) that $h^2(N_{Z/{\Bbb{P}}^{n-1}}|_{X_j}\otimes {\cal O}_{X_j}(-E))=0$ we find that $h^1(N_{Z/{\Bbb{P}}^{n-1}})>0$, contrary to the degree $n$ odd case. Moreover sequences (9)--(11) show that $h^0(N_{Z/{\Bbb{P}}^{n-1}})=n^2+1$ or $n^2+2$ and $h^1(N_{Z/{\Bbb{P}} ^{n-1}})=1$ or $2$ respectively. In fact we shall see later (Corollary \ref {cor3.3}) that $h^0(N_{Z/{\Bbb{P}}^{n-1}})=n^2+1$.\\ We now want to construct an explicit embedded smoothing of the singular surface $Z$ to a bielliptic surface. Since $\omega^2_Z={\cal O}_Z$ it is natural to look at bielliptic surfaces of type 1) or 2) in the Bagnera-de {}Franchis list \cite [List VI.20]{B}. It is easy to see by Reider's method (cf.\cite {Se}) that bielliptic surfaces of type 1) cannot be embedded in ${\Bbb{P}}^{n-1}$ for $n\le 8$. Hence we shall now turn our attention to bielliptic surfaces of type 2). Recall that these surfaces are of the form $S=E\times F/G$ where $G={\Bbb{Z}}_2\times {\Bbb{Z}}_2$ acts on $E$ by translation with 2-torsion points and on $F$ by $x\mapsto -x, x\mapsto x+\varepsilon, \varepsilon$ a 2-torsion point of $F$. We shall first show that these surfaces can be embedded as surfaces of degree $2n$ in ${\Bbb{P}}^{n-1}$. This will then give us the right idea for the construction of the degenerations. \begin{proposition}\label{pro3.1} Every bielliptic surface $S$ of type 2) can be embedded as a linearly normal surface of degree $2n$ in ${\Bbb{P}}^{n-1}(n\ge 6)$. \end{proposition} \begin{Proof} Let $\pi:E\times F\rightarrow S$ be the projection map and set $A=\pi(E), B=\pi(F)$. Then $A.B=4$. By \cite[Proposition 1.7]{Se} the element $B/2$ is in $\operatorname{NS}(S)$ and we can consider the divisor $$ H=A+\frac n 4 B. $$ (Since $n$ is even this is indeed a divisor on $S$). Then $H^2=2n, H.A=n$ and $H.B=4$. It is easy to check that $H$ is ample and Riemann-Roch together with Kodaira vanishing gives $h^0({\cal O}_S(H))=n$. It is a straightforward application of Reider's theorem to prove that $H$ is very ample. Hence the complete linear system defined by $H$ embeds $S$ as a linearly normal surface of degree $2n$ in ${\Bbb{P}}^{n-1}$. \end{Proof} \begin{uremark} The line bundle $\pi^*{\cal O}_S(H)$ has degree $n$ on $E$ and degree $4$ on $F$. \end{uremark} \begin{theorem}\label{theo3.2} (Smoothing) Let $Z=X_i\cup X_j$ be a union of two degree $n$ elliptic scrolls in ${\Bbb{P}}^{n-1}(n\ge 6$, even). Then there exists a flat family of surfaces $(Z_t)_{t\in T}$ in ${\Bbb{P}}^{n-1}$ such that $Z_0=Z$ and $Z_t$ for $t\neq 0$ is a linearly normal smooth bielliptic surface of degree $2n$. \end{theorem} \begin{Proof} We fix the elliptic curve $E=X_i\cap X_j$ and the two non-zero 2-torsion points $P_i$ and $P_j$ which define $X_i$ and $X_j$. Let $F={\Bbb{C}}/({\Bbb{Z}}+{\Bbb{Z}}\tau)$ be another elliptic curve which we consider variable. Let $S(4)\rightarrow X(4)$ be the Shioda modular surface of level 4. We consider the family ${\cal F}=(F_t)_{t\in T}$ where $t$ varies in some neighbourhood of a cusp of $X(4)$, say $i\infty$, where $t=e^{2\pi i {\tau}/4}$. Then $F_0$ is a 4-gon of rational curves and the 2-torsion points $Q_0=0, Q_1=1/2, Q_2=\tau/2$ and $Q_3=(1+\tau)/2$ of $F_t={\Bbb{C}}/({\Bbb{Z}}+{\Bbb{Z}}\tau)$ define 4 sections of ${\cal F}$ which intersect the singular fibre $F_0$ as indicated below \begin{figure}[h] $$ \unitlength1cm \begin{picture}(5,4.5) \put(0.9,0.7){\line(1,0){3.2}} \put(0.9,3.7){\line(1,0){3.2}} \put(1.0,0.6){\line(0,1){3.2}} \put(4.0,0.6){\line(0,1){3.2}} \put(0.865,1.6){$\times$} \put(0.865,2.6){$\times$} \put(3.865,1.6){$\times$} \put(3.865,2.6){$\times$} \put(1.9,3.615){$\times$} \put(2.9,3.615){$\times$} \put(1.9,0.615){$\times$} \put(2.9,0.615){$\times$} \put(1.8,0.9){$Q_0$} \put(2.8,0.9){$Q_1$} \put(1.8,3.3){$Q_2$} \put(2.8,3.3){$Q_3$} \put(0.45,1.55){${\hat Q}_1$} \put(0.45,2.55){${\hat Q}_0$} \put(4.15,1.55){${\hat Q}_3$} \put(4.15,2.55){${\hat Q}_2$} \put(0,2){$F^1$} \put(4.6,2){$F^3$} \put(2.3,0){$F^0$} \put(2.3,4.2){$F^2$} \end{picture} $$ \caption{Singular fibre of $\cal F$} \end{figure} \noindent The action of the 2-torsion points on smooth fibres extends to an action on~${\cal F}$. Similarly the involution $ \iota : x\mapsto-x$ on smooth fibres extends to an involution on ${\cal F}$. We denote the section of ${\cal F}$ given by $Q_2$ by $\varepsilon$. Then $\varepsilon$ acts on $F_0$ by rotation with $180^{\circ}$, i.e. it identifies $F^0$ and $F^2$, resp. $F^1$ and $F^3$. The involution $\iota$ interchanges $F^1$ and $F^3$, resp.\ induces involutions on $F^0$ and $F^2$ with fixed points $Q_0, Q_1$ and $Q_2, Q_3$.\\ We now consider the product ${\cal X}=E\times {\cal F}$ which is naturally fibred over $T$ with fibre $X_t=E\times F_t$. We define an action of $G={\Bbb{Z}}_2\times {\Bbb{Z}}_2$ on ${\cal X}$ as follows: The element $g_1=(1,0)$ acts on $E$ by $x\mapsto x + P_i$ and on $F$ by $x\mapsto x+\varepsilon$, whereas $g_2=(0,1)$ acts on $E$ by $x\mapsto x+P_j$ and on $F$ by $x\mapsto -x$. Then $g_1 g_2=(1,1)$ acts on $E$ by $x\mapsto x+(P_i+P_j)$ and on $F$ by $x\mapsto-x+\varepsilon$. The total space ${\cal F}$ and hence ${\cal X}$ is smooth and $G$ acts freely on ${\cal X}$. Then ${\cal Z}={\cal X}/G$ is smooth and $Z_t=X_t/G$ is a bielliptic surface of type 2) for $t\neq 0$. The singular surface $Z_0$ has the following properties: \begin{itemize} \item $Z_0$ consists of two components $Z_0^0$ and $Z_0^1$, namely the images of $E\times F^0$ (resp.\ $E\times F^2$) and $E\times F^1$ (resp.\ $E\times F^3$). \item The singular locus $E\times \operatorname{ Sing } F_0$ of $X_0$ is mapped to an irreducible curve isomorphic to $E$ (and again denoted by $E$). This curve $E=Z^0_0\cap Z^1_0$ is the singular locus of $Z_0$. \item $Z^0_0$ and $Z^1_0$ are ${\Bbb{P}}^1$ -- bundles over the elliptic curves $E/\langle P_i\rangle$, resp. $E/\langle P_j\rangle$ and the singular curve $E$ is a bisection of both ${\Bbb{P}}^1$ -- bundles. \item The curves $E\times \{Q_i\}$, $i=0,\ldots , 3$ are mapped to two disjoint sections $C^0_0$ and $C^0_1$ of $Z^0_0$ with $(C^0_0)^2=(C^0_1)^2=0$. \item Similarly we can consider the sections of ${\cal F}$ given by the points ${\hat Q}_i=Q_i+ {\tau}/4$. These sections again intersect $F_0$ in 4 points, which this time lie on $F^1$ and $F^3$ (see again figure 1). These curves $E\times \{{\hat Q}_i\}$ map to two sections $C^1_0$ and $C^1_1$ on $Z^1_0$ with $(C^1_0)^2=(C^1_1)^2=0$. \end{itemize} The next step is to construct a suitable line bundle ${\cal L}$ on ${\cal X}$ which descends to ${\cal Z}$. First consider the degree $n$ line bundle ${\cal L}_0={\cal O}_E(n0)$ on $E$. Then the group $G$, which operates on $E$ by translation with 2-torsion points leaves ${\cal L}_0$ fixed as a line bundle. However, if we want to lift the action of $G$ to the bundle ${\cal L}$ itself we might have to extend the group $G$ depending on the commutator of lifts $g^{{\cal L}_0}_1$ and $g^{{\cal L}_0}_ 2$ of $g_1$ and $g_2$ to ${\cal L}_0$. By general Heisenberg theory $$ \left [g^{{\cal L}_0}_1, g^{{\cal L}_0}_2\right ]=\left( e^{2\pi i/n}\right)^{n^2/4}=e^{2\pi i n/4} $$ which is either $1$ or $-1$ depending on whether $n\equiv 0\mod 4$ or not. Hence if ${n\equiv 0\mod 4}$ then the action of $G$ on $E$ lifts to an action on ${\cal L}_0$,whereas if $n\equiv 2\mod 4$ we have to extend $G$ to the level 2 Heisenberg group $H$ which is a central extension $$ 1\rightarrow \{\pm 1\}\rightarrow H\rightarrow G\rightarrow \{0\}. $$ Next we consider the sections $D_i$, resp.\ $\hat {D}_i$ of ${\cal F}$ given by the points $Q_i$, resp. ${\hat Q}_i$. Let ${\cal L}_1={\cal O}_{\cal F}(D_0+D_2+{\hat D}_0+{\hat D}_2)$. We claim that the action of $G$ on ${\cal F}$ lifts to~${\cal L}_1$, i.e. that $\left [ g_1^{{\cal L}_1}, g_2^{{\cal L}_2}\right ]=1$. It is enough to check this on a general fibre $F_t$ of~${\cal F}$. For this let $s$ be a section of ${\cal L}_1$ vanishing on $D_0+D_2+{\hat D}_0+{\hat D}_2$. Since this divisor is invariant under $G$ it follows that both $g_1^{{\cal L}_1}$ and $g_2^{{\cal L}_2}$ map $s_t=s|_{F_t}$ to a multiple of itself and hence commute. If $n\equiv 0\mod 4$ we can set ${\cal L}={\cal L}_0\boxtimes {\cal L}_1$. Then $G$ acts on ${\cal L}$ and since $G$ acts freely on ${\cal X}=E\times {\cal F}$ we obtain a line bundle ${\bar {\cal L}}={\cal L}/G$ on ${\cal Z}$. In the case $n\equiv 2\mod 4$ we must replace ${\cal L}$ by some suitable other line bundle. Let ${\cal M}_1={\cal O}_{\cal F}(D_0-D_1)$. Then ${\cal M}_1$ is invariant under $G$, but $G$ does not lift to an action on ${\cal M}_1$. In fact we claim that $\left [ g_1^{{\cal M}_1}, g_2^{{\cal M}_2}\right]=-1$. For this consider the function $$ f(\tau,z)=\frac{\vartheta(\tau,z+\frac 12 (\tau+1))}{\vartheta(\tau, z+\frac 12\tau)} $$ where $\vartheta(\tau, z) = \sum\limits_{n\in{\Bbb{Z}}} e^{2\pi i(\frac 12 n^2\tau+nz)}$ is the standard theta function. This is a meromorphic section of ${\cal M}_1$ at least for $t=e^{2\pi i/4}\neq 0$. The claim then follows from the identity $$ \frac{\vartheta(\tau,-z-\frac 12(\tau+1)+\frac{\tau}2)}{\vartheta(\tau,-z-\frac{\tau}2+\frac{\tau}2)}=- \frac{\vartheta(\tau,-z-\frac 12(\tau+1)-\frac{\tau}2)}{\vartheta(\tau,-z-\frac{\tau}2-\frac{\tau}2)}. $$ which follows immediately from \cite [pp. 49,50]{I}, formulae ($\Theta$1)-($\Theta$3). Now consider ${\cal L}={\cal L}_0 \boxtimes ({\cal L}_1\otimes{\cal M}_1)$. This time the action of $G$ on ${\cal X}$ lifts a priori to an action of $H$ on ${\cal L}$. But by construction the centre of $H$ acts by $(-1)^2=1$, i.e. trivially. Hence we obtain again an action of $G$ on ${\cal L}$ and we can take ${\bar{\cal L}}={\cal L}/G$.\\ It remains to verify that ${\bar{\cal L}}$ has the desired properties. Let ${\bar{\cal L}}_t= {\bar{\cal L}}|_{Z_t}$. For $t\neq 0$ by Proposition \ref {pro3.1} ${\bar{\cal L}}_t$ embeds $Z_t$ as a linearly normal bielliptic surface (which by construction is of type 2)\ ). We have to show that ${\bar{\cal L}}_0$ embeds $Z_0$ as the union of the two scrolls $X_i$ and $X_j$. Let ${\bar{\cal L}}_0^i= {\bar{\cal L}}_0|_{Z^i_0}$ for $i=0,1$. By construction ${\bar{\cal L}}_0^i$ has degree~$n/2$ on the sections $C_0^i$ and $C^i_1$, degree $1$ on the rulings and degree $n$ on the bisection $E$. Hence ${\bar{\cal L}}_0^i\equiv {\cal O}_{Z_0^i}(C^i_0+\frac n2 f)$. Thus $h^0( {\bar{\cal L}}_0^i)=n$ and the restriction map $H^0(Z^i_0, {\bar{\cal L}}_0^i)\rightarrow H^0(E, {\bar{\cal L}}_0^i|_E)$ is an isomorphism. In particular we find that $$ h^0(Z_0,{\bar{\cal L}}_0)=h^0(Z^0_0,{\bar{\cal L}}^0_0)+h^0(Z^1_0, {\bar{\cal L}}^1_0)-h^0(E,{\bar{\cal L}}_0|_E)=n=h^0(Z_t,{\bar{\cal L}}_t). $$ Moreover the restriction map $H^0(Z_0,{\bar{\cal L}}_0)\rightarrow H^0(Z^i_0,{\bar{\cal L}}^i_0)$ is an isomorphism and ${\bar{\cal L}}_0$ embeds each of the component $\bar{Z}^i_0$ as a degree $n$ elliptic scroll. By construction the image scrolls are the translation scrolls of the embedded elliptic normal curve $E$ defined by the 2-torsion points $P_i$ and $P_j$. This gives the claim. \end{Proof} \begin{uremark} The difference between the cases $n\equiv 0\mod 4$ and $n\equiv 0\mod 2$ is easily understood in terms of the geometry of bielliptic surfaces. In the first case $\frac n4B$ is an integer multiple of $B$ and hence effective. In the second case $B/2$ is an element of the Neron-Severi group of $S$, but is not effective. \end{uremark} \begin{cor}\label{cor3.3} If $n$ is even, then \begin{enumerate} \item[{\rm(i)}] $h^0(N_{Z/{\Bbb{P}}^{n-1}})=n^2+1$ (and hence $h^1(N_{Z/{\Bbb{P}}^{n-1}})=1 $), \item[{\rm(ii)}] the Hilbert scheme containing the surface $Z$ is smooth at $[Z]$ where it has dimension $n^2+1$. \end{enumerate} \end{cor} \begin{Proof} The bielliptic surfaces of type 2) define a component of the Hilbert scheme containing $Z$ which is of dimension at least $n^2+1$. Hence, if we can prove (i), then assertion (ii) is an immediate consequence. In view of our earlier computations it is, therefore, enough to exclude the case $h^0(N_{Z/{\Bbb{P}}^{n-1}})=n^2+2$. Consider the diagram $$ \begin{array}{rc} & 0{\ \ }\\ & \downarrow{\ }\\ & H^0(N_{X_i/{\Bbb{P}}^{n-1}})\\[1mm] & \downarrow^{\beta}\\[1mm] 0\rightarrow H^0(N_{Z/{\Bbb{P}}^{n-1}}|_{X_j}\otimes {\cal O}_{X_j}(-E))\rightarrow H^0(N_{Z/{\Bbb{P}}^{n-1}}) \stackrel{\alpha}{\rightarrow} & H^0(N_{Z/{\Bbb{P}}^{n-1}}|_{X_i})\\[1mm] & \downarrow{\ }\\[1mm] & H^0(T).{\ } \end{array} $$ It follows from (12) and sequence (10) that $h^0(N_{Z/{\Bbb{P}}^{n-1}}|_{X_j} \otimes{\cal O}_{X_j}(-E))=1$. Hence, if $h^0(N_{Z/{\Bbb{P}}^{n-1}})=n^2+2$ then, since $h^0(N_{Z/{\Bbb{P}}^{n-1}}|_{X_j})=n^2+1$ (by (14) and sequence (9)), the map $\alpha$ must be surjective. In particular im $(\alpha) \supset \mbox{ im } (\beta)$. On the other hand the sequence $$ 0\rightarrow \begin{array}[t]{c}N_{E/X_i}\\\Vert\\{\cal O}_E\end{array} \rightarrow N_{E/{\Bbb{P}}^{n-1}} \rightarrow N_{X_i/{\Bbb{P}}^{n-1}}|_E\rightarrow 0 $$ gives rise to a diagram $$ \begin{array}{cccccc} H^0(N_{E/X_i}) & \rightarrow H^0(N_{E/{\Bbb{P}}^{n-1}}) \rightarrow &H^0(N_{X_i/{\Bbb{P}}^{n-1}}|_E)& \rightarrow &H^1(N_{E/X_i})& \rightarrow 0.\\[1mm] ||& &\uparrow{\cong}& & || &\\[1mm] {\Bbb{C}}&&H^0(N_{X_i/{\Bbb{P}}^{n-1}}).&&{\Bbb{C}} \end{array} $$ Here the vertical isomorphism is a consequence of (12). In particular we can find a section $s\in H^0(N_{X_i/{\Bbb{P}}^{n-1}})$ which does not lift to $H^0(N_{E/{\Bbb{P}}^{n-1}})$. We claim that such a section cannot be in the image of $\alpha$. To see this, assume that there exists a section ${\tilde s}\in H^0(N_{Z/{\Bbb{P}}^{n-1}})$ with $\alpha({\tilde s})=\beta(s)$. Then $s$ and ${\tilde s}$ define first order deformations ${\cal X}_i$ and ${\cal Z}$ of $X_i$ and $Z$ over $\mbox{ Spec } ({\Bbb{C}}[\epsilon]/\epsilon^2)$ such that ${\cal X}_i \subset {\cal Z}$. A straightforward local calculation then shows that ${\cal Z}={\cal X}_i \cup {\cal X}_j$ where ${\cal X}_j$ is a first order deformation of $X_j$. Moreover ${\cal X}_i \cap {\cal X}_j={\cal E}$ is a first order deformation of $E$. In particular ${\cal X}_i$ contains a first order deformation of $E$ which contradicts our choice of the section s. This proves the claim. \end{Proof}

9703

alg-geom/9703015

en

https://arxiv.org/abs/alg-geom/9703015

alg-geom/9703015

Andrew Kresch

Andrew Kresch

Associativity relations in quantum cohomology

LaTeX2e, 22 pages

We describe interdependencies among the quantum cohomology associativity relations. We strengthen the first reconstruction theorem of Kontsevich and Manin by identifying a subcollection of the associativity relations which implies the full system of WDVV equations. This provides a tool for identifying non-geometric solutions to WDVV.

alg-geom

\section{Introduction} \label{intro} The geometry of moduli spaces of stable maps of genus 0 curves into a complex projective manifold $X$ leads to a system of quadratic equations in the tree-level (genus 0) {\em Gromov-Witten numbers} of $X$. These quadratic equations were written down by physicists before the geometry was worked out rigorously. In all the worked examples, the equations were seen to determine a solution, uniquely and consistently, from starting data. The beautiful paper of Di Francesco and Itzykson, \cite{a}, presents a number of examples in this context. In one of the foundational papers in the area of quantum cohomology, \cite{b}, the authors remark on the overdeterminedness of the system of equations, and pose the question whether the seemingly redundant equations follow algebraically from the useful ones. This same paper presents the {\em first reconstruction theorem}, which applies to manifolds $X$ such that $H^*(X,{\mathbb Q})$ is generated by $H^2(X)$. This result gives an effective procedure for solving for genus 0 Gromov-Witten numbers from starting data using the quadratic relations (since this entire paper is concerned only with the genus 0 invariants, we omit explicit mention of genus from now on). This paper continues in the spirit of these early investigations into the structure of these relations. After a review in which notation is introduced and the basic problem is set up (section \ref{review}), some results are presented (sections \ref{threesi} and \ref{fivesi}) explicitly showing interdependencies among these relations. Then follows a generalization of the first reconstruction theorem. In this, we are forced to keep the hypothesis that the cohomology ring of $X$ be generated by divisors. We then show that an initial collection of numbers and relations determines, purely algebraically, the entire system of relations ({\em strong reconstruction}) as well as the Gromov-Witten numbers. Finally, some examples are worked out. The structure of the associativity relations in the case $X$ has dimension 2 is particularly easy to describe. Section \ref{srttwo} gives an explicit picture of strong reconstruction in this case. Section \ref{examples} presents some higher-dimensional examples. Focusing on the algebra, rather than the geometry, of quantum cohomology leads to some generalizations. First, we can do away with $X$ entirely and work just with its cohomology ring, or more generally any {\em positively graded Gorenstein ${\mathbb Q}$-algebra} with socle in degree $n\ge2$ ($n$ plays the role of the dimension of $X$, and we also must assume that the zeroth graded piece is isomorphic to ${\mathbb Q}$). The canonical class $K_X$ plays an important role in geometry since it determines the expected dimension of the moduli spaces, but is less important to us: we can change $K_X$ at will, or drop it entirely from the discussion. Finally, the very equations lead us to solutions which do not come from geometry. Some of the examples exhibit that the geometric solution, with its family of obvious rescalings, may live in an even larger solution space. In an appendix (section \ref{geoground}), we sketch some of the geometry that the rest of the paper ignores. This is used to motivate the result in section \ref{fivesi}, although this result is purely algebraic. This is in no sense a substitute for a survey of quantum cohomology; we recommend \cite{c} as starting point to reader unacquainted with the subject. The author, who spent the 1996--97 year at the Mittag-Leffler Institute, would like to take this opportunity to thank the staff and organizers there for providing a comfortable and stimulating atmosphere for research. The author would also like to express his appreciation to P. Belorousski, C. Faber, W. Fulton, T. Graber, B. Kim, S. Kleiman, and R. Pandharipande for insightful conversation. \section{The basic problem} \label{review} Setting up and solving the system of quadratic equations in the Gromov-Witten numbers requires a ring which, in an appropriate sense, looks like a cohomology ring of a manifold. Precisely, let $A$ be a positively graded Gorenstein ${\mathbb Q}$-algebra with socle in degree $n\ge2$ such that the zeroth graded piece of $A$ is isomorphic to ${\mathbb Q}$, and assume a nonzero choice of $\varphi\in\Hom_A({\mathbb Q},A)\simeq{\mathbb Q}$ is made. We call the degree of an element its {\em codimension}, and denote the $k^{\rm th}$ graded piece by $A^k$. Then $\varphi$ is an isomorphism of ${\mathbb Q}$ with $A^n$. Denote by $\int$ the composite of projection with this isomorphism: $$A\to A^n\stackrel{\varphi^{-1}}\to{\mathbb Q}.$$ If $\{T_i\,|\,i\in I\}$ is a basis for $A$ as a ${\mathbb Q}$-module, where $I$ is a (necessarily finite) indexing set, let $g_{ij}=\int T_i\cdot T_j$. This is a nondegenerate pairing; we denote by $(g^{ij})$ the inverse matrix. We make the further assumption that $A^1$ is nonzero. In order to get a well-defined system of equations, we need two more pieces of data: a maximal-rank integral lattice $\Lambda$ and a strongly convex polyhedral cone $\Theta$, both contained in ${(A^1)}^*\otimes{\mathbb R}$. Let $B=\{T_1,T_{\sigma_1},\ldots,T_{\sigma_r}, T_{\tau_1},\ldots,T_{\tau_s}\}$ be a basis of $A$ as a ${\mathbb Q}$-module, consisting of homogeneous elements, with $T_1=1$, $T_{\sigma_i}\in A^1$ for each $i$, $\codim T_{\tau_j}\ge2$ for each $j$, and $\codim T_{\tau_j}\le\codim T_{\tau_k}$ whenever $j\le k$. This defines $r$ and $s$ as the ranks of the first and higher graded pieces of $A$, respectively; recall also that $n$ is the top grading, necessarily equal to $\codim T_{\tau_s}$. Assume the choice of basis is made so that $\Lambda$ is dual to the integral span of $T_{\sigma_1},\ldots,T_{\sigma_r}$. Geometry provides many examples of such data: $A^*=H^{2{*}}(X,{\mathbb Q})$, where $X$ is a complex projective manifold with homology only in even dimensions, $\int=\int_X$, $\Lambda=H_2(X,{\mathbb Z})/{\rm torsion}$, $\Theta=$ dual to ample cone. Let $C=\Theta\cap\Lambda\setminus\{0\}$. We define the set of unknown numbers to be the collection of all $N(\beta;d_1,\ldots,d_s)$ with $\beta\in C$ and $d_j\ge0$ for all $j$. For any $\beta\in C$, denote by $c_i$ the pairing of $\beta$ with $T_{\sigma_i}$. We now define a system of equations in these unknowns, one for each 4-tuple $(i,j,k,l)$ of elements of the basis indexing set $I=\{1,\sigma_1,\ldots,\sigma_r,\tau_1,\ldots,\tau_s\}$ and each {\em degree} $(\beta;d_1,\ldots,d_s)$. To do this, introduce the {\em potential function}, a formal Laurent series $$\Phi=\Phi^{\rm cl} + \Gamma$$ in variables $y_i$, $i\in I$, composed of the {\em classical part} $$\Phi^{\rm cl}=\frac{1}{6}\sum_{i,j,k\in I} \left(\int T_i\cdot T_j\cdot T_k\right) y_iy_jy_k$$ and the {\em quantum correction} $$\Gamma=\sum_{\fra{\scriptstyle\beta\in C}{\scriptstyle d_1,\ldots,d_s\ge0}} N(\beta;d_1,\ldots,d_s) e^{c_1y_{\sigma_1}}\cdots e^{c_ry_{\sigma_r}} \frac{y_{\tau_1}^{d_1}}{d_1!}\cdots\frac{y_{\tau_s}^{d_s}}{d_s!}.$$ Then any $(i,j,k,l)$ determines a differential equation \begin{equation} \label{assrel} \sum_{e,f}\Phi_{ije}g^{ef}\Phi_{fkl}= \sum_{e,f}\Phi_{jke}g^{ef}\Phi_{fil}, \end{equation} where we have used subscripts to denote partial differentiation. Isolating the coefficient of $e^{c_1y_{\sigma_1}}\cdots e^{c_ry_{\sigma_r}} y_{\tau_1}^{d_1}\cdots y_{\tau_s}^{d_s}$ on each side produces a quadratic equation in $N$'s, which we call an {\em associativity relation} (they imply associativity of the so-called quantum product; see \cite{c}). We follow Dubrovin in calling the system of equations (\ref{assrel}) the {\em WDVV equations} (after E. Witten, R. Dijkgraaf, H. Verlinde and E. Verlinde). A particular WDVV equation is represented symbolically by an equivalence of Feynman diagrams $$\feyn ijkl \ \sim\ \Bigl( \begin{picture}(40,10)(-20,17) \put(0,16){\line(0,1){8}} \put(0,16){\line(3,-1){10}} \put(0,16){\line(-3,-1){10}} \put(0,24){\line(3,1){10}} \put(0,24){\line(-3,1){10}} \put(12,8){\em k} \put(-17,8){\em j} \put(12,26){\em l} \put(-17,26){\em i} \end{picture}\Bigr)$$ We adopt the notation $\feyn ijkl$ to refer to the WDVV equation (\ref{assrel}) and $\dfeyn ijkl{(\beta;d_1,\ldots,d_s)}$ to refer to a particular associativity relation. We would like to rewrite (\ref{assrel}) in which we split $\Phi$ into its classical and quantum parts. If $T_i\cdot T_j=\sum_h q_h T_h$ then denote $\sum_h q_h\Gamma_{hkl}$ by $\Gamma_{(ij)kl}$. Then by rewriting (\ref{assrel}) we get \begin{equation} \label{altassrel} \Gamma_{ij(kl)}+\Gamma_{(ij)kl}-\Gamma_{jk(il)}-\Gamma_{(jk)il} = \sum_{e,f}\Gamma_{jke}g^{ef}\Gamma_{fil}- \sum_{e,f}\Gamma_{ije}g^{ef}\Gamma_{fkl}. \end{equation} Thus we have split the WDVV equation into the {\em linear contribution} (left-hand side) and {\em quadratic contribution} (right-hand side). We emphasize that $\dfeyn ijkl{(\beta;d_1,\ldots,d_s)}$ is an equation; it can be scaled or combined linearly with other equations, and it can imply other equations. We will sometimes need to distinguish the linear and quadratic contributions to a particular associativity relation, each being the polynomial appearing on the appropriate side of (\ref{altassrel}). With this set-up, we can now state \begin{pb} \label{mainprob} Given $A$, $\int$, $\Lambda$, $\Theta$ as above, find solutions in rational numbers $N(\beta;d_1,\ldots,d_s)$ to the full set of WDVV equations (\ref{assrel}). \end{pb} For the purposes of ordering elements of $C$, we let $\omega$ be an element of the interior of the dual cone to $\Theta$ (which must be nonempty since $\Theta$ is strongly convex). Then $\langle\beta,\omega\rangle > 0$ for all $\beta\in C$, and for each $k$, the set $\{\beta\in{\mathbb Z}\langle\sigma^*_1,\ldots,\sigma^*_r\rangle\,|\, \langle\beta,\omega\rangle < k\}$ is finite. We must remark on four distinct uses of the Feynman symbols. First, with $i,j,k,l\in I$, $\feyn ijkl$ is as above. Second, for $\xi, \pi, \rho, \sigma\in A$, $\feyn\xi\pi\rho\sigma$ refers to the equation obtained by writing each element in terms of the basis and summing in a multilinear fashion. Third, for subsets $\Xi, \Pi, {\rm P}, \Sigma$ of $A$, $\feyn \Xi\Pi{\rm P}\Sigma$ refers to the collection of all $\feyn\xi\pi\rho\sigma$ with $\xi$ in $\Xi$, etc. Finally, as a special case, for integers $w, x, y, z$, $\feyn wxyz$ refers to $\feyn {A^w}{A^x}{A^y}{A^z}$. Also, if $i,j,k,l,m\in I$, we write $\feyn{ij}klm$ as shorthand for $\feyn{T_i\cdot T_j}{T_k}{T_l}{T_m}$. \section{How many relations?} \label{howmany} As we have seen, to each choice of $i,j,k,l\in I$ there corresponds a WDVV equation. Thus, if the vector space $A$ has dimension $r$ then the number of distinct WDVV equations is of the order of magnitude $r^4$. It is an exercise in combinatorics to provide a precise count. First, if any of $i$, $j$, $k$, $l$ is 1 (i.e., indexes the identity element of $A$), then the corresponding WDVV equation is a trivial identity. If $k=i$ or $l=j$ then the equation is trivial. There are symmetries. If we swap $i$ and $j$ and swap $k$ and $l$ then the WDVV equation remains unchanged. Swapping two symbols on a diagonal of the Feynman diagram only changes the WDVV equation by a sign. If $i$, $j$, $k$, and $l$ are all distinct, then the three relations one obtains by cyclically permuting $i$, $j$, $k$ are linearly related: \begin{equation} \label{twooutofthree} \feyn ijkl + \feyn jkil + \feyn kijl = 0. \end{equation} A tally shows that the number of distinct nontrivial WDVV equations (modulo sign) is $$\frac{r^4-6r^3+15r^2-18r+8}{8},$$ while if we count only two out of the three distinct WDVV equations involving four distinct symbols --- since by (\ref{twooutofthree}) any two imply the third, a fact we refer to as the {\em two-out-of-three} implication --- the count is $$\frac{r^4-4r^3+5r^2-2r}{12}.$$ Since there is only (at most) one unknown number in each degree, the count above is, na\"{\i}vely, the factor of overdeterminedness in the system of associativity relations. \section{The three symbols relation} \label{threesi} Let $i,j,k,l,m\in I$. Let $\Phi=\Phi^{\rm cl} + \Gamma$ be the potential function. The following algebraic identity, called the {\em three symbols identity}, holds: \begin{eqnarray} \lefteqn{\frac{\partial}{\partial y_m} \Bigl(\sum_{e,f} \Phi_{ije}g^{ef}\Phi_{fkl} - \sum_{e,f} \Phi_{jke}g^{ef}\Phi_{fil}\Bigr) + {}} \nonumber \\ & & \frac{\partial}{\partial y_j} \Bigl(\sum_{e,f} \Phi_{ile}g^{ef}\Phi_{fkm} - \sum_{e,f} \Phi_{lke}g^{ef}\Phi_{fim}\Bigr) + {} \label{threesymbol}\\ & & \frac{\partial}{\partial y_l} \Bigl(\sum_{e,f} \Phi_{ime}g^{ef}\Phi_{fkj} - \sum_{e,f} \Phi_{mke}g^{ef}\Phi_{fij}\Bigr) = 0. \nonumber \end{eqnarray} Let $(\beta,d)$ be a degree. Define $e_{\sigma_i}=0$ and $e_{\tau_j}=(0,\ldots,1,\ldots,0)$ with the 1 in the $j^{\rm th}$ place. Then (\ref{threesymbol}) gives us \begin{pr} \label{threesymrel} Suppose $i,j,k,l,m\in I$ with $\codim T_m\ge2$, and let $(\beta,d)$ be a degree with $d_m\ge1$. Then relations $\dfeyn ilkm{(\beta;d+e_j-e_m)}$ and $\dfeyn imkj{(\beta;d+e_l-e_m)}$ together imply $\dfeyn ijkl{(\beta;d)}$. \end{pr} This we call the {\em three symbols relation} (\tsr), and denote by the diagram $\dthree ikjlm{(\beta;d)}$. We now record one application of \tsr. \begin{lm} \label{lemma} With the notation of Problem \ref{mainprob}, suppose $(\beta;d)$ is a degree with $d\ne0$. Then the collection of all relations in degrees $(\beta;d')$ with $\sum_id'_i=\left(\sum_id_i\right)-1$ implies $\dfeyn A{A^1}A{A^1} {(\beta;d)}$. \end{lm} Indeed, if $d_m\ge1$ then $\three ij11m$ yields $\dfeyn i1j1{(\beta;d)}$ for any $i,j\in I$. \section{The five symbols relation} \label{fivesi} Given $i,j,k,l,m\in I$, the following algebraic identity holds: \begin{equation} \label{mdiag} \sum_{e,f} \Gamma_{ij(me)}g^{ef}\Gamma_{fkl} = \sum_{e,f} \Gamma_{kl(me)}g^{ef}\Gamma_{fij}. \end{equation} Recall, if $T_m\cdot T_e=\sum_q t_q T_q$ then by $\Gamma_{ij(me)}$ we mean $\sum_q t_q\Gamma_{ijq}$. Now (\ref{mdiag}) follows by observing that with $g_{abc}=\int T_a\cdot T_b\cdot T_c$ we have $t_q=\sum_p g_{mep} g^{pq}$, and now the coefficient $\sum_{e,p} g^{ef}g_{mep}g^{pq}$ of $\Gamma_{ijq}\Gamma_{fkl}$ on the left-hand side is symmetric in $f$ and $q$. The appendix gives geometric motivation for (\ref{mdiag}). We write the expression $\sum \Gamma_{ij(me)}g^{ef}\Gamma_{fkl} - \sum \Gamma_{kl(me)}g^{ef}\Gamma_{fij}$ and add to it the four additional expressions obtained by permuting the variables $i,j,k,l,m$ cyclically. We use the identity coming from the associativity relation $\feyn emij$ and its cyclic translates to obtain \begin{eqnarray*} \lefteqn{0 = \Gamma_{ij(me)}\Gamma_{fkl} + \Gamma_{jk(ie)}\Gamma_{flm} + \Gamma_{kl(je)}\Gamma_{fmi} + \Gamma_{lm(ke)}\Gamma_{fij} + \Gamma_{mi(le)}\Gamma_{fjk} } \\ & & {} - \Gamma_{mi(je)}\Gamma_{fkl} - \Gamma_{ij(ke)}\Gamma_{flm} - \Gamma_{jk(le)}\Gamma_{fmi} - \Gamma_{kl(me)}\Gamma_{fij} - \Gamma_{lm(ie)}\Gamma_{fjk} \\ \lefteqn{\phantom0 = \Gamma_{(mi)je}\Gamma_{fkl} + \Gamma_{(ij)ke}\Gamma_{flm} + \Gamma_{(jk)le}\Gamma_{fmi} + \Gamma_{(kl)me}\Gamma_{fij} + \Gamma_{(lm)ie}\Gamma_{fjk} } \\ & & {} - \Gamma_{(ij)me}\Gamma_{fkl} - \Gamma_{(jk)ie}\Gamma_{flm} - \Gamma_{(kl)je}\Gamma_{fmi} - \Gamma_{(lm)ke}\Gamma_{fij} - \Gamma_{(mi)le}\Gamma_{fjk}. \end{eqnarray*} We have omitted summations symbols and $g^{ef}$'s to save space. We have also omitted the (cubic) terms obtained by substituting the quadratic contributions of the associativity relations, but the key point is that these cancel. The final expression above is the quadratic contribution of a sum of associativity relations, conveniently written \begin{equation} \label{fivearound} \feyn{mi}jkl - \feyn m{ij}kl + \feyn mi{jk}l - \feyn mij{kl} + \feyn{lm}ijk. \end{equation} Since the linear contribution of (\ref{fivearound}) vanishes, as may be checked, we have, at least formally, that the indicated associativity relations imply the vanishing of (\ref{fivearound}). We turn this into a precise, practical statement by grading the terms in $\Gamma$ by degree. We use the notation of Problem \ref{mainprob}. Rewrite the above, isolate the coefficient of some degree $(\beta;d)$, and note that then every quadratic term is a sum over $\beta_1+\beta_2=\beta$ with $\langle \beta_i,\omega \rangle < \langle \beta,\omega \rangle$ for $i=1,2$. This establishes \begin{pr} \label{fivesymrel} Suppose $i,j,k,l,m\in I$, and let $(\beta;d)$ be a degree. The collection of relations consisting of $\dfeyn ijke{(\beta';d')}$ and its cyclic translates through $\{i,j,k,l,m\},$ for all $e\in I$ and all degrees $(\beta',d')$ with $\langle \beta',\omega\rangle < \langle \beta,\omega\rangle$ and $d'\le d$ (componentwise), implies the relation \begin{eqnarray*} \lefteqn{\dfeyn{mi}jkl{(\beta;d)} - \dfeyn m{ij}kl{(\beta;d)} + \dfeyn mi{jk}l{(\beta;d)}}\hspace{100pt} \\ & & {} - \dfeyn mij{kl}{(\beta;d)} + \dfeyn{lm}ijk{(\beta;d)}. \end{eqnarray*} \end{pr} We call this the {\em five symbols relation} (\fsr). We employ the notation $\dfiveard mijkl{(\beta;d)}$ to describe the above relation. \section{Strong reconstruction for $n=2$} \label{srttwo} To illustrate an application of the three and five symbols relations, we work out a strong reconstruction theorem for $n=2$, where the associativity relations are simple to organize. When $n=2$, there is only one $\tau$, and there are three types of associativity relations: \begin{enumerate} \item[(i)] $\feyn \tau \tau {\sigma_j} {\sigma_i}$; \item[(ii)] $\feyn \tau {\sigma_i} {\sigma_j} {\sigma_k}$; \item[(iii)] $\feyn {\sigma_i} {\sigma_j} {\sigma_k} {\sigma_l}$. \end{enumerate} Assume for simplicity that, as in the geometric situation, there is a canonical class $K$ which dictates that there is at most one nontrivial number $N(\beta;d)$ in each curve class $\beta$, namely when $d=\langle \beta,-K \rangle - 1 \ge 0$ (for the general case, see the next section). The potential function is composed of \begin{eqnarray*} \Phi^{\rm cl} &=& \frac{1}{2} y_1^2 y_\tau + \frac{1}{2} \sum_{e,f=1}^r g_{ef} y_1 y_{\sigma_e} y_{\sigma_f}, \\ \Gamma &=& \sum_{\fra{\scriptstyle\langle\beta,-K\rangle\ge1} {\fra{\scriptstyle\beta=\sum c_i\sigma^*_i\in C} {\scriptstyle d=\langle \beta,-K \rangle - 1}}} N(\beta;d)e^{c_1 y_{\sigma_1}}\cdots e^{c_r y_{\sigma_r}} \frac{y_\tau^d}{d!} \end{eqnarray*} (we write $g_{ef}$ for $g_{\sigma_e\sigma_f}$). Suppose we are given all $N(\beta;d)$ with $d\le2$, and suppose that these satisfy $$\dfeyn \tau {\sigma_i} {\sigma_j} {\sigma_k} {(\beta;0)}{\rm\ and\ }\dfeyn {\sigma_i} {\sigma_j} {\sigma_k} {\sigma_l} {(\beta;0)}$$ for all $i,j,k,l\in\{1,\ldots,r\}$ and all $\beta$. We claim that relations of type (i) allow us to solve for all further $N(\beta;d)$ ({\em reconstruction}) and that the numbers thus obtained satisfy the full system of WDVV equations ({\em strong reconstruction}). Indeed, by the three symbols relation, $$ \dfeyn \tau {\sigma_j} {\sigma_i} {\sigma_l} {(\beta;0)} {\rm\,and\,\ } \dfeyn \tau {\sigma_j} {\sigma_k} {\sigma_l} {(\beta;0)} \,\Rightarrow\,\dfeyn {\sigma_i} {\sigma_j} {\sigma_k} {\sigma_l} {(\beta;1)}, $$ and thus the hypothesis implies relations (ii) $(\beta;1)$. Similarly, relations (i) $(\beta;d)$ imply (ii) $(\beta;d+1)$ and (iii) $(\beta;d+2)$. Inductively on $d$, assume all $N(\beta;d')$ known and all relations satisfied for $\langle \beta, -K \rangle < d+4$. Relation $\dfeyn \tau \tau {\sigma_j} {\sigma_i} {(\beta;d)}$ reads $g_{ij}\Gamma_{zzz}^{(\beta;d)}=Q_{ij}^{(\beta;d)}$, where $Q_{ij}^{(\beta;d)}$ is a quadratic expression in known quantities. Now $\fivebrd\tau {\sigma_k} {\sigma_l} {\sigma_j} {\sigma_i}$ tells us $g_{kl}Q_{ij}^{(\beta;d)} = g_{ij}Q_{kl}^{(\beta;d)}$, which says we can solve for $\Gamma_{zzz}^{(\beta;d)}$ (that is, $N(\beta;d+3)$) satisfying (i), and we have just seen that the relations of type (i) imply all the relations in degree $\beta$. \section{Strong reconstruction theorem} \label{srt} The main result here is a generalization of the first reconstruction theorem of Kontsevich and Manin \cite{b}. Working in the same class of cohomology rings, namely those generated by elements in codimension 1, we prove that an identifyable collection of numbers satisfying an identifyable collection of relations gives us strong reconstruction, i.e.\ allows us to solve uniquely for numbers satisfying the complete system of associativity relations. In case $-K$ is ample, we need only a finite collection of numbers and relations as starting data. The result is \begin{tm} \label{strongr} With the notation of Problem \ref{mainprob}, suppose $A$ is generated by $A^1$. Then the collection of $N(\beta;d)$ with $\sum_{i=1}^s d_i\le2$ extends to a solution to WDVV if and only if $\dfeyn A{A^1}A{A^1} {(\beta;0)}$ is satisfied for all $\beta$. \end{tm} We begin by organizing notation. By hypothesis, we may assume the basis $B$ chosen such that for each $j$, $1\le j\le s$, there exists $i_j\in\{1,\ldots,r\}$ and $\mu_j\in I$ such that $T_{\tau_j}=T_{\sigma_{i_j}}\cdot T_{\mu_j}$. We wish to impose a partial order on the collection of degrees $d=(d_1,\ldots,d_s)$ with fixed $|d|:=\sum_{i=1}^s d_s$, such that $(d_1,\ldots,d_s)$ precedes $(d_1,\ldots,d_i+1,\ldots,d_j-1,\ldots,d_s)$ for any $i < j$. A convenient way is to order by $\sum id_i$. Let us give an outline of the proof of the theorem. Inducting on $\langle \beta, \omega \rangle$, then on $|d|$, then downwards on $\sum id_i$, we verify all associativity relations in degree $(\beta;d)$, showing that those of the form $$\dfeyn {\mu_j}{\sigma_{i_j}}{\tau_k}{\tau_l}{(\beta;d)}$$ with $\codim T_{\tau_j}\le\codim T_{\tau_k}\le\codim T_{\tau_l}$ and $\max(j,k,l)\le\min\{m\,|\,d_m\ne0\}$ determine the numbers $N(\beta;d+e_j+e_k+e_l)$ (here $e_i=(0,\ldots,1,\ldots,0)$ with 1 in the $i^{\rm th}$ position). \section{Proof of the strong reconstruction theorem} \label{srtproof} The induction breaks up into an outer induction on degrees and an inner induction within each degree. The outer induction proceeds with respect to the partial order: $(\beta',d')\prec(\beta,d)$ if $\langle \beta',\omega\rangle < \langle \beta,\omega\rangle$ and $|d'|\le|d|$; $\beta'=\beta$ and $|d'|<|d|$; or $\beta'=\beta$, $|d'|=|d|$, and $\sum i d'_i > \sum i d_i$. The inner induction is on $(u,c,a,b)$ with $u$ (corresponding to $\codim T_{\mu_j}$ above) up from 1, $c$ ($=\codim T_{\tau_k}+\codim T_{\tau_l}$) up from $2(u+1)$, and $a$ ($=\codim T_{\tau_k}$) up from $u+1$ to $[c/2]$. Define $b=c-a$; then we always have $a\le b$. The induction hypothesis, at a given step $(\beta,d,u,c,a,b)$, consists of all relations in previous degrees plus all numbers they refer to (i.e., all $N(\beta';d'+e')$ with $(\beta',d')\prec(\beta,d)$, $|e'|\le3$), plus, in the current degree, all $\feyn z1xy$ with $\min(x,y,z)<u$, all $\feyn u1xy$ and $\feyn x1yu$ with $x\ge u+1$, $y\ge u+1$, and either $x+y<c$ or $x+y=c$ with $\min(x,y)<a$, together with all numbers these relations refer to. In any degree, for any integers $x$ and $y$, $\feyn x1y1$ follows either by hypothesis ($d=0$) or by the induction hypothesis and Lemma \ref{lemma} ($|d|\ge1$). When $u\ge2$ we obtain $\feyn x1yu$ for $x\ge u$ and $y\ge u$ from $\fivedlu x1y{u{-}1}1$, and now $\feyn u1ux$ for $x>u$ from $\fivearu {u{-}1}11ux$. The main step is to deduce $\feyn u1ab$. Here the linear terms coming from the associativity relation possibly involve new $N$'s. We divide this into two steps. First, we show it suffices to prove a distinguished set of $\feyn u1ab$. Let $S$ be the set of relations $\feyn {\mu_j}{\sigma_{i_j}}{\tau_k}{\tau_l}$ with $\codim T_{\tau_j}=u+1$, $\codim T_{\tau_k}=a$, and $\codim T_{\tau_l}=b$. We claim that $S$ (and the new $N$'s referred to) implies $\feyn u1ab$. Indeed, if $\codim T_\mu=u$ and $\codim T_\sigma=1$ with $T_\sigma\cdot T_\mu=\sum \lambda_j T_{\tau_j}$, then comparing $\fiveclu \mu\sigma{\sigma_{i_k}}{\mu_k}{\tau_l}$ with $\sum \lambda_j \fiveclu {\mu_j}{\sigma_{i_j}}{\sigma_{i_k}}{\mu_k}{\tau_l}$ establishes $\feyn \mu\sigma{\tau_k}{\tau_l}$ from the relations in $S$. For the second step, we establish all relations in $S$. Each $\feyn {\mu_j}{\sigma_{i_j}}{\tau_k}{\tau_l}$ in $S$ involves the variable $N(\beta;d+e_j+e_k+e_l)$. For $a$, $b$, $u+1$ distinct, there is a one-to-one correspondence between elements of $S$ and such variables. In other cases, we shall need symmetrizing arguments to show any two relations in $S$ sharing a common such variable are equivalent. In case $a=b$, the two-out-of-three implication gives $\feyn {\mu_j}{\sigma_{i_j}}{\tau_k}{\tau_l}\Leftrightarrow \feyn {\mu_j}{\sigma_{i_j}}{\tau_l}{\tau_k}$. In case $a=u+1$, we get $\feyn {\mu_j}{\sigma_{i_j}}{\tau_k}{\tau_l}\Leftrightarrow \feyn {\sigma_{i_j}}{\mu_j}{\tau_k}{\tau_l}\Leftrightarrow \feyn {\mu_k}{\sigma_{i_k}}{\tau_j}{\tau_l}$ by two-out-of-three and $\fivecru {\sigma_{i_j}}{\mu_j}{\sigma_{i_k}}{\mu_k}{\tau_l}$. Thus, it suffices to establish only those $\feyn{\mu_j}{\sigma_{i_j}}{\tau_k}{\tau_l}\in S$ such that $j\le k\le l$. In case $d_m\ge1$ for some $m<l$, $\three {\mu_j}{\tau_k}{\sigma_{i_j}}{\tau_l}{\tau_m}$ establishes $\feyn{\mu_j}{\sigma_{i_j}}{\tau_k}{\tau_l}$. Otherwise, $N(\beta;d+e_j+e_k+e_l)$ is actually an unknown, so solving $\feyn{\mu_j}{\sigma_{i_j}}{\tau_k}{\tau_l}$ establishes simultaneously the number and the relation. Finally, two-out-of-three establishes $\feyn u1ba$ from $\feyn u1ab$. Having finished the inner induction, to establish general $\feyn wxyz$ is an easy induction on $\min(w,x,y,z)$, using \fsr{} by decomposing the entry of lowest codimension. \section{Examples} \label{examples} Since we wish not to stray far from geometry, we focus mainly on rings of the form $A=A^*_{\mathbb Q} X:=A^*X\otimes{\mathbb Q}$, where $X$ is a projective manifold, and with a dimension restriction on numbers coming from a class $K\in A^1$: $N(\beta;d)=0$ unless $\sum_j d_j (\codim T_{\tau_j}-1)=\langle\beta, -K\rangle + n - 3,$ where $n=\dim X$. Unless otherwise stated, $K=K_X$, the canonical class on $X$. {\bf Example 1.} $X={\mathbb P}^n$. The ring $A=A^*_{\mathbb Q}{\mathbb P}^n$ has $\rk A^1=1$, so strong reconstruction dictates a vacuous set of relations (a Feynman diagram with a symbol appearing twice, in opposite corners, determines a trivial relation), and the solutions to WDVV correspond exactly to choices of $N(\beta;d)$ with $|d|\le2$. With $K=K_X$, there is only one such $N$, so there are no solutions beyond the geometric solution and its rescalings. Let $h=c_1({\mathcal O}(1))$. When $K=-bh$ with $1\le b\le n-1$, there is more than one $N(\beta;d)$ with $|d|\le2$, and hence there is a family of solutions to WDVV of dimension $>1$. Some of these solutions have geometric significance. If $V\stackrel{i}\hookrightarrow{\mathbb P}^{n+k}$ is a $(s_1,\ldots,s_k)$-complete intersection with $b=n+k+1-\sum s_j\ge 1$, and $K=-bh$, then $K_V=i^*K$, and we get a solution to WDVV for $A=A^*_{\mathbb Q}{\mathbb P}^n$ with this ``wrong'' canonical class by setting $N(\beta;d)$ equal to the sum over ${\beta'}\in H_2V$ with $i_*{\beta'}=\beta$ of the (geometric) Gromov-Witten number for $V$ corresponding to $\bigotimes (i^*T_{\tau_j})^{\otimes d_j}$ in curve class $\beta'$ (cf.\ \cite{g}, sec.\ 4). {\bf Example 2.} A toric manifold. In the integral lattice of rank 3, let $v_1=(1,0,0)$, $v_2=(0,1,0)$, $v_3=(0,0,1)$, $v_4=(-2,-1,-1)$, $v_5=(-1,0,0)$, and let the toric manifold $X$ be defined by the fan consisting of the cones $\langle v_1,v_2,v_3\rangle $, $\langle v_1,v_2,v_4\rangle$, $\langle v_1,v_3,v_4\rangle$, $\langle v_2,v_3,v_5\rangle$, $\langle v_2,v_4,v_5\rangle$, $\langle v_3,v_4,v_5\rangle$. If $D_i$ denotes the divisor corresponding to vector $v_i$, then $D_1$, $D_2$ span $A^1X$ with the dual to the ample cone generated by the dual basis elements $D^*_1$, $D^*_2$. We have $A=A_{\mathbb Q}^*X={\mathbb Q}[D_1,D_2]/(D_1^2-2D_1D_2, D_2^3)$ with ${\mathbb Q}$-basis $\{ 1, D_1, D_2, D_1 D_2, D_2^2, D_1D_2^2 \}$ and $-K=2 D_1 + D_2$. Given this setup, the strong reconstruction theorem dictates 21 relations involving 17 variables. When written out, this system of equations reduces to \begin{eqnarray*} N(D^*_1;2,0,0) &\hskip-4pt=\hskip-4pt& N(D^*_1;0,0,1); \\ N(D^*_1+D^*_2;0,1,1) &\hskip-4pt=\hskip-4pt& - N(D^*_1;0,0,1) N(D^*_2;0,1,0); \\ 0 &\hskip-4pt=\hskip-4pt& N(D^*_1;0,0,1)\left[N(2D^*_2;0,2,0)+N(D^*_2;0,1,0)^2\right] \end{eqnarray*} (with the rest of the variables all zero). Thus the solution space to WDVV has two irreducible components: one, containing the geometric solution, with its expected two-dimensional family of rescalings, and another, supported in curve classes lying along one edge of the dual to the ample cone, with a one-dimensional family of rescalings plus dependence on another free parameter. {\bf Example 3.} $X=G(2,4)$, $X=\Sym^2{\mathbb P}^2$. Both have isomorphic cohomology rings (up to scale factors), not generated by divisors, so we are outside the scope of the strong reconstruction theorem. We illustrate here a technique which allows one to prove, subject to a genericity hypothesis on starting data, a strong reconstruction result in this case. The starting data according to the statement of strong reconstruction consists of just one number when $X=G(2,4)$ and 3 numbers when $X=\Sym^2{\mathbb P}^2$. The result we seek is that modulo a genericity hypothesis (some quantity in the starting data being nonzero) any choice of starting data extends to a solution to WDVV. It is helpful to recall the dimension condition on relations. For $\dfeyn \xi\pi\rho\sigma{(\beta;d)}$ to be nontrivial requires \begin{eqnarray*} \lefteqn{\langle\beta, -K\rangle - \sum_{j=1}^s d_j(\codim T_{\tau_j}-1) = } \hspace{80pt} \\ & & \codim \xi + \codim \pi + \codim \rho + \codim \sigma - n. \end{eqnarray*} One may take as cohomology basis the powers of the ample generator $h$ of $A^1X$, plus an extra codimension 2 element, chosen orthogonal to $h$. So $B=\{ 1,h,c,h^2,h^3,h^4 \}$ with $c\cdot h=0$, $\int h^4\ne 0$, $\int c^2\ne 0$. We have $K=-4h$, $K=-3h$ in the cases of the two respective varieties; set $\kappa=4$, $\kappa=3$ accordingly. The relations in curve class $\beta$ never involve the number $N(\beta; \kappa\beta+1,0,0,0)$. We recover this exceptional number from a particular degree $\beta+1$ relation in which it appears in a quadratic term, for which, to be able to solve, we must add the hypothesis that $N(1;0,0,1,1)\ne0$ (resp.\ $N(1;1,0,0,1)\ne0$) when $\kappa=4$ (resp.\ $\kappa=3$). For each $\beta$, there are 10 numbers $N(\beta;d)$ which are not of the form $N(\beta;t,u,v,w)$ with $u+v+w\ge3$; these are the numbers unreachable by the proof of strong reconstruction applied to the subring of $A$ generated by $h$. We must show how to solve for 9 of these (all except $N(\beta; \kappa\beta+1,0,0,0)$) as well as the leftover degree $\beta-1$ number. Table \ref{gtwofour} outlines how to do this. \begin{table} \begin{center} \begin{tabular}{lcclr} \smallskip (a) & $N(\beta; \kappa\beta-5,0,0,2)$ & by & $\dfeyn {h^4}h{h^3}c{(\beta;\kappa\beta-6,0,0,0)}$ & $(\beta\ge2)$ \\ \smallskip (b) & $\fra{\displaystyle N(\beta-1;\hfill} {\displaystyle\ \kappa\beta-\kappa+1,0,0,0)}$ & by & $\dfeyn ch{h^4}c{(\beta;\kappa\beta-5,0,0,0)}$ & $(\beta\ge2)$ \\ \smallskip (c) & $\fra{\displaystyle N(\beta; t,u,v,w)\hfill} {\displaystyle{\rm with\ }u+v+w\ge3}$ & by & $\dfeyn {\langle h \rangle}{\langle h \rangle}{\langle h \rangle} {\langle h \rangle}{(\beta;d)}$ & \\ \smallskip (d) & $N(\beta; \kappa\beta-4,0,1,1)$ & by & $\dfeyn {h^4}h{h^2}c{(\beta;\kappa\beta-5,0,0,0)}$ & $(\beta\ge2)$ \\ \smallskip (e) & $N(\beta; \kappa\beta-3,1,0,1)$ & by & $\dfeyn {h^4}hhc{(\beta;\kappa\beta-4,0,0,0)}$ & $(\kappa+\beta\ge5)$ \\ \smallskip (f) & $N(\beta; \kappa\beta-3,0,2,0)$ & by & $\dfeyn {h^3}h{h^2}c{(\beta;\kappa\beta-4,0,0,0)}$ & $(\kappa+\beta\ge5)$ \\ \smallskip (g) & $N(\beta; \kappa\beta-2,0,0,1)$ & by & $\dfeyn {h^3}hcc{(\beta;\kappa\beta-4,0,0,0)}$ & $(\kappa+\beta\ge5)$ \\ \smallskip (h) & $N(\beta; \kappa\beta-2,1,1,0)$ & by & $\dfeyn {h^3}hhc{(\beta;\kappa\beta-3,0,0,0)}$ & \\ \smallskip (i) & $N(\beta; \kappa\beta-1,0,1,0)$ & by & $\dfeyn {h^2}hcc{(\beta;\kappa\beta-3,0,0,0)}$ & \\ \smallskip (j) & $N(\beta; \kappa\beta-1,2,0,0)$ & by & $\dfeyn {h^2}hhc{(\beta;\kappa\beta-2,0,0,0)}$ & \\ \smallskip (k) & $N(\beta; \kappa\beta,1,0,0)$ & by & $\dfeyn hhcc{(\beta;\kappa\beta-2,0,0,0)}$ & \end{tabular} \end{center} \caption{Order, within the outer induction, for the proof of strong reconstruction for $G(2,4)$/$\Sym^2{\mathbb P}^2$.} \label{gtwofour} \end{table} We induct first on curve class $\beta$. The induction hypothesis consists of all relations and all numbers in degrees less than $\beta-1$, and all relations and all numbers except $N(\beta-1; \kappa\beta-\kappa+1,0,0,0)$ in degree $\beta-1$. The relations indicated in entries (a), (b) of Table \ref{gtwofour} give us two new numbers, including $N(\beta-1; \kappa\beta-\kappa+1,0,0,0)$. Thus from now on we assume all relations and all numbers in degrees less than or equal to $\beta-1$. Now, inductively on $d$ via the partial ordering $d'=(t',u',v',w')\prec d=(t,u,v,w)\Leftrightarrow |d'| < |d|$ or $|d'|=|d|$, $t'>t$ or $|d'|=|d|$, $t'=t$, $u'+2v'+3w' > u+2v+3w$, we establish the relations indicated in (c)--(k) of the table, as applicable to the current degree. Step (c) is the inner induction of the proof of the strong reconstruction theorem, applied to the subring of $A$ generated by $h$. Finally, it follows from \fsr{} (this takes a bit of checking) that in any degree, the (31 out of 55 total) relations indicated in Table \ref{gtwofour} imply all the relations. The 21 encoded by step (c) follow by the proof of strong reconstruction, so we are reduced to establishing the remaining 10. When $u=v=w=0$, we are done by Table \ref{gtwofour} (each remaining relation solves for an unknown number). Otherwise, since each of the Feynman diagrams indicated in (a), (b), (d)--(k) of the table has a diagonal containing $c$ and $h$, an application of \tsr{} reduces us to relations obtained in previous degrees (via the partial ordering above). Summarizing, in a neighborhood of the geometric solution, the solution space to WDVV consists of only the expected rescalings for $X=G(2,4)$, but has a dependence on two extra free parameters in case $X=\Sym^2{\mathbb P}^2$ (where the geometric solution comes from viewing $X$ as a global quotient of a homogeneous variety by a finite group). When $N(1;0,0,1,1)=0$ (resp.\ $N(1;1,0,0,1)=0$) the situation is considerably more complicated. For instance, for $X=G(2,4)$, setting $N(1;5,0,0,0)=1$ and all other $N(1;t,u,v,w)=0$ yields a consistent solution through $\beta=4$ (with $N(\beta;4\beta+1,0,0,0)$ indeterminate for $\beta=2,3,4$), but fails to satisfy the relations consistently starting in degree 5. {\bf Example 4.} $X=G(2,5)$. The same technique as that of Example 3 establishes strong reconstruction for $G(2,5)$. The statement of strong reconstruction dictates 3 starting numbers and no relationships, but as we shall see, we need to assume one relationship. As in the previous example, we will also have to make a genericity assumption. We shall find that the geometric solution to WDVV lives in a two-parameter family of solutions. \begin{table} \begin{center} \begin{tabular}{c|cccccccc} & $t_1$ & $t_2$ & $t_3$ & $t_4$ & $t_5$ & $t_6$ & $t_7$ & $t_8$ \\ \hline $t_1$ & $t_2$ & $t_4$ & $t_5$ & $t_6$ & $(1/3)t_7$ & $5t_8$ & $0$ & $t_9$ \\ $t_2$ & $t_4$ & $t_6$ & $(1/3)t_7$ & $5t_8$ & $0$ & $5t_9$ & $0$ & $0$ \\ $t_3$ & $t_5$ & $(1/3)t_7$ & $t_6-(11/3)t_7$ & $0$ & $5t_8$ & $0$ & $15t_9$ & $0$ \\ $t_4$ & $t_6$ & $5t_8$ & $0$ & $5t_9$ & $0$ & $0$ & $0$ & $0$ \\ $t_5$ & $(1/3)t_7$ & $0$ & $5t_8$ & $0$ & $5t_9$ & $0$ & $0$ & $0$ \\ $t_6$ & $5t_8$ & $5t_9$ & $0$ & $0$ & $0$ & $0$ & $0$ & $0$ \\ $t_7$ & $0$ & $0$ & $15t_9$ & $0$ & $0$ & $0$ & $0$ & $0$ \\ $t_8$ & $t_9$ & $0$ & $0$ & $0$ & $0$ & $0$ & $0$ & $0$ \end{tabular} \end{center} \caption{Multiplication table for $A^*_{\mathbb Q} G(2,5)$ with respect to the basis of T. Graber.} \label{multtable} \end{table} Thinking of $X$ as the space of rank 2 quotients of ${\mathbb C}^5$, let $Q$ be the universal quotient bundle and $c_i=c_i(Q)$. We use the following basis for $A^*_{\mathbb Q} X$, suggested by T. Graber, \begin{center} \begin{tabular}{llll} codim 0:&& $t_0=1$ \\ codim 1:&& $t_1=c_1$ \\ codim 2:&& $t_2=c_1^2$ & $t_3=2c_1^2-5c_2$ \\ codim 3:&& $t_4=c_1^3$ & $t_5=2c_1^3-5c_1c_2$ \\ codim 4:&& $t_6=c_1^4$ & $t_7=c_1^4-5c_2^2$ \\ codim 5:&& $t_8=c_1c_2^2$ \\ codim 6:&& \multicolumn{2}{l}{ $t_9=c_2^3$ (point class)} \\ \end{tabular} \end{center} with multiplication table given in Table \ref{multtable}. We denote a typical unknown by $N(\beta;d_2,d_3,d_4,d_6,d_8,d_9,d_3,d_5,d_7)$ (note special order). Inductively on degree $\beta$, we show that degree $\beta$ relations solve consistently for all but 8 degree $\beta$ numbers, plus the 5 linear expressions shown in Table \ref{linexpr}. The 8 exceptions are the 7 numbers appearing in Table \ref{linexpr} as well as $N(\beta;0,0,0,0,0,5\beta+3,0,0)$. The genericity assumption is $N(1;0,0,0,0,1,0,0,1) \ne 0$. Given the induction hypothesis, we solve for the remaining degree $(\beta-1)$ numbers according to Table \ref{gtwofivea}, first by the path shown with $(0,0,0,0,0,-4,2,0)$ added to all degrees, then by the path shown with $(0,0,0,0,0,-2,1,0)$ added to all degrees, and then by the path as shown. Only for $\beta=2$, during the first pass, we must substitute $\dfeyn {t_9}{t_1}{t_6}{t_5}{(\beta;0,0,0,0,0,5\beta-10,1,0)}$ for step (c). Once we have all the numbers in degree $\beta-1$, we then induct on $d$ with respect to the partial ordering $d'\prec d\Leftrightarrow$ \begin{itemize} \item[(i)] $|d'|<|d|$, or \item[(ii)] $|d'|=|d|$ and $d'_2+d'_4+d'_6+d'_8+d'_9<d_2+d_4+d_6+d_8+d_9$, or \item[(iii)] $|d'|=|d|$ and $d'_2+d'_4+d'_6+d'_8+d'_9=d_2+d_4+d_6+d_8+d_9$, but $d'_2+2d'_4+3d'_6+4d'_8+5d'_9>d_2+2d_4+3d_6+4d_8+5d_9$, or \item[(iv)] $|d'|=|d|$, $(d'_2,d'_4,d'_6,d'_8,d'_9)=(d_2,d_4,d_6,d_8,d_9)$, and $d'_3+2d'_5+3d'_7<d_3+2d_5+3d_7$. \end{itemize} For each $d$, we perform the inner induction of the proof of strong reconstruction to obtain all relations involving only powers of $t_1$. Next, we obtain all of $$ \begin{tabular}{lll} \smallskip $\fra{\displaystyle N(\beta;0,0,0,0,2,u,v,w)} {\displaystyle \hfill u+2v+3w=5\beta-7}$ & by & $\dfeyn {t_9}{t_1}{t_8}{t_7}{(\beta;0,0,0,0,0,u,v,w-1)}$ \\ \smallskip & or & $\dfeyn {t_9}{t_1}{t_8}{t_5}{(\beta;0,0,0,0,0,u,v-1,w)}$ \\ \smallskip & or & $\dfeyn {t_9}{t_1}{t_8}{t_3}{(\beta;0,0,0,0,0,u-1,v,w)}$ \\ \smallskip $\fra{\displaystyle N(\beta;0,0,0,1,1,u,v,w)} {\displaystyle \hfill u+2v+3w=5\beta-6}$ & by & $\dfeyn {t_9}{t_1}{t_6}{t_7}{(\beta;0,0,0,0,0,u,v,w-1)}$ \\ & or & etc. \\ $\ldots$ \\ $\fra{\displaystyle N(\beta;2,0,0,0,0,u,v,w)} {\displaystyle \hfill u+2v+3w=5\beta+1}$ & by & $\dfeyn {t_2}{t_1}{t_1}{t_7}{(\beta;0,0,0,0,0,u,v,w-1)}$ etc.\\ \end{tabular} $$ coming from relations in degree $(\beta;d)$. The exception to be noted occurs in attempting to solve for $N(\beta;2,0,0,0,0,5\beta+1,0,0)$: we get a value for $(L5)$ of Table \ref{linexpr} rather than a single $N$. \begin{table} $$\begin{array}{ll} \medskip (L1) & N(\beta;0,0,0,0,0,5\beta,0,1) - 3 N(\beta;0,0,0,0,0,5\beta-1,2,0) \\ (L2) & N(\beta;1,0,0,0,0,5\beta,1,0) - N(\beta;0,1,0,0,0,5\beta+1,0,0) \\ \medskip & \hspace{130pt} {} + \beta N(\beta;0,0,0,0,0,5\beta-1,2,0) \\ \medskip (L3) & N(\beta;1,0,0,0,0,5\beta,1,0) - 2 \beta N(\beta;0,0,0,0,0,5\beta-1,2,0) \\ (L4) & N(\beta;1,0,0,0,0,5\beta+2,0,0) - 2\beta N(\beta;0,0,0,0,0,5\beta+1,1,0) \\ \medskip & \hspace{130pt} {} - 11 \beta^2 N(\beta;0,0,0,0,0,5\beta-1,2,0) \\ (L5) & N(\beta;2,0,0,0,0,5\beta+1,0,0) - 4 \beta^2 N(\beta;0,0,0,0,0,5\beta-1,2,0) \end{array}$$ \caption{The linear expressions obtained by degree $\beta$ relations.} \label{linexpr} \end{table} Still in a particular degree, we obtain $$ \begin{tabular}{cll} \smallskip $\fra{\displaystyle N(\beta;0,0,0,0,1,u,v,w)} {\displaystyle \hfill u+2v+3w=5\beta-2}$ & by & $\dfeyn{t_8}{t_1}{t_7}{t_7}{(\beta;0,0,0,0,0,u,v,w-2)}$ \\ & or & $\dfeyn{t_8}{t_1}{t_7}{t_5}{(\beta;0,0,0,0,0,u,v-1,w-1)}$ \\ & etc. \end{tabular} $$ with exceptions noted below: $$ \begin{tabular}{ccl} \smallskip \hspace{80pt}$(L2)$ & by & $\dfeyn{t_2}{t_1}{t_3}{t_3}{(\beta;0,0,0,0,0,5\beta-1,0,0)}$ \\ \smallskip \hspace{80pt}$(L3)$ & by & $\dfeyn{t_1}{t_1}{t_5}{t_3}{(\beta;0,0,0,0,0,5\beta-1,0,0)}$ \\ \hspace{80pt}$(L4)$ & by & $\dfeyn{t_1}{t_1}{t_3}{t_3}{(\beta;0,0,0,0,0,5\beta,0,0)}$ \end{tabular} $$ \begin{table} \begin{center} \begin{tabular}{lccl} \smallskip (a) & $N(\beta; 0,0,1,0,1,5\beta-5,0,0)$ & by & $\dfeyn {t_9}{t_1}{t_4}{t_3}{(\beta;0,0,0,0,0,5\beta-6,0,0)}$ \\ \smallskip (b) & $N(\beta; 0,0,0,2,0,5\beta-5,0,0)$ & by & $\dfeyn {t_6}{t_1}{t_8}{t_3}{(\beta;0,0,0,0,0,5\beta-6,0,0)}$ \\ \smallskip (c) & $N(\beta; 0,0,0,1,1,5\beta-6,0,0)$ & by & $\dfeyn {t_9}{t_1}{t_6}{t_3}{(\beta;0,0,0,0,0,5\beta-7,0,0)}$ \\ \smallskip (d) & $N(\beta; 0,0,0,0,1,5\beta-5,0,1)$ & by & $\dfeyn {t_8}{t_1}{t_7}{t_3}{(\beta;0,0,0,0,0,5\beta-6,0,0)}$ \\ \smallskip (e) & $N(\beta; 0,0,0,1,0,5\beta-4,0,1)$ & by & $\dfeyn {t_6}{t_1}{t_7}{t_3}{(\beta;0,0,0,0,0,5\beta-5,0,0)}$ \\ \smallskip (f) & $N(\beta; 0,0,0,0,1,5\beta-4,1,0)$ & by & $\dfeyn {t_8}{t_1}{t_5}{t_3}{(\beta;0,0,0,0,0,5\beta-5,0,0)}$ \\ \smallskip (g) & $N(\beta{-}1; 0,0,0,0,0,5\beta-2,0,0)$ & by & $\dfeyn {t_3}{t_1}{t_9}{t_3}{(\beta;0,0,0,0,0,5\beta-5,0,0)}$ \end{tabular} \end{center} \caption{Path to the remaining degree $(\beta-1)$ numbers for $G(2,5)$.} \label{gtwofivea} \end{table} Lastly, we have numbers of the form $N(\beta;0,0,0,0,0,u,v,w)$ and the relations that produce these: $$ \begin{tabular}{cccclcccc} relation & \multicolumn{3}{c}{for cases} & & relation & \multicolumn{3}{c}{for cases} \\ \smallskip $\feyn{t_5}{t_1}{t_7}{t_7}$ & & & $w\ge3$ & & $\feyn{t_3}{t_1}{t_7}{t_3}$ & $u\ge1$ & $v\ge1$ & $w\ge1$ \\ \smallskip $\feyn{t_5}{t_1}{t_7}{t_5}$ & & $v\ge1$ & $w\ge2$ & & $\feyn{t_5}{t_1}{t_3}{t_5}$ & & $v\ge3$ \\ \smallskip $\feyn{t_5}{t_1}{t_7}{t_3}$ & $u\ge1$ & & $w\ge2$ & & $\feyn{t_3}{t_1}{t_5}{t_3}$ & \multicolumn{3}{r}{(only to get $(L1)$)} \\ $\feyn{t_3}{t_1}{t_7}{t_5}$ & & $v\ge2$ & $w\ge1$ \end{tabular} $$ Finally, we obtain $\feyn {t_3}{t_1}{t_9}{t_3}$, $\feyn {t_5}{t_1}{t_9}{t_3}$, and $\feyn {t_7}{t_1}{t_9}{t_3}$. When $(d_2,d_4,d_6,d_8,d_9)=(0,0,0,0,0)$ and $d_5=d_7=0$, these are established by the three passes through the path in Table \ref{gtwofivea}. For the details see Table \ref{gtwofiveb}. Note that each pass through Table \ref{gtwofivea} is used to solve for an unknown when $\beta\ge2$; when $\beta=1$ we actually get a condition on starting data, described below. \begin{table} \begin{center} \begin{tabular}{crcl} Assumptions & \multicolumn{3}{c}{Implications} \\ $\dfeyn{t_8}{t_1}{t_7}{t_3}{(5\beta-8,1,0)}$ & $\dfeyn{t_3}{t_1}{t_9}{t_3}{(5\beta-9,2,0)} + \mbox{\tsr}$ & $\Rightarrow$ & $\dfeyn{t_3}{t_1}{t_9}{t_5}{(5\beta-8,1,0)}$ \\ $\dfeyn{t_5}{t_1}{t_8}{t_5}{(5\beta-8,1,0)}$ & $\fiveblu{t_1}{t_6}{t_1}{t_7}{t_3}$ & $\Rightarrow$ & $\dfeyn{t_8}{t_1}{t_3}{t_7}{(5\beta-8,1,0)}$ \\ $\dfeyn{t_6}{t_1}{t_7}{t_5}{(5\beta-8,1,0)}$ & $\fivecru{t_5}{t_1}{t_8}{t_1}{t_3}$ & $\Rightarrow$ & $\dfeyn{t_5}{t_1}{t_9}{t_3}{(5\beta-8,1,0)}$ \\ $\dfeyn{t_8}{t_1}{t_7}{t_5}{(5\beta-7,0,0)}$ & \tsr & $\Rightarrow$ & $\dfeyn{t_5}{t_1}{t_9}{t_5}{(5\beta-7,0,0)}$ \\ & $\fivecru{t_1}{t_7}{t_1}{t_8}{t_3}$ & $\Rightarrow$ & $\dfeyn{t_1}{t_7}{t_9}{t_3}{(5\beta-7,0,0)}$ \\ & $\fiveblu{t_1}{t_5}{t_1}{t_9}{t_3}$ & $\Rightarrow$ & $\dfeyn{t_7}{t_1}{t_9}{t_3}{(5\beta-7,0,0)}$ \end{tabular} \end{center} \caption{How $\dfeyn {t_3}{t_1}{t_9}{t_3}{(5\beta-9,2,0)}\Rightarrow \dfeyn {t_7}{t_1}{t_9}{t_3}{(5\beta-7,0,0)}$ follows from \tsr{} and \fsr{}. We use $(u,v,w)$ as shorthand for degree $(\beta;0,0,0,0,0,u,v,w)$. Each relation listed as an assumption solves for an unknown number and for that reason is satisfied. Note that $\dfeyn {t_3}{t_1}{t_9}{t_3}{(5\beta-7,1,0)}\Rightarrow \dfeyn {t_5}{t_1}{t_9}{t_3}{(5\beta-6,0,0)}$ is just the first three steps above.} \label{gtwofiveb} \end{table} When $(d_2,d_4,d_6,d_8,d_9)=(0,0,0,0,0)$ but $d_5\ne0$ or $d_7\ne0$, then because of the induction order, some of the (85 total) relations listed as determining numbers will determine numbers that have already been solved for. But in each such case, \tsr{} allows us to deduce the relation in question. Finally, when $(d_2,d_4,d_6,d_8,d_9)\ne(0,0,0,0,0)$ then all 85 relations follow by \tsr{} just as in Example 3. As in Example 3, we now note that the 85 relations in the above lists plus the 120 only involving powers of $t_1$ imply the remaining 461 relations by \fsr. Because of the number of relations involved, the author used a computer to complete this verification. Relation $\dfeyn {t_3}{t_1}{t_9}{t_3}{(\beta;0,0,0,0,0,5\beta-5,0,0)}$ from (i) in Table \ref{gtwofivea} solves for an unknown number whenever $\beta\ge2$, but still needs to be verified when $\beta=1$. This is what imposes the one condition on starting data. When written out for $\beta=1$, (e)--(g) of Table \ref{gtwofivea} translate as \begin{eqnarray*} \lefteqn{11 N(1;0,0,0,0,1,0,0,1)=} \hspace{30pt} \\ & & 6 N(1;0,0,1,0,1,0,0,0)+15 N(1;0,0,0,2,0,0,0,0). \end{eqnarray*} \section{Appendix: geometric groundwork} \label{geoground} Let $X$ be a complex projective manifold, and for simplicity, assume $X$ has homology only in even dimensions. To avoid writing doubled indices, set $A_dX=H_{2d}(X,{\mathbb Z})$ and $A^dX=H^{2d}(X,{\mathbb Z})$. Set $A^d_{\mathbb Q} X=A^dX\otimes{\mathbb Q}$. For $\beta\in A_1X$, we denote by $M_{0,n}(X,\beta)$ the moduli space of $n$-pointed rational curves $f\colon {\mathbb P}^1\to X$ such that $f_*[{\mathbb P}^1]=\beta$. The Kontsevich space $\overline{M}_{0,n}(X,\beta)$ is a compactification of $M_{0,n}(X,\beta)$ whose points correspond to $n$-pointed trees of ${\mathbb P}^1$'s mapping into $X$ satisfying a stability hypothesis. Evaluation maps $\rho_i\colon\overline{M}_{0,n}(X,\beta)\to X$ send a given map $f$ to the image under $f$ of the $i^{\rm th}$ marked point. If $X$ is a homogeneous variety (a quotient of a complex reductive Lie group by a parabolic subgroup), then the {\em tree-level system of Gromov-Witten numbers} is the system of maps $I_\beta\colon \bigoplus \Sym^n A^*_{\mathbb Q} X\to{\mathbb Q}$ for each $\beta\in A_1X$ given by the formula \begin{equation} \label{gwformula} I_\beta(\gamma_1\cdots\gamma_n)=\int_{\overline{M}_{0,n}(X,\beta)} \rho_1^*(\gamma_1)\mathbin{\text{\scriptsize$\cup$}}\cdots\mathbin{\text{\scriptsize$\cup$}}\rho_n^*(\gamma_n). \end{equation} With the notation of Problem \ref{mainprob}, the {\em geometric solution} to WDVV is given by $N(\beta;d)=I_\beta(\bigotimes (T_{\tau_j})^{\otimes d_j})$. These numbers have enumerative significance: $N(\beta;d)$ is the number of rational curves on $X$ in homology class $\beta$ meeting $d_j$ general translates of a cycle Poincar\'e dual to $T_{\tau_j}$, for each $j$. For general $X$, there is still a geometric solution to WDVV, given by formula (\ref{gwformula}) but with the integration performed over a virtual fundamental cycle, although the enumerative significance of these numbers is less clear. It is still the case for any $\beta\ne0$ that $I_\beta=0$ unless $\int_\beta\omega>0$ for every ample divisor $\omega$. There exist forgetful maps forgetting $X$ and forgetting any subset of the marked points, so in particular to any $\{i,j,k,l\}\subset\{1,\ldots,n\}$ there corresponds a map $\overline{M}_{0,n}(X,\beta)\to\overline{M}_{0,\{i,j,k,l\}}$. Pulling back rational equivalences on $\overline{M}_{0,4}\cong{\mathbb P}^1$ leads to the associativity relations that the Gromov-Witten numbers must satisfy. Since the differential equation (\ref{assrel}) is invariant under translations $y_{\sigma_i}\mapsto y_{\sigma_i}+\alpha_i$, there is an $r$-dimensional family of rescalings acting on the geometric solution to WDVV, where $r=\rk A^1X$. Much of the contents of this paper was motivated by a search for solutions to WDVV besides the geometric solution and its translates under these rescalings. The formal computation of section \ref{fivesi} is motivated by considering a typical component of the boundary of one of the Kontsevich spaces $\overline{M}_{0,n}(X,\beta)$. Say $A\cup B=\{1,\ldots,n\}$, $A\cap B=\emptyset$, $|A|\ge2$, $|B|\ge2$, and $\beta_1+\beta_2=\beta$. Then there is a component $D(A,B;\beta_1,\beta_2)$ of the boundary of $\overline{M}_{0,n}(X,\beta)$, which fits into a fiber diagram (see \cite{c}) \begin{equation*} \begin{CD} D(A,B;\beta_1,\beta_2) @>>> X^n\times X \\ @VVV @V{\delta}VV \\ \overline{M}_{0,A\cup\{{*}\}}(X,\beta_1)\times \overline{M}_{0,B\cup\{{*}\}}(X,\beta_2) @>{\rho}>> X^n\times X\times X \\ \end{CD} \end{equation*} where $\delta$ is given by the diagonal embedding of $X$ in $X\times X$. Given cohomology classes $\gamma_1,\ldots,\gamma_n,\xi\in A^*X$, we have \begin{eqnarray*} \delta_*(\gamma_1\times\cdots\times\gamma_n\times\xi) &=& \sum_{e,f} g^{ef} \gamma_1\times\cdots\times\gamma_n \times(\xi\mathbin{\text{\scriptsize$\cup$}} T_e)\times T_f \\ &=& \sum_{e,f} g^{ef} \gamma_1\times\cdots\times\gamma_n \times T_e\times (\xi\mathbin{\text{\scriptsize$\cup$}} T_f) \end{eqnarray*} corresponding to two ways of writing the class $\xi$ on the diagonal. Pulling back by $\rho$ and integrating gives the identity \begin{eqnarray*} \lefteqn{ \sum_{e,f} g^{ef} I_{\beta_1}\bigl(\prod_{a\in A} \gamma_a \cdot (\xi\mathbin{\text{\scriptsize$\cup$}} T_e)\bigr) I_{\beta_2}\bigl(\prod_{b\in B} \gamma_b\cdot T_f\bigr) = } \hspace{20pt} \\ & & \sum_{e,f} g^{ef} I_{\beta_1} \bigl(\prod_{a\in A}\gamma_a \cdot T_e\bigr) I_{\beta_2}\bigl(\prod_{b\in B}\gamma_b\cdot (\xi\mathbin{\text{\scriptsize$\cup$}} T_f)\bigr). \end{eqnarray*} If $\gamma_1=T_i$, $\gamma_2=T_j$, $\gamma_3=T_k$, $\gamma_4=T_l$, and $\xi=T_m$, and if we sum over partitions $(\beta_1,\beta_2,A,B)$ such that $A$ contains 1 and 2 and $B$ contains 3 and 4, we get a special case of (\ref{mdiag}), namely the case where the potential function $\Gamma$ is given by the geometric solution to WDVV.

9703

alg-geom/9703004

en

https://arxiv.org/abs/alg-geom/9703004

alg-geom/9703004

Philip A. Foth

Philip A. Foth

Geometry of Moduli Spaces of Flat Bundles on Punctured Surfaces

LaTeX, 12 pages

We consider the moduli spaces of flat $SL(n, C)$-bundles on Riemann surfaces with one puncture when we fix the conjugacy class ${\cal C}$ of the monodromy transformation around the puncture. We show that under a certain condition on the class ${\cal C}$ (namely the product of $k<n$ eigenvalues is not equal to $1$) that we call property P the moduli space in question is smooth and its natural closure is a normal algebraic variety with rational singularities. The set of conjugacy classes having property P constitutes a Zariski open subset of $SL(n, C)$ and it is also possible to define property P for the groups $SO(n, C)}$ and $Sp(2n, C)$ to prove similar results. There are a few other applications of our techniques, one of which is that if $G$ is a classical reductive algebraic group and $A_1, A_2, ..., A_p\in G$, $p>1$ and $A_1A_2\cdots A_pA_1^{-1}A_2^{-1}\cdots A_p^{-1}$ belongs to a class which satisfies property P then the $p$ -tuple $(A_1, ..., A_p)$ algebraically generates the whole group $G$.

alg-geom

\section{Introduction} \setcounter{equation}{0} Let $X$ be a Riemann surface of the genus $g>0$ with one puncture. We consider the moduli space of flat $GL(n, {\Bbb C})$-bundles such that the monodromy transformation around the puncture belongs to a given conjugacy class ${\cal C}\in SL(n, {\Bbb C})$. We further assume that the class ${\cal C}$ has property P, meaning that for the set of its eigenvalues $({\lambda}_1, {\lambda}_2, ..., {\lambda}_n)$ we have ${\lambda}_{i_1}{\lambda}_{i_2}\cdots {\lambda}_{i_m}\ne 1$ for any $i_1< i_2<\dots <i_m$,\ $1\le m < n$. We carry out all the proofs for the genus $1$ and later show that all the result easily generalize for higher genera. Due to the well-known correspondence between flat bundles on $X$ and representations of its fundamental group, the problem (for $g=1$) is reduced to the consideration of moduli spaces of pairs of matrices $(B,D)$ from $GL(n, {\Bbb C})$ such that $BDB^{-1}D^{-1}\in{\cal C}$. The main result (presented in Theorem 3.2 and Proposition 3.4) is \begin{th} Let $X$ be a Riemann surface with one puncture and ${\cal C}$ - a conjugacy class in $SL(n, {\Bbb C})$ with property P. (a) The moduli space ${{\cal M}_{\cal C}}$ of flat $\frak{gl}(n, {\Bbb C})$-connections over $X$ with the monodromy transformations around the puncture in ${\cal C}$ is smooth. (b) Let $\bar{\cal C}$ is the closure of ${\cal C}$ in $SL(n, {\Bbb C})$. The variety ${{\cal M}_{\bar{\cal C}}}$ (defined in (a) by changing ${\cal C}$ to $\bar{\cal C}$) is normal with rational singularities. \end{th} We extend the definition of property P to the case of orthogonal or symplectic groups to establish similar results. We also show that if $BDB^{-1}D^{-1}$ has property P, then the pair $(B,D)$ algebraically generates the whole group. The moduli spaces in question are of great importance by a number of reasons. A theorem of Mehta-Seshadri \cite{MS} identifies two moduli spaces: the space of unitary representations of fundamental group and the space of parabolic bundles on $X$. When ${\cal C}=\exp(2\pi{\sqrt{-1}} d/n)Id$ is (the class of) a central element, the space ${{\cal M}_{\cal C}}$ is a smooth K\"{a}hler manifold and appears in algebraic geometry as the space of holomorphic vector bundles on the closed surface of rank $n$, degree $d$ and fixed determinant \cite{AB}. Also those moduli spaces appear in topological and quantum field theories; they are related to Jones-Witten invariants (see \cite{A} for details). They are closely related to Yang-Mills theory and geometric quantization. It is necessary to mention the results due to Simpson \cite{S2}, which provide natural correspondence between ${\cal D}_X$-modules, Higgs bundles and local systems on $X$ (with extra conditions). This allows one to identify moduli spaces of those objects with the moduli spaces we consider in the present paper. I express deep gratitude to Jean-Luc Brylinski for helpful and valuable advices. He kindly guided me and generously shared his knowledge. \section{Common stabilizer of two matrices} \setcounter{equation}{0} We say that a matrix $C$ or a conjugacy class ${\cal C}$ in $SL(n, {\Bbb C})$ with the set of eigenvalues $\{{\lambda}_1,\dots,{\lambda}_n\}$ has {\it property P} if the following condition holds. For any $m < n$ distinct numbers $1\le i_1\le i_2\le\dots\le i_m\le n$ the product ${\lambda}_{i_1}{\lambda}_{i_2}\cdots {\lambda}_{i_m}$ is not equal to $1$. One notices that the set of matrices with property P is Zariski open in $SL(n,{\Bbb C})$. \begin{th} Let $B, \ D \in GL(n, {\Bbb C})$ be such that $[B,D]$ satisfies property P. Then the common stabilizer of $B$ and $D$ consists of scalar matrices only. \end{th} {\noindent{\it Proof.\ \ }} Let $K\in GL(n, {\Bbb C})$ be non-central matrix commuting with both $B$ and $D$. Let ${\lambda}$ be an eigenvalue of $K$ and let $W\subset {\Bbb C}^n$ be the kernel of $K-{\lambda}.Id$. It follows that both $B$ and $D$ stabilize $W$. Hence the product of eigenvalues of $[B,D]$ which correspond to $W$ is equal to $1$. This means that $W={\Bbb C}^n$ and $K$ is scalar. $\bigcirc$ \ We will denote by $SL(n, {\Bbb C})^2$ or $GL(n, {\Bbb C})^2$ the Cartesian product of two copies of $SL(n, {\Bbb C})$ or $GL(n, {\Bbb C})$ respectively. Also denote by $$\kappa=[,]: \ GL(n, {\Bbb C})^2\to SL(n, {\Bbb C})$$ the commutator map. \ \noindent{\bf Remark.} For $n>2$ the author can prove the following statement converse to the above Theorem. Let ${\cal C}$ be a conjugacy class in $SL(n, {\Bbb C})$. If for any $(B,D)\in GL(n, {\Bbb C})^2$ the condition $[B,D]\in{\cal C}$ implies $\dim Z(B,D)=1$, then ${\cal C}$ has property P. We do not include the proof since it is long and computational and we will not use it in the persent paper. The situation, however, is different for $n=2$. If we take as ${\cal C}$ the class of $ \pmatrix{ 1 & 1 \cr 0 & 1 } $, then for all $\{ (B,D)\in GL(2,{\Bbb C})^2;\ [B,D]\in{\cal C}\}$ the common stabilizer of $B$ and $D$ denoted by $Z(B,D)$ is the center of $GL(n,{\Bbb C})$, i.e. $\dim Z(B,D)=1$. So, when $n=2$, if $B$ and $D$ do not commute, then their common stabilizer is the center of $GL(n,{\Bbb C})$. \ One can think of another interpretation of property P. Let $V={\Bbb C}^n$ be the tautological representation space of $SL(n,{\Bbb C})$. The spaces $\wedge^iV$ are also naturally representation spaces for $SL(n, {\Bbb C})$. Property P for the matrix $C$ means that $C$ doesn't stabilize any $\ne 0$ vector in $\wedge^iV$ for $0<i<n$. The next proposition was first proven in \cite{Shoda}. \begin{proposition} The map $\kappa$ is onto. \end{proposition} For instance, the commutator of two matrices from $SL(n, {\Bbb C})$ $$ \pmatrix{ 0 & e_1 & 0 & \dots & 0 \cr 0 & 0 & e_2 & \dots & 0 \cr . & . & . & \dots & . \cr 0 & 0 & 0 & \dots & e_{n-1} \cr e_n & 0 & 0 & \dots & 0 } \ \ \ and \ \ \ \pmatrix{ 0 & 0 & 0 & \dots & f_n \cr f_1 & 0 & 0 & \dots & 0 \cr 0 & f_2 & 0 & \dots & 0 \cr 0 & 0 & f_3 & \dots & 0 \cr . & . & . & \dots & . } $$ can be conjugate to any semisimple element of the group. Also for any unipotent element $U$, $U^{-1}$ is conjugate to $U$, hence $U^2$ is also in the image of $\kappa$. Any unipotent element is a square of another unipotent element. This proves the proposition for unipotent elements. \section{Moduli spaces of flat bundles} \setcounter{equation}{0} Let us fix a conjugacy class ${\cal C}$ in $SL(n, {\Bbb C})$. Further we consider the variety of pairs and the moduli spaces of pairs of matrices with their commutator in ${\cal C}$. We denote ${{X}_{\cal C}}=\{ (B,D)\in GL(n,{\Bbb C})^2; [B,D]\in {\cal C}\}$. It is well-known that if ${\cal C}=1$ then the variety of commuting pairs is irreducible (\cite{MT}), but is not a smooth variety. \begin{lem} The orbit of every element $(B,D)$, such that $[B,D] \in {\cal C}$ is closed in ${{X}_{\cal C}}$ if the class ${\cal C}$ satisfies property P. \end{lem} {\noindent{\it Proof.\ \ }} All orbits in the closure of ${{X}_{\cal C}}$ have the same dimension when ${\cal C}$ has property P. (Because all the conjugacy classes in the closure of ${\cal C}$ still have property P, and whenever $[B,D]$ has property P the stabilizer $Z(B,D)$ coincides with the center of the group.) This implies that orbits are closed in ${{X}_{\cal C}}$. $\bigcirc$ \ Actually, we have proved a stronger result, namely that the orbit of $(B,D)$ is closed {\it in $GL(n, {\Bbb C})^2$} if property P holds for $[B, D]$. We consider the moduli space $$ {{\cal M}_{\cal C}}=\{ (B,D)\in GL(n,{\Bbb C})^2; [B,D]\in {\cal C}\}/SL(n,{\Bbb C}), $$ where factoring occurs by the adjoint action of the special linear group. In order to define ${{\cal M}_{\cal C}}$ properly one takes the quotient in the sense of the GIT (\cite{GIT}). In the case of an affine variety $X$ there is a natural definition of the quotient variety $Y=X/G$. The algebra of the regular functions ${\cal O}(Y)$ is just the subalgebra of ${\cal O}(X)$ of $G$-invariant regular functions on $X$. But when all orbits are closed, each point of the quotient ${{\cal M}_{\cal C}}$ correspond to an orbit in ${{X}_{\cal C}}$. Using the above lemma we see that when ${\cal C}$ is semisimple with property P, the variety ${{X}_{\cal C}}$ is closed affine, the quotient ${{X}_{\cal C}}/SL(n,{\Bbb C})$ exists and its points correspond exactly to the orbits. We can construct a nice quotient when ${\cal C}$ is not semisimple, but still satisfies property P. Let ${\bar{\cal C}}$ be the closure of ${\cal C}$ in $SL(n,{\Bbb C})$ and ${{X}_{\bar{\cal C}}}=\{(B,D)\in GL(n,{\Bbb C})^2;[B,D]\in{\bar{\cal C}}\}$. It is affine so the quotient ${{\cal M}_{\bar{\cal C}}}$ by $SL(n,{\Bbb C})$ exists. Since all $SL(n,{\Bbb C})$-orbits have the same dimension, points of ${{\cal M}_{\bar{\cal C}}}$ again correspond exactly to orbits. It is well-known that ${\bar{\cal C}}{\setminus}{\cal C}$ is the union of {\it finite} number of conjugacy classes ${\cal C}_i$. So we have the corresponding finite set of closed subvarieties ${\cal M}_i$ in ${{\cal M}_{\bar{\cal C}}}$. We define now the algebraic variety ${{\cal M}_{\cal C}}$ which is the complement of the union $\cup_i{\cal M}_i$ in ${{\cal M}_{\bar{\cal C}}}$. \ {\it Example.} Here we consider the simple but important case $n=2$. We have five types of conjugacy classes in $SL(2, {\Bbb C})$ (here we mention just a representative of each): $I=\pmatrix{ 1 & 0 \cr 0 & 1}$ ($\dim (I)=0$), $-I=\pmatrix{ -1 & 0 \cr 0 & -1}$ ($\dim (-I)=0$), $R_2=$ the class of $\pmatrix{ 1 & 1 \cr 0 & 1}$, ($\dim (R_2)=2$), $R_e=$ the class of $\pmatrix{ -1 & 1 \cr 0 & -1}$, ($\dim (R_e)=2$), $R_{{\lambda}}=$ the class of $\pmatrix{ {\lambda} & 0 \cr 0 & {\lambda}^{-1}}$, ${\lambda}^2\ne 1$, ($\dim (R_{{\lambda}})=2$). One has $$ GL(2, {\Bbb C})^2=X_I\cup X_{-I}\cup X_{R_2}\cup X_{R_e}\cup \bigcup_{{\lambda}^2\ne 1}X_{R_{{\lambda}}}. $$ Each ${{X}_{\cal C}}$ except for $X_I$ is a connected smooth variety. Their dimensions are $6,\ 5,\ 7,\ 7,\ 7$ respectively. The varieties ${{X}_{\cal C}}$ corresponding to semisimple classes are closed. Of course, $X_I$ and $X_{-I}$ lie in the closure of $X_{R_2}$ and $X_{R_e}$ respectively. Also, they both are limit varieties of $X_{R_{{\lambda}}}$ for ${\lambda}\to\pm 1$, so they both lie in the closure of $\cup_{{\lambda}^2\ne 1}X_{R_{{\lambda}}}$. As it was mentioned above, $X_I$ is an irreducible variety. Also it has singularities only in codimension 2. The space ${\cal M}_{\bar{R_e}}$ identifies with the space of pairs of matrices $(B,D)$ from $GL(2,{\Bbb C})$ of the form $B=\pmatrix{-{\lambda} & x \cr 0 & {\lambda}}$, $D=\pmatrix{y & a \cr a & 0}$ and it has dimension $4$. It is isomorphic to ${\Bbb C}^*\times {\Bbb C}^*\times {\Bbb C}^2$ as an algebraic variety. Clearly ${\cal M}_{\bar{R_e}}={\cal M}_{R_e}\cup {\cal M}_{-I}$. The space ${\cal M}_{-I}$ identifies with the space of pairs of matrices $(B,D)$ from $GL(2,{\Bbb C})$ of the form $B=\pmatrix{-{\lambda} & 0 \cr 0 & {\lambda}}$, $D=\pmatrix{0 & a \cr a & 0}$, which is isomorphic to ${\Bbb C}^*\times {\Bbb C}^*$ and has dimension $2$. (So, ${{\cal M}_{\cal C}}$ is isomorphic to to ${\Bbb C}^*\times {\Bbb C}^*\times {\Bbb C}^2{\setminus} {0}$, and hence is not affine.) In general, property P for the class ${\cal C}$ implies that the affine variety ${{\cal M}_{\bar{\cal C}}}$ is stratified by the smooth locally closed subvarieties ${{\cal M}_{\cal C}}$ and ${\cal M}_{{\cal C}_i}$ and the codimension of ${\cal M}_{{\cal C}_i}$ in ${{\cal M}_{\bar{\cal C}}}$ is equal to the codimension of ${\cal C}_i$ in ${\bar{\cal C}}$. This means, first of all, that all the singularities of ${{\cal M}_{\bar{\cal C}}}$ are in codimension at least $2$. \begin{th} If ${\cal C}$ satisfies property P then ${{\cal M}_{\bar{\cal C}}}$ is a normal affine algebraic variety. It is Cohen-Macaulay and has rational singularities. \end{th} Before we prove this assertion, we exhibit an auxiliary result. \begin{lem} If $[B,D]=A$ and $\dim Z(B,D)=1$ then the differential map $d{\kappa}: T_{(B,D)}GL(n,{\Bbb C})^2\to T_ASL(n,{\Bbb C})$ is surjective${}^1$. \end{lem} \footnotetext[1]{Jean-Luc Brylinski noticed that it should be true in every characteristic.} {\noindent{\it Proof.\ \ }} We identify the first tangent space with $\frak{gl}(n,{\Bbb C})^2$ and the second one with $\frak{sl}(n,{\Bbb C})$ via the corresponding left multiplications. Computations show that with these identifications $d{\kappa}$ sends $(x,y)\in \frak{gl}(n,{\Bbb C})^2$ to $DB(D^{-1}xD-x+y-B^{-1}yB)B^{-1}D^{-1}$. So, it is enough to show that $R(AdD-1)+R(AdB-1)=\frak{sl}(n,{\Bbb C})$, where $R(L)$ denotes the range of a linear operator $L$. With respect to the bilinear form $Tr(XY)$ one has $R(AdB-1)=Ker(AdB-1)^{\perp}$. We notice that $Ker(AdB-1)=z(B)$ - the centralizer of $B$ in the Lie algebra. Now the condition of the lemma implies that $$ R(AdD-1)+R(AdB-1)=Ker(AdB-1)^{\perp} + Ker(AdD-1)^{\perp}= $$ $$ =(z(B)\cap z(D))^{\perp}=z(B,D)^{\perp}=\frak{sl}(n,{\Bbb C}). \ \ \ \bigcirc $$ \noindent{\it Proof of the theorem.} Let us consider the morphism of algebraic varieties ${\kappa} : {{X}_{\bar{\cal C}}}\to{\bar{\cal C}}$. We notice that every conjugacy class in the closure of a conjugacy class with property P also satisfies property P. To see that ${\kappa}$ is actually a smooth morphism, one notices that smoothness is preserved by base extensions. Let $U$ be the open set of matrices with property P and $U^{(2)}\subset GL(n, {\Bbb C})^2$ its preimage under ${\kappa}$. We saw in Lemma 3.3 that ${\kappa} : U^{(2)}\to U$ is smooth. For a class ${\cal C}$ with property P we have the inclusion ${\bar{\cal C}}\hookrightarrow U$. Now we make the base change : $U^{(2)}\times_U {\bar{\cal C}}\to{\bar{\cal C}}$. We conclude that ${\kappa} : {{X}_{\bar{\cal C}}}\to{\bar{\cal C}}$ is a smooth morphism. Now we invoke the theorem of Kraft and Procesi (\cite{KP}) which tells us that ${\bar{\cal C}}$ is normal, Cohen-Macaulay with rational singularities. As a consequence we obtain the fact that ${{X}_{\bar{\cal C}}}$ has rational singularities. Now we use the theorem of Boutot \cite{Bou}, which implies that ${{\cal M}_{\bar{\cal C}}}$ has also rational singularities. In particular, ${{\cal M}_{\bar{\cal C}}}$ is normal and Cohen-Macaulay. $\bigcirc$ \ Actually, we have proved that every connected component of ${{X}_{\bar{\cal C}}}$ (and ${{\cal M}_{\bar{\cal C}}}$) is normal. But it seems likely that those varieties are connected. As far as we know this is an open problem. It is very possible that whenever a class ${\cal C}$ in the image of ${\kappa}$ is not the identity, we can find a pair $(B,D)\in{{X}_{\cal C}}$ with trivial $Z(B,D)$. If it would be so, one could reprove the results of Shoda and Ree (Propositions 2.2 and 5.2) in a nice algebraic way as follows. Let $X_I\subset G\times G$ be the variety of commuting matrices. The above result on the surjectivity of $d{\kappa}$ implies that the image of ${\kappa}$ restricted to $(G\times G){\setminus} X_I$ is open in $G{\setminus} \{I\}$. We know from previous explicit constructions that every semisimple or unipotent element is in the image of ${\kappa}$. One needs only to remark that each conjugacy class in $G{\setminus} \{ I\}$ is either unipotent or contains in its closure a semisimple element $\ne I$. \begin{proposition} If ${\cal C}$ has property P then ${{\cal M}_{\cal C}}$ is a smooth algebraic variety. \end{proposition} {\noindent{\it Proof.\ \ }} The statement follows from the above theorem, because the fibers of the map ${\kappa}: {{X}_{\bar{\cal C}}}\to{\bar{\cal C}}$ are smooth varieties. $\bigcirc$ \ By a procedure similar to the one we described above, it is possible to define the space ${{\cal M}_{\cal C}}$ for {\it any} conjugacy class ${\cal C}$. (First, we define ${{\cal M}_{\bar{\cal C}}}$ and then using its natural stratification we throw out irrelevant pieces.) The following lemma which was pointed out to me by J.-L. Brylinski calculates the dimension of the tangent space to ${{\cal M}_{\cal C}}$ at the class $(B,D)$. A point $(B,D)\in{{X}_{\cal C}}$ is called a general point if ${{X}_{\cal C}}$ is smooth at $(B,D)$ and all stabilizers of elements in some neighbourhood of $(B,D)$ in ${{X}_{\cal C}}$ form a smooth group bundle. \begin{lem} Let ${\cal C}$ be a conjugacy class in $SL(n, {\Bbb C})$. For a general point $(B, D)\in{{X}_{\cal C}}$ one has $\dim (T_{(B,D)}{{\cal M}_{\cal C}})=\dim {\cal C} +2\dim Z(B,D).$ \end{lem} {\noindent{\it Proof.\ \ }} The loops $a$ and $b$ generate freely the fundamental group of the elliptic curve with one puncture ${\Sigma}\setminus \{O\}$. The corresponding monodromy transformations $B$ and $D$ define the local system $V$ on ${{\Sigma}}$ of the dimension $n$. Let us define a manifold $X$ as consisting of all conjugacy classes of homomorphisms $\rho: \pi_1({\Sigma}{\setminus}\{O\})\to SL(n,{\Bbb C})$ such that the image of $\rho$ is Zariski dense in $SL(n,{\Bbb C})$. (So that ${{\cal M}_{\cal C}}$ is a subspace of $X$.) The tangent space to $X$ in the class of $\rho$ identifies, by a well-known theorem of A. Weil, with the group cohomology $H^1(\pi_1({\Sigma}{\setminus}\{O\}),{\frak{g}})$, where ${\frak{g}}=\frak{sl}(n, {\Bbb C})$ is a $\pi_1({\Sigma}{\setminus}\{O\})$-module via the adjoint action followed by $\rho$. For any $A\in SL(n,{\Bbb C})$ we identify the tangent space $T_ASL(n, {\Bbb C})$ to ${\frak{g}}$ via the action of the left translation by $A$. The tangent space to the conjugacy class ${\cal C}_A$ of $A$ is the subspace of ${\frak{g}}$ given as the range of $Ad(A)-1: {\frak{g}}\to{\frak{g}}$. If $\Gamma$ is the cyclic subgroup generated by ${\gamma}=aba^{-1}b^{-1}$ then the cohomology group $H^1(\Gamma,{\frak{g}})$ is the cokernel of the map $Ad(A)-1$. So the tangent space $T_{[\rho ]}{{\cal M}_{\cal C}}$ identifies with the subspace $Ker(H^1(\pi_1({\Sigma}{\setminus}\{O\}),{\frak{g}})\to H^1(\Gamma,{\frak{g}}))$ of $T_{[\rho ]}X=H^1(\pi_1({\Sigma}{\setminus}\{O\}),{\frak{g}})$. Thus, \begin{equation} T_{(B,D)}{{\cal M}_{\cal C}}=Ker[H^1({{\Sigma}}\setminus \{O\}, End(V))\to H^1(D^*, End(V))], \end{equation} where $D^*$ is a small disk around the puncture. (It is the same as $$ Ker[H^1(\pi_1({{\Sigma}}\setminus \{O\}), End(V))\to H^1({\Bbb Z} , End(V))], $$ because $\pi_1(D^*)={\Bbb Z}$.) The Euler characteristic of the punctured elliptic curve is $\chi({{\Sigma}}\setminus \{O\})=-1$ and whence, $$ \dim [H^0({{\Sigma}}\setminus \{O\}, End(V))]-\dim [H^1({{\Sigma}}\setminus \{O\}, End(V))]= $$ $$ =n^2\chi({{\Sigma}}\setminus \{O\})=-n^2. $$ But $H^0({{\Sigma}}\setminus \{O\}, End(V))=End(V)^{B,D}=Z(B,D)$ in $GL(n, {\Bbb C})$. So, \break $\dim H^1({{\Sigma}}\setminus \{O\}, End(V))=n^2+\dim Z(B, D)$. Also we notice that the map in the equation (3.1) is onto due to the exact sequence${}^1$ \footnotetext[1]{The fact that $T_{(B,D)}{{\cal M}_{\cal C}} = Im(H^1_c({\Sigma}\setminus\{0\}, End(V))\to H^1({\Sigma}\setminus\{0\}, End(V)))$ was also used in \cite{BG}.} $$ H^1({{\Sigma}}{\setminus} \{O\}, End(V))\to H^1(D^*, End(V))\to H^2_c({{\Sigma}}{\setminus}\{O\}, End(V)) \to $$ $$ \to H^2({{\Sigma}}{\setminus}\{0\}, End(V))\to 0. $$ The group $H^2 ({{\Sigma}}{\setminus} \{ 0\}, End(V))$ is $0$ for dimension reasons. Besides, one sees that the group $H^2_c ({{\Sigma}} {\setminus} \{ 0\}, End (V))$ is dual to $H^0(({{\Sigma}}{\setminus} \{ 0\}, End(V))$, because the local system $End(V)$ is self-dual. The group $H^0(({{\Sigma}}{\setminus} \{ 0\}, End(V))$ is the group of matrices which commute with $B$ and $D$. It follows that the image of the linear map $H^1({{\Sigma}}{\setminus} \{ 0\}, End(V)) \to H^1(D^*, End(V))$ has dimension equal to $$\dim H^1(D^*,End(V)) - \dim Z(B,D) = \dim Z([B,D]) - \dim Z(B,D).$$ Therefore the dimension of the kernel of this same map is equal to $$\dim H^1({{\Sigma}}{\setminus} \{ 0\},End(V))-\dim Z([B,D])+\dim Z(B,D)$$ which is equal to $$n^2+\dim Z(B,D)-\dim Z([B,D])+\dim Z(B,D)=\dim ({\cal C})+2\dim Z(B,D).$$ Here we used the fact that $n^2 = \dim GL(n, {\Bbb C}) = \dim ({\cal C}) + \dim Z(g)$, $g\in {\cal C}$. $\bigcirc$ \ First, we notice that the space ${{\cal M}_{\cal C}}$ has even dimension, (the smooth loci are actually known to be hyper-K\"{a}hler manifolds). Also we have \begin{cor} $\dim ({{X}_{\cal C}})=n^2+\dim {\cal C} +\dim Z(B,D)$, where $(B,D)$ is a generic element of ${{X}_{\cal C}}$. \end{cor} \section{On subalgebras generated by pairs} \setcounter{equation}{0} Let $p$ be an integer, $p>1$, and let $G^p=\underbrace{G\times\dots\times G}_p$, where $G$ is a reductive algebraic group (over ${\Bbb C}$) with Lie algebra ${\frak{g}}$. Let ${\bf x}=(x_1,\dots,x_p)\in G^p$, $G.{\bf x}$ - its orbit, and $A({\bf x})$ - the algebraic subgroup of $G$ generated by the set $\{x_1,\dots,x_p\}$. (So $A({\bf x})$ is the Zariski closure of the subgroup of $G$ in the abstract sense generated by the set $\{x_1,\dots,x_p\}$.) Let also $\pi: G^p\to G^p/G$ stand for the quotient morphism. Following Richardson \cite{Rich} we call ${\bf x}$ a {\it semisimple $p$-tuple} if $A({\bf x})$ is linearly reductive. (Since we are in characteristic zero this it is equivalent to reductive.) We cite from \cite{Rich} the following \begin{th} The orbit $G.{\bf x}$ is closed if and only if ${\bf x}$ is semisimple. \end{th} This allows us to apply our knowledge to the situation of the group $GL(n,{\Bbb C})$ and $p=2$. \begin{lem} If ${\cal C}$ satisfies property P and $[B,D]\in{\cal C}$ then the algebraic subgroup $A(B,D)$ of $GL(n,{\Bbb C})$ is reductive. \end{lem} {\noindent{\it Proof.\ \ }} We saw before (sections 2 and 3) that property P for ${\cal C}$ implies that the orbit of the element $(B,D)$ is closed. Now the above theorem finishes the job. $\bigcirc$ \begin{prop} If ${\cal C}$ satisfies property P and $[B,D]\in{\cal C}$, then $A(B,D)=GL(n,{\Bbb C})$. \end{prop} {\noindent{\it Proof.\ \ }} Let $\frak a\subset \frak{gl}(n,{\Bbb C})$ be the Lie algebra of $A(B,D)$. The lemma above implies that $\frak a$ is reductive and its centralizer inside $\frak{gl}(n,{\Bbb C})$ is the center of the algebra. Now we apply the double commutant theorem to obtain that $\frak a$ actually coincides with $\frak{gl}(n,{\Bbb C})$. $\bigcirc$ \section{Symplectic and orthogonal groups} \setcounter{equation}{0} Here we will formulate property P for orthogonal and symplectic groups. It turns out that many of the results that we proved remain valid for other semisimple algebraic groups. Let $G$ be either $SO(2n, {\Bbb C})$, $SO(2n+1, {\Bbb C})$, or $Sp(2n, {\Bbb C})$. The group $G$ naturally acts on ${\Bbb C}^{2n(+1)}$ preserving the bilinear form $\langle , \rangle$. It is well-known that if ${\lambda}$ is an eigenvalue of $A\in G$ then ${\lambda}^{-1}$ is an eigenvalue too of the same multiplicity and partition. Let ${\lambda}_1^{\pm 1}, {\lambda}_2^{\pm 1}, ..., {\lambda}_n^{\pm 1}$ (and $1$ in the case of $SO(2n+1, {\Bbb C})$) be the set of eigenvalues of a conjugacy class ${\cal C}\subset G$. We say that ${\cal C}$ has {\it property P} if no product of its eigenvalues of the form $\prod_{j\in S}{\lambda}_j^{e_j}$ is equal to one, where $S$ is non-empty subset of $\{ 1,2,...,n\}$ and $e_j=\pm 1$. For an element $A\in {\cal C}$ this is equivalent to the condition that for any isotropic subspace $V$ preserved by $A$ the product of eigenvalues of $A$ in $V$ is not equal to $1$. \ \noindent{\it Remark.${}^1$} One can formulate property P for an element $C$ for any group $G$ as follows. Let $C=C_sC_u$ be a Jordan decomposition into a product of commuting unipotent and semisimple elements and let ${\Bbb C}^r$ be the standard representation space of $G$. Consider $V=\oplus_{i=0}^r\wedge^i{\Bbb C}^r$, which is naturally a representation space of $G$ too. Now we say that $C$ (or its conjugacy class in $G$) has property P if the following two stable subspaces of $V$ have the same dimension: $\dim(V^{C_s})=\dim(V^T)$, where $T$ is a maximal torus of $G$. \ \footnotetext[1]{The idea of this nice remark is due to Ranee Brylinski.} Suppose that an element $K\in G$ belongs to the common stabilizer of elements $B$ and $D$. Also we make an assumption that $K^2\ne I$ - the identity matrix. We will show that we may always find an isotropic subspace $V$ preserved by both $B$ and $D$. It means in turn that the product of eigenvalues of $[B,D]$ in $V$ is equal to $1$. At first, we consider the case when $K$ has an eigenvalue ${\lambda}\ne\pm 1$. Here we may take as $V$ the kernel of $K-{\lambda}.Id$. It remains to assume that $K$ has only $\pm 1$ as the set of eigenvalues. The condition that $K^2\ne I$ implies that there exists a vector $y\in {\Bbb C}^{2n(+1)}$ such that $y$ belongs to the generalized eigenspace of $1$ or $-1$ and $Ky\ne\pm y$. It is equivalent to the condition that there is an eigenvalue $\mu$ of $K$ (which is, of course, $1$ or $-1$), and a $K$-irreducible and $K$-invariant subspace $Y$ such that $\dim Y >1$. Let us consider the subspace $W=Ker(K-\mu.Id)$. Define the subspace $V\subset W$ consisting of all $v\in W$ such that the equation $K^{-1}x=\mu x+v$ has a solution. One notices that $V$ is the intersection of $Ker(K-\mu)$ with $Im(K^{-1}-\mu)$. Then for $v\in V$ we have $$ \langle v,v\rangle =\langle v, K^{-1}x-\mu x\rangle =\langle Kv-\mu v,x\rangle =0. $$ Moreover, if $B$ commutes with $K$, then $V$ is also $B$-invariant. The fact that $V\ne 0$ follows from the choice of $\mu$. Now we will simply rewrite some of the statements which we proved before for the group $SL(n, {\Bbb C})$ and which continue to be true in the case when $G=SO(2n, {\Bbb C})$, $SO(2n+1, {\Bbb C})$, or $Sp(2n, {\Bbb C})$. We just saw that \begin{th} Let $B, D\in G$ be such that $[B,D]$ satisfies property P. Then the common stabilizer of $B$ and $D$ is finite of exponent $2$. \end{th} Let $G^2$ stand for the cartesian product of two copies of $G$ and let ${\kappa}=[,]: G^2\to G$ be the commutator map. The next proposition is quoted from \cite{Ree}: \begin{prop} The map ${\kappa}$ is onto. \end{prop} Analogously to Lemma 2.3 one may prove the following \begin{lem} For $n>2$ and a conjugacy class ${\cal C}\subset G$ the following two conditions are equivalent: (i)\ \ \ ${\cal C}$ has property P. (ii)\ For any $(B,D)\in G^2$ such that $[B,D]\in {\cal C}$ we have $\dim Z(B,D)=0$. \end{lem} Also we have \begin{lem} If ${\cal C}$ has property P and $[B,D]\in {\cal C}$ then the orbit of the element $(B,D)\in G^2$ is closed. \end{lem} Just as in section 3, we may define the space ${{X}_{\cal C}}$ and the moduli space ${{\cal M}_{\cal C}}$. \begin{prop} (a) $\dim({{\cal M}_{\cal C}})=\dim{\cal C}+2\dim Z(B,D)$ and $\dim {{X}_{\cal C}}=\dim(G)+\dim{\cal C}+\dim Z(B,D)$, for a generic element $(B,D)\in {{X}_{\cal C}}$. (b) If ${\cal C}$ has property P, then ${{\cal M}_{\cal C}}$ has at worst quotient singularities. \end{prop} Unfortunately, we cannot say anything about the normality of ${{\cal M}_{\bar{\cal C}}}$, because the closures of conjugacy classes in $G$ are not always normal (cf. \cite{KP}). The result from the section 4 about the algebraic subgroup $A(B,D)$ generated by a pair $(B,D)\in G^2$ still generalizes: \begin{prop} If the commutator $[B,D]$ belongs to a conjugacy class ${\cal C}$ with property P, then $A(B,D)=G$. \end{prop} \noindent{\bf Remark.} We saw that the questions of normality of variety ${{\cal M}_{\bar{\cal C}}}$ and rationality of its singularities depend only on the geometry of ${\bar{\cal C}}$ if the class ${\cal C}$ has propetry P. In the case of orthogonal or symplectic group there occur different phenomena which are described by Kraft and Procesi in \cite{KrP}. Using their results one can easily prove the analog of the part {\it (b)} of the above theorem for the group $SO(n, {\Bbb C})$ or $Sp(2n, {\Bbb C})$ according to whenever ${\bar{\cal C}}$ is normal. \section{Generalizations for $p$-tuples.} \setcounter{equation}{0} Here we briefly discuss how results of the paper can be extended to the case of $p$-tuples. Let us consider the operation ${\kappa}:=[,]: G^p\to G$, $$ [A_1, A_2, ..., A_p]=A_1A_2\cdots A_pA_1^{-1}A_2^{-1}\cdots A_p^{-1}. $$ It is clear what property P means in this case. It is still true that if \break $[A_1, A_2, ..., A_p]\in{\cal C}$ has property P, then the stabilizer $Z({\bf x})$ (where ${\bf x} :=\break(A_1, A_2, ... A_p)$) is the center of $G$ in the case $G=SL(n, {\Bbb C})$ and is finite of exponent $2$ when $G$ is orthogonal or symplectic group. Taking $A_3=A_4=\cdots =A_p=1$ we see that this fact actually characterizes the class ${\cal C}$ and that the map ${\kappa}$ is onto. If $[A_1, A_2, ..., A_p]$ has property P, then its orbit is closed. As before, we can define the spaces ${{X}_{\cal C}}$ and ${{\cal M}_{\cal C}}$. They are of dimensions $(p-1)\dim(G)+\dim{\cal C}+\dim Z({\bf x})$ and $(p-2)\dim(G)+\dim{\cal C}+2\dim Z({\bf x})$. When $G=GL(n, {\Bbb C})$ and the class ${\cal C}$ has property P, both ${{X}_{\cal C}}$ and ${{\cal M}_{\cal C}}$ are smooth varieties. Notice, that in the application to Riemann surfaces $p$ is even and so is the dimension of ${{\cal M}_{\cal C}}$. In the case $G=SL(n, {\Bbb C})$ the same method as in section 3 proves the normality of ${{\cal M}_{\bar{\cal C}}}$. We also deduce that if ${\bf x}$ has property P, then the set $\{ A_1, A_2, ..., A_p\}$ generates the whole group $G$. \ \noindent{\bf Remark.} Let ${\bar X}$ be a compact oriented surface of genus $g>0$, let $s_1, ..., s_k\in {\bar X}$ and let $X={\bar X}{\setminus}\{ s_1, ..., s_k\}$ be the punctured surface. We consider local systems of rank $n$ on $X$. Assume that $a_1, a_2, ..., a_{2p}$ are loops in $X$ generating the fundamental group $\pi_1(\bar X, x_0)$. Also let ${\gamma}_i$ be a loop which comes from $x_0$, goes once around $s_i$ counterclockwise and then goes back to $x_0$. One can take them in such a way that ${\gamma}_1{\gamma}_2\cdots {\gamma}_k = [a_1, a_2, ..., a_{2p}]$. Let us have $k$ matrices $C_1, ...,C_k$ defined up to simultaneous conjugation, such that $C_i$ is the monodromy transformations of a local system $V$ on $X$ corresponding to ${\gamma}_i$. So we have the corresponding matrix equation: $C_1C_2\cdots C_k=[A_1, A_2, ..., A_{2p}]$. Here $A_1, ..., A_{2p}$ are the monodromy transformations corresponding to the generators of the fundamental group of the compactified curve. (The local system $V$ is comletely determined by the $2p+k-1$ matrices $C_1, C_2, ..., C_{k-1},A_1, A_2, ..., A_{2p}$.) Under the assumption that $\det(C_1\cdots C_k)=1$ Proposition 2.2 (and 5.2) shows that we always can find a solution of this equation. When the genus is zero this is not always the case (see \cite{Simmat} for details). \thebibliography{123} \bibitem{AB}{M. Atiyah and R. Bott. The Yang-Mills Equations over a Riemann Surface. {\it Phil. Trans. Roy. Soc}, {\bf A308}, 523 (1982)} \bibitem{A}{M. Atiyah, The geometry and physics of knots, {\it Cambridge Univ. Press}, 1990} \bibitem{BG}{I. Biswas and K. Guruprasad, Principal bundles on open surfaces and invariant functions on Lie groups, {\it Int. J. Math.}, {\bf 4}, 1993, 535-544} \bibitem{Bou}{J.-F. Boutot, Singularites rationnelles et quotients par les groupes reductifs, {\it Invent. Math.}, {\bf 88}, 1987, 65-68} \bibitem{KP}{H. Kraft and C. Procesi, Closures of Conjugacy Classes of Matrices are Normal, {\it Inventiones math.} {\bf 53}, 1979, 227-247.} \bibitem{KrP}{H. Kraft and C. Procesi, On the geometry of conjugacy classes in classical groups, {\it Comm. Math. Helv.} {\bf 57}, 1982, 539-602.} \bibitem{MS}{V. Mehta and C. Seshadri, Moduli of Vector Bundles on Curves with Parabolic Structures, {\it Math. Ann.}, {\bf 248}, 1980, 205-239} \bibitem{MT}{T. Motzkin and O. Taussky, Pairs of matrices with property L II, {\it Trans. Amer. Math. Soc.}, {\bf 80}, 1955, 387-401} \bibitem{GIT}{D. Mumford, Geometric Invariant Theory, Springer-Verlag, 1965} \bibitem{Ree}{R. Ree, Commutators in semi-simple algebraic groups, {\it Proc. Amer. Math. Soc.}, {\bf 15}, 1964, 457-460} \bibitem{Rich}{R. W. Richardson, Conjugacy Classes of $n$-tuples in Lie Algebras and Algebraic Groups, {\it Duke Math. J.}, {\bf 57}, 1988, 1-35} \bibitem{Shoda}{K. Shoda, Einige S\"{a}tze \"{u}ber Matrizen, {\it Jap. J. Mat.}, {\bf 13}, 1936, 361-365.} \bibitem{S2}{C. Simpson, Harmonic Bundles on Noncompact Curves, {\it J. Amer. Math. Soc.}, {\bf 3}, 1990, 713-770} \bibitem{Simmat}{C. Simpson, Product of Matrices, {\it Diff. Geom., Global Anal. \& Top.}, CMS Conf. Proc., {\bf 12}, 1992, 157-185} \vskip 0.3in {Dept. of Mathematics, Penn State University, University Park, PA 16802, [email protected]} \end{document}

9703

alg-geom/9703008

en

https://arxiv.org/abs/alg-geom/9703008

alg-geom/9703008

Angelo Vistoli

Angelo Vistoli

The deformation theory of local complete intersections

52 pages. Plain TeX file, with AMS fonts and Eplain macro package (included). Many typos have been corrected, and some material has been added

This is an expository paper on the subject of the title. It assumes basic scheme theory, commutative and homological algebra.

alg-geom

9703

alg-geom/9703030

en

https://arxiv.org/abs/alg-geom/9703030

alg-geom/9703030

Dan Cohen

Daniel C. Cohen, Alexander I. Suciu

Alexander Invariants of Complex Hyperplane Arrangements

26 pages; LaTeX2e with amscd, amssymb packages

Trans. Amer. Math. Soc. 351 (1999), no. 10, 4043-4067

10.1090/S0002-9947-99-02206-0

Let A be an arrangement of complex hyperplanes. The fundamental group of the complement of A is determined by a braid monodromy homomorphism from a finitely generated free group to the pure braid group. Using the Gassner representation of the pure braid group, we find an explicit presentation for the Alexander invariant of A. From this presentation, we obtain combinatorial lower bounds for the ranks of the Chen groups of A. We also provide a combinatorial criterion for when these lower bounds are attained.

alg-geom

\section*{Introduction}\label{sec:intro} Let ${\mathcal A} =\{H_{1},\dots ,H_{n}\}$ be an arrangement of hyperplanes in ${\mathbb C} ^{d}$, with complement $M={\mathbb C} ^{d} \setminus \cup_{i=1}^n H_{i}$, and group $G=\pi _{1}(M)$. Let ${M'}$ be the maximal abelian cover, corresponding to the abelianization $\ab: G \to {\mathbb Z}^n$. The action of ${\mathbb Z}^n$ on ${M'}$ puts on $H_*({M'})$ the structure of a module over the group ring ${\mathbb Z}\Z^n$. This ring can be identified with the ring of Laurent polynomials $\L ={\mathbb Z} [t_{1}^{\pm 1},\dots ,t_{n}^{\pm 1}]$, with $t_i$ corresponding to a standardly oriented meridional loop around $H_i$. The object of our study is the Alexander invariant, $B({\mathcal A})=H_{1}({M'})$, viewed as a module over the ring $\L $. Let $L({\mathcal A})$ denote the intersection lattice of ${\mathcal A}$, with rank function given by codimension (see \cite{OT} as a general reference for arrangements). Let $s$ denote the cardinality of $L_2({\mathcal A})$, the set of rank two elements in $L({\mathcal A})$. From the defining polynomial of ${\mathcal A}$, one can compute the Moishezon-Libgober braid monodromy homomorphism, $\a:F_s\to P_n$, see~\cite{CS3}. This homomorphism determines a finite presentation for the group of the arrangement: $G=\langle t_{1},\dots ,t_{n}\mid \a _{k}(t_{i})=t_{i} \rangle$, where $\a_1,\dots ,\a_s$ generate the image of $\a$. The braid monodromy may also be used to obtain a finite presentation for the Alexander invariant $B({\mathcal A})$. We accomplish this here, by means of the Gassner representation, $\Theta :P_{n} \to \GL (n,\Lambda )$, the Fox calculus, and homological algebra. Surprisingly, the size of the presentation depends only on the first two betti numbers of the complement: there are $\binom{n}{2}$ generators and $\binom{n}{3}+b_{2}(M)$ relations. When ${\mathcal A} $ is the complexification of a real arrangement, the presentation of $B({\mathcal A})$ can be simplified to $\binom{n}{2} - b_{2}(M)$ generators and $\binom{n}{3}$ relations. More generally, if $G$ is the group of a collection of $s$ basis-conjugating automorphisms of a finitely generated free group $F_n$, our methods yield a presentation of $B(G)$ with $\binom{n}{2}$ generators and $\binom{n}{3}+ns$ relations. In particular, the Alexander invariant of any pure link has such a presentation (with $s=1$). This should be compared with the general situation for links in $S^3$, where there is no upper bound on the number of relations, see~\cite{Ma}. Note that the Alexander invariant is isomorphic to $G'/G''$ (with the usual $G/G'$ action), and so depends only on the isomorphism type of $G$. Consequently, we may obtain invariants of an arrangement ${\mathcal A}$ from the module $B=B({\mathcal A})$ and its presentation. For instance, if $\Delta$ is a presentation matrix for $B$, the elementary ideal $E_k(B)$ is defined to be the ideal generated by the codimension $k$ minors of $\Delta$. It is well-known that these ideals depend only on the module $B$. These ideals, and the closely related characteristic varieties, arise in the study of plane algebraic curves; see for instance the recent works of Hironaka~\cite{eko} and Libgober~\cite{L2}. The structure of the elementary ideals and characteristic varieties of the Alexander invariant an arrangement will be the subject of a future work. In this paper, we focus on Chen groups. The Chen groups of $G$ are the lower central series quotients of the maximal metabelian quotient $G/G''$. Using an observation of Massey~\cite{Ma} relating the Chen groups and the Alexander invariant, together with Mora's tangent cone algorithm, we obtain an algorithm for computing the Chen groups of an arrangement from the presentation of the Alexander invariant $B$. The ranks of the Chen groups often serve to distinguish the groups of combinatorially ``similar'' arrangements. This is particularly useful for fiber-type arrangements, where the ranks of the lower central series quotients of $G$ itself are determined by the exponents of the arrangement. On the other hand, we know of no combinatorially equivalent arrangements whose Chen groups differ. The precise relation between the Chen groups and the intersection lattice of a central arrangement ${\mathcal A}$ is not known. We obtain partial results toward this end here. To each element $V\in L_2({\mathcal A})$, we associate a ``local'' Alexander invariant $B_V$. Algebraic considerations yield a surjective homomorphism $B \to {B^{\operatorname{cc}}}$, where ${B^{\operatorname{cc}}}=\oplus_V B_V$ is the ``coarse combinatorial Alexander invariant'' of ${\mathcal A}$, determined by (only) the multiplicities of the elements of $L_2({\mathcal A})$. From this map, we obtain combinatorial lower bounds on the ranks of the Chen groups of ${\mathcal A}$. These ranks are determined by the $I$-adic completion, $\widehat B$, of the Alexander invariant $B$, where $I$ is the augmentation ideal of $\Lambda$. We find a combinatorial criterion for when the completion of the Alexander invariant of ${\mathcal A}$ decomposes as a direct sum, i.e.,~ $\widehat B\xrightarrow{\sim}{\widehat{B}^{\operatorname{cc}}}$. We also obtain a combinatorial formula for the rank of the third Chen group of any arrangement. The above results may be viewed as evidence that the ranks, $\theta_k$, of the Chen groups of ${\mathcal A}$ are combinatorially determined. In~\cite{CS1}, we conjectured an explicit combinatorial formula for $\theta_k$, for sufficiently large $k$. This formula involved the number $\beta$ of subarrangements of ${\mathcal A}$ lattice-isomorphic to the braid arrangement ${\mathcal A}_4~\subset~{\mathbb C}^4$. In the present context, we show by example that if $\beta>1$, then the map $\widehat B \to {\widehat{B}^{\operatorname{cc}}}$ is not an isomorphism. Other examples exhibit combinatorially different ways this map can fail to be an isomorphism. These provide counterexamples to the aforementioned formula, and illustrate the subtlety of the relationship between the Chen groups and the lattice of an arrangement. Our results on Chen groups parallel a portion of Falk's work on the LCS quotients of an arrangement group. The combinatorial lower bounds we obtain for the ranks of the Chen groups are analogous to those for the ranks, $\phi_k$, of the LCS quotients found in~\cite{F2}. Moreover, the formula we obtain for $\theta_3 = \phi_3$ may be viewed as dual to the description of $\phi_3$ found in~\cite{F1}, \cite{F2}. The precise relationship between the Chen groups and LCS quotients of an arrangement will be explored elsewhere. The structure of the paper is as follows: \begin{itemize} \item In section~\ref{sec:AlexanderChen}, we review Alexander invariants and Chen groups, and present a Groebner basis algorithm for determining the latter. The section concludes with an analysis of the Alexander invariant and Chen groups of a product of spaces. \item In section~\ref{sec:FoxCalc}, we introduce our basic computational tools: the Fox free differential calculus and the Magnus representations. \item In section~\ref{sec:AlexFreeAuto}, we study the Alexander invariant of the group of a free automorphism. An explicit presentation is given when the automorphism is basis-conjugating. \item In section~\ref{sec:LocalAlexInv}, we find presentations for the local Alexander invariants of an arrangement. \item In section~\ref{sec:AlexInvArr}, the presentation for the Alexander invariant of an arrangement is obtained. \item In section~\ref{sec:DecompAlex}, the homomorphism $B \to {B^{\operatorname{cc}}}$ is defined, and its completion proven to be an isomorphism when a certain criterion is satisfied. \item In section~\ref{sec:Combinatorics}, the aforementioned criterion is shown to be combinatorial, and lower bounds on the ranks of the Chen groups of an arrangement are obtained. \item In section~\ref{sec:Examples}, we illustrate our results by means of several explicit examples. \end{itemize} \begin{conv} Given a group $G$, we will denote by $\Aut(G)$ the group of {\em right} automorphisms of $G$, with multiplication $\alpha\cdot \beta=\beta\circ\alpha$. We will regard all modules over the group ring ${\mathbb Z} G$ as {\em left} modules. Elements of the free module $({\mathbb Z} G)^{n}$ are viewed as {\em row} vectors, and ${\mathbb Z} G$-linear maps $({\mathbb Z} G)^{n}\to ({\mathbb Z} G)^{m}$ are viewed as $n\times m$ matrices which act on the {\em right} (so that the matrix of $B\circ A$ is $A\cdot B$). We will write $A^{\top }$ for the transpose of $A$, and $\left( A_1\ \cdots\ A_s \right)^{\top}$ for $\bigl(\begin{smallmatrix}A_1 \\ \dots \\ A_s \end{smallmatrix}\bigr)$. If $\phi :G\to H$ is a homomorphism, $\tilde {\phi }:{\mathbb Z} G\to {\mathbb Z} H$ denotes its ${\mathbb Z}$-linear extension to group rings. We will abuse notation and also write $\tilde {\phi }:({\mathbb Z} G)^{n}\to ({\mathbb Z} H)^{n}$ for the map $\oplus _{1}^{n} \tilde \phi $. \end{conv} \section{Alexander Invariants, Chen Groups, and Products} \label{sec:AlexanderChen} We start by reviewing the definition of the Alexander invariant of a finite complex. We then present an algorithm for computing the ranks of the Chen groups of a group, based on a presentation of this module. Finally, we determine the structure of the Alexander invariant of a product of spaces in terms of those of the factors. \subsection{Alexander Invariants}\label{subsec:Alexander} Let $M$ be a path-connected space that has the homotopy type of a finite CW-complex. Let $G=\pi _{1}(M,*)$ be the fundamental group, and $K=H_{1}(M)$ its abelianization. Let $p:{M'} \to M$ be the maximal abelian cover. The action of $K$ on ${M'} $ passes to an action of $K$ on the homology groups $H_*({M'})$. This defines on $H_*({M'})$ the structure of a module over the group ring ${\mathbb Z} K$. The ${\mathbb Z} K$-module $B=H_{1}({M'})$ is called the (first) {\em Alexander invariant} of $M$. Closely related to it is the (first) {\em Alexander module}, $A=H_{1}({M'} , p^{-1}(*))$. These two modules, together with the augmentation ideal $I=IK=\ker (\epsilon :{\mathbb Z} K \to {\mathbb Z} )$, comprise the Crowell exact sequence, $0\to B\to A \to I\to 0$, of \cite{Cr1}. The two Alexander modules depend only on the group $G$. Indeed, $A={\mathbb Z} K \otimes_{{\mathbb Z} G} IG$, with $K=G/G'$ acting by multiplication on the left factor, and $B=G'/G''$, with the action of $K$ defined by the extension $1\to G'/G''\to G/G'' \to G/G'\to 1$. Since $M$ is by assumption a finite complex, $G$ is a finitely presented group. Hence, the ${\mathbb Z} K$-module $A$ is finitely presented; Fox's free differential calculus provides an explicit presentation (see \cite{Cr2}, and also sections \ref{sec:FoxCalc} and \ref{sec:AlexFreeAuto}). Less evident, but still true, is the fact that $B$ also admits a finite presentation as a ${\mathbb Z} K$-module (see \cite{Cr1}, \cite{Ma}, and also section \ref{sec:AlexFreeAuto}). \subsection{Chen Groups}\label{subsec:Chen} Let $\Gamma _{k}(G)$ denote the $k^{\text{th}}$ lower central series subgroup of $G$, defined inductively by $\Gamma _{1}(G)=G$ and $\Gamma _{k+1}(G)=[\Gamma _{k}(G), G]$ for $k \ge 1$. The projection of $G$ onto its maximal metabelian quotient $G/G''$ induces an epimorphism \begin{equation*} \label{eq:Gamma} {\frac{\Gamma _{k}(G)}{\Gamma _{k+1}(G)}} \twoheadrightarrow{\frac{\Gamma _{k}(G/G'')} {\Gamma _{k+1}(G/G'')}} \end{equation*} from the $k^{\text{th}}$ lower central series quotient of $G$ to the $k^{\text{th}}$ Chen group of $G$. Since $G$ is finitely presented, these quotients are finitely generated abelian groups, whose ranks we will denote by $\phi_k$, respectively $\theta_k$. It is readily seen that $\phi_k=\theta_k$ for $k\le 3$, and $\phi_k \ge \theta_k$ for $k> 3$. The Chen groups of $G$ can be determined from the Alexander invariant of $G$. Indeed, Massey \cite{Ma} noted the following isomorphism, for $k\ge 2$: \begin{equation*} \label{eq:massey} {\frac{\Gamma _{k}(G/G'')}{\Gamma _{k+1}(G/G'')}}= {\frac{I^{k-2} B}{I^{k-1}B}}. \end{equation*} Thus, the Chen groups are determined by $\gr B=\bigoplus _{k\ge 0} I^{k}B/I^{k+1}B$, viewed as a graded module over the graded ring $\gr {\mathbb Z} K =\bigoplus _{k\ge 0} I^{k}/I^{k+1}$. Now assume $K$ is free abelian, and fix a system of generators, $t_{1},\dots , t_{n}$. The group ring ${\mathbb Z} K$ can be identified with the ring of Laurent polynomials in $n$ variables, $\L ={\mathbb Z} [t_{1}^{\pm 1},\dots ,t_{n}^{\pm 1}]$. The ring $\L$ can be viewed as a subring of the formal power series ring $P={\mathbb Z} [[x_{1},\dots ,x_{n}]]$ via the ``Magnus embedding,'' given by $t_{i}\mapsto 1-x_{i}$ and $t_{i}^{-1}\mapsto \sum _{k=0}^{\infty }x_{i}^{k}$. Let $\widehat {\Lambda } =\varprojlim \L /I^{k}$ be the completion of $\L $ relative to the $I$-adic topology. Then, the Magnus embedding extends to a ring isomorphism $\widehat {\Lambda } \xrightarrow{\sim} P$. Consider the $\mathfrak{m}$-adic filtration on $P$, where $\mathfrak{m}=\langle x_{1},\dots , x_{n} \rangle $, and its associated graded ring, $\gr P=\bigoplus _{k\ge 0}\mathfrak{m}^{k}/\mathfrak{m}^{k+1}$. As is well-known, this ring is isomorphic to the polynomial ring $R={\mathbb Z}[x_1,\dots,x_n]$. Moreover, the Magnus embedding induces a graded ring isomorphism $\gr \L \xrightarrow{\sim} \gr P=R$. Let $\widehat B $ be the $I$-adic completion of $B$, and $\gr \widehat B =\bigoplus _{k\ge 0} \mathfrak{m}^{k} \widehat B /\mathfrak{m}^{k+1}\widehat B $ the associated graded module. Then, the canonical map $B\to \widehat B $ induces an isomorphism $\gr B \xrightarrow{\sim}\gr \widehat B $ of graded modules over the ring $R$. Combining these facts, we can restate Massey's result as follows: \begin{thm}[{\cite{Ma}}]\label{thm:massey} The generating series for the ranks of the Chen groups of $G$, $\sum _{k=0}^{\infty }\theta _{k+2}t^{k},$ is equal to the Hilbert series of the graded module associated to the $I$-adic completion of $B(G)$, $\sum _{k=0}^{\infty }\operatorname{rank} (\mathfrak{m}^{k} \widehat B /\mathfrak{m}^{k+1}\widehat B )t^{k}.$ \end{thm} An immediate consequence of this theorem is that, for $k$ sufficiently large, $\theta _{k}$ is given by a polynomial in $k$. Indeed, this is just the Hilbert-Serre polynomial of $\gr \widehat B $, see \cite{ZS}. \subsection{Groebner Bases}\label{subsec:Groebner} Let $\L ^{a} \xrightarrow{\Delta} \L ^{b} \rightarrow B \rightarrow 0$ be a (finite) presentation of the Alexander invariant. Note that, by replacing the generators of the free module $\L ^{a}$ by suitable multiples if necessary, we may assume that the entries of the matrix of $\Delta $ are polynomials in the variables $t_{i}$. Let $J=\im \Delta $. A presentation for the $I$-adic completion of $B$ is given by $\displaystyle {\widehat {\Lambda } ^{a} \xrightarrow{\widehat {\Delta }}\widehat {\Lambda } ^{b} \rightarrow \widehat B \rightarrow 0}$, where $\widehat {\Delta } $ is obtained from $\Delta $ via the Magnus embedding. Clearly, $\im \widehat {\Delta } =\widehat J $. Since all the entries of the matrix for $\widehat {\Delta } $ belong to the subring $R\subset P$, we may restrict $\widehat {\Delta } $ to a map $\boldsymbol \Delta : R^{a}\to R^{b}$, whose image, $\widehat J \cap R ^{b}$, we denote by $\mathbf{J}$. We must find a presentation for the associated graded module $\gr \widehat B =\gr (P^{b}/\widehat J )$. This module is isomorphic to $R^{b}/\text {\scshape lt} (\mathbf{J} )$, where $\text {\scshape lt} (\mathbf{J} )$ is the submodule of $R^{b}$ consisting of lowest degree homogeneous forms of elements in $\mathbf{J}$, see \cite{ZS}. We are left with finding a finite generating set for $\text {\scshape lt} (\mathbf{J})$. Such a set is provided by Mora's algorithm for obtaining the tangent cone of an affine variety at the origin, see \cite{CLO}, \cite{BW}. Essentially, we must determine a (minimal) {\em Groebner basis} ${\mathcal G} =\{g_{1},\dots ,g_{c}\}$ for the module $\mathbf{J}$, with respect to a suitable monomial ordering. Then, $\text {\scshape lt} (\mathbf{J})$ has Groebner basis $\text {\scshape lt} ({\mathcal G} )=\{\text {\scshape lt} g_{1},\dots ,\text {\scshape lt} g_{c}\}$, from which we can extract a minimal Groebner basis $\H =\{h_{1},\dots ,h_{d}\}$. Putting all these facts together, we obtain the following. \begin{thm}\label{thm:grB} The module $\gr \widehat B $ has presentation $R^{d} \xrightarrow{\gr \widehat {\Delta }} R^{b} \rightarrow \gr \widehat B \rightarrow 0$, where the rows of $\gr \widehat {\Delta } $ constitute a minimal Groebner basis for the module generated by the rows of the matrix $\widehat {\Delta } $, obtained from a presentation matrix ${\Delta } $ for $B$ by replacing $t_{i}$ by $1-x_{i}$. \end{thm} \begin{exm}\label{exm:Fn} Let $G = F_n $ be a finitely generated free group. A presentation for the Alexander invariant $B$ of $G$ is given by $\L^a \xrightarrow{d_3} \L^b \to B \to 0$, where $a=\binom{n}{3}$, $b=\binom{n}{2}$, and $d_3$ is the differential in the standard $\L$-resolution of ${\mathbb Z}$. In this instance, it is readily checked that the rows of the matrix $\hat d_3$ form a Groebner basis for the module $\mathbf{J}$. A standard argument then yields the ranks of the Chen groups of $F_n$: $\theta_1 = n$ and $\theta_k=(k-1)\binom{k+n-2}{k}$ for $k\ge 2$, a calculation originally due to Murasugi \cite{Mu}. \end{exm} \subsection{Products}\label{subsec:Products} Let $M_{1}$ and $M_{2}$ be two path connected finite CW-complexes, with $K_{i}=H_{1}(M_{i})$ free abelian, and let $M'_{i}$ be the corresponding maximal abelian covers. Then $M=M_{1}\times M_{2}$ has maximal abelian cover ${M'} =M'_{1}\times M'_{2}$, corresponding to $K=H_{1}(M)=K_{1} \times K_{2}$. \begin{prop}\label{prop:kunneth} There is an isomorphism of ${\mathbb Z} K$-modules, \begin{equation*} \label{eq:kunneth} H_{1}({M'} )\cong \bigl ( (H_{1}(M'_{1})\otimes _{{\mathbb Z} K_{1}} {\mathbb Z} K) \otimes _{{\mathbb Z} K_{2}} {\mathbb Z} \bigr ) \oplus \bigl ( (H_{1}(M'_{2}) \otimes _{{\mathbb Z} K_{2}} {\mathbb Z} K) \otimes _{{\mathbb Z} K_{1}} {\mathbb Z} \bigr ) . \end{equation*} \end{prop} \begin{proof} By the K\"{u}nneth formula, the group $H_{1}({M'} )$ is isomorphic to $H_{1}(M'_{1})\otimes H_{0}(M'_{2}) \oplus H_{0}(M'_{1})\otimes H_{1}(M'_{2}) .$ When viewed as a ${\mathbb Z} K$-module, the first summand is isomorphic to \begin{equation*} \label{eq:summand} \begin{split} &( (H_{1}(M'_{1})\otimes _{{\mathbb Z} K_{1}} {\mathbb Z} K)\otimes _{{\mathbb Z} K} {\mathbb Z} K_{1} ) \otimes ( ({\mathbb Z} \otimes _{{\mathbb Z} K_{2}} {\mathbb Z} K)\otimes _{{\mathbb Z} K} {\mathbb Z} K_{2} ) \\ &\qquad = (H_{1}(M'_{1})\otimes _{{\mathbb Z} K_{1}} {\mathbb Z} K) \otimes _{{\mathbb Z} K} ({\mathbb Z} K_{1} \otimes {\mathbb Z} K_{2}) \otimes _{{\mathbb Z} K} ({\mathbb Z} \otimes _{{\mathbb Z} K_{2}} {\mathbb Z} K) \\ &\qquad = (H_{1}(M'_{1})\otimes _{{\mathbb Z} K_{1}} {\mathbb Z} K) \otimes _{{\mathbb Z} K_{2}} {\mathbb Z} , \end{split} \end{equation*} where we made use of the obvious isomorphism $({\mathbb Z} K_{1} \otimes {\mathbb Z} K_{2})\cong {\mathbb Z} (K_{1}\times K_{2})$, and viewed the induced module $H_{1}(M'_{1})\otimes _{{\mathbb Z} K_{1}} {\mathbb Z} K$ as a ${\mathbb Z} K_{2}$-module by restriction of scalars. The second summand is treated exactly the same way. \end{proof} We want to find now a presentation for the Alexander invariant $B(M)=H_{1}(M' )$, given presentations for the Alexander invariants $B(M_{i})=H_{1}(M'_{i})$. Fix generators $t_{1}^{(i)},\dots ,t_{n_{i}}^{(i)}$ for $K_{i}$, and use them to identify ${\mathbb Z} K_{i}$ with $\L _{i}$. \begin{thm}\label{thm:alexprod} If the Alexander invariants of $M_{1}$ and $M_{2}$ have presentations $\L _{i}^{a_{i}} \xrightarrow{{\Delta } _{i}} \L _{i}^{b_{i}} \rightarrow B(M_{i}) \rightarrow 0$, then the Alexander invariant of $M=M_{1}\times M_{2}$ has presentation \begin{equation*} \label{eq:alexprod} \L ^{a} \xrightarrow{\begin{pmatrix}{\Delta } _{1}\\ D_{2}^{b_{1}} \end{pmatrix} \oplus \begin{pmatrix}{\Delta } _{2}\\ D_{1}^{b_{2}}\end{pmatrix}} \L ^{b} \rightarrow B(M) \rightarrow 0, \end{equation*} where $a=a_{1}+n_{2}b_{1}+a_{2}+n_{1}b_{2},\, b=b_{1}+b_{2}$, and $D_{i}=(t_{1}^{(i)}-1,\dots ,t_{n_{i}}^{(i)}-1)^{\top }$. \end{thm} \begin{proof} Let us look at the first summand in the direct sum decomposition of $H_{1}({M'} )$ from Proposition~\ref{prop:kunneth}. It is the tensor product over ${\mathbb Z} K$ of two induced modules. The first one is the ${\mathbb Z} K$-module induced from the ${\mathbb Z} K_{1}$-module $H_{1}(M'_{1})$, and has presentation \begin{equation}\label{eq:firstmod} ({\mathbb Z} K)^{a_{1}} \xrightarrow{{\Delta } _{1}} ({\mathbb Z} K)^{b_{1}} \rightarrow H_{1}(X_{1})\otimes _{{\mathbb Z} K_{1}} {\mathbb Z} K \to 0. \end{equation} The second one is the ${\mathbb Z} K$-module induced from the trivial ${\mathbb Z} K_{2}$-module ${\mathbb Z} $, and has presentation \begin{equation}\label{eq:secmod} ({\mathbb Z} K)^{n_{2}} \xrightarrow{D_{2}} {\mathbb Z} K \rightarrow {\mathbb Z} \otimes _{{\mathbb Z} K_{2}} {\mathbb Z} K \rightarrow 0. \end{equation} Taking the tensor product (over ${\mathbb Z} K$) of the complexes \eqref{eq:firstmod} and \eqref{eq:secmod} and truncating yields the following presentation for the first summand of $H_{1}({M'} )$: \begin{equation*}\label{eq:firstsum} ({\mathbb Z} K)^{a_{1}+n_{2}b_{1}} \xrightarrow{\begin{pmatrix}{\Delta } _{1}\{\Delta }_{2}^{b_{1}}\end{pmatrix}} ({\mathbb Z} K)^{b_{1}} \rightarrow (H_{1}(X_{1})\otimes _{{\mathbb Z} K_{1}} {\mathbb Z} K) \otimes _{{\mathbb Z} K_{2}} {\mathbb Z} \rightarrow 0. \end{equation*} The second summand is handled the same way, and that finishes the proof. \end{proof} \begin{cor}\label{cor:chenprod} The ranks of the Chen groups of $G=\pi_1(M_1 \times M_2)$ are given by \begin{equation*} \label{eq:chenprod} \theta_k(G) = \theta_k(G_1)+\theta_k(G_2), \end{equation*} where $G_i=\pi_1(M_i)$. \end{cor} \begin{exm}\label{exm:prodFn} Let $G=F_{d_{1}}\times \dots \times F_{d_{\ell }}$ be a direct product of finitely generated free groups. Using the above result, and the calculation in Example~\ref{exm:Fn}, one can easily recover the ranks of the Chen groups of $G$ announced in \cite{CS1}: $\theta _{1}=\sum _{i=1}^{\ell }d_{i}$ and $\theta _{k}=(k-1)\cdot \sum _{i=1}^{\ell }\binom {k+d_{i}-2}k$ for $k\ge 2$. \end{exm} \begin{exm}\label{exm:cone} Let ${\mathcal A}$ be an affine arrangement of $n$ hyperplanes in ${\mathbb C}^d$, and let $\overline {\mathcal A}$ be the {\em cone} of ${\mathcal A}$, a central arrangement of $n+1$ hyperplanes in ${\mathbb C}^{d+1}$ (see \cite{OT}). It is well-known that the complement $\overline M$ of $\overline {\mathcal A}$ is homeomorphic to the product of the complement $M$ of ${\mathcal A}$ and ${\mathbb C}^*$, $\overline M = M \times{\mathbb C}^*$. Fix a generator $x$ for $\pi_1({\mathbb C}^*) = {\mathbb Z}$. Let $\L={\mathbb Z}[t_1^{\pm 1},\dots,t_n^{\pm 1}]$, and suppose that the Alexander invariant $B$ of ${\mathcal A}$ has presentation $\L^a \xrightarrow{\Delta} \L^b \to B \to 0$ (see section~\ref{sec:AlexInvArr}). Using Theorem~\ref{thm:alexprod}, we obtain a presentation $\overline \L^{a+b} \xrightarrow{\overline \Delta} \overline \L^b \to \overline B \to 0$ for the Alexander invariant of $\overline {\mathcal A}$, where $\overline \L = \L[x^{\pm 1}]$ and $\overline \Delta = \begin{pmatrix}{\Delta } & (x-1)\cdot \id\end{pmatrix}^{\top}$. Thus the ranks of the Chen groups of $G(\overline {\mathcal A})$ coincide with those of $G({\mathcal A})$ for $k \ge 2$. Note that $\theta_1(\overline {\mathcal A}) = \theta_1({\mathcal A}) + 1 = n+1$. \end{exm} \section{A Quick Trip through Fox Calculus} \label{sec:FoxCalc} In this section we review the basics of Fox's free differential calculus, as introduced in \cite{Fo}, and developed in \cite{Bi}, and derive some consequences. \subsection{Fox Gradient}\label{subsec:FoxGrad} Let $F_{n}$ be the free group on generators $t_{1},\dots ,t_{n}$, and ${\mathbb Z} F_{n}$ its group ring. Let $W_{n}=\bigvee _{1}^{n} S^{1}$ be a wedge of $n$ circles, with basepoint $*$ at the wedge point. Let $\widetilde W_{n}$ be the universal cover, with basepoint $\widetilde *$, and let $\widetilde C_{\bullet }(\widetilde W_{n})$ be the augmented, equivariant chain complex of $\widetilde W_{n}$. Identifying $C_{0}(\widetilde W_{n})$ with ${\mathbb Z} F_{n}$, and $C_{1}(\widetilde W_{n})$ with $({\mathbb Z} F_{n})^{n}$ (with basis $e_{1},\dots ,e_{n}$ given by the lifts of the 1-cells at $\widetilde *$), we obtain the standard free ${\mathbb Z} F_{n}$-resolution of ${\mathbb Z} $, \begin{equation*} \label{eq:fnres} 0\rightarrow ({\mathbb Z} F_{n})^{n} \xrightarrow{\partial _{1}} {\mathbb Z} F_{n} \xrightarrow{\epsilon} {\mathbb Z} \rightarrow 0, \end{equation*} where $\partial _{1}(e_{i})=t_{i}-1$ and $\epsilon (t_{i})=1$. The Fox Calculus is based on the observation that the augmentation ideal, $IF_{n}=\ker \epsilon $, is a free ${\mathbb Z} F_{n}$-module of rank $n$, generated by the entries of the matrix of $\partial _{1}$. This can be rephrased as follows: Given any $w\in {\mathbb Z} F_{n}$, there exist unique elements $\frac{\partial w}{\partial t_{i}} \in {\mathbb Z} F_{n}$ (called the Fox derivatives of $w$) such that the following ``fundamental formula of Fox Calculus '' holds: \begin{equation}\label{eq:FTC} w-\epsilon (w)=\sum _{i=1}^{n} \frac{\partial w} {\partial t_{i}} (t_{i}-1). \end{equation} Let us define the {\em Fox gradient} to be the ${\mathbb Z} $-linear homomorphism $\nabla :{\mathbb Z} F_{n} \to ({\mathbb Z} F_{n})^{n}$ given by \begin{equation*}\label{eq:nabla} \nabla (w) = \sum _{i=1}^{n} \frac{\partial w}{\partial t_{i}}\, e_{i}. \end{equation*} Then, formula \eqref{eq:FTC} takes the form $\partial _{1}(\nabla (w)) = w - \epsilon (w).$ From this can be deduced the following ``product rule'' for the Fox gradient: $\nabla (uv) = \nabla (u)\cdot \epsilon (v)+u\cdot \nabla (v).$ In particular, $\nabla (z^{-1})=-z^{-1}\nabla (z)$, for $z\in F_{n}$. Now consider an endomorphism $\alpha :F_{n}\to F_{n}$. This defines a map $\a :W_{n}\to W_{n}$ (unique up to homotopy). The induced chain map $\alpha _{\bullet }:C_{\bullet } (\widetilde W_{n})\to C_{\bullet }(\widetilde W_{n})$ can be written as \begin{equation}\label{eq:jack} \begin{CD} C_{1}(\widetilde W_{n}) @>\partial _{1}>> C_{0}(\widetilde W_{n})\\ @VVJ(\alpha )\circ \tilde \alpha V @VV\tilde \alpha V \\ C_{1}(\widetilde W_{n}) @>\partial _{1}>> C_{0}(\widetilde W_{n}) \end{CD} \end{equation} where $J(\alpha ):({\mathbb Z} F_{n})^{n}\to ({\mathbb Z} F_{n})^{n}$ is the {\em Fox Jacobian} of $\a$; namely, the ${\mathbb Z} F_{n}$-linear homomorphism given by $J(\a)(e_{i}) = \nabla (\a (t_{i})).$ If $\b :F_{n}\to F_{n}$ is another endomorphism, the fact that $(\b\circ\a)_{\bullet }=\b _{\bullet }\circ\a _{\bullet}$ may be rephrased as the ``chain rule of Fox Calculus:'' $J(\a\cdot\b)=\tilde \b (J(\a))\cdot J(\b).$ In particular, $J(\a)^{-1}=\tilde\a\circ J(\a^{-1})\circ\tilde\a^{-1}$. \subsection{Abelianized Fox Jacobian} \label{AbelJack} Let ${\mathbb Z} ^{n}$ be the free abelian group on generators $t_{1},\dots ,\linebreak[0] t_{n} $, and identify the group ring ${\mathbb Z} {\mathbb Z} ^{n}$ with $\L ={\mathbb Z} [t_{1}^{\pm 1},\dots ,t_{n}^{\pm 1}]$. Let $T^{n}=\times _{1}^{n} S^{1}$ be the $n$-torus. The augmented, equivariant chain complex, $\widetilde C_{\bullet }=\widetilde C_{\bullet }(\widetilde T^{n})$, of the universal (abelian) cover can be written as \begin{equation}\label{eq:znres} 0\rightarrow C_{n} \xrightarrow{d_{n}} \cdots \rightarrow C_{3} \xrightarrow{d_{3}} C_{2} \xrightarrow{d_{2}} C_{1} \xrightarrow{d_{1}} C_{0} \xrightarrow{\epsilon} {\mathbb Z} \rightarrow 0. \end{equation} Identifying $C_{0}$ with $\L $, $C_{1}$ with $\L ^{n}$, and $C_{k}$ with $\bigwedge ^{k} C_{1}=\L^{\binom {n}k}$, we obtain the standard free $\L$-resolution of ${\mathbb Z} $, with differentials given by $d_{k}(e_J)=\sum _{r=1}^{k} (-1)^{k+r} (t_{j_{r}}-1) \cdot e_{J\setminus\{j_r\}}$, where $e_J=e_{j_1}\wedge\dots\wedge e_{j_k}$ if $J=\{j_1,\dots,j_k\}$. Let $\ab :F_{n}\to {\mathbb Z} ^{n}$, $x\mapsto x^{\ab }$, be the abelianization homomorphism. For an element $w\in {\mathbb Z} F_{n}$, let $\nabla ^{\ab }(w):=\widetilde{\ab}(\nabla (w))\in \L ^{n}$ be its abelianized Fox gradient. This defines a $\L $-linear homomorphism \begin{equation*}\label{eq:nab} \nabla ^{\ab }(w):C_{0}\to C_{1}, \qquad u\mapsto \nabla ^{\ab }(w)\cdot u. \end{equation*} For an endomorphism $\a$ of $F_{n}$, let $\Theta (\a ):=\widetilde{\ab}(J(\alpha )):C_{1}\to C_{1}$ be its abelianized Fox Jacobian. This is a $\L $-linear map, whose matrix has rows $\Theta (\a )(e_{i})=\nabla ^{\ab }(\alpha (t_{i}))$. Abelianizing diagram~\eqref{eq:jack} yields the chain map: \begin{equation}\label{eq:Theta} \begin{CD} C_{1} @>d_{1}>> C_{0}\\ @VV\Theta (\a )\circ\tilde\a V @VV\tilde\a V \\ C_{1} @>d_{1}>> C_{0} \end{CD} \end{equation} Set $\Theta _{k}(\a ) = \bigwedge ^{k} \Theta (\a ):C_k \to C_k$ (in particular, $\Theta_0=\id$). A computation in the exterior algebra $C_{\bullet}=\bigwedge^{\bullet} C_1$ shows that $\Theta _{k-1}(\a )\circ\tilde\a \circ d_k = d_k \circ \Theta _{k}(\a )\circ\tilde\a$ for each $k$, $1\le k \le n$. Thus, \eqref{eq:Theta} extends to a chain map \begin{equation}\label{eq:chainTheta} \Theta_{\bullet}(\a)\circ \tilde\alpha : C_{\bullet}\to C_{\bullet} \end{equation} This chain map is the composite of two chain maps. The first is the (non-$\L$-linear) map $\tilde\a: (C_{\bullet},d_{\bullet}) \to (C_{\bullet},\tilde\a\circ d_{\bullet} \circ\tilde\a^{-1})$. The second is the ($\L$-linear) map $\Theta_{\bullet}(\a): (C_{\bullet},\tilde\a\circ d_{\bullet} \circ\tilde\a^{-1})\to (C_{\bullet},d_{\bullet})$. \subsection{Magnus representations} \label{subsec:MagnusRep} An automorphism $\a \in \Aut(F_{n})$ is called an {\em IA-automorphism} if its abelianization, $\ab (\a ):{\mathbb Z} ^{n}\to {\mathbb Z} ^{n}$, is the identity map. In this case, $\tilde\a=\id$, and so $\Theta _{\bullet }(\a ):C_{\bullet }\to C_{\bullet }$ is a chain map. The set of IA-automorphisms forms a subgroup of $\Aut (F_{n})$, denoted by $\IA (F_{n})$. By the chain rule, $\Theta (\a \cdot \b) =\Theta (\a)\cdot \Theta (\b)$, for $\a,\b \in \IA (F_{n})$. Thus $\Theta : \IA (F_{n}) \to \Aut _{\L }(C_{1})\cong \GL (n,\L )$ is a linear representation of $\IA (F_{n})$, called the {\em Magnus representation}, see \cite{Bi}. From the above discussion, we see that this representation generalizes to $\Theta _{k}: \IA (F_{n}) \to \Aut _{\L }(C_{k}) \cong \GL (\binom {n}{k},\L )$. \begin{rem} For $\a \in \IA (F_{n})$, the chain automorphism $\Theta _{\bullet }(\a ):C_{\bullet }\to C_{\bullet }$ admits the following topological interpretation. The map $\a : W_{n} \to W_{n}$ lifts to a map of the maximal abelian covers, $\a ':W_{n}' \to W_{n}'$. View $W'_{n}$ as the $1$-skeleton of $\widetilde T^{n}$. The map $\a '$ extends to a ${\mathbb Z}^n$-equivariant map $\bar \a : \widetilde T^{n} \to \widetilde T^{n}$. The induced chain map, $\bar\a_{\bullet}: C_{\bullet }(\widetilde T^{n})\to C_{\bullet }(\widetilde T^{n})$, is chain-equivalent to $\Theta _{\bullet }(\a )$. \end{rem} \section{The Alexander Invariant of a Free Automorphism} \label{sec:AlexFreeAuto} In this section, we find presentations for the Alexander module and the Alexander invariant of the group of an IA-automorphism. A more explicit presentation for the latter is given in case the automorphism is basis-conjugating. \subsection{The Group of a Free Automorphism} \label{subsec:GroupFreeAuto} Associated to an automorphism $\a$ of the free group $F_{n}=\langle t_{1},\dots ,t_{n}\rangle $ is the group \begin{equation*}\label{eq:Galpha} G(\a )=\langle t_{1},\dots ,t_{n} \mid t_{1}=\a (t_{1}),\dots , t_{n}=\a (t_{n})\rangle . \end{equation*} Notice that $\a$ induces the identity automorphism on $G(\a)$. In fact, $G(\a)$ is the maximal quotient of $F_{n}$ with this property. Also, note that $G(\a )$ is independent of the choice of free generators for $F_{n}$: If $x_{1},\dots , x_{n}$ is another such choice, then $\langle t_{1},\dots ,t_{n} \mid t_{i}=\a (t_{i})\rangle \cong \langle x_{1},\dots ,x_{n} \mid x_{i}=\a (x_{i})\rangle$. Finally, notice that the group of a free automorphism depends only on the conjugacy class of that automorphism: If $\b \in \Aut (F_{n})$, then $G(\b^{-1} \circ \a \circ \b ) = \langle t_{i} \mid t_{i}= \b^{-1} \circ \a \circ \b (t_{i})\rangle = \langle t_{i} \mid \b (t_{i})=\a (\b (t_{i}))\rangle =\langle x_{i} \mid x_{i}=\a (x_{i})\rangle \cong G(\a )$. See~\cite{Mo} for details. Topologically, the group $G(\a )$ can be interpreted as follows. Recall that $W_{n}$ denotes a wedge of $n$ circles, and that $\a :W_{n}\to W_{n}$ also denotes a basepoint preserving homotopy equivalence that induces $\a :F_{n} \to F_{n}$ on fundamental groups. Let $Y(\a )=W_{n} \times _{\a } S^{1}$ be the mapping torus of $\a $; its fundamental group is the semidirect product $F_{n}\rtimes _{\a } {\mathbb Z} =\langle t_{1},\dots , t_{n}, x\mid x^{-1} t_{i} x = \a (t_{i})\rangle $. Let $X(\a ) = W_{n} \times _{\a } S^{1} \bigcup _{*\times S^{1}} *\times D^{2}$. Then $\pi _{1}(X(\a ))=G(\a )$, and, in fact, $X(\a )$ is homotopy equivalent to the 2-complex associated to the above presentation of $G(\a )$. \subsection{Alexander Invariants} \label{subsec:AlexInvIA} Let $\a $ be an $\IA $-automorphism of $F_{n}$, and $G=G(\a )$ the associated group. Then $H_{1}(G)={\mathbb Z} ^{n}$, the free abelian group generated by $t_{1},\dots , t_{n}$. Let $p:{X'} \to X$ be the corresponding (maximal abelian) cover of $X=X(\a )$. We call the $\L $-modules $A(\a )=H_{1}({X'},p^{-1}(*) )$, resp. $B(\a )=H_{1}({X'} )$ the {\em Alexander module}, resp. {\em Alexander invariant} of $G(\a )$. We wish to find presentations for these modules. First consider $Y=W_{n} \times _{\a } S^{1}$. The chain complex of its maximal abelian cover is obtained using the Fox calculus as in \cite{CS2}: \begin{equation*}\label{eq:chainY} C_{\bullet }(Y'):\qquad C_{2}(Y') \xrightarrow{\begin{pmatrix}\id - x\cdot \Theta (\a ) & d_{1} \end{pmatrix}} C_{1}(Y') \xrightarrow{\begin{pmatrix}d_{1} \\ x-1 \end{pmatrix}} C_{0}(Y') \xrightarrow{\epsilon}{\mathbb Z} \rightarrow 0 \end{equation*} where the chain groups are the modules over $\overline{\L}=\L [x^{\pm 1}]$ given by $C_{2}(Y')=C_{1}\otimes _{\L }\overline{\L}$,\ $C_{1}(Y')=(C_{1}\oplus C_{0})\otimes _{\L }\overline{\L}$, and $C_{0}(Y')=C_{0}\otimes _{\L }\overline{\L}$. It follows that the chain complex of the maximal abelian cover of $X=X(\a )$ is \begin{equation}\label{eq:alexmod} C_{\bullet }({X'} ):\quad C_{1} \xrightarrow{\id -\Theta (\a )} C_{1} \xrightarrow{d_{1}} C_{0} \xrightarrow{\epsilon} {\mathbb Z} \rightarrow 0. \end{equation} Hence, $A(\a )=\coker (\id -\Theta (\a ))$ and $B(\a )=\ker (d_{1})/\im (\id -\Theta (\a )).$ By homological algebra, there exists a chain map from the chain complex \eqref{eq:alexmod} to the free $\L $-resolution \eqref{eq:znres}, extending the identity map of ${\mathbb Z} $: \begin{equation*}\label{eq:CDPhi} \begin{CD} @. C_{1} @>\id -\Theta (\a )>> C_{1} @>d_{1}>> C_{0} @>\epsilon >>{\mathbb Z} @>>> 0\\ @. @VV{\Phi (\a )}V @VV{=}V @VV{=}V @VV{=}V \\ C_{3} @>d_{3}>> C_{2} @>d_{2}>> C_{1} @>d_{1}>> C_{0} @>\epsilon >>{\mathbb Z} @>>> 0\ \end{CD} \end{equation*} A diagram chase shows that $B(\a )=\coker \bigl(\begin{smallmatrix}\Phi (\a ) \\ d_{3}\end{smallmatrix}\bigr).$ To summarize, we have: \begin{prop}\label{prop:IAB} If $\a \in \IA (F_{n})$, the Alexander invariants of $G(\a )$ have presentation \begin{equation*} \label{eq:IAB} C_{1} \xrightarrow{\id - \Theta(\a)} C_{1} \rightarrow{A(\a )} \rightarrow 0, \quad\text{resp.}\quad C_{1}\oplus C_{3} \xrightarrow{\begin{pmatrix}\Phi (\a ) & d_{3}\end{pmatrix}^\top} C_{2} \rightarrow{B(\a )} \rightarrow 0. \end{equation*} \end{prop} \begin{rem}\label{rem:modd3} The map $\Phi (\a )$ is not unique, but rather, it is unique up to chain homotopy: Given two choices, $\Phi _{1}(\a )$ and $\Phi _{2}(\a )$, there is a homomorphism $D:C_{1}\to C_{3}$ such that $\Phi _{1}(\a )-\Phi _{2}(\a )= d_{3}\circ D$. We abbreviate this by saying that $\Phi_{1}(\a)=\Phi_{2}(\a)\mod d_{3}$. Of course, any two choices yield equivalent presentations for $B(\a)$. \end{rem} As noted previously, the group $G(\b \circ \a \circ \b ^{-1})$ is isomorphic to $G(\a )$. The relationship between the corresponding chain maps is as follows: \begin{prop}\label{prop:Phiconj} For $\a , \b \in \IA (F_{n})$, $ \Phi (\b \circ \a \circ \b ^{-1})= \Theta _{2}(\b )\circ \Phi (\a )\circ \Theta (\b ^{-1}) \mod d_{3}. $ \end{prop} \begin{proof} By Remark~\ref{rem:modd3}, it is enough to show that \begin{equation*}\label{eq:bab'} d_{2}\circ \Theta _{2}(\b )\circ \Phi (\a )\circ \Theta (\b ^{-1}) = \id -\Theta (\b \circ \a \circ \b ^{-1}). \end{equation*} Since the right-hand side equals $\Theta (\b )\circ (\id -\Theta (\a ))\circ \Theta (\b ^{-1})$, the claim follows from the equalities $d_{2}\circ \Theta _{2}(\b )=\Theta (\b )\circ d_{2}$ and $d_{2}\circ \Phi (\a )=\id -\Theta (\a )$. \end{proof} \subsection{Basis-Conjugating Automorphisms}\label{subsec:BasisConj} An automorphism $\a $ of $F_{n}=\langle t_{1},\dots ,t_{n}\rangle $ is called a {\em basis-conjugating automorphism} if there exists an $n$-tuple $\mathbf{z} = (z_{1}, \dots , z_{n})$, with $z_{i}\in F_{n}$, such that $\a =\c _{\mathbf{z}}$, where $\c _{\mathbf{z}}(t_{i})=z_{i}t_{i}z_{i}^{-1}.$ The basis-conjugating automorphisms of $F_{n}$ form a subgroup, $\CA (F_{n})$, of $\Aut (F_{n})$. For $\a \in \CA (F_n)$, the following definition/proposition gives an explicit formula for $\Phi (\a )$. \begin{prop}\label{prop:Phidef} For $\c _{\mathbf{z}}\in \CA (F_{n})$, define the $\L $-homomorphism $\Phi (\c _{\mathbf{z}}):C_{1}\to C_{2}$ by \begin{equation} \label{eq:Phidef} \Phi (\c _{\mathbf{z}})(e_{i}) = \nabla ^{\ab }(z_{i})\wedge e_{i}. \end{equation} Then $\id - \Theta (\c _{\mathbf{z}})=d_{2}\circ \Phi (\c _{\mathbf{z}}).$ \end{prop} \begin{proof} First, note that the Magnus representation of $\c _{\mathbf{z}}$ is given by: \begin{equation*}\label{eq:magnus} \Theta (\gamma _{\mathbf{z}})(e_{i})=(1-t_{i})\cdot \nabla ^{\ab }(z_{i}) + z_{i}^{\ab }\cdot e_{i}. \end{equation*} Hence: \begin{align*} \left ( \id - \Theta (\gamma _{\mathbf{z}}) \right ) (e_{i}) &=(t_{i}-1)\cdot \nabla ^{\ab }(z_{i}) + (1-z_{i}^{\ab })\cdot e_{i}\\ &=d_{1}(e_{i})\cdot \nabla ^{\ab }(z_{i}) - d_{1}(\nabla ^{\ab }(z_{i}))\cdot e_{i}\\ &=d_{2}(\Phi (\gamma _{\mathbf{z}})(e_{i})). \qquad\qquad \qed \end{align*} \renewcommand{\qed}{}\end{proof} \begin{rem} \label{rem:Bpurelink} As mentioned before, an explicit formula for the Alexander invariant $B(L)$ of an arbitrary link $L\subset S^3$ is lacking. If $L$ is a pure link of $n$ components, though, Propositions~\ref{prop:IAB} and ~\ref{prop:Phidef} provide a presentation for $B(L)$, with $\binom{n}{2}$ generators and $\binom{n}{3}+n$ relations. Indeed, as shown by Artin, the braid group $B_n$ admits a faithful representation $B_n \hookrightarrow \Aut(F_n)$, which restricts to $P_n\hookrightarrow \CA(F_n)$. Moreover, any link $L$ is the closure, $\hat\alpha$, of a braid $\a\in B_n$, and $\pi_1(S^3\setminus L) = G(\a)$. Now assume $L$ is a pure link, i.e,~$L=\hat\a$, for some $\a\in P_n$. Then $\a=\gamma_{\mathbf{z}}$, where $z_i$ is the longitude corresponding to the meridian $t_i$, and we get $B(L)=\coker \bigl(\begin{smallmatrix}\Phi (\gamma_{\mathbf{z}}) \\ d_{3}\end{smallmatrix}\bigr)$, with $\Phi (\gamma_{\mathbf{z}})$ given by~\eqref{eq:Phidef}. \end{rem} \subsection{Alexander Invariant of Several Automorphisms} \label{subsec:Generalization} The above notions generalize in a straightforward manner, from a single automorphism $\a $ to several automorphisms $\a _{1},\dots ,\a _{s}$ of $F_{n}$. Namely, let \begin{equation*}\label{eq:Gmono} G(\a _{1},\dots ,\a _{s})=\langle t_{1},\dots , t_{n} \mid t_{i}=\a _{k}(t_{i}),\; 1\le k\le s\rangle \end{equation*} be the maximal quotient of $F_{n}$ on which all $\a _{k}$ act trivially. This group can also be characterized as the quotient of the semidirect product $F_{n} \rtimes F_{s} = \langle t_{1},\dots ,t_{n}, x_{1},\dots ,x_{s} \mid x_{k}^{-1}t_{i}x_{k} =\a _{k}(t_{i})\rangle $ by the normal closure of $F_{s}=\langle x_{1},\dots ,x_{s}\rangle $. Assume $\a _{k}\in \IA (F_{n})$, for $1\le k\le s$. Let $Y$ be the presentation $2$-complex for $F_{n} \rtimes F_{s}$, and $X$ that of $G=G(\a _{1},\dots ,\a _{s})$. The chain complex of the maximal abelian cover of $Y$ has the following form: \begin{equation*} \label{eq:chainbigY} C_{\bullet}(Y'):\quad C_{2}(Y') \xrightarrow{% \begin{pmatrix}\id -x_{1}\cdot \Theta (\a _{1}) & d_{1} &\cdots & 0 \\ \vdots & & \ddots \\ \id -x_{s}\cdot \Theta (\a _{s}) & 0 &\cdots & d_{1} \end{pmatrix}} C_{1}(Y') \xrightarrow{\begin{pmatrix}d_{1} \\ \overline{d}_{1}\end{pmatrix}} C_{0}(Y') \xrightarrow{\epsilon} {\mathbb Z} \rightarrow 0 \end{equation*} where the chain groups are the modules over $\overline{\L}=\L [x_{1}^{\pm 1},\dots ,x_{s}^{\pm 1}]$ given by $C_{2}(Y')=\oplus _{1}^{s} C_{1}\otimes _{\L }\overline{\L}$,\ $C_{1}(Y')=(C_{1}\oplus \oplus _{1}^{s} C_{0})\otimes _{\L }\overline{\L}$, and $C_{0}(Y')=C_{0} \otimes _{\L }\overline{\L}$, and where $\overline{d}_{1}=\begin{pmatrix}x_{1}-1 & \cdots & x_{s}-1 \end{pmatrix}^{\top }$. The chain complex of the maximal abelian cover of $X$ is then \begin{equation*}\label{eq:chainbigX} C_{\bullet}({X'} ):\quad C_{1}^{s} \xrightarrow{\begin{pmatrix}\id - \Theta (\a _{1})\\ \cdots \\ \id - \Theta (\a _{s})\end{pmatrix}} C_{1} \xrightarrow{d_{1}} C_{0} \xrightarrow{\epsilon} {\mathbb Z} \rightarrow 0. \end{equation*} This chain complex provides a presentation matrix---the so-called {\em Alexander matrix}, $\left( \id-\Theta(\a _{1}) \ \cdots \ \id-\Theta(\a _{s}) \right) ^{\top }$---for the Alexander module $A=A(\a_1,\dots,\a_s)$. Furthermore, if $\Phi (\a _{k}):C_{1}\to C_{2}$ satisfy $d_{2}\circ \Phi (\a _{k})=\id - \Theta (\a _{k}) $, then the Alexander invariant $B=B(\a_1,\dots,\a_s)$ has presentation matrix $\begin{pmatrix}\Phi (\a _{1}) &\cdots &\Phi (\a _{s}) & d_{3} \end{pmatrix}^{\top }$. When $\a_k\in\CA(F_n)$, we obtain an explicit presentation for $B$. \begin{thm}\label{thm:Bpres} Let $\c _{\mathbf{z}^{1}},\dots ,\c _{\mathbf{z}^{s}}$ be a collection of basis-conjugating automorphisms of $F_{n}$. Let $\Phi(\a_k):C_1 \to C_2$ be the homomorphisms defined by \eqref{eq:Phidef}. Then the Alexander invariant of $G(\c _{\mathbf{z}^{1}},\dots ,\c _{\mathbf{z}^{s}})$ has presentation \begin{equation*} \label{eq:Bpres} C_{1}^{s}\oplus C_{3} \xrightarrow{\begin{pmatrix}\Phi (\c _{\mathbf{z}^{1}}) &\cdots & \Phi (\c _{\mathbf{z}^{s}}) & d_{3} \end{pmatrix}^{\top } } C_{2} \rightarrow B \rightarrow 0. \end{equation*} \end{thm} \section{Local Alexander Invariants} \label{sec:LocalAlexInv} We now find presentations for the Alexander invariant of the group of a full-twist braid automorphism, and that of a related ``vertex" group. These presentations are given in terms of the Gassner representation, $\Theta:P_n\to \GL(n,\L)$, which is simply the restriction of the Magnus representation, $\Theta:\IA(F_n)\to \GL(n,\L)$, to the pure braid group $P_n$. \subsection{Alexander Invariant of a Twist Automorphism} \label{subsec:TwistAuto} Let $V=\{i_{1},\dots ,i_{r}\}$ be an increasingly ordered subset of $[n]=\{1,\dots,n\}$. Let $A_V$ the pure braid in $P_n$ which performs a full twist on the strands corresponding to $V$, leaving the other strands fixed. Let $\sigma_i$ ($1\le i <n$) be the standard generators of $B_n$, and $A_{i,j}=\sigma _{j-1}\cdots \sigma _{i+1} \sigma _{i}^{2}\sigma _{i+1}^{-1}\cdots \sigma _{j-1}^{-1}$ ($1\le i<j\le n$) the standard generators of $P_n$, see~\cite{Bi}. The twist on $V$ is given by \begin{equation}\label{eq:A_V} A_{V} = (A_{i_{1},i_{2}})(A_{i_{1},i_{3}}A_{i_{2},i_{3}})(A_{i_{1},i_{4}} A_{i_{2},i_{4}}A_{i_{3},i_{4}})\cdots (A_{i_{1},i_{r}}\cdots A_{i_{r-1},i_{r}}). \end{equation} A computation with the Artin representation reveals that $A_{V}=\c _{\mathbf{w}}$, where $\mathbf{w} = (w_{1},\dots ,w_{n})$ is defined as follows: \begin{equation}\label{eq:w_i} w_{i}=\begin{cases}t_{V}^{} &\text{ if } i\in V,\\ \relax [t_{V^{i}}^{},t_{^{i}V}^{}] &\text{ if } i\in \overline{V}\setminus V,\\ 1 &\text{ otherwise,} \end{cases} \end{equation} where $\overline{V} = \{i\in [n] \mid i_{1}\le i \le i_{r}\}$, $V^{i}=\{ j\in V \mid j<i\}$, $^{i}V=\{j\in V\mid i<j\}$, and $t_{V}^{}=\prod _{j\in V} t_{j}=t_{i_1}\cdots t_{i_r}$. Let $G(A_V)=\langle t_1,\dots,t_n \mid A_V(t_i)=t_i\rangle$ be the group associated to $A_V\in\Aut(F_n)$. A computation with~\eqref{eq:w_i} shows that \begin{equation*}\label{eq:GAV} G(A_V)=\langle t_1,\dots,t_n \mid t_V^{} t_i t_V^{-1} = t_i\ \ , i\in V\rangle, \end{equation*} and so $G(A_V)\cong (F_{r-1}\times F_1)*F_{n-r}$. By Proposition~\ref{prop:IAB}, the Alexander invariant of $G(A_V)$ has presentation \begin{equation} \label{eq:BAV} C_{1}\oplus C_{3} \xrightarrow{\begin{pmatrix}\Phi(A_V) & d_{3}\end{pmatrix}^\top} C_{2} \rightarrow{B(A_V)} \rightarrow 0. \end{equation} By Proposition~\ref{prop:Phidef}, and a Fox calculus computation, the map $\Phi _{V}^{}:=\Phi (A_{V}):C_{1} \to C_{2}$ is given by \begin{equation}\label{eq:PhiV} \Phi _{V}^{}(e_{i}) =\begin{cases}\nabla _{V} \wedge e_{i} &\text{ if } i\in V\\ (1-t_{i})\nabla _{V}\wedge \nabla _{^{i}V} &\text{ if } i\in \overline{V}\setminus V\\ 0 &\text{ otherwise.} \end{cases} \end{equation} where $\nabla _{V}:=\nabla ^{\ab }(t_{V})=\sum _{i\in V}t_{V^{i}}e_{i}$. \subsection{Simplified Presentation for $B(A_V)$} \label{subsec:BAV} Set $C_{k}(V)=\Span \{ e_{J}\mid J\subset V\}$, and let $\iota_{V}:C_k(V)\to C_k$ be the inclusion, and $\pi_{V}:C_k\to C_k(V)$ the natural projection. Write $V':=V\setminus \{\min V\}=\{i_{2},\dots , i_{r}\}$. From~\eqref{eq:PhiV} it is apparent that $\Phi_V(C_1)=\Phi_V(C_1(V))\subset C_2(V)$. Since $0=\nabla_V \wedge \nabla_V = \nabla_V \wedge e_{i_1} + \nabla_V \wedge \sum _{i\in V'}t_{V^{i}}e_{i}$, we see that $\Phi_V(e_{i_1}) \in \Phi_V(C_1(V'))$. Thus, \begin{equation} \label{eq:imPhi} \Phi _{V}(C_{1})=\Phi _{V}(C_{1}(V'))\subset C_{2}(V). \end{equation} Define an automorphism $\mu_V: F_n \to F_n$ by: \begin{equation*} \label{eq:muV} \mu_{V}(t_i)= \begin{cases} t_{V} &\text{ if } i=\min V,\\ t_i &\text{ otherwise.} \end{cases} \end{equation*} Note that $\Theta(\mu_V)(e_{i_1})=\nabla_V$, and $\Theta_2(\mu_V)(e_{i_1}\wedge e_i) =\Phi_V(e_i)$, for $i\in V'$. Thus, \begin{equation} \label{eq:muPhi} \Theta_2(\mu _{V})^{-1}\circ \Phi _{V}(e_{i}) = e_{i_{1}}\wedge e_{i} \quad \text{for } i\in V'. \end{equation} Let $C'_2(V)$ be the direct summand of $C_2$ spanned by $\{e_{i_{1}}\wedge e_{i} \mid i\in V'\}$, let $C_2^{\perp}(V)$ be the complementary summand, and let $\pi^V:C_2 \to C_2^{\perp}(V)$ be the canonical projection. Putting together \eqref{eq:BAV}, \eqref{eq:imPhi}, and \eqref{eq:muPhi}, we obtain: \begin{prop}\label{prop:alexGAV} The Alexander invariant of $G(A_V)$ has presentation \begin{equation} \label{eq:alexGAV} C_{3} \xrightarrow{\Delta(V)} C_2^{\perp}(V) \rightarrow B(A_V) \rightarrow 0, \end{equation} where $\Delta(V)=\pi^V \circ \Theta_2(\mu_V)^{-1} \circ d_{3}$. \end{prop} \subsection{Alexander Invariant of a Vertex Group} \label{subsec:BV} To the twist automorphism $A_V$, we also associate a ``vertex group," $G_V:=G(\{A_V ,A_{i,j} \mid |\{i,j\}\cap V| \le 1\})$. Using \eqref{eq:w_i}, we obtain the following presentation: \begin{equation} \label{eq:GV} G_V= \langle t_1,\dots ,t_n \mid t_V^{} t_i t_V^{-1} = t_i \text{ if } i\in V, \ \ t_j^{} t_i t_j^{-1} = t_i \text{ if } \{i,j\}\not\subset V\rangle. \end{equation} A (minimal) presentation for the Alexander invariant $B_V=B(G_V)$ may be obtained from~\eqref{eq:alexGAV} by restricting the map $\Delta(V)$ to a map $C_2(V') \wedge C_1 \to C_2(V')$, and some further matrix operations. Alternatively, it may be obtained by applying Theorem~\ref{thm:alexprod} to the direct product decomposition $G_V\cong F_{r-1}\times{\mathbb Z}^{n-r+1}$, apparent from \eqref{eq:GV}. The result is as follows: \begin{prop} The Alexander invariant of $G_V$ has presentation \begin{equation*} \label{eq:presBV} C_2(V')\wedge C_1 \xrightarrow{\Delta_V} C_2(V') \rightarrow B_V \rightarrow 0, \end{equation*} where $\Delta_V=\pi_{V'}\circ \tilde\mu_V^{} \circ d_3 \circ \tilde\mu_V^{-1}\circ (\iota_{V'}\wedge \id)$. \end{prop} \noindent (This will be useful only when $|V|\ge 3$; if $|V|=2$, then $G_V={\mathbb Z}^n$, $B_V=0$, and $C_2(V')=0$.) The above presentation may be extended to a free resolution, \begin{equation} \label{eq:BVres} \dots \rightarrow C_2(V') \wedge C_{2} \xrightarrow{\Delta^2_V} C_2(V') \wedge C_{1} \xrightarrow{\Delta_V} C_2(V') \rightarrow B_V \rightarrow 0, \end{equation} with boundary maps $\Delta^{\bullet}_V$ given by \begin{equation*} \label{eq:DeltakV} \Delta^k_V = (\pi_{V'}\wedge \id)\circ\tilde\mu_V\circ d_{k+2}\circ \tilde\mu_V^{-1}\circ (\iota_{V'}\wedge \id) : C_2(V')\wedge C_{k} \to C_2(V')\wedge C_{k-1}. \end{equation*} Furthermore, by the discussion following \eqref{eq:chainTheta}, there exists a naturally defined chain map $\Psi_{V,\bullet}:(C_{\bullet}, d_{\bullet}) \to (C_2(V')\wedge C_{\bullet-2}, \Delta_V^{\bullet-2})$, given by \begin{equation} \label{eq:ChainV} \Psi_{V,k}=(\pi_{V'}\wedge \id)\circ \Theta_k(\mu_V)^{-1}: C_k \to C_2(V')\wedge C_{k-2}, \ \ \text{ for } k\ge 2. \end{equation} \section{The Alexander Invariant of an Arrangement} \label{sec:AlexInvArr} In this section, we use the results of the previous sections to obtain a presentation for the Alexander invariant of the group of a hyperplane arrangement. \subsection{Braid Monodromy} \label{subsec:bmono} The fundamental group of the complement of an arrangement of complex hyperplanes is, by a Lefschetz-type theorem of Zariski, isomorphic to that of a generic two-dimensional section. So, for the purpose of computing the Alexander invariant, it is enough to consider affine line arrangements in ${\mathbb C} ^{2}$. Let ${\mathcal A} =\{H_{1},\dots ,H_{n}\}$ be such an arrangement, with vertices $\mathcal V = \{v_{1},\dots ,v_{s}\}$. If $v_k = H_{i_1}\cap \dots \cap H_{i_r}$, let $V_k=\{i_1,\dots,i_r\}$ denote the corresponding ``vertex set.'' We identify the set $L_2({\mathcal A})$ of rank two elements in the lattice of ${\mathcal A}$ and the collection $\{V_1,\dots,V_s\}$ of vertex sets of ${\mathcal A}$. The braid monodromy of ${\mathcal A}$ is determined as follows (see~\cite{CS3} for details). Choose coordinates $(x,z)$ in ${\mathbb C} ^{2}$ so that the projection $\pr _{1}:{\mathbb C} ^{2}\to {\mathbb C} $ is generic with respect to ${\mathcal A} $. Let $f(x,z)=\prod _{i=1}^{n} (z-a_{i}(x))$ be a defining polynomial for ${\mathcal A} $. The root map $a=(a_{1},\dots , a_{n}):{\mathbb C} \to {\mathbb C} ^{n}$ restricts to a map from the complement of $\mathcal{Y}=\pr _{1}(\mathcal{V})$ to the complement of the braid arrangement ${\mathcal A} _n=\{\ker (y_{i}-y_{j})\}_{1\le i <j \le n}$. Identify $\pi _{1}({\mathbb C} \setminus \mathcal{Y})$ with the free group $F_{s}=\langle x_{1},\dots ,x_{s}\rangle $, and $\pi _{1}({\mathbb C} ^{n} \setminus {\mathcal A} _{n})$ with the pure braid group $P_{n}$. Then, the {\em braid monodromy} of ${\mathcal A} $ is the induced homomorphism on fundamental groups, $\a :F_{s} \to P_{n}$. The braid monodromy generators $\a _{k}=\a (x_{k})$ can be written explicitly using a {\em braided wiring diagram} ${\mathcal W}$ associated to ${\mathcal A}$. Such a diagram, determined by the choices made above, may be (abstractly) specified by a sequence of vertex sets and braids, ${\mathcal W} ={\mathcal W}_s= \{ V_1,\b_1,V_2,\b_2,\dots,\b_{s-1},V_s\}$. The braid monodromy generators are given by $\a _{k}=A_{V_{k}}^{\d _{k}}$, where $A_{V_k}$ is the twist braid defined in~\eqref{eq:A_V} and $\d_k$ is a pure braid determined by the subdiagram ${\mathcal W}_k$. \subsection{The Presentation for $B({\mathcal A})$} \label{subsec:PresB} Let $M=M({\mathcal A})$ be the complement of ${\mathcal A}$. Let $G=G(\a_1,\dots , \a_s)$ be the fundamental group of $M$, with Alexander invariant $B=B({\mathcal A})$. Theorem~\ref{thm:Bpres} provides the following presentation for $B$: \begin{equation*}\label{eq:Barrangement} C_{1}^{s}\oplus C_{3} \xrightarrow{\begin{pmatrix}\Phi _{V_{1}}^{\d _{1}} &\cdots & \Phi _{V_{s}}^{\d _{s}} & d_{3} \end{pmatrix}^{\top }} C_{2} \rightarrow B \rightarrow 0, \end{equation*} where $\Phi _{V}^{\d}:=\Phi (A_{V}^{\d })=\Theta _{2}(\d )\circ \Phi _{V}^{} \circ \Theta (\d ^{-1}):C_{1} \to C_{2}$, and $\Phi_V$ is given by \eqref{eq:PhiV}. This presentation can be simplified, based on the following elementary observation: If $R$ is a ring, and $B$ is an $R$-module, with presentation $R^{p} \xrightarrow{\Delta} R^{q} \rightarrow B\rightarrow 0$, where $\Delta =\Upsilon \circ \Xi $, or $\Delta =\Xi \circ \Upsilon $, with $\Xi $ invertible, then $B$ can also be presented as $R^{p} \ \xrightarrow{\Upsilon} R^{q} \rightarrow B \rightarrow 0$. Since the maps $\Theta (\d _{k}^{-1})$ are invertible we may replace $\Phi_{V_k}^{\d_k}$ by $\Theta_{2}(\d _{k})\circ \Phi _{V_{k}}$. Furthermore, by~\eqref{eq:imPhi}, we may subsequently restrict each of the maps $\Phi_{V_k}$ to $\Phi_k:C_1(V_k') \to C_2$. Thus, we obtain the following: \begin{thm}\label{thm:alexarr} The Alexander invariant of an arrangement ${\mathcal A} $, with braid monodromy generators $A_{V_{1}}^{\d _{1}},\dots, A_{V_{s}}^{\d _{s}}$, has presentation \begin{equation*} \label{eq:alexinv} K_1 \xrightarrow{\Delta} K_0 \rightarrow B({\mathcal A} ) \rightarrow 0, \end{equation*} where $\displaystyle{K_1=\bigoplus_{k=1}^s C_{1}(V_k') \oplus C_{3}}$, $K_0=C_2$, and $\Delta = \begin{pmatrix} \Phi & d_3 \end{pmatrix}^{\top}$, with $\Phi|_{C_1(V_k')}=\Theta _{2}(\d _{k})\circ \Phi_k$. \end{thm} Note that this presentation has $\binom {n}{2}$ generators and $\sum _{k=1}^{s}(|V_{k}|-1)+\binom {n}{3}$ relations, and that $\sum _{k=1}^{s}(|V_{k}|-1)=b_{2}(M)$. \subsection{Real Arrangements} \label{subsec:RealArrangements} The presentation can be simplified in the case where ${\mathcal A} $ is the complexification of a line arrangement ${\mathcal A}_{\mathbb R}$ in ${\mathbb R}^2$. In this instance, the wiring diagram ${\mathcal W}$ can be chosen so that it contains no intermediary braids, and each ``conjugating braid,'' $\d_k$, is a subword of the full twist, $A_{[n]}$, on $n$ strands. Let $U_k$ denote the set of indices of wires of ${\mathcal W}$ which lie above the vertex $v_k$ in $\pr_1^{-1}(y_k)$, and let $J_k = (\overline V_k \setminus V_k) \cap U_k$. Then the conjugating braids may be written as $\d _{k}=\prod_{j<i} A_{j,i}$, where the product is over all $i\in V_{k}$ and $j\in J_k$, see~\cite{CF}, \cite{CS3}. Define a homomorphism $\boldsymbol{\Theta}_2(\mu):C_2\to C_2$ by \begin{equation*} \label{eq:Thetamu} \boldsymbol{\Theta}_2(\mu) (e_{i}\wedge e_{j}) =\begin{cases}\Theta _{2}(\mu_{V_k})(e_{i}\wedge e_{j}) &\text{ if } \{i,j\}\subset V_{k}\\ e_{i}\wedge e_{j} &\text{ otherwise}. \end{cases} \end{equation*} It is readily seen that $\boldsymbol{\Theta}_2(\mu)$ is invertible. Similarly, define $\boldsymbol{\Theta}_2(\d):C_2\to C_2$. A computation shows that $\d_k(t_i) = t_{J_k^i}\cdot t_i\cdot t_{J_k^i}^{-1}$, and that $\Theta (\d_k)(e_{i})= (1-t_{i})\cdot \nabla _{J_{k}^{i}} + t_{J_{k}^{i}}\cdot e_{i}$, for $i \in V_k$. Thus, $\boldsymbol\Theta_2(\d)$ is also invertible. Proceeding as in~\ref{subsec:BAV}, we obtain the following. \begin{thm}\label{thm:alexreal} The Alexander invariant of a complexified real arrangement ${\mathcal A} $ has presentation \begin{equation*} \label{eq:alexreal} C_3 \xrightarrow{\Delta'} L_0 \rightarrow B({\mathcal A} ) \rightarrow 0, \end{equation*} where $L_0$ is the complementary summand to $K'_0=\oplus_V C_2'(V)$ in $C_2$, $\pi_0:C_2\to L_0$ is the canonical projection, and $\Delta' = \pi_0\circ \boldsymbol{\Theta}_2(\mu)^{-1} \circ \boldsymbol{\Theta}_2(\d)^{-1}\circ d_{3}$. \end{thm} Note that this presentation has only $\binom{n}{2}-b_2(M)$ generators, and $\binom {n}{3}$ relations. \begin{rem} For an arbitrary complex arrangement, the map $\boldsymbol\Theta_2(\d)$ need not be invertible. Thus the simplification of the presentation of the Alexander invariant afforded by the above result may not be available. However, for any arrangement, we obtain an analogous simplified presentation for the $I$-adic completion, $\widehat B({\mathcal A})$, of the Alexander invariant of ${\mathcal A}$ in Corollary~\ref{cor:hBpres}. \end{rem} \section{Decomposition of the Alexander Invariant} \label{sec:DecompAlex} We now relate the Alexander invariant of an arrangement ${\mathcal A}$ to a ``combinatorial'' Alexander invariant, determined by the intersection lattice of ${\mathcal A}$. For these purposes, we restrict our attention to central arrangements and their generic sections. It is enough to consider an affine arrangement, ${\mathcal A}=\{H_1,\dots,H_n\}$, of $n$ lines in ${\mathbb C}^2$ that is transverse to infinity. Recall that we identify set of rank two elements in the lattice of ${\mathcal A}$ and the collection of vertex sets of ${\mathcal A}$: $L_2({\mathcal A})=\{V_1,\dots,V_s\}$. \subsection{The Coarse Combinatorial Alexander Invariant} \label{subsec:ChainMap} For each $V\in L_2({\mathcal A})$, let $G_V$ be the corresponding vertex group, and $B_V$ the corresponding Alexander invariant. Define the {\it coarse combinatorial Alexander invariant} of ${\mathcal A}$ to be \begin{equation*} \label{eq:localB} {B^{\operatorname{cc}}} ({\mathcal A})=\oplus_V B_V. \end{equation*} Notice that the module $B_V$ depends only on the cardinality $|V|$ of the vertex set $V$. Consequently, the module ${B^{\operatorname{cc}}}$ depends only on the number and multiplicities of the elements of $L_2({\mathcal A})$. This $\L$-module admits a free resolution \begin{equation*} \label{eq:localres} \dots\rightarrow L_2 \xrightarrow{D_2} L_1 \xrightarrow{D_1} L_0 \rightarrow {B^{\operatorname{cc}}} \rightarrow 0 \end{equation*} obtained by taking the direct sum of the resolutions \eqref{eq:BVres}: $L_k=\oplus_V C_2(V')\wedge C_{k}$, $D_k=\oplus_V\Delta_V^{k}$. (Since ${\mathcal A}$ is assumed to be transverse to infinity, $L_0=\oplus_V C_2(V')$ is indeed the complementary summand of $K'_0=\oplus_V C_2'(V)$ in $K_0 = C_2 = \oplus_V C_2(V)$.) Let $\Psi_{V,\bullet}:C_{\bullet} \to C_2(V')\wedge C_{{\bullet}-2}$ be the chain map introduced in \eqref{eq:ChainV}. Define a chain map $\Psi_{\bullet}:C_{\bullet} \to L_{{\bullet}-2}$ by $\Psi_k = \sum_V\Psi_{V,k}$, for $k\ge 2$. \begin{prop} \label{prop:ups21} The image of the composition $\Psi_2\circ \Phi: \bigoplus_V C_1(V') \to L_0$ is contained in the image of the map $D_1: L_1 \to L_0$. Therefore, there exists a map $\Gamma:\bigoplus_V C_1(V') \to L_1$ such that $D_1 \circ \Gamma = \Psi_2 \circ \Phi$. \end{prop} \begin{proof} Let $A_V^\delta$ be a braid monodromy generator of ${\mathcal A}$, where $V=\{i_1,\dots,i_r\}$ and $\delta$ is some pure braid. Using the pure braid relations to rewrite $\delta$ if necessary, we may assume that this pure braid is a word in the generators $\{A_{r,s} \mid \{r,s\} \not\subset V\}$. For $j\in V'$, we have $\Psi_2\circ \Phi(e_j) = \Psi_2 \circ \Theta_2(\delta)(\nabla_V \wedge e_j)$. Since $\delta\in\IA(F_n)$, we have $\im(\Theta_k(\delta)-\id)\subset I\cdot C_k$. Hence, $\Theta_2(\d)(e_i\wedge e_j) = e_i\wedge e_j + W^\d_{i,j}$, where $W^\d_{i,j}=\sum w_{p,q} e_p\wedge e_q$, with $w_{p,q}\in I$. Thus $\Psi_2\circ \Phi(e_j) = \Psi_2(\nabla_V \wedge e_j) + \sum_{i\in V} t_{V^i}\cdot \Psi_2(W^\delta_{i,j})$. Since $\Psi_2 (\nabla_V \wedge e_j)=\pi_{V'}\circ\Theta_2(\mu_V)^{-1} (\Theta(\mu_V)(e_{i_1}\wedge e_j))=\pi_{V'}(e_{i_1}\wedge e_j)=0$, it suffices to show that \begin{equation} \label{eq:imups1} \Psi_2(W^\d_{i,j}) \in \im(D_1). \end{equation} For a vertex set $U \in L_2({\mathcal A})$, recall the natural projection $\pi_U:C_k\to C_k(U)$, and denote by $I_U^\perp$ the ideal in $\L$ generated by $\langle 1-t_k \mid k \notin U \rangle$. \begin{claim} For each vertex set $U \in L_2({\mathcal A})$, we have $\pi_U(W^\d_{i,j}) \in I_U^\perp \cdot C_2(U)$. \end{claim} Before proving this claim, let us show that it implies \eqref{eq:imups1}. For $1\le p<q\le n$, let $V(p,q)$ denote the unique vertex set of ${\mathcal A}$ with $p,q\in V(p,q)$. If $w\cdot e_p\wedge e_q$ is a summand of $W^\d_{i,j}$, write $U=V(p,q)$. Then, by the claim, we have $w\in I_{U}^\perp$. Now $\Psi_2(e_p\wedge e_q) = \Psi_{U,2}(e_p\wedge e_q) \in C_2(U')$, and it is readily checked that $I_{U}^\perp \cdot C_2(U') \subset \im(\Delta_U)$. It follows that $\Psi_2(w\cdot e_p\wedge e_q) \in \im(\Delta_U)$. Thus it suffices to prove the claim. This may be accomplished by induction on the length of the word $\delta$. If $\delta = 1$, then $W^\d_{i,j}=0 $, and there is nothing to prove. If $\delta=A_{r,s}^{\pm 1}$, a computation shows that $\Theta_2(A_{r,s})(e_i\wedge e_j) = e_i\wedge e_j + W^{r,s}_{i,j}$ and $\Theta_2(A_{r,s}^{-1})(e_i\wedge e_j) = e_i\wedge e_j -(t_rt_s)^{-1} W^{r,s}_{i,j}$, where \begin{equation*} \label{eq:T2rs} W^{r,s}_{i,j}= \begin{cases} t_r(t_s-1)e_i\wedge e_r+t_r(1-t_r)e_i\wedge e_s & \text{if $i<j=r<s$,}\\ (1-t_j)[(1-t_s)e_i\wedge e_r+(t_r-1)e_i\wedge e_s] & \text{if $i<r<j<s$,}\\ (t_r-1)e_i\wedge e_s+(1-t_s)e_i\wedge e_r & \text{if $i<r<j=s$,}\\ (t_rt_s-1)e_r\wedge e_s & \text{if $i=r<j=s$,}\\ (t_r-1)e_i\wedge e_s+(t_s-1)[e_r\wedge e_i+(t_i-1)e_r\wedge e_s] & \text{if $r<i<j=s$,}\\ (1-t_i)[(1-t_s)e_r\wedge e_j+(t_r-1)e_s\wedge e_j] & \text{if $r<i<s<j$,}\\ t_r(t_s-1)e_r\wedge e_j+t_r(1-t_r)e_s\wedge e_j & \text{if $r=i<s<j$,}\\ (t_r-1)e_s\wedge e_j+(1-t_s)e_r\wedge e_j & \text{if $r<i=s<j$,}\\ t_r(t_s-1)e_r\wedge e_j+(t_r-1)[t_re_j\wedge e_s+(1-t_j)e_r\wedge e_s] & \text{if $r=i<j<s$,}\\ (t_j-1)[(1-t_s)e_r\wedge e_i+(1-t_r)e_i\wedge e_s] & \text{if $r<i<j<s$,}\\ \quad + (1-t_i)[(1-t_s)e_r\wedge e_j+(1-t_r)e_j\wedge e_s] \\ 0 & \text{otherwise.} \end{cases} \end{equation*} If $\{r,s\}\not\subset V$, it is readily checked that $\pi_U(W^{\d}_{i,j}) \in I_U^\perp\cdot C_2(U)$ for each $U \in L_2({\mathcal A})$. In general, write $\d$ as the product of $A_{r,s}^{\pm 1}$ and $\d'$, and assume inductively that $\Theta_2(\d')(e_i\wedge e_j) = e_i\wedge e_j +W'_{i,j}$ satisfies $\pi_U(W'_{i,j}) \in I_U^\perp\cdot C_2(U)$ for each $U$. Then $\Theta_2(\d)(e_i\wedge e_j)=\Theta_2(A_{r,s}^{\pm 1})(e_i\wedge e_j)+ \Theta_2(A_{r,s}^{\pm 1})(W'_{i,j})$, and by the above, it remains to analyze the latter summand. If $w\cdot e_p\wedge e_q$ is a summand of $W'_{i,j}$, then $w\in I_{V(p,q)}^\perp$ by induction. Case-by-case analysis then shows that each summand $x\cdot e_\ll\wedge e_m$ of $\Theta_2(A_{r,s}^{\pm 1})(w\cdot e_p\wedge e_q)$ satisfies $x \in I_{V(\ll,m)}^\perp$. This completes the proof of the claim, and hence that of the proposition. \end{proof} We can now formulate the main result of this subsection. \begin{thm} \label{thm:BtoBlocal} There exists a chain map $\Upsilon_\bullet$ from the presentation $K_{\bullet} \to B({\mathcal A})$ to the resolution $L_{\bullet}\to {B^{\operatorname{cc}}} ({\mathcal A})$, \begin{equation*}\label{eq:chainmap} \begin{CD} &&&& K_1 @>\Delta>> K_0@>>> B@>>>0\\ &&&& @VV{\Upsilon_1}V @VV{\Upsilon_0}V @VV{\Pi}V \\ \dots@>>> L_2 @>D_2>> L_1 @>D_1>> L_0 @>>> {B^{\operatorname{cc}}} @>>> 0, \end{CD} \end{equation*} given by $\Upsilon_0=\Psi_2$, and $\Upsilon_1(x,y) = \Gamma(x) + \Psi_3(y)$. Furthermore, the resulting map $\Pi:B\to {B^{\operatorname{cc}}}$ is surjective. \end{thm} \begin{proof} It is immediate from the above that $\Upsilon_\bullet$ is a chain map. Thus it suffices to show that the map $\Upsilon_0:K_0\to L_0$, which by definition equals $\Psi_2=\sum_V\pi_{V'}\circ\Theta_2(\mu_V)^{-1}: C_2\to \oplus_V C_2(V')$, is surjective. With respect to the decomposition $K_0=K_0'\oplus L_0 =\left( \oplus_V C_2'(V)\right) \oplus \left( \oplus_V C_2(V')\right)$, we have $\Upsilon_0=\pmatrix \Upsilon'_0 & \id_{L_0}\endpmatrix^{\top}: K_0'\oplus L_0\to L_0$, where $\Upsilon'_0(e_i\wedge x) = (e_i-\nabla_V)\wedge x$, for $i=\min V$ and $x \in C_1(V')$. Thus $\Upsilon_0:K_0\to L_0$ is surjective. \end{proof} \subsection{Decomposition of the Completion} \label{subsec:Decomp} Recall that if $B$ is a $\L$-module, then $\widehat B$ denotes its $I$-adic completion, and that if $f:A\to B$ is a map of $\L$-modules, we write $\widehat f:\widehat A \to \widehat B$ for the extension of $f$ to the completions. The $I$-adic completion functor takes chain complexes to chain complexes, and chain maps to chain maps. \begin{thm} \label{thm:decomp} The chain map $\widehat\Upsilon_\bullet:\widehat K_\bullet\to \widehat L_\bullet$ induces an isomorphism $\widehat B \xrightarrow{\sim}{\widehat{B}^{\operatorname{cc}}}$ if and only if the map $\widehat\Psi_3:\widehat C_3 \to \widehat L_1$ is surjective. \end{thm} \begin{proof} Consider the mapping cone, $K_\bullet(\widehat \Upsilon)$, of $\widehat\Upsilon_\bullet$, given by \begin{equation} \label{eq:mapcone} \dots \to \widehat L_2 \oplus \widehat K_1 \xrightarrow{\partial_2} \widehat L_1 \oplus \widehat K_0 \xrightarrow{\partial_1} \widehat L_0, \end{equation} where $\partial_2(x,y) = (\widehat{D}_2(x) - \widehat \Upsilon_1(y),\widehat\Delta(y))$ and $\partial_1(x,y) = \widehat{D}_1(x) + \widehat\Upsilon_0(y)$, and the short exact sequence of chain complexes \begin{equation} \label{eq:seq} 0 \to \widehat L_\bullet \xrightarrow{\iota_\bullet} K_\bullet(\widehat \Upsilon) \xrightarrow{\pi_\bullet} \widehat K_{\bullet - 1} \to 0, \end{equation} where $\iota_\bullet$ and $\pi_\bullet$ denote the natural inclusion and projection. Since $\widehat\Upsilon_0$ is surjective, we have $H_0(K_\bullet(\widehat \Upsilon))=0$. Also, since $\widehat L_\bullet$ is a resolution, $\widetilde H_*(\widehat L_\bullet)=0$. Thus the associated long exact sequence in homology reduces to \begin{equation} \label{eq:Hseq} 0 \to H_1(K_\bullet(\widehat \Upsilon)) \to H_0(\widehat K_\bullet) \xrightarrow{\widehat\Upsilon_{*}} H_0(\widehat L_\bullet) \to 0. \end{equation} The map $\widehat\Upsilon_{*}:H_0(\widehat K_\bullet) = H_0(\widehat L_\bullet)$ identifies canonically with $\widehat\Pi:\widehat B \to{\widehat{B}^{\operatorname{cc}}}$. Thus, it suffices to show that $\coker(\widehat\Psi_3)=0$ if and only if $H_1(K_\bullet(\widehat \Upsilon))=0$. Recall the map $\Phi:\oplus_V C_1(V') \to C_2$ from Theorem~\ref{thm:alexarr}. Recall also (from the proof of Proposition~\ref{prop:ups21}) that $\Phi|_{C_1(V')}=\Phi_V\mod I$. Thus, $\widehat\Phi|_{\widehat C_1(V')} = \widehat\Phi_V \mod \mathfrak{m}$. Using the identification $C_1(V')\xrightarrow{\sim} C'_2(V)$, $x\mapsto e_i\wedge x$, where $i=\min V$, and the projection onto first factor $p':K_0=K_0'\oplus L_0\to K_0'$, define $\Phi':= p' \circ \Phi : K'_0 \to K'_0$. Since $\widehat\nabla_V=\sum_{i\in V} e_i$, the map $\widehat p'\circ \widehat\Phi_V:\widehat C_1(V')\to \widehat C'_2(V)$ coincides with the above identification. Hence, $\widehat\Phi'=\id\mod \mathfrak{m}$. Consequently, $\widehat\Phi'$ is an isomorphism. We now alter the short exact sequence~\eqref{eq:seq}. Write $K_1(\widehat\Upsilon) = \widehat L_1 \oplus \widehat K'_0 \oplus \widehat L_0$ and $K_2(\widehat\Upsilon) = \widehat L_2 \oplus \widehat K'_0 \oplus \widehat C_3$, and define $\rho \in \Aut K_1(\widehat\Upsilon)$ and $\psi \in \Aut K_2(\widehat\Upsilon)$ by \begin{align*} \rho(x,y,z) & = \left( x + \widehat\Gamma \circ \widehat{\Phi'}^{-1}(y), y, D_1(x)+\widehat\Upsilon'_0(y) +z\right),\\ \psi(x,y,z) &= \left( x, \widehat{\Phi'}^{-1}(y) - \widehat{\Phi'}^{-1} \circ p'\circ\widehat d_3(z), z\right). \end{align*} Note that the restriction of $\psi$ to $\widehat L_2$ is the identity, and let $\bar\psi$ denote the restriction of $\psi$ to $\widehat K_1 = \widehat K'_0 \oplus \widehat C_3$. We modify the sequence~\eqref{eq:seq} as indicated below. \begin{equation}\label{eq:MapConeSeq} \begin{CD} \dots @>>> \widehat L_2 @>\widehat D_2>> \widehat L_1 @>\widehat D_1>> \widehat L_0\\ &&@VV{\psi^{-1} \circ \iota_2}V @VV{\rho \circ \iota_1}V @VV{\iota_0=\id}V \\ \dots @>>> \widehat L_2 \oplus \widehat K_1 @>{\rho \circ \partial_2 \circ \psi}>> \widehat L_1 \oplus \widehat K_0 @>{\partial_1 \circ \rho^{-1}}>> \widehat L_0\\ &&@VV{\pi_2}V @VV{\pi_1 \circ \rho^{-1}}V @VVV\\ &&\widehat K_1 @>{\widehat\Delta \circ \bar\psi}>> \widehat K_0 @>>> 0 \end{CD} \end{equation} Since $\bar\psi\circ\pi_2 = \pi_2\circ\psi$, this diagram commutes. Consider the map $\Xi:\widehat L_2 \oplus \widehat C_3 \to \widehat L_1$ defined by \begin{equation} \label{eq:H1pres} \Xi(x,z)=\widehat D_2(x)+ \widehat\Gamma \circ \widehat{\Phi'}^{-1} \circ p' \circ \widehat d_3(z) - \widehat\Psi_3(z). \end{equation} Computations with the definitions (making use of the fact that $\widehat\Upsilon_\bullet$ is a chain map) reveal that $\partial_1 \circ \rho^{-1}(x,y,z)=z$ and $\rho \circ \partial_2 \circ \psi(x,y,z)=(\Xi(x,z),y,0)$. Thus, $\Xi$ provides a presentation for the module $H_1(K_\bullet(\widehat \Upsilon))$, and this module is trivial if and only if $\Xi$ is surjective. Since $\im(\widehat D_2)\subset {\mathfrak m} \cdot \widehat L_1$ and $\im(\widehat\Gamma \circ \widehat{\Phi'}^{-1} \circ p' \circ \widehat d_3)\subset {\mathfrak m} \cdot \widehat L_1$, the map $\Xi$ is surjective if and only if $\widehat\Psi_3$ is surjective. \end{proof} The above proof has several consequences, even in the instance when the map $\widehat\Psi_3$ is not surjective, see below and Theorem~\ref{thm:theta3}. These results hold for an arbitrary arrangement (real or complex, compare~\ref{thm:alexarr} and~\ref{thm:alexreal}) that is transverse to infinity. \begin{cor} \label{cor:hBpres} The $I$-adic completion $\widehat B$ of the Alexander invariant an arrangement ${\mathcal A}$ has a presentation with $\binom {n}{2}-b_2(M({\mathcal A}))$ generators, and $\binom {n}{3}$ relations. \end{cor} \begin{proof} From the commutative diagram~\eqref{eq:MapConeSeq}, we have the presentation $\widehat\Delta \circ \bar\psi:\widehat K_1 \to \widehat K_0$ for $\widehat B$. Let $p'':K_0 = K'_0 \oplus L_0 \to L_0$ denote the projection onto second factor. With respect to the decompositions $\widehat K_1 = \widehat K'_0 \oplus \widehat C_3$ and $\widehat K_0 = \widehat K'_0 \oplus \widehat L_0$, the map $\widehat\Delta \circ \bar\psi:\widehat K_1\to \widehat K_0$ is given by \begin{equation*} \label{eq:deltapsi} \widehat\Delta \circ \bar\psi(x,y) = \left(x, \widehat p''\left(\widehat\Phi\circ\widehat{\Phi'}^{-1}(x) - \widehat\Phi\circ\widehat{\Phi'}^{-1}\circ\widehat p'\circ \widehat d_3(y) + \widehat d_3(y) \right)\right). \end{equation*} Define \begin{equation} \label{eq:deltasharp} \widehat\Delta^\sharp= \widehat p''\circ \left( \id - \widehat \Phi\circ \widehat {\Phi'}^{-1} \circ \widehat p'\right)\circ \widehat d_3:\widehat C_3 \to \widehat L_0, \end{equation} and define $\chi \in \Aut \widehat K_0$ by $\chi(x,y) = (x,y - \widehat p''\circ\widehat\Phi\circ \widehat{\Phi'}^{-1}(x))$. Then $\chi\circ \widehat\Delta \circ \bar\psi(x,y) = (x, \widehat\Delta^\sharp(y))$. Thus, $\widehat\Delta^\sharp$ provides a presentation for $\widehat B$ with the specified numbers of generators and relations. \end{proof} \section{Combinatorics and the Chen Groups} \label{sec:Combinatorics} In this section, we examine the relationship between the results obtained in the previous sections and the combinatorics of the arrangement ${\mathcal A}$. An invariant of ${\mathcal A}$ is called {\it combinatorial} if it is determined by the isomorphism type of the lattice $L({\mathcal A})$. As is well-known from \cite{F1}, the ranks $\phi_k$ of the LCS quotients of the group of ${\mathcal A}$ are combinatorially determined. Thus, the ranks of the first three Chen groups of ${\mathcal A}$ are combinatorial. We now describe some explicit combinatorial bounds and formulas for the ranks $\theta_k$ of the Chen groups of ${\mathcal A}$. \subsection{A Bound on Chen Ranks} \label{subsec:ChenBound} Recall that the coarse combinatorial Alexander invariant ${B^{\operatorname{cc}}}$ of ${\mathcal A}$ is the direct sum $\oplus_V B_V$ of the Alexander invariants of the vertex groups $G_V$, indexed by $V\in L_2({\mathcal A})$, the rank two elements of $L({\mathcal A})$. For $k\ge 2$, define the {\it coarse combinatorial Chen ranks} by \begin{equation*} \label{eq:thetacc} \theta^{\operatorname{cc}}_k({\mathcal A}) = \sum_{V\in L_2({\mathcal A})} \theta_k(G_V). \end{equation*} From Theorem~\ref{thm:massey} and Theorem~\ref{thm:BtoBlocal}, we obtain the following. \begin{cor}\label{cor:chenlowerbound} For $k\ge 2$, the ranks of the Chen groups of ${\mathcal A}$ are bounded below by the coarse combinatorial Chen ranks: $\theta_k({\mathcal A}) \ge \theta^{\operatorname{cc}}_k({\mathcal A})$. \end{cor} To compute these lower bounds explicitly, recall that $G_V\cong F_{r-1} \times {\mathbb Z}^{n-r}$, where $r=|V|$. By Corollary~\ref{cor:chenprod}, the ranks of the Chen groups of $G_V$ are given by $\theta_k(G_V) = (k-1) \binom{k+r-3}{k}$ for $k \ge 2$. Let $c_r$ denote the number of elements of $L_2({\mathcal A})$ of multiplicity $r$, and write $\binom{m}{k}=0$ if $m<k$. Then, \begin{equation*} \label{eq:LocalComb} \theta_k^{\operatorname{cc}} =\sum_{V \in L_2({\mathcal A})} (k-1) \binom{k+|V|-3}{k} = \sum_{r\ge 3} c_r (k-1) \binom{k+r-3}{k}, \end{equation*} and so $\theta_k^{\operatorname{cc}}$ is determined by (only) the multiplicities of the elements of $L_2({\mathcal A})$. \begin{rem} The ranks of the lower central series quotients of the group $G$ of ${\mathcal A}$ satisfy analogous lower bounds: $\phi_k({\mathcal A}) \ge \phi^{\operatorname{cc}}_k({\mathcal A}) =\sum_{V \in L_2({\mathcal A})} \phi_k(G_V)$, see~\cite{F2}~Proposition~3.8. \end{rem} \begin{rem} \label{rem:mobius} The lower bounds for the ranks of the Chen groups of ${\mathcal A}$ may be expressed in terms of the M\"obius function $\mu:L({\mathcal A})\to{\mathbb Z}$: $\theta_k({\mathcal A}) \ge \sum_{V\in L_2({\mathcal A})} (k-1)\binom{k+\mu(V)-2}{k}$. \end{rem} We now analyze the difference $\theta_k-\theta_k^{\operatorname{cc}}$. Recall that $\theta_k({\mathcal A}) = \rank({\mathfrak m}^{k-2}\widehat B/{\mathfrak m}^{k-1}\widehat B)$. Checking that the image of the map $\widehat\Delta^\sharp:\widehat C_3 \to \widehat L_0$ defined in~\eqref{eq:deltasharp} is contained in ${\mathfrak m} \cdot \widehat L_0$, we see that $\theta_2({\mathcal A}) = \rank(\widehat B/{\mathfrak m} \widehat B) = \binom {n}{2}-b_2(M({\mathcal A}))$, and thus $\theta_2({\mathcal A})=\theta^{\operatorname{cc}}_2({\mathcal A})$. Recall the mapping cone $K_{\bullet}(\widehat\Upsilon)$ from~\eqref{eq:mapcone}, and set $H=H_1(K_\bullet(\widehat \Upsilon))$. \begin{thm} \label{thm:theta3} For $k \ge 3$, the rank of the $k^{\text{th}}$ Chen group of ${\mathcal A}$ is given by $\theta_k({\mathcal A}) = \rank({\mathfrak m}^{k-3}H/{\mathfrak m}^{k-2}H) + \theta^{\operatorname{cc}}_k({\mathcal A})$. In particular, $\theta_3({\mathcal A}) = \rank (\coker\widehat\Psi_3) + \theta^{\operatorname{cc}}_{3}({\mathcal A})$. \end{thm} \begin{proof} Consider the short exact sequence~\eqref{eq:Hseq}, rewritten as \begin{equation} \label{eq:PiB} 0 \to H \xrightarrow{\tau} \widehat B \xrightarrow{\widehat\Pi} {\widehat{B}^{\operatorname{cc}}} \to 0. \end{equation} Altering the commutative diagram~\eqref{eq:MapConeSeq} using the isomorphism $\chi \in \Aut \widehat K_0$ defined in the proof of Corollary~\ref{cor:hBpres}, we see that $\tau$ is induced by $\chi \circ \pi_1 \circ \rho^{-1}: \widehat L_1 \oplus \widehat K'_0 \oplus \widehat L_0 \to \widehat K_0$, $(x,y,z) \mapsto (y, z - \widehat D_1(x))$. Thus the restriction of this map to $\ker(\partial_1 \circ\rho^{-1})= \widehat L_1 \oplus \widehat K'_0$ is given by $(x,y) \mapsto (y, - \widehat D_1(x))$. Since $\widehat B = \coker(\chi\circ \widehat\Delta \circ \bar\psi)$ and $\chi\circ \widehat\Delta \circ \bar\psi(y,0) = (y,0)$, the map $\tau:H\to \widehat B$ in homology is induced by $\widehat D_1:\widehat L_1 \to \widehat L_0$. Since $\im(\widehat D_1(z)) \subset{\mathfrak m} \cdot \widehat L_0$, from the exact sequence~\eqref{eq:PiB} we have \begin{equation*} \rank({\mathfrak m}^{\ll}\widehat B/{\mathfrak m}^{\ll-1}\widehat B) = \rank({\mathfrak m}^{\ll-1}H/{\mathfrak m}^{\ll-2}H) + \rank({\mathfrak m}^{\ll} {\widehat{B}^{\operatorname{cc}}}/{\mathfrak m}^{\ll-1}{\widehat{B}^{\operatorname{cc}}}). \end{equation*} It follows from Theorem~\ref{thm:massey} that the ranks of the Chen groups of ${\mathcal A}$ are as asserted. In particular, the third Chen group of ${\mathcal A}$ has rank $\theta_3({\mathcal A}) = \rank(H/{\mathfrak m} H) + \theta^{\operatorname{cc}}_3({\mathcal A})$. Recall the presentation, $\Xi:\widehat L_2 \oplus \widehat C_3 \to \widehat L_1$, for the module $H$ from~\eqref{eq:H1pres}. Using elementary row and column operations, we obtain a presentation $\Xi':\widehat {\Lambda }^a \to \widehat {\Lambda }^b$ from this with $b=\rank \widehat L_1 - \rank\widehat\Psi_3$ generators. Checking that $\im(\Xi') \subset {\mathfrak m}\cdot \widehat {\Lambda }^b$, we get $\rank(H/{\mathfrak m} H) = \rank(\coker \widehat\Psi_3)$. \end{proof} \subsection{Decomposition is Combinatorial} \label{subsec:theta3} Let $\hat\epsilon:\widehat {\Lambda } \to {\mathbb Z}$ be the augmentation map, which takes a power series to its constant coefficient. If $\widehat F=\widehat {\Lambda }^p$ is a free module, denote its image under $\hat\epsilon$ by $\overline F={\mathbb Z}^p$, and if $\widehat f:\widehat F\to\widehat F'$ is a $\widehat {\Lambda }$-linear map, denote its image by $\overline f:\overline F \to \overline F'$. \begin{lem} The rank of $\widehat f$ is equal to the rank of $\overline f$. \end{lem} \begin{proof} Suppose the rank of $\overline f:{\mathbb Z}^p \to {\mathbb Z}^q$ is $r$. Then there are integral matrices $X \in \GL(p,{\mathbb Z})$ and $Y \in \GL(q,{\mathbb Z})$ so that $X \cdot \overline f \cdot Y = \begin{pmatrix} I_r&0\\0&0 \end{pmatrix}$, where $I_r$ denotes the $r\times r$ identity matrix. By definition, $\widehat f = \overline f + Z$, where the entries of $Z$ are in $\mathfrak m$. Thus, $X \cdot \widehat f \cdot Y = X \cdot \overline f \cdot Y+ X \cdot Z \cdot Y$. Clearly, the rank of $X \cdot \widehat f \cdot Y$, and hence that of $\widehat f$, is $r$. The converse follows from the functoriality of the construction. \end{proof} We now show that the rank of the map $\widehat\Psi_3:\widehat C_3\to \widehat L_1$ is combinatorially determined. Thus, the criterion for decomposition of the $I$-adic completion of the Alexander invariant of Theorem~\ref{thm:decomp}---the surjectivity of $\widehat\Psi_3$---is combinatorial as well. By the lemma, it suffices to show that the rank of $\overline \Psi_3:\overline C_3\to \overline L_1$ is combinatorially determined. For this, let ${\mathcal A}$ and ${\mathcal A}^*$ be lattice-isomorphic arrangements of $n$ lines in ${\mathbb C}^2$ (which are transverse to the line at infinity). Let ${\mathcal W}$ be a braided wiring diagram associated to ${\mathcal A}$, and let $\overline \Psi_3:\overline C_3\to \overline L_1= \bigoplus_V \overline C_2(V') \wedge \overline C_1$ be the map defined by the vertex sets $\{V_1,\dots,V_s\}$ of ${\mathcal W}$. Choose arbitrary orderings of the hyperplanes and rank two lattice elements of ${\mathcal A}^*$, and denote the elements of $L_2({\mathcal A}^*)$ by $\{U_1,\dots,U_s\}$. Then formally construct the map $\overline\Psi^*_{3} = \sum_U \overline\Psi_{U,3}: \overline C_3 \to \overline L_1^*= \bigoplus_U \overline C_2(U') \wedge \overline C_1$ using~\eqref{eq:ChainV}, the Magnus embedding, and the augmentation map~$\hat\epsilon$. Since ${\mathcal A}$ and ${\mathcal A}^*$ are lattice-isomorphic, there are permutations $\omega\in\Sigma_n$ and $\nu\in\Sigma_s$ so that $\omega(V_k) = U_{\nu(k)}$ for each $k$, $1\le k\le s$. The permutation $\omega$ induces an isomorphism $\omega_k: C_k\to C_k$ defined by $\omega_k(e_J) = e_{\omega(J)}$. The map $\overline\Psi_3$ is combinatorially determined in the sense of the following. \begin{prop} There is an isomorphism $\overline\xi : \overline L_1 \to \overline L_1^*$ so that $ \overline\xi \circ \overline{\Psi}_{3} = \overline\Psi^*_{3} \circ \overline\omega_3$. \end{prop} \begin{proof} Let $V$ be a vertex set of ${\mathcal A}$, and $U = \omega(V)$ be the corresponding vertex set of ${\mathcal A}^*$. Define a map $\xi^U_V:C_2(V') \wedge C_1 \to C_2(U') \wedge C_1$ by \begin{equation} \label{eq:latticeiso} \xi^U_V=(\pi_{U'}\wedge\id) \circ \Theta_2(\mu_U)^{-1}\circ \omega_3 \circ \Theta_2(\mu_V)\circ (\iota_{V'}\wedge \id). \end{equation} Clearly, $\xi^U_V$ is an isomorphism, with inverse $\xi^V_U$. Moreover, $\xi^U_V \circ\Psi_{V,3} = \Psi_{U,3} \circ \omega_3$. The collection $\{\xi^V_U\}$ defines a map $\xi : L_1 \to L_1^*$, which yields the desired isomorphism $\overline\xi$. \end{proof} Combining these results with those of the previous section, we obtain \begin{thm} \label{thm:ChenComb} The rank of the third Chen group of the arrangement ${\mathcal A}$ is given by the combinatorial formula $\theta_3({\mathcal A}) = \rank(\coker\overline\Psi_3) + \theta^{\operatorname{cc}}_3({\mathcal A})$. Furthermore, if the map $\overline \Psi_3:\overline C_3\to \overline L_1$ is surjective, then the $I$-adic completion of the Alexander invariant of ${\mathcal A}$ decomposes as a direct sum: $\widehat B \cong {\widehat{B}^{\operatorname{cc}}}=\oplus_V \widehat B_V$, and the ranks of the Chen groups of ${\mathcal A}$ are given by $\theta_k({\mathcal A})=\theta_k^{\operatorname{cc}}({\mathcal A})$ for all $k\ge 2$. \end{thm} \begin{rem} If ${\mathcal A}$ is an arrangement for which the map $\overline \Psi_3:\overline C_3\to \overline L_1$ is {\em not} surjective, the ranks $\theta_k({\mathcal A})$ of the Chen groups of ${\mathcal A}$ for $k \ge 4$ may be computed using the Groebner basis algorithm described in~\ref{subsec:Chen}. Alternatively, in light of Theorem~\ref{thm:theta3}, one can apply this algorithm to the presentation~\eqref{eq:H1pres} of the module $H$ (or the smaller presentation described in the proof of~\ref{thm:theta3}) to determine $\rank({\mathfrak m}^{k-3}H/{\mathfrak m}^{k-2}H)= \theta_k({\mathcal A}) - \theta^{\operatorname{cc}}_k({\mathcal A})$. \end{rem} \section{Examples} \label{sec:Examples} In this section, we illustrate the results of the previous sections by means of several explicit examples. We order the hyperplanes of an arrangement ${\mathcal A}=\{H_1,\dots,H_n\}$ in the order indicated by the defining polynomial $Q({\mathcal A}) = \prod_{k=1}^n \ll_k$ (so $H_k = \ker\ll_k$). \begin{exm}\label{exm:6lines} Consider the central 3-arrangement ${\mathcal A}$ with defining polynomial $Q=xyz(y+z)(x-z)(2x+y)$. Randell \cite{Ra} noted that this arrangement is not $K(\pi ,1)$, and that there is no aspherical arrangement with the same lattice in ranks one and two. Arvola \cite{Ar} further showed that the group of this arrangement is not of type FL. The rank two elements of the lattice of ${\mathcal A}$ are \begin{equation*} L_2({\mathcal A}) = \{\{1, 2, 6\}, \{1, 3, 5\}, \{2, 3, 4\}, \{1, 4\}, \{2, 5\}, \{4, 5\}, \{3, 6\}, \{4, 6\}, \{5, 6\}\}. \end{equation*} It is readily checked that the map $\overline\Psi_3:{\mathbb Z}^{20} \to {\mathbb Z}^{12}$ is surjective. By Theorem~\ref{thm:ChenComb}, $\widehat B \cong{\widehat{B}^{\operatorname{cc}}}=\widehat B_{\{1,2,6\}} \oplus \widehat B_{\{1,3,5\}} \oplus \widehat B_{\{2,3,4\}}$. It follows that the ranks of the Chen groups of ${\mathcal A} $ are $\theta_{1}=6$ and $\theta _{k}=3(k-1)$ for $k\ge 2$. Notice that these ranks coincide with those of the Chen groups of a direct product of three free groups on two generators, though clearly $G \not\cong F_{2}\times F_{2}\times F_{2}$. Using Theorem~\ref{thm:alexreal} and elementary row operations, one can show that the Alexander invariant itself decomposes as a direct sum, $B \cong {B^{\operatorname{cc}}}= B_{\{1,2,6\}} \oplus B_{\{1,3,5\}} \oplus B_{\{2,3,4\}}$. \end{exm} \begin{exm}\label{exm:braid} The braid arrangement ${\mathcal A}_4$ is the smallest arrangement for which the completion of the Alexander invariant does not decompose. The polynomial $Q=xyz(x-y)(x-z)(y-z)$ defines a central 3-arrangement whose complement is homotopy equivalent to that of ${\mathcal A}_4$. The rank two elements of $L({\mathcal A}_4)$ (the partition lattice) are \begin{equation*} L_2({\mathcal A}_4) =\{\{1, 2, 4\}, \{1, 3, 5\}, \{2, 3, 6\}, \{3, 4\}, \{2, 5\}, \{4, 5, 6\}, \{1, 6\}\}. \end{equation*} The map $\overline\Psi_3:{\mathbb Z}^{20}\to {\mathbb Z}^{16}$ is {\it not} surjective. Thus $\widehat B({\mathcal A}_4)$ does not decompose. A basis for $\coker\overline\Psi_3$ is given by the two elements \begin{align*} \kappa_1&=e_{\{2,4\}}\wedge(e_6-e_3)+e_{\{3,5\}}\wedge(e_4-e_6)+ e_{\{3,6\}}\wedge(e_1-e_4)+e_{\{5,6\}}\wedge(e_3-e_1),\\ \kappa_2&=e_{\{2,4\}}\wedge(e_6-e_5)+e_{\{3,5\}}\wedge(e_2-e_6)+ e_{\{3,6\}}\wedge(e_1-e_5)+e_{\{5,6\}}\wedge(e_2-e_1). \end{align*} Since $\rank\overline\Psi_3=14$, we have $\theta_3({\mathcal A}_4)=10$. \end{exm} \begin{rem} Note that the rank of the third Chen group of ${\mathcal A}_4$ is equal to that of the product arrangement defined by $xy(y-x)z(z-x)(z-2x)$. In general, by the LCS formula~\cite{FR1}, the ranks of the lower central series quotients of the pure braid group $P_n$ are equal to those of the direct product of free groups $\Pi_n = F_{n-1}\times\dots\times F_1$. These groups are distinguished by their Chen groups. For $k \ge 4$, we have $\theta_k(\Pi_n) = (k-1)\binom{n+k-2}{k+1}$, by Example~\ref{exm:prodFn}, and $\theta_k(P_n) = (k-1)\binom{n+1}{4}$, by the main result of~\cite{CS1}. Thus, $\theta_k(P_n) \neq \theta_k(\Pi_n)$ for $n\ge 4$, and the groups $P_n$ and $\Pi_n$ are not isomorphic. \end{rem} \begin{rem}\label{rem:suba4} Example~\ref{exm:braid} provides an easy means for detecting when the completion of the Alexander invariant of an arrangement ${\mathcal A}$ does not decompose. If ${\mathcal S} \subset {\mathcal A}$ is a subarrangement which is lattice-isomorphic to the braid arrangement ${\mathcal A}_4$, one can use the above elements of $\coker\overline\Psi_3({\mathcal A}_4)$ and maps of the form~\eqref{eq:latticeiso} to generate non-trivial elements of $\coker \overline\Psi_3({\mathcal A})$. It is interesting to note that (the matroid of) such an arrangement ${\mathcal A}$ has ``non-local decomposable relations,'' see~\cite{F3}. \end{rem} \begin{exm}[Diamond]\label{exm:diamond} Let $\mathcal D$ be the central $3$-arrangement with defining polynomial $Q(\mathcal D)=x(x+y+z)(x+y-z)y(x-y-z)(x-y+z)z$. This is a free, simplicial arrangement for which the LCS formula does not hold, and the Orlik-Solomon algebra is not quadratic, see \cite{F1}, \cite{FR2}, \cite{SY}. The rank two elements of $L(\mathcal D)$ are \begin{equation*} L_2(\mathcal D) =\{\{3,4,5\}, \{1,2,5\}, \{1,4\}, \{1,3,6\}, \{2,4,6\}, \{1,7\}, \{2,3,7\}, \{4,7\}, \{5,6,7\}\}. \end{equation*} This arrangement has three distinct subarrangements lattice-isomorphic to ${\mathcal A}_4$. One such subarrangement is ${\mathcal S}=\mathcal D \setminus \{H_1\}$. Define $\omega:[6]\hookrightarrow [7]$ by $1\mapsto 2$, $2\mapsto 3$, $3\mapsto 4$, $4\mapsto 7$, $5\mapsto 6$, $6\mapsto 5$. The map $\omega$ gives rise to a lattice-isomorphism $L({\mathcal A}_4) \xrightarrow{\sim} L({\mathcal S})$. Define $\overline\xi:\bigoplus_{V\in L_2({\mathcal A}_4)} \overline C_2(V')\wedge \overline C_1 \to \bigoplus_{U\in L_2(\mathcal D)} \overline C_2(U')\wedge \overline C_1$ as in~\eqref{eq:latticeiso}: $\overline\xi(e_{\{5,6\}}\wedge e_j) = -e_{\{6,7\}}\wedge \overline\omega_1(e_j)$ and $\overline\xi(e_K\wedge e_j) = \overline\omega_3(e_K \wedge e_j)$ for $K \neq \{5,6\}$. Then \begin{align*} \overline\xi(\kappa_1)&=e_{\{3,7\}}\wedge(e_5-e_4)+ e_{\{4,6\}}\wedge(e_7-e_5)+ e_{\{4,5\}}\wedge(e_2-e_7)+e_{\{6,7\}}\wedge(e_2-e_4),\\ \overline\xi(\kappa_2)&=e_{\{3,7\}}\wedge(e_5-e_6)+ e_{\{4,6\}}\wedge(e_3-e_5)+ e_{\{4,5\}}\wedge(e_2-e_6)+e_{\{6,7\}}\wedge(e_2-e_3) \end{align*} are in the cokernel of $\overline\Psi_3: {\mathbb Z}^{35}\to{\mathbb Z}^{30}$. We obtain 6 distinct elements of~$\coker \overline\Psi_3$ in this way. However, there is a relation among them. We have $\rank\overline\Psi_3=25$, and~$\theta_3(\mathcal D)=17$. The ranks of the higher Chen groups may be found via the Groebner basis algorithm of~Theorem~\ref{thm:grB}. By Example~\ref{exm:cone}, we can simplify the computation by working with the decone of $\mathcal D$ defined by $Q(\mathcal D)|_{z=1}$. Rotating this arrangement counterclockwise to insure that first coordinate projection is generic, we obtain a wiring diagram ${\mathcal W}=\{\{3,4,5\},\{1,2,5\},\{1,4\},\{1,3,6\},\{2,4,6\}\}$. The image of the associated braid monodromy $\a :F_{5}\to P_{6}$ is generated by $\{A_{3,4,5}, A_{1,2,5}, A_{1,4}^{A_{3,4}}, A_{1,3,6}, A_{2,4,6}^{A_{3,4}A_{3,6}}\}$. From the presentation $\L ^{20} \xrightarrow{\Delta} \L ^{6} \rightarrow B \rightarrow 0$ provided by Theorem~\ref{thm:alexreal}, we find $\theta_k({\mathcal A})=9(k-1)$, for $k\ge 4$. \end{exm} We have found a number of other arrangements for which the completion of the Alexander invariant does not decompose as a direct sum. For example, for the Coxeter arrangement of type $\text{B}_3$, we have $\overline C_3={\mathbb Z}^{84}$, $\overline L_1={\mathbb Z}^{85}$, and $\overline\Psi_3:\overline C_3\to \overline L_1$ is obviously not surjective. (This arrangement also has (many) subarrangements lattice-isomorphic to ${\mathcal A}_4$.) More subtle examples include the following. \begin{exm}[MacLane]\label{exm:maclane} The polynomials \begin{equation*} Q^{\pm}=x y (y-x) z (z-x-\omega^2 y) (z+\omega y) (z-x) (z+\omega^2 x+\omega y) \end{equation*} where $\omega=(-1\pm \sqrt{-3})/2$, define complex conjugate realizations ${\mathcal A}^\pm$ of the MacLane matroid (the $8_3$ configuration). These arrangements were used by Rybnikov \cite{Ry} in his construction of lattice-isomorphic arrangements with distinct fundamental groups. Rybnikov's arrangements are not distinguished by their Chen groups. Complex conjugation induces an isomorphism of the groups of ${\mathcal A}^+$ and ${\mathcal A}^-$, and thus an isomorphism of the Alexander invariants, $B^+\cong B^-$. Neither of these arrangements has a subarrangement lattice-isomorphic to the braid arrangement ${\mathcal A}_4$. Nevertheless, the maps $\overline\Psi_3^{\pm}$ are not surjective, and the modules $\widehat B^{\pm}$ do not decompose. The hyperplane $H=\{7x-5y+z=1\}$ is generic with respect to both these arrangements. Moreover, the projection $\pr :{\mathbb C} ^{2}\to {\mathbb C}$ defined by $\pr (x,y)=13x-4y$ is generic with respect to both affine $2$-arrangements $H\cap{\mathcal A}^\pm$. Changing coordinates accordingly, we obtain braided wiring diagrams ${\mathcal W}^\pm=\{ V_1,\b_1^\pm,V_2,\b_2^\pm,\dots,\b_{11}^\pm,V_{12}\}$ with vertex sets and intermediary braids given by: \begin{equation*} \label{eq:bwire} \matrix V_{1}=\{3,4\},\hfill&\b_{1}^{\pm} = 1,\hfill&\qquad V_{7\,\,\,}=\{2,5,7\},\hfill&\b_{7}^{\pm}={\sigma }_3^{\mp 1},\hfill\\ V_2=\{3,5,6\},\hfill&\b_{2}^{\pm} = 1,\hfill&\qquad V_{8\,\,\,}=\{6,7\},\hfill&\b_{8}^{\pm} = {\sigma }_4^{\pm 1}{\sigma }_3^{\pm 1},\hfill\\ V_3=\{3,7,8\},\hfill&\b_{3}^{\pm}=1,\hfill&\qquad V_{9\,\,\,}=\{1,6,8\},\hfill&\b_{9}^{\pm}={\sigma }_1^{\mp 1},\hfill\\ V_4=\{2,4,6\},\hfill&\b_{4}^{\pm}={\sigma }_4^{\mp 1}{\sigma }_3^{\mp 1},\hfill&\qquad V_{10}=\{1,4,7\},\hfill&\b_{10}^{\pm}={\sigma }_2^{\mp 1},\hfill\\ V_5=\{2,8\},\hfill&\b_{5}^{\pm} = 1,\hfill&\qquad V_{11}=\{1,5\},\hfill&\b_{11}^{\pm} = {\sigma }_4^{\pm 1},\hfill\\ V_6=\{4,5,8\},\hfill&\b_{6}^{\pm}={\sigma }_2^{\pm 1}{\sigma }_5^{\pm 1},\hfill&\qquad V_{12}=\{1,2,3\}.\hfill \endmatrix \end{equation*} An argument as in \cite{CS3} shows that the braid monodromies associated to ${\mathcal W}^+$ and ${\mathcal W}^-$ are equivalent, but not braid equivalent. Calculations with these monodromies reveal that $\theta^{{\operatorname{cc}}}_3({\mathcal A}^\pm) = 16$, $\theta_3({\mathcal A}^\pm) = 21$, and $\theta_k({\mathcal A}^\pm) = \theta^{\operatorname{cc}}_k({\mathcal A}^\pm) = 8(k-1)$ for $k\ge 4$. Thus the failure of $\overline\Psi_3$ to be surjective is detected only by the third Chen group. \end{exm} \begin{exm}[$9_3$ Configurations]\label{exm:pappus} The relationship between the (completion of the) Alexander invariant and the combinatorics of an arrangement appears to be quite delicate. As an illustration, consider the arrangements $\mathcal P_1$ and $\mathcal P_2$ defined by \begin{align*} Q(\mathcal P_1)&=xyz(x-y)(y-z)(x-y-z)(2x+y+z)(2x+y-z)(2x-5y+z),\\ Q(\mathcal P_2)&=xyz(x+y)(y+z)(x+3z)(x+2y+z)(x+2y+3z)(2x+3y+3z). \end{align*} The arrangement $\mathcal P_1$ is a realization of the Pappus configuration $(9_3)_1$, while $\mathcal P_2$ is a realization of the configuration $(9_3)_2$. Note that neither of these arrangements has a subarrangement lattice-isomorphic to the braid arrangement. The combinatorial distinction between these arrangements (resp.,~their underlying matroids) is detected by the maps $\overline\Psi_3(\mathcal P_k):{\mathbb Z}^{84}\to{\mathbb Z}^{63}$. The map $\overline\Psi_3(\mathcal P_2)$ is surjective, and consequently the module $\widehat B(\mathcal P_2)$ decomposes as a direct sum. Thus, $\theta_k(\mathcal P_2)=9 (k-1)$ for $k\ge 2$. However, the map $\overline\Psi_3(\mathcal P_1)$ is not surjective, and $\widehat B(\mathcal P_1)$ does not decompose. A calculation shows that $\theta_2(\mathcal P_1)=9$, and $\theta_k(\mathcal P_1) = 10 (k-1)$ for $k\ge 3$. It was conjectured in \cite{CS1} that, for $k$ sufficiently large, one has $\theta_k({\mathcal A})=\theta_k^{\operatorname{cc}}({\mathcal A}) + (k-1)\beta({\mathcal A})$, where $\beta({\mathcal A})$ is the number of subarrangements of ${\mathcal A}$ that are lattice-isomorphic to ${\mathcal A}_4$. The arrangement $\mathcal P_1$ has $\beta=0$ and $\theta_k^{\operatorname{cc}} = 9(k-1)$, and hence provides a counterexample to that conjecture. \end{exm} \bibliographystyle{amsalpha}

9703

alg-geom/9703001

en

https://arxiv.org/abs/alg-geom/9703001

alg-geom/9703001

Frank Sottile

Nantel Bergeron and Frank Sottile

Schubert polynomials, the Bruhat order, and the geometry of flag manifolds

Revised version of MSRI preprint \# 1996 - 083, 61 pages with 36 figures, where 15 of the pages and 26 of the figures are in an appendix containing examples of the major geometric and combinatorial results LaTeX 2e

Duke Math. J., Vol 95 (1998) pp. 373-423

10.1215/S0012-7094-98-09511-4

MSRI 1996-083

We illuminate the relation between the Bruhat order on the symmetric group and structure constants (Littlewood-Richardson coefficients) for the cohomology of the flag manifold in terms of its basis of Schubert classes. Equivalently, the structure constants for the ring of polynomials in variables $x_1,x_2,...$ in terms of its basis of Schubert polynomials. We use combinatorial, algebraic, and geometric methods, notably a study of intersections of Schubert varieties and maps between flag manifolds. We establish a number of new identities among these structure constants. This leads to formulas for some of these constants and new results on the enumeration of chains in the Bruhat order. A new graded partial order on the symmetric group which contains Young's lattice arises from these investigations. We also derive formulas for certain specializations of Schubert polynomials.

alg-geom

\section*{Introduction} Extending work of Demazure~\cite{Demazure} and of Bernstein, Gelfand, and Gelfand~\cite{BGG}, in 1982 Lascoux and Sch\"utzenberger~\cite{Lascoux_Schutzenberger_polynomes_schubert} defined remarkable polynomial representatives for Schubert classes in the cohomology of a flag manifold, which they called Schubert polynomials. For each permutation $w$ in ${\cal S}_\infty$, there is a Schubert polynomial ${\frak S}_w\in{\Bbb Z}[x_1,x_2,\ldots]$. The collection of all Schubert polynomials forms an additive homogeneous basis for this polynomial ring. Thus the identity \begin{equation}\label{eq:structure} {\frak S}_u \cdot {\frak S}_v \quad =\quad \sum_w c^w_{u\, v} {\frak S}_w \end{equation} defines integral {\em structure constants} $c^w_{u\, v}$ for the ring of polynomials with respect to its Schubert basis. Littlewood-Richardson coefficients are a special case of the $c^w_{u\, v}$ as every Schur symmetric polynomial is a Schubert polynomial. The $c^w_{u\, v}$ are positive integers: Evaluating a Schubert polynomial at certain Chern classes gives a Schubert class in the cohomology of the flag manifold. Hence, $c^w_{u\, v}$ enumerates the flags in a suitable triple intersection of Schubert varieties. This evaluation exhibits the cohomology of the flag manifold as the quotient: $$ {\Bbb Z}[x_1,x_2,\ldots]/\Span{{\frak S}_w\,|\, w\not\in {\cal S}_n}. $$ These constants, $c^w_{u\, v}$, are readily computed: The MAPLE libraries ACE~\cite{ACE} include routines for manipulating Schubert polynomials, Gr{\"o}bner basis methods are applied in~\cite {Fomin_Gelfand_Postnikov}, and a new approach, using orbit values of Kostant polynomials, is developed in~\cite{Billey}. However, it remains an open problem to understand these constants combinatorially. By this we mean a combinatorial interpretation or a bijective formula for these constants. We expect such a formula will have the form \begin{equation}\label{eq:expected_rule} c^w_{u\, v}\quad =\quad\# \left\{ \mbox{\begin{minipage}[c]{3.6in} (saturated) chains in the Bruhat order on ${\cal S}_\infty$ from \mbox{\ $u$ to $\,w$} \,satisfying {\em \,some} \,condition \,imposed \,by $\,v$ \end{minipage}} \right\}. \end{equation} The Littlewood-Richardson rule~\cite{Littlewood_Richardson}, which this would generalize, may be expressed in this form ({\em cf.}~\S\ref{sec:chain_description}), as standard Young tableaux are chains in Young's Lattice, a suborder of the Bruhat order. A relation between chains in the Bruhat order and the multiplication of Schubert polynomials has previously been noted~\cite{Hiller_intersections}. A new proof of the classical Pieri's formula for Grassmannians~\cite{sottile_explicit_pieri} suggests a geometric rationale for such `chain-theoretic' formulas. Lastly, known formulas for these constants, particularly Monk's formula~\cite{Monk}, Pieri-type formulas (first stated by Lascoux and Sch\"utzenberger~\cite{Lascoux_Schutzenberger_polynomes_schubert} but only recently given proofs using geometry~\cite{sottile_pieri_schubert} and algebra~\cite{Winkel_multiplication}), and other formulas of~\cite{sottile_pieri_schubert}, are all of this form. Recently, Ciocan-Fontanine~\cite{Ciocan_partial} has generalized these chain-theoretic Pieri-type formulas to the quantum cohomology rings of manifolds of partial flags. This has also been announced~\cite{Kirillov_Maeno} in the special case of quantum Schubert polynomials~\cite{Fomin_Gelfand_Postnikov}. In \S\ref{sec:chains_and_orders}, we give a refinement of~(\ref{eq:expected_rule}). We establish several new identities for the $c^w_{u\,v}$, including Theorems~\ref{thm:B} ({\em ii}) and~\ref{thm:C} ({\em ii}). One, Theorem~\ref{thm:A} ({\em i})({\em b}), gives a recursion for $c^w_{u\,v}$ when one of the permutations $wu^{-1}, wv^{-1}$, or $w_0uv^{-1}$ has a fixed point and a condition on its inversions holds. When ${\frak S}_v$ is a Schur polynomial, we give a chain-theoretic interpretation (Theorem~\ref{thm:skew_shape}) for some $c^w_{u\,v}$, determine many more (Theorem~\ref{thm:skew_permutation}) in terms of the classical Littlewood-Richardson coefficients, and show how a map that takes certain chains in the Bruhat order to standard Young tableaux and satisfies some additional properties would give a combinatorial interpretation for these $c^w_{u\,v}$ (Theorem~\ref{thm:combinatorial}). Most of these identities have an order-theoretic companion which could imply them, were a description such as (\ref{eq:expected_rule}) known. The one identity (Theorem~\ref{thm:D}) lacking such a companion yields a new result about the enumeration of chains in the Bruhat order (Corollary~\ref{cor:equal_chains}). In \S\ref{sec:orders}, we study a suborder called the $k$-Bruhat order, which is relevant in (\ref{eq:expected_rule}) when ${\frak S}_v$ is Schur symmetric polynomial in $x_1,\ldots,x_k$. This leads to a new graded partial order on ${\cal S}_\infty$ containing every interval in Young's lattice as an induced suborder for which many group homomorphisms are order preserving (Theorem~\ref{thm:new_order}). Some of these identities require the computation of maps on the cohomology of flag manifolds induced by certain embeddings, including Theorem~\ref{thm:projection}, Lemma~\ref{lemma:fixed_pts}, and Theorem~\ref{thm:many_identities}. We use these to determine the effect of some homomorphisms of ${\Bbb Z}[x_1,x_2,\ldots]$ on its Schubert basis. For example, let $P\subset {\Bbb N}$ and list the elements of $P$ and ${\Bbb N}-P$ in order: $$ P\ :\ p_1<p_2<\cdots\qquad \qquad {\Bbb N}-P\ :\ p^c_1<p^c_2<\cdots $$ Define $\Psi_P:{\Bbb Z}[x_1,x_2,\ldots]\rightarrow {\Bbb Z}[y_1,y_2,\ldots,z_1,z_2,\ldots]$ by: $$ \Psi_P(x_{p_j}) \ =\ y_j\qquad\mbox{and}\qquad \Psi_P(x_{p^c_j}) \ =\ z_j. $$ Then there exist integers $d^{u\,v}_w$ such that $$ \Psi_P({\frak S}_w(x))\ =\ \sum_{u,v} d^{u\,v}_w{\frak S}_u(y){\frak S}_v(z). $$ We show (Theorem~\ref{thm:substitution}) there exist $\pi\in{\cal S}_\infty$ (depending upon $w$ and $P$) such that $d^{u\,v}_w = c^{(u\times v)\pi}_{\pi\;w}$. In particular, the coefficients $d^{u\,v}_w$ are nonnegative. This generalizes 1.5 of~\cite{Lascoux_Schutzenberger_structure_de_Hopf}, where it is shown that the $d^{u\,v}_w$ are non-negative when $P=\{1,2,\ldots,n\}$. Algebraic structures in the cohomology of a flag manifold also yield identities among the $c^w_{u\,v}$ such as $c^w_{u\,v}=c^w_{v\,u}$ (imposed by commutativity) or $c^w_{u\,v}=c^{w_0u}_{w_0w\:v}= c^{\overline{w}}_{\overline{u}\:\overline{v}}$, where $\overline{w}:= w_0 w w_0$, (imposed by Poincar{\'e} duality among the Schubert classes). Such `algebraic' identities for the classical Littlewood-Richardson coefficients were studied combinatorially in~\cite{Zelevinsky,% Berenstein_Zelevinsky,Hanlon_Sundaram,% benkart_sottile_stroomer_switching,Fulton_tableaux}. We expect the identities established here will similarly lead to some beautiful combinatorics, once a combinatorial interpretation for the $c^w_{u\,v}$ is known. These identities impose stringent conditions on the form of any combinatorial interpretation and should be useful in guiding the search for such an interpretation. \tableofcontents \section{Description of results} \subsection{Suborders of the Bruhat order and the $c^w_{u\,v}$}\label{sec:chains_and_orders} Suppose the Schubert polynomial ${\frak S}_v$ in (\ref{eq:structure}) is replaced by the Schur polynomial $S_\lambda(x_1,\ldots,x_k)$. The resulting identity $$ {\frak S}_u \cdot S_\lambda(x_1,\ldots,x_k) \quad =\quad \sum_w c^w_{u\, v(\lambda,k)}\, {\frak S}_w \eqno(1.1.1) $ defines integer constants $c^w_{u\, v(\lambda,k)}$, which we call {\em Littlewood-Richardson coefficients for Schubert polynomials}, as we show they share many properties with the classical Littlewood-Richardson coefficients. They are related to chains in a suborder of the Bruhat order called the {\em $k$-Bruhat order}, $\leq_k$. Its covers coincide with the index of summation in Monk's formula~\cite{Monk}: $$ {\frak S}_u \cdot {\frak S}_{(k,\,k{+}1)} \quad =\quad {\frak S}_u \cdot (x_1+\cdots+x_k) \quad =\quad \sum {\frak S}_{u(a, b)}, $$ the sum over all $a\leq k<b$ where $\ell(u(a,b))=\ell(u)+1$. The set of permutations comparable to the identity in the $k$-Bruhat order is isomorphic to Young's lattice of partitions with at most $k$ parts. This is the set of Grassmannian permutations with descent $k$, those permutations whose Schubert polynomials are Schur symmetric polynomials in $x_1,\ldots,x_k$. If $f^\lambda$ is the number of standard Young tableaux of shape $\lambda$, then~\cite[ I.5, Example 2]{Macdonald_symmetric}, $$ (x_1+\cdots+x_k)^m \quad =\quad \sum_\lambda f^\lambda S_\lambda(x_1,\ldots,x_k), $$ the sum over all $\lambda$ which partition the integer $m$. Considering the coefficient of ${\frak S}_w$ in the product ${\frak S}_u\cdot (x_1+\cdots+x_k)^m$ and the definition (1.1.1) of $c^w_{u\,v(\lambda,k)}$, we obtain: \begin{prop}\label{prop:chains} The number of chains in the $k$-Bruhat order from $u$ to $w$ is $$ \sum_\lambda f^\lambda c^w_{u\,v(\lambda,k)}. $$ \end{prop} In particular, $c^w_{u\,v(\lambda,k)}=0$ unless $u\leq_k w$. A chain-theoretic description of the constants $c^w_{u\,v(\lambda,k)}$ should provide a bijective proof of Proposition~\ref{prop:chains}. By this we mean a function $\tau$ from the set of chains in $[u,w]_k$ to the set of standard Young tableaux $T$ whose shape is a partition of $\ell(w)-\ell(u)$ with the further condition that whenever $T$ has shape $\lambda$, then $\#\tau^{-1}(T)=c^w_{u\,v(\lambda,k)}$. For the classical Littlewood-Richardson coefficients, Schensted insertion~\cite{Schensted} furnishes a proof~\cite{Thomas_schensted_construction} ({\em cf.}~\S\ref{sec:chain_description}), as does Sch{\"u}tzenberger's {\em jeu de taquin}~\cite{Schutzenberger_jeu_de_taquin}. In~\S\ref{sec:further} we show (Theorem~\ref{thm:combinatorial}) that if $\tau$ is a function where $\#\tau^{-1}(T)$ depends only upon the shape of $T$ and satisfies a further condition, then $\#\tau^{-1}(T)=c^w_{u\,v(\lambda,k)}$. Such a function $\tau$ would be a generalization of Schensted insertion to this setting. The $k$-Bruhat order has a more intrinsic formulation, which we establish in \S\ref{sec:k-bruhat}: \begin{thm}\label{thm:k-length} Let $u,w\in {\cal S}_\infty$. Then $u\leq _k w$ if and only if \begin{enumerate} \item[I.] $a\leq k < b$ implies $u(a)\leq w(a)$ and $u(b)\geq w(b)$. \item[II.] If $\/a<b$, $u(a)<u(b)$, and $w(a)>w(b)$, then $a\leq k< b$. \end{enumerate} \end{thm} The $k$-Bruhat order and its connection to the Littlewood-Richardson coefficients $c^w_{u\,v(\lambda,k)}$ may be generalized, which leads to a refinement of (\ref{eq:expected_rule}). A {\em parabolic subgroup} $P$ of ${\cal S}_\infty$~\cite{Bourbaki_Groupes_IV} is a subgroup generated by some adjacent transpositions, $(i,i{+}1)$. Given a parabolic subgroup $P$ of ${\cal S}_\infty$, define the {\em P-Bruhat order} by its covers. A cover $u\lessdot_P w$ in the $P$-Bruhat order is a cover in the Bruhat order where $u^{-1}w \not\in P$. When $P$ is generated by all adjacent transpositions except $(k,k{+}1)$, this is the $k$-Bruhat order. Let $I\subset \{1,2,\ldots,n{-}1\}$ index the adjacent transpositions {\em not} in $P$. A {\em coloured chain} in the $P$-Bruhat order is a chain together with an element of $I\bigcap \{a,a{+}1,\ldots,b{-}1\}$ for each cover $u\lessdot_P u(a,b)$ in the chain. This notion of colouring the Bruhat order was introduced in~\cite{Lascoux_Schutzenberger_symmetry}. Iterating Monk's rule, we obtain: $$ \left(\sum_{i\in I}\:{\frak S}_{(i,i{+}1)}\right)^m \quad =\quad \sum_v\, f^v_e(P)\; {\frak S}_v, \eqno(1.1.2) $$ where $f^v_e(P)$ counts the coloured chains in the $P$-Bruhat order from $e$ to $v$. This number, $f^v_e(P)$, is nonzero only for those $v$ which are minimal in their coset $vP$. More generally, let $f^w_u(P)$ count the coloured chains in the $P$-Bruhat order from $u$ to $w$. Multiplying (1.1.2) by ${\frak S}_u$ and equating coefficients of ${\frak S}_w$, gives a generalization of Proposition~\ref{prop:chains}: \begin{thm}\label{thm:chains} Let $u,w\in{\cal S}_\infty$ and $P$ be any parabolic subgroup of ${\cal S}_\infty$. Then $$ f^w_u(P)\quad =\quad \sum_v\, c^w_{u\,v}\, f^v_e(P). $$ \end{thm} This also shows $c^w_{u\,v}=0$ unless $u\leq_P w$, whenever $v$ is minimal in $vP$. Theorem~\ref{thm:chains} suggests a refinement of (\ref{eq:expected_rule}): Let $u,v,w\in{\cal S}_\infty$, and let $P$ be any parabolic subgroup such that $v$ is minimal in $vP$. (There always is such a $P$.) Then, for every coloured chain $\gamma$ in the $P$-Bruhat order from $e$ to $v$, we expect that $$ c^w_{u\, v}\quad =\quad\# \left\{ \mbox{\begin{minipage}[c]{3.6in} coloured chains in the $P$-Bruhat order on ${\cal S}_\infty$ from $u$ to $w$ which satisfy {\em some} condition imposed by $\gamma$ \end{minipage}} \right\}. \eqno(1.1.3 $$ Moreover, this rule should give a bijective proof of Theorem~\ref{thm:chains}. This $P$-Bruhat order may be defined for every parabolic subgroup of every Coxeter group. Likewise, the problem of finding the structure constants for a Schubert basis also generalizes. For Weyl groups, the basis is the Schubert classes in the cohomology of a generalized flag manifold $G/B$ or the analogues of Schubert polynomials in this case~\cite{Billey_Haiman_Schubert,Fomin_Kirillov_Bn,% Fulton_orthogonal,Pragacz_Ratajski_formulas}. For finite Coxeter groups, the basis is the `Schubert classes' of Hiller~\cite{Hiller_schubert} in the coinvariant algebra. Likewise, Theorem~\ref{thm:chains} and the expectation (1.1.3) have analogues in this more general setting. Of the known formulas in this setting~\cite{Chevalley91,Hiller_Boe,Pragacz_S-Q,% Stembridge_shifted,Pragacz_Ratajski_Pieri_Odd_I,% Pragacz_Ratajski_Pieri_Odd_II,Pragacz_Ratajski_Pieri_Even} (see also the survey~\cite{Pragacz_divided}), few~\cite{Chevalley91,Hiller_Boe,Pragacz_S-Q,% Stembridge_shifted} have been expressed in such a chain-theoretic manner. \subsection{Substitutions and the Schubert basis} In \S\S\ref{sec:endomorphism} and \ref{sec:fixed_point_identities}, we study the $c^w_{u\, v}$ when $w(p)=u(p)$ for some $p$. This leads to a formula for the substitution of 0 for $x_p$ in terms of the Schubert basis, a recursion for some $c^w_{u\, v}$, and new identities. For $w\in{\cal S}_{n+1}$ and $1\leq p\leq n+1$, let $w/_p\in {\cal S}_n$ be defined by deleting the $p$th row and $w(p)$th column from the permutation matrix of $w$. If $y\in{\cal S}_n$ and $1\leq q\leq n+1$, then $\varepsilon_{p,q}(y)\in {\cal S}_{n+1}$ is the permutation such that $\varepsilon_{p,q}(y)/_p = y$ and $\varepsilon_{p,q}(y)(p)=q$. The index of summation in a particular case of the Pieri-type formula~\cite{Lascoux_Schutzenberger_polynomes_schubert,% sottile_pieri_schubert,Winkel_multiplication}, $$ {\frak S}_v \cdot (x_1\cdots x_{p-1}) \quad =\quad \sum_{v \cpp w} {\frak S}_w, $$ defines the relation $v\cpp w$, which is described in more detail before Theorem~\ref{thm:projection}. Define $\Psi_p:{\Bbb Z}[x_1,x_2,\ldots] \rightarrow {\Bbb Z}[x_1,x_2,\ldots]$ by $$ \Psi_p(x_j)\ =\ \left\{\begin{array}{ll} x_j&\mbox{ if } j<p\\ 0& \mbox{ if } j=p\\ x_{j-1}&\mbox{ if } j>p\end{array}\right.. $$ \begin{thm}\label{thm:A} Let $u,w\in {\cal S}_\infty$ and $p\in{\Bbb N}$. \begin{enumerate} \item[({\em i})] Suppose $w(p)=u(p)$ and $\ell(w)-\ell(u)=\ell(w/_p)-\ell(u/_p)$. Then \begin{enumerate} \item[({\em a})] $\varepsilon_{p,u(p)} : [u/_p,w/_p] \stackrel{\sim}{\longrightarrow} [u,w]$. \item[({\em b})] For every $v\in {\cal S}_\infty$, $$ c^w_{u\, v}\quad =\quad \sum_{\stackrel{\mbox{\scriptsize $y\in{\cal S}_\infty$}}% {v \cpp \varepsilon_{p,1}(y)}} c^{w/_p}_{u/_p\: y}. $$ \end{enumerate} \item[({\em ii})] For every $v\in {\cal S}_\infty$, $$ \Psi_p({\frak S}_v) = \sum_{\stackrel{\mbox{\scriptsize $y\in{\cal S}_\infty$}}% {v \cpp \varepsilon_{p,1}(y)}} {\frak S}_y. $$ \end{enumerate} \end{thm} The first statement (Lemma~\ref{lem:expanding_bruhat} ({\em ii})) is proven using combinatorial arguments, while the second (Theorem~\ref{thm:coeff_sum}) and third (Theorem~\ref{thm:theorem_A_iii}) are proven by computing certain maps on cohomology. Since $c^w_{u\,v}=c^w_{v\,u}=c^{w_0 u}_{v\:w_0w}$, Theorem~\ref{thm:A} ({\em i})({\em b}) gives a recursion for $c^w_{u\, v}$ when one of $wu^{-1}, wv^{-1}$, or $w_0uv^{-1}$ has a fixed point and the condition on lengths is satisfied. We also compute the effect of other substitutions of the variables in terms of the Schubert basis: Let $P\subset {\Bbb N}$ and list the elements of $P$ and ${\Bbb N}-P$ in order: $$ P\ :\ p_1<p_2<\cdots\qquad \qquad {\Bbb N}-P\ :\ p^c_1<p^c_2<\cdots $$ Define $\Psi_P:{\Bbb Z}[x_1,x_2,\ldots]\rightarrow {\Bbb Z}[y_1,y_2,\ldots, z_1,z_2,\ldots]$ by: $$ \Psi_P(x_{p_j}) \ =\ y_j\qquad\mbox{and}\qquad \Psi_P(x_{p^c_j}) \ =\ z_j. $$ In Remark~\ref{rem:I_P}, we define an infinite set $I_P$ of permutations with the following property: \begin{thm}\label{thm:substitution} For every $w\in {\cal S}_\infty$, there exists an integer $N$ such that if $\pi\in I_P$ and $\pi\not\in{\cal S}_N$, then $$ \Psi_P({\frak S}_w)\ =\ \sum_{u,\, v} c^{(u\times v)\cdot \pi}_{\pi\; w}\; {\frak S}_u(y)\;{\frak S}_v(z). $$ \end{thm} A precise version of Theorem~\ref{thm:substitution} (Theorem~\ref{thm:substitution_constants}) is proven in \S\ref{sec:substitution}. Theorem~\ref{thm:substitution} gives infinitely many identities of the form $c^{(u\times v)\cdot \pi}_{\pi\; w} = c^{(u\times v)\cdot \sigma}_{\sigma\; w}$ for $\pi,\sigma\in I_P$. Moreover, for these $u,v$, and $\pi$ with $c^{(u\times v)\pi}_{\pi\: w}\neq 0$, we have $[\pi,\; (u\times v)\cdot \pi]\simeq [e,u]\times[e,\,v]$, which is suggestive of a chain-theoretic basis for these identities. Theorem~\ref{thm:substitution} extends 1.5 of~\cite{Lascoux_Schutzenberger_structure_de_Hopf}, where it is shown that the $d^{u\,v}_w([n])$ are non-negative. A combinatorial proof of the non-negativity of these coefficients $d^{u\;v}_w(P)$ and of Theorem~\ref{thm:A} ({\em ii}) using, perhaps, one of the combinatorial constructions of Schubert polynomials~\cite{% Kohnert,Bergeron,BJS,Fomin_Kirillov_YB,% Fomin_Stanley,Bergeron_Billey,Winkel_kohnert_rule} may provide insight into the problem of determining the $c^w_{u\,v}$. Theorems~\ref{thm:A} ({\em ii}) and ~\ref{thm:substitution} enable the computation of rather general substitutions: Let ${P\!_{\DOT}}:= (P_0,P_1,\ldots)$ be any (finite or infinite) partition of ${\Bbb N}$. For $i>0$, let $\underline{x}^{(i)}:= x^{(i)}_1,x^{(i)}_2,\ldots$ be a set of variables in bijection with $P_i$. Define $\Psi_{{P\!_{\DOT}}}:{\Bbb Z}[x_1,x_2,\ldots]\rightarrow {\Bbb Z}[\underline{x}^{(1)},\underline{x}^{(2)},\ldots]$ by $$ \Psi_{{P\!_{\DOT}}}(x_j)\ =\ \left\{\begin{array}{ccl} 0&&\mbox{if } j\in P_0\\ x_l^{(i)}&&\mbox{if $j$ is the $l$th element of $P_i$} \end{array}\right.. $$ \begin{cor}\label{cor:general_substitution} For every partition ${P\!_{\DOT}}$ of\/ ${\Bbb N}$ and $w\in {\cal S}_\infty$, $$ \Psi_{{P\!_{\DOT}}}({\frak S}_w(x))\ =\ \sum_{u_1,u_2,\ldots} d^{u_1,u_2,\ldots}_w({P\!_{\DOT}})\: {\frak S}_{u_1}(\underline{x}^{(1)}) {\frak S}_{u_1}(\underline{x}^{(2)})\cdots, $$ where each $d^{u_1,u_2,\ldots}_w({P\!_{\DOT}})$ is a(n explicit) sum of products of the $c^z_{v\,y}$, hence non-negative. \end{cor} A {\em ballot sequence}~\cite[\S 4.9]{Sagan} $A = (a_1,a_2,\ldots)$ is a sequence of non-negative integers where, for each $i,j \geq 1$, $$ \#\{k\leq j\;|\; a_k=i\}\quad \geq \quad \#\{k\leq j\;|\; a_k=i+1\}. $$ (Traditionally, the $a_i>0$. One should consider $a_i=0$ as a vote for `none of the above'.) Given a ballot sequence $A$, define $\Psi_A: {\Bbb Z}[x_1,x_2,\ldots]\rightarrow {\Bbb Z}[x_1,x_2,\ldots]$ by $$ \Psi_A(x_i)\quad =\quad \left\{\begin{array}{ll} 0& a_i=0\\ x_{a_i}& a_i \neq 0\end{array}\right.. $$ \begin{cor} For every ballot sequence $A$ and $w\in{\cal S}_n$, there exist non-negative integers $d^u_w(A)$ for $u,w\in {\cal S}_\infty$ such that $$ \Psi_A({\frak S}_w(x))\quad=\quad \sum_u d^u_w(A)\:{\frak S}_u(x). $$ Moreover, each $d^u_w(A)$ is a(n explicit) sum of products of the $c^z_{v\,y}$. \end{cor} \noindent{\bf Proof. } If $P_0:=\{i\;|\; a_i=0\}$ and for $j>0$ $$ P_j \ :=\ \{i\;|\;a_i\mbox{ is the $j$th occurrence of some integer in $A$}\}, $$ then $\Psi_A = \Delta \circ \Psi_{(P_0,P_1,\ldots)}$, where $\Delta$ is the diagonal map, $\Delta(x^{(i)}_j)= x_j$. \QEDnoskip \subsection{Identities when ${\frak S}_v$ is a Schur polynomial}\label{sec:Schur_identities} If $\lambda, \mu$, and $\nu$ are partitions with at most $k$ parts then the classical Littlewood-Richardson coefficients $c^\nu_{\mu\,\lambda}$ are defined by the identity $$ S_\mu(x_1,\ldots,x_k)\cdot S_\lambda(x_1,\ldots,x_k) \quad = \quad \sum_\nu c^\nu_{\mu\,\lambda} S_\nu(x_1,\ldots,x_k). $$ The $ c^\nu_{\mu\,\lambda}$ depend only upon $\lambda$ and the skew partition $\nu/\mu$. That is, if $\kappa$ and $\rho$ are partitions with at most $l$ parts, and $\kappa/\rho = \nu/\mu$, then for all partitions $\lambda$, $$ c^\nu_{\mu\,\lambda} \quad=\quad c^\kappa_{\rho\,\lambda}. $$ Moreover, $c^\kappa_{\rho\,\lambda}$ is the coefficient of $S_\kappa(x_1,\ldots,x_l)$ when $S_\rho(x_1,\ldots,x_l)\cdot S_\lambda(x_1,\ldots,x_l)$ is expressed as a sum of Schur polynomials. The order type of the interval in Young's lattice from $\mu$ to $\nu$ is determined by $\nu/\mu$. These facts generalize to the Littlewood-Richardson coefficients $c^w_{u\,v(\lambda,k)}$. If $u\leq_k w$, let $[u,w]_k$ be the interval between $u$ and $w$ in the $k$-Bruhat order, a graded poset. Permutations $\zeta$ and $\eta$ are {\em shape equivalent} if there exist sets of integers $P=\{p_1<\cdots<p_n\}$ and $Q=\{q_1<\cdots<q_n\}$, where $\zeta$ (respectively $\eta$) acts as the identity on ${\Bbb N}- P$ (respectively ${\Bbb N}- Q$), and $$ \zeta(p_i)\ =\ p_j \quad \Longleftrightarrow\quad \eta(q_i)\ =\ q_j. $$ \begin{thm}\label{thm:B} Suppose $u\leq_kw$ and $x\leq_l z$ where $wu^{-1}$ is shape equivalent to $zx^{-1}$. Then \begin{enumerate} \item[({\em i})] $[u,w]_k\simeq[x,z]_l$. When $wu^{-1}=zx^{-1}$, this isomorphism is given by $v\mapsto vu^{-1}x$. \item[({\em ii})] For all partitions $\lambda$, $c^w_{u\,v(\lambda,k)} = c^z_{x\,v(\lambda,l)}$. \end{enumerate} \end{thm} Theorem~\ref{thm:B} ({\em i}) is a consequence of Theorems~\ref{thm:k-order} and~\ref{thm:new_order}, which are proven using combinatorial arguments. Theorem~\ref{thm:B} ({\em ii}) is proven in \S\ref{sec:proof_thm_B} using geometric arguments. By Theorem~\ref{thm:B}, we may define the constant $c^\zeta_\lambda$ for $\zeta\in {\cal S}_\infty$ and $\lambda$ a partition by $c^\zeta_\lambda := c^{\zeta u}_{u\,v(\lambda,k)}$ and also define $|\zeta| := \ell(\zeta u) - \ell(u)$ for any $u \in {\cal S}_\infty$ with $u\leq_k \zeta u$. In \S\ref{sec:new:order}, this analysis leads to a graded partial order $\preceq$ on ${\cal S}_\infty$ with rank function $|\zeta|$ which has the defining property: Let $[e,\zeta]_\preceq$ be the interval in the $\preceq$-order from the identity to $\zeta$. If $u\leq_k \zeta u$, then the map $[e,\zeta]_\preceq \rightarrow [u,\zeta u]_k$ defined by $$ \eta \quad \longmapsto \quad \eta \, u $$ is an order isomorphism. Each interval in Young's lattice is an induced suborder in $({\cal S}_\infty,\preceq)$ rooted at the identity, as is the lattice of partitions with at most $l$ parts (these are embedded differently for different $l$). Proposition~\ref{prop:chains} may be stated in terms of this order: $\sum_\lambda f^\lambda c^\zeta_\lambda$ counts the chains in $[e,\zeta]_\preceq$. This order is studied further in~\cite{bergeron_sottile_order}, where an upper bound is given for $c^\zeta_\lambda$. For $\eta\in{\cal S}_n$, the map $\eta\mapsto\overline{\eta}$ induces an order isomorphism $[e,\zeta]_\preceq \stackrel{\sim}{\rightarrow} [e,\overline{\zeta}]_\preceq$. The involution ${\frak S}_w \mapsto {\frak S}_{\overline{w}}$ shows $c^\zeta_\lambda = c^{\overline{\zeta}}_{\lambda^t}$, where $\lambda^t$ is the conjugate or transpose of $\lambda$. Thus $[e,\zeta]_\preceq \simeq [e,\eta]_\preceq$ is not sufficient to guarantee $c^\zeta_\lambda = c^\eta_\lambda$. We express some of the Littlewood-Richardson coefficients in terms of chains in the Bruhat order. If $u\lessdot_k u(a,b)$ is a cover in the $k$-Bruhat order, label that edge of the Hasse diagram with the integer $u(b)$. The {\em word} of a chain in the $k$-Bruhat order is the sequence of labels of edges in the chain. \begin{thm}\label{thm:skew_shape} Suppose $u\leq_k w$ and $wu^{-1}$ is shape equivalent to $v(\mu,l)\cdot v(\nu,l)^{-1}$, for some $l$ and partitions $\mu,\nu$. Then, for all partitions $\lambda$ and standard Young tableaux $T$ of shape $\lambda$, $$ c^w_{u\,v(\lambda,k)}\ =\ \#\left\{\begin{array}{cc}\mbox{Chains in $k$-Bruhat order from $u$ to $w$ whose word} \\\mbox{has recording tableau $T$ for Schensted insertion} \end{array}\right\}. $$ \end{thm} Theorem~\ref{thm:skew_shape} gives a combinatorial proof of Proposition~\ref{prop:chains} for some $u,w$. We prove this in \S\ref{sec:chain_description} using Theorem~\ref{thm:B} and combinatorial arguments. If a skew partition $\theta=\rho\coprod\sigma$ is the union of incomparable skew partitions $\rho$ and $\sigma$, then $$ \rho\coprod\sigma\ \simeq\ \rho\times\sigma, $$ as graded posets. The skew Schur function $S_\theta$ is defined~\cite[I.5]{Macdonald_symmetric} to be $\sum_\lambda c^\theta_\lambda\, S_\lambda$ and $S_{\rho\coprod \sigma}=S_\rho\cdot S_\sigma$~\cite[I.5.7]{Macdonald_symmetric}. Thus $$ c^{\rho\coprod\sigma}_\lambda\quad =\quad \sum_{\mu,\,\nu} c^\lambda_{\mu\,\nu}\, c^\rho_\mu\, c^\sigma_\nu. \eqno(1.3.1 $$ Permutations $\zeta$ and $\eta$ are {\em disjoint} if $\zeta$ and $\eta$ have disjoint supports and $|\zeta\eta|=|\zeta|+|\eta|$. \begin{thm}\label{thm:C} Let $\zeta$ and $\eta$ be disjoint permutations. Then \begin{enumerate} \item[({\em i})] The map $(\xi,\chi)\mapsto \xi\chi$ induces an isomorphism of graded posets $$ [e,\zeta]_\preceq\times[e,\eta]_\preceq \stackrel{\sim}{\longrightarrow} [e,\zeta\eta]_\preceq. $$ \item[({\em ii})] For every partition $\lambda$,\ ${\displaystyle c^{\zeta\eta}_\lambda\ =\ \sum_{\mu,\,\nu} c^\lambda_{\mu\,\nu}\, c^\zeta_\mu\, c^\eta_\nu. }$ \end{enumerate} \end{thm} The first statement is proven in \S\ref{sec:disjoint_permutations} (Theorem~\ref{thm:disjoint_iso}), using a characterization of disjointness related to non-crossing partitions~\cite{Kreweras}, and the second in \S\ref{sec:proof_C} using geometry. Our last identity has no analogy with the classical Littlewood-Richardson coefficients. Let $(1\,2\,\ldots\, n)$ be the permutation which cyclicly permutes $[n]$. \begin{thm}[Cyclic Shift]\label{thm:D} Suppose $\zeta\in S_n$ and $\eta = \zeta^{(1\,2\,\ldots\,n)}$. Then, for every partition $\lambda$, $c^\zeta_\lambda = c^\eta_\lambda$. \end{thm} This is proven in \S\ref{sec:thmd} using geometry. Combined with Proposition~\ref{prop:chains}, we obtain: \begin{cor}\label{cor:equal_chains} If $u\leq_k w$ and $x\leq_k z$ with $wu^{-1},zx^{-1}\in{\cal S}_n$ and $(wu^{-1})^{(1\,2\,\ldots\,n)} = zx^{-1}$, then the two intervals $[u,w]_k$ and $[x,z]_k$ each have the same number of chains. \end{cor} The two intervals $[u,w]_k$ and $[x,z]_k$ of Corollary~\ref{cor:equal_chains} are typically non-isomorphic: For example, in ${\cal S}_4$ let $u=1234$, $x=2134$, and $v=1324$. If $\zeta=(1243)$, $\eta=(1423)= \zeta^{(1234)}$, and $\xi=(1342)=\eta^{(1234)}$, then $$ u \ \leq_2\ \zeta u,\quad x\ \leq_2\ \eta x,\quad\mbox{and} \quad v\ \leq_2 \ \xi v. $$ Here are the intervals $[u,\zeta u]_2$, $[x,\eta x]_2$, and $[v,\xi v]_2$. \begin{figure}[htb] $$\epsfxsize=4in \epsfbox{figures/fig1.eps}$$ \caption{Effect of cyclic shift on intervals\label{fig:fig1}} \end{figure} The Theorems of this section, together with the `algebraic' identities $c^w_{u\,v}=c^{w_0w}_{w_0u\:v}= c^{\overline{w}}_{\overline{u}\:\overline{v}}$, greatly reduce the number of distinct Littlewood-Richardson coefficients $c^w_{u\; v(\lambda,k)}$ which need to be determined. We make this precise. For $\lambda$ a partition, let $\lambda^t$ be its conjugate, or transpose. Note $\overline{v(\lambda,k)}= v(\lambda^t,n-k)$. \begin{thm}[Symmetries of the $c^w_{u\; v(\lambda,k)}$] Let $\lambda$ be a partition and $\zeta\in{\cal S}_n$. Then $$ c^\zeta_\lambda \ =\ c^{\zeta^{-1}}_{\lambda^t}\ =\ c^{\overline{\zeta}}_{\lambda^t}\ =\ c^{\zeta^{(1\,2\,\ldots\,n)}}_\lambda. $$ \end{thm} Let $D_n$ be the dihedral group with $2n$ elements. These identities show that the action of ${\Bbb Z}/2{\Bbb Z}\times D_n$ on these coefficients leaves their values invariant. \section{Preliminaries} \subsection{Permutations}\label{sec:permutations} Let ${\cal S}_n$ be the group of permutations of $[n] := \{1,2,\ldots,n\}$. Let $(a,b)$ be the transposition interchanging $a<b$. The {\em length\/} $\ell(w)$ of a permutation $w\in {\cal S}_n$ counts the {\em inversions\/}, $\{i<j\,|\, w(i)>w(j)\}$, of $w$. The Bruhat order $\leq$ on ${\cal S}_n$ is the partial order whose cover relation is $w \lessdot w(a,b)$ if $w(a)<w(b)$ and whenever $a<c<b$, either $w(c)<w(a)$, or $w(b)<w(c)$. Thus $\ell(w)+1=\ell(w(a,b))$, so the Bruhat order is graded by length with minimal element the identity, $e$. If $u\leq w$, let $[u,w]:=\{v\,|\, u\leq v\leq w\}$ be the interval between $u$ and $w$ in ${\cal S}_n$, a poset graded by $\ell(v)-\ell(u)$. Let $w_0^{(n)}\in {\cal S}_n$ (or simply $w_0$) be defined by $w_0(j)=n+1-j$. A permutation $w\in {\cal S}_n$ acts on $[n{+}1]$, fixing $n+1$. Thus ${\cal S}_n\subset{\cal S}_{n+1}$. Define ${\cal S}_\infty:= \bigcup_n {\cal S}_n$, the permutations of the positive integers ${\Bbb N}$ fixing all but finitely many integers. For $P=\{p_1<p_2<\cdots\}\subset {\Bbb N}$, define $\phi_P: {\cal S}_{\#P} \rightarrow {\cal S}_\infty$ by requiring that $\phi_P$ act as the identity on ${\Bbb N}-P$ and $\phi_P(\zeta)(p_i)=p_{\zeta(i)}$. This injective homomorphism is a map of posets, but not of graded posets, as $\phi_P$ typically does not preserve length. If $P=\{n+1,n+2,\ldots\}$, then $\phi_P$ does preserve length. For this $P$, set $1^n\times w:= \phi_P(w)$. If there exist permutations $\xi,\zeta$, and $\eta$ and sets of positive integers $P,Q$ such that $\phi_P(\xi)=\zeta$ and $\phi_Q(\xi)=\eta$, then $\zeta$ and $\eta$ are {\em shape equivalent}. \subsection{Schubert polynomials}\label{sec:schubert} Lascoux and Sch\"utzenberger invented and then developed the elementary theory of Schubert polynomials in a series of papers~\cite{Lascoux_Schutzenberger_polynomes_schubert,% Lascoux_Schutzenberger_structure_de_Hopf,% Lascoux_Schutzenberger_symmetry,% Lascoux_Schutzenberger_interpolation,% Lascoux_Schutzenberger_schub_LR_rule,% Lascoux_Schutzenberger_operators}. A self-contained exposition of some of this elegant theory is found in~\cite{Macdonald_schubert}. For an interesting historical survey, see~\cite{Lascoux_historique}. ${\cal S}_n$ acts on polynomials in $x_1,\ldots,x_n$ by permuting the variables. For a polynomial $f$, $f - (i,i{+}1) f$ is antisymmetric in $x_i$ and $x_{i+1}$, hence divisible by $x_i - x_{i+1}$. Define the {\em divided difference} operator $$ \partial_i \ :=\ (x_i-x_{i+1})^{-1} (e - (i,i{+}1)). $$ If $w = (a_1, a_1{+}1)\cdots (a_p,a_p{+}1)$ is a factorization of $w$ into adjacent transpositions with minimal length ($p = \ell(w)$), then $\partial_{a_1}\circ\cdots\circ\partial_{a_p}$ depends only upon $w$, defining an operator $\partial_{w}$ for each $w\in {\cal S}_n$. For $w \in {\cal S}_n$, Lascoux and Sch\"utzenberger~\cite{Lascoux_Schutzenberger_polynomes_schubert} defined the {\em Schubert polynomial} ${\frak S}_{w}$ by $$ {\frak S}_{w}\ :=\ \partial_{w^{-1}w_0} \left( x_1^{n-1} x_2^{n-2}\cdots x_{n-1} \right). $$ The degree of $\partial_i$ is $-1$, so ${\frak S}_{w}$ is homogeneous of degree ${n\choose 2} - \ell(w^{-1}w_0) = \ell(w)$. Since $w_0^{(n)}= (n,n{+}1)\cdots(2,3)(1,2)w_0^{(n+1)}$ and $x_1^{n-1}\cdots x_{n-1}= \partial_n\circ\cdots\circ\partial_1 (x_1^n \cdots x_{n-1}^2 x_n)$, ${\frak S}_w$ is independent of $n$ (when $w\in {\cal S}_n$). This defines polynomials ${\frak S}_w\in{\Bbb Z}[x_1,x_2,\ldots]$ for $w\in{\cal S}_\infty$. A {\em partition} $\lambda$ is a decreasing sequence $\lambda_1\geq\lambda_2\geq\cdots\geq\lambda_k\geq 0$ of integers. Each $\lambda_j$ is a {\em part} of $\lambda$. For partitions $\lambda$ and $\mu$, write $\mu\subset \lambda$ if $\mu_i\leq \lambda_i$ for all $i$. {\em Young's lattice} is the set of partitions ordered by $\subset$. The partition with $n$ parts each equal to $m$ is written $m^n$. For a partition $\lambda$ with $\lambda_{k+1}=0$, the Schur polynomial $S_\lambda(x_1,\ldots,x_k)$ is $$ S_\lambda(x_1,\ldots,x_k) \ :=\ \frac{\det\left|x_j^{k-i+\lambda_i}\right|_{i,j=1}^k}% {\det\left|x_j^{k-i}\right|_{i,j=1}^k}. $$ $S_\lambda(x_1,\ldots,x_k)$ is symmetric in $x_1,\ldots,x_k$ and homogeneous of degree $|\lambda|:=\lambda_1+\cdots+\lambda_k$. A permutation $w$ is {\em Grassmannian of descent $k$} if $j\neq k \Rightarrow w(j)<w(j+1)$. A Grassmannian permutation $w$ with descent $k$ defines, and is defined by, a partition $\lambda$ with $\lambda_{k+1}=0$: $$ \lambda_{k+1-j}\ =\ w(j) - j\qquad j=1,\ldots, k. $$ (The condition $w(k{+}1)<w(k{+}2)<\cdots$ determines the remaining values of $w$.) In this case, write $w = v(\lambda,k)$. The {\em raison d'etre} for this definition is that ${\frak S}_{v(\lambda,k)} = S_\lambda(x_1,\ldots,x_k)$. Thus the Schubert polynomials form a basis for ${\Bbb Z}[x_1,x_2,\ldots]$ which contains all Schur symmetric polynomials $S_\lambda(x_1,\ldots,x_k)$ for all $\lambda$ and $k$. \subsection{The flag manifold}\label{sec:flag} Let $V\simeq {\Bbb C}^n$. A {\em flag} ${F\!_{\DOT}}$ in $V$ is a sequence $$ \{0\}\ =\ F_0 \subset F_1 \subset F_2\subset \cdots \subset F_{n-1} \subset F_n\ =\ V, $$ of subspaces with $\dim_{{\Bbb C}} F_i = i$. Flags ${F\!_{\DOT}}$ and ${{F\!_{\DOT}}'}$ are {\em opposite} if $F_{n-j}\cap F'_j = \{0\}$ for all $j$. The set of all flags is an ${n\choose 2}$-dimensional complex manifold, ${\Bbb F}\ell V$ (or ${\Bbb F}\ell_n$), called the {\em flag manifold}. There is a tautological flag ${{\mathcal F}\!_{\DOT}}$ of bundles over ${\Bbb F}\ell V$ whose fibre at ${F\!_{\DOT}}$ is ${F\!_{\DOT}}$. Let $-x_i$ be the first Chern class of the line bundle ${\cal F}_i/{\cal F}_{i-1}$. Borel~\cite{Borel_cohomology} showed the cohomology ring of ${\Bbb F}\ell V$ to be $$ {\Bbb Z}[x_1,\ldots,x_n]/\Span{e_i(x_1,\ldots,x_n)\,|\,i=1,\ldots,n}, $$ where $e_i(x_1,\ldots,x_n)$ is the $i$th elementary symmetric polynomial in $x_1,\ldots, x_n$. Given a subset $S \subset V$, let $\Span{S}$ be its linear span. For subspaces $W\subset U$, let $U- W$ be their set-theoretic difference. An ordered basis $f_1,f_2,\ldots,f_n$ for $V$ determines a flag ${E_{\DOT}}:=\SPan{f_1,\ldots,f_n}$, where $E_i = \Span{f_1,\ldots,f_i}$ for $1\leq i \leq n$. A fixed flag ${F\!_{\DOT}}$ gives a decomposition due to Ehresmann~\cite{Ehresmann} of ${\Bbb F}\ell V$ into affine cells indexed by permutations $w$ of $S_n$. The cell determined by $w$ has two equivalent descriptions: $ \label{eq:Schubert_cell_definition} X^{\circ}_w {F\!_{\DOT}} \quad := \quad \left\{\begin{array}{ll} \{ {E_{\DOT}}\in {\Bbb F}\ell V\,|\, \dim E_i\bigcap F_j = \#\{p\leq i\,|\, w(p)>n-j\}\}, \\ \{ {E_{\DOT}}=\SPan{f_1,\ldots,f_n}\,|\, f_i \in F_{n+1-w(i)}- F_{n-w(i)}, \,1\leq i\leq n\}.\rule{0pt}{15pt} \end{array}\right. $$ Its closure is the Schubert subvariety $X_w{F\!_{\DOT}}$, which has complex codimension $\ell(w)$. Also, $u\leq w \Leftrightarrow X_u{F\!_{\DOT}} \supset X_w{F\!_{\DOT}}$. The {\em Schubert class} $[X_w{F\!_{\DOT}}]$ is the cohomology class Poincar{\'e} dual to the fundamental cycle of $X_w{F\!_{\DOT}}$. These Schubert classes form a basis for the cohomology. Schubert polynomials were defined so that ${\frak S}_w(x_1,\ldots,x_n)=[X_w{F\!_{\DOT}}]$. We write ${\frak S}_w$ for $[X_w{F\!_{\DOT}}]$. If ${F\!_{\DOT}}$ and ${{F\!_{\DOT}}'}$ are opposite flags, then $X_u{F\!_{\DOT}}\bigcap X_v{{F\!_{\DOT}}'}$ is an irreducible, generically transverse intersection, a consequence of~\cite{Deodhar} ({\em cf.}~\cite[\S5]{sottile_pieri_schubert}). Thus its codimension is $\ell(u)+\ell(v)$, and the fundamental cycle of $X_u{F\!_{\DOT}}\bigcap X_v{{F\!_{\DOT}}'}$ is Poincar\'e dual to ${\frak S}_u\cdot{\frak S}_v$. Since $$ {\Bbb Z}[x_1,\ldots,x_n]\ \longrightarrow \ {\Bbb Z}[x_1,\ldots,x_{n+m}]/\Span{e_i(x_1,\ldots,x_{n+m}) \,|\,i=1,\ldots,n+m}, $$ is an isomorphism on ${\Bbb Z}\Span{x_1^{a_1}\cdots x_n^{a_n}\,|\, a_i<m}$, identities of Schubert polynomials follow from product formulas for Schubert classes. The Schubert basis is self-dual for the intersection pairing: If $\ell(w) + \ell(v) = {n\choose 2}$, then $$ {\frak S}_w\cdot {\frak S}_v \quad=\quad\left\{ \begin{array}{lll} {\frak S}_{w_0}&\ & v=w_0 w\\0&&\mbox{otherwise} \end{array}\right.. $$ Let $\mbox{\it Grass}_kV$ be the Grassmannian of $k$-dimensional subspaces of $V$, a $k(n{-}k)$-dimensional manifold. A flag ${F\!_{\DOT}}$ induces a cellular decomposition indexed by partitions $\lambda\subset (n{-}k)^k$. The closure of the cell indexed by $\lambda$ is the Schubert variety $\Omega_\lambda{F\!_{\DOT}}$: $$ \Omega_\lambda{F\!_{\DOT}}\quad:=\quad \{H\in \mbox{\it Grass}_kV\,|\, \dim H\bigcap F_{n+j-k-\lambda_j} \geq j,\ j=1,\ldots,k\}. $$ The cohomology class Poincar{\'e} dual to the fundamental cycle of $\Omega_\lambda{F\!_{\DOT}}$ is $S_\lambda(x_1,\ldots,x_k)$, where $x_1,\ldots,x_k$ are negative Chern roots of the tautological $k$-plane bundle on $\mbox{\it Grass}_kV$. Write $S_\lambda$ for $S_\lambda(x_1,\ldots,x_k)$, if $k$ is understood. As with the flag manifold, these {\em Schubert classes} form a basis for cohomology, $\mu\subset\lambda\Leftrightarrow \Omega_\mu{F\!_{\DOT}}\supset\Omega_\lambda{F\!_{\DOT}}$, and if ${F\!_{\DOT}},{{F\!_{\DOT}}'}$ are opposite flags, then $$ [\Omega_\mu{F\!_{\DOT}} \bigcap \Omega_\nu{{F\!_{\DOT}}'}]\ =\ [\Omega_\mu{F\!_{\DOT}}]\ \cdot\ [\Omega_\nu{{F\!_{\DOT}}'}]\ =\ \sum_{\lambda\subset (n-k)^k} c^\lambda_{\mu\,\nu}\, S_\lambda, $$ where the $c^\lambda_{\mu\,\nu}$ are the Littlewood-Richardson coefficients~\cite{Fulton_tableaux}. This Schubert basis is self-dual: If $\lambda\subset (n-k)^k$, then let $\lambda^c$, the {\em complement} of $\lambda$, be the partition $(n-k-\lambda_k,\ldots,n-k-\lambda_1)$. Suppose $|\lambda|+|\mu|=k(n-k)$, then $$ S_\lambda(x_1,\ldots,x_k)\cdot S_\mu(x_1,\ldots,x_k) \quad=\quad \left\{ \begin{array}{ll} S_{(n-k)^k} & \mbox{ if } \mu=\lambda^c\\ 0 & \mbox{ otherwise} \end{array} \right. . $$ We suppress the dependence of $\lambda^c$ on $n$ and $k$, which may be determined by context. A map $f: X\rightarrow Y$ between manifolds induces a homomorphism $f_* : H^*X \rightarrow H^*Y$ of abelian groups via the functorial map on homology and the Poincar{\'e} duality isomorphism between homology and cohomology. While $f_*$ is not a map of graded rings, it does satisfy the projection formula ({\em cf.}~\cite[8.1.7]{Fulton_intersection}): Let $\alpha\in H^*X$ and $\beta\in H^*Y$, then $$ f_*(f^*\alpha \cap \beta)\quad=\quad \alpha\cap f_*\beta. \eqno(2.3.1) $$ For a(n oriented) manifold $X$ of dimension $d$, $H^dX={\Bbb Z}\cdot[\mbox{pt}]$ is generated by the class of a point. Let $\deg : H^*X\rightarrow {\Bbb Z}$ be the map which selects the coefficient of $[\mbox{pt}]$ Then $\deg(f_*\beta)=\deg(\beta)$. Let $\pi_k:{\Bbb F}\ell V\twoheadrightarrow \mbox{\it Grass}_kV$ be defined by $\pi_k({E_{\DOT}})=E_k$. Then $\pi_k^{-1}\Omega_\lambda{F\!_{\DOT}} = X_{v(\lambda,k)}{F\!_{\DOT}}$ and $\pi_k: X_{w_0v(\lambda^c,k)}{F\!_{\DOT}}\twoheadrightarrow\Omega_\lambda{F\!_{\DOT}}$ is generically one-to-one. Thus on cohomology, \begin{eqnarray*} \pi^* S_\lambda&=& {\frak S}_{v(\lambda,k)}\\ (\pi_k)_* {\frak S}_w &=&\left\{\begin{array}{lll} S_\lambda&\ & \mbox{if }w = w_0 v(\lambda^c,k)\\ 0&& \mbox{otherwise}\end{array}\right.. \end{eqnarray*} By the K{\"u}nneth formula, the cohomology of ${\Bbb F}\ell V\times{\Bbb F}\ell W$ ($\dim W=m$) has an integral basis of classes ${\frak S}_u\otimes{\frak S}_x$ for $u\in{\cal S}_n$ and $x\in {\cal S}_m$. Likewise the cohomology of $\mbox{\it Grass}_kV\times \mbox{\it Grass}_l W$ has a basis $S_\lambda\otimes S_\mu$ for $\lambda\subset(n-k)^k$ and $\mu\subset(m-l)^l$. While we use the cohomology rings of complex varieties, our results and methods are valid for the Chow rings~\cite{Fulton_intersection} and $l$-adic ({\'e}tale) cohomology~\cite{Deligne_SGA4.5} of these same varieties over any field. \section{Orders on ${\cal S}_\infty$}\label{sec:orders} \subsection{The $k$-Bruhat order}\label{sec:k-bruhat} The $k$-Bruhat order, $\leq_k$, is a suborder of the Bruhat order on ${\cal S}_\infty$ which is linked to the Littlewood-Richardson coefficients $c^w_{u\,v(\lambda,k)}$. It appeared in~\cite{Lascoux_Schutzenberger_symmetry}, where it was called the $k$-coloured Ehresmano{\"e}dre. Its covers are given by the index of summation in Monk's formula~\cite{Monk}: $$ {\frak S}_u \cdot (x_1+\cdots+x_k)\quad =\quad \sum_{\stackrel{\mbox{\scriptsize $u\leq_k w$}}{\ell(w)=\ell(u)+1}} {\frak S}_w $$ Thus $w$ covers $u$ in the $k$-Bruhat order if $w$ covers $u$ in the Bruhat order, so that $w = u(a,b)$ and $\ell(w)=\ell(u)+1$, with the additional requirement that $a\leq k<b$. The $k$-Bruhat order has the following non-recursive characterization. \medskip \noindent{\bf Theorem~\ref{thm:k-length}.}\ {\em Let $u,w\in {\cal S}_\infty$. Then $u\leq_k w$ if and only if \begin{enumerate} \item[I.] $a\leq k < b$ implies $u(a)\leq w(a)$ and $u(b)\geq w(b)$. \item[II.] If $a<b$, $u(a)<u(b)$, and $w(a)>w(b)$, then $a\leq k<b$. \end{enumerate} }\medskip \noindent{\bf Proof. } The idea is to show that the transitive relation $u\trianglelefteq_k w$ defined by the conditions I and II coincides with the $k$-Bruhat order. If $u\lessdot _k u(a,b)$ is a cover, then $u\trianglelefteq_k u(a,b)$. Thus $u\leq _k w$ implies $u\trianglelefteq_k w$. Algorithm~\ref{alg:chain}, which, given $u\trianglelefteq_k w$ produces a chain in the $k$-Bruhat order from $u$ to $w$, completes the proof. \QEDnoskip \begin{alg}[Produces a chain in the $k$-Bruhat order]\label{alg:chain} \mbox{ } \noindent{\tt input: }Permutations $u,w\in {\cal S}_\infty$ with $u\trianglelefteq_k w$. \noindent{\tt output: }A chain in the $k$-Bruhat order from $w$ to $u$. Output $w$. While $u\neq w$, do \begin{enumerate} \item[1] Choose $a\leq k$ with $u(a)$ minimal subject to $u(a)< w(a)$. \item[2] Choose $k< b$ with $u(b)$ maximal subject to $w(b)<w(a)\leq u(b)$. \item[3] $w:=w(a,b)$, output $w$. \end{enumerate} At every iteration of\/ {\rm 1}, $u\trianglelefteq_k w$. Moreover, this algorithm terminates in $\ell(w)-\ell(u)$ iterations and the sequence of permutations produced is a chain in the $k$-Bruhat order from $w$ to $u$. \end{alg} \noindent{\bf Proof. } It suffices to consider a single iteration. We first show it is possible to choose $a$ and $b$, then $u\trianglelefteq_k w(a,b)$, and lastly $w(a,b)\lessdot_kw$ is a cover in the $k$-Bruhat order. In 1, $u\neq w$, so one may always choose $a$. Suppose $u\trianglelefteq_k w\in {\cal S}_n$ and it is not possible to choose $b$. In that case, if $j>k$ and $w(j)<w(a)$, then also $u(j)<w(a)$. Similarly, if $j\leq k$ and $w(j)<w(a)$, then $u(j)\leq w(j)<w(a)$. Thus $\alpha<w(a)\Leftrightarrow uw^{-1}(\alpha)<w(a)$, which contradicts $uw^{-1}(w(a))=u(a)< w(a)$. Let $w':= w(a,b)$. Note that $w(b)\geq u(a)$ implies Condition I for $(u,w')$. Suppose $w(b)< u(a)$. Set $b_1:= u^{-1}w(b)$. Then $w(b_1)\neq u(b_1)$ and the minimality of $u(a)$ shows that $b_1>k$ and $w(b_1)<u(b_1)$. Similarly, if $b_2:= u^{-1}w(b_1)$, then $b_2>k$ and $w(b_2)<u(b_2)$. Continuing, we obtain a sequence $b_1,b_2,\ldots$ with $u(a)>u(b_1)>u(b_2)>\cdots$, a contradiction. We show $(u,w')$ satisfies Condition II. Suppose $i<j$ and $u(i)<u(j)$. If $j\leq k$, then $w(i)<w(j)$. To show $w'(i)<w'(j)$, it suffices to consider the case $j=a$. But then $u(i)<u(a)$, and thus $u(i)=w(i)=w'(i)$, by the minimality of $u(a)$. Then $w'(i)<u(a)\leq w(b)=w'(a)$. Similarly, if $k<i$, then $w'(i)<w'(j)$. Finally, suppose $w$ does not cover $w'$ in the $k$-Bruhat order. Since $w(a)>w(b)$, there exists a $c$ with $a<c<b$ and $w(a)>w(c)>w(b)$. If $k<c$, then Condition II implies $u(c)>u(b)$ and then the maximality of $u(b)$ implies $w(a)<w(c)$, a contradiction. The case $c\leq k$ similarly leads to a contradiction. \QEDnoskip \begin{rem} {\em Algorithm~\ref{alg:chain} depends only upon $\zeta=wu^{-1}$. } \noindent{\tt input: }A permutation $\zeta\in {\cal S}_\infty$. \noindent{\tt output: }Permutations $\zeta,\zeta_1,\ldots,\zeta_m=e$ such that if $u\leq_k \zeta u$, then $$ u\ \lessdot_k\ \zeta_{m-1}u\ \lessdot_k\ \cdots\ \lessdot_k\ \zeta_1u\ \lessdot_k\ \zeta u\ (=w) $$ is a saturated chain in the $k$-Bruhat order. Output $\zeta$. While $\zeta\neq e$, do \begin{enumerate} \item[1] Choose $\alpha$ minimal subject to $\alpha< \zeta(\alpha)$. \item[2] Choose $\beta$ maximal subject to $\zeta(\beta)<\zeta(\alpha)\leq\beta$. \item[3] $\zeta:=\zeta(\alpha,\beta)$, output $\zeta$. \end{enumerate} {\em To see this is equivalent to Algorithm~\ref{alg:chain}, set $\alpha=u(a)$ and $\beta=u(b)$ so that $w(a)=\zeta(\alpha)$ and $w(b)=\zeta(\beta)$. Thus $w(a, b) = \zeta u(a, b) = \zeta(\alpha,\beta) u$. } \end{rem} More is true, the full interval $[u,w]_k$ depends only upon $wu^{-1}$: \begin{thm}\label{thm:k-order} If $u\leq_kw$ and $x\leq_ky$ with $wu^{-1}=zx^{-1}$, then the map $v\mapsto v u^{-1}x$ induces an isomorphism of graded posets $[u,w]_k \stackrel{\sim}{\longrightarrow}[x,z]_k$. \end{thm} This is a consequence of the following lemma. \begin{lem}\label{lem:k-order} Let $u\leq_kw$ and $x\leq_k z$ with $wu^{-1}=zx^{-1}$. If $u\lessdot_k (\alpha,\beta)u$ is a cover with $(\alpha,\beta)u \leq_k w$, then $x\lessdot_k (\alpha,\beta)x$ is a cover with $(\alpha,\beta)x \leq_k z$ \end{lem} \noindent{\bf Proof. } Let $\zeta=wu^{-1}=zx^{-1}$. By the position of $\gamma$ in $u$, we mean $u^{-1}(\gamma)$. Suppose $(\alpha,\beta)x$ does not cover $x$ in the $k$-Bruhat order, so there is a $\gamma$ with $\alpha<\gamma<\beta$ and $x^{-1}(\alpha)<x^{-1}(\gamma)<x^{-1}(\beta)$. Then, in one line notation, $x$ and $z$ are as illustrated: $$ \begin{array}{lc} x:\ &\ldots\ \ \alpha\ \ \ldots\ \ \gamma\ \ \ldots\ \ \beta\ \ \ldots\\ z:\ &\ldots\zeta(\alpha)\ldots\zeta(\gamma)\ldots\zeta(\beta)\ldots \end{array} $$ Since $u\lessdot_k (\alpha,\beta)u$ is a cover in the $k$-Bruhat order, either $k<u^{-1}(\beta)<u^{-1}(\gamma)$ or else $u^{-1}(\gamma)<u^{-1}(\alpha) \leq k$. We illustrate $u$, $(\alpha,\beta)u$, and $w$ for each possibility: $$ \begin{array}{rccc} &k<u^{-1}(\beta)<u^{-1}(\gamma)&&u^{-1}(\gamma)<u^{-1}(\alpha)\leq k\\ u:\ & \ldots\ \ \alpha\ \ \ldots\ \ \beta\ \ \ldots\ \ \gamma\ \ \ldots &\qquad & \ldots\ \ \gamma\ \ \ldots\ \ \alpha\ \ \ldots\ \ \beta\ \ \ldots\\ (\alpha,\beta)u:\ & \ldots\ \ \beta\ \ \ldots\ \ \alpha\ \ \ldots\ \ \gamma\ \ \ldots&& \ldots\ \ \gamma\ \ \ldots\ \ \beta\ \ \ldots\ \ \alpha\ \ \ldots\\ w:\ & \ldots\zeta(\alpha)\ldots\zeta(\beta)\ldots\zeta(\gamma)\ldots&& \ldots\zeta(\gamma)\ldots\zeta(\alpha)\ldots\zeta(\beta)\ldots \end{array} $$ Assume $k<u^{-1}(\beta)<u^{-1}(\gamma)$ . Then Theorem~\ref{thm:k-length} and $(\alpha,\beta)u\leq_k w$ imply $\gamma \geq \zeta(\gamma)$ and $\zeta(\beta)<\zeta(\gamma)$, since $\alpha<\gamma$ and both have positions greater than $k$ in $(\alpha\,\beta)u$. Let $c:=x^{-1}(\gamma)$. If $c\leq k$, then $x\leq_k z$ implies $\gamma \leq \zeta(\gamma)$ so $\gamma=\zeta(\gamma)$. Also, $\alpha<\gamma$ implies $\zeta(\alpha)<\zeta(\gamma)$ and thus $\zeta(\gamma)=\gamma <\beta\leq \zeta(\alpha)$, a contradiction. Similarly, if $c>k$, then $\gamma<\beta$ implies $\zeta(\gamma)<\zeta(\beta)$, another contradiction. The other possibility, $u^{-1}(\gamma)<u^{-1}(\alpha)$, leads to a similar contradiction. Thus $x\lessdot_k (\alpha,\beta)x$ is a cover in the $k$-Bruhat order. To show $y:= (\alpha,\beta)x\leq_k z$, first note that the pair $(y,z)$ satisfy condition I of Theorem~\ref{thm:k-length}, because $(\alpha,\beta)u\leq_k w$. For condition II, we need only show: \begin{enumerate} \item[a)] If $\alpha<\gamma<\beta$ and $x^{-1}(\gamma)<x^{-1}(\alpha)$, so that $\gamma=yx^{-1}(\gamma)<yx^{-1}(\alpha)=\beta$, then $zx^{-1}(\gamma) = \zeta(\gamma) <\zeta(\beta) = zx^{-1}(\alpha)$, and \item[b)] If $\alpha<\gamma<\beta$ and $x^{-1}(\beta)<x^{-1}(\gamma)$, so that $\alpha=yx^{-1}(\beta)<yx^{-1}(\gamma)=\gamma$, then $\zeta(\alpha) <\zeta(\gamma)$. \end{enumerate} If $\alpha<\gamma<\beta$, then one of these two possibilities does occur, as $x\lessdot_k (\alpha,\beta) x = y$ is a cover in the $k$-Bruhat order. Suppose $x^{-1}(\gamma)<x^{-1}(\alpha)$, as the other case is similar. Since $x^{-1}(\gamma)<k$ and $x\leq_k z$, we have $\gamma\leq \zeta(\gamma)$, by condition I. If $u^{-1}(\gamma)<u^{-1}(\alpha)$, then $(\alpha\,\beta)u\leq_k w\Rightarrow \zeta(\gamma)<\zeta(\alpha)$. If $u^{-1}(\beta)<u^{-1}(\gamma)$, then $\gamma=\zeta(\gamma)$, and so $\zeta(\gamma)=\gamma<\beta\leq\zeta(\alpha)$. Since $u\leq_k (\alpha\,\beta)u$, we cannot have $u^{-1}(\alpha)<u^{-1}(\gamma)<u^{-1}(\beta)$. \QED Define $\mathrm{up}_\zeta := \{ \alpha\,|\, \alpha<\zeta(\alpha)\}$ and $\mathrm{down}_\zeta := \{ \beta\,|\, \beta>\zeta(\beta)\}$. \begin{thm}\label{thm:lengthenings_exist} Let $\zeta\in {\cal S}_\infty$. \begin{enumerate} \item[({\em i})] For $u\in {\cal S}_\infty$, $u\leq_k \zeta u$ if and only if the following conditions are satisfied. \begin{enumerate} \item $u^{-1} \mathrm{up}_\zeta \subset \{1,\ldots,k\}$, \item $u^{-1} \mathrm{down}_\zeta \subset \{k+1,k+2,\ldots\}$, and \item For all $\alpha,\beta\in \mathrm{up}_\zeta$ (respectively $\alpha,\beta\in \mathrm{down}_\zeta$), $\alpha<\beta$ and $u^{-1}(\alpha) < u^{-1}(\beta)$ together imply $\zeta(\alpha)<\zeta(\beta)$. \end{enumerate} \item[({\em ii})] If $\# \mathrm{up}_\zeta \leq k$, then there is a permutation $u$ such that $u\leq_k \zeta u$. \end{enumerate} \end{thm} \noindent{\bf Proof. } Statement ({\em i}) is a consequence of Theorem~\ref{thm:k-length}. For ({\em ii}), let $\{a_1,\ldots,a_k\}\subset{\Bbb N}$ contain $\mathrm{up}_\zeta$ and possibly some fixed points of $\zeta$, and let $\{a_{k+1},a_{k+2},\ldots\}$ be the complementary set in ${\Bbb N}$. Suppose these sets are indexed so that $\zeta(a_i)<\zeta(a_{i+1})$ for $i\neq k$. Define $u\in {\cal S}_\infty$ by $u(i) = a_i$ Then $\zeta u$ is Grassmannian with descent $k$, and Theorem~\ref{thm:k-length} implies $u\leq_k\zeta u$. \QEDnoskip \subsection{A new partial order on ${\cal S}_\infty$}\label{sec:new:order} For $\zeta\in {\cal S}_\infty$, define $|\zeta|$ to be the quantity: $$ \begin{array}{c} \#\{(\alpha,\beta)\in \zeta(\mathrm{up}_\zeta)\times\zeta(\mathrm{down}_\zeta)\,|\, \alpha>\beta\}\ -\ \#\{a,b\in \mathrm{up}_\zeta\,|\, a>b \mbox{ and } \zeta(a)<\zeta(b)\}\\ -\ \#\{a,b\in \mathrm{down}\zeta\,|\, a>b \mbox{ and } \zeta(a)<\zeta(b)\}\ -\ \#\{(a,b)\in \mathrm{up}_\zeta\times\mathrm{down}_\zeta\,|\, a>b\}. \end{array} $$ \begin{lem} If $u\leq_k \zeta u$, then $\ell(u) + |\zeta|= \ell(\zeta u)$. \end{lem} \noindent{\bf Proof. } By Theorem~\ref{thm:k-order}, $\ell(\zeta u)-\ell(u)$ depends only upon $\zeta$. Computing this for the permutation $u$ defined in the proof of Theorem~\ref{thm:lengthenings_exist}, shows it equals $|\zeta|$: If $c=\zeta(c)$, then the number of inversions involving $c$ in $u$ equals the number involving $c$ in $\zeta u$. The first term of the expression for $|\zeta|$ counts the remaining inversions in $\zeta u$ and the last three terms the remaining inversions in $u$. \QED By Theorem~\ref{thm:k-order}, the interval $[u,\zeta u]_k$ depends only upon $\zeta$ if $u\leq_k \zeta u$. A closer examination of our arguments shows it is independent of $k$, as well. That is, if $x\leq_l \zeta x$, then the map $v \mapsto xu^{-1}v$ defines an isomorphism $[u,\,\zeta u]_k\stackrel{\sim}{\longrightarrow}[x,\,\zeta x]_l$. This motivates the following definition. \begin{defn} {\em For $\zeta,\eta\in{\cal S}_\infty$, let $\eta\preceq \zeta$ if there exists $u\in {\cal S}_\infty$ and a positive integer $k$ such that $u\leq_k\eta u\leq_k\zeta u$. If $u$ is chosen as in the proof of Theorem~\ref{thm:lengthenings_exist}, then we see that $\eta\preceq\zeta$ if \begin{enumerate} \item if $\alpha<\eta(\alpha)$, then $\eta(\alpha)\leq\zeta(\alpha)$, \item if $\alpha>\eta(\alpha)$, then $\eta(\alpha)\geq\zeta(\alpha)$, and \item if $\alpha,\beta\in\mathrm{up}_\zeta$ (respectively, $\alpha,\beta\in\mathrm{down}_\zeta$) with $\alpha<\beta$ and $\zeta(\alpha)<\zeta(\beta)$, then $\eta(\alpha)<\eta(\beta)$. \end{enumerate} } \end{defn} Figure~\ref{fig:new_order.S_4} illustrates $\preceq$ on ${\cal S}_4$. \begin{figure}[htb] $$\epsfxsize=4.in \epsfbox{figures/new_order.S_4.eps}$$ \caption{ $\preceq$ on ${\cal S}_4$\label{fig:new_order.S_4}} \end{figure} \begin{thm}\label{thm:new_order} Suppose $u,\zeta,\eta,\xi\in{\cal S}_\infty$. \begin{enumerate} \item[({\em i})] $({\cal S}_\infty,\preceq)$ is a graded poset with rank function $|\zeta|$. \item[({\em ii})] The map $\lambda \mapsto v(\lambda,k)$ exhibits Young's lattice of partitions with at most $k$ parts as an induced suborder of $({\cal S}_\infty,\preceq)$. \item[({\em iii})] If $u\leq_k \zeta u$, then the map $\eta\mapsto \eta u$ induces an isomorphism $[e,\,\zeta]_\preceq \stackrel{\sim}{\longrightarrow} [u,\,\zeta u]_k$. \item[({\em iv})] If $\eta\preceq\zeta$, then the map $\xi\mapsto \xi\eta^{-1}$ induces an isomorphism $[\eta,\zeta]_\preceq \stackrel{\sim}{\longrightarrow} [e,\zeta\eta^{-1}]_\preceq$. \item[({\em v})] For every infinite set $P\subset {\Bbb N}$, $\phi_P:{\cal S}_\infty \rightarrow {\cal S}_\infty$ is an injection of graded posets. Thus, if $\zeta, \eta\in{\cal S}_\infty$ are shape equivalent, then $[e,\,\zeta]_\preceq \simeq [e,\,\eta]_\preceq$. \item[({\em vi})] The map $\eta\mapsto \eta\zeta^{-1}$ induces an order reversing isomorphism between $[e,\zeta]_\preceq$ and $[e,\zeta^{-1}]_\preceq$. \item[({\em vii})] The homomorphism $\zeta\mapsto \overline{\zeta}$ on ${\cal S}_n$ induces an automorphism of $({\cal S}_n,\preceq)$. \end{enumerate} \end{thm} Theorem~\ref{thm:B} ({\em i}) is an immediate consequence of the definition of $\preceq$ and ({\em v}). \noindent{\bf Proof. } Statements ({\em i})--({\em v}) follow from the definitions. Suppose $u\leq_k \eta u\leq_k \zeta u$ with $u,\eta u,\zeta u\in{\cal S}_n$. If $w:= \zeta u$, then $ w w_0 \leq_{n-k} \eta\zeta^{-1}w w_0 \leq_{n-k} \zeta^{-1}w w_0$, which proves ({\em vi}). Similarly, $u\leq_k w \Leftrightarrow \overline{u}\leq_{n-k}\overline{w}$ implies ({\em vii}). \QEDnoskip \begin{ex}\label{ex:neworder}{\em Let $\zeta = (24)(153)$ and $\eta=(35)(174)=\phi_{\{1,3,4,5,7\}}(\zeta)$. Then $21345 \leq_2 45123 = \zeta\cdot 21345$ and $3215764 \leq_3 5273461 = \eta\cdot 3215764$. Figure~\ref{fig:new_order} illustrates the intervals $[21342,\,45123]_2$, $[3215764,\,5273461]_3$, and $[e,\zeta]_\preceq$. \begin{figure}[htb] $$\epsfxsize=5.3in \epsfbox{figures/new_order.eps}$$ \caption{Isomorphic intervals in $\leq_2, \leq_3$, and $\preceq$\label{fig:new_order}} \end{figure} }\end{ex} \subsection{Disjoint permutations}\label{sec:disjoint_permutations} Let $\zeta\in {\cal S}_n$ and $1,\ldots,n$ be the vertices of a convex planar $n$-gon numbered consecutively. Define $\Gamma_\zeta$, a directed geometric graph to be the union of directed chords $\Span{\alpha,\zeta(\alpha)}$ for $\alpha$ in the support, $\mbox{supp}_\zeta$, of $\zeta$. Permutations $\zeta$ and $\eta$ are {\em disjoint} if the edge sets of their geometric graphs $\Gamma_\zeta$ and $\Gamma_\eta$ (drawn on the same $n$-gon) are disjoint as {\em subsets of the plane}. This implies (but is not equivalent to) supp$_\zeta\bigcap\:$supp$_\eta=\emptyset$. Disjointness may be rephrased in terms of partitions~\cite{Stanley_enumerative} of $[n]$. Suppose for simplicity, that supp$_\zeta$ and supp$_\eta$ partition $[n]$. Then $\zeta$ and $\eta$ are disjoint if and only if there is a non-crossing partition~\cite{Kreweras} $\pi$ of $[n]$ refining the partition (supp$_\zeta$, supp$_\eta$), and which is itself refined by the partition given by the cycles of $\zeta$ and $\eta$. We compare the graphs of the pairs of permutations $(1782), (345)$ and $(13), (24)$. $$ \epsfxsize=2.5in \epsfbox{figures/disjoint.eps} $$ The first pair is disjoint and the second is not. We relate this definition to that given in \S\ref{sec:Schur_identities}. \begin{lem} Let $\zeta,\eta\in {\cal S_\infty}$. Then the edges of $\/\Gamma_\zeta$ are disjoint from the edges of $\/\Gamma_\eta$ if and only if $\/\zeta$ and $\eta$ have disjoint support and $|\zeta|+|\eta|=|\zeta\eta|$. \end{lem} \noindent{\bf Proof. } Suppose $\zeta$ and $\eta$ have disjoint support and let $\Span{a,\zeta(a)}$ be an edge of $\Gamma_\zeta$ and $\Span{b,\eta(b)}$ be an edge of $\Gamma_\eta$. Consider the contribution of the endpoints of these edges to $|\zeta\eta|-|\zeta|-|\eta|$. This contribution is zero if the edges do not cross, which proves the forward implication. For the reverse, suppose these edges cross. Then the contribution is 1 if $a<\zeta(a)$ and $b>\eta(b)$ (or vice-versa), and 0 otherwise. Because each edge is part of a directed cycle in the graph, if one edge of $\Gamma_\zeta$ crosses an edge of $\Gamma_\eta$, then there are at least four crossings, one of each type illustrated. $$ \begin{picture}(100,80)\setlength{\unitlength}{.7pt}\thicklines \put(25,25){\line(1,1){50}} \put(37.5,37.5){\line(1,0){5}}\put(37.5,37.5){\line(0,1){5}} \put(25,75){\line(1,-1){50}} \put(35,65){\line(1,0){5}}\put(35,65){\line(0,-1){5}} \put(-5,10){$\zeta(a)$}\put(-5,80){$\eta(b)$} \put(80,10){$b$}\put(80,80){$a$} \put(25,25){\circle*{5}}\put(25,75){\circle*{5}} \put(75,25){\circle*{5}}\put(75,75){\circle*{5}} \end{picture \begin{picture}(100,85)\setlength{\unitlength}{.75pt}\thicklines \put(25,25){\line(1,1){50}} \put(35,35){\line(1,0){5}}\put(35,35){\line(0,1){5}} \put(25,75){\line(1,-1){50}} \put(65,35){\line(-1,0){5}}\put(65,35){\line(0,1){5}} \put(-5,10){$\zeta(a)$}\put(12,80){$b$} \put(78,10){$\eta(b)$}\put(80,80){$a$} \put(25,25){\circle*{5}}\put(25,75){\circle*{5}} \put(75,25){\circle*{5}}\put(75,75){\circle*{5}} \end{picture} \begin{picture}(100,85)\setlength{\unitlength}{.75pt}\thicklines \put(25,25){\line(1,1){50}} \put(65,65){\line(-1,0){5}}\put(65,65){\line(0,-1){5}} \put(25,75){\line(1,-1){50}} \put(35,65){\line(1,0){5}}\put(35,65){\line(0,-1){5}} \put(12,10){$a$}\put(-5,80){$\eta(b)$} \put(80,10){$b$}\put(80,80){$\zeta(a)$} \put(25,25){\circle*{5}}\put(25,75){\circle*{5}} \put(75,25){\circle*{5}}\put(75,75){\circle*{5}} \end{picture} \begin{picture}(100,85)\setlength{\unitlength}{.75pt}\thicklines \put(25,25){\line(1,1){50}} \put(65,65){\line(-1,0){5}}\put(65,65){\line(0,-1){5}} \put(25,75){\line(1,-1){50}} \put(65,35){\line(-1,0){5}}\put(65,35){\line(0,1){5}} \put(12,10){$a$}\put(12,80){$b$} \put(78,10){$\eta(b)$}\put(80,80){$\zeta(a)$} \put(25,25){\circle*{5}}\put(25,75){\circle*{5}} \put(75,25){\circle*{5}}\put(75,75){\circle*{5}} \end{picture} $$ Here, the numbers increase in a clockwise direction, with the least number in the northeast ($\nearrow$). Thus $|\zeta\eta|>|\zeta|+|\eta|$. \QEDnoskip \begin{lem}\label{lem:disjoint_one} Let $\alpha<\beta$ and $\zeta\in{\cal S}_\infty$ and suppose $\zeta{\prec\!\!\!\cdot\,}(\alpha,\beta)\zeta$ is a cover. Then \begin{enumerate} \item[({\em i})] $\alpha$ and $\beta$ are connected in the geometric graph $\Gamma_{(\alpha,\beta)\zeta}$. \item[({\em ii})] If $\Span{c,d}$ is any chord of the $n$-gon meeting $\Gamma_\zeta$, then $\Span{c,d}$ meets $\Gamma_{(\alpha,\beta)\zeta}$. \item[({\em iii})] If $p$ and $q$ are connnected in $\Gamma_\zeta$, then they are connected in $\Gamma_{(\alpha,\beta)\zeta}$. \item[({\em iv})] If $\zeta$ and $\eta$ are disjoint permutations and $\zeta'\preceq\zeta$, then $\zeta'$ and $\eta$ are disjoint. \end{enumerate} \end{lem} \noindent{\bf Proof. } Suppose $u\in {\cal S}_\infty$ with $u\leq_k \zeta u\lessdot_k (\alpha,\beta)\zeta u$. Define $i$ and $j$ by $\zeta u(i)=\alpha$ and $\zeta u(j)=\beta$, and set $a=u(i)$ and $b=u(j)$. Since $\zeta u\lessdot_k (\alpha,\beta)\zeta u$ is a cover, $i\leq k <j$, and thus $a\leq \alpha<\beta\leq b$, since $u\leq_k \zeta u$. Thus the edges $\Span{a,\beta}$ and $\Span{b,\alpha}$ of $\Gamma_{(\alpha,\beta)\zeta}$ meet, proving ({\em i}). For ({\em ii}), note that $\Gamma_{(\alpha,\beta)\zeta}$ differs from $\Gamma_\zeta$ only by the (possible) deletion of edges $\Span{a,\alpha}$ and $\Span{b,\beta}$ and the addition of the edges $\Span{a,\beta}$ and $\Span{b,\alpha}$. Checking all possibilities for the chords $\Span{c,d}$, $\Span{a,\alpha}$, and $\Span{b,\beta}$ shows ({\em ii}). Statement ({\em iii}) follows from ({\em ii}) by considering edges of $\Gamma_\zeta-\Span{a,\alpha}-\Span{b,\beta}$. The contrapositive of ({\em iv}) is also a consequence of ({\em ii}); If $\zeta'$ and $\eta$ are not disjoint and $\zeta'\preceq\zeta$, then $\zeta$ and $\eta$ are not disjoint. \QEDnoskip \begin{lem}\label{lem:disjoint_two} Suppose $\zeta$ and $\eta$ are disjoint permutations. For every $u\in {\cal S}_\infty$, $$ u\ \leq_k\ \zeta \eta u \quad \Longleftrightarrow\quad u\ \leq_k\ \zeta u \quad\mbox{and}\quad u\ \leq_k\ \eta u. $$ \end{lem} \noindent{\bf Proof. } Suppose $u \leq_k \zeta \eta u$. Let $i\leq k$ so that $u(i)\leq \zeta\eta u(i)$. Since $\zeta$ and $\eta$ have disjoint supports, $u(i)\leq \zeta u(i)$. Similarly, if $k<j$, then $u(j)\geq \zeta u(j)$, showing Condition I of Theorem~\ref{thm:k-length} holds for the pair $(u,\zeta u)$. For Condition II, suppose $i<j$, with $u(i)<u(j)$ and $\zeta u(i)>\zeta u(j)$. If $j\leq k$, this implies $u(i)\in \mbox{supp}_\zeta$. Since $u \leq_k \zeta \eta u$, and $\zeta,\eta$ have disjoint supports, we have $\zeta u(i) = \eta\zeta u(i)<\eta\zeta u(j)$, which implies $u(j)\in\mbox{supp}_\eta$ and so $$ u(i) \ <\ u(j)\ <\ \zeta u(i) \ <\ \eta u(j). $$ But then the edge $\Span{u(i),\,\zeta u(i)}$ of $\Gamma_\zeta$ meets the edge $\Span{u(j),\,\eta u(j)}$ of $\Gamma_\eta$, a contradiction. The assumption that $k<i$ leads similarly to a contradiction. Thus $u\leq_k \zeta u$ and similarly, $u\leq_k \eta u$. Suppose now that $u\leq_k \zeta u$ and $u\leq_k \eta u$. Condition I of Theorem~\ref{thm:k-length} for $(u,\zeta\eta u)$ holds as $\zeta$ and $\eta$ have disjoint support. For condition II, let $i<j$ with $u(i)<u(j)$ and suppose that $j\leq k$. If the set $\{u(i),u(j)\}$ meets at most one of supp$_\zeta$ or supp$_\eta$ , say supp$_\zeta$, then $u\leq_k \zeta u$ implies $\zeta\eta u(i) < \zeta \eta u(j)$. Suppose now that $u(i)$ is in the support of $\zeta$ and $u(j)$ is in the support of $\eta$. Since $u\leq_k \zeta u$, we have $\zeta u(i)<\zeta u(j) =u(j)$. But $u\leq_k \eta u$ implies $u(j)\leq \eta u(j)$. Thus $\eta \zeta u(i) = \zeta u(i) < u(j)\leq \eta u(j) = \eta \zeta u(j)$. Similar arguments suffice when $k<i$. \QEDnoskip \begin{thm}\label{thm:disjoint_iso} Suppose $\zeta$ and $\eta$ are disjoint. Then the map $[e,\zeta]_\preceq\times [e,\eta]_\preceq \rightarrow [e,\zeta\eta]_\preceq$ defined by $(\zeta',\eta') \mapsto \zeta'\eta'$ is an isomorphism of graded posets. \end{thm} \noindent{\bf Proof. } By Lemmas~\ref{lem:disjoint_one} and~\ref{lem:disjoint_two}, this map is an injection of graded posets. For surjectivity, let $\xi\preceq\zeta\eta$. By Lemma~\ref{lem:disjoint_one} ({\em iii}) and downward induction from $\zeta\eta$ to $\xi$, $\Gamma_\xi$ has no edges connecting supp$_\zeta$ to supp$_\eta$. Set $\xi':= \xi|_{\mbox{\scriptsize supp}_\zeta}$, and $\xi'':= \xi|_{\mbox{\scriptsize supp}_\eta}$. Then $\xi = \xi' \xi''$ and $\xi'$ and $\xi''$ are disjoint. Surjectivity will follow by showing $\xi'\preceq\zeta$ and $\xi''\preceq\eta$. It suffices to consider the case $\xi{\prec\!\!\!\cdot\,} (\alpha,\beta)\xi = \zeta\eta$ is a cover. By Lemma~\ref{lem:disjoint_one} ({\em i}), $\alpha$ and $\beta$ are connected in $\Gamma_{\zeta\eta}$, so we may assume that $\alpha,\beta$ are both in the support of $\zeta$. Then $\xi''=\eta$ and $(\alpha,\beta)\xi'=\zeta$. We show that $\xi' \preceq (\alpha,\beta)\xi'=\zeta$ is a cover, which will complete the proof. Choose $u\in {\cal S}_\infty$ with $u\leq_k\xi u\leq_k\zeta\eta u$. Let $a:=(\xi'u)^{-1}(\alpha)$ and $b:=(\xi' u)^{-1}(\beta)$. Since $\xi'$ and $\eta$ are disjoint, $\alpha,\beta\notin\mbox{supp}_\eta$ and so $a,b\notin\mbox{supp}_\eta$. Thus $(\alpha,\beta)\xi'\eta u = \xi'\eta u(a,b)$, showing $a\leq k <b$, as $\xi'\eta u\lessdot_k (\alpha,\beta)\xi'\eta u$. Since $\xi'$ and $\eta$ are disjoint and $\xi=\xi'\eta$, Lemma~\ref{lem:disjoint_two} implies $u\leq_k \xi' u$. Thus $|\xi'|+\ell(u)=\ell(\xi' u)$. But since $\xi'$ and $\eta$ are disjoint and $\xi'\eta {\prec\!\!\!\cdot\,} \zeta\eta$ is a cover, we have $$ |\zeta|+|\eta|\ =\ |\zeta\eta|\ =\ 1+ |\xi'\eta|\ =\ 1 + |\xi'| + |\eta|, $$ so $\ell(\xi'u)+1 = \ell(\xi' u(a,b))$. Since $a\leq k<b$ and $\zeta u=\xi'u(a,b)$, this implies $\xi' u\lessdot_k \zeta u$. \QEDnoskip \begin{ex} {\em Let $\zeta=(2354)$ and $\eta=(176)$, which are disjoint. Let $u= 2316745$. Then $$ u\leq_3 \zeta\eta u = 3571624, \quad u\leq_3 \zeta u = 3516724, \quad \mbox{ and } u\leq_3 \eta u = 2371645. $$ The intervals $[u,\zeta u]_3$, $[u,\eta u]_3$, and $[u,\;\zeta\eta u]_3$ are illustrated in Figure~\ref{fig:disjoint_interval}. \begin{figure}[htb] $$\epsfxsize=5.2in \epsfbox{figures/disjoint_interval.eps}$$ \caption{Intervals of disjoint permutations\label{fig:disjoint_interval}} \end{figure} }\end{ex} \section{Cohomological formulas and identities for the $c^w_{u\,v}$} \subsection{Two maps on ${\cal S}_\infty$}\label{sec:fixed_points} For positive integers $p,q$ and $w\in {\cal S}_\infty$, define $\varepsilon_{p,q}(w)\in {\cal S}_\infty$ by $$ \varepsilon_{p,q} (w)(j) \quad =\quad \left\{\begin{array}{lcl} w(j) && j < p \mbox{ and } w(j) < q\\ w(j)+1 && j < p \mbox{ and } w(j)\geq q\\ q && j = p\\ w(j-1) && j > p \mbox{ and } w(j) < q\\ w(j-1)+1 && j > p \mbox{ and } w(j)\geq q \end{array}\right.. $$ Note that $\varepsilon_{p,p}=\phi_{{\Bbb N}-\{p\}}$. However, if $p\neq q$, then this injection, $\varepsilon_{p,q}: {\cal S}_\infty \hookrightarrow {\cal S}_\infty$, is not a group homomorphism. The map $\varepsilon_{p,q}$ has a left inverse $/_p : {\cal S}_\infty\rightarrow {\cal S}_\infty$: For $x\in {\cal S}_\infty$, define $x/_p$ by $$ x/_p(j)\quad =\quad \left\{\begin{array}{lcl} x(j) &&j < p\mbox{ and }x(j) < x(p)\\ x(j)-1 &&j < p\mbox{ and }x(j) > x(p)\\ x(j+1) &&j\geq p\mbox{ and }x(j) < x(p)\\ x(j+1)-1&&j\geq p\mbox{ and }x(j) > x(p) \end{array}\right.. $$ Representing permutations as matrices, the effect of $/_p$ on $x$ is to erase the $p$th row and $x(p)$th column. The effect of $\varepsilon_{p,q}$ is to expand the matrix by adding a new $p$th row and $q$th column consisting mostly of zeroes, but with a 1 in the $(p,q)$th position. For example, $$ \varepsilon_{3,3}(23154)\ =\ 243165 \quad\mbox{and}\quad 264351/_3\ =\ 25341 $$ These maps have some order-theoretic properties. \begin{lem}\label{lem:expanding_bruhat} Suppose $x\leq z$ and $p,q$ are positive integers. Then \begin{enumerate} \item[({\em i})] $\varepsilon_{p,q}(x)\leq \varepsilon_{p,q}(z)$. \item[({\em ii})] If $\ell(z)-\ell(x) = \ell(\varepsilon_{p,q}(z))-\ell(\varepsilon_{p,q}(x))$, then $$ \varepsilon_{p,q}\ :\ [x,z] \ \stackrel{\sim}{\longrightarrow} [\varepsilon_{p,q}(x),\varepsilon_{p,q}(z)]. $$ \item[({\em iii})] If $x,z\in{\cal S}_n$ and either of $p$ or $q$ is equal to either 1 or $n+1$, then $\ell(z)-\ell(x) = \ell(\varepsilon_{p,q}(z))-\ell(\varepsilon_{p,q}(x))$. \item[({\em iv})]If $x\leq_k z$ and $x(p)=z(p)$, then $x/_p \leq_{k'} z/_p$ and $[x,z]_k\simeq[x/_p,z/_p]_{k'}$, where $k'$ is equal to $k$ if $k<p$ and $k-1$ otherwise. Furthermore, $zx^{-1} = \varepsilon_{x(p),x(p)}(z/_p(x/_p)^{-1})$. \end{enumerate} \end{lem} \noindent{\bf Proof. } Suppose $x\lessdot x(a,b)$ is a cover. Then $\varepsilon_{p,q}(x)< \varepsilon_{p,q}(x(a,b))$ is a cover if either $p\leq a$ or $b<p$, or else $a<p\leq b$ and either $q\leq x(a)$ or $x(b)<q$. If however, $a<p\leq b$ and $x(a)<q\leq x(b)$, then there is a chain of length 3 from $\varepsilon_{p,q}(x)$ to $\varepsilon_{p,q}(x(a,b))=\varepsilon_{p,q}(x) (a,b{+}1)$: $$ \varepsilon_{p,q}(x)\ \lessdot \ \varepsilon_{p,q}(x)(a,p)\ \lessdot \ \varepsilon_{p,q}(x) (a,b{+}1,p) \ \lessdot \ \varepsilon_{p,q}(x) (a,b{+}1). $$ The lemma follows from this observation. For example, under the hypothesis of ({\em ii}), the number of inversions in $\varepsilon_{p,q}(z)$ involving $q$ equals the number of inversion in $\varepsilon_{p,q}(x)$ involving $q$. Thus, if $\varepsilon_{p,q}(x)\leq u\leq \varepsilon_{p,q}(z)$, then $u(p)=q$. \QEDnoskip \subsection{An embedding of flag manifolds}\label{sec:emdedd} Let $W\subset V$ with $W\simeq {\Bbb C\,}^n$ and $V\simeq {\Bbb C\,}^{n+1}$. Suppose $f\in V- W$ so that $V=\Span{W,f}$. For $p\in [n{+}1]$ define the injection $\psi_p: {\Bbb F}\ell W\hookrightarrow {\Bbb F}\ell V$ by $$ \left( \psi_p {E_{\DOT}}\right)_j\quad =\quad \left\{ \begin{array}{lcl} E_j &&\mbox{if } j<p\\ \Span{E_{j-1},f} &&\mbox{if }j\geq p \end{array}\right. $$ \begin{prop}[\cite{sottile_pieri_schubert}, Lemma~12]\label{prop:embedding} Let ${E_{\DOT}}\in {\Bbb F}\ell W$ and $w\in {\cal S}_n$. Then, for every $p,q\in [n{+}1]$, $$ \psi_p X_w {E_{\DOT}} \quad \subset \quad X_{\varepsilon_{p,q}(w)} \psi_{n+2-q}{E_{\DOT}}. $$ \end{prop} Recall that $e$ is the identity permutation. \begin{cor}\label{cor:geometric_pushforward} Let $w\in {\cal S}_n$ and ${E_{\DOT}},{E_{\DOT}}\!' \in {\Bbb F}\ell W$ be opposite flags. Then $\psi_1 {E_{\DOT}}$ and $\psi_{n+1}{E_{\DOT}}\!'$ are opposite flags in ${\Bbb F}\ell V$ and $$ \psi_p X_w{E_{\DOT}} \ =\ X_{\varepsilon_{p,1}(w)}\psi_{n+1}{E_{\DOT}} \bigcap X_{\varepsilon_{p,n+1}(e )}\psi_1{E_{\DOT}}\!' \ =\ X_{\varepsilon_{p,1}(e )}\psi_{n+1}{E_{\DOT}}\!' \bigcap X_{\varepsilon_{p,n+1}(w)}\psi_{1}{E_{\DOT}}. $$ \end{cor} \noindent{\bf Proof. } Since $X_e {E_{\DOT}}\!' = {\Bbb F}\ell W$, Proposition~\ref{prop:embedding} with $q=1$ or $n+1$ implies $\psi_p X_w{E_{\DOT}}$ is a subset of either intersection: $$ X_{\varepsilon_{p,1}(w)}\psi_{n+1}{E_{\DOT}} \bigcap X_{\varepsilon_{p,n+1}(e )}\psi_1{E_{\DOT}}\!' \qquad\mbox{or}\qquad X_{\varepsilon_{p,1}(e )}\psi_{n+1}{E_{\DOT}}\!' \bigcap X_{\varepsilon_{p,n+1}(w)}\psi_{1}{E_{\DOT}}. $$ Since ${E_{\DOT}}$ and ${E_{\DOT}}\!'$ are opposite flags, $\psi_{n+1}{E_{\DOT}}$ and $\psi_1{E_{\DOT}}\!'$ are opposite flags, so both intersections are generically transverse and irreducible. Since $$ \ell(\varepsilon_{p,1}(w))\ =\ \ell(w) + p-1\ \qquad \mbox{and}\ \qquad \ell(\varepsilon_{p,n+1}(w))\ =\ \ell(w) + n+1-p, $$ both intersections have the same dimension as $\psi_p X_w{E_{\DOT}}$, proving equality. \QED Since $\varepsilon_{p,n+1}(e)=v(n{+}1{-}p,\,p)$, where $n+1-p$ is the partition of $n+1-p$ into a single part, we see that ${\frak S}_{\varepsilon_{p,n+1}(e )}=h_{n+1-p}(x_1,\ldots,x_p)$, the complete symmetric polynomial of degree $n+1-p$ in $x_1,\ldots,x_p$. Similarly, ${\frak S}_{\varepsilon_{p,1}(e )}=e_{p-1}(x_1,\ldots,x_{p-1}) = x_1\cdots x_{p-1}$, as $\varepsilon_{p,1}=v(1^{p-1},p-1)$, where $1^{p-1}$ is the partition of $p{-}1$ into $p{-}1$ equal parts, each of size 1. \begin{cor}\label{cor:equal_products} Let $w\in {\cal S}_n$. In $H^*{\Bbb F}\ell V$, $$ {\frak S}_{\varepsilon_{p,1}(w)} \cdot h_{n+1-p}(x_1,\ldots,x_p)\quad =\quad {\frak S}_{\varepsilon_{p,n+1}(w)} \cdot x_1\cdots x_{p-1} $$ and these products are equal to $(\psi_p)_* {\frak S}_w$. \end{cor} We use this to compute $\psi_p^*$. The Pieri-type formulas of~\cite{sottile_pieri_schubert} show that if $u\in {\cal S}_n$ and $k,m\leq n$ positive integers, then \begin{eqnarray*} \hspace{.8in} {\frak S}_u \cdot{\frak S}_{w_0w} \cdot e_m(x_1\cdots x_k) &=& \left\{\begin{array}{ll}1&u\stackrel{c_{k,m}}{\relbar\joinrel\longrightarrow} w\\ 0&\mbox{otherwise} \end{array}\right. \makebox[.1in]{\qquad\qquad} \hspace{1.35in}(4.2.1)\\ {\frak S}_u\cdot{\frak S}_{w_0w} \cdot h_{n+1-m}(x_1,\ldots,x_k) &=& \left\{\begin{array}{ll}1&u\stackrel{r_{k,m}}{\relbar\joinrel\longrightarrow} w\\ 0&\mbox{otherwise} \end{array}\right., \hspace{1.38in}(4.2.2) \end{eqnarray*} where $u\stackrel{c_{k,m}}{\relbar\joinrel\longrightarrow} w$ if there is a (saturated) chain in the $k$-Bruhat order from $u$ to $w$: $$ u\ \lessdot_k\ (\alpha_1,\beta_1)u\ \lessdot_k\ \cdots\ \lessdot_k \ (\alpha_m,\beta_m)\cdots(\alpha_1,\beta_1)u\ =\ w $$ such that $\beta_1>\cdots>\beta_m$. When $k=m$, it follows that $\{\alpha_1,\ldots,\alpha_k\}=\{u(1),\ldots,u(k)\}$. When $k=m=p-1$, we write $\cpp$ for this relation. Similarly, $u\stackrel{r_{k,m}}{\relbar\joinrel\longrightarrow} w$ if there is a chain in the $k$-Bruhat order: $$ u\ \lessdot_k\ (\alpha_1,\beta_1)u\ \lessdot_k\ \cdots\ \lessdot_k\ (\alpha_{n+1-m},\beta_{n+1-m})\cdots(\alpha_1,\beta_1)u\ =\ w $$ such that $\beta_1<\beta_2<\cdots<\beta_{n+1-m}$. \begin{thm}\label{thm:projection} Let $x\in {\cal S}_{n+1}$. In $H^*{\Bbb F}\ell_n$, \begin{enumerate} \item[({\em i})] ${\displaystyle \psi_p^*{\frak S}_x \quad=\quad \sum_{\stackrel{\mbox{\scriptsize$w\in {\cal S}_n$}}% {x\cpp\varepsilon_{p,1}(w)}} {\frak S}_w \quad=\quad \sum_{\stackrel{\mbox{\scriptsize$w\in {\cal S}_n$}}% {x\rpp\varepsilon_{p,n+1}(w)}} {\frak S}_w}$. \item[({\em ii})] ${\displaystyle \psi_p^*(x_i)\ =\ \left\{ \begin{array}{lll}x_i&& i<p\\ 0&& i=p\\x_{i-1}&& i>p\end{array}\right.}$. \end{enumerate} \end{thm} \noindent{\bf Proof. } In $H^*{\Bbb F}\ell_n$, $$ \psi_p^* {\frak S}_x \quad =\quad \sum_{w\in {\cal S}_n}\, \deg({\frak S}_{w_0w}\cdot\psi_p^* {\frak S}_x)\,{\frak S}_w. $$ By the projection formula~(2.3.1) and Corollary~\ref{cor:equal_products}, we have $$ \deg({\frak S}_{w_0w}\cdot\psi_p^* {\frak S}_x)\quad=\quad \deg( {\frak S}_x\cdot(\psi_{p})_*{\frak S}_{w_0w})\quad=\quad \deg({\frak S}_x\cdot {\frak S}_{\varepsilon_{p,n+1}(w_0w)}\cdot x_1\cdots x_{p-1}). $$ Note that $\varepsilon_{p,n+1}(w_0w)=w_0^{(n+1)}\varepsilon_{p,1}(w)$. By~(4.2.1), the triple product $$ {\frak S}_x\cdot {\frak S}_{\varepsilon_{p,n+1}(w_0w)}\cdot x_1\cdots x_{p-1} $$ is zero unless $x\cpp \varepsilon_{p,1}(w)$, and in this case it equals ${\frak S}_{w_0^{(n+1)}}$. This establishes the first equality of ({\em i}). For the second, use the other formula for $(\psi_p)_*{\frak S}_w$ from Corollary~\ref{cor:equal_products} and~(4.2.2). For ({\em ii}), let ${\cal F}_{\DOT}$ be the tautological flag on ${\Bbb F}\ell_{n+1}$, ${\cal E}_{\DOT}$ the tautological flag on ${\Bbb F}\ell_n$, and 1 the trivial line bundle. Then $$ \psi_p^*({\cal F}_i/{\cal F}_{i-1})\ =\ \left\{\begin{array}{lll}{\cal E}_i/{\cal E}_{i-1}&&\mbox{ if } i<p\\ 1&&\mbox{ if } i=p\\ {\cal E}_{i-1}/{\cal E}_{i-2}&&\mbox{ if } i>p\end{array}\right., $$ Since $-x_i$ is the Chern class of both ${\cal F}_i/{\cal F}_{i-1}$ and ${\cal E}_i/{\cal E}_{i-1}$, we are done. \QEDnoskip \subsection{The endomorphism $x_p\mapsto 0$}\label{sec:endomorphism} For $p\in{\Bbb N}$ and $x\in {\cal S}_\infty$, define $$ A_p(x)\ :=\ \{u\in{\cal S}_\infty\,|\, x\cpp\varepsilon_{p,1}(u)\}. $$ \begin{lem}\label{lem:index_sets} If $x\in {\cal S}_n$ and $p\leq n$, then ${\displaystyle A_p(x)\ =\ \{u\in{\cal S}_n\,|\, x\stackrel{r_{p,n+1-p}}{\relbar\joinrel\relbar\joinrel\llra}\varepsilon_{p,n+1}(u)\}}$. \end{lem} \noindent{\bf Proof. } If $x\in{\cal S}_n$, $p\leq n$, and $x\cpp w$, then $w\in{\cal S}_{n+1}$, so $A_p(x)\subset{\cal S}_n$. But then $A_p(x)$ and $\{u\in{\cal S}_n\,|\, x\stackrel{r_{p,n+1-p}}{\relbar\joinrel\relbar\joinrel\llra}\varepsilon_{p,n+1}(u)\}$ index the two equal sums in Theorem~\ref{thm:projection}({\em i}). \QED Let $\Psi_p:{\Bbb Z}[x_1,x_2,\ldots]\rightarrow{\Bbb Z}[x_1,x_2,\ldots]$ be defined by $$ \Psi_p(x_i)\ =\ \left\{ \begin{array}{lll}x_i&&\mbox{ if } i<p\\ 0&&\mbox{ if } i=p\\x_{i-1}&&\mbox{ if } i>p\end{array}\right.. $$ \begin{thm}\label{thm:theorem_A_iii} For $x\in {\cal S}_\infty$, and $p\in{\Bbb N}$, ${\displaystyle \Psi_p{\frak S}_x = \sum_{u\in A_p(x)} {\frak S}_u}$. \end{thm} \noindent{\bf Proof. } For $p\leq n+1$, the homomorphism $\Psi_p$ induces the map $\psi_p^*: H^*{\Bbb F}\ell_{n+1} \rightarrow H^*{\Bbb F}\ell_n$, by Theorem~\ref{thm:projection} ({\em ii}). Choosing $n$ large enough completes the proof. \QEDnoskip \begin{cor}\label{cor:more_identities} For $w, x, y\in {\cal S}_\infty$ and $p\in{\Bbb N}$, $$ \sum_{u\in A_p(x)} \quad \sum_{v\in A_p(y)} c^w_{u\,v} \qquad = \qquad \sum_{\stackrel{\mbox{\scriptsize $z$}}{w \in A_p(z)}}c^z_{x\, y}. $$ \end{cor} \noindent{\bf Proof. } Apply $\Psi_p$ to the identity ${\frak S}_x\cdot{\frak S}_y = \sum_zc^z_{x\,y}{\frak S}_z$ to obtain: $$ \sum_{u\in A_p(x)} \quad \sum_{v\in A_p(y)} {\frak S}_u \cdot{\frak S}_v \quad=\quad \sum_z c^z_{x\, y} \sum_{w \in A_p(z)}c^z_{x\, y}\, {\frak S}_w. $$ Expanding the product ${\frak S}_u \cdot{\frak S}_v$ and equating the coefficients of ${\frak S}_w$ proves the identity. \QEDnoskip \begin{ex} {\em We illustrate the effect of $\Psi_3$ with an example. Since \begin{eqnarray*} {\frak S}_{413652} &=& x_1^4x_2x_4x_5 + x_1^3x_2^2x_4x_5 + x_1^3x_2x_4^2x_5 +\\ &\ & x_1^4x_2x_3x_4 + x_1^4x_2x_3x_5 + x_1^4x_3x_4x_5 + x_1^3x_2^2x_3x_4 + x_1^3x_2^2x_3x_5 + x_1^3x_2x_3^2x_4 +\\ &\ & x_1^3x_2x_3^2x_5 + x_1^3x_2x_3x_4^2 + x_1^3x_3^2x_4x_5 + x_1^3x_3x_4^2x_5 + 2\cdot x_1^3x_2x_3x_4x_5, \end{eqnarray*} we have $$ \Psi_3({\frak S}_{413652}) \ =\ x_1^4x_2x_3x_4 + x_1^3x_2^2x_3x_4 + x_1^3x_2x_3^2x_4. $$ However, \begin{eqnarray*} {\frak S}_{52341} &=& x_1^4x_2x_3x_4\\ {\frak S}_{42531} &=& x_1^3x_2^2x_3x_4 + x_1^3x_2x_3^2x_4, \end{eqnarray*} which shows $$ \Psi_3({\frak S}_{413652})\ =\ {\frak S}_{52341}+{\frak S}_{42531}. $$ To see this agrees with Theorem~\ref{thm:theorem_A_iii}, compute the permutations $w$ such that $x\stackrel{c_3}{\longrightarrow}w$: $$ \epsfxsize=2.in \epsfbox{figures/psi_p-ex.eps} $$ Of these, only the two underlined permutations are of the form $\varepsilon_{3,1}(u)$: $$ 631452\ =\ \varepsilon_{3,1}(52341)\quad\mbox{and}\quad 531642\ =\ \varepsilon_{3,1}(42531). $$ }\end{ex} \begin{lem}\label{lem:restriction} Let $\lambda$ be a partition and $p,k$ positive integers. Then $A_p(v(\lambda,k)) = \{(v(\lambda,k')\}$, where $k'=k-1$ if $p\leq k$ and $k$ otherwise. \end{lem} \noindent{\bf Proof. } By the combinatorial definition of Schur functions~\cite[\S 4.4]{Sagan}, $\Psi_p({\frak S}_{v(\lambda,k)}) = {\frak S}_{v(\lambda,k')}$. \QED Lemma~\ref{lem:restriction} implies that $v(\lambda,k')$ is the only solution $x$ to the equation $v(\lambda,k)\cpp\varepsilon_{p,1}(x)$, a statement about chains in the Bruhat order. \subsection{Identities for $c^z_{x\,y}$ when $x(p)=z(p)$}\label{sec:fixed_point_identities} \begin{lem}\label{lemma:fixed_pts} Let $x,z\in {\cal S}_{n+1}$ with $x(p)=z(p)$ for some $p\in [m+1]$ and suppose $\ell(z)-\ell(x)= \ell(z/_p)-\ell(x/_p)$. In $H^*{\Bbb F}\ell_{n+1}$, $$ (\psi_p)_* \left({\frak S}_{x/_p}\cdot {\frak S}_{w_0^{(n)}(z/_p)}\right) \quad = \quad {\frak S}_x \cdot {\frak S}_{w_0^{(n+1)}z}. $$ \end{lem} \noindent{\bf Proof. } Let ${E_{\DOT}},{E_{\DOT}}\!'$ be opposite flags in $W$. By Proposition~\ref{prop:embedding}, $$ \psi_p\left( X_{w_0^{(n)}(z/_p)}{E_{\DOT}} \bigcap X_{x/_p} {E_{\DOT}}\!' \right) \quad = \quad X_{w_0^{(n+1)}z}\psi_{z(p)}{E_{\DOT}} \bigcap X_x \psi_{n+2-x(p)}{E_{\DOT}}\!' \eqno(4.4.1) $$ Note that $w_0^{(n)}(z/_p)= (w_0^{(n+1)}z)/_p$. Since $x(p)=z(p)$, the flags $\psi_{z(p)}{E_{\DOT}}$ and $\psi_{n+2-x(p)}{E_{\DOT}}\!'$ are opposite in $V$. Moreover, as $\ell(z)-\ell(x)= \ell(z/_p)-\ell(x/_p)$, both sides of (4.4.1) have the same dimension, so they are equal, proving the lemma. \QEDnoskip \begin{thm}\label{thm:coeff_sum} Let $x,z\in {\cal S}_\infty$ with $x(p)=z(p)$ and suppose that $\ell(z)-\ell(x)= \ell(z/_p)-\ell(x/_p)$. Then, for every $y\in {\cal S}_\infty$ and positive integer $p$, $$ c^z_{x\, y}\quad =\quad \sum_{v\in A_p(y)} c^{z/_p}_{x/_p\: v} $$ \end{thm} \noindent{\bf Proof. } It suffices to compute this in $H^*{\Bbb F}\ell_{n+1}$, for $n$ such that $p\leq n$, $y\in {\cal S}_{n+1}$ and $A_p(y)\subset {\cal S}_n$. By Lemma~\ref{lemma:fixed_pts}, $$ {\frak S}_x\cdot {\frak S}_{w_0^{(n+1)}z} \ =\ (\psi_p)_* \left({\frak S}_{x/_p}\cdot {\frak S}_{w_0^{(n)}(z/_p)}\right) \ =\ (\psi_p)_* \left(\sum_{v\in {\cal S}_n} c^{w_0^{(n)}v}_{x/_p\ \,w_0^{(n)}(z/_p)}\:{\frak S}_{w_0^{(n)}v}\right). $$ Since $c^{w_0^{(n)}v}_{u\:w_0^{(n)}w} = c^w_{u\,v}$ for $u,v,w\in {\cal S}_n$ and $\varepsilon_{p,1}(w_0^{(n)}v)=w_0^{(n+1)}\varepsilon_{p,1}(v)$, \begin{eqnarray*} {\frak S}_x\cdot {\frak S}_{w_0^{(n+1)}z} &=& \sum_{v\in {\cal S}_n} c^{z/_p}_{x/_p\:v} (\psi_p)_*\left({\frak S}_{w_0^{(n)}v}\right)\\ &=& \sum_{v\in {\cal S}_n} c^{z/_p}_{x/_p\:v} {\frak S}_{w_0^{(n+1)}\varepsilon_{p,1}(v)} \cdot x_1\cdots x_{p-1}, \end{eqnarray*} by Corollary~\ref{cor:equal_products}. Thus \begin{eqnarray*} c^z_{x\,y}&=& \deg\left({\frak S}_x\cdot {\frak S}_{w_0^{(n+1)}z} \cdot{\frak S}_y\right)\\ &=& \sum_{v\in {\cal S}_n}c^{z/_p}_{x/_p\: v} \cdot \deg\left({\frak S}_{w_0^{(n+1)}\varepsilon_{p,1}(v)} \cdot (x_1\cdots x_{p-1})\cdot {\frak S}_y\right)\\ &=& \sum_{v\in A_p(y)}c^{z/_p}_{x/_p\: v}.\ \ \ \ \ \ \QEDnoskip \end{eqnarray*} When $p=1$, this has the following consequence: \begin{cor}\label{cor:1_restrict} If $x(1)=z(1)$, then $c^z_{x\,y}=0$ unless $y=1\times v$. In that case, $c^z_{x\;1\times v}=c^{z/\!_1}_{x/\!_1\: v}$. \end{cor} \subsection{Products of flag manifolds} \label{sec:flag_products} Let $P,Q\in {[n+m]\choose n}$, that is, $P,Q\subset [n+m]$ and each has order $n$. Index the sets $P,Q$ and their complements $P^c,Q^c$ as follows: $$ \begin{array}{lcl} P\ =\ p_1<\cdots<p_n &\qquad& P^c\ :=\ [n+m]-P\ =\ p^c_1<\cdots<p^c_m\\ Q\ =\ q_1<\cdots<q_n && Q^c\ :=\ [n+m]-Q\ =\ q^c_1<\cdots<q^c_m \end{array} $$ Define a function $\varepsilon_{P,Q}: {\cal S}_n\times{\cal S}_m \hookrightarrow{\cal S}_{m+n}$ by: \begin{eqnarray*} \varepsilon_{P,Q}(v,w)(p_i) \ =\ q_{v(i)} && i=1,\ldots,n\\ \varepsilon_{P,Q}(v,w)(p^c_j) \ =\ q^c_{w(j)} && i=1,\ldots,m. \end{eqnarray*} As permutation matrices, $\varepsilon_{P,Q}(v,w)$ is obtained from $v$ and $w$ by placing the entries of $v$ in the blocks $P\times Q$ and those of $w$ in the blocks $P^c\times Q^c$. If $P=[n+1]- \{p\}$ and $Q=[n+1]-\{q\}$, then $\varepsilon_{P,Q}(v,e ) = \varepsilon_{p,q}(v)$. Suppose $V\simeq {\Bbb C}\,^n$, $W\simeq {\Bbb C}\,^m$, and $P\in{[n+m]\choose n}$. Define a map $$ \psi_P\ :\ {\Bbb F}\ell V\times{\Bbb F}\ell W \quad \hookrightarrow \quad{\Bbb F}\ell(V\oplus W) $$ by $\psi_P({E_{\DOT}},{F\!_{\DOT}})_j = \Span{E_i,F_{i'}\,|\, p_i, p^c_{i'} \leq j}$. Equivalently, if $e_1,\ldots,e_n$ is a basis for $V$ and $f_1,\ldots,f_m$ a basis for $W$, then $\psi_P(\SPan{e_1,\ldots,e_n},\SPan{f_1,\ldots,f_m})= \SPan{g_1,\ldots,g_{n+m}}$, where $g_{p_i}=e_i$ and $g_{p^c_i}=f_i$. From this, it follows that if ${E_{\DOT}},{E_{\DOT}}\!'\in{\Bbb F}\ell V$ and ${F\!_{\DOT}},{{F\!_{\DOT}}'}\in{\Bbb F}\ell W$ are pairs of opposite flags, then $\psi_P({E_{\DOT}},{F\!_{\DOT}})$ and $\psi_{w_0^{(n+m)}P}({E_{\DOT}}\!',{{F\!_{\DOT}}'})$ are opposite flags in $V\oplus W$. \begin{lem}\label{lem:product_subset} Let $P,Q\in{[n+m]\choose n}$, $v\in {\cal S}_n$, and $w\in {\cal S}_m$. Then, for ${E_{\DOT}}\in{\Bbb F}\ell V$ and ${F\!_{\DOT}}\in{\Bbb F}\ell W$, \begin{eqnarray*} \psi_P\left(X_{w_0^{(n)}v}{E_{\DOT}} \times X_{w_0^{(m)}w}{F\!_{\DOT}}\right) &\subset& X_{w_0^{(n+m)}\varepsilon_{P,Q}(v,w)}\psi_Q({E_{\DOT}},{F\!_{\DOT}})\\ \psi_P\left(X_v{E_{\DOT}} \times X_w{F\!_{\DOT}}\rule{0pt}{13pt}\right) &\subset& X_{\varepsilon_{P,Q}(v,w)}\psi_{w_0^{(n+m)}Q}({E_{\DOT}},{F\!_{\DOT}}). \end{eqnarray*} \end{lem} \noindent{\bf Proof. } For a flag ${G_{\DOT}}$, define $G^\circ_j:= G_j-G_{j-1}$. By the definition of $\psi_Q$, we have $E^\circ_i\subset \psi_Q({E_{\DOT}},{F\!_{\DOT}})^\circ_{q_i}$ and $F^\circ_i\subset \psi_Q({E_{\DOT}},{F\!_{\DOT}})^\circ_{q^c_i}$. Since $$ w_0^{(n+m)}Q\quad =\quad n+m+1-q_n\ <\ \cdots\ <\ n+m+1-q_1, $$ $E_{n+1-j} \subset \psi_{w_0^{(n+m)}Q}({E_{\DOT}},{F\!_{\DOT}})_{n+m+1-q_j}$, and $F_{n+1-j} \subset \psi_{w_0^{(n+m)}Q}({E_{\DOT}},{F\!_{\DOT}})_{n+m+1-q'_j}$, the lemma is a consequence of the definitions of Schubert varieties and $\psi_P$. \QEDnoskip \begin{cor}\label{cor:geometry_product} Let ${E_{\DOT}},{E_{\DOT}}\!'\in{\Bbb F}\ell V$ and ${F\!_{\DOT}},{{F\!_{\DOT}}'}\in{\Bbb F}\ell W$ be pairs of opposite flags and let $P\in{[n+m]\choose n}$. Set $Q=\{m+1,\ldots,m+n\}$. Then, for every $v\in {\cal S}_n$ and $w\in {\cal S}_m$, \begin{eqnarray*} \psi_P\left(X_v{E_{\DOT}}\times X_w{F\!_{\DOT}}\right) &=& X_{\varepsilon_{P,[n]}(v,w)}\psi_{Q}({E_{\DOT}},{F\!_{\DOT}}) \bigcap X_{\varepsilon_{P,Q}(e ,e )} \psi_{[n]}({E_{\DOT}}\!',{{F\!_{\DOT}}'})\\ &=& X_{\varepsilon_{P,[n]}(v,e )}\psi_{Q}({E_{\DOT}},{{F\!_{\DOT}}'}) \bigcap X_{\varepsilon_{P,Q}(e ,w)} \psi_{[n]}({E_{\DOT}}\!',{F\!_{\DOT}})\\ &=& X_{\varepsilon_{P,[n]}(e ,w)}\psi_{Q}({E_{\DOT}}\!',{F\!_{\DOT}}) \bigcap X_{\varepsilon_{P,Q}(v,e )} \psi_{[n]}({E_{\DOT}},{{F\!_{\DOT}}'})\\ &=& X_{\varepsilon_{P,[n]}(e ,e )}\psi_{Q}({E_{\DOT}}\!',{{F\!_{\DOT}}'}) \bigcap X_{\varepsilon_{P,Q}(v,w)} \psi_{[n]}({E_{\DOT}},{F\!_{\DOT}}). \end{eqnarray*} \end{cor} \noindent{\bf Proof. } Since $w_0^{(n+m)}[n] = Q$, $X_e {E_{\DOT}}={\Bbb F}\ell V$, and $X_e {F\!_{\DOT}}={\Bbb F}\ell W$, Lemma~\ref{lem:product_subset} shows that $\psi_P\left(X_v{E_{\DOT}}\times X_w{F\!_{\DOT}}\right)$ is a subset of any of the four intersections. Equality follows as they have the same dimension. Indeed, for $x,z\in{\cal S}_n$ and $y,u\in{\cal S}_m$, \begin{eqnarray*} \ell(\varepsilon_{P,[n]}(x,y))&=& \ell(x)+\ell(y)+\#\{i\in[n],j\in[m]\,|\,p_i>p^c_j\}\\ \ell(\varepsilon_{P,Q}(z,u))&=& \ell(z)+\ell(u)+\#\{i\in[n],j\in[m]\,|\,p^c_j>p_i\}. \end{eqnarray*} Thus $\ell(\varepsilon_{P,[n]}(x,y)) + \ell(\varepsilon_{P,Q}(z,u)) = \ell(x)+\ell(y)+\ell(z)+\ell(u) + n\cdot m$ and so $$ {n+m\choose 2} -\ell(\varepsilon_{P,[n]}(x,y)) - \ell(\varepsilon_{P,Q}(z,u))\ =\ {n\choose 2}+{m\choose 2}-\ell(x)-\ell(y)-\ell(z)-\ell(u). $$ If $(x,y,z,u)$ is one of $(v,w,e ,e ), (v,e ,e ,w), (e ,w,v,e ), (e ,e ,v,w)$, then these are, respectively, the the dimension of one of the intersections and the dimension of $X_v{E_{\DOT}}\times X_w{F\!_{\DOT}}$. \QEDnoskip \begin{cor}\label{cor:product_push_forward} Let $Q=\{m+1,\ldots,m+n\}=w_0^{(n+m)}[n]$. For every $v\in{\cal S}_n$, $w\in{\cal S}_m$, and $P\in{[n+m]\choose n}$, the following identities hold in $H^*{\Bbb F}\ell_{n+m}$: \medskip \noindent$\hspace{.8in} {\frak S}_{\varepsilon_{P,[n]}(v,w)}\cdot {\frak S}_{\varepsilon_{P,Q}(e ,e )} \quad=\quad {\frak S}_{\varepsilon_{P,[n]}(v,e )}\cdot {\frak S}_{\varepsilon_{P,Q}(e ,w)} \quad=\quad$\smallskip \hfill$ {\frak S}_{\varepsilon_{P,[n]}(e ,w)}\cdot {\frak S}_{\varepsilon_{P,Q}(v,e )} \quad=\quad {\frak S}_{\varepsilon_{P,[n]}(e ,e )}\cdot {\frak S}_{\varepsilon_{P,Q}(v,w)},\hspace{.8in} $\medskip \noindent and this common cohomology class is $(\psi_P)_*({\frak S}_v\otimes{\frak S}_w)$. \end{cor} \begin{thm}\label{thm:many_identities} Let $x\in {\cal S}_{n+m}$ and $P\in{[n+m]\choose n}$. Then \begin{enumerate} \item[({\em i})]\begin{minipage}[t]{5in} \mbox{ }\vspace{-18pt} \begin{eqnarray*} \psi_P^*{\frak S}_x &=& \sum_{v\in{\cal S}_n,\ w\in{\cal S}_m} c^{\varepsilon_{P,[n]}(v,w)}_{\varepsilon_{P,[n]}(e ,e )\ x} \ {\frak S}_v\otimes{\frak S}_w\\ &=& \sum_{v\in{\cal S}_n,\ w\in{\cal S}_m} c^{\varepsilon_{P,[n]}(v,w_0^{(m)})}_% {\varepsilon_{P,[n]}(e ,w_0^{(m)}w)\ x} \ {\frak S}_v\otimes{\frak S}_w\\ &=& \sum_{v\in{\cal S}_n,\ w\in{\cal S}_m} c^{\varepsilon_{P,[n]}(w_0^{(n)},w)}_% {\varepsilon_{P,[n]}(w_0^{(n)}v,e )\ x} \ {\frak S}_v\otimes{\frak S}_w\\ &=& \sum_{v\in{\cal S}_n,\ w\in{\cal S}_m} c^{\varepsilon_{P,[n]}(w_0^{(n)},w_0^{(m)})}_% {\varepsilon_{P,[n]}(w_0^{(n)}v,w_0^{(m)}w)\ x} \ {\frak S}_v\otimes{\frak S}_w \end{eqnarray*}\end{minipage}\medskip \item[({\em ii})] Let $Q=\{m+1,\ldots,m+n\}$. For every $v\in{\cal S}_n$ and $w\in {\cal S}_m$, we have $$ \begin{array}{ccccccc} c^{\varepsilon_{P,[n]}(v,w)}_{\varepsilon_{P,[n]}(e ,e )\ x} & = & c^{\varepsilon_{P,[n]}(v,w_0^{(m)})}_% {\varepsilon_{P,[n]}(e ,w_0^{(m)}w)\ x} & = & c^{\varepsilon_{P,[n]}(w_0^{(n)},w)}_% {\varepsilon_{P,[n]}(w_0^{(n)}v,e )\ x} & = & c^{\varepsilon_{P,[n]}(w_0^{(n)},w_0^{(m)})}_% {\varepsilon_{P,[n]}(w_0^{(n)}v,w_0^{(m)}w)\ x} \\ ||&&||&&||&&||\\ c^{\varepsilon_{P,Q}(v,w)}_{\varepsilon_{P,Q}(e ,e )\ x} & = & c^{\varepsilon_{P,Q}(v,w_0^{(m)})}_% {\varepsilon_{P,Q}(e ,w_0^{(m)}w)\ x} & = & c^{\varepsilon_{P,Q}(w_0^{(n)},w)}_% {\varepsilon_{P,Q}(w_0^{(n)}v,e )\ x} & = & c^{\varepsilon_{P,Q}(w_0^{(n)},w_0^{(m)})}_% {\varepsilon_{P,Q}(w_0^{(n)}v,w_0^{(m)}w)\ x} \end{array} $$ \end{enumerate} \end{thm} \begin{rem} {\em Each structure constant in ({\em ii}) is of the form $c^{\zeta y}_{y\, x}$, where $\zeta$ is, respectively, $v\times w, v\times \overline{w}^{-1}, \overline{v}^{-1}\times w$, and $\overline{v}^{-1}\times\overline{w}^{-1}$. Each interval $[y,\zeta y]$ is isomorphic to $[e ,v]\times[e ,w]$. This is consistent with the expectation that the $c^z_{y\,x}$ should only depend upon $[y,z]$ and $x$. }\end{rem} \noindent{\bf Proof. } In ({\em ii}), the second row is a consequence of the first as $c^z_{y\,x} = c^{w_0^{(n+m)}y}_{w_0^{(n+m)}z\: x}$, for $x,y,z\in {\cal S}_{n+m}$. The first row of equalities is a consequence of the identities in ({\em i}). For ({\em i}), there exist integral constants $d^{v\,w}_x$ defined by the identity $$ \psi^*_P {\frak S}_x \quad=\quad \sum d^{v\,w}_x\ {\frak S}_v\otimes{\frak S}_w. $$ Since the Schubert basis is self-dual with respect to the intersection pairing, we have \begin{eqnarray*} d^{v\,w}_x &=& \deg \left(\psi^*_P {\frak S}_x\cdot ({\frak S}_{w_0^{(n)}v}\otimes{\frak S}_{w_0^{(m)}w})\right)\\ &=& \deg\left({\frak S}_x\cdot (\psi_P)_*({\frak S}_{w_0^{(n)}v}\otimes{\frak S}_{w_0^{(m)}w})\right). \end{eqnarray*} Each expression for $(\psi_P)_*({\frak S}_{w_0^{(n)}v}\otimes{\frak S}_{w_0^{(m)}w})$ of Corollary~\ref{cor:product_push_forward} yields one of the sums in ({\em i}). For example, the last expression in Corollary~\ref{cor:product_push_forward} yields \begin{eqnarray*} d^{v\,w}_x &=& \deg\left({\frak S}_x\cdot {\frak S}_{\varepsilon_{P,[n]}(e ,e )}\cdot {\frak S}_{w_0^{(n+m)}\varepsilon_{P,[n]}(v,w)}\right)\\ &=& c^{\varepsilon_{P,[n]}(v,w)}_{\varepsilon_{P,[n]}(e ,e )\ x}, \end{eqnarray*} since $w_0^{(n+m)}\varepsilon_{P,[n]}(v,w) = \varepsilon_{P,w_0^{(n+m)}[n]}\left(w_0^{(n)}v,w_0^{(m)}w\right)$. \QEDnoskip \begin{cor}\label{cor:v_times_w} Let $u,v,w\in {\cal S}_n$ and $x,y,z\in {\cal S}_\infty$. Then $c^{ w\times z}_{u\times x\ \,v\times y} = c^w_{u\,v}\cdot c^z_{x\,y}$. \end{cor} \noindent{\bf Proof. } Choose $m$ so that $x,y,z\in{\cal S}_m$. Since $\varepsilon_{[n],[n]}(u,x)=u\times x$, the first identity of Theorem~\ref{thm:many_identities} ({\em i}) implies $\psi^*_{[n]} {\frak S}_{u\times x} = {\frak S}_u \otimes {\frak S}_x$. Then \begin{eqnarray*} c^w_{u\,v}\cdot c^z_{x\,y} &=& \deg\left( ({\frak S}_u \otimes {\frak S}_x)\cdot ({\frak S}_v \otimes {\frak S}_y)\cdot ({\frak S}_{w_0^{(n)}w} \otimes {\frak S}_{w_0^{(m)}z})\right)\\ &=& \deg\left( \psi^*_{[n]}({\frak S}_{u\times x}\cdot {\frak S}_{v\times y})\cdot ({\frak S}_{w_0^{(n)}w} \otimes {\frak S}_{w_0^{(m)}z})\right)\\ &=& \deg\left( {\frak S}_{u\times x}\cdot {\frak S}_{v\times y}\cdot {\frak S}_{w_0^{(n+m)}(w\times z)}\right)\\ &=&c^{ w\times z}_{u\times x\ \,v\times y}, \end{eqnarray*} as $(\psi_{[n]})_*{\frak S}_{w_0^{(n+m)}(w\times z)} = {\frak S}_{w_0^{(n)}w} \otimes {\frak S}_{w_0^{(m)}z}$, by Corollary~\ref{cor:product_push_forward}. \QEDnoskip \subsection{Maps ${\Bbb Z}[x_1,x_2,\ldots]\rightarrow {\Bbb Z}[y_1,y_2,\ldots,z_1,z_2,\ldots]$} \label{sec:substitution} Let $P\subset {\Bbb N}$, define $P^c:= {\Bbb N}-P$, and suppose $P^c$ is infinite. Enumerate $P$ and $P^c$ as follows: $$ \begin{array}{rcl} P&:& p_1\ <\ p_2\ <\ \left\{\begin{array}{lll} \cdots <p_s&\quad&\mbox{if } \#P=s\\ \cdots &&\mbox{otherwise}\end{array}\right.\\ P^c&:&p^c_1\ <\ p^c_2\ <\ \cdots\end{array} $$ Define $\Psi_P: {\Bbb Z}[x_1,x_2,\ldots]\rightarrow {\Bbb Z}[y_1,y_2,\ldots,z_1,z_2,\ldots]$ by $$ x_{p_i}\ \longmapsto\ y_i \qquad x_{p^c_i}\ \longmapsto\ z_i. $$ Then there exist integer constants $d_w^{u\,v}(P)$ for $u,v,w\in {\cal S}_\infty$ defined by the identity: $$ \Psi_P({\frak S}_w(x))\ =\ \sum_{u,v} d_w^{u\,v}(P)\:{\frak S}_u(y)\;{\frak S}_v(z). $$ For $l,d\in{\Bbb N}$ and $R\subset \{d+1,\ldots,d+2l\}$ with $\#R=l$, define $\overline{P}(l,d,R):= (P\bigcap [d])\bigcup R$. \begin{thm}\label{thm:substitution_constants} Let $P\subset{\Bbb N}$ and $w\in {\cal S}_\infty$. For any integers $l>\ell(w)$ and $d$ exceeding the last descent of $w$ and any subset $R$ of $\{d+1,\ldots,d+2l\}$ of cardinality $l$, set $n:=\#\overline{P}(l,d,R)$, $m:=d+2l-n$, and $\pi :=\varepsilon_{\overline{P}(l,d,R),\:[n]}(e,e)$. Then $d_w^{u\,v}(P) = 0$ unless $u\in {\cal S}_n$ and $\in {\cal S}_m$, and in that case, $$ d_w^{u\,v}(P)\ =\ c^{(u\times v)\pi}_{\pi\; w}. $$ Moreover, $d_w^{u\,v}(P) \neq 0$ implies that $a:= \# P\bigcap [d]$ exceeds the last descent of $u$ and $b:=d-a$ exceeds the last descent of $v$. \end{thm} \begin{rem}\label{rem:I_P} {\em Theorem~\ref{thm:substitution_constants} generalizes~\cite[1.5]{Lascoux_Schutzenberger_structure_de_Hopf} (see also~\cite[4.19]{Macdonald_schubert}) where it is shown that $d_w^{u\,v}([a])\geq 0$. Define $I_P$ to be $$ \{\varepsilon_{\overline{P}(l,l,R),[n]}(e,e)\;|\; l\in {\Bbb N},\ n=l+\#(P\bigcap [l]),\mbox{ and } R\subset\{l+1,\ldots,3l\},\# R=l\}. $$ For $w\in {\cal S}_n$, let $N$ be an integer such that $N/3$ exceeds both the last descent the length of $w$. If $\pi\in I_P$ with $\pi\not\in{\cal S}_N$, then $\pi = \varepsilon_{\overline{P}(l,d,R),[n]}(e,e)$ for $l,d,R$ satisfying the conditions of Theorem~\ref{thm:substitution_constants} and so $d_w^{u\,v}(P)\ =\ c^{(u\times v)\pi}_{\pi\; w}$ for every $\pi \in I_P-{\cal S}_N$, which establishes Theorem~\ref{thm:substitution}.} \end{rem} Apply the ring homomorphism $\Psi_P$ to both sides of the product: $$ {\frak S}_w(x)\;{\frak S}_\gamma(x)\quad=\quad \sum_\zeta c^\zeta_{w\,\gamma}\,{\frak S}_\zeta(x). $$ If we expand this in terms of ${\frak S}_\eta(y)\; {\frak S}_\xi(z)$ and equate the coefficients, we get a corollary. \begin{cor}\label{cor:brutally_complicated} Let $w,\gamma,\eta,\xi\in{\cal S}_\infty$, and $P\subset{\Bbb N}$. Then there exists an integer $N\in {\Bbb N}$ such that if $\pi\in I_P-{\cal S}_N$, then $$ \sum_{\zeta} c^{(\eta\times\xi)\pi}_{\pi\ \zeta}\: c^\zeta_{w\;\gamma} \quad = \quad \sum_{u,v,\alpha,\beta} c^{(u\times v)\pi}_{\pi\ w}\: c^{(\alpha\times\beta)\pi}_{\pi\ \gamma}\: c^\eta_{u\;\alpha}\:c^\xi_{v\;\beta}. $$ \end{cor} \noindent{\bf Proof of Theorem~\ref{thm:substitution_constants}. } First, a Schubert polynomial ${\frak S}_\pi(x)\in{\Bbb Z}[x_1,\ldots,x_s]$ if and only if $s$ exceeds the last descent of $\pi$~\cite{Lascoux_Schutzenberger_polynomes_schubert} (see also~\cite[4.13]{Macdonald_schubert}). Thus, ${\frak S}_w(x)\in{\Bbb Z}[x_1,\ldots,x_d]$, and if $d^{u\,v}_w(P)\neq 0$, then ${\frak S}_u(y)\in{\Bbb Z}[y_1,\ldots,y_a]$ and ${\frak S}_v(z)\in{\Bbb Z}[z_1,\ldots,z_b]$, hence $a$, respectively, $b$, exceeds the last descent of $u$, respectively $v$. Since $\deg{\frak S}_w(x)\leq l$, both $\deg{\frak S}_u(y)$ and $\deg{\frak S}_v(z)$ are at most $l$. Consider the commutative diagram $$ \begin{picture}(395,70) \put(0,50){${\Bbb Z}[x_1,\ldots,x_d]$} \put(110,50){${\Bbb Z}[x_1,\ldots,x_{n+m}]$} \put(130,5){$H^* {\Bbb F}\ell_{n+m}$} \put(258,50){${\Bbb Z}[y_1,\ldots,y_n,z_1,\ldots,z_m]$} \put(280,5){$H^*{\Bbb F}\ell_n\otimes H^*{\Bbb F}\ell_m$} \put(72,54){\oval(4,4)[l]}\put(72,52){\vector(1,0){33}} \put(86,57){$\iota$} \put(195,52){\vector(1,0){58}}\put(215,57){$\overline{\Psi_P}$} \put(185,7){\vector(1,0){90}}\put(215,14){$\psi^*_{\overline{P}}$} \put(150,43){\vector(0,-1){20}}\put(150,43){\vector(0,-1){25}} \put(320,43){\vector(0,-1){20}}\put(320,43){\vector(0,-1){25}} \end{picture} $$ Here, $\overline{\Psi_P}$ is the restriction of $\Psi_{\overline{P}}$ to ${\Bbb Z}[x_1,\ldots,x_{n+m}]$. The vertical arrows are injective on the module ${\Bbb Z}\Span{x_1^{\alpha_1}\cdots x_d^{\alpha_d}\;|\; \alpha_i \leq l}$ and its image $$ {\Bbb Z}\Span{y_1^{\beta_1}\cdots y_a^{\beta_a} z_1^{\gamma_1}\cdots z_b^{\gamma_b} \;|\; \beta_i,\gamma_j \leq l} \quad\subset\quad{\Bbb Z}[y_1,\ldots,y_n,z_1,\ldots,z_m]. $$ Moreover, since $P\bigcap[d]=\overline{P}\bigcap [d]$, the composition, $\Psi_{\overline{P}}\circ\iota$, of the top row coincides with $\Psi_{P}\circ\iota$. Since ${\frak S}_w(x)\in {\Bbb Z} \Span{x_1^{\alpha_1}\cdots x_d^{\alpha_d}\;|\; \alpha_i \leq l}$, the cohomological formula for $\psi_{\overline{P}}^*({\frak S}_w)$ in Theorem~\ref{thm:many_identities} computes $\Psi_P({\frak S}_w(x))$. \QED In the statement of the Theorem~\ref{thm:substitution_constants}, $\ell(w)$ could be replaced by $\max_i\{\deg_{x_i}({\frak S}_w(x))\}$. \subsection{Products of Grassmannians} Let $k\leq n$ and $l\leq m$ be integers, $V\simeq {\Bbb C}^n$, and $W\simeq{\Bbb C}^m$. Define $\varphi_{k,l}:\mbox{\it Grass}_k V\times \mbox{\it Grass}_l W \hookrightarrow \mbox{\it Grass}_{k+l}(V\oplus W)$ by $$ \varphi_{k,l}\ :\ (H,K)\quad \longmapsto\quad H\oplus K. $$ \begin{thm}\label{thm:grassmann_product} \mbox{ } \begin{enumerate} \item[({\em i})] For every Schubert class $S_\lambda\in H^*\mbox{\it Grass}_{k+l}V\oplus W$, $$ \varphi^*_{k,l}(S_\lambda) \quad=\quad \sum_{\mu,\nu} c^\lambda_{\mu\,\nu} S_\mu\otimes S_\nu. $$ \item[({\em ii})] If $S_{\mu^c}\otimes S_{\nu^c}\in H^*\mbox{\it Grass}_k V\otimes H^*\mbox{\it Grass}_l W$, then $$ (\varphi_{k,l})_*(S_{\mu^c}\otimes S_{\nu^c}) \quad =\quad \sum_\lambda c^\lambda_{\mu\,\nu} S_{\lambda^c}, $$ where $\lambda^c,\mu^c$, and $\nu^c$ are defined by $\mu^c_i=n-k-\mu_{k+1-i}$, $\nu^c_i=m-l-\nu_{l+1-i}$, and $\lambda^c_i=m+n-k-l-\lambda_{k+l+1-i}$. \end{enumerate} \end{thm} \begin{rem}{\em If $-x_1,\ldots,-x_k$ are the Chern roots of the tautological $k$-plane bundle over $\mbox{\it Grass}_k V$, and $-y_1,\ldots,-y_l$ those of the tautological $l$-plane bundle over $\mbox{\it Grass}_l W$, and $f\in H^*\mbox{\it Grass}_{k+l}V\oplus W$ (which is a symmetric polynomial in the negative Chern roots of the tautological bundle over $\mbox{\it Grass}_{k+l}V\oplus W$). Then $$ \varphi_{k,l}^* f\ = \ f(x_1,\ldots,x_k,y_1,\ldots,y_l). $$ Let $\Lambda=\Lambda(z)$ be the ring of symmetric functions, which is the inverse limit (in the category of graded rings) of the rings of symmetric polynomials in the variables $z_1,\ldots,z_n$. Fixing $\lambda$ and choosing $k,l,n$, and $m$ large enough gives a new proof of~\cite[I.5.9]{Macdonald_symmetric}: }\end{rem} \begin{prop}[{\cite[I.5.9]{Macdonald_symmetric}}] Let $\lambda$ be a partition and $x$, $y$ be infinite sets of variables. Then $$ S_\lambda(x,y)\ =\ \sum_{\mu,\nu} c^\lambda_{\mu\,\nu}\; S_\mu(x)\cdot S_\nu(y), $$ where $S_\mu$ denotes the Schur function basis of the ring $\Lambda$ of symmetric functions. \end{prop} If we define a linear map $\Delta:\Lambda(z) \rightarrow \Lambda(x)\otimes_{\Bbb Z}\Lambda(y)$ by $\Delta(f(z))= f(x,y)$, then $\Delta$ is induced by the maps $\varphi^*_{k,l}$. Moreover, the obvious commutative diagrams of spaces give a new proof of~\cite[I.5.24]{Macdonald_symmetric}, that $\Lambda$ is a cocommutative Hopf algebra with comultiplication $\Delta$.\medskip \noindent{\bf Proof of Theorem~\ref{thm:grassmann_product}.} The first statement is a consequence of the second: Schubert classes form a basis for the cohomology ring, so there exist integral constants $d^{\mu\,\nu}_\lambda$ such that $$ \varphi^*_{k,l}(S_\lambda)\quad=\quad \sum_{\mu,\nu} d^{\mu\,\nu}_\lambda S_\mu\otimes S_\nu. $$ Since the Schubert basis diagonalizes the intersection pairing, $$ d^{\mu\,\nu}_\lambda \quad=\quad \deg(\varphi^*_{k,l}(S_\lambda)\cdot (S_{\mu^c}\otimes S_{\nu^c})). $$ Apply $(\varphi_{k,l})_*$ and use the second assertion to obtain \begin{eqnarray*} d^{\mu\,\nu}_\lambda&=& \deg(S_\lambda\cdot(\varphi_{k,l})_*(S_{\mu^c}\otimes S_{\nu^c}))\\ &=& S_\lambda \cdot \sum_\kappa c^\kappa_{\mu\,\nu} S_{\kappa^c}\\ &=& c^\lambda_{\mu\,\nu}. \end{eqnarray*} The second assertion is a consequence of the following lemma. \begin{lem}\label{lem:grassmann_computation} Suppose $\mu,\nu$ are partitions with $\mu\subset (n-k)^k$ and $\nu\subset(m-l)^l$. Let ${E_{\DOT}}\in {\Bbb F}\ell V$ and ${F\!_{\DOT}}\in{\Bbb F}\ell W$ and let ${{G_{\DOT}}'}$ be any flag opposite to $\psi_{[n]}({E_{\DOT}},{F\!_{\DOT}})$ with $G'_m=W$. Then $$ \varphi_{k,l}\left(\Omega_{\mu^c}{E_{\DOT}}\times \Omega_{\nu^c}{F\!_{\DOT}}\right) \quad =\quad \Omega_{\rho^c}\psi_{[n]}({E_{\DOT}},{F\!_{\DOT}}) \bigcap \Omega_{(m-l)^k}{{G_{\DOT}}'}, \eqno(4.7.1) $$ where $\rho$ is the partition $$ \nu_1+(n-k)\ \geq\ \cdots\ \geq\ \nu_l+(n-k)\ \geq \ \mu_1\ \geq \cdots\ \geq \mu_k. $$ \end{lem} We finish the proof of Theorem~\ref{thm:grassmann_product}. Lemma~\ref{lem:grassmann_computation} implies \begin{eqnarray*} \left(\varphi_{k,l}\right)_*\left(S_{\mu^c}\otimes S_{\nu^c}\right) &=& \left[ \Omega_{\rho^c}\psi_{[n]}({E_{\DOT}},{F\!_{\DOT}}) \bigcap \Omega_{(n-k)^l}{{G_{\DOT}}'}\right]\\ &=& \sum_\lambda c^{\lambda^c}_{\rho^c\: (n-k)^l}\; S_{\lambda^c}. \end{eqnarray*} Since $\deg(S_\alpha\cdot S_\beta\cdot S_\gamma) = c^{\alpha^c}_{\beta\,\gamma}$, we see that $$ c^{\lambda^c}_{\rho^c\: (n-k)^l} \ =\ c^\rho_{(n-k)^l\; \lambda} \ =\ c^{\rho/(n-k)^l}_\lambda\ =\ c^{\mu\coprod \nu}_\lambda \ =\ c^\lambda_{\mu\,\nu}. $$ Here, $\mu \coprod\nu$ is a skew partition with two components $\mu$ and $\nu$ and the last equality is a special case of~(1.3.1) in \S\ref{sec:Schur_identities}. \QED \noindent{\bf Proof of Lemma~\ref{lem:grassmann_computation}. } Since $\Omega_{(n-k)^l}{{G_{\DOT}}'} = \{M\in \mbox{\it Grass}_{k+l}V\oplus W\,|\, \dim M\bigcap G'_m\geq l\}$ and $G'_m=W$, we see that $\varphi_{k,l}(\mbox{\it Grass}_kV\times\mbox{\it Grass}_lW)\subset\Omega_{(n-k)^l}{{G_{\DOT}}'}$. It is an exercise in the definition of the Schubert varieties involved and of $\psi_{[n]}({E_{\DOT}},{F\!_{\DOT}})$ to see that $$ \varphi_{k,l}(\Omega_{\mu^c}{E_{\DOT}}\times\Omega_{\nu^c}{F\!_{\DOT}}) \ \subset\ \Omega_{\rho^c}\psi_{[n]}({E_{\DOT}},{F\!_{\DOT}}), $$ which shows the inclusion $\subset$ in~(4.7.1). Equality follows as they have the same dimension: The intersection has dimension $|\rho|-|(n-k)^l|\ =\ |\mu|+|\nu|$, the dimension of $\Omega_{\mu^c}{E_{\DOT}}\times\Omega_{\nu^c}{F\!_{\DOT}}$. \QEDnoskip \section{Symmetries of the Littlewood-Richardson coefficients} \label{sec:geometry} \subsection{Proof of Theorem~\ref{thm:B} ({\em ii})}\label{sec:proof_thm_B} Combining Lemma~\ref{lem:restriction} with Theorem~\ref{thm:coeff_sum}, we deduce: \begin{lem}\label{lem:shape_reduction} Suppose $x\leq_k z$ and $x(p)=z(p)$. Let $k'=k-1$ if $p<k$ and $k'=k$ otherwise. Then for all partitions $\lambda$, we have $$ c^z_{x\;v(\lambda,k)}\quad =\quad c^{z/_p}_{x/_p\:v(\lambda,k')}. $$ \end{lem} Note that $zx^{-1}$ and $z/_p(x/_p)^{-1}$ are shape-equivalent, by Lemma~\ref{lem:expanding_bruhat}~({\em iv}). \begin{lem}\label{lem:skew_critical} Let $x,z,u,w\in{\cal S}_n$. Suppose $x\leq_kz$, $u\leq_kw$, and $zx^{-1}=wu^{-1}$. Further suppose that $w$ is Grassmannian with descent $k$, the permutation $wu^{-1}$ has no fixed points, and, for $k<i\leq n$, $u(i)=x(i)$. Then, for all partitions $\lambda$ with at most $k$ parts, $$ c^w_{u\,v(\lambda,k)}\quad =\quad c^z_{x\,v(\lambda,k)}. $$ \end{lem} \noindent{\bf Proof of Theorem~\ref{thm:B}~({\em ii}) using Lemma~\ref{lem:skew_critical}. } We reduce Theorem~\ref{thm:B}~({\em ii}) to the special case of Lemma~\ref{lem:skew_critical}. First, by Lemma~\ref{lem:shape_reduction}, it suffices to prove Theorem~\ref{thm:B}~({\em ii}) when $x,z,u,w\in{\cal S}_n$, $k=l$, with $wu^{-1}=zx^{-1}$ and the permutation $wu^{-1}$ has no fixed points. Define $s\in {\cal S}_n$ by $$ s(i)\quad:=\quad\left\{\begin{array}{lcl}u(i)&&1\leq i\leq k\\ x(i)&& k<i\leq n\end{array}\right. $$ and set $t:=wu^{-1}s$. Then $s\leq_k t$ and $$ t(i)\quad=\quad\left\{\begin{array}{lcl}w(i)&&1\leq i\leq k\\ z(i)&& k<i\leq n\end{array}\right.. $$ It suffices to show separately that $c^w_{u\:v(\lambda,k)}$ and $c^t_{s\:v(\lambda,k)}$ each equal $c^z_{t\:v(\lambda,k)}$. Thus we may further assume $u(i)=x(i)$ for $1\leq i\leq k$ or $u(i)=x(i)$ for $k<i\leq n$. Suppose that $u(i)=x(i)$ for $1\leq i\leq k$. If for $v\in {\cal S}_n$, $\overline{v}:=w_0vw_0$, $$ c^z_{x\:v(\lambda,k)}\quad=\quad c^w_{u\:v(\lambda,k)} \quad\Longleftrightarrow\quad c^{\overline{z}}_{\overline{x}\:\overline{v(\lambda,k)}} \quad=\quad c^{\overline{w}}_{\overline{u}\:\overline{v(\lambda,k)}}. $$ Set $l=n-k$ and $\lambda^t$ the partition conjugate to $\lambda$. Then $\overline{x}\leq_{k'}\overline{z}$, $\overline{u}\leq_{k'}\overline{w}$, $\overline{z}(\overline{x}^{-1})=\overline{w}{\overline{u}^{-1}}$, $\overline{v(\lambda,k)}= v(\lambda^t,l)$, and $\overline{x}(i)=\overline{u}(i)$ for $l<i\leq n$. Thus we may assume $x(i)=u(i)$ for $1\leq i\leq k$. Finally, there is a Grassmannian permutation $t\in {\cal S}_n$ with descent $k$ and a permutation $s\in{\cal S}_n$ such that $t=wu^{-1}s$. Thus it suffices to further assume that $w$ is Grassmannian with descent $k$, the situation of Lemma~\ref{lem:skew_critical}. \QED We prove Lemma~\ref{lem:skew_critical} by studying two intersections of Schubert varieties and their image under the projection ${\Bbb F}\ell V\twoheadrightarrow \mbox{\it Grass}_k V$. Let $e_1,\ldots,e_n$ be a basis for $V$ and set ${F\!_{\DOT}}=\SPan{e_1,\ldots,e_n}$. Let $M(w)\subset M_{n\times n}{\Bbb C}$ be the set of matrices satisfying the conditions: \begin{enumerate} \item[(a)] $M(w)_{i,w(i)}=1$ \item[(b)] $M(w)_{i,j}=0$ if either $w(i)<j$ or else $w^{-1}(j)<i$. \end{enumerate} Then $M(w)\simeq{\Bbb C\,}^{\ell(w)}$ as the only unconstrained entries of $M(w)$ are $M(w)_{i,j}$ when $j<w(i)$ and $i<w^{-1}(j)$, and there are $\ell(w)$ such entries. \begin{ex} {\em Let $w=25134\in {\cal S}_5$, a Grassmannian permutation with descent 2. Then $M(w)$ is the set of matrices $$ \left\{ \left[\begin{array}{ccccc} a & 1 & 0 & 0 & 0\\ b & 0 & c & d & 1\\ 1 & 0 & 0 & 0 & 0\\ 0 & 0 & 1 & 0 & 0\\ 0 & 0 & 0 & 1 & 0\end{array}\right] \ \raisebox{-30pt}{\rule{.5pt}{68pt}}\, \mbox{ for every } (a,b,c,d)\in{\Bbb C}\,^4\right\} $$ }\end{ex} Fix a basis $e_1,\ldots,e_n$ for $V$. For $\alpha\in M(w)$, and $1\leq i\leq n$, define the vector $f_i(\alpha) := \sum_j \alpha_{i,j}e_j$. Then $ f_1(\alpha),\ldots,f_n(\alpha)$ are the `row vectors' of the matrix $\alpha$ and they form a basis for $V$ as $\alpha$ has determinant $(-1)^{\ell(w)}$. Set ${E_{\DOT}}(\alpha)=\SPan{f_1(\alpha),\ldots,f_n(\alpha)}$. Since $f_i(\alpha)\in F_{w(i)}- F_{w(i)-1}$, we see that ${E_{\DOT}}(\alpha)\in X^\circ_{w_0w}{F\!_{\DOT}}$. Moreover, given ${E_{\DOT}}\in X^\circ_{w_0w}{F\!_{\DOT}}$, restricted row reduction on a basis for ${E_{\DOT}}$ shows there is a unique $\alpha\in M(w)$ with ${E_{\DOT}}={E_{\DOT}}(\alpha)$. In the case that $w$ is Grassmannian with descent $k$, matrices in $M(w)$ have a simple form: if $k<i$, then $f_i(\alpha) = e_{w(i)}$. For opposite flags ${F\!_{\DOT}},{{F\!_{\DOT}}'}$, ${\frak S}_{w_0w}\cdot{\frak S}_u$ is the class Poincar\'e dual to the fundamental cycle of $X_{w_0w}{F\!_{\DOT}}\bigcap X_u{{F\!_{\DOT}}'}$. We use the projection formula~(2.3.1) to compute the coefficient $c^w_{u\,v(\lambda,k)}$: \begin{eqnarray*} c^w_{u\,v(\lambda,k)} &=& \deg(S_\lambda(x_1,\ldots,x_k)\cdot{\frak S}_{w_0w}\cdot{\frak S}_u)\\ &=& \deg (\pi_k)_*(S_\lambda(x_1,\ldots,x_k) \cdot{\frak S}_{w_0w}\cdot{\frak S}_u)\\ &=& \deg(S_\lambda \cdot (\pi_k)_*({\frak S}_{w_0w}\cdot{\frak S}_u)) \end{eqnarray*} Thus Lemma~\ref{lem:skew_critical} is a consequence of the following calculation: \begin{lem}\label{lem:skew_calculation} Let $u,w,x,z$ satisfy the hypotheses of Lemma~\ref{lem:skew_critical}. Then, if ${F\!_{\DOT}}$ and ${{F\!_{\DOT}}'}$ are opposite flags in $V$, $$ \pi_k \left(X_{w_0w}{F\!_{\DOT}}\bigcap X_u{{F\!_{\DOT}}'}\right) \quad =\quad \pi_k \left(X_{w_0z}{F\!_{\DOT}}\bigcap X_x{{F\!_{\DOT}}'}\right). $$ \end{lem} \noindent{\bf Proof. } Let $e_1,\ldots,e_n$ be a basis for $V$ such that ${F\!_{\DOT}}=\SPan{e_1,\ldots,e_n}$ and ${{F\!_{\DOT}}'}=\SPan{e_n,\ldots,e_1}$, and define $M(w)$ as before. Let $A\subset M(w)$ consist of those matrices $\alpha$ such that ${E_{\DOT}}(\alpha)\in X^\circ_u{{F\!_{\DOT}}'}$. If $j>k$, set $g_j(\alpha)=f_j(\alpha)=e_{w(j)}$. For $j\leq k$ construct $g_j(\alpha)$ inductively, setting $g_j(\alpha)$ to be the intersection of $F'_{n+1-u(j)}$ and the affine space $f_j(\alpha) + \Span{g_i(\alpha)\,|\,i<j\mbox{ and } u(i)<u(j)}$. Since ${E_{\DOT}}(\alpha)\in X_u{F\!_{\DOT}}$ and $\dim E_j(\alpha)\bigcap F'_{n+1-u(j)}= \#\{i\leq j\,|\, u(i)>u(j)\}$, this intersection consists of a single, non-zero vector, $g_j(\alpha)$. Then the algebraic map $A\ni \alpha \mapsto (g_1(\alpha),\ldots,g_n(\alpha))\in V^n$ gives a parameterized basis of $V$. Moreover, ${E_{\DOT}}(\alpha)=\SPan{g_1(\alpha),\ldots,g_n(\alpha)}$ for all $\alpha\in A$, and if $1\leq j\leq k$, then $g_j(\alpha)\in F'_{n+1-u(j)}\bigcap F_{w(j)}$. Note that for $\alpha\in A$, $$ {G_{\DOT}}(\alpha) \ :=\ \SPan{g_{u^{-1}x(1)}(\alpha),\ldots,g_{u^{-1}x(n)}(\alpha)} \ \in\ X_{w_0z}{F\!_{\DOT}}\bigcap X_x{{F\!_{\DOT}}'}. \eqno(5.1.1) $$ and thus $A$ also parameterizes a subset of $X_{w_0z}{F\!_{\DOT}}\bigcap X_x{{F\!_{\DOT}}'}$. Indeed, for $1\leq j\leq k$, $g_{u^{-1}x(j)}\in F'_{n+1-x(j)}\bigcap F_{y(j)}$. Also for $j>k$, we have $u^{-1}x(j)=j=w^{-1}y(j)$, thus $g_j(\alpha)=f_j(\alpha)=e_{y(j)}$ and $G_j(\alpha)=E_j(\alpha)$. The definition of Schubert cells for the flag manifold in \S\ref{sec:flag} then implies (5.1.1). Both cycles $X_{w_0w}{F\!_{\DOT}}\bigcap X_u{{F\!_{\DOT}}'}$ and $X_{w_0 z}{F\!_{\DOT}}\bigcap X_x{{F\!_{\DOT}}'}$ are irreducible and have the same dimension, $\ell(w)-\ell(u)= | w u^{-1}|$. Since ${G_{\DOT}}(\alpha)={G_{\DOT}}(\beta)$ if and only if $\alpha=\beta$, the loci of flags $\{{G_{\DOT}}(\alpha)\,|\, \alpha\in A\}$ is dense in $X_x{{F\!_{\DOT}}'}\bigcap X_{w_0y}{F\!_{\DOT}}$. Finally, for $\alpha\in A$, we have $G_k(\alpha)=E_k(\alpha)$, as $u^{-1}x$ permutes $\{1,\ldots,k\}$, which completes the proof. \QEDnoskip \subsection{Proof of Theorem~\ref{thm:C}~({\em ii}). }\label{sec:proof_C} We show that if $\zeta$ and $\eta$ are disjoint permutations and $\lambda$ any partition, then $$ c^{\zeta\eta}_\lambda\quad=\quad \sum_{\mu,\nu} c^\lambda_{\mu\,\nu}\;c^\zeta_\mu\; c^\eta_\nu. $$ \begin{lem}\label{lem:long} Let $\zeta,\eta\in{\cal S}_{n+m}$ be disjoint permutations. Suppose $k\geq \#\mbox{up}_\zeta$, $l\geq \#\mbox{up}_\eta$, $n\geq\#\mbox{supp}_\zeta$, and $m\geq \#\mbox{supp}_\eta$. Let $u\in{\cal S}_{n+m}$ be a permutation such that $u\leq _{k+l}\zeta\eta u$. Let $Q$ be any element of ${[n+m]-\mbox{supp}_\eta\choose n}$ which contains $\mbox{supp}_\zeta$ for which $k=\# u^{-1}(Q)\bigcap [k+l]$. Set $Q^c:= [n+m]-Q$. Define $\zeta'\in{\cal S}_n$ and $\eta'\in{\cal S}_m$ by $\phi_Q(\zeta')=\zeta$ and $\phi_{Q^c}(\eta')=\eta$. Set $P=u^{-1}(Q)$, $P^c=u^{-1}(Q^c)$, and define $v\in{\cal S}_n$ and $w\in{\cal S}_m$ by $u(p_i)=q_{v(i)}$ and $u(p^c_i)=q^c_{w(i)}$, where $$ \begin{array}{lll} P\ =\ p_1\ <\ p_2\ <\ \cdots\ <\ p_n &\qquad& P^c\ =\ p^c_1\ <\ p^c_2\ <\ \cdots\ <\ p^c_m\\ Q\ =\ q_1\ <\ q_2\ <\ \cdots\ <\ q_n &\qquad& Q^c\ =\ q^c_1\ <\ q^c_2\ <\ \cdots\ <\ q^c_m\end{array} $$ Then \begin{enumerate} \item[({\em i})] $v\leq_k \zeta' v$ and $w\leq_l \eta' w$, \item[({\em ii})] $u=\varepsilon_{P,Q}(v,w)$ and $\zeta\eta u =\varepsilon_{P,Q}(\zeta' v,\eta' w)$, and \item[({\em iii})] For all pairs of opposite flags ${E_{\DOT}},{E_{\DOT}}\!'\in{\Bbb F}\ell_n$ and ${F\!_{\DOT}},{{F\!_{\DOT}}'}\in{\Bbb F}\ell_m$, \smallskip ${\displaystyle \psi_P\left[ \left(X_{w_0^{(n)}\zeta' v}{E_{\DOT}} \bigcap X_v{E_{\DOT}}\!'\right) \times \left(X_{w_0^{(m)}\eta' w}{F\!_{\DOT}} \bigcap X_w{{F\!_{\DOT}}'}\right)\right]\ =}$ \hfill ${\displaystyle X_{w_0^{(n+m)}\zeta\eta u}\psi_Q({E_{\DOT}},{F\!_{\DOT}}) \bigcap X_u\psi_{w_0^{(m+n)}Q}({E_{\DOT}}\!',{{F\!_{\DOT}}'})}$.\qquad \end{enumerate} \end{lem} \noindent{\bf Proof. } Since $u\leq_{k+l} \zeta\eta u$, ({\em i}) follows from Theorem~\ref{thm:k-length} and the definitions. The second statement is also immediate. For ({\em iii}), Lemma~\ref{lem:product_subset} shows the inclusion $\subset$. Since $\zeta'$ is shape equivalent to $\zeta$, $\eta'$ to $\eta$, and $\zeta$ and $\eta$ are disjoint, $|\zeta\eta|=|\zeta'|+|\eta'|$, showing both cycles have the same dimension, and hence are equal, as $\psi_Q({E_{\DOT}},{F\!_{\DOT}})$ and $\psi_{w_0^{(m+n)}Q}({E_{\DOT}}\!',{{F\!_{\DOT}}'})$ are opposite flags. \QED Note that if $u\leq_k \zeta u$, then $$ c^\zeta_\lambda \quad=\quad \deg(S_\lambda \cdot (\pi_k)_*({\frak S}_{w_0\zeta u}\cdot{\frak S}_u)). $$ Thus the skew Littlewood-Richardson coefficients $c^\zeta_\lambda$ are defined by the identity in $H^* \mbox{\it Grass}_k V$: $$ (\pi_k)_*({\frak S}_{w_0\zeta u}\cdot{\frak S}_u) \quad=\quad \sum_{\lambda\subset (n-k)^k} c^\zeta_\lambda\, S_{\lambda^c}. \eqno(5.2.1) $$ \noindent{\bf Proof of Theorem~\ref{thm:C}~({\em ii}). } We use the notation of Lemma~\ref{lem:long}. The following diagram commutes since $[k+l]=\{p_1,\ldots,p_k,p^c_1,\ldots,p^c_l\}$. $$ \setlength{\unitlength}{2.2pt} \begin{picture}(107,34) \put(0,3){$\mbox{\it Grass}_k{\Bbb C}^n\times\mbox{\it Grass}_l{\Bbb C}^m$} \put(75,3){$\mbox{\it Grass}_{k+l}{\Bbb C}^{n+m}$} \put(13.8,26){${\Bbb F}\ell_n\times{\Bbb F}\ell_m$} \put(82,26){${\Bbb F}\ell_{n+m}$} \put(58,7){$\varphi_{k,l}$} \put(60,30){$\psi_P$} \put(6.5,16){$\pi_k\times\pi_l$} \put(91,16){$\pi_{k+l}$} \put(53,4.5){\vector(1,0){20}} \put(40,27.5){\vector(1,0){40}} \put(25.5,23){\vector(0,-1){14}} \put(89,23){\vector(0,-1){14}} \end{picture} $$ {}From this and Lemma~\ref{lem:long}, we see that $$ \pi_{k+l}\left(X_{w_0^{(n+m)}\zeta\eta u}\psi_Q({E_{\DOT}},{F\!_{\DOT}}) \bigcap X_u\psi_{w_0^{(m+n)}Q}({E_{\DOT}}\!',{{F\!_{\DOT}}'})\right) $$ is equal to $$ \varphi_{k,l}\left( \pi_k\left(X_{w_0^{(n)}\zeta' v}{E_{\DOT}}\bigcap X_v{E_{\DOT}}\!'\right) \times \pi_l\left(X_{w_0^{(m)}\eta' w}{F\!_{\DOT}}\bigcap X_w{{F\!_{\DOT}}'}\right) \right). $$ Thus $(\pi_{k+l})_*\left({\frak S}_{w_0^{(n+m)}\zeta\eta u} \cdot {\frak S}_u\right)$ is equal to $$ (\varphi_{k,l})_*\left( (\pi_k)_*\left({\frak S}_{w_0^{(n)}\zeta' v} \cdot{\frak S}_v\right)\otimes (\pi_l)_*\left({\frak S}_{w_0^{(m)}\eta' w} \cdot{\frak S}_w\right)\right). $$ This, together with (5.2.1), gives \begin{eqnarray*} \sum_\lambda c^{\zeta\eta}_\lambda S_{\lambda^c} &=& (\pi_{k+l})_*\left({\frak S}_{w_0^{(n+m)}\zeta\eta u} \cdot {\frak S}_u\right) \\ &=& (\varphi_{k,l})_*\left( \sum_\mu c^{\zeta'}_\mu S_{\mu^c} \otimes \sum_\nu c^{\eta'}_\nu S_{\nu^c}\right) \\ &=& \sum_{\mu,\nu} c^{\zeta'}_\mu c^{\eta'}_\nu (\varphi_{k,l})_*\left(S_{\mu^c} \otimes \ S_{\nu^c}\right) \\ &=& \sum_{\mu,\nu} c^{\zeta'}_\mu c^{\eta'}_\nu \sum_\lambda c^\lambda_{\mu\,\nu} S_{\lambda^c}. \end{eqnarray*} This completes the proof, as $\zeta',\zeta$ and $\eta',\eta$ are shape equivalent pairs. \QEDnoskip \subsection{Theorem~\ref{thm:D}$'$ (Cyclic Shift) }\label{sec:thmd}\ {\em Let $u,w,x,z\in{\cal S}_\infty$ with $u\leq_k w$ and $x\leq_l z$. Suppose $wu^{-1}\in {\cal S}_n$ and $zx^{-1}$ is shape equivalent to $(wu^{-1})^{(1\,2\,\ldots\,n)^t}$, for some $t$. For every partition $\lambda$, $$ c^w_{u\,v(\lambda,k)}\quad =\quad c^z_{x\,v(\lambda,l)}. $$ }\medskip \noindent{\bf Proof. } By Theorem~\ref{thm:B}~({\em ii}), it suffices to prove a restricted case. Suppose $u,w\in{\cal S}_n$, $u\leq_k w$, and $w$ is Grassmannian with descent $k$. The idea is to construct permutations $x,z\in {\cal S}_n$ with $x\leq_k z$ and $zx^{-1}=(wu^{-1})^{(1\,2\,\ldots\,n)}$ for which $$ \pi_k\left(X_{w_0w}{F\!_{\DOT}}\bigcap X_u{{F\!_{\DOT}}'}\right) \quad=\quad \pi_k\left(X_{w_0z}{G_{\DOT}}\bigcap X_x{{G_{\DOT}}'}\right), \eqno(5.3.1) $$ where $e_1,\ldots,e_n$ be a basis for $V$ and the flags ${F\!_{\DOT}},{{F\!_{\DOT}}'},{G_{\DOT}}$, and ${{G_{\DOT}}'}$ are $$ \begin{array}{rclcrcl} {F\!_{\DOT}} & = & \SPan{e_1,\ldots,e_n} && {{F\!_{\DOT}}'} & = & \SPan{e_n,\ldots,e_1}\\ {G_{\DOT}} & = & \SPan{e_n,e_1,\ldots,e_{n-1}} && {{G_{\DOT}}'} & = & \SPan{e_{n-1},\ldots,e_1,e_n}. \end{array} $$ Then (5.3.1) implies the identity $c^w_{u\: v(\lambda,k)} = c^z_{x\: v(\lambda,k)}$, which completes the proof. If $wu^{-1}(n)= n$, then $zx^{-1} = 1\times wu^{-1}$, which is shape equivalent to $wu^{-1}$, and the result follows by Theorem~\ref{thm:B}~({\em ii}). Assume $wu^{-1}(n)\neq n$. Then $w(k)=n$ and $u(k)<n$, as $w$ is Grassmannian with descent $k$. Set $m:=u(k)$, $p:= u^{-1}(n) (>k)$ , and $l:=w(p)$. Define $x\in {\cal S}_n$ by: $$ x(j)\quad=\quad\left\{\begin{array}{lcl} u(j)+1 && 1\leq j< k\mbox{ or } p<j\\ 1 && j=k\\ m+1 && j=k+1\\ u(j-1)+1&& k+1<j\leq p \end{array}\right.. $$ Then $x\leq _k z:= (wu^{-1})^{(1\,2\,\ldots\,n)}x$ where $$ z(j)\quad=\quad\left\{\begin{array}{lcl} w(j)+1 && 1\leq j<k\mbox{ or } p<j\\ l+1 && j=k\\ 1 && j=k+1\\ w(j-1)+1&& k+1<j\leq p \end{array}\right.. $$ To show (5.3.1), let $g_1(\alpha),\ldots,g_n(\alpha)$ for $\alpha\in A$ be the parameterized basis for flags ${E_{\DOT}}(\alpha)\in X^\circ_u{{F\!_{\DOT}}'}\bigcap X^\circ_{w_0w}{F\!_{\DOT}}$ constructed in the proof of Lemma~\ref{lem:skew_calculation}. Since $g_k(\alpha)\in F'_{n+1-u(k)}\bigcap F_{w(k)}$, $u(k)=m$, and $w(k)=n$, there exist regular functions $\beta_j(\alpha)$ on $A$ such that $$ g_k(\alpha)= e_n + \sum_{j=m}^{n-1} \beta_j(\alpha)e_j. $$ Since $F'_1=\Span{e_n}\subset E_p(\alpha)- E_{p-1}(\alpha)$ and $g_p(\alpha)=e_l$, there exist regular functions $\delta_j(\alpha)$ on $A$ with $\delta_p(\alpha)$ nowhere vanishing such that \begin{eqnarray*} e_n &=& \sum_{j=1}^p \delta_j(\alpha)g_j(\alpha)\\ &=& g_k(\alpha)+ \sum_{j=1}^{k-1} \delta_j(\alpha)g_j(\alpha) +\sum_{j=k+1}^p \delta_j(\alpha)e_{w(j)}, \end{eqnarray*} as $g_k(\alpha)$ is the only vector among the $g_j(\alpha)$ in which $e_n$ has a non-zero coefficient. Thus $$ e_n- \sum_{j=k+1}^p \delta_j(\alpha) e_{w_j}\ =\ g_k(\alpha) + \sum_{j=1}^{k-1}\delta(\alpha)g_j(\alpha) $$ is a vector in $E_k(\alpha) - E_{k-1}(\alpha)$. Define a basis $h_1(\alpha),\ldots,h_n(\alpha)$ for $V$ by $$ h_j(\alpha)\quad=\quad \left\{\begin{array}{lcl} g_j(\alpha) && 1\leq j< k \mbox{ or } p<j\\ e_n-\left( \sum_{j=k+1}^p \delta_j(\alpha)e_{w(j)}\right) && j=k\\ e_n && j=k+1\\ g_{j-1}(\alpha) && k+1<j\leq p \end{array}\right.. $$ We claim ${E_{\DOT}}\!'(\alpha):=\SPan{h_1(\alpha),\ldots,h_n(\alpha)}$ is a flag in $X_{w_0z}{G_{\DOT}}\bigcap X_x{{G_{\DOT}}'}$, which implies (5.3.1): Since $h_k(\alpha)\in E_k(\alpha) - E_{k-1}(\alpha)$ and $h_j(\alpha)=g_j(\alpha)$ for $j<k$, we have $$ E'_k(\alpha)\quad=\quad \Span{h_1(\alpha),\ldots,h_k(\alpha)} \quad=\quad E_k(\alpha). $$ Thus if $\alpha\neq\alpha'$, then ${E_{\DOT}}\!'(\alpha)\neq{E_{\DOT}}\!'(\alpha')$ and so $\{{E_{\DOT}}\!'(\alpha)\,|\,\alpha\in A\}$ is a subset of the intersection $X_{w_0z}{G_{\DOT}}\bigcap X_x{{G_{\DOT}}'}$ of dimension equal to $\dim A=\ell(w)-\ell(u)=\ell(z)-\ell(x)$, the dimension of $X_{w_0z}{G_{\DOT}}\bigcap X_x{{G_{\DOT}}'}$. Thus $\{{E_{\DOT}}\!'(\alpha)\,|\,\alpha\in A\}$ is dense, and so $E'_k(\alpha)=E_k(\alpha)$ implies (5.3.1). For notational convenience, set $G^\circ_j:=G_j-G_{j-1}$, and similarly for $F^\circ_j$. To establish this claim, we first show that $h_j(\alpha)\in G^\circ_{z(j)}$ for $j=1,\ldots,n$, which shows $h_1(\alpha),\ldots,h_n(\alpha)$ is a parameterized basis for $V$ and ${E_{\DOT}}\!'(\alpha)\in X_{w_0z}{G_{\DOT}}$. Then, for a fixed $\alpha\in A$, we construct $h'_1,\ldots,h'_n$ which satisfy ${E_{\DOT}}\!'(\alpha)=\SPan{h'_1,\ldots,h'_n}$ and $h'_j\in G'_{n+1-x(j)}$ for $j=1,\ldots,n$, showing ${E_{\DOT}}\!'(\alpha)\in X_x{{G_{\DOT}}'}$. Note that if $i<n$, then $G_{i+1}=\Span{e_n,F_i}$. Thus $h_j(\alpha)\in F^\circ_{w(j)}\subset G^\circ_{z(j)}$ for $1\leq j<k$ and $p<j$, and if $k+1<j\leq p$, then $h_j(\alpha)\in F^\circ_{w(j-1)}\subset G^\circ_{z(j)}$. Then, since $G_1=\Span{e_n}$, we see that $h_{k+1}(\alpha)=e_n\in G'_1=G'_{n+1-x(k+1)}$. Finally, since $w$ is Grassmannian of descent $k$, if $k+1\leq i\leq p$, then $w(i)\leq w(p)=l$, which shows $h_k(\alpha)\in G^\circ_{l+1}=G^\circ_{z(k)}$. Thus ${E_{\DOT}}\!'(\alpha)\in X^\circ_{w_0z}{G_{\DOT}}$. We now show that ${E_{\DOT}}\!'(\alpha)\in X_x{{G_{\DOT}}'}$. Note that if $a\leq b<n$, then $F'_{n+1-a}\bigcap F_b\subset G'_{n-a}\cap G_{b+1}$. Thus if $1\leq j< k$, $h_j(\alpha)=g_j(\alpha)\in F'_{n+1-u(j)}\bigcap F_{w(j)} \subset G'_{n+1-x(j)}$. Since $x(k)=1$, we see that $h_k(\alpha)\in G'_{n+1-x(k)}=V$. Fix $\alpha\in A$ and set $h'_j=h_j(\alpha)$ for $1\leq j\leq k$. Define $$ h'_{k+1}\ :=\ g_k(\alpha) - e_n\ =\ \sum_{j=m}^{n-1} \beta_j(\alpha)e_j\ \in\ G'_{n+1-(m+1)}\ =\ G'_{n+1-x(k+1)}. $$ Since $h'_{k+1} + h_{k+1}(\alpha)= g_k(\alpha)$, we see that $E'_{k+1}(\alpha) = \Span{E'_k(\alpha),H'_{k+1}}$. Finally, since ${E_{\DOT}}(\alpha)\in X_u{{F\!_{\DOT}}'}$, if $k<j$ there exists a vector $$ g'_j\ :=\ \sum_{i\leq j}\gamma_{i,j}g_i(\alpha) \ \in\ F'_{n+1-u(j)} $$ such that $\Span{E_{j-1}(\alpha),g'_j}= E_j(\alpha)$. For $k+1<j\leq p$, set $$ h'_j\ =\ g'_{j-1}-\gamma_{k,j-1}e_n\ \in\ \Span{e_{n-1}\ldots,e_{n+1-u(j-1)}}\ =\ G'_{n+1-x(j)}, $$ as as $g_k(\alpha)$ is the only vector among $\{g_1(\alpha),\ldots,g_n(\alpha)\}$ which is not in the span of $e_1,\ldots,e_{n-1}$. If $p<j$, set $h'_j=g'_j-\gamma_{k,j}e_n\in G'_{n+1-x(j)}$. Then $\SPan{h'_1,\ldots,h'_n}={E_{\DOT}}\!'(\alpha)$, completing the proof. \QEDnoskip \section{Formulas for some Littlewood-Richardson coefficients} \subsection{A chain-theoretic interpretation}\label{sec:chain_description} We give a chain-theoretic interpretation for some Littlewood-Richardson coefficients $c^\zeta_\lambda$ in terms chains in either the $k$-Bruhat order or the $\preceq$-order, similar to the main results of~\cite{sottile_pieri_schubert}. If either $u\lessdot_k (\alpha, \beta) u$ or $\zeta{\prec\!\!\!\cdot\,}(\alpha,\beta)\zeta$ is a cover, label that edge in the Hasse diagram with the integer $\beta=\max\{\alpha,\beta\}$. Given a saturated chain in the $k$-Bruhat order from $u$ to $\zeta u$, equivalently, a saturated $\preceq$-chain from $e$ to $\zeta$, the {\em word} of that chain is its sequence of edge labels. Given a word $\omega=a_1.a_2\ldots a_m$, Schensted insertion~\cite{Schensted} or~\cite[\S 3.3]{Sagan} of $\omega$ into the empty tableau gives a pair $(S,T)$ of Young tableaux, where $S$ is the {\em insertion tableau} and $T$ the {\em recording tableau} of $\omega$. Let $\mu\subset \lambda$ be partitions. A permutation $\zeta$ is {\em shape-equivalent} to a skew Young diagram $\lambda/\mu$ if there is a $k$ such that $\zeta$ is shape-equivalent to $v(\lambda,k)\cdot v(\mu,k)^{-1}$. It follows that $\zeta$ is shape equivalent to some skew partition $\lambda/\mu$ if and only if whenever $\alpha,\beta\in \mbox{\rm up}_\zeta$ or $\alpha,\beta\in \mbox{\rm down}_\zeta$, $$ \alpha\ <\ \beta \quad \Longleftrightarrow \quad \zeta(\alpha)\ <\ \zeta(\beta). $$ We prove a stronger version of Theorem~\ref{thm:skew_shape}: \begin{thm}\label{thm:skew_shape_prime} Let $\mu\subset\lambda$ be partitions and suppose $\zeta\in {\cal S}_\infty$ is shape equivalent to $\lambda/\mu$. Then, for every partition $\nu$ \begin{enumerate} \item[({\em i})] $c^\zeta_\nu = c^{\lambda/\mu}_\nu$, and \item[({\em ii})] For every standard Young tableau $T$ of shape $\nu$, $$ c^\zeta_\nu\ =\ \#\left\{\begin{array}{cc}\mbox{$\preceq$-chains from $e $ to $\zeta$ whose} \\ \mbox{word has recording tableau $T$} \end{array} \right\} $$ Equivalently, if $u\leq_k w$ and $wu^{-1}=\zeta$, then $$ c^w_{u\,v(\nu,k)}\ =\ \#\left\{\begin{array}{cc}\mbox{Chains in $k$-Bruhat order from $u$ to} \\\mbox{$w$ whose word has recording tableau $T$} \end{array}\right\} $$ \end{enumerate} \end{thm} \begin{rem}{\em Theorem~\ref{thm:skew_shape_prime} ({\em ii}) gives a combinatorial proof of Proposition~\ref{prop:chains}, when $wu^{-1}$ is shape equivalent to a skew partition. Theorem~\ref{thm:skew_shape_prime} ({\em ii}) is similar in form to Theorem 8 of~\cite{sottile_pieri_schubert}: \medskip \noindent{\bf Theorem 8~\cite{sottile_pieri_schubert}. }} Suppose $\nu=(p,1^{q-1})$, a partition of `hook' shape. Then for every $u,w\in {\cal S}_\infty$ and $k\in{\Bbb N}$, the constant $c^w_{u\, v(\nu,k)}$ counts either set \begin{enumerate} \item[({\em i})] ${\displaystyle \left\{\begin{array}{c}\makebox[3.1in][c]{Chains in $k$-Bruhat order from $u$ to $w$ with}\\\makebox[3.1in][c]{\ word \ $a_1<\cdots<a_p>a_{p+1}>\cdots>a_{p+q-1}$. }\end{array}\right\} }$. \item[({\em ii})]${\displaystyle \left\{\begin{array}{c}\makebox[3.1in][c]{Chains in $k$-Bruhat order from $u$ to $w$ with}\\\makebox[3.1in][c]{word \ $a_1>\cdots>a_q<a_{q+1}<\cdots<a_{p+q-1}$.} \end{array}\right\} }$. \end{enumerate} \medskip {\em The recording tableaux of words in ({\em i}) each have the integers $1,2,\ldots, p$ in the first row and $1,p+1,\ldots,p{+}q{-}1$ in the first column. Furthermore, these are the only words with this recording tableau. Similarly, the recording tableaux of words in ({\em ii}) all have the integers $1,2,\ldots,q$ in the first column and $1,q{+}1,\ldots,p{+}q{-}1$ in the first row. However, Theorem~\ref{thm:skew_shape} is {\em not} a generalization of this result: The permutation $\zeta:=(143652)$ is not shape equivalent to any skew partition as $4,5\in \mbox{down}_\zeta$ but $\zeta(4)>\zeta(5)$. Nevertheless, $c^\zeta_{(4,1)}=1$. Interestingly, $\zeta$ satisfies the conclusions of Theorem~\ref{thm:skew_shape}. While the hypothesis of Theorem~\ref{thm:skew_shape} is not necessary for the conclusion to hold, some hypotheses are necessary: Let $\zeta=(162)(354)$, a product of two disjoint 3-cycles. Then $\zeta^{(1\,\ldots\,6)}=(132)(465)=v(\,% \begin{picture}(12,6) \put(0,0){\line(0,1){6}} \put(0,0){\line(1,0){12}}\put(3,0){\line(0,1){6}} \put(0,3){\line(1,0){12}}\put(6,0){\line(0,1){6}} \put(0,6){\line(1,0){6}}\put(9,0){\line(0,1){3}} \put(12,0){\line(0,1){3}}\end{picture}\,,\, 2)\cdot v(% \begin{picture}(6,3) \put(0,0){\line(0,1){3}} \put(0,0){\line(1,0){6}}\put(3,0){\line(0,1){3}} \put(0,3){\line(1,0){6}}\put(6,0){\line(0,1){3}} \end{picture}\,,\, 2)^{-1}$. Hence, by Theorem~\ref{thm:D}, we have: $$ c^\zeta_{\begin{picture}(12,3) \put(0,0){\line(0,1){3}} \put(0,0){\line(1,0){12}}\put(3,0){\line(0,1){3}} \put(0,3){\line(1,0){12}}\put(6,0){\line(0,1){3}} \put(9,0){\line(0,1){3}}\put(12,0){\line(0,1){3}} \end{picture}} \quad=\quad c^\zeta_{\,\begin{picture}(9,6) \put(0,0){\line(0,1){6}} \put(0,0){\line(1,0){9}}\put(3,0){\line(0,1){6}} \put(0,3){\line(1,0){9}}\put(6,0){\line(0,1){3}} \put(0,6){\line(1,0){3}}\put(9,0){\line(0,1){3}} \end{picture}} \quad=\quad c^\zeta_{\,\begin{picture}(6,6) \put(0,0){\line(0,1){6}} \put(0,0){\line(1,0){6}}\put(3,0){\line(0,1){6}} \put(0,3){\line(1,0){6}}\put(6,0){\line(0,1){6}} \put(0,6){\line(1,0){6}}\end{picture}} \quad=\quad 1. $$ (This may also be seen as a consequence of Theorem~\ref{thm:C} and the form of the Pieri-type formula in~\cite{Lascoux_Schutzenberger_polynomes_schubert}, or of the main result, Theorem~5, of~\cite{sottile_pieri_schubert}.) If $u=312645$, then $\zeta u=561234$ and the labeled Hasse diagram of $[u,\zeta u]_2$ is: $$ \epsfxsize=1.8in \epsfbox{figures/bad_Schensted.eps} $$ The labels of the six chains are: $$ 2456,\ 2465,\ 2645,\ 4526,\ 4256,\ 4265 $$ and these have (respective) recording tableaux: $$ \epsfxsize=4.4in \epsfbox{figures/recording.eps}\ \raisebox{6pt}{.} $$ This list omits the tableau \begin{picture}(15,15)(0,2) \put(0,0){\line(0,1){15}} \put(0,0){\line(1,0){15}}\put(7.5,0){\line(0,1){15}} \put(0,7.5){\line(1,0){15}}\put(15,0){\line(0,1){15}} \put(0,15){\line(1,0){15}} \put(1.5,1.5){\tiny 1} \put(9,1.5){\tiny 2} \put(1.5,9){\tiny 3} \put(9,9){\tiny 4} \end{picture}, and the third and fourth tableaux are identical. } \end{rem} \noindent{\bf Proof of Theorem~\ref{thm:skew_shape}. } Suppose first that $\zeta=v(\lambda,k)\cdot v(\mu,k)^{-1}$. Then $$ [e,\zeta]_\preceq\quad \simeq \quad [v(\mu,k),\:v(\lambda,k)]_k \quad \simeq \quad [\mu,\lambda]_\subset. $$ The first isomorphism preserves the edge labeling of the Hasse diagrams, and in the second the labels of the $k$-Bruhat order correspond to diagonals in a Young diagram: If $\nu\;{\subset\!\!\!\!\cdot\,\,}\nu'$ is a cover in Young's lattice, there is a unique $i$ such that $\nu_i \neq\nu'_i$. In that case $\nu_i+1=\nu'_i$ and the label of the corresponding edge in the $k$-Bruhat order is $k-i+\nu'_i$, the diagonal on which the new box in $\nu'$ lies. A chain in Young's lattice from $\mu$ to $\lambda$ is a standard skew tableau $R$ of shape $\lambda/\mu$. Consider the word, $a_1\ldots a_m$, of that chain as a two-rowed array: $$ w\quad = \quad \left(\begin{array}{cccc}1&2&\cdots&m\\ a_1&a_2&\cdots&a_m\end{array}\right). $$ Then the entry $i$ of $R$ is in the $a_i$th diagonal. Let $S$ and $T$ be, respectively, the insertion and recording tableaux for that two-rowed array. Consider the two-rowed array consisting of the columns ${a_i\choose i}$ arranged in lexicographic order: that is, ${a_i\choose i}$ is to the left of ${a_j\choose j}$ if either $a_i<a_j$ or $a_i=a_j$ and $i<j$. Then the insertion and recording tableaux of this new array are $T$ and $S$, respectively~\cite{Schutzenberger_insertion,Knuth}. The second row of this new array, the word inserted to obtain $T$, is the `diagonal' word of the skew tableau $R$. That is, the entries of $R$ read lexicographically by diagonal. By Lemma~\ref{lem:diagonal_word} (proven below), the diagonal word is Knuth-equivalent to the original word. Thus $T$ is the unique tableau of partition shape Knuth-equivalent to $R$. This gives a combinatorial bijection $$ \left\{\begin{array}{cc}\mbox{$\preceq$-chains from $e $ to $\zeta$ whose} \\ \mbox{word has recording tableau $T$} \end{array} \right\} \quad\Longleftrightarrow\quad \left\{\begin{array}{cc}\mbox{Skew tableaux $R$ of shape}\\ \mbox{$\lambda/\mu$ Knuth-equivalent to $T$} \end{array} \right\}, $$ proving the theorem in this case, as it is well-known that (see, for example~\cite[\S 4.9]{Sagan}), $$ c^{\lambda/\mu}_\nu \quad =\quad \#\left\{\begin{array}{cc}\mbox{Skew tableaux $R$ of shape}\\ \mbox{$\lambda/\mu$ Knuth-equivalent to $T$} \end{array} \right\}. $$ Now suppose $\zeta$ is shape-equivalent to $v(\lambda,k)\cdot v(\mu,k)^{-1}$. By Theorem~\ref{thm:B}~({\em ii}), $c^\zeta_\nu=c^{\lambda/\mu}_\nu$, proving ({\em i}). Assume $\lambda, \mu$, and $k$ have been chosen so that $\zeta=\phi_P\left(v(\lambda,k)\cdot v(\mu,k)^{-1}\right)$, for some $P$. By Theorem~\ref{thm:new_order}~({\em iii}), $\phi_P$ induces an isomorphism $$ \phi_P\ :\ [e ,\,v(\lambda,k)\cdot v(\mu,k)^{-1}]_\preceq \ \stackrel{\sim}{\longrightarrow}\ [e ,\,\zeta]_\preceq. $$ Moreover, if $\eta{\prec\!\!\!\cdot\,}(\alpha,\beta)\eta$ is a cover in $[e ,\,v(\lambda,k)\cdot v(\mu,k)^{-1}]_\preceq$, then $\phi_P\eta{\prec\!\!\!\cdot\,}(\phi_P(\alpha\,\beta))\phi_P\eta$ is a cover in $[e,\zeta]_\preceq$ which has label $p_\beta$, where $P=p_1<p_2<\cdots$. Thus, if $\gamma$ is a chain in $[e,\,v(\lambda,k)\cdot v(\mu,k)^{-1}]_\preceq$ whose word $a_1,\ldots,a_m$ has recording tableau $T$, then $\phi_P(\gamma)$ is a chain in $[e ,\zeta]_\preceq$ with word $p_{a_1},\ldots,p_{a_m}$, which also has recording tableau $T$. \QED Order the diagonals of a skew Young tableau $R$ beginning with the diagonal incident to the the end of the first column of $R$. The {\em diagonal word} of $R$ is the entries of $R$ listed in lexicographic order by diagonal, with magnitude breaking ties. The tableau on the left below has diagonal word $7\,58\,379\,148\,26\,26\,5\,8$. If we apply Schensted insertion to the initial segment $7\,58\,379\,148$, (those diagonals incident upon the first column), we obtain the tableau on the right, whose row word equals this initial segment. $$ \epsfxsize=1.5in \epsfbox{figures/diagonal.eps} $$ This observation is the key to the proof of the following lemma. \begin{lem}\label{lem:diagonal_word} The diagonal word of a skew tableau is Knuth-equivalent to its column word. \end{lem} \noindent{\bf Proof. } For a skew tableau $R$, let $d(R)$ be its diagonal word. We show that $d(R)$ is Knuth equivalent to the word $c.d(R')$, the concatenation of the first column $c$ of $R$ and $d(R')$, where $R'$ is obtained from $R$ by the removal of its first column. An induction completes the proof. Suppose the first column of $R$ has length $b$ and $R$ has $r$ diagonals. For $1\leq j\leq b$ let $w_j:= a^j_1\ldots a^j_{s_j}$ be the subword of $d(R)$ consisting of the $j$th diagonal. Then $a^j_1<\cdots<a^j_{s_j}$, $s_1\leq s_2\leq \cdots\leq s_b$, and if $k\leq s_j$, then $a^j_k>a^{j+1}_k>\cdots>a^b_k$, as these are consecutive entries in the $k$th column of $R$. Consider the insertion tableau $T_l$ of the word $w_1.w_2.\ldots w_l$ for $1\leq l\leq b$. Then the $k$th column of $T_l$ is $a^j_k>\cdots>a^l_k$, where $s_{j-1}<k\leq s_j$. Hence $c.d(R') = c.\mbox{\em row}(T'). w_{b+1}\ldots w_r$, where $\mbox{\em row}(T')$ is the row word of the tableau obtained from $T_b$ by removing its first column, which is $c$. Since the column word of a tableau is Knuth-equivalent to its row word, we are done. \QEDnoskip \subsection{Skew permutations} Define the set of {\em skew permutations} to be the smallest set of permutations containing all permutations $v(\lambda,k)\cdot v(\mu,k)^{-1}$ which is closed under: \begin{enumerate} \item[1.] Shape equivalence. If $\eta$ is shape equivalent to a skew permutation $\zeta$, then $\eta$ is skew. \item[2.] Cyclic shift. If $\zeta\in{\cal S}_n$ is skew, then so is $\zeta^{(1\,2\,\ldots\,n)}$. \item[3.] Products of disjoint permutations. If $\zeta,\eta$ are disjoint and skew, then $\zeta\eta$ is skew. \end{enumerate} A {\em shape} of a skew permutation $\zeta$ is a (non-unique!) skew partition $\theta$ which is defined inductively. If $\zeta$ is shape equivalent to $\lambda/\mu$, then $\zeta$ has shape $\lambda/\mu$. If $\zeta\in {\cal S}_n$ is a skew permutation with shape $\theta$, then $\zeta^{(1\,2\,\ldots\,n)}$ has shape $\theta$. If $\zeta$ and $\eta$ are disjoint skew permutations with respective shapes $\rho$ and $\sigma$, then $\zeta\eta$ has skew shape $\rho\coprod \sigma$. \begin{thm}\label{thm:skew_permutation} Let $\zeta$ be a skew permutation with shape $\theta$, then \begin{enumerate} \item[({\em i})] For all partitions $\nu$, $$ c^\zeta_\nu\quad=\quad c^\theta_\nu. $$ \item[({\em ii})] The number of chains in the interval $[e,\,\zeta]_\preceq$ is equal to the number of standard Young tableaux of shape $\theta$. \end{enumerate} \end{thm} \noindent{\bf Proof. } The number of standard skew tableaux of shape $\theta$ is $\sum_\lambda f^\lambda c^\theta_\lambda$, hence ({\em ii}) is consequence of ({\em i}) and Proposition~\ref{prop:chains}. To show ({\em i}), we need only consider the last part (3.) of the recursive definition of skew permutations, by Theorems~\ref{thm:B}~({\em ii}) and~\ref{thm:D}. Suppose $\zeta$ and $\eta$ are disjoint skew permutations with respective shapes $\rho$ and $\sigma$, and for all partitions $\nu$, $c^\zeta_\nu=c^\rho_\nu$ and $c^\eta_\nu=c^\sigma_\nu$. Then by Theorem~\ref{thm:C}~({\em ii}), \begin{eqnarray*} c^{\zeta\eta}_\nu &=& \sum_{\lambda,\mu} c^\nu_{\lambda\,\mu}\; c^\zeta_\lambda\;c^\eta_\mu\\ &=&\sum_{\lambda,\mu} c^\nu_{\lambda\,\mu}\; c^\rho_\lambda\;c^\sigma_\mu\\ &=& c^{\rho \coprod \sigma}_\nu. \makebox[.1in]{\hspace{1in}\QEDnoskip} \end{eqnarray*} \begin{ex}\label{example:disjoint} {\em Consider the geometric graph of the permutation $(1978)(26354)$: $$ \epsfxsize=1.1in \epsfbox{figures/skew_grass_ex.eps} $$ Thus the two cycles $\zeta=(1978)$ and $\eta=(26354)$ are disjoint. Note that $\zeta$ is shape equivalent to $(1423)$ and $(1423)^{(1234)} = (1342)$. Similarly, $\eta$ is shape equivalent to $(15243)$ and $(15243)^{(12345)} = (13542)$. Both of these cycles, $(1423)$ and $(15243)$, are skew partitions: Let $\lambda=\;$% \setlength{\unitlength}{1.3pt}\begin{picture}(3,6) \put(0,0){\line(1,0){3}}\put(3,0){\line(0,1){3}} \put(3,3){\line(-1,0){3}}\put(0,3){\line(0,-1){3}} \end{picture}\,, $\mu=\;$% \begin{picture}(6,6) \put(0,0){\line(1,0){6}}\put(6,0){\line(0,1){6}} \put(6,6){\line(-1,0){6}}\put(0,6){\line(0,-1){6}} \put(3,0){\line(0,1){6}}\put(0,3){\line(1,0){6}} \end{picture}\,, $\nu=\;$% \begin{picture}(9,6) \put(0,0){\line(0,1){6}}\put(0,0){\line(1,0){9}} \put(3,0){\line(0,1){6}}\put(0,6){\line(1,0){6}} \put(6,0){\line(0,1){6}}\put(0,3){\line(1,0){9}} \put(9,0){\line(0,1){3}} \end{picture}\,. Then $$ v(\lambda,2)\quad=\quad 13245,\qquad v(\mu,2)\quad=\quad 34125,\qquad v(\nu,2)\quad=\quad 35124 $$ and $$ \begin{array}{rcccl} v(\lambda,2)&\leq_2&(1342)\cdot v(\lambda,2)&=& v(\mu,2),\\ v(\lambda,2)&\leq_2&(13542)\cdot v(\lambda,2)&=& v(\nu,2).\\ \end{array} $$ Hence, for every partition $\kappa$, $c^\zeta_\kappa=c^{\mu/\lambda}_\kappa$ and $c^\eta_\kappa=c^{\nu/\lambda}_\kappa$. Thus it follows that $c^{\zeta\eta}_\kappa = c^\rho_\kappa$, where $\rho$ is any of the four skew partitions: $$ \setlength{\unitlength}{.75pt}\begin{picture}(50,40)\thicklines \put(30, 0){\line(1,0){20}} \put(20,10){\line(1,0){30}} \put( 0,20){\line(1,0){40}} \put( 0,30){\line(1,0){20}} \put( 0,40){\line(1,0){10}} \put( 0,20){\line(0,1){20}} \put(10,20){\line(0,1){20}} \put(20,10){\line(0,1){20}} \put(30, 0){\line(0,1){20}} \put(40, 0){\line(0,1){20}} \put(50, 0){\line(0,1){10}} \end{picture}\qquad \begin{picture}(50,40)\thicklines \put(30, 0){\line(1,0){20}} \put(20,10){\line(1,0){30}} \put(10,20){\line(1,0){30}} \put( 0,30){\line(1,0){20}} \put( 0,40){\line(1,0){20}} \put( 0,30){\line(0,1){10}} \put(10,20){\line(0,1){20}} \put(20,10){\line(0,1){30}} \put(30, 0){\line(0,1){20}} \put(40, 0){\line(0,1){20}} \put(50, 0){\line(0,1){10}} \end{picture}\qquad \begin{picture}(50,40)\thicklines \put(40, 0){\line(1,0){10}} \put(30,10){\line(1,0){20}} \put(10,20){\line(1,0){40}} \put( 0,30){\line(1,0){30}} \put( 0,40){\line(1,0){20}} \put( 0,30){\line(0,1){10}} \put(10,20){\line(0,1){20}} \put(20,20){\line(0,1){20}} \put(30,10){\line(0,1){20}} \put(40, 0){\line(0,1){20}} \put(50, 0){\line(0,1){20}} \end{picture}\qquad \begin{picture}(50,40)\thicklines \put(30, 0){\line(1,0){20}} \put(30,10){\line(1,0){20}} \put(10,20){\line(1,0){30}} \put( 0,30){\line(1,0){30}} \put( 0,40){\line(1,0){20}} \put( 0,30){\line(0,1){10}} \put(10,20){\line(0,1){20}} \put(20,20){\line(0,1){20}} \put(30, 0){\line(0,1){30}} \put(40, 0){\line(0,1){20}} \put(50, 0){\line(0,1){10}} \end{picture} $$ }\end{ex} \subsection{Further remarks}\label{sec:further} For small symmetric groups, it is instructive to examine all permutations and determine to which class they belong. Here, we enumerate each class in ${\mathcal S}_4$, ${\mathcal S}_5$, and ${\mathcal S}_6$: \begin{center} \begin{tabular}{|l||c|c|c|} \hline &\begin{tabular}{c}skew\\ partitions\end{tabular}& \begin{tabular}{c}shape equivalent to\\ a skew partition\end{tabular} & \begin{tabular}{c}skew\\ permutation\end{tabular}\\ \hline \hline ${\cal S}_4$ & 14 & 21 & 24\\ \hline ${\cal S}_5$ & 42 & 79 & 120 \\ \hline ${\cal S}_6$ & 132 & 311 & 678\\ \hline \end{tabular} \end{center} If $\zeta$ is one of the 42 permutations in ${\cal S}_6$ which are not skew permutations, and $\zeta$ is not among $$ (125634),\ (145236),\ (143652),\ (163254),\ (153)(246),\ \mbox{or}\ (135)(264), $$ then there is a skew partition $\theta$ such that $c^\zeta_\nu = c^\theta_\nu$ for all partitions $\nu$. It would be interesting to understand why this occurs for all but these 6 permutations. Is there a wider class of permutations $\zeta$ such that there exists a skew partition $\theta$ with $c^\zeta_\nu = c^\theta_\nu$ for all partitions $\nu$? For the six `exceptional' permutations $\zeta$, there is a skew partition $\theta$ for which $c^\zeta_\nu = c^\theta_\nu$ for all $\nu\subset a^b$, where $a=\#$up$_\zeta$ and $b=\#$down$_\zeta$. For these, $\theta\not\subset a^b$. For example, let $\zeta=(153)(246)$. If $u=214365$, then $u\leq_3 \zeta u$ and there are 42 chains in $[u,\,\zeta u]_3$. Also $$ c^\zeta_{\,\begin{picture}(6,4) \put(0,0){\line(0,1){4}} \put(0,0){\line(1,0){6}} \put(2,0){\line(0,1){4}} \put(0,2){\line(1,0){6}} \put(4,0){\line(0,1){4}} \put(0,4){\line(1,0){6}} \put(6,0){\line(0,1){4}} \end{picture}}\ =\ 1, \qquad c^\zeta_{\,\begin{picture}(6,6) \put(0,0){\line(0,1){6}} \put(0,0){\line(1,0){6}} \put(2,0){\line(0,1){6}} \put(0,2){\line(1,0){6}} \put(4,0){\line(0,1){4}} \put(0,4){\line(1,0){4}} \put(6,0){\line(0,1){2}} \put(0,6){\line(1,0){2}} \end{picture}}\ =\ 2, \qquad\mbox{and}\qquad c^\zeta_{\,\begin{picture}(4,6)(0,0) \put(0,0){\line(1,0){4}} \put(0,0){\line(0,1){6}} \put(0,2){\line(1,0){4}} \put(2,0){\line(0,1){6}} \put(0,4){\line(1,0){4}} \put(4,0){\line(0,1){6}} \put(0,6){\line(1,0){4}} \end{picture}}\ =\ 1, $$ which verifies Proposition~\ref{prop:chains} as $f^{\,\begin{picture}(6,4) \put(0,0){\line(0,1){4}} \put(0,0){\line(1,0){6}} \put(2,0){\line(0,1){4}} \put(0,2){\line(1,0){6}} \put(4,0){\line(0,1){4}} \put(0,4){\line(1,0){6}} \put(6,0){\line(0,1){4}} \end{picture}}=5$, $f^{\,\begin{picture}(6,6) \put(0,0){\line(0,1){6}} \put(0,0){\line(1,0){6}} \put(2,0){\line(0,1){6}} \put(0,2){\line(1,0){6}} \put(4,0){\line(0,1){4}} \put(0,4){\line(1,0){4}} \put(6,0){\line(0,1){2}} \put(0,6){\line(1,0){2}} \end{picture}}=16$, and $f^{\,\begin{picture}(6,4) \put(0,0){\line(0,1){4}} \put(0,0){\line(1,0){6}} \put(2,0){\line(0,1){4}} \put(0,2){\line(1,0){6}} \put(4,0){\line(0,1){4}} \put(0,4){\line(1,0){6}} \put(6,0){\line(0,1){4}} \end{picture}}=5$. In this case, $\theta = \begin{picture}(12,12) \put(6,0){\line(1,0){6}} \put(0,6){\line(0,1){6}} \put(6,3){\line(1,0){6}} \put(3,6){\line(0,1){6}} \put(0,6){\line(1,0){9}} \put(6,0){\line(0,1){9}} \put(0,9){\line(1,0){6}} \put(9,0){\line(0,1){6}} \put(0,12){\line(1,0){3}} \put(12,0){\line(0,1){3}} \end{picture}\,$. Since ${\rm up}_\zeta=\{1,2,4\}$ and ${\rm down}_\zeta=\{6,5,3\}$, we see that $a=b=3$, however $\theta\not\subset \begin{picture}(9,9) \put(0,0){\line(1,0){9}} \put(0,0){\line(0,1){9}} \put(0,3){\line(1,0){9}} \put(3,0){\line(0,1){9}} \put(0,6){\line(1,0){9}} \put(6,0){\line(0,1){9}} \put(0,9){\line(1,0){9}} \put(9,0){\line(0,1){9}} \end{picture}\, = a^b$. A combinatorial interpretation of the Littlewood-Richardson coefficients $c^w_{u\,v(\lambda,k)}$ should also give a bijective proof of Proposition~\ref{prop:chains}. We show a partial converse to this, that a function $\tau$ from chains to standard Young tableaux satisfying some extra conditions will provide a combinatorial interpretation of the Littlewood-Richardson coefficients $c^w_{u\,v(\lambda,k)}$. Let $\mbox{ch}[u,w]_k$ denote the set of (saturated) chains in the interval $[u,w]_k$. For a partition $\mu$ and integer $m$, let $\mu*m$ be the set of partitions $\lambda$ with $\lambda-\mu$ a horizontal strip of length $m$. These partitions arise in the classical Pieri's formula: $$ S_\mu(x_1,\ldots,x_k)\cdot h_m(x_1,\ldots,x_k) \ =\ \sum_{\lambda\in \mu*m} S_\lambda(x_1,\ldots,x_k). $$ If $T$ is a standard tableau of shape $\mu$ and $m$ and integer, let $T*m$ be the set of tableaux $U$ which contain $T$ as an initial segment such that $U-T$ is a horizontal strip whose entries increase from left to right. \begin{thm}\label{thm:combinatorial} Suppose that for every $u\leq_k w$, there is a map \begin{eqnarray*} \mbox{ch}[u,w]_k& \longrightarrow& \left\{ \mbox{\begin{minipage}[c]{2.5in} \begin{center} Standard Young tableau $T$ whose shape is a partition of $\ell(w)-\ell(u)$\end{center}\end{minipage}}\right\}\\ \gamma&\longmapsto& \tau(\gamma) \end{eqnarray*} such that \begin{enumerate} \item $d^w_{u\:v(\lambda,k)}:= \#\{\gamma\in\mbox{ch}[u,w]_k\:|\: \tau(\gamma)=T\}$ depends only upon the shape $\lambda$ of the standard tableau $T$. \item If $\gamma = \delta.\varepsilon$ is the concatenation of two chains $\delta$ and $\varepsilon$, then $\tau(\delta)$ is a subtableau of $\tau(\gamma)$. (This means that $\tau(\gamma)$ is a recording tableau.) \item Suppose $\gamma = \delta.\varepsilon$ with $\delta\in\mbox{ch}[u,x]_k$, and hence $\varepsilon\in\mbox{ch}[x,w]_k$. Then $\tau(\delta.\varepsilon)\in \tau(\delta)*m$ only if $x\rkm w$, and $\varepsilon(\delta):=\varepsilon\in\mbox{ch}[x,w]_k$ is unique for this to occur. \end{enumerate} Then, for every standard tableau $T$ of shape $\lambda$ and $u\leq_k w$, $$ c^w_{u\;v(\lambda,k)}\ =\ d^w_{u\:v(\lambda,k)}. $$ \end{thm} Such a map $\tau$ is a generalization of Schensted insertion. In that respect, the existence of such a map would generalize Theorem~\ref{thm:skew_shape_prime}. \medskip \noindent{\bf Proof. } We induct on $\lambda$. Assume the theorem holds for all $u,w$, and partitions $\pi$ either with fewer rows than $\lambda$, or if $\lambda$ and $\pi$ have the same number of rows, then the last row of $\pi$ is shorter than the last row of $\lambda$. The form of the Pieri-type formulas expressed in~\cite{sottile_pieri_schubert,Winkel_multiplication} (also \S\ref{sec:emdedd}) and condition (3) prove the theorem when $\lambda$ consists of a single row. Assume that $\lambda$ has more than one row and set $\mu$ to be $\lambda$ with its last row removed. Let $m$ be the length of the last row of $\lambda$ and $T$ be any tableau of shape $\mu$. Recall that $U\mapsto \mbox{shape}(U)$ gives a one-to-one correspondence between $T*m$ and $\mu*m$. By the definition of $c^y_{u\; v(\mu,k)}$, we have $$ {\frak S}_u\cdot S_\mu(x_1,\ldots,x_k)\ =\ \sum_{u\leq_k y}c^y_{u\; v(\mu,k)}\: {\frak S}_y. $$ By the Pieri formula for Schubert polynomials, $$ {\frak S}_u\cdot S_\mu(x_1,\ldots,x_k)\cdot h_m(x_1,\ldots,x_k)\ =\ \sum_w\left( \sum_{\stackrel{\mbox{\scriptsize $u\leq_k y$}}{y\rrkm w}} c^y_{u\; v(\mu,k)}\right) {\frak S}_w. $$ By the classical Pieri formula, this also equals $$ {\frak S}_u\cdot \sum_{\pi\in \mu*m}S_{\pi}(x_1,\ldots,x_k) \ =\ \sum_w\left(\sum_{\pi\in \mu*m} c^w_{u\; v(\pi,k)}\right) {\frak S}_w. $$ Hence $$ \sum_{\pi\in \mu*m}c^w_{u\; v(\pi,k)}\ =\ \sum_{\stackrel{\mbox{\scriptsize $u\leq_k y$}}{y\rrkm w}} c^y_{u\; v(\mu,k)}. $$ We exhibit a bijection between the two sets $$ M_{T,{k,m}}\ :=\ \coprod_{\stackrel{\mbox{\scriptsize $u\leq_k y$}}{y\rrkm w}} \{ \delta\in\mbox{ch}[u,y]_k\: |\: \tau(\delta)=T\} $$ and $\coprod_{\pi\in\mu*m}L_{\pi}$, where $$ L_{\pi}\ :=\ \{ \gamma\in\mbox{ch}[u,w]_k\: |\; \tau(\gamma)\in T*m \mbox{ and $\tau(\gamma)$ has shape $\pi$}\}. $$ This will complete the proof. Indeed, by the induction hypothesis $$ \#M_{T,{k,m}}\ =\ \sum_{\stackrel{\mbox{\scriptsize $u\leq_k y$}}{y\rrkm w}} c^y_{u\:v(\mu,k)} $$ and for $\pi\in \mu*m$ with $\pi\neq \lambda$, $$ \#L_{\pi}\ = \ c^w_{u\:v(\pi,k)}. $$ Thus the bijection shows $$ c^w_{u\:v(\lambda,k)}\ =\ \sum_{\stackrel{\mbox{\scriptsize $u\leq_k y$}}{y\rrkm w}} c^y_{u\:v(\mu,k)} - \sum_{\pi\in\mu*m,\ \pi\neq \lambda} c^w_{u\:v(\pi,k)}\ =\ \#L_{\lambda}, $$ which is $\#\tau^{-1}(U)$, for any $U$ of shape $\lambda$. To construct the desired bijection, consider first the map $$ M_{T,{k,m}}\ \longrightarrow\ \coprod_{\pi\in\mu*m} L_\pi $$ defined by $\delta\in\mbox{ch}[u,y]_k \mapsto \delta.\varepsilon(\delta)$. By property 3, $\tau(\delta.\varepsilon(\delta))\in T*m$, so this injective map has the stated range. To see it is surjective, let $\pi\in \mu*m$ and $\gamma\in L_\pi$. Let $\delta$ be the first $|\mu|$ steps in the chain $\gamma$, so that $\gamma = \delta.\varepsilon$ and suppose $\delta\in\mbox{ch}[u,y]_k$. Then $\tau(\delta)=T$ so $\tau(\delta.\varepsilon)\in \tau(\delta)*m$. By 3, this implies $y\rkm w$, and hence $\delta\in M_{T,{k,m}}$. \QEDnoskip

9703

alg-geom/9703028

en

https://arxiv.org/abs/alg-geom/9703028

alg-geom/9703028

Jim Alexander

J. Alexander, A. Hirschowitz

Interpolation on Jets

5 pages, Latex2e

We show that for any finite generic union of pairs $(x_i,L_i)_i$ where $x_i\in L_i$ is a point of the line $L_i$ in projective $n$-space, the divisors $m_ix_i$ on the $L_i$ have maximal rank with respect to homogeneous $d$ forms for all $d\geq 0$ and all $m_i\geq 0$ modulo the expected numerical restrictions.

alg-geom

\section{Introduction} We work over a field $k$ of characteristic zero. A {\em jet} in $n$-dimensional projective space $\mbox{{\tenmsbm P}}_{k}^{\, n}$ over $k$ will be any divisor on a one dimensional linear subspace (i.e. a line in $\mbox{{\tenmsbm P}}_{k}^{\, n}$) with support a point. The {\em length} of a jet will be its degree as a divisor and we will say $r$-{\em jet} for a jet of length $r$. When $1< r$, an $r$-jet is contained in a unique line, which we call its {\em axis}. As for $r = 1$, we will distinguish $1$-jets, which are furnished with an axis (namely a line containing them) from (free) points. Finally, by convention, a $0$-jet is the empty subscheme. We say that a closed sub-scheme $Y\hookrightarrow \mbox{{\tenmsbm P}}_{k}^{n}$ has {\em maximal rank in degree} $d\geq 0$ if the canonical map $H^{0}(\mbox{{\tenmsbm P}}_{k}^{\, n},\mbox{{\teneusm O}}_{\mbox{{\tenmsbm P}}^{\, n}}(d)) \longrightarrow H^{0}(Y,\mbox{{\teneusm O}}_{Y}(d))$ has maximal rank as a linear map. Our theorem 1.2 slightly refines a result proved by A. Eastwood in [E1,2] characterizing those generic unions of jets having maximal rank. The proof in [1, 2] used complex specialisation arguments and his proof runs to some sixty pages. Our proof is independent of this earlier one and much simpler. It works by an induction arguement based on the simple proposition 2.1, which allows us to reduce to the extremal case where the sequence of lengths is maximal (among the allowed ones) with respect to the lexicographical order. This case can then be treated by elementary techniques when $n=2$ and for $n\geq 3$, using, as was done in [1, 2], using an old theorem of Hartshorne and the second author [3]. This theorem says that generic unions of lines have maximal rank in any degree. \begin{defn}We will say that a set $S$ of lines is {\em in general position} if for any subset of $S$, the corresponding union has maximal rank in any degree.\end{defn} Observe that in $\mbox{{\tenmsbm P}}^{\, 2}$, any finite set of lines is in general position, while in $\mbox{{\tenmsbm P}}^{\, n}$, it follows from the above mentioned result that any generic finite set of lines is in general position. \begin{thm} Let $L = (L_{1},\ldots , L_{m})$ be a sequence of $m$ lines in general position in $\mbox{{\tenmsbm P}}_{k}^{\, n}$, $r = (r_1, ..., r_m)$ be a sequence of positive integers in non-increasing order and $d$ an integer such that the sum $r_{1}+\cdots + r_{m}$ is at most $\scriptsize{\left(\begin{array}{c} n+d \\ d\end{array}\right)}$. Then the union $J_1 \cup ... \cup J_m$ where $J_i$ is the generic $r_i$-jet on $L_i$ has maximal rank in degree $d$ if and only if the following (necessary) numerical condition $C(n,d)$ holds: $r_{1}\leq d+1$ and, if $n=2$, then for any $1\leq s \leq d+1$, $$\mbox{$r_{1}+\cdots + r_{s}\leq d\, .\, s+1-$}\scriptsize{\left(\begin{array}{c} s-1 \\ 2 \end{array}\right)}.$$ \end{thm} In other words, if one fixes the degree $d$, then for given lines $L_1,\ldots ,L_m\in \mbox{{\tenmsbm P}}^n$ in general position, the $\Sigma_i r_i$ conditions imposed on hypersurfaces of degree $d$ by requiring a contact of order $r_i$ at the generic point $x_i$ of $L_i$ for $i=1,\ldots ,m$, are linearly independent precisely under the expected numerical conditions given in the theorem. Before passing on to the proof, we introduce the following notation. \begin{defn} For a fixed $n$, we say that a sequence of non-negative integers $(\raisebox{.4ex}{$\chi$} ,r_1,\ldots ,r_m)$ is $d$-{\em admissible} if \newcounter{mist0} \begin{list}{\textbf{ (\roman{mist0}) }} {\setlength{\leftmargin}{1cm} \setlength{\rightmargin}{1cm} \usecounter{mist0} } \item $\mbox{$\raisebox{.4ex}{$\chi$} + r_1\cdots + r_m$}=\scriptsize{\left(\begin{array}{c}n+d \\d\end{array}\right)}$ \item $r_1\geq \cdots \geq r_m \geq 1$ \item if $n=2$ then $$r_1+\cdots r_s\leq ds+1-(s-1)(s-2)/2$$ for $s=1,\ldots ,d+1$. \end{list} and we denote by $S_d$ the set of all $d$-admissible sequences, which we give the total lexicographical order (i.e. $(\alpha_i)_i> (\beta_i)_i$ if the first non-zero term in the sequence $(\alpha_i-\beta_i)_i$ is strictly positive). \end{defn} \begin{defn} Given a $d$-admissible sequence $\mbox{{\boldmath $r$}}=(\raisebox{.4ex}{$\chi$} , r_1,\ldots ,r_m)$ and a sequence of lines in general position $L = (L_{1},\ldots , L_{m})$ it is clear that the union $J_1 \cup \cdots \cup J_m$ of the generic $r_i$-jets $J_i$ in $L_i$, has maximal rank in degree $d$ if and only if the union $J := J_{L,r}=J_1 \cup \cdots. \cup J_m \cup R$ has maximal rank, where $R$ is the union of $\raisebox{.4ex}{$\chi$}$ generic (free) points. We call such a union $J$ {\em a $d$-admissible union of jets} and we say that $$\mbox{{\boldmath $r$}}=(\raisebox{.4ex}{$\chi$} , r_1,\ldots ,r_m)$$ is the weight of $J$. \end{defn} In this language, theorem 1.2 simply says that every $d$-admissible union of jets has maximal rank in degree $d$. Without further comment, we will freely use the fact that `` maximal rank in degree $d$ '' is stable by generisation. We express our gratitude to the referee for having suggested changes which have improved the presentation of the proof. \section{The proof} \begin{prop} Fix a closed subscheme $Y\subset \mbox{{\tenmsbm P}}_{k}^{\, n}$ and let $D$ be a line not contained in $Y$. Suppose that for some $d, v >0$ the union of $Y$ with the generic jet of length $v+1$ (resp $v-1$) in $D$ has maximal rank in degree $d$, then the union of $Y$ with the generic $v$-jet in $D$ has maximal rank in degree $d$. \end{prop} \noindent{\bf Proof.} We denote by $Y_r$ the union of $Y$ with the generic $r$-jet; denoted $J_{r}$; in $D$ and by $m_r$ the canonical map $$H^0(\mbox{{\tenmsbm P}}^n,\mbox{{\teneusm O}}_{\mbox{{\ninemsbm P}}^n}(d)) \longrightarrow H^0(\mbox{{\tenmsbm P}}^n,\mbox{{\teneusm O}}_{Y}(d))\times H^0(\mbox{{\tenmsbm P}}^n,\mbox{{\teneusm O}}_{J_{r}}(d))\, =\, H^0(\mbox{{\tenmsbm P}}^n,\mbox{{\teneusm O}}_{Y_r}(d))$$ If $m_{v+1}$ is surjective, then so is $m_v$, and if $m_{v-1}$ is injective, then so is $m_v$. In the remaining case, $m_{v+1}$ is injective and not surjective, and $m_{v-1}$ is surjective and not injective and we have to prove that $m_v$ is bijective. What is clear is that $h^0(\mbox{{\tenmsbm P}}^n, I_Y(d))=v$, where $I_Y$ is the ideal sheaf of $Y$ as a subscheme of $\mbox{{\tenmsbm P}}^n$, and that the canonical map $$H^0(\mbox{{\tenmsbm P}}^n, I_Y(d))\hookrightarrow H^0(\mbox{{\tenmsbm P}}^n, \mbox{{\teneusm O}}_{J_{v+1}}(d))$$ is injective. It follows that the map $$H^0(\mbox{{\tenmsbm P}}^n, I_Y(d))\hookrightarrow H^0(D, \mbox{{\teneusm O}}_{D}(d))$$ is injective. If $W$ is the image subspace of this latter map we need only show that the generic $v$-jet in $D$ imposes linearly independent conditions on $W$, but this is true for any $v$ dimensional subspace of $H^0(D,\mbox{{\teneusm O}}_{D}(d))$. This follows from the well known fact that a familly of polynomials $f_1,\ldots ,f_v\in k[t]$ is linearly independent if and only if the $v\times v$ Wronskian $$\begin{array}{lll}W(f_1,\ldots ,f_v)&=&\left[\begin{array}{l}\partial^i f_j \vspace{.5ex}\\ \hline\vspace{-2ex} \\ \partial t^i \end{array}\right]\end{array}$$ has rank $v$. \subsection{Proof of theorem 1.2 for $\mbox{{\tenmsbm P}}^{2}$} As indicated after definition 1.4, we will prove that every $d$-admissible union of jets has maximal rank in degree $d$. We will write a $d$-admissible weight $\mbox{{\boldmath $r$}}=(\raisebox{.4ex}{$\chi$},\sigma_1,\ldots ,\sigma_m)$ in the form $$\mbox{{\boldmath $r$}}=(\raisebox{.4ex}{$\chi$},m_1,\ldots ,m_p,r_1,\ldots ,r_q)$$ where $m_1,\ldots ,m_p$ is the extremal sequence $d+1,\ldots ,d+2-p$ given by the condition 1.3 (iii) and $r_1< d+1-p$. Fix a $d$-admissible union of jets $J$ of weight $$\mbox{\boldmath $r$}=(\raisebox{.4ex}{$\chi$},m_1,\ldots , m_p,r_1,\ldots , r_q)$$ and suppose that every $d$-admissible union of jets $J^{\prime}$ of weight $\mbox{{\boldmath $r$}}^{\prime}>\mbox{{\boldmath $r$}}$ has maximal rank in degree $d$. We will show that this implies that $J$ has maximal rank in degree $d$ completing the proof. If $q\leq 1$ (i.e. $r_2=0$) then by specialising the $\raisebox{.4ex}{$\chi$}$ free points, we can specialise $J$ to the $d$-admissible union of jets $J_d$ of weight $(0,d+1,\ldots ,1)$. A simple induction then shows that any homogeneous form of degree $d$ on $\mbox{{\tenmsbm P}}^2$ which vanishes on $J_d$ vanishes on all $d+1$ axes and is thus identically zero. This shows that $J$ has maximal rank in degree $d$ if $r_2=0$. Now suppose that $r_2>0$. We will apply 2.1 with $$Y=J_1\cup\cdots \cup J_p\cup J^{\prime\prime}_2\cup \cdots \cup J_q^{\prime\prime}\cup R$$ where $J_i$ is the $m_i$-jet, $J_i^{\prime\prime}$ is the $r_i$-jet and $R$ is the union of the $\raisebox{.4ex}{$\chi$}$ free points. We must show that the unions $J_{(+)}$ and $J_{(-)}$ of $Y$ with the jet of length $r_1+1$ (resp. $r_1-1$) extending (resp. contracting) $J_1^{\prime\prime}$ have maximal rank in degree $d$. On the one hand, $J_{(-)}$ is contained in the union $\widetilde{J}_{(-)}$ obtained by adding a generic free point to $J_{(-)}$. Now $\widetilde{J}_{(-)}$ is clearly a $d$-admissible union of jets of weight $\widetilde{\mbox{{\boldmath$r$}}}_{(-)}> \mbox{{\boldmath$r$}}$ so that $J_{(-)}$ has maximal rank in degree $d$ by the induction hypothesis. On the other hand, $J_{(+)}$ contains the union of jets $\widetilde{J}_{(+)}$ obtained by contracting $J_2^{\prime\prime}$ to length $r_2-1$ and it will suffice to show that the weight $$\widetilde{\mbox{{\boldmath$r$}}}_{(+)} =(\raisebox{.4ex}{$\chi$},m_1,\ldots ,m_p,r_1+1,r_2,\ldots ,r_{t-1},r_t-1,r_{t+1},\ldots ,r_q)$$ of $\widetilde{J}_{(+)}$ is $d$-admissible, where $r_2=\cdots = r_t>r_{t+1}$. This is equivalent to showing that the series of inequalities $$r_1 + (s-1)r_2 \leq (d-p)s+1-(s-1)(s-2)/2$$ for $1\leq s\leq \mbox{min}\, (t,d-p+1)$ implies strict inequality for $1\leq s\leq \mbox{min}\, (t-1,d-p+1)$. Since the associated quadratic form $$Q(s)=s^2 -(3+2(d-p-r_2)) s + 2(r_1-r_2)$$ vanishes on the interval $[ 0,1[$, there is only one root $\geq 1$. This completes the proof if $t\leq d-p+1$ or if $Q$ has no integer roots $\leq d-p+1$. We will now finish by showing that if $t> d-p+1$ then $Q$ does not have integer roots. In fact if $Q$ has integer roots, they must be $0$ and $s_1=3+2(d-p-r_2)\leq d-p+1$. In this case $r_2\geq (d-p+2)/2$. However we have $$tr_2\leq r_1+(t-1)r_2\leq r_1+\ldots + r_t\leq (d-p+2)(d-p+1)/2$$ showing that $t\leq d-p+1$.$\;\;\;\;\;\;\;\Box$ \subsection{Proof of theorem 1 for $\mbox{{\tenmsbm P}}^{n}$, $n\geq 3$.} The proof is similar to the previous one. We write the weight $\mbox{{\boldmath $r$}}$ of a $d$-admissible union of jets $J$ in the form $$\mbox{{\boldmath $r$}}=(\raisebox{.4ex}{$\chi$}, m_1,\ldots , m_p,r_1,\ldots , r_q)$$ where $m_i=d+1$ and $r_1\leq d$. Now let $J$ be a $d$-admissible union of jets of weight $\mbox{{\boldmath $r$}}$ and suppose that every $d$-admissible union of jets $J^{\prime}$ of weight $\mbox{{\boldmath $r$}}^{\prime}> \mbox{{\boldmath $r$}}$ has maximal rank in degree $d$. We will show that this implies that $J$ has maximal rank in degree $d$. If $r_2\neq 0$ one easily applies 2.1 as before. So suppose that $r_2=0$. By specialising the $\raisebox{.4ex}{$\chi$}$ free points firstly to the $r_1$-jet then to further lines in general position we can specialise $J$ to a $d$-admissible union of jets of weight $\mbox{{\boldmath $r$}}=(0,m_1,\ldots ,m_p,\delta)$ where $0\leq \delta\leq d$. This is equivalent to the corresponding problem obtained by replacing the $p$, $(d+1)$-jets by their corresponding axes. That is to say, we must show that if $D_1,\ldots,D_{p+1}$ are $p+1$ lines in general position, then the union of $D_1,\ldots ,D_p$ with the generic $\delta$-jet in $D_{p+1}$ has maximal rank in degree $d$. The general position hypothesis implies that $h^0(\mbox{{\tenmsbm P}}^n,I_{D_1\cup\, \cdots \, \cup D_p}(d))\; =\; \delta$ and that the canonical map $$H^0(\mbox{{\tenmsbm P}}^n,I_{D_1\cup\, \cdots\, \cup D_p}(d))\longrightarrow H^0(D_{p+1}, \mbox{{\teneusm O}}_{D_{p+1}}(d))$$ is injectif. This completes the proof as in the concluding arguement of the proof of proposition 2.1. $\;\;\;\;\;\;\;\;\Box$\vspace{2ex}\\ Note: our method would work equally well for any union of curves: from a maximal rank statement concerning a union of curves, we derive a maximal rank statement for unions of generic jets on these curves, under natural necessary numerical conditions. \noindent{\Large\bf Bibliography.}\\ \begin{enumerate} \item Eastwood A.: {\em Collision de biais et interpolation} Manuscr. Math. 67 (1990), 227-249. \item Eastwood A. : {\em Interpolation \`a N variables } J. of Algebra, 139 (1991) 273-310. \item Hartshorne R., Hirschowitz A. : {\em Droites en position g\'en\'erale}, in Proceedings La Rabida 1981, LNM 961, 169-189. \item Hirschowitz A. : {\em Probl\`emes de Brill-Noether en rang sup\'erieur} Pr\'eprint Nice 1986. \end{enumerate} $\;$\vspace{1cm}\\ \begin{minipage}[t]{.4\linewidth}{James Alexander\\ Universit\'e d'Angers\\ [email protected]}\end{minipage}\hfill \begin{minipage}[t]{.4\linewidth}{Andr\'e Hirschowitz\\ Universit\'e de Nice\\ [email protected]}\end{minipage} \end{document}

9703

alg-geom/9703013

en

https://arxiv.org/abs/alg-geom/9703013

alg-geom/9703013

Lars Ernstr{\o}m

Lars Ernstr\"om and Gary Kennedy

Contact Cohomology of the Projective Plane

18 pages AMSLaTeX v 2e with xy-pic v 3.2; minor revison

We construct an associative ring which is a deformation of the quantum cohomology ring of the projective plane. Just as the quantum cohomology encodes the incidence characteristic numbers of rational plane curves, the contact cohomology encodes the tangency characteristic numbers.

alg-geom

\section{Introduction} \label{intro} \par In this paper we construct an associative ring which we call the {\em contact cohomology ring} of the projective plane. We believe that an analogous construction should work for all homogeneous varieties, but in our proof of associativity we rely on certain technical results from our earlier paper on characteristic numbers of rational plane curves \cite{ErnstromKennedy}. \par As we formulate it in section \ref{qtp} of the present paper, the quantum product is actually a whole family of products parametrized by elements of the Chow ring $\ap$. Each product is defined on the formal power series ring $\apt$ in one variable, and encodes the characteristic numbers \begin{equation*} N_d= \text{ the number of rational plane curves of degree $d$ through $3d-1$ general points}. \end{equation*} The remarkable fact about the quantum product is that it is associative. This fact, together with the triviality $N_1=1$, suffices to determine all values of $N_d$. \par The contact products are defined on the same formal power series ring. But now the parameter space is the Chow ring $A^*I$ of the incidence variety of points and lines in $\pp$, and these products encode (as we explain in section \ref{rrc}) a larger collection of characteristic numbers: \begin{align*} N_d(a,b,c)=&\text{ the number of rational plane curves of degree $d$ through $a$ general points, tangent to}\\ &\text{$b$ general lines, and tangent to $c$ general lines at a specified general point on each line}\\ &\quad\text{(where $a+b+2c=3d-1$).} \end{align*} Section \ref{qcp} is devoted to defining these products precisely and to showing that they are likewise associative. As we explain in section \ref{rrc}, the associativity implies a remarkable recursive relation among the characteristic numbers, but it is insufficient to determine all their values unless one already knows all values $N_d(a,b,c)$ for which $a<3$. (Our previous paper \cite{ErnstromKennedy} explains how to obtain this additional information.) \par The associativity of the quantum product is a consequence of the recursive structure of the boundary divisors on the moduli space of stable maps to $\pp$ (of a given degree and with a given number of markings). In brief, each boundary divisor is a fiber product of simpler instances of the same sort of moduli space. But in studying questions of tangency it is natural to look at a moduli space of stable lifts (defined in section \ref{siv}), whose boundary divisors have a somewhat more complicated structure. To understand these divisors, consider a family of immersions $\pl \to \pp$. Associated to each immersion is its lift, a map from $\pl$ to the incidence correspondence $I$. Now suppose that the family of immersions degenerates to a map from a two-component curve onto the union of two plane curves of lower degree. Then one can show that the family of lifts degenerates to a map from a three-component curve, with the central component mapping two-to-one onto the fiber of $I$ over a point of $\pp$. Thus to create a moduli space with a recursive structure we form a fiber product of the space of stable lifts with a space of two-to-one covers of fibers of $I$ over $\pp$; we then exploit this recursive structure to define our associative product. \par Our recursive relation specializes to that of Di~Francesco and Itzykson \cite{FrancescoItzykson}, who also interpret their formula as the associativity of a certain product. We thank S.~Colley for carefully reading earlier versions of this paper and making many helpful suggestions. We also want to indicate our indebtedness to B.~Fantechi, W.~Fulton, L.~G{\"o}ttsche and R.~Pandharipande. \par We should warn the reader of a potentially confusing clash in terminology: in symplectic geometry Y.~Eliashberg and H.~Hofer have introduced a new invariant of contact manifolds called ``contact homology.'' The Gromov-Witten invariants of a symplectic manifold with contact boundary take values in the contact homology of the boundary. \par \section{A ``push-pull'' formula} \label{ppf} \par For an algebraic variety, scheme, or algebraic stack $X$, we write $A_*X$ for the rational equivalence group with coefficients in $\q$, and call it the {\em homology}; we write $A^*X$ for the operational cohomology ring. (See \cite{Fulton} for the case of a variety or scheme, \cite{Vistoli} for intersection theory on stacks.) We will often use the fact that $A_*(X)$ and $A^*(X)$ of a nonsingular variety $X$ are naturally isomorphic. We record here a formula which we will use repeatedly. It is the stack version of \cite[Proposition 1.7]{Fulton}. \begin{lem} \label{pushpull} Suppose that $$ \xymatrix{ & M_1\times_X M_2 \ar[dl]^{q_1} \ar[dr]_{q_2} \\ M_1 \ar[dr]^{g_1} & & M_2 \ar[dl]_{g_2} \\ & X } $$ is a fiber square of stacks over a nonsingular variety $X$, with $g_2$ flat and $g_1$ proper. Then $q_1$ is flat, $q_2$ is proper, and for each class $\alpha \in A^*M_1$ we have $$ q_{2*}\left(q_1^*\alpha\cap\lbrack M_1 \times_X M_2\rbrack\right) =g_2^*g_{1*}(\alpha\cap\lbrack M_1 \rbrack)\cap\lbrack M_2 \rbrack. $$ \end{lem} \par \section{The quantum product} \label{qtp} Here we recall the basic definitions of quantum cohomology and prove that the quantum product on $\pp$ is associative. Our proof is basically the same as in section 8 of \cite{FultonP}, except that we avoid the use of coordinates; we use this proof as a prototype for the longer proof in section \ref{qcp}. As general references for the material in this section we suggest \cite{BehrendManin}, \cite{ErnstromKennedy}, \cite{FultonP}, \cite{Kontsevich}, \cite{KontsevichManin}, and \cite{LiTian}. \par Let $\pmoduli{n+3}{d}$ denote the stack of stable maps from curves of arithmetic genus $0$, with $n+3$ markings, to the projective plane. Let $i,j,k:\pmoduli{n+3}{d}\to \pp$ be the evaluation maps associated to the first three markings; let $e_1,\dots,e_n$ be the others. Let $\ap$ be the Chow cohomology ring of the plane with rational coefficients, and let $\apt$ be the ring of formal power series in one variable. Suppose that $\alpha$, $\beta$, and $\delta$ are elements of $\ap$. Then the {\em quantum product of $\alpha$ and $\beta$, deformed by $\delta$}, is the element of $\apt$ whose $n$th coefficient is \begin{equation} \label{qtpn} \frac{1}{n!} \sum_{d \geq 0} k_*(i^*\alpha \cup j^*\beta \cup {\bigcup_{t=1}^n} e_t^*\delta \cap \lbrack \pmoduli{n+3}{d} \rbrack ). \end{equation} Note that, since the dimension of $\pmoduli{n+3}{d}$ is $3d+2+n$, the sum is finite. We will denote the quantum product by $(\alpha * \beta)_{\delta}$ or simply $\alpha * \beta$. Extending by $\q[[T]]$-linearity, we have a product on $\apt$. We call this ring the {\em quantum cohomology} of $\pp$, and denote it by $QH^*(\pp)$. \begin{prop} For each $\delta$ in $\ap$, the quantum product is commutative and associative. The identity element $1 \in \ap$ for the ordinary cup product also serves as the identity element for the quantum product. \end{prop} \begin{proof} The commutativity is obvious. Unless $n=d=0$ the class $$ i^*1 \cup j^*\beta \cup {\bigcup_{t=1}^n} e_t^*\delta $$ is the pullback via the forgetful morphism $\pmoduli{n+3}{d} \to \pmoduli{n+2}{d}$ of a class on the latter space. Since the fibers of this morphism have positive dimension, the projection formula tells us that the corresponding term of (\ref{qtpn}) vanishes. As for the remaining term $n=d=0$, note that in this case $i$, $j$, and $k$ are all the same isomorphism from $\pmoduli{3}{0}$ to $\pp$. Hence $1$ is the identity element for the quantum product. \par To see that the quantum product is associative, consider the moduli space $\pmoduli{n+4}{d}$. Denote the first four evaluation maps $\pmoduli{n+4}{d} \to \pp$ by $i$, $j$, $k$, and $l$, and the remaining $n$ maps by $e_t$. Consider the ``forgetful'' morphism from $\pmoduli{n+4}{d}$ to $\stablefour$, the space of stable 4-pointed curves, which associates to a stable map its source curve together with the first four markings, with any unstable components contracted to a point. The space $\stablefour$ is isomorphic to $\pl$. It has a distinguished point $P(12\mid 34)$ representing the two-component curve having the first two markings on one component and the latter two on the other; similarly there are two other distinguished points $P(13\mid 24)$ and $P(14\mid 23)$. Hence on $\pmoduli{n+4}{d}$ there are three linearly equivalent divisors $D(12\mid 34)$, $D(13\mid 24)$, and $D(14\mid 23)$, which we call the {\em special boundary divisors}. \par Kontsevich \cite{Kontsevich} (cf. \cite{FultonP}) identifies the components of $D(12\mid 34)$. For a finite set $A$, let $\pmoduli{A}{d}$ denote the stack of stable maps with markings labeled by $A$. Suppose that $A_1 \cup A_2$ is a partition of $\{1,\dots,n+4\}$ and that $d_1+d_2=d$. Suppose that $\{\star\}$ is a single-element set. Then the fiber product $$ D(A_1,A_2 ,d_1,d_2) = \pmoduli{A_1\cup\{\star\}}{d_1} \times_{\pp} \pmoduli{A_2\cup\{\star\}}{d_2} $$ is naturally a substack of $\pmoduli{n+4}{d}$; the typical point represents a map from a curve with two components, with the point of attachment corresponding to the point labeled by $\star$, as indicated in Figure 3.1. The divisor $D(12\mid 34)$ is the sum \begin{equation} \label{donetwo} D(12\mid 34)=\sum D(A_1,A_2 ,d_1,d_2) \end{equation} over all partitions $A_1 \cup A_2$ in which $1$ and $2$ belong to $A_1$, and $3$ and $4$ belong to $A_2$. There are corresponding statements for the other two special boundary divisors. \par $$ \xy 0;<1cm,0cm>: (-1,-1);(1,0.25)**\dir{-}; (-1,1);(1,-0.25)**\dir{-}; (0.9,0)*{\star}; (0.6,0)*{\bullet}; (-0.467,0.667)*{\bullet}; (0.067,0.333)*{\bullet}; (-0.6,-0.75)*{\bullet}; (-0.2,-0.5)*{\bullet}; (0.2,-0.25)*{\bullet}; (-2.5,0.5)*{\text{markings labeled by }A_1}; (-2.5,-0.5)*{\text{markings labeled by }A_2}; (1.5,0);(3.5,0)**\dir{-}; ?>*\dir{>}; (4,-0.5);(6.5,0.75)**\dir{-}; (5,0)*{\bullet}; (6.25,0.625)*{\bullet}; (4,1);(6,1)**\crv{(5,-3)}\POS?(0.333)*{\bullet}\POS?(0.65)*{\bullet}\POS?(0.8)*{\bullet}; (4.41,-0.3)*{\bullet}; \endxy $$ \begin{center} {\bf Figure 3.1} Typical member of $D(A_1,A_2 ,d_1,d_2)$. \end{center} \smallskip \par Since the divisors $D(12\mid 34)$ and $D(14\mid 23)$ are linearly equivalent, the equation \begin{multline} \label{transeq} \frac{1}{n!} \sum_{d \geq 0} l_*(i^*\alpha \cup j^*\beta \cup k^*\gamma \cup {\bigcup_{t=1}^n} e_t^*\delta \cap \lbrack D(12\mid 34) \rbrack ) \\ = \frac{1}{n!} \sum_{d \geq 0} l_*(i^*\alpha \cup j^*\beta \cup k^*\gamma \cup {\bigcup_{t=1}^n} e_t^*\delta \cap \lbrack D(14\mid 23) \rbrack ). \end{multline} is valid for each triple $\alpha$, $\beta$, $\gamma$ of classes in $\ap$ and for each $n$. According to (\ref{donetwo}), the divisor $D(12\mid 34)$ is a sum of fiber products, each of which fits into a fiber diagram $$ \xymatrix{ & & \pmoduli{n_1+3}{d_1}\times_{\pp}\pmoduli{n_2+3}{d_2} \ar[ddl]^{q_1} \ar[ddr]_{q_2} & \\ \\ & \pmoduli{n_1+3}{d_1} \ar@/^/[dd]^{e_{1t}} \ar@/_/[dd]_{i_1}\ar[dd]|{j_1} \ar[ddr]^{g_1} & & \pmoduli{n_2+3}{d_2} \ar@/^/[dd]^{e_{2t}} \ar@/_/[dd]_{k_2} \ar[dd]|{l_2} \ar[ddl]_{g_2} & \\ \\ & \pp & \pp & \pp & } $$ in which $i=i_1 \circ q_1$, $j=j_1 \circ q_1$, $k=k_2 \circ q_2$, and $l=l_2 \circ q_2$. Furthermore each $e_t$ equals either $e_{1t} \circ q_1$ or $e_{2t} \circ q_2$; by relabeling we may assume that the former equation holds for $1 \leq t \leq n_1$. Note that, for specified partitions $n=n_1+n_2$ and $d=d_1+d_2$, the number of such fiber diagrams is $\binom{n}{n_1}$. Let $M_1=\pmoduli{n_1+3}{d_1}$ and $M_2=\pmoduli{n_2+3}{d_2}$. By Lemma~\ref{pushpull} and the projection formula we have \begin{align*} l_*&\left(i^*\alpha\cup j^*\beta\cup k^*\gamma\cup\bigcup_{t=1}^n e_t^*\delta\cap[M_1\times_{\pp}M_2]\right) \\ &=l_{2*}q_{2*}\left(q_1^*(i_1^*\alpha\cup j_1^*\beta\cup\bigcup_{t=1}^{n_1}e_{1t}^*\delta)\cup q_2^*(k_2^*\gamma\cup\bigcup_{t=n_1+1}^{n}e_{2t}^*\delta) \cap[M_1\times_ {\pp}M_2]\right) \\ &=l_{2*}\left( \left(k_2^*\gamma\cup\bigcup_{t=n_1+1}^n e_{2t}^*\delta\right) \cap q_{2*} \left(q_1^*(i_1^*\alpha\cup j_1^*\beta\cup\bigcup_{t=1}^{n_1}e_{1t}^*\delta) \cap[M_1\times_{\pp}M_2] \right) \right) \\ &=l_{2*}\left( \left(k_2^*\gamma\cup\bigcup_{t=n_1+1}^ne_{2t}^*\delta\right) \cap \left(g_2^*g_{1*}(i_1^*\alpha\cup j_1^*\beta\cup\bigcup_{t=1}^{n_1}e_{1t}^*\delta \cap[M_1])\cap[M_2] \right) \right) \\ &=l_{2*}\left( g_2^*\left( g_{1*}(i_1^*\alpha\cup j_1^*\beta\cup\bigcup_{t=1}^{n_1}e_{1t}^*\delta\cap[M_1]) \right) \cup k_2^*\gamma\cup\bigcup_{t=n_1+1}^ne_{2t}^*\delta \cap[M_2] \right). \end{align*} Summing---for fixed $n$---over all $d \geq 0$ and all components of $D(12\mid 34)$, we find that the left side of (\ref{transeq}) equals the $n$th coefficient of $(\alpha*\beta) *\gamma$. A similar argument shows that the right side of (\ref{transeq}) equals the $n$th coefficient of $\alpha*(\beta*\gamma)$. \par Invoking the $\q[[T]]$-linearity, we conclude that the quantum product is associative. \end{proof} \section{Stable maps to the incidence variety} \label{siv} \par Let $I$ be the incidence variety of points and lines in $\pp$. Its cohomology ring $A^*(I)$ is generated by two classes $h$ and $\hd$ representing the pullbacks of, respectively, the class of a line in $\pp$ and the class of a dual line in $\dpp$. The fundamental class of a curve is determined by its intersection numbers $d$ and $\dd$ with the two classes; we denote this class by $(d,\dd)$. The incidence variety is a projective line bundle over the plane; we denote the projection map by $p:I \to \pp$. For sake of simplicity in notation, we will consider $\ap$ as a subring of $A^*(I)$, embedded by $$ p^*\colon \ap \hookrightarrow A^*(I). $$ \begin{lem} \label{lemmachar} $ \ap=\{\alpha \in A^*(I) \mid p_*\alpha=0\}. $ \end{lem} (Note that whenever we use a push-forward of a cohomology class on a nonsingular space, it is defined via the push-forward of the dual homology class.) \par As we explain in our earlier paper \cite{ErnstromKennedy}, an $n$-pointed immersion from $\pl$ to $\pp$ can be lifted to an $n$-pointed map from $\pl$ to $I$; if the immersion has degree $d$ then the class of the lift is $(d,2d-2))$. Thus there is a substack $M^1_{0,n}(\pp,d)$ of $\imoduli{n}{(d,2d-2))}$ representing the lifts of immersions. We call its closure $\lamoduli{n}{d}$ the {\em stack of stable lifts}; it is birationally isomorphic to the stack $\pmoduli{n}{d}$ and thus has dimension $3d-1+n$. \par We will be working with two other special stacks of stable maps to the incidence variety. For $n\geq2$, consider the stack ${\overline M}_{0,n}(I,(0,1))$, whose typical member is an $n$-pointed isomorphism from $\pl$ to a fiber of $I$ over $\pp$. Let $e_1,\dots,e_n$ be the evaluation maps. \begin{lem} \label{disappear} Let $\alpha$ be a class in $\ap$ and let $\delta_2,\dots,\delta_n$ be classes in $A^*(I)$. Then $$ e_{ 2*}\left(e_1^*p^*\alpha\cup e_2^*\delta_2\cup \dots \cup e_n^*\delta_n\cap [{\overline M}_{0,n}(I,(0,1))]\right) =0. $$ \end{lem} \begin{proof} Since each element of ${\overline M}_{0,n}(I,(0,1))$ is a map to a fiber of $I$ over $\pp$, we have that $p\circ e_1=p\circ e_2$. Thus $e_1^*p^*=e_2^*p^*$, and by the projection formula \begin{align*} &e_{ 2*}\left(e_2^*p^*\alpha\cup e_2^*\delta_2\cup \dots \cup e_n^*\delta_n\cap [{\overline M}_{0,n}(I,(0,1))] \right) =\\ &p^*\alpha\cap e_{2*}\bigl(e_2^*\delta_2\cup\dots\cup e_n^*\delta_n \cap[{\overline M}_{0,n}(I,(0,1))] \bigr). \end{align*} The class in parentheses is the pullback of a class on ${\overline M}_{0,n-1}(I,(0,1))$ via the morphism $$ {\overline M}_{0,n}(I,(0,1))\to {\overline M}_{0,n-1}(I,(0,1)) $$ which forgets the first point. Furthermore $e_2$ factors through this forgetful morphism. Hence by the projection formula its pushforward via $e_2$ vanishes. \end{proof} \par Similarly, let $M_{0,n,2}(I,(0,2))$ be the substack of $\imoduli{n+2}{(0,2)}$ representing degree $2$ maps from $\pl$ to a fiber of $I$ over $\pp$, with the two ramification points specially marked and with an additional $n$ markings. Let $\double{n}$ be its closure, a stack of dimension $4+n$. Denote by $e_1,\dots,e_{n+2}$ the evaluation maps of $\double{n}$. It does not matter which two are evaluation maps at the ramification; this flexibility is convenient in proving the next three lemmas. (Later we will use a different notation.) \begin{lem} \label{lemmafundclass} Let $\delta_2,\dots,\delta_{n+2}$ be classes in $A^*(I)$. Then $$ e_{2*}\bigl(e_2^*\delta_2\cup\dots\cup e_{n+2}^*\delta_{n+2} \cap[\double{n}]\bigr)=0. $$ \end{lem} (Note that there is no condition on the marking corresponding to $e_1$.) \begin{proof} Each evaluation map $e_i$ ($i=2,\dots,n+2$) can be factored into the product map $f=e_2\times\dots\times e_{n+2}$ followed by projection $f_i$ onto the appropriate factor. $$ \xymatrix{ \double{n} \ar[d]^{f} \\ I\times_\pp\dots \times_\pp I \ar[d]^{f_i} \\ I} $$ Therefore \begin{align*} &e_{2*}\bigl(e_2^*\delta_2\cup\dots\cup e_{n+2}^*\delta_{n+2} \cap[\double{n}]\bigr)=\\ &f_{2*}f_*\bigl(f^*(f_2^*\delta_2\cup\dots\cup f_{n+2}^*\delta_{n+2}) \cap[\double{n}]\bigr)=\\ &f_{2*}\bigl(f_2^*\delta_2\cup\dots\cup f_{n+2}^*\delta_{n+2} \cap f_*[\double{n}]\bigr), \end{align*} which is zero because $\dim(\double{n})=4+n$ and $\dim(I\times_\pp\dots \times_\pp I)=3+n$. \end{proof} \begin{lem}\label{lemmapbpp} Let $\alpha$ be a class in $\ap$ and let $\delta_2,\dots,\delta_{n+2}$ be classes in $A^*(I)$. Then $$ e_{2*}\left(e_1^*p^*\alpha\cup e_2^*\delta_2\cup \dots \cup e_{n+2}^*\delta_{n+2}\cap [\double{n}]\right)=0. $$ \end{lem} \begin{proof} Since each element of $\double{n}$ is a map to a fiber of $I$ over $\pp$, we have that $p\circ e_1=p\circ e_2$. Thus $e_1^*p^*=e_2^*p^*$, and by the projection formula \begin{align*} &e_{2*}\left(e_{2}^*p^*\alpha\cup e_2^*\delta_2\cup \dots \cup e_{n+2}^*\delta_{n+2}\cap [\double{n}]\right)=\\ &p^*\alpha\cap e_{2*}\bigl(e_2^*\delta_2\cup\dots\cup e_{n+2}^*\delta_{n+2} \cap [\double{n}]\bigr), \end{align*} which is zero by Lemma~\ref{lemmafundclass}. \end{proof} \begin{lem}\label{lemmapfp} For any classes $\delta_2,\dots,\delta_{n+2}$ in $A^*(I)$, the class $$ e_{1*}\left(e_2^*\delta_2\cup \dots \cup e_{n+2}^*\delta_{n+2}\cap [\double{n}]\right) $$ is in $\ap$. \end{lem} \begin{proof} Since $p\circ e_1=p\circ e_2$, we have \begin{align*} &p_*e_{1*}\left(e_2^*\delta_2\cup \dots \cup e_{n+2}^*\delta_{n+2}\cap [\double{n}]\right)=\\ &p_*e_{2*}\bigl(e_2^*\delta_2\cup \dots \cup e_{n+2}^*\delta_{n+2}\cap [\double{n}], \end{align*} which is zero by Lemma~\ref{lemmafundclass}. Now apply Lemma~\ref{lemmachar}. \end{proof} \par \section{The contact product} \label{qcp} \par Denote by ${\overline M}(m,n,d)$ the fiber product $$ \lamoduli{m+3}{d} \times_I \double{n}. $$ Denote the evaluation maps ${\overline M}(m,n,d) \to I$ coming from $\lamoduli{m+3}{d}$ by $i,j,e_1,\dots,e_{m}$ and $e_{\star}$; denote those coming from $\double{n}$ by $f_1,\dots,f_{n},f_{\star}$ and $k$, with the latter two being at the points of ramification. The fiber product over $I$ is defined using the maps $e_{\star}$ and $f_{\star}$. Thus the typical point of ${\overline M}(m,n,d)$ represents a map from a curve with two components, as depicted in Figure 5.1; the vertical component maps two-to-one to a fiber of $I$ and is ramified at the points marked $\times$. In addition to the indicated markings, there are $m$ additional markings on the horizontal component and $n$ on the vertical component. $$ \xy 0;<1cm,0cm>: (0,1);(2.5,1)**\dir{-}; (2,0);(2,1.25)**\dir{-}; (0.5,1)*{\bullet}+U(4)*{i}; (1,1)*{\bullet}+U(4)*{j}; (2,1)*{\bullet}+UL(2)*{e_\star}; (2,1)*{\bullet}*{\times}+DR(2.5)*{f_\star}; (2,0.25)*{\bullet}*{\times}+R(2)*{k}; \endxy $$ \begin{center} {\bf Figure 5.1} Typical member of ${\overline M}(m,n,d)$. \end{center} \smallskip \par Given cohomology classes $\alpha$ and $\beta$ in $\ap$ and a class $\delta$ in $A^*(I)$, let $(\alpha \bullet \beta)_\delta$ (or simply $\alpha \bullet \beta$) be the element of $\ait$ whose $q$th coefficient is \begin{equation}\label{contactproduct} \sum_{\substack{m + n = q\\d>0}} \frac{2}{m!n!} k_* \bigl(i^*\alpha\cup j^*\beta\cup\bigcup_{t=1}^m e_t^*\delta\cup\bigcup_{t=1}^nf_t^*\delta\cap [{\overline M}(m,n,d)]\bigr). \end{equation} Note that with our convention $\ap$ is a subring of $\ai$, so it makes sense to pull back classes of $\ap$ via evaluation maps to $I$. The product formula (\ref{contactproduct}) makes sense for any two classes $\alpha$ and $\beta$ in $A^*(I)$. However, to prove that the product is associative we will need to assume that the classes are in $\ap$. \par We now prove that $\alpha \bullet \beta$ is in $\apt$. (For this proof we don't need to assume that $\alpha$ and $\beta$ are in $\ap$.) Let $i_1$, $j_1$, $e_{1\star}$ and $e_{1t}$ indicate the evaluation maps from $\lamoduli{m+3}{d}$ to $I$; thus if $q_1$ is the projection of ${\overline M}(m,n,d)$ onto its first factor, we have $i=i_1 \circ q_1$, etc. Similarly let $q_2$ be the projection onto the second factor and $k_2,f_{2\star},f_{2t}:\double{n} \to I$ be the evaluation maps, so that $k=k_2 \circ q_2$, etc. $$ \xymatrix{ & & {\overline M}(m,n,d) \ar[ddl]^{q_1} \ar[ddr]_{q_2} \\ \\ & \lamoduli{m+3}{d} \ar[dd]|{i_1, j_1, e_{1t}} \ar[ddr]^{e_{1\star}} & & \double{n} \ar[dd]|{k_2, f_{2t}} \ar[ddl]_{f_{2\star}} \\ \\ & I & I & I } $$ Lemma~\ref{pushpull} and the projection formula applied to the above diagram yield the following expression for the corresponding term of (\ref{contactproduct}): \begin{equation}\label{factoredcp} \frac{2}{m!n!} k_{2*}\biggl(f_{2\star}^*\bigl(e_{1\star *} (i_1^*\alpha \cup j_1^*\beta \cup\bigcup_{t=1}^m e_{1t}^*\delta \cap [\lamoduli{m+3}{d}]) \bigr) \cup \bigcup_{t=1}^n f_{2t}^*\delta \cap [\double{n}] \biggr). \end{equation} It follows from Lemma~\ref{lemmapfp} that $\alpha\bullet\beta$ is in $\apt$. \par Extending by $\q[[T]]$-linearity, we have a product $\bullet$ on $\apt$. We now define the {\em contact product of $\alpha$ and $\beta$, deformed by $\delta$}, to be $$ \alpha * \beta =\alpha\cup\beta +\alpha \bullet \beta. $$ We call $\apt$, together with this product, the {\em contact cohomology ring} of $\pp$, and denote it by $Q^1H^*(\pp)$. \begin{thm} For each $\delta$, the contact product is commutative and associative. The identity element $1\in A^0(\pp)$ for the ordinary cup product also serves as the identity element for the contact product. \end{thm} \begin{proof} The commutativity is obvious. The class $$ i_1^*1 \cup j_1^*\beta \cup\bigcup_{t=1}^m e_{1t}^*\delta \cap [\lamoduli{m+3}{d}] $$ is the pullback via the forgetful morphism $$ \lamoduli{m+3}{d}\to \lamoduli{m+2}{d} $$ of a class on the latter, and $e_{1\star}$ factors through the forgetful morphism. Hence if $\alpha=1$ then (\ref{factoredcp}) vanishes. Hence $1$ is the identity for the contact product. \par Now, let $\alpha,\beta,\gamma$ be classes in $\ap$. The cup product is associative. Thus we must show that \begin{equation} \label{qcpeqn} (\alpha \cup \beta )\bullet \gamma +(\alpha \bullet \beta )\cup \gamma +(\alpha \bullet \beta )\bullet \gamma =\alpha \cup (\beta \bullet \gamma) +\alpha \bullet (\beta \cup\gamma) +\alpha \bullet (\beta \bullet \gamma). \end{equation} As in the proof of associativity of the quantum product in section \ref{qtp}, we use a linear equivalence of divisors $$ D(ij \mid kl) \simeq D(il \mid jk), $$ this time on the stack of stable lifts $\lamoduli{m+4}{d}$. This linear equivalence, which is derived from the forgetful map $\lamoduli{m+4}{d}\to \stablefour$, induces a linear equivalence on the fiber product $\lamoduli{m+4}{d}\times_I \double{n}$: $$ D(ij\mid kl)\times_I \double{n} \simeq D(il \mid jk)\times_I \double{n}. $$ For these two fiber products, denote the evaluation maps to $I$ coming from the first factor by $i,j,k,l$ and $e_1,\dots,e_m$; denote the evaluation maps coming from the second factor by $r$ and $s$ (for evaluation at the ramification points) and by $f_1,\dots,f_n$. The fiber product is defined via the map $l$ on the first factor and $r$ on the second. \par The linear equivalence implies the equality \begin{multline} \label{qccdiveqn} \sum_{\substack{m + n = q\\d>0}} \frac{2}{m!n!} s_*\bigl(i^*\alpha \cup j^*\beta \cup k^*\gamma \cup {\bigcup_{t=1}^m} e_t^*\delta \cup {\bigcup_{t=1}^n} f_t^*\delta \cap \lbrack D(ij \mid kl)\times_I \double{n} \rbrack \bigr) \\ = \sum_{\substack{m + n = q\\d>0}} \frac{2}{m!n!} s_*\bigl(i^*\alpha \cup j^*\beta \cup k^*\gamma \cup {\bigcup_{t=1}^m} e_t^*\delta \cup {\bigcup_{t=1}^n} f_t^*\delta \cap \lbrack D(il \mid jk)\times_I \double{n} \rbrack \bigr). \end{multline} We will show that the left side of (\ref{qccdiveqn}) equals the $q$th coefficient of the left side of (\ref{qcpeqn}); an entirely similar argument shows that the right side of (\ref{qccdiveqn}) equals the $q$th coefficient of the right side of (\ref{qcpeqn}). \par In \cite[Section 5]{ErnstromKennedy} we have analyzed the components of $D(ij \mid kl)$. We call such a component $D$ {\em numerically irrelevant} if $$ \int i_1^*\gamma_i\cup j_1^*\gamma_j \cup k_1^*\gamma_k\cup l_1^*\gamma_l \cup {\bigcup_{t=1}^m}e_{1t}^*\delta_t \cap [D]=0. $$ for all choices of cohomology classes $\gamma_i,\gamma_j,\gamma_k,\gamma_l,$ and $\delta_1, \dots, \delta_m$ in $A^*(I)$. Here $i_1,j_1$, etc. indicate the maps in the following diagram, for which $i_1 \circ q_1=i$, etc. \par $$ \xymatrix{ & & \lamoduli{m+4}{d}\times_I \double{n} \ar[ddl]^{q_1} \ar[ddr]_{q_2} \\ \\ & \lamoduli{m+4}{d} \ar[dd]|{i_1, j_1, k_1, e_{1t}} \ar[ddr]^{l_1} & & \double{n} \ar[dd]|{s_2,f_{2t}} \ar[ddl]_{r_2} \\ \\ & I & I & I } $$ \par To understand the contribution of a numerically irrelevant component to (\ref{qccdiveqn}), consider the class \begin{equation} \label{vanish} s_*\bigl(i^*\alpha \cup j^*\beta \cup k^*\gamma \cup {\bigcup_{t=1}^m} e_t^*\delta \cup {\bigcup_{t=1}^n} f_t^*\delta \cap \lbrack D \times_I \double{n} \rbrack \bigr). \end{equation} Let $\tau$ be a test class in $A_*(I)$. Then the degree of its intersection with (\ref{vanish}) equals, by repeated use of the projection formula, the degree of the class \begin{equation*} i_1^*\alpha \cup j_1^*\beta \cup k_1^*\gamma \cup {\bigcup_{t=1}^m} e_{1t}^*\delta \cap q_{1*}\left( q_2^*(s_2^*\tau \cup{\bigcup_{t=1}^n} f_{2t}^*\delta) \cap \lbrack D \times_I \double{n} \rbrack\right). \end{equation*} By Lemma~\ref{pushpull}, this class equals \begin{equation*} i_1^*\alpha \cup j_1^*\beta \cup k_1^*\gamma \cup {\bigcup_{t=1}^m} e_{1t}^*\delta \cup l_1^*r_{2*}\left(s_2^*\tau \cup {\bigcup_{t=1}^n}f_{2t}^*\delta\cap[\double{n}])\right) \cap \lbrack D \rbrack. \end{equation*} Since $D$ is numerically irrelevant, the degree is 0. And since this is true for every $\tau$, the class (\ref{vanish}) is zero. \par Besides the numerically irrelevant components, the divisor $D(ij \mid kl)$ on $\lamoduli{m+4}{d}$ has three types of components \cite[Proposition 5.9]{ErnstromKennedy}. Thus there are three corresponding types of components of $D(ij \mid kl)\times_I \double{n}$. A component of the first type is of the form \begin{equation} \label{type1a} \imoduli{m_1+3}{(0,0)} \times_I \lamoduli{m_2+3}{d} \times_I \double{n} \end{equation} or \begin{equation} \label{type1b} \lamoduli{m_1+3}{d} \times_I \imoduli{m_2+3}{(0,0)} \times_I \double{n}, \end{equation} where $m_1+m_2=m$. A typical point of such a component represents a map from a curve with three components, as depicted in Figure 5.2. In each case the component on the right maps two-to-one to a fiber of $I$ and is ramified at the points marked $\times$; the horizontal component maps to $I$ via the lift of an immersion; and the remaining component maps to a point of $I$. In addition to the indicated markings, there are $m_1$ markings on the left component, $m_2$ on the middle component, and $n$ on the right component. \par $$ \xy 0;<1cm,0cm>: (0,1);(2.5,1)**\dir{-}; (0.5,1)*{\bullet}; (1,1)*{\bullet}+U(4)*{k}; (2,1)*{\times}*{\bullet}+UL(2.5)*{l}+R(9)*{r}; (0.5,0.75);(0.5,2.5)**\dir{-}; (0.5,1.5)*{\bullet}+L(4)*{j}; (0.5,2)*{\bullet}+L(4)*{i}; (2,0.75);(2,2.5)**\dir{-}; (2,2)*{\times}*{\bullet}+R(4)*{s}; (5,2.25);(7.5,2.25)**\dir{-}; (5.75,2.25)*{\bullet}+U(4)*{i}; (6.5,2.25)*{\bullet}+U(4)*{j}; (7.25,2.5);(7.25,1.5)**\dir{-}; (7.75,1)**\crv{(7.25,1)}*{\times}*{\bullet}+DR(2.5)*{r}+L(9)*{l}; (7.75,1);(8,1)**\dir{-}; (7.75,0.75);(7.75,2.5)**\dir{-}; (7.25,1.875)*{\bullet}+L(4)*{k}; (7.25,2.25)*{\bullet}; (7.75,1.875)*{\bullet}*{\times}+R(2.5)*{s}; \endxy $$ \begin{center} {\bf Figure 5.2} Typical members of a component of the first type, (\ref{type1a}) (left) and (\ref{type1b}) (right.) \end{center} \smallskip \par Consider a component of type (\ref{type1a}). Let us denote the three factors simply by $M_1,M_2,M_3$. We may relabel the markings so that the evaluation maps $e_1,\dots,e_{m_1}$, as well as $i$, $j$, and the map $g_1$ used to create the fiber product, factor through projection onto $M_1$. We also note that all of these evaluation maps coincide. Hence we have the following fiber diagram. $$ \xymatrix{ & & M_1\times_I M_2\times_I M_3 \ar[ddl]^{q_{1}} \ar[ddr]_{q_{23}} & \\ \\ & M_1 \ar[ddr]|{g_{1}=i_{1}=j_{1}=e_{1t}} & & M_2\times_I M_3 \ar[dd]|{k_{23},e_{23t},f_{23t},s_{23}} \ar[ddl]_{g_{23}} & \\ \\ & {} & I & I & } $$ By Lemma~\ref{pushpull} and the projection formula, the contribution of our component to the left side of (\ref{qccdiveqn}) is \begin{align*} \frac{2}{m!n!}& s_*\bigl(i^*\alpha\cup j^*\beta\cup k^*\gamma\cup \bigcup_{t=1}^me_t^*\delta\cup\bigcup_{t=1}^nf_t^*\delta \cap [M_1\times_IM_2\times_I M_3]\bigr)\\ &=\frac{2}{m!n!} s_{23*}\biggl( g_{23}^*g_{1*}\biggl( g_{1}^*(\alpha\cup \beta \cup \bigcup_{t=1}^{m_1}\delta) \cap [M_1] \biggr) \cup k_{23}^*\gamma \cup \bigcup_{t=m_1+1}^{m}e_{23t}^*\delta \cup \bigcup_{t=1}^n f_{23t}^*\delta \cap [M_2\times_IM_3] \biggr). \end{align*} If $m_1 > 0$ then the fibers of the map from $M_1$ to $I$ have positive dimension; hence \begin{equation*} g_{1*}\biggl( g_{1}^*(\alpha\cup \beta \cup \bigcup_{t=1}^{m_1}\delta) \cap [M_1] \biggr) = \alpha\cup\beta\cup\bigcup_{t=1}^{m_1}\delta\cap g_{1*}[M_1]=0, \end{equation*} and the component makes no contribution to (\ref{qccdiveqn}). If $m_1 = 0$ then $M_1$ and $I$ are isomorphic, and \begin{equation*} g_{23}^*g_{1*}\biggl( g_{1}^*(\alpha\cup \beta \cup \bigcup_{t=1}^{m_1}\delta) \cap [M_1] \biggr) = g_{23}^*\biggl(\alpha\cup \beta \cap [I] \biggr). \end{equation*} Thus if we sum the contributions from all such components with $m+n=q$ we obtain the $q$th coefficient of $(\alpha \cup \beta )\bullet \gamma$. \par Entirely similar arguments show that a component of type (\ref{type1b}) makes no contribution to (\ref{qccdiveqn}) unless $m_2=0$, and that the sum of contributions from all such components with $m+n=q$ is the $q$th coefficient of $(\alpha \bullet \beta )\cup \gamma$. \par The second type of component of $D(ij \mid kl)\times_I \double{n}$ is one of the form \begin{equation}\label{type2a} {\overline M}_{0,m_1+3}(I,(0,1)) \times_I {\overline C}^1_{0,m_2+2,\{\star\}}(\pp,d) \times_I \double{n}, \end{equation} or \begin{equation}\label{type2b} {\overline C}^1_{0,m_1+2,\{\star\}}(\pp,d) \times_I {\overline M}_{0,m_2+3}(I,(0,1)) \times_I \double{n} \end{equation} where $m_1+m_2=m$, and where the second (respectively first) factor is the stack of cuspidal stable lifts \cite[Section 4]{ErnstromKennedy}. A general point of such a component represents a map from a curve with the same configuration of components and markings as in Figure 5.2. The left and middle components map, in either order, to the lift of a degree $d$ rational curve with one cusp, and to the fiber of $I$ over the cusp, and the right component maps two-to-one to a fiber of $I$. \par Consider a component of type (\ref{type2a}). Denote the three factors by $M_1, M_2, M_3$ and consider the following fiber diagram. \par $$ \xymatrix{ & M_1\times_I M_2 \times_I M_3 \ar[ddl]^{q_1} \ar[ddr]_{q_{23}} \\ \\ M_1 \ar[dd]|{i_1, j_1, e_{1t}} \ar[ddr]^{g_1} & & M_2 \times_I M_3 \ar[dd]|{k_{23},s_{23},e_{23t},f_{23t}} \ar[ddl]_{g_{23}} \\ \\ I & I & I } $$ By Lemma~\ref{pushpull} and the projection formula, the contribution of our component to the left side of (\ref{qccdiveqn}) is $$ \frac{2}{m! n!}s_{23*}\biggl(g_{23}^*g_{1*}\bigl(i_1^*\alpha\cup j_1^*\beta\cup\bigcup_{t=1}^{m_1}e_{1t}^*\delta \cap [M_1]\bigr) \cup k_{23}^*\gamma\cup\bigcup_{t=m_1+1}^m e_{23t}^* \delta\cup \bigcup_{t=1}^nf_{23t}^*\delta\cap [M_2\times_I M_3]\biggr). $$ By Lemma~\ref{disappear}, the class $$ g_{1*}\bigl(i_1^*\alpha\cup j_1^*\beta\cup\bigcup_{t=1}^{m_1}e_{1t}^*\delta \cap [M_1]\bigr)=0. $$ Hence the component makes no contribution. A similar argument applies to a component of type (\ref{type2b}). \par We come finally to the third type of component of $D(ij \mid kl)\times_I \double{n}$. Among them are components of the form \begin{equation} \label{type3a} \lamoduli{m_1+3}{d_1}\times_I \double{m_2} \times_I\lamoduli{m_3+3}{d_3}\times_I\double{n}, \end{equation} with $m_1+m_2+m_3=m$, $d_1,d_3>0$ and $d_1+d_3=d$. A typical point of such a component represents a map from a curve with four components, as depicted at the top of Figure 5.3. The vertical components map two-to-one to fibers of $I$ and are ramified at the points marked $\times$; the horizontal components map to $I$ via the lifts of immersions. In addition to the indicated markings, there are $m+n$ others. \par $$ \xy 0;<1cm,0cm>: (0,1.75);(1.5,1.75)**\dir{-}; (0.25,1.75)*{\bullet}+U(4)*{i}; (0.75,1.75)*{\bullet}+U(4)*{j}; (1.25,2);(1.25,0)**\dir{-}; (1.25,0.25)*{\times}*{\bullet}; (1.25,1.75)*{\times}*{\bullet}; (1.25,0.75)*{\bullet}+L(4)*{k}; (1,0.25);(3,0.25)**\dir{-}; (2.75,0.25)*{\times}*{\bullet}+UL(2.5)*{l}+R(9)*{r}; (2.75,0);(2.75,1.75)**\dir{-}; (2.75,1.5)*{\times}*{\bullet}+R(4)*{s}; (0,4.75);(1.5,4.75)**\dir{-}; (0.25,4.75)*{\bullet}+U(4)*{i}; (1.25,5);(1.25,3)**\dir{-}; (1.25,3.25)*{\times}*{\bullet}; (1.25,4.75)*{\times}*{\bullet}; (1.25,3.75)*{\bullet}+L(4)*{j}; (1,3.25);(3,3.25)**\dir{-}; (2,3.25)*{\bullet}+U(4)*{k}; (2.75,3.25)*{\times}*{\bullet}+UL(2.5)*{l}+R(9)*{r}; (2.75,3);(2.75,4.75)**\dir{-}; (2.75,4.5)*{\times}*{\bullet}+R(4)*{s}; (4,4.75);(5.5,4.75)**\dir{-}; (5.25,5);(5.25,3)**\dir{-}; (5.25,3.25)*{\times}*{\bullet}; (5.25,4.75)*{\times}*{\bullet}; (5.25,3.75)*{\bullet}+L(4)*{j}; (5.25,4.25)*{\bullet}+L(4)*{i}; (5,3.25);(7,3.25)**\dir{-}; (6,3.25)*{\bullet}+U(4)*{k}; (6.75,3.25)*{\times}*{\bullet}+UL(2.5)*{l}+R(9)*{r}; (6.75,3);(6.75,4.75)**\dir{-}; (6.75,4.5)*{\times}*{\bullet}+R(4)*{s}; (8,4.75);(9.5,4.75)**\dir{-}; (8.25,4.75)*{\bullet}+U(4)*{j}; (9.25,5);(9.25,3)**\dir{-}; (9.25,3.25)*{\times}*{\bullet}; (9.25,4.75)*{\times}*{\bullet}; (9.25,3.75)*{\bullet}+L(4)*{i}; (9,3.25);(11,3.25)**\dir{-}; (10,3.25)*{\bullet}+U(4)*{k}; (10.75,3.25)*{\times}*{\bullet}+UL(2.5)*{l}+R(9)*{r}; (10.75,3);(10.75,4.75)**\dir{-}; (10.75,4.5)*{\times}*{\bullet}+R(4)*{s}; (4,7.75);(5.5,7.75)**\dir{-}; (4.25,7.75)*{\bullet}+U(4)*{i}; (4.75,7.75)*{\bullet}+U(4)*{j}; (5.25,8);(5.25,6)**\dir{-}; (5.25,6.25)*{\times}*{\bullet}; (5.25,7.75)*{\times}*{\bullet}; (6,6.25)*{\bullet}+U(4)*{k}; (5,6.25);(7,6.25)**\dir{-}; (6.75,6.25)*{\times}*{\bullet}+UL(2.5)*{l}+R(9)*{r}; (6.75,6);(6.75,7.75)**\dir{-}; (6.75,7.5)*{\times}*{\bullet}+R(4)*{s}; (4,1.75);(6,1.75)**\dir{-}; (4.25,1.75)*{\bullet}+U(4)*{i}; (4.75,1.75)*{\bullet}+U(4)*{j}; (5.75,1.75)*{\times}*{\bullet}; (5.75,2);(5.75,0)**\dir{-}; (5.75,1)*{\times}*{\bullet}; (5.5,1);(7,1)**\dir{-}; (5.25,1.5);(5.25,1)**\dir{-}*{\times}*{\bullet}+L(4)*{s}; (5.25,1);(5.75,0.5)**\crv{(5.25,0.5)}; (6,0.5);(5.75,0.5)**\dir{-}*{\times}*{\bullet}+DR(2.5)*{l}+L(9)*{r}; (5.75,1.4)*{\bullet}+R(4)*{k}; (8,1.75);(10,1.75)**\dir{-}; (8.25,1.75)*{\bullet}+U(4)*{i}; (8.75,1.75)*{\bullet}+U(4)*{j}; (9.75,1.75)*{\times}*{\bullet}; (9.75,2);(9.75,0)**\dir{-}; (9.75,1)*{\times}*{\bullet}; (9.5,1);(11,1)**\dir{-}; (9.75,1.75)*{\bullet}; (9.25,1.5);(9.25,1)**\dir{-}*{\times}*{\bullet}+L(4)*{s}; (9.25,1);(9.75,0.5)**\crv{(9.25,0.5)}; (10,0.5);(9.75,0.5)**\dir{-}*{\times}*{\bullet}+DR(2.5)*{l}+L(9)*{r}; (10.3625,1)*{\bullet}+U(4)*{k} \endxy $$ \begin{center} {\bf Figure 5.3} The different configurations for components of the third type. \end{center} \smallskip \par There are six other possibilities for a component of the third type, corresponding to the six different possible ways in which the four special markings $i,j,k,l$ can lie on the first three components of the typical curve. (The markings $i$ and $j$ may lie on either of the first two components; $k$ and $l$ may lie on either the second or third component; and the pair $i,j$ must be separated from the pair $k,l$ by a node.) We claim that in each of these six cases the component makes no contribution to (\ref{qccdiveqn}). In five cases the argument is identical to that for a component of the second type: a configuration with $i,j$ or $k$ on a component mapping to a fiber (vertical component in Figure 5.3) does not contribute, as a consequence of Lemma~\ref{lemmapbpp}. In the sixth case we have a component of the form \begin{equation} \label{type3f} \biggl(\lamoduli{m_1+3}{d_1}\times_I \double{m_2+1} \times_I\lamoduli{m_3+2}{d_3}\biggr)\times_I\double{n}, \end{equation} A typical point of such a component represents a map from the sort of curve shown at the bottom right of Figure 5.3. Denote the four factors by $M_1, M_2, M_3$ and $M_4$. Note that the fiber product of $M_1\times_I M_2 \times_I M_3$ and $M_4$ is created by using an evaluation map from $M_1\times_I M_2 \times_I M_3$ which comes from its second factor. Consider the following fiber diagram. \par $$ \xymatrix{ & M_1\times_I M_2 \times_I M_3\times_I M_4 \ar[ddl]^{q_{123}} \ar[ddr]_{q_4} \\ \\ M_1\times_I M_2 \times_I M_3 \ar[dd]|{i_{123}, j_{123}, k_{123}, e_{123t}} \ar[ddr]^{l_{123}} & & M_4 \ar[dd]|{s_4,f_{4t}} \ar[ddl]_{r_4} \\ \\ I & I & I } $$ Using Lemma~\ref{pushpull} one finds that the contribution to (\ref{qccdiveqn}) is $$ \frac{2}{m!n!}s_{4*}\biggl(\bigcup_{t=1}^{n}f_{4t}^*\delta \cup r_4^* l_{123*}\bigl(i_{123}^*\alpha\cup j_{123}^*\beta\cup k_{123}^*\gamma \cup \bigcup_{t=1}^{m}e_{123t}^*\delta \cap [M_1\times_I M_2 \times_I M_3]\bigr) \cap [M_4]\biggr). $$ We claim that the class \begin{equation}\label{ppclass} l_{{123}*}\bigl(i_{123}^*\alpha\cup j_{123}^*\beta\cup k_{123}^*\gamma \cup\bigcup_{t=1}^{m}e_{{123}t}^*\delta \cap [M_1\times_I M_2 \times_I M_3]\bigr) \end{equation} is in $\ap$. It will then follow from Lemma~\ref{lemmapbpp} that the contribution is zero. To prove the claim we use the following diagram, in which there are three fiber squares. $$ \xymatrix{ & & M_1\times_I M_2 \times_I M_3 \ar[dl] \ar[dr] \\ & M_1\times_I M_2 \ar[dl] \ar[dr] & & M_2\times_I M_3 \ar[dl] \ar[dr]\\ M_1 \ar[d]|{i_{1}, j_{1}, e_{{1}t}} \ar[dr]^{g_1} & & M_2 \ar[dl]_{g_2} \ar[d]|{l_2, e_{2t}} \ar[dr]^{h_2} & & M_3 \ar[dl]_{h_3} \ar[d]|{k_3,e_{3t}}\\ I & I & I & I & I } $$ By repeated use of Lemma~\ref{pushpull} and the projection formula, the class (\ref{ppclass}) can be expressed as $$ l_{2*}\biggl( g_2^*g_{1*}\bigl(i_1^*\alpha\cup j_1^*\beta\cup \bigcup_{t=1}^{m_1}e_{1t}^*\delta\cap [M_1]\bigr) \cup h_2^* h_{3*}\bigl(k_3^*\gamma\cup \bigcup_{t=m_1+m_2+1}^{m_1+m_2+m_3}e_{3t}^*\delta\cap [M_3]\bigr) \cup\bigcup_{t=m_1+1}^{m_1+m_2}e_{2t}^*\delta\cap [M_2]\biggr). $$ Now $M_2=\double{n}$, so by Lemma~\ref{lemmapfp} this is a class in $\ap$ as we claimed. \par It remains to consider the contribution of the component with the configuration on the top of Figure 5.3. Note that the product (\ref{type3a}) can be written as the following fiber product: \par $$ \xymatrix{ & & \spamoduli{m_1}{m_2}{d_1}\times_I\spamoduli{m_3}{n}{d_3} \ar[ddl]^{q_1} \ar[ddr]_{q_2} \\ \\ & \spamoduli{m_1}{m_2}{d_1} \ar[dd]|{i_1, j_1, e_{1t}} \ar[ddr]^{g_1} & & \spamoduli{m_3}{n}{d_3} \ar[dd]|{k_2,s_2,e_{2t},f_{2t}} \ar[ddl]_{g_2} \\ \\ & I & I & I } $$ Using Lemma~\ref{pushpull} one finds that the contribution to the left side of (\ref{qccdiveqn}) is \begin{multline*} \frac{2}{n!m!}s_{2*}\biggl(k_2^*\gamma\cup\bigcup_{t=m_1+m_2+1}^{m_3}e_{2t}^*\delta \cup\bigcup_{t=1}^n f_{2t}^*\delta\cup g_2^*g_{1*}\bigl(i_1^*\alpha\cup j_1^*\beta\cup \\ \bigcup_{t=1}^{m_1}e_{1t}^*\delta\cup\bigcup_{t=m_1+1}^{m_2}e_{1t}^*\delta \cap [\spamoduli{m_1}{m_2}{d_1}]\bigr) \cap [\spamoduli{m_3}{n}{d_3}]\biggr). \end{multline*} According to \cite{ErnstromKennedy} each divisor component of this type appears in $D(ij \mid kl)$ with a multiplicity of two. The reason is that $(d,2d-2)$ can be partitioned in two ways, either as $(d_1,2d_1-2)+(d_2,2d_2)$ or as $(d_1,2d_1)+(d_2,2d_2-2)$. If we keep $q$ fixed, and sum the contributions for all $d$, $d_1+d_3=d$, $m+n=q$ and all partitions of the $m$ markings into three disjoint subsets with cardinalities $m_1$, $m_2$ and $m_3$, the result is the $q$th coefficient of $(\alpha\bullet\beta)\bullet\gamma$. Thus we have shown that $(\alpha * \beta)*\gamma=\alpha *(\beta * \gamma)$ when $\alpha, \beta, \gamma$ are elements of $\ap$. Invoking the $\q[[T]]$-linearity, we conclude that the contact product is associative. \par \end{proof} \section{The recursive relation among characteristic numbers} \label{rrc} The associativity of the quantum product implies Kontsevich's recursive formula for the characteristic numbers \begin{equation*} N_d= \text{ the number of rational plane curves of degree $d$ through $3d-1$ general points.} \end{equation*} For details of this story, see \cite{FrancescoItzykson}, \cite{FultonP}, \cite{KontsevichManin}. Here we show that, in a similar fashion, the associativity of the contact product implies a recursive formula for the numbers \begin{align*} N_d(a,b,c)=&\text{ the number of rational plane curves of degree $d$ through $a$ general points, tangent to}\\ &\text{$b$ general lines, and tangent to $c$ general lines at a specified general point on each line}\\ &\quad\text{(where $a+b+2c=3d-1$).} \end{align*} Our formula will specialize both to Kontsevich's formula and the more general formula of Di~Francesco and Itzykson \cite[Equation 2.95]{FrancescoItzykson}. \par We will use the following ordered basis for the cohomology of the incidence correspondence: $$ \{T_0,T_1,T_2,T_3,T_4,T_5\}=\{1,h,h^2,\hd,\hd^2,h^2\hd\}. $$ With respect to this basis the fundamental class of the diagonal $\Delta$ in $I \times I$ has the simple decomposition \begin{equation*} [\Delta] = \sum_{s=0}^5 [T_s] \times [T_{5-s}]. \end{equation*} Suppose that $d$ is a positive integer, and that $\gamma_1,\dots,\gamma_n$ are elements of $A^*I$. Then the {\em first-order Gromov-Witten invariant} is $$ N_d(\gamma_1\cdots\gamma_n)=\int e_1^*(\gamma_1) \cup \dots \cup e_n^*(\gamma_n) \cap [\lamoduli{n}{d}]. $$ According to \cite[Section 4]{ErnstromKennedy}, we have the following interpretations: \begin{trivlist} \item[(1)] Suppose that $a$ of the $\gamma_t$'s equal the class $h^2$, that $b$ of them equal the class $\hd^2$, and that the remaining $c$ of them equal $h^2\hd$, where $a+b+2c=3d-1$. Then the Gromov-Witten invariant is the characteristic number $N_d(a,b,c)$. \item[(2)] For all $d$ and all $\gamma_1,\dots,\gamma_{n-1}$, $$ N_d(\gamma_1\cdots\gamma_{n-1}\cdot 1)=0. $$ \par \item[(3)] If $\gamma_n$ is the class of a divisor, then for all $d$ and all $\gamma_1,\dots,\gamma_{n-1}$, $$ N_d(\gamma_1\cdots\gamma_n)= N_d(\gamma_1\cdots\gamma_{n-1})\int\gamma_n\cap [C], $$ where $[C]=d \hd^2 +(2d-2) h^2$. \end{trivlist} \par Using a general element $$ \gamma=y_0T_0+\dots+y_5T_5 $$ of $A^*I$, we define the {\em quantum potential} to be the following formal power series in $y_0,\dots,y_5$: $$ {\mathcal N}=\sum_{\substack{m \geq 0 \\ d\geq 1}} \frac{1}{m!} \int e^*_1(\gamma)\cup \dots \cup e^*_m(\gamma) \cap [\lamoduli{m}{d}], $$ where $e_1,\dots,e_m$ are the evaluation maps. By the previous remarks $$ {\mathcal N}=\sum_{\substack{d\geq 1 \\a +b+2c=3d-1\\ a,b,c\geq 0}} \frac{N_d(a,b,c)y_2^ay_4^by_5^c\exp(dy_1+(2d-2)y_3)}{a!b!c!}. $$ In a similar way, we define a potential associated to the stacks $\double{n}$. This will be a formal power series in two sets of indeterminates. Let $\delta=z_0T_0+\dots+z_5T_5$ be a second general element of $A^*I$. Then \begin{equation*} {\mathcal R}=\sum_{n \geq 0} \frac{1}{2n!} \int e^*_1(\gamma) \cup \dots \cup e^*_n(\gamma) \cup e^*_{n+1}(\delta) \cup e^*_{n+2}(\delta) \cap [\double{n}], \end{equation*} where $e_{n+1}$ and $e_{n+2}$ are evaluation at the points of ramification. As we show in \cite[Section 6]{ErnstromKennedy}, \begin{equation} \label{potr} {\mathcal R}=\left\{\frac{z_3^2}{2}(y_4^2+y_5)+ z_3z_4y_4+\frac{z_3z_5}{2}+\frac{z_4^2}{4}\right\}\exp(2y_3). \end{equation} \par Similarly, we define a potential associated to the stacks $$ \lamoduli{m+1}{d} \times_I \double{n}. $$ Let $e_1,\dots,e_m$ and $e_{\star}$ be the evaluation maps coming from the first factor; let $f_{\star}$ and $k$ be evaluation at the points of ramification; let $f_1,\dots,f_n$ be the other evaluations coming from the second factor; let $e_{\star}$ and $f_{\star}$ be the maps used to create the fiber product. We define ${\mathcal K}$ to be \begin{equation*} \sum_{\substack{m, n \geq 0 \\ d\geq 1}} \frac{2}{m!n!} \int e^*_1(\gamma) \cup \dots \cup e^*_m(\gamma) \cup f^*_1(\gamma) \cup \dots \cup f^*_n(\gamma) \cup k^*(\delta) \cap [\lamoduli{m+1}{d} \times_I \double{n}]. \end{equation*} Then by Propositions 6.2 and 6.3 of \cite{ErnstromKennedy}, the three potentials are related by the differential equation \begin{equation} \label{knr} {\mathcal K}= 2 \sum_{s=0}^5 \frac{\partial {\mathcal N}}{\partial y_s} \frac{\partial {\mathcal R}}{\partial z_{5-s}}; \end{equation} from the explicit form (\ref{potr}) of ${\mathcal R}$, however, we see that the sum needs to run only from $s=0$ to $2$. \par Proposition 6.3 also tells us how the product $\bullet$ is related to the potential ${\mathcal K}$: for $0 \leq i,j \leq 2$ we have $$ T_i \bullet T_j = \sum_{s=0}^5 \kp{i}{j}{5-s} T_s. $$ Again we note that the sum needs to run only from $s=0$ to $2$, since the product of two elements of $\ap$ is a formal power series whose coefficients are likewise in $\ap$. Thus $$ T_i \bullet T_j = \kp{i}{j}{5} T_0 + \kp{i}{j}{4} T_1 + \kp{i}{j}{3} T_2. $$ \par We have shown that the contact product is associative. In particular $(T_1*T_1) * T_2 = T_1*(T_1*T_2)$. Equating the coefficients of $T_0$ on the two sides of this equation, we find that \begin{multline*} \kp{1}{1}{4}\kp{1}{2}{5}+\kp{2}{2}{5}+\kp{1}{1}{3}\kp{2}{2}{5} \\ =\kp{1}{1}{5}\kp{1}{2}{4}+\kp{1}{2}{5}\kp{1}{2}{3}. \end{multline*} Applying (\ref{knr}) throughout and simplifying, we obtain the following partial differential equation for the potential ${\mathcal N}$: \begin{equation} \label{npde} {\mathcal N}_{222}=\exp(2y_3)\biggl({\mathcal N}_{112}^2-{\mathcal N}_{111}{\mathcal N}_{122} +2y_4\bigl({\mathcal N}_{112}{\mathcal N}_{122}-{\mathcal N}_{111}{\mathcal N}_{222}\bigr) +(2y_4^2+2y_5)\bigl({\mathcal N}_{122}^2-{\mathcal N}_{112}{\mathcal N}_{222}\bigr)\biggr). \end{equation} Note that if we set $y_3=y_5=0$ we recover equation (2.95) of \cite{FrancescoItzykson}, and that if furthermore we set $y_4=0$ we recover equation (5.16) of \cite{KontsevichManin}. \par Our equation (\ref{npde}) can be rewritten as a formula for the characteristic number $N_d(a,b,c)$. Note we must assume that $a\geq 3$. In this formula $d_1$ and $d_2$ are greater than zero; thus it determines our characteristic number if we assume we already know those for curves of lower degree. \begin{equation} \label{recurs} \begin{aligned} N_d(a,b&,c)=\sum_{\substack{d_1+d_2=d\\ a_1+a_2=a-1\\b_1+b_2=b\\c_1+c_2=c}} N_{d_1}(a_1,b_1,c_1)N_{d_2}(a_2,b_2,c_2) \left[d_1^2d_2^2\binom{a-3}{a_1-1} -d_1^3d_2\binom{a-3}{a_1} \right]\binom b{b_1}\binom c{c_1}\\ &+2\cdot\sum_{\substack{d_1+d_2=d\\ a_1+a_2=a\\b_1+b_2=b-1\\c_1+c_2=c}} N_{d_1}(a_1,b_1,c_1)N_{d_2}(a_2,b_2,c_2) \left[d_1^2d_2\binom{a-3}{a_1-1} -d_1^3\binom{a-3}{a_1} \right]\binom b{b_1\, b_2\, 1}\binom c{c_1}\\ &+4\cdot\sum_{\substack{d_1+d_2=d\\ a_1+a_2=a+1\\b_1+b_2=b-2\\c_1+c_2=c}} N_{d_1}(a_1,b_1,c_1)N_{d_2}(a_2,b_2,c_2) \left[d_1d_2\binom{a-3}{a_1-2} -d_1^2\binom{a-3}{a_1-1} \right]\binom b{b_1\, b_2\, 2}\binom c{c_1}\\ &+2\cdot\sum_{\substack{ d_1+d_2=d\\ a_1+a_2=a+1\\b_1+b_2=b\\c_1+c_2=c-1}} N_{d_1}(a_1,b_1,c_1)N_{d_2}(a_2,b_2,c_2) \left[d_1d_2\binom{a-3}{a_1-2} -d_1^2\binom{a-3}{a_1-1} \right]\binom b{b_1}\binom c{c_1\, c_2\, 1}. \end{aligned} \end{equation} If we set $c=0$ we recover the recursive formula of Di~Francesco and Itzykson \cite{FrancescoItzykson}. They never state the formula explicitly, but it is an immediate consequence of their equation (2.95), and they have clearly used it in calculating the table of values (2.97). If we also set $a=3d-1$ and $b=0$ we recover the formula (5.17) of \cite[Claim 5.2.1]{KontsevichManin}. \par Formula \ref{recurs} by itself is not enough to determine all characteristic numbers, since we need to supply it with all those values for which $a \leq 2$. Now it may appear that we could perhaps derive additional information by writing down the associativity equations for three other basis elements (rather than $T_1, T_1, T_2$) or by extracting the coefficients with respect to some other basis element (rather than $T_0$). But in fact all such derivations lead---if not to a triviality---to the same differential equation (\ref{npde}). We will omit the proof of this fact. To determine all characteristic numbers it is necessary to use the other relations of \cite[Section 7]{ErnstromKennedy}. \par Using the formalism of Fulton and Pandharipande \cite[9 Prop.10]{FultonP}, it is possible to give a presentation of the contact cohomology ring, based on the standard presentation $$ 0\to (z^3)\to \q[z]\to A^*(\pp)\to 0 $$ of $\ap$ as an algebra over $\q$, in which $z^i$ is sent to the class $h^i=h\cup\dots\cup h$. Using their arguments, we see that $1,h$ and $h*h$ form a $\q[[y_0,\dots,y_5]]$-basis for $Q^1H^*(\pp)$, and that we have a similar presentation for the $\q[[y_0,\dots,y_5]]$-algebra $Q^1H^*(\pp)$: $$ 0\to (z^3-\xi_2z^2-\xi_1z-\xi_0)\to\q[[y_0,\dots,y_5]][z]\to Q^1H^*(\pp)\to 0. $$ Here $z^i$ is sent to $h^{*i}=h * \dots * h$, and thus the elements $\xi_i$ are the coefficients of $h*h*h$ with respect to this basis. Explicitly, \begin{multline} Q^1H^*(\pp)=\q[[y_0,\dots,y_5]][z]/\biggl(z^3-\exp(2y_3)\biggl( \bigl({\mathcal N}_{111}+4y_4{\mathcal N}_{112}+(2y_4^2+2y_5){\mathcal N}_{122}\bigr)z^2 \\ +\bigl(2{\mathcal N}_{112}+2y_4{\mathcal N}_{122}\bigr)z+{\mathcal N}_{122}\biggr) -\exp(4y_3)\bigl({\mathcal N}_{111}{\mathcal N}_{122}-{\mathcal N}_{112}^2\bigr) \bigl((2y_4^2+2y_5)z+2y_4\bigr)\biggr). \end{multline} If we set $y_3=y_4=y_5=0$ we recover their presentation of $QH^*(\pp)$ \cite[9 Eqn.64]{FultonP}. \bibliographystyle{nyjalpha} \ifx\undefined\leavevmode\hbox to3em{\hrulefill}\, \newcommand{\leavevmode\hbox to3em{\hrulefill}\,}{\leavevmode\hbox to3em{\hrulefill}\,} \fi

9703

alg-geom/9703037

en

https://arxiv.org/abs/alg-geom/9703037

alg-geom/9703037

Jim Alexander

J. Alexander, A. Hirschowitz

An Assympotic Vanishing Theorem for Generic Unions of Multiple Points

26 pages, Latex 2e, using diagrams.tex. Revised edition

In this revised form, the proof of the principal lemma has been simplified and the main theorem has been extended to all characteristics for those varieties which are smooth in codimension one. This principal theorem essentially says the following : given an ample line bundle O(1) on a projective variety X and a fixed upper bound M on the multiplicities, there exists a lower bound D such that any generic union of multiple points of multiplicity at most M imposes independent conditions on the sections of O(d) for d>D. Here a multiple point is the closed subscheme defined by a power m of the ideal of a smooth point in X and m is its multiplicity.

alg-geom

\section{Introduction} This work is devoted to the following asymptotic statement : \begin{thm}\label{thmm} Let $X$ be a projective geometrically reduced and irreducible scheme over a field $k$ of (arbitrary) characteristic $p$ and let $\fasm M\, ,\, \fasm L$ be line bundles on $X$ with $\fasm L$ ample. If $p$ is positive then suppose further that $X$ is smooth in codimension one. For fixed $m\geq 0$ there exists $d_0=d_0(m)$, depending also on $X,\fasm L,\fasm M$, such that for any $d\geq d_0$ and any generic union $Z$ of (fat) points of multiplicity $\leq m$ the canonical map $$H^0(X,\fasm M\otimes\fasm L^d)\longrightarrow H^0(Z,\fasm O_Z \otimes\fasm M\otimes \fasm L^d)$$ has maximal rank.\end{thm} Here, as usual, we call (fat) point of multiplicity $m$ in $X$, any subscheme defined by $\fasm I_z^m$, where $\fasm I_z$ is the ideal sheaf of a point $z$ in the smooth locus of $X$. The reader may prefer the following statement, which is more or less equivalent to the preceding one: \begin {cor}Let $X$, $\fasm M\, ,\, \fasm L$, $m$ be as above. There exists an integer $\ell$ such that for any generic union $Z$ of (fat) points of multiplicity at most $ m$ and of total degree (i.e length) at least $\ell$, all the canonical maps $$H^0(X,\fasm M\otimes\fasm L^d)\longrightarrow H^0(Z,\fasm O_Z\otimes\fasm M\otimes \fasm L^d)$$ have maximal rank. \end {cor} Note that the above statement applies as soon as the number of points is at least $\ell$. To simplify the presentation and highlight the essential elements, a detailed proof will only be given in the case $\fasm M=\fasm O_X$. The easy modifications needed to prove the general result are then outlined in \S 7 along with another variant. \begin{rem} The statement of the theorem is false for $p>0$ if we allow $X$ to be singular in codimension one. This is illustrated in the example \ref{cp}. \end{rem} These results are already new in the case where $X$ is the projective plane (with $\fasm M = \fasm O$ and $ \fasm L=\fasm O (1)$). Indeed, even in that case, the expected vanishing theorem for generic unions of fat points [S, Ha, Hi2] is still unproven, see a survey in [G] and more recent contributions in [Xu, ShT, CM1, CM2, M]. Reformulations of the general problem and its relation to other topics have been considered at length in [N, I1, I2, MP]. Much attention has been paid to the ``homogeneous'' case on $\sbm P^n$, namely when all the points have the same multiplicity $m$: see [AC] or [Hi1] for $n=m=2$, [A, AHi1,2,3] for $m=2$ and $n$ arbitrary, [Hi1] for $m=3, n=2, 3$, [LL2] for $m=4, n=2$ in perfectly adjusted cases, and finally [CM1, CM2], where they have settled completely the equal multiplicity cases $m\leq 12, n=2$ by a new and promising method. Concerning the heterogeneous plane case, we can just mention the recent work of Th. Mignon [M], where the case of multiplicities at most four is completely elucidated, using our differential Horace lemma presented below. In [AHi2] we developed a new technique of a differential nature for the case $m=2$ which, in that and later papers, made it possible not only to solve some delicate low-degree cases in [AHi2,3], but also to simplify the proof for the high-degree case [AHi4][C]. The main new ingredient in the proofs of the present paper is an extension of this technique applicable to higher-order fat points ($m> 2$), see lemma \ref{diff} and \ref{prin.cor}. The new lemma does not imply the multiplicity two lemma of [AHi2], and an entirely new proof is needed. In sections 2-7, we present the proof of the theorem. Sections 8-9 are devoted to our differential Horace lemmas. Indeed, the results presented there (see \S 9) are substantially more general than \ref{diff}. While for the present asymptotic statement, \ref{diff} is perfectly sufficient, the full strength of \ref{prin.cor} will be much more efficient for concrete cases with small $n$ and $m$. The proof of \ref{prin.cor} is achieved by an ideal theoretic argument. We would like to point out that our original proof of the lemmas computed the first non-zero derivative of a determinant in a way which owed much to [LL1,2]. \noindent{\bf Outline of the proof of the theorem} In the remainder of this introduction we will try to illustrate the general ideas in the proof of the main theorem in the particular case of the projective plane. We start with a given maximum multiplicity $m$ and a sufficiently large degree $d$. We want to prove a maximal rank statement for a generic union $Z$ of multiple points, which, by adding simple points we can suppose to be of total degree at least $(d+2)(d+1)/2$. Horace's method amounts to specialising some of these points to the generic curve $\Gamma$ of some intermediate degree $\gamma$. Modulo an analogous maximal rank statement on $\Gamma$ which we suppose to hold inductively, our differential lemma can then be applied under certain numerical conditions (holding for large $d$) and we reduce to a new subscheme $\gotm D^{(1)}(Z)$ (the {\it derivative} of $Z$, see \S 4) and a new degree $d-\gamma$. This can be safely applied as long as the current degree, $d_c$, is not too small, say $d_c > \overline{d}$. But when $d_c$ becomes smaller than or equal to $ \overline{d}$, we have to backtrack in order to complete the proof. Our trick is to modify the procedure early on so as to generate in the current subscheme $Z_c$ a sufficient number of unconstrained points of multiplicity at most $m-1$ (of total degree at least $(\overline{d}+2)(\overline{d}+1)/2$). So that when the degree of the current subscheme has been lowered under $\overline{d}$, only points of multiplicity at most $m-1$ remain. Having chosen $\overline{d}$ large enough (i.e. $\overline{d}\geq d_0(m-1)$ in the notation of the theorem), we conclude by induction on $m$. It remains to explain how we generate these free points (see \S 6): our differential lemma generates in $\gotm D^{(1)}(Z)$ a lot of points of multiplicity smaller than $m$, but all of them lie on the exploited divisor $\Gamma$ of degree $\gamma$. The trick here consists in specialising $\Gamma$ to the union of two generic divisors $\Gamma^{\prime}$ and $\Gamma^{\prime\prime}$ of degrees $\gamma^{\prime}$ and $\gamma^{\prime\prime}$, with the desired number of points specialised to say $\Gamma^{\prime}$. If this number of points is sufficiently small with respect to $\gamma^{\prime}$, these points suffer no constraint by being supported on a curve of degree $\gamma^{\prime}$ and are thus freed. Of course, the points remaining on $\Gamma^{\prime\prime}$ should not be too numerous, and we have to find numbers $d_0, \gamma^{\prime}$ and $\gamma^{\prime\prime}$ satisfying all the necessary inequalities. A slight complication arises with the degree of the current divisor $\Gamma_c$. Indeed, the number of free points to be generated is computed in terms of the degree of the divisor which appears at the final stage of the procedure (this degree must be sufficiently large to comply with the induction hypothesis). On the other hand, the initial degree $\gamma$ of the current divisor must be significantly larger to allow the production of enough free points. This compels us to lower the degree of the current divisor, by specialization, at each stage of the procedure (see \S 5). \section{The simplified differential lemma} Throughout this section, $X$ stands for a quasi-projective variety which is geometrically reduced and irreducible, of dimension $n+1$ over a field $k$ of arbitrary characteristic. Since all statements are ``generic'' one can safely suppose $k$ algebraically closed. The hypothesis `$X$ is smooth in codimension one if $char(k)>0$' will not come into play until the proof of the theorem in \S 7. In this section, we present a weakened form of the differential lemma which we prove in \S 9, this form being sufficient for our main theorem. As we already outlined in the previous section, the theorem is proved by a Horace induction argument. In such an argument, specialisation techniques are used to place a certain number of points on a chosen divisor $H$, then the induction hypotheses are applied to the trace and the residual as defined in the \begin{defn}\label{trace-residual} Let $H$ be a Cartier divisor on $X$ and let $W$ be a closed subscheme of $X$. The schematic intersection $$ W^{\prime\prime} =H\cap W $$ defined by the ideal $\fasm I_{H,W^{\prime\prime}}=(\fasm I_H+\fasm I_W) /\fasm I_H$ of $\fasm O_H$ is called the trace of $W$ on $H$ and denoted by $\mbox{Tr}_H(W)$ or simply $W^{\prime\prime}$ if no confusion is possible. The closed subscheme of $X$ defined by the conductor ideal $\fasm I_{W^{\prime}}=(I_W :I_H)$ is called the residual of $W$ with respect to $H$ and denoted by $\mbox{Res}_H(W)$ or $W^{\prime}$. The canonical exact sequence \begin{equation}\label{res.exact}0\longrightarrow \fasm I_{W^{\prime}}(-H)\longrightarrow \fasm I_W \longrightarrow \fasm I_{H,W^{\prime\prime}}\longrightarrow 0\end{equation} is called the residual exact sequence of $W$ with respect to $H$. \end{defn} \subsection{Geometric intuition for the differential lemma} Here we try to share with the reader our intuition for our differential lemma. Suppose that $X$ is projective and let $\fasm L$ be a line bundle on $X$. We will keep the notation of the definition in the remainder of the discussion. The first thing one needs to take note of is that any basic Horace type argument is based on the following trivial consequence of the residual exact sequence (\ref{res.exact}): $$ \mbox {if $h^0(X,\fasm I_{H,W^{\prime\prime}}\otimes \fasm L|_H)=0$ and $h^0(X,\fasm I_{W^{\prime}}\otimes \fasm L(-H))=0$ then $h^0(X,\fasm I_W\otimes \fasm L)=0$}$$ For this to apply, one must have aprori $\mbox{deg}W^{\prime\prime} \ge h^{0}(H,\fasm L_H)$ and $\mbox{deg}W^{\prime} \ge h^{0}(X,\fasm L(-H))$. In fact to be generally applicable in an induction argument, the stronger requirement $\mbox{deg}W^{\prime\prime} = h^{0}(H,\fasm L_H)$ is needed. We will therefore say that $h^{0}(H,\fasm L_H)$ is the {\em critical degree}. In practice one starts with some general union $G$ of multiple points, then, by specialising them one by one to the chosen divisor $H$ one hopes to obtain a specialisation $W$ for which the trace has the critical degree. Since each point specialised to $H$ increases the degree of the trace by at least the multiplicity of the point, it is not always possible to get exactly the critical degree using this process. This is the technical obstacle that the differential lemmas \ref{prin.lem} and \ref{diff} are designed to overcome. To see how this comes about, it is enough to consider that $H$ is a line in the affine plane $X$. The ideal of a point $Z$ of multiplicity $r$ at the origin is then \begin{equation}\label{ideal}\gotm (x,y)^r =\gotm n^r \oplus\gotm n^{r-1} y \oplus \cdots \oplus \gotm n y^{r-1}\oplus (y^r)\end{equation} where $\gotm n$ is the ideal $(x)\subset k[x]$, and each $\gotm n^i$ is the ideal of a point of multiplicity $i$ in $H$. In particular the trace corresponds to $\gotm n^r$. One can then consider that $Z$ is formed by infinitesimally piling up the subschemes of $H$ with ideals $\gotm n^i$. We then refer to these subschemes of $H$ as the layers of $Z$. Of course only the trace given by $\gotm n^r$ is actually contained in $H$, the others only appear in successive infinitesimal neighbourhoods of $H$. Now if we consider $Z$ as the limit of a multiple point that is translated to the origin along the $y$-axis, it's the layer of highest multiplicity that arrives in $H$ (or, as might be said, arrives first) and the degree of the trace increases by $r$. In the differential approach, one or more points are translated to as many distinct points supported in $H$. The rate of approach may differ, but all arrive at the same time. Our corollary \ref{prin.lem} says that if some sequence of layers, one from each point, have degrees adding up to the critical degree, then one can consider that these arrive first and then take their union as the (differential) trace , while the subsequent remainder becomes the (differential) residual. Precisely, with respect to the ideal (\ref{ideal}), if the layer corresponding to $\gotm n^p$ is taken at that point to be its contribution to the (differential) trace, then the (differential) residual at that point is the subscheme of the plane defined by the ideal $$\gotm n^r\oplus\gotm n^{r-1}y\oplus \cdots \oplus\gotm n^{p+1}y^{r-p-1}\oplus\gotm n^{p-1}y^{r-p}\oplus\cdots \gotm n y^{r-2}\oplus (y^{r-1}).$$ obtained by {\em slicing off} the corresponding layer. If the cohomology vanishes as before when the trace and residual are replaced by the chosen differential versions, then the lemma says that the cohomology vanishes for $\fasm I_G\otimes \fasm L$. The conclusion now concerns the general union $G$ and not the specialisation $W$. \begin{center}\includegraphics{figure1.eps}\end{center} Figure 1. illustrates an example where $X$ is the affine plane and $H$ is a line, while the critical degree is supposed to be five. Example A shows two points of multiplicity four in the plane. The shaded region represents the trace while the unshaded region represents the residual with respect to the line $H$. From the standard specialisation point of view, these points are translated, one by one, to $H$ giving a trace of degree four, then eight, so that five is unattainable. The examples B and C show two possibilities for choosing the differential traces (shaded part) so that the critical degree is obtained. The differential residuals correspond to the unshaded part. \subsection{The simplified lemma} The simplication of the following lemma with respect to that in \S 9, is as follows : in the process just described, instead of choosing arbitrary slices, we will systematically take the smallest non-trivial one, which is just a simple point of $H$. In this case, the (differential) residual falls within the bounds of the following definition. \begin{defn}\label{simple residue} Let $H$ be a reduced Cartier divisor on $X$ and let $z$ be a non-singular point of $H$. We define the $m^{th}$ simple residue; denoted $D_{H,m}(z)$ or $D_{m}(z)$ if no confusion can arise; to be the trace of $z^m$ on $(m-1)H$; $$D_m(z)=z^m\cap H^{m-1}.$$ We will say that $m$ is the {\em multiplicity} of the simple residue. \end{defn} With this definition, our simplified lemma, which will be proved in \S9, reads as follows. \begin{lem}\label{diff} Suppose $X$ is projective and furnished with a line bundle $\fasm L$, and let $H$ be a reduced and irreducible effective Cartier divisor on $X$. Let $Z_0$ be a zero-dimensional subscheme of $X$, and let $a$, $d$ be positive integers. We suppose that $$r\; =\; h^0\left(H,\fasm L|_H\right) - \mbox{deg}\left(Tr_{H}(Z_0)\right) \geq 0$$ and that $m_1,\ldots ,m_r$ are positive integers satisfying $$\mbox{deg}(Z_0)+\sum_{i=1}^r\,\mbox{$\binom{m_i+n}{n+1}$}\; \geq \; h^0(X,\fasm L)$$ Let $P_1,\ldots ,P_r$ be generic points in $X$ and $Q_1,\ldots ,Q_r$ be generic points in $H$. In the notation of \ref{trace-residual} and \ref{simple residue}, set $$T=Z_0\cup P_1^{m_1}\cup \cdots \cup P_r^{m_r}\quad ;\quad T^{\prime}_{\star}=Z_0^{\prime}\cup D_{m_1}(Q_1)\cup \cdots \cup D_{m_r}(Q_r) \quad ; \quad T^{\prime\prime}_{\star} = Z_0^{\prime\prime}\cup Q_1\cup \cdots \cup Q_r$$ Then $H^0(X , \fasm I_{T}\otimes\fasm L)=0$ holds as soon as the following two conditions are satisfied :\vspace{2ex}\\ \makebox[5cm][l]{(dime)}\mbox{$\displaystyle\begin{array}{lll}H^0(H,\fasm I_{T^{\prime\prime}_{\star}}\otimes \fasm L|_H)&=&0\end{array}$}\vspace{2ex}\\ \makebox[5cm][l]{(degue)}\mbox{$\displaystyle \begin{array}{lll} H^0(X,\fasm I_{T^{\prime}_{\star}}\otimes \fasm L(-H))&=&0.\end{array}$} \end{lem} \begin{rem} The dime and degue concern respectively the differential trace and the differential residual as discussed above. \end{rem} \section{\label{CC}Configurations and candidates} Here we introduce the general class of subschemes of $X$ which we will be dealing with. From here on, $X$ is projective of dimension $n+1$ and furnished with an ample line bundle $\fasm O(1)$ of degree $\nu$. We let $\alpha_0$ be the least integer such that $\fasm O(a)$ is very ample for $a\geq \alpha_0$ and it will henceforth be understood that $a\geq \alpha_0$. \begin{defn}\label{candidates} Let $G_a$ be the generic effective divisor in the linear system $\bigl|H^0(X,\fasm O(a))\bigr|$. A $G_a$-residue or just {\bf residue}, will be any point or simple residue (see \ref{simple residue}) with support in $G_a$. The {\bf multiplicity} of a residue will be its multiplicity as a point, or as a simple residue \ref{simple residue}, respectively. \newline Given positive integers $a,m$ an $(a,m)$-{\bf configuration} will be any subscheme $Z$ of $X$ which is a generic union of points of multiplicity at most $m$ in $X$, called the free part of $Z$ and denoted $\mbox{Free}(Z)$, with a generic set of $G_a$-residues equally of multiplicity at most $m$, called the constrained part of $Z$ and denoted $\mbox{Const}(Z)$. \newline Given a positive integer $d$, we say that an $(a,m)$-configuration $Z$ is a $(d,m,a)$-{\bf candidate} if the following two conditions hold: $$h^0(X, \fasm O(d))\leq \mbox{deg} (Z)$$ and $$\mbox{deg}\left( Tr_{G_a}(Z)\right)\leq h^0(G_a,\fasm O_{G_a}(d)).$$ We consider a $(d,m,a)$-candidate $Z$ to be a candidate for the property $h^0(X,\fasm I_Z(d))=0$ and we say that $Z$ is {\bf winning} if this property holds.\end{defn} The bound for $\mbox{deg} (Z)$ in the definition of candidates is for convenience: a vanishing statement for a more general configuration will be treated by considering the candidate obtained by adding the right number of simple points. The following easy lemma says that for large $d$, candidates contain sufficiently many free points. \begin{lem}\label{free} Let $m$ and $a$ be positive integers. For any $(d,m,a)$-candidate $Z$, we have $$\mbox{deg}(\mbox{Free}(Z))\geq \nu \frap{d^{n+1}}{(n+1)!}- O(d^n),$$ where $\nu$ is the degree of $X$. \end{lem} For presentation purposes we introduce the \begin{defn}\label{dvm} Given a polarised pair $(V,\fasm O(1))$ and $m>0$ we define $\mbox{\bf d}(V,m)$ to be the least degree (a-priori possibly infinite, and a-posteriori finite by our theorem) such that for $d\geq \mbox{\bf d}(V,m)$ any $(d,m,0)$-candidate is winning.\end{defn} \section{Derivatives} In practice, when we apply lemma \ref{diff}, we think of the condition (dime) as being satisfied. This is easily justified by an induction hypothesis on the dimension (i.e. precisely that $d(G_a,m)$ is finite). Lemma \ref{diff} is then a justification for replacing $T$ by $T_{\star}^{\prime}$. This leads us to introduce a formal operator $\gotm D$ sending one $(a,m)$-configuration to another which we call the derivative (see \ref{D1}). Of course, we are especially interested in the case where this operator takes $(d,m,a)$-candidates to $(d-a,m,a)$-candidates. In the present section, we define the derivative and show that it behaves well for large $d$. Here is the idea behind the definition of a derivative. Given a $(d,m,a)$-candidate $Z$, we wish to apply our lemma \ref{diff} as follows. We specialize the $s$ biggest free points of $Z$ onto the divisor $G_a$, with $s$ as large as possible. Still a few conditions (say $r$) are missing in $G_a$, and we require that $r$ further free points be available in $Z$ so that we may apply \ref{diff}. In that case, the derivative of $Z$ is the subscheme $T^{\prime}_{\star}$ involved in the degue condition of \ref{diff}. \begin{defn}\label{D1} Let $Z$ be a $(d,m,a)$-candidate on $X$ with $t=t(Z)$ free points $P_1^{m_1},\ldots ,P_t^{m_t}$, where the multiplicities appear in non-decreasing order. Let $s=s(Z)\leq t$ be the greatest integer such that $$\mbox{deg} (Tr_{G_a}(Z))+\mbox{$\binom{m_1+n-1}{n}+ \cdots + \binom{m_s+n-1}{n}$}\leq h^0(G_a,\fasm O_{G_a}(d)).$$ and set $$ r=r(Z) = h^0(G_a,\fasm O_{G_a}(d)) - \mbox{deg} (Tr_{G_a}(Z))-\mbox{$\binom{m_1+n-1}{n}- \cdots - \binom{m_s+n-1}{n}$}.$$ We say that $Z$ is {\bf derivable} with respect to $G_a$ if $$ r+s\leq t. $$ If $Z$ is a derivable $(d,m,a)$-candidate, its {\bf derivative} with respect to $G_a$, denoted $\gotm D^{(1)}(Z)$, is defined to be the $(a,m)$-configuration $$\begin{array}{lll} \gotm D^{(1)}(Z)&=&P_{s+r+1}^{m_{s+r+1}}\; \cup\; \cdots\; \cup \; P_t^{m_t}\; \cup\; \mbox{Const}(Z)^{\prime}\; \cup \\ &&\\ &&\hspace{3cm} Q_1^{m_1-1}\; \cup \;\cdots\; \cup \; Q_s^{m_s-1}\; \cup\\ &&\\ &&\hspace{4.5cm} D_{m_{s+1}}(Q_{s+1})\; \cup\; \cdots\; \cup\; D_{m_{s+r}}(Q_{s+r}) \end{array}$$ where $Q_1,\ldots ,Q_{s+r}$ are generic points of $G_a$ and the notation is that of \ref{trace-residual} and \ref{simple residue}. \end{defn} Recall that $\alpha_0$ is an integer such that $\fasm O(a)$ is very ample for $a \ge \alpha_0$. What we need to know about the derivative is the following : \begin{lem}\label{der} Let $a\geq \alpha_0$ and $m$ be positive integers. Then there exists an integer $\mbox{\bf der}(a,m)$ such that for any $d\geq \mbox{\bf der}(a,m)$ and any $(d,m,a)$-candidate $Z$ on $X$: \begin{enumerate} \item $Z$ admits a derivative $\gotm D^{(1)}(Z)$; \item for any $N$, if $Z$ has either no free point of multiplicity $m$ or at least $N$ free points of multiplicity less than $m$, then so does $\gotm D^{(1)}(Z)$; \item The degree of the trace of $\gotm D^{(1)}(Z)$ satisfies the following estimate, where, as above, $\nu = deg (\fasm O(1))$:\\ $\begin{array}{lll} \mbox{deg Tr}_{G_a}\left( \gotm D^{(1)} (Z)\right) &\;=\;& \left(\frap{(m-1)\, a\, \nu}{m+n-1}\right)\; \frap{d^n}{n!}+O(d^{n-1})\\ &\;=\;& h^0(G_a,\fasm O_{G_a}(d-a))- \left(\frap{n\, a\, \nu}{m+n-1}\right)\; \frap{d^n}{n!}+O(d^{n-1});\\ \end{array}$ \item $\gotm D^{(1)}(Z)$ is a $(d-a,m,a)$-candidate; \item if $d(G_a,m)$ is finite and $\gotm D^{(1)}(Z)$ is winning, then so is $Z$. \end{enumerate} \end{lem} \noindent{\bf Proof.} In order to prove 1., it is enough to prove that the number of free points in $Z$ is larger than $2h^0\bigl(G_a,\fasm O_{G_a}(d)\bigr)$. The latter is bounded by $Cd^n$ for some constant $C$, so we may conclude by \ref {free}. As for 2., it is an immediate consequence of the definition of the derivative. For 3., let $r,s, t$ and $m_i$ be as in \ref{D1}. Then $$\frap{m+n-1}{n}\left(\mbox{$\sum_{i=1}^{s}$}\mbox{$\binom{m_i +n-2}{n-1}$}\right)\geq \mbox{$\sum_{i=1}^{s}$}\mbox{$\binom{m_i +n-1}{n}$}=h^0(G_a,\fasm O_{G_a}(d))-r$$ and $$\begin{array}{ccl}\mbox{deg Tr}_{G_a}\left(\gotm D^{(1)}(Z)\right)&\leq& \sum_{i=1}^{s}\mbox{$\binom{m_i +n-2}{n}$}+r\, \mbox{$\binom{m+n-1}{n}$} \vspace{1ex}\\ & = & \left( \sum_{i=1}^{s} \mbox{$\binom{m_i+n-1}{n}$}+r\right) - \sum_{i=1}^{s}\mbox{$\binom{m_i+n-2}{n-1}$}\vspace{1ex}\\ &&- r+r \,\mbox{$\binom{m+n-1}{n}$} \vspace{1ex}\\ & \leq & h^0(G_a,\fasm O_{G_a}(d)) -\frap{m-1}{m+n-1}\biggl( h^0 (G_a,\fasm O_{G_a}(d)) - r)\biggr)\vspace{1ex}\\ &&- r +r\, \mbox{$\binom{m+n-1}{n}$}\vspace{.5ex}\\ & \leq & \frap{n}{m-1}\left( h^0 (G_a,\fasm O_{G_a}(d))\right)+\mbox{$\binom{m+n-1}{n}$}^2\vspace{1ex}\\ &= & \left(\frap{(m-1)\, a\, \nu}{m+n-1}\right)\; \frap{d^n}{n!}+O(d^{n-1}). \end{array}$$ Finally, we have $$\begin{array}{ccl} h^0(G_a,\fasm O_{G_a}(d-a))-\mbox{deg Tr}_{G_a}\left(\gotm D^{(1)}(Z)\right) & \geq & h^0(G_a,\fasm O_{G_a}(d))-\frap{m-1}{m+n-1}h^0(G_a,\fasm O_{G_a}(d)) \vspace{1ex}\\&&-\mbox{$\binom{m+n-1}{n}^2$} \vspace{1ex}\\ &= & \left(\frap{n\, a\, \nu}{m+n-1}\right)\; \frap{d^n}{n!}+O(d^{n-1}). \end{array}$$ For 4., we first note that, when $\gotm D^{(1)}(Z)$ is defined and $H^1(X,\fasm O(d-a))=0$, one has $$h^0(X,\fasm O(d-a))\leq \mbox{deg}\left(\gotm D^{(1)}(Z)\right) = \mbox{deg}(Z)-h^0(G_a,\fasm O_{G_a}(d)).$$ This means that, for sufficiently large $d$, the $(a,m)$-configuration $\gotm D^{(1)}(Z)$ is a $(d-a,m,a)$-candidate, since by 3., its trace on $G_a$ has degree at most $h^0(G_a,\fasm O_{G_a}(d-a))$. For 5., using the notation of \ref{D1}, we apply \ref{diff}, with $Z_0$ the closed subscheme $$\mbox{Const}(Z)\; \cup \; Q_1^{m_1}\; \cup \cdots \cup \; Q_s^{m_s}\;\cup \; P_{s+r+1}^{m_{s+r+1}}\; \cup \cdots \cup\; P_t^{m_t}.$$ Let $W=Q_1^{m_{s+1}}\cup \cdots \cup Q_r^{m_{s+r}}$. The dime of \ref{diff} holds for $d\geq d(G_a,m)$, while the degue of \ref{diff} is just the hypothesis that $\gotm D^{(1)}(Z)$ is winning, so the lemma follows from \ref{diff}. $\hfill \square$ \section{Concentrated derivatives} \label {CD} If theorem \ref{thmm} were true for low degrees, then repeated applications of lemma \ref{diff}, hence of the derivative, would suffice to prove the theorem by induction on the degree. Instead one must modify the process and try to reduce the multiplicities of the free points, thus ending the proof by induction on the multiplicity. This is done using a specialisation of the second derivative (see \ref{D(2)}): bearing in mind the semi-continuity of the cohomology, one easily sees that the (degue) of \ref{diff} holds if it holds for some specialisation of $T^{\prime}_{\star}$. A complication arises with the degree of the base divisor $G_a$ which must be lowered during the induction on $d$ before an induction hypothesis on $m$ allows one to finish the proof. We get around this problem using a specialisation of the first derivative which we call a concentrated derivative. In this section we introduce this concentrated derivative and prove results analogous to those for derivatives. \begin{defn}\label{defcd} Let $d\, ,m\, , a$ be positive integers with $a>1$, and let $Z$ be a derivable $(d,m,a)$-candidate. We define the {\bf concentrated derivative} of $Z$ with respect to $G_a$, denoted $\gotm D^{(1)}_c(Z)$, to be the $(a-1,m)$-configuration obtained from $\gotm D^{(1)}(Z)$ by degenerating $G_a$ to the generic union $G_1+ G_{a-1}$ and specialising all $G_a$-residues of $\gotm D^{(1)}(Z)$ to have generic support in $G_{a-1}$. \end{defn} What we need to know about the concentrated derivative is concentrated in the following: \begin{lem}\label{derc} Given $m>0$ there exists an integer $A(m)$ such that for all $a\geq A(m)$ there exists an integer $\mbox{\bf derc}(a,m)$ such that for any $d\geq \mbox{\bf derc}(a,m)$ and any $(d,m,a)$-candidate $Z$: \begin{enumerate} \item $Z$ admits a concentrated derivative $\gotm D^{(1)}_c(Z)$ which is a $(d-a,m,a-1)$-candidate; \item for any $N$, if $Z$ has either no free point of multiplicity $m$ or at least $N$ free points of multiplicity less than $m$, then so does $\gotm D^{(1)}_c(Z)$; \item if $d(G_a,m)$ is finite and $\gotm D^{(1)}_c(Z)$ is winning, then so is $Z$. \end{enumerate} \end{lem} \noindent{\bf Proof.} For 1., let $A(m)$ be an integer $a$ satisfying $$\frap{m-1}{m+n-1}A(m) < A(m)-1.$$ Then, for $a\geq A(m)$ and $d$ sufficiently large, for any $(d,m,a)$-candidate $Z$, we have, by (\ref{der}.3), $$\mbox{deg}\, \mbox{Tr}_{G_{a-1}}\left( \gotm D^{(1)}_c(Z)\right) = \mbox{deg}\, \mbox{Tr}_{G_a}\left( \gotm D^{(1)}(Z) \right)\leq h^0(G_{a-1}, \fasm O_{G_{a-1}}(d-a))$$ so that $\gotm D^{(1)}_c(Z)$ is a $(d-a,m,a-1)$-candidate. As for 2., it follows from the similar statement for the derivative, since the derivative and the concentrated derivative have the same free points. For 3., if $\gotm D_c^{(1)}(Z)$ is winning then so is $\gotm D^{(1)}(Z)$, since the former is a specialisation of the latter. We conclude that $Z$ is winning for $d\geq \mbox{\bf der}(a,m)$ by (\ref{der}.5).$\hfill \square$ \section {Special second derivative} In this section, we explain the construction which generates free points. This corresponds to a modified second derivative, which we denote by $\gotm D^{(2)}[\alpha]$, where $\alpha$ is an integer. \begin{defn}\label{D(2)}Let $m,a>0 $, and let $Z$ be a twice derivable $(d,m,a)$-candidate. Let $r^{(2)}(Z)$ be the number of residues of $\gotm D^{(2)}(Z)$ which are points, necessarily of multiplicity at most $m-1$. For $0<\alpha <a$, we set $$r^{(2)}[\alpha](Z)=\mbox{min}\left(h^0(X,\fasm O(\alpha))-1,r^{(2)}(Z)\right)$$ and define $\gotm D^{(2)}[\alpha](Z)$ to be the specialisation of the second derivative $\gotm D^{(2)}(Z)$ obtained by degenerating $G_a$ and its residues to the generic union $G_{\alpha}+G_{a-\alpha}$ with $r^{(2)}[\alpha](Z)$ of the residues which are points specialised to have generic support in $G_{\alpha}$, and all other residues specialised to $G_{a-\alpha}$.\end{defn} Here is what we need to know about this construction. \begin{lem}\label{sder} Given $m,N$, there exist integers $\alpha$ and $a_0>\alpha $ such that for all $a\geq a_0$ there exists $d'_0=d'_0(m,N,a)$ such that for $d\geq d'_0$ and any $(d,m,a)$-candidate $Z$: \begin{enumerate} \item $Z$ is twice derivable and $\gotm D^{(2)}[\alpha](Z)$ is a $(d-2a , m ,a-\alpha)$-candidate having either no free point of multiplicity $m$ or at least $N$ free points of multiplicity at most $m-1$; \item if $d(G_a,m)$ and $d(G_{a-\alpha},m)$ are finite and $\gotm D^{(2)}[\alpha](Z)$ is winning, then so is $Z$. \end{enumerate} \end{lem} \noindent{\bf Proof.} Let $\alpha=\alpha(N)$ be an integer satisfying $h^0(X,\fasm O(\alpha)) > N.$ For 1., let $a_0> \alpha $ be such that for $a-\alpha > \frac{m-1}{m+n-1}a$ for $a\geq a_0$. Then for $a \geq a_0$, and $d>\!\!> 0$, we have $$\mbox{deg}\, \mbox{Tr}_{G_a}\left(\gotm D^{(2)}(Z)\right) < h^0(G_{a-\alpha}, \fasm O_{G_{a-\alpha}}(d-2a))$$ because by (\ref{der}.3) $$\mbox{deg}\, \mbox{Tr}_{G_{a}}\left(\gotm D^{(2)}(Z)\right)\leq \frap{m-1}{m+n-1}a\nu \frap{d^n}{n!}+ O(d^{n-1})$$ while $$h^0(G_{a-\alpha} , \fasm O_{G_{a-\alpha}}(d-2a))\geq (a-\alpha)\nu\frap{d^n}{n!}+ O(d^{n-1}).$$ This implies that $\gotm D^{(2)}[\alpha](Z)$ is a $(d-2a , m ,a-\alpha)$-candidate. Now for $d>\!\!> 0$, by (\ref{der}.3), we have $$h^0(G_{a},\fasm O_{G_a}(d-a))-\mbox{deg}\, \mbox{Tr}_{G_a}\, (\gotm D^{(1)}(Z))\geq N\mbox{$\binom{m+n-1}{n}$}$$ so that, in the notation of \ref{D1}, $s(\gotm D^{(1)}(Z))\geq N$. If $\gotm D^{(1)}(Z)$ has at least $N$ free points of multiplicity $m$, then $\gotm D^{(2)}[\alpha](Z)$ has $N$ free points of multiplicity $m-1$: indeed, in that case, \mbox{$r^{(2)}[\alpha](Z)$} is larger than $N$, and the \mbox{$r^{(2)}[\alpha](Z)$} points specialised to \mbox{$G_{\alpha}$} are without constraint, since any set of $N$ points lie on an effective divisor in the linear system \mbox{$\bigl|H^0(X,\fasm O(\alpha))\bigr|$}. Otherwise, $\gotm D^{(2)}[\alpha](Z)$ has no more free points of multiplicity $m$. For 2., we observe that the second derivative $\gotm D^{(2)}(Z)$ is also a winning candidate, and conclude by applying twice (\ref {der}.5). $\hfill \square$ \section{Proof of the theorem} \subsection{A proposition implying the theorem} The following proposition (which we prove below \ref{prp}) sums up the efforts of the previous sections and, as we willl now show, easily implies our theorem. \begin{prop}\label {mainp} Let $X$ be a projective, geometrically reduced and irreducible variety of dimension $n+1$ over a field $k$ of arbitrary characteristic $p$. If $p > 0$, suppose further that $X$ is smooth in codimension one if $p>0$. Let $\fasm O(1)$ be an invertible ample bundle on $X$. Given $m>0$, there exists $a_0(m)$ such that for any $a\geq a_0(m)$ there exists $d_0(a,m)$ such that for all $d\geq d_0(a,m)$ any $(d,m,a)$-candidate is winning.\end{prop} \noindent{\bf Proof of \ref {thmm}.} We first handle the case where $\fasm M = \fasm O$. We take $d_0(m)=d_0(a_0(m),m)$ and consider some $d\geq d_0$ and some generic union $Z$ of (fat) points of multiplicity $\leq m$. If the degree of $Z$ is smaller than $h^0(\fasm O(d))$, we reduce to the case with equality by adding generic simple points. Since the trace on $G_{a_0}$ is empty, we may consider $Z$ as a $(d,m,a_0)$-candidate (\ref{candidates}), and conclude by \ref {mainp}. As announced, we only gloss over the proof in the case where $\fasm M$ is arbitrary. Firstly, replacing $\fasm M$ by $\fasm M \otimes \fasm L ^b$, we may suppose that $\fasm M$ is effective. Next, we can suppose as above $deg \, Z \geq h^0(\fasm M \otimes \fasm L^d)$ and we have to prove that $H^0(X, \fasm I_Z \otimes \fasm M \otimes \fasm L^d)=0$. The idea of the proof is then to choose a suitable $a$ and to apply \ref {diff} as in \ref{der} using the generic divisor $G^{\star}_a$ in $\vert H^0(X,\fasm M \otimes \fasm L ^a) \vert$. By induction on the dimension we can suppose that the dime condition holds. To prove the degue condition, we degenerate $G^{\star}_a$ to $M \cap G_a$, where $M$ is in $\vert H^0(X,\fasm M )\vert$ and $G_a$ is the generic divisor in $\vert H^0(X, \fasm L^a) ,\vert$; specializing all the residues onto $G_a$. In this way, we get a $(a,m)$-configuration $Z^{\prime}_c$ and, if this is a $(d-a, m, a)$-candidate, we can end with the particular case ($\fasm M= \fasm O$) since such a candidate is winning for sufficiently large $d$. To see that $Z_c^{\prime}$ is a $(d-a,m,a)$-candidate for suitable large $a$ and $d$, one persues an argument analogous to \ref{der}.1 and one shows that for any $a$, and all $d$ sufficiently large with respect to $a$, $Z$ has enough free points to make \ref {diff} applicable. Then, as in \ref{derc}.1, one shows that for sufficiently large $a$ and all $d$ sufficiently large with respect to $a$, the $(a,m)$-configuration $Z^{\prime}_c$ is a $(d-a, m, a)$-candidate.$\hfill \square$ \begin {rem} A further generalisation would be to take a fixed closed (zero-dimensional) subscheme $V_0$ and its union with points of multiplicity $\leq m$. The union of $V_0$ with sufficiently many generic simple points has maximal rank in all degrees giving the initial case for an induction on the multiplicity, while the proof of the dimension one case is virtually unchanged. \end{rem} \subsection{\label{prp}Proof of the proposition} To prove the proposition, we argue by induction on the dimension $n+1$. Note that in all characteristics, the generic effective divisor in a very ample linear system on a variety $X$ of dimension $>1$ is a variety which is smooth outside the singular locus of $X$ (see [L] VII 13). Thanks to the initial cases \ref{ci} and \ref{cpi} below, we may suppose that the proposition has been proven for multiplicity $m$ in dimension $n$ and for multiplicity $m-1$ in dimension $n+1$. This implies that $\mbox{\bf d}(G_a,m)$ is finite for all $a\geq 1$ and that there exists $a_0(m-1)$ such that for $a\geq a_0(m-1)$ there exists $d_0(a,m-1)$ such that for $d\geq d_0(a,m-1)$ any $(d,m-1,a)$-candidate is winning. We proceed in three steps. \noindent{\bf First step.} With the notation of \ref{der} and \ref{derc}, we define $$b_0=\mbox{max}\bigl(A(m), a_0(m-1)\bigr),$$ $$\Delta=\mbox{max}\bigl(\mbox{\bf der}(b_0,m),d_0(b_0,m-1)+b_0 \bigr)$$ and $$N=h^0(X,\fasm O(\Delta+b_0 -1)) + \binom{n+m-1}{n},$$ and prove by induction that, for any $d \geq \Delta$, any $(d,m,b_0)$-candidate with either no free point of multiplicity $m$ or at least $N$ free points of multiplicity less than $m$ is winning. We start with the case $\Delta \leq d < \Delta + b_0$, and consider a $(d,m,b_0)$-candidate $Z$ with either no free point of multiplicity $m$ or at least $N$ free points of multiplicity less than $m$. If $\mbox{deg}(Z)\geq h^0(X,\fasm O(d)) + \binom{n+m-1}{n}$, and $Z$ has a free point of multiplicity $m$, we may replace $Z$ by the subscheme obtained by diminishing by one the multiplicity of this free point, which still has at least $N$ free points of multiplicity less than $m$. In other words, we may suppose either that $Z$ has no free point of multiplicity $m$, or that $\mbox{deg}(Z) < h^0(X,\fasm O(d)) + \binom{n+m-1}{n}$ holds. In the latter case, there is no room for $N$ free points of multiplicity less than $m$. Summing up, we can suppose that $Z$ has no free point of multiplicity $m$. Thanks to $d \geq \mbox {\bf der}(b_0,m)$ and \ref {der}, $Z$ has a first derivative $\gotm D^{(1) }(Z)$ which is a $(d-b_0,m-1,b_0)$-candidate. Thanks to $d-b_0 \geq d_0(b_0,m-1)$, this candidate is winning. Thanks to $d \geq \mbox {\bf der}(b_0,m)$ and \ref {der} again, $Z$ is winning too. For $d\geq \Delta + b_0$ let $Z$ be a $(d,m,b_0)$-candidate having either no free point of multiplicity $m$, or at least $N$ free points of multiplicity at most $m-1$. Thanks to $d \geq \mbox {\bf der}(b_0,m)$ and \ref {der}, $Z$ has a first derivative $\gotm D^{(1) }(Z)$ which is a $(d-b_0,m,b_0)$-candidate having either no free point of multiplicity $m$ or at least $N$ free points of multiplicity at most $m-1$. Thanks to the inductive assumption, $\gotm D^{(1) }(Z)$ is winning. Again thanks to $d \geq \mbox{\bf der}(b_0,m)$ and \ref {der}, $Z$ is winning too. \noindent{\bf Second step.} Here we prove that for any $b\geq b_0$ there exists $\delta =\delta(b,m)$ such that for $d\geq \delta $ any $(d,m, b)$-candidate having either no free point of multiplicity $m$, or at least $N$ free points of multiplicity at most $m-1$ is winning. The proof is by induction on $b$. The initial case $b=b_0$ is the previous step. For the induction step, we take $\delta (b) = max ( \mbox {\bf derc}(b,m), \delta (b-1)+b)$. The statement then follows by \ref {derc}, which applies because $b_0 \geq A(m)$. \noindent{\bf Final step.} Here we set $a_0=a_0(m)=max (b_0+\alpha(N), a_0 (m,N))$ where $\alpha=\alpha(N)$ and $a_0 (m,N))$ are defined in \ref {sder}, and, for $a \geq a_0$, $d_0=d_0(a,m)= max (d'_0 (m,N,a), \delta(a-\alpha,m) +2a)$, and we prove the full statement, namely that, for $d\geq d_0(a,m)$, any $(d,m,a)$-candidate $Z$ is winning. Indeed, by \ref {sder} applied to $n,N$, $Z$ is twice derivable and $\gotm D^{(2)}[\alpha](Z)$ is a $(d-2a , m ,a-\alpha)$-candidate having either no free point of multiplicity $m$ or at least $N$ free points of multiplicity at most $m-1$. Since $d-2a \geq \delta(a-\alpha,m)$, this candidate is winning by the second step. This implies that $Z$ itself is winning by \ref {sder}. $\hfill \square$ \subsection{The proposition in dimension one} The initial case $n=0$ can be deduced from the following general results for curves. We first treat the characteristic zero case with the \begin{prop} \label{ci} Let $C$ be a geometrically irreducible quasi-projective curve over a field $k$ of characteristic zero. Let $V\subset H^0(C,\fasm L)$ be a linear subspace of finite dimension $v$ of global sections of the invertible sheaf $\fasm L$ on $C$. Let $x_1,\ldots ,x_r$ be the generic set of $r$ closed points of $C$ defined over the function field $K$ of $C\times \cdots \times C$ ($r$ factors), and let $m_1,\ldots ,m_r$ be positive integers. Let $D$ be the divisor $m_1x_1 + \cdots + m_rx_r$ on $C_K=C\times_k K$. Then the canonical map $$V\longrightarrow H^0(C_K,\fasm O_D\otimes \fasm L)$$ has maximal rank.\end{prop} \noindent{\bf Proof.} If $v\neq m=\sum_i m_i$ , one can either diminish the multiplicities or add (generic) free points and suppose that $v=m$. Since the property is open, we can specialise to the case of a single point $x$ and the divisor $D=mx$. In this case the proposition is equivalent to showing that the determinant of the canonical map \begin{equation}\label{pp}V\otimes \fasm O_C \longrightarrow \mbox{P}^v(\fasm L)\end{equation} is not identically zero, where $\mbox{P}^v(\fasm L)$ is the sheaf of $v^{th}$ order principal parts of $\fasm L$. For this we can suppose that the base field is algebraically closed and, since this map commutes with localisation and the completion at a closed point of $C$, it is sufficient to show that the canonical map $$V\otimes k[[t]]\longrightarrow k[[t,x]]/((x-t)^{v})$$ $$ f \mapsto f(t)+f^{\prime}(t)(x-t)+ f^{\prime\prime}(t)\frac{(x-t)^2}{2!} +\cdots +f^{(v-1)}(t)\frac{(x-t)^{v-1}}{(v-1)!}$$ has maximal rank. Choosing a basis $f_1,\ldots ,f_v$ for $V$, the determinant of this map is just the Wronskian $$W(f_1,\cdots ,f_v)=\mbox{det}\left[\frac{\partial^i f_j}{\partial t^{i}}\right]$$ which, as is well known, has maximal rank for $f_1,\ldots ,f_v$ linearly independent.$\hfill \square$ We now give the initial case for smooth curves in arbitrary characteristic. \begin{prop}\label{cpi} Let $C$ be a smooth, geometrically connected, projective curve of genus $g$ over an arbitrary field. Let $\fasm M$, $\fasm L$ be line bundles on $C$ with $\fasm L$ ample, let $m>0$ be an integer and let $d_0(m)$ be the least integer $d$ such that $h^0(C,\fasm M\otimes \fasm L^d) > m(m-1)(g-1)/2$ and $\fasm M\otimes \fasm L^d$ is non-special. Let $x_1,\ldots ,x_r$ be generic points on $C$ and let $Z$ be the divisor $m_1x_1+\cdots +m_rx_r$ where $0<m_i\leq m$ for $i=1,\ldots ,r$. Then the canonical map $$H^0(C,\fasm M\otimes \fasm L^d)\longrightarrow H^0(C,\fasm O_Z\otimes \fasm M\otimes \fasm L^d)$$ has maximal rank for $d\geq d_0(m)$. \end{prop} \noindent{\bf Proof.} Adding points if necessary, we can suppose $\mbox{deg}(Z)\geq h^0(C,\fasm M\otimes \fasm L^d)$. By hypothesis we then have $\mbox{deg}\, Z > m(m-1)(g-1)/2$ so that some set of $g$ points amongst the $x_i$ have the same multiplicity $m_0$. Renumbering, we can write $m_1x_1+\cdots + m_rx_r =m_0(y_1+\cdots +y_g) + D=Z+D$, where $D$ has support away from the $y_i$. Since the natural map $C^g\longrightarrow \mbox{Pic}^g(C)$ and the power map $\mbox{Pic}^g(C)\longrightarrow \mbox{Pic}^{m_0g}(C)$ are surjective, it follows that for $y_1,\ldots ,y_g$ generic, the sheaf $\fasm O(Z)$ and hence $\fasm L^d\otimes \fasm M\otimes \fasm O(-Z)$ is the generic sheaf in its component of the Picard scheme so that either $h^0(C,\fasm L^d\otimes \fasm M\otimes \fasm O(-Z))=0$ or $h^1(C,\fasm L^d\otimes \fasm M\otimes \fasm O(-Z))=0$. \hfill$\hfill \square$ This completes the proof of the cases in dimension 1. We end with the following example showing that the `smooth in codimension one' hypothesis cannot be dropped in characteristic $p>0$. \begin{rem}\label{cp} Let $p$ be an odd prime and $C$ the plane curve defined by the equation $y^2-x^p=0$ over an algebraically closed field of characteristic $p$. The tangent line at $z=(t^2,t^p)$, $t\neq 0$, is given by $y=t^p$ and has a contact of order $p$ with $C$ at $z$. It follows that for any choice $z_1,\ldots ,z_d$ of points on the smooth locus of $C$, the divisor $Z=pz_1 +\cdots +pz_d$ is an effective divisor associated to $\fasm O_C(d)$, whereas $h^0(C,\fasm O_C(d))= dp+1-(p-1)(p-2)/2\leq dp$ for $d\geq p-2$. \end{rem} \section{\label{formallemma}The formal lemma} In this section, we prove the formal part of our differential lemma, the rest of the proof being in the next section. We would like to point out that the original motivation and proof of the following results owed much to the work [LL1,2]. \subsection{Preliminaries} Consider the algebra of formal functions $k[[\bm x,y]]$, where $\bm x=(x_1,\ldots ,x_{n-1})$, which we furnish with an ideal $I$ of the form $$I=I_0\oplus I_1y\oplus \cdots \oplus I_{m-1}y^{m-1}\oplus (y^m)$$ where, for $\alpha=0,..., m-1$, $I_{\alpha}\subset k[[\bm x]]$ is an ideal. We call such ideals {\em vertically graded ideals}. Note that \begin{equation}\label{inclusions}I_0 \subset I_1 \subset \cdots \subset I_{m-1}\end{equation} An ideal $$I_t=I_0[[t]]\oplus I_1[[t]](y-t^r)\oplus \cdots\oplus I_{m-1}[[t]](y-t^r)^{m-1}\oplus ((y-t^r)^m)$$ in the algebra $k[[t,\bm x,y]]$ is called a {\em standard deformation } of the vertically graded ideal $I$. For $i\geq m$ we let $I_i=k[[\bm x]]$. Given a function $F_0+F_1t+\cdots $ in $I_t$, the functions $F_i(\bm x ,y)$ must satisfy certain residual conditions. If $r=1$ and $I=(\bm x,y)^m$, the residual condition is just that $F_i(\bm x,y)$ must vanish to the order $m-i$, and can be compared with [Xu]. This is the sense of the following statement. \begin{prop}\label{gettingstarted} Let $F =\sum_{\alpha\geq 0}F_{\alpha}(\bm x,y)t^{\alpha} =\sum_{\alpha, \beta\geq 0} F_{\alpha,\beta}(\bm x)t^{\alpha}y^{\beta}$ be a function in $I_t$. Then $$\begin{array}{lll}F_{\alpha,\beta}(\bm x) &\in &I_{\beta+[\![ \frac{\alpha}{r}]\!]}\end{array}.$$ If $y$ divides $F_{\alpha}$ for $\alpha = 0,r,2r,\ldots ,pr$ then $F_0(\bm x,y)$ is in the ideal $$I_0y\oplus I_1y^2\oplus \cdots \oplus I_{p-1} y^{p}\oplus I_{p+1}y^{p+1} \oplus\cdots \oplus I_{m-1}y^{m-1}\oplus ((y^m)).$$ \end{prop} \noindent{\bf Proof.} Write $F$ in the following form $$F = a_0(\bm x,t) + a_1(\bm x,t)(y-t^r)+\cdots +a_{m-1}(\bm x,t)(y-t^r)^{m-1} + a_m(\bm x,t)(y-t^r)^m + \cdots $$ with $$a_i(\bm x,t)=\sum_{j\geq 0}a_{ij}(\bm x)t^j$$ hence $a_{ij}(\bm x)\in I_i$. Developping out we find $$\begin{array}{lll}F_{\alpha ,\beta}&=&\sum_{\nu =0}^{[\![ \frac{\alpha}{r}]\!]} \; (-1)^{\nu}\, \binom{\beta+\nu}{\beta} a_{\beta+\nu,\, \alpha-\nu r}(\bm x)\\ &\in &I_{\beta+[\![ \frac{\alpha}{r}]\!]}\end{array}$$ where $[\![ z]\!]$ is the greatest integer part of $z$. This proves the first part. Now suppose that $y$ divides $F_{\alpha}$ for $\alpha= \lambda r$ and $\lambda=0,1,\ldots ,p$. Then we have $$0=F_{\lambda r,\, 0} = a_{0,\lambda r}-a_{1,(\lambda - 1)r}+\cdots +(-1)^{\lambda -1}a_{\lambda -1,r}+ (-1)^{\lambda}a_{\lambda,0}$$ so that $a_{0,0}=0$ and $a_{\lambda,0}\in I_{\lambda -1}$ for $\lambda =1,\ldots ,p$ as one sees using $a_{\mu , \nu}\in I_{\mu}$ and (\ref{inclusions}). This gives the last part of the proposition. $\hfill \square$ \subsection{The formal lemma} Throughout this subsection we will use the following notation. For $i=1,\ldots ,\ell$, let $B^{(i)}=k[[\bm x_i,y_i]]$ be an algebra of formal functions in $n$ variables where $\bm x_i=(x_{i,1},\ldots ,x_{i,n-1})$ and let $$I^{(i)}=I_0^{(i)}\oplus I_1^{(i)}y_i \oplus \cdots\oplus I_{m_i-1}^{(i)}y_i^{m_i-1}\oplus (y_i^{m_i})$$ be a vertically graded ideal in $B^{(i)}$. Let $$I=I^{(1)}\times \cdots \times I^{(\ell)}\subset B^{(1)}\times \cdots \times B^{(\ell)}=B.$$ Let $k[[\bm t]]=k[[t_1,\ldots ,t_{\ell}]]$ and let $I_{\bm t}$ in $B[[\bm t]]$ be the product of the ideals $$I^{(i)}_{\bm t}=I_0^{(i)}[[\bm t]]\oplus I_1^{(i)}[[\bm t]](y_i-t_i) \oplus \cdots \oplus I_{m_i-1}^{(i)}[[\bm t]](y_i-t_i)^{m_i-1}\oplus ((y_i-t_i)^{m_i}).$$ Let $y=(y_1,\ldots ,y_{\ell})$ and for any linear subspace $V\subset B$, let $V_{\mbox{res}(y)}=\{ v\in B\, |\, vy\in V\,\}$. Since $y$ is not a zero-divisor, we get a residual exact sequence \begin{equation}\label{res.of.v}0\longrightarrow V_{\mbox{res}(y)}\stackrel{y}{\longrightarrow}V\longrightarrow V/V\cap (y)\longrightarrow 0\end{equation} \begin{prop}\label{fl} Let $V\subset B$ be a $k$-linear subspace. Suppose that for $i=1,\ldots ,\ell$ there exist nonnegative integers $p_i$ such that the following two conditions are satisfied \begin{enumerate} \item the canonical map $$ V/V\cap (y)\longrightarrow k[[\bm x_1]]/I^{(1)}_{p_1}\times \cdots \times k[[\bm x_{\ell}]]/I^{(\ell)}_{p_{\ell}}$$ is injective \item The canonical map $$V_{\mbox{res}(y)}\longrightarrow B/J$$ is injective where $J=J^{(1)}\times \cdots \times J^{(\ell)}$ and $$J^{(i)}=I^{(i)}_{0}\oplus I^{(i)}_{1}y_i\oplus \cdots \oplus I^{(i)}_{p_i-1}y_i^{p_i-1}\oplus I^{(i)}_{p_i+1}y_i^{p_i}\oplus \cdots \oplus I^{(i)}_{m_i-1}y_i^{m_i-2}\oplus (y_i^{m_i-1})$$ \end{enumerate} Then the canonical map $$\varphi_{\bm t} : V\otimes k[[\bm t]]\longrightarrow B_{\bm t}/I_{\bm t}$$ is (generically) injective. \end{prop} \noindent{\bf Proof.} We first reduce to the case where the $p_i$ are positive. Let us suppose for simplicity that $p_1,\ldots,p_s$ are positive and $p_{s+1},\ldots ,p_{\ell}$ are all zero. We denote by $V_0$ the subspace of $V$ formed by the elements vanishing in each of the $k[[\bm x_i,y_i]]/I^{(i)}$ for $i=s+1 , \ldots ,\ell$. Conditions 1. and 2. of the proposition imply the corresponding conditions for $V_0$ when only the first $s$ factors on the right hand side are present. If we write $\bm t'$ for $(t_1,\ldots,t_s)$, the conclusion of the proposition in the case where all $p_i$ are positive then implies that $V_0\otimes k[\bm t']]$ injects into $B_{\bm t'}/I_{\bm t'}$. Since $\varphi_{\bm t}$ is a map of free $k[[\bm t]]$-modules, it is enough to prove that its restriction $\varphi_{\bm t'}$ over $Spec \, k[[\bm t']]$ is injective. We write $$\varphi_{\bm t'}= (\varphi_{\bm t'}^{\prime}, \varphi_{\bm t'}^{\prime\prime}): V\otimes k[\bm t']] \rightarrow B_{\bm t'}/I_{\bm t'} \times R$$ with $$R= (B^{(s+1)}/I^{(s+1)} \times \cdots \times B^{(\ell )}/I^{(\ell )}) \otimes k[[\bm t']]$$ The kernel of $\varphi_{\bm t'}^{\prime\prime}$ is $V_0\otimes k[\bm t']]$ and the restriction of $\varphi_{\bm t'}^{\prime}$ to this kernel is injective, thus so is $\varphi_{\bm t'}$. Henceforth we suppose that the $p_i$ are positive and we let $$h=lcm(p_1,\ldots ,p_{\ell})=r_ip_i$$ be the least common multiple of the $p_i$ and consider the one-parameter deformation obtained by setting $t_i=t^{r_i}$. Since the rank of $\varphi_{\bm t}$ is semi-continuous, we need only show that the canonical map $$\varphi_t : V\otimes k[[t]]\longrightarrow B[[t]]/I_t$$ obtained by the formal base change $k[[t_1,\ldots ,t_{\ell}]]\longrightarrow k[[t]]; \; t_i\mapsto t^{r_i}\, $; is injective. Let $$F_t =(F^{(1)}_t,\ldots ,F^{(\ell)}_t)\in \ker\varphi_t\; =\; V_t\cap I_t,$$ where $V_t$ is the image of $V\otimes k[[\bm t]]$ and $I_t$ is the image of $I_{\bm t}$ in $B[[t]]$. In case $F_0=0$, we may replace $F_t$ by $F_t/t$ since $B[[t]]/I_t$ is a torsion free $k[[t]]$-module, Thus we only have to prove $F_0=0$. Since $F^{(i)}_t=\sum_{\alpha\geq 0}\, F^{(i)}_{\alpha}(\bm x_i,y_i)t^{\alpha}\in I^{(i)}_t$, where $I^{(i)}_t$ is the image of $I^{(i)}_{\bm t}$ in $B^{(i)}[[t]]$, the first part of proposition \ref{gettingstarted} implies $$(F^{(1)}_{\alpha}(\bm x_1,0),\ldots ,F^{(\ell)}_{\alpha}(\bm x_{\ell},0))\in I^{(1)}_{p_1}\times \cdots \times I^{(\ell)}_{p_{\ell}}$$ for $\alpha = 0,1,\ldots ,h$. Applying hypothesis 1. of the proposition, we conclude that $y$ divides $(F^{(1)}_{\alpha}(\bm x_1,y_1),\ldots ,F^{(\ell)}_{\alpha}(\bm x_{\ell},y_{\ell}))$ for $\alpha = 0,1,\ldots ,h$. Now applying the second part of proposition \ref{gettingstarted} we obtain $$F^{(i)}_0(\bm x_i,y_i) =y_i G_0^{(i)}(\bm x_i,y_i)$$ with $G_0^{(i)}(\bm x_i,y_i)\in J^{(i)}$. Letting $G_0=(G_0^{(1)},\ldots ,G_0^{(r)})$ we see that $G_0$ is in $V_{\mbox{res}(y)}\cap J$, but the second hypothesis of the proposition simply says that $V_{\mbox{res}(y)}\cap J=0$, giving > $F_0=0$ as required. $\hfill \square$ \section{The differential lemma} Throughout this section, $X$ denotes an irreducible algebraic variety, $H$ a reduced irreducible positive Cartier divisor on $X$, $X^0$ a dense open nonsingular subscheme of $X$ such that $H^0:=H \cap X^0$ is the nonsingular locus of $H$. Finally $\fasm I_H$ denotes the ideal sheaf of $H$. For $M$ another $k$-scheme, we denote by $Hom(M, X)$ (resp. $Hom(M, X^0)$) the set of morphisms from $M$ to $X$ (resp. $X^0$) as well as, in case $M$ is projective, the corresponding Hilbert scheme. If $M$ is algebraic, zero-dimensional and connected, it is easy to check that the natural morphism from $Hom(M, X^0)$ to $X^0$ is smooth with smooth irreducible fibers. Thus $Hom(M, X^0)$ is also irreducible. Its generic point represents an embedding whose image in $X$ we denote by $M_X$. Now let $M$ be a subscheme of $\mbox{Spec}\, k[[x_1, \dots, x_n]]$. We denote by $Hom(M, X, H)$ the set (or Hilbert scheme) of morphisms $f$ from $M$ to $X^0$ such that the ideal $f^*({\cal I}_H)$ is contained in $(x_n)$. We call these morphisms $H$-morphisms from $M$ to $X$, and if a $H$-morphism is an embedding, we say that it is a $H$-embedding. If $M$ is algebraic, thus zero-dimensional and connected, it is easy to check that the natural (restriction) morphism from $Hom(M, X, H)$ to $H^0$ is smooth. Furthermore, its fiber $Hom(M, X, H, z)$ over a point $z$ of $H^0$ is a vector space, thus smooth and irreducible. As a consequence, $Hom(M, X, H)$ is again irreducible and smooth. Its generic point is a $H$-embedding whose image in $X$ we denote by $M_{X,H}$. We say that the subscheme $M$ of $\mbox{Spec}\, k[[x_1, \dots, x_n]]$ is a model of dimension $n$ if its ideal is a vertically graded ideal as in \S 8: $$I=I_0\oplus I_1x_n\oplus \cdots \oplus I_mx_n^m\oplus \dots$$ where $I_m$ is a non-decreasing sequence of ideals in $k[[x_1, \dots, x_{n-1}]]$ with $I_m = k[[x_1, \dots, x_{n-1}]]$ for large $m$. For $M$ a model of dimension $n$, we denote by $TrM$ its trace on the hyperplane defined by $x_n$, and by $Res\, M$ the corresponding residual scheme, which is again a model of dimension $n$. We define more generally $Tr^{(p)}M$ and $Res^{(p)}M$ for any nonnegative integer $p$: with the notations introduced above, we set $Tr^{(p)}M := I_p$, and define $Res^{(p)}M$ to be the model corresponding to the ideal $$I_0\oplus I_1x_n\oplus \cdots \oplus I_{p-1}x_n^{p-1}\oplus I_{p+1}x_n^{p}\oplus\dots \oplus I_{q+1}x_n^{q}\oplus\dots$$ If $M_1, \dots , M_{\ell}$ are models, we say that their disjoint union ${\bf M}$ is a multi-model. If ${\bf p}=(p_1, \dots, p_{\ell})$ is a multi-integer, we define $Tr^{\bf p}{\bf M}$ to be the disjoint union of the $Tr^{p_i}M_i$ and $Res^{\bf p}{\bf M}$ to be the disjoint union of the $Res^{p_i}M_i$. The Hilbert scheme $Hom({\bf M}, X^0)$ is the product $Hom(M_1, X^0)\times \dots \times Hom(M_{\ell},X^0)$, thus irreducible (and smooth). Its generic point represents an embedding whose image in $X$ we denote by ${\bf M}_X$. We denote by $Hom({\bf M}, X, H)$ the Hilbert scheme of morphisms $f$ from ${\bf M}$ to $X^0$ whose restrictions to the components $M_i$ are $H$-morphisms. We call these morphisms $H$-morphisms from ${\bf M}$ to $X$, and we call $H$-embeddings those $H$-morphisms which are embeddings. The scheme $Hom({\bf M}, X, H)$ is the product $Hom(M_1, X, H)\times \dots \times Hom(M_{\ell},X, H)$, thus irreducible (and smooth). Its generic point is a $H$-embedding whose image in $X$ we denote by ${\bf M}_{X,H}$. We are now ready to state and prove our differential Horace lemma: \begin{prop}\label{prin.cor} Let $X$ be, as above, a reduced projective variety of dimension $n$, furnished with a line bundle $\fasm L$, and $H$ a reduced irreducible positive Cartier divisor on $X$ not contained in the singular locus of $X$. Let $W\subset X$ be a closed subscheme of $X$ not containing $H$. We denote by $W^{\prime\prime}$ and $W'$ the trace and residual of $W$ with respect to $H$. Let ${\bf M}$ be a multi-model of dimension $n$. and ${\bf p}$ a multi-integer (of the same length). Suppose \begin{enumerate} \item \makebox[2.5cm][l]{{\bf Dime}} \hspace{5ex} $H^0(H,\fasm I_{W^{\prime\prime}\cup Tr^{\bf p}{\bf M}_H}\otimes \fasm L\vert H)=0$ \item \makebox[2.5cm][l]{{\bf Degue}} \hspace{5ex } $H^0(X,\fasm I_{W^{\prime}\cup Res^{\bf p}{\bf M}_{X,H}}\otimes \fasm L(-H))=0.$ \end{enumerate} Then $H^0(X,\fasm I_{W\cup {\bf M}_X}\otimes \fasm L)=0$. \end{prop} \noindent{\bf Proof.} We may suppose that $k$ is algebraically closed. By semi-continuity, there exist rational points $z_1, \dots, z_{\ell}$ in $H$, and corresponding $H$-embeddings $e_i: \mbox{Spec}\, k[[x_1, \dots, x_n]] \rightarrow X$, with $z_i$ as image of the closed point, allowing us to rewrite our first assumption as follows: \makebox[1.5cm][l]{{\bf Dime}} \hspace{5ex} $H^0(H,\fasm I_{W^{\prime\prime}\cup e_1(Tr^{p_1}{M_1}) \cup \dots \cup e_{\ell}(Tr^{p_{\ell}}M_{\ell})}\otimes \fasm L\vert H)=0.$ Similarly, there exist rational points $z'_1, \dots, z'_{\ell}$ in $H$, and corresponding $H$-embeddings $e'_i: \mbox{Spec}\, k[[x_1, \dots, x_n]] \rightarrow X$, with $z'_i$ as image of the closed point, allowing us to rewrite our second assumption as follows: \makebox[1.5cm][l]{{\bf Degue}} \hspace{5ex } $H^0(X,\fasm I_{W^{\prime}\cup e'_1(Res^{p_1}{M_1}) \cup \dots \cup e'_{\ell}(Res^{p_{\ell}}M_{\ell})}\otimes \fasm L(-H))=0.$ Since the two conditions do not interfere with one another, we may even suppose $z_i=z'_i$. Let us now show that it's possible to obtain $e_i=e'_i$. Using induction on $i$, all we need to prove is the following statement: given such a point $z$, a vertically graded model $M$ in $\mbox{Spec}\, k[[x_1, \dots, x_n]]$, an integer $p$ and two non-empty open subschemes $S^{\prime\prime}$ in $Hom(Tr^pM, H, z)$ and $S^{\prime}$ in $Hom(Res^pM, X, H, z)$, there exists a $H$-embedding $e: \mbox{Spec}\, k[[x_1, \dots, x_n]] \rightarrow \mbox{Spec}\, \hat {\fasm O}_{X, z} \rightarrow X$ whose restriction to $Tr^pM$ is in $S^{\prime\prime}$ and whose restriction to $Res^pM$ is in $S^{\prime}$. This statement is easily proven, using the fact that the restriction maps $$Hom (M, X, H, z) \rightarrow Hom (Res^p\,M, X, H, z)$$ and $$Hom (M, X, H, z) \rightarrow Hom (Tr\,M, H, z) \rightarrow Hom (Tr^p\,M, H, z)$$ are dominant, and the fact that any element of $Hom (M, X, H, z)$ extends as a $H$-morphism from $\mbox{Spec}\, k[[x_1, \dots, x_n]]$ to $ X$. The proposition then follows by applying our formal lemma \ref{fl} with $V= H^0(X, \fasm I_W \otimes \fasm L)$. $\hfill \square$ \begin{rem} Our formal lemma can give more accurate information. For instance if $X$ is a projective space and $H$ a hyperplane, we may handle {\em linear} embeddings of (multi-)models in a similar way to that used for general embeddings of (multi-)models. \end{rem} We now make explicit the particular case of the proposition where the $M_i$ are points (of various multiplicities). \begin{cor} \label {prin.lem} Let $X$ be, as above, a reduced projective variety of dimension $n$, furnished with a line bundle $\fasm L$, and $H$ a reduced irreducible positive Cartier divisor on $X$ not contained in the singular locus of $X$. Let $W$ be a closed subscheme of $X$ not containing $H$. Let $P_1,\ldots ,P_r$ be generic points of $X$, $Q_1, \ldots ,Q_r$ generic points in $H$ and $m_1,\ldots ,m_r$ a sequence of positive integers. Then $H^0(\fasm I_{W\cup P_1^{m_1}\cup \cdots \cup P_r^{m_r}}\otimes \fasm L)=0$ if the following two conditions are satisfied (see 2.2 for the notation $D_m$): \begin{enumerate} \item \makebox[2.5cm][l]{{\bf Dime}} \hspace{5ex} $H^0(X,\fasm I_{W^{\prime\prime}\cup Q_1\cup \cdots \cup Q_r}\, \otimes \fasm L|_H)=0$ \item \makebox[2.5cm][l]{{\bf Degue}} \hspace{5ex } $H^0(X,\fasm I_{W^{\prime}\cup D_{m_1}(Q_1)\cup \cdots \cup D_{m_r}(Q_r)}\otimes \fasm L(-H))=0$ \end{enumerate} \end{cor} \noindent{\bf Proof.} This is just proposition 9.1 with $p_{i}=m_{i}-1$.$\hfill \square$ \begin{rem} The lemma \ref{diff} is obtained by taking $H=G_a$ in the previous corollary. \end{rem}

9703

alg-geom/9703021

en

https://arxiv.org/abs/alg-geom/9703021

alg-geom/9703021

Alexander Polishchuk

Alexander Polishchuk

Determinant bundles for abelian schemes

26 pages, AMSLatex. One proof is shortened, a new linear relation between determinant bundles is proven

To a symmetric, relatively ample line bundle on an abelian scheme one can associate a linear combination of the determinant bundle and the relative canonical bundle, which is a torsion element in the Picard group of the base. We improve the bound on the order of this element found by Faltings and Chai. In particular, we obtain an optimal bound when the degree of the line bundle d is odd and the set of residue characteristics of the base does not intersect the set of primes p dividing d, such that $p\equiv -1\mod(4)$ and p<2g, where g is the relative dimension of the abelian scheme. Also, we show that in some cases these torsion elements generate the entire torsion subgroup in the Picard group of the corresponding moduli stack.

alg-geom

\section{The behavior under isogenies} \label{behisog} In this section we study the relation between $\Delta(L)$ and $\Delta(\a^*L)$ where $\a:A\rightarrow B$ is an isogeny of abelian schemes, $L$ is a relatively ample, symmetric, line bundle on $B$ trivialized along the zero section. Let $d=\operatorname{rk}\pi_*L$. The main result of this section is the following theorem. \begin{thm}\label{isogmain} \begin{enumerate} \item One has $$gcd(12,\deg(\a))\cdot(\Delta(\a^*L)-\deg(\a)\cdot\Delta(L))=0.$$ \item If $\deg(\a)$ is odd then $$\det\pi_*(\a^*L)=\deg(\a)\cdot\det\pi_*L +d\cdot\det\pi_*\O_{\ker{\a}}+\zeta$$ where $gcd(3,\deg(\a))\cdot\zeta=0$. \end{enumerate} More precisely, these equalities in $\operatorname{Pic}(S)$ are realized by canonical isomorphisms of line bundles, compatible with arbitrary base changes. \end{thm} \begin{rem} In the case $d=1$ this follows from the result of Moret-Bailly in \cite{MB}, VIII, 1.1.3. When $d$ is even the second equality of the theorem can be rewritten as $$gcd(3,\deg(\a))\cdot(\Delta'(\a^*L)-\deg(\a)\cdot\Delta'(L))=0$$ where $\Delta'(L):=\det\pi_*L+\frac{d}{2}\cdot\overline{\omega}_A$. \end{rem} Outside of characteristic $3$ we can improve Theorem \ref{isogmain} for some isogenies of degree divisible by $3$. \begin{thm}\label{isogmain2} Assume that $3$ is not among residue characteristics of $S$ and that $d$ is relatively prime to $3$. Assume that $3^k\cdot\ker(\a)=0$, that $K(\a^*L)^{(3)}$ is annihilated by $\frac{1}{3}\cdot|K(\a^*L)^{(3)}|^{\frac{1}{2}}$, and that $|K(\a^*L)^{(3)}/(\ker(\a)+3K(\a^*L)^{(3)})|>3$. Then one has a canonical isomorphism of line bundles on $S$ realizing the equality $$\Delta(\a^*L)=\deg(\a)\cdot\Delta(L)$$ in $\operatorname{Pic}(S)$. \end{thm} \begin{rem} When $d$ is even the similar relation holds for elements $\Delta'(\cdot)$ instead of $\Delta(\cdot)$. \end{rem} \begin{cor} Let $L$ be an ample symmetric line bundle on an abelian scheme $A/S$, trivialized along the zero section. Then for any odd integer $n>0$ which is not divisible by $3$ one has $$\Delta(L^{n^2})=n^{2g}\cdot\Delta(L).$$ Also one has $$\Delta(L^4)^{(3)}=4^{g}\cdot\Delta(L)^{(3)}$$ and $$\Delta(L^9)^{(2)}=9^{g}\cdot\Delta(L)^{(2)}.$$ Furthermore, if $g>1$ and $3$ is prime to $d=\operatorname{rk}\pi_*L$ and to the residue characteristics of $S$ then $$\Delta(L^9)=9^{g}\cdot\Delta(L).$$ \end{cor} We are going to use the relative Fourier-Mukai transform, so let us briefly recall some of its properties. For details the reader should consult \cite{Muk2} and \cite{La}. The Fourier-Mukai transform is the functor $${\cal F}_A={\cal F}_{A/S}:{\cal D}^b(A)\rightarrow{\cal D}^b(\hat{A}): X\mapsto p_2^*(p_1^*X\otimes\cal P),$$ where $\cal P$ is the (normalized) relative Poincar\'e bundle on $A\times_S\hat{A}$. It is compatible with arbitrary base changes and satisfies the following fundamental property: $${\cal F}_{\hat{A}}\circ {\cal F}_{A}\simeq (-\operatorname{id}_A)^*(\cdot)\otimes\pi^*\overline{\omega}_A^{-1}[-g]$$ where $g$ is the relative dimension of $\pi$. Because of this the relative canonical bundles of $A$ and $\hat{A}$ often appear when working with the Fourier-Mukai transform so it is useful to know that there is a canonical isomorphism $\overline{\omega}_{\hat{A}}\simeq\overline{\omega}_A$ (see \cite{La}, 1.1.3). Also for any homomorphism $f:A\rightarrow B$ of abelian schemes over $S$ one has the following canonical isomorphisms \begin{equation}\label{hom1} {\cal F}_B\circ f_*\simeq \hat{f}^*\circ{\cal F}_A, \end{equation} \begin{equation}\label{hom2} {\cal F}_A\circ f^!\simeq \hat{f}_*\circ{\cal F}_B. \end{equation} The particular case of (\ref{hom1}) is the isomorphism $$e^*\circ{\cal F}_A\simeq\pi_*$$ where $e:S\rightarrow A$ is the zero section, $\pi:A\rightarrow S$ is the projection. The following lemma will be used in the proof of Theorem \ref{isogmain2}. \begin{lem}\label{fur2} For any relatively ample line bundle $L$ on $B$ trivialized along the zero section one has a canonical isomorphism $$\phi_L^*{\cal F}_B(L)\simeq\pi^*\pi_*L\otimes L^{-1}.$$ \end{lem} \noindent {\it Proof}. Making the base change $\phi_L:B\rightarrow\hat{B}$ of the projection $p_2:B\times\hat{B}\rightarrow\hat{B}$ we can write $$\phi_L^*{\cal F}_B(L)\simeq\phi_L^*p_{2*}(p_1^*L\otimes\cal P) \simeq p_{2*}(p_1^*L\otimes (\operatorname{id}_B,\phi_L)^*\cal P)$$ where in the latter expression $p_2$ denotes the projection of the product $B\times_S B$ on the second factor. But we have an isomorphism $$\mu^*L\simeq p_1^*L\otimes p_2^*L\otimes (\operatorname{id}_B,\phi_L)^*\cal P$$ since a trivialization of $L$ along the zero section is equivalent to a cube structure on it (see \cite{Br}). Hence, $$\phi_L^*{\cal F}_B(L)\simeq p_{2*}(p_2^*L^{-1}\otimes\mu^*L) \simeq L^{-1}\otimes p_{2*}\mu^*(L)\simeq L^{-1}\otimes\pi^*\pi_*L$$ as required. \qed\vspace{3mm} \vspace{3mm} \noindent {\it Proof of Theorem \ref{isogmain}}. Applying (\ref{hom2}) to the isogeny $\a:A\rightarrow B$ we obtain $${\cal F}_A(\a^!L)\simeq \hat{\a}_*{\cal F}_B(L).$$ Restricting this isomorphism to the zero section we obtain $$\pi_*(\a^!L)\simeq e^*{\cal F}_A(\a^!L)\simeq \pi_*({\cal F}_B(L)|_H)$$ where $H=\ker(\hat{\a})\subset\hat{B}$. Taking the determinant of this isomorphism we get $$\det\pi_*(\a^!L)=\det\pi_*({\cal F}_B(L)|_H).$$ Using the fact that $\a^!L\simeq\a^*L\otimes\omega_{A/S}\otimes\omega_{B/S}^{-1}$ we can rewrite the left hand side as follows: $$\det\pi_*(\a^!L)=\det\pi_*(\a^*L)+n\cdot d\cdot\omega_{\a}$$ where $\omega_a=\overline{\omega}_A-\overline{\omega}_B$, $n=\deg\a$. Recall (see e.~g. \cite{De}) that for any vector bundle $E$ of rank $r$ on $H$ one has \begin{equation}\label{detpi} {\det}\pi_*(E)=r\cdot{\det}\pi_*\O_H +\operatorname{N}_{H/S}({\det} E) \end{equation} where $\operatorname{N}_{H/S}:\operatorname{Pic}(H)\rightarrow\operatorname{Pic}(S)$ is the corresponding norm homomorphism. Consider the line bundle $M=\det{\cal F}_B(L)\otimes\pi^*(\det\pi_*L)^{-1}$ on $\hat{B}$. Then $M$ has a canonical trivialization along the zero section. Furthermore, since the Fourier transform commutes with $[-1]^*$, we have an isomorphism $[-1]_{\hat{B}}^*M\simeq M$ compatible with the symmetry structure on $L$. Applying the equation (\ref{detpi}) to $E={\cal F}_B(L)|_H$ we get \begin{align*} &\det\pi_*({\cal F}_B(L)|_H)= d\cdot\det\pi_*\O_H+\operatorname{N}_{H/S}(\pi^*\det\pi_*L+M|_H)=\\ &d\cdot\det\pi^*\O_H+ n\cdot\det\pi_*L+\operatorname{N}_{H/S}(M|_H). \end{align*} Hence, \begin{equation}\label{is1} \det\pi_*(\a^*L) +n\cdot d\cdot\omega_{\a}= n\cdot\det\pi_*L+\operatorname{N}_{H/S}(M|_H)+d\cdot\det\pi_*\O_H. \end{equation} This implies that $$\Delta(\a^*L)-n\cdot\Delta(L)=2\cdot\operatorname{N}_{H/S}(M|_H)+ 2\cdot d\cdot\det\pi_*\O_H - n\cdot d\cdot\omega_{\a}.$$ Recall that one has the canonical isomorphism $$(\pi_*\O_H)^{\vee}\simeq\omega_H\otimes\pi_*\O_H$$ where $\omega_H=(\pi_*\O_{\hat{H}})^H$ is the line bundle of (relative) invariant measures on $H$. Passing to determinants we get $$2\det\pi_*\O_H=-n\cdot\omega_H.$$ Hence, $$\Delta(\a^*L)-n\cdot\Delta(L)=2\cdot\operatorname{N}_{H/S}(M|_H)- n\cdot d\cdot\omega_H - n\cdot d\cdot\omega_{\a}= 2\cdot\operatorname{N}_{H/S}(M|_H)$$ since $\omega_H\simeq\omega_{\hat{\a}}\simeq\omega_{\a}^{-1}$. Also using that $\ker(\a)$ is Cartier dual to $H$ we deduce from (\ref{is1}) the following equality $$\det\pi_*(\a^*L)=n\cdot\det\pi_*L+ d\cdot\det\pi_*\O_{\ker\a}+\zeta$$ where $\zeta=\operatorname{N}_{H/S}(M|_H)$. It remains to show that $gcd(24,2n)\cdot\operatorname{N}_{H/S}(M|_H)=0$ and that $gcd(3,n)\cdot\operatorname{N}_{H/S}(M|_H)=0$ if $\deg(\a)$ is odd. To this end let us decompose $H$ into a product of two group schemes $H\simeq H'\times_S H''$ such that the order of $H'$ is odd, while the order of $H''$ is a power of 2. Then the cube structure on $M$ induces the decomposition of $M|_H$ into the external tensor product of $M|_{H'}$ and $M|_{H''}$. Hence, we obtain $$\operatorname{N}_{H/S}(M|_H)=|H''|\cdot\operatorname{N}_{H'/S}(M|_{H'})+ |H'|\cdot\operatorname{N}_{H''/S}(M|_{H''}).$$ Recall that since $M$ has a cube structure and $[-1]^*M\simeq M$, it follows that $[n]^*M\simeq M^{n^2}$ for any $n$. Now the multiplication by $3$ is an automorphism of $H''$, hence $$\operatorname{N}_{H''/S}(M|_{H''})=\operatorname{N}_{H''/S}([3]_{\hat{B}}^*M|_{H''})= \operatorname{N}_{H''/S}(M^{9}|_{H''})=9\cdot\operatorname{N}_{H''/S}(M|_{H''}).$$ Thus, $8\cdot\operatorname{N}_{H''/S}(M|_{H''})=0$. Similarly, using the multiplication by $2$ we obtain $$\operatorname{N}_{H'/S}(M|_{H'})=\operatorname{N}_{H'/S}([2]_{\hat{B}}^*M|_{H'})= \operatorname{N}_{H'/S}(M^{4}|_{H'})=4\cdot\operatorname{N}_{H'/S}(M|_{H'}),$$ hence $3\cdot\operatorname{N}_{H'/S}(M|_{H'})=0$. It remains to note that \begin{align*} &2\cdot\operatorname{N}_{H/S}(M|_{H})=\operatorname{N}_{H/S}(M|_{H})+ \operatorname{N}_{H/S}([-1]_{\hat{B}}^*M|_{H})=\\ &\operatorname{N}_{H/S}((M\otimes [-1]_{\hat{B}}^*M)|_{H})= \operatorname{N}_{H/S}((\operatorname{id}_{\hat{B}},\phi_M)^*\cal P|_{H}) \end{align*} due to an isomorphism $M\otimes [-1]_{\hat{B}}^*M\simeq(\operatorname{id}_{\hat{B}},\phi_M)^*\cal P$. In particular, since $H$ is annihilated by $n$ it follows that $$2n\cdot\operatorname{N}_{H/S}(M|_{H})=0.$$ \qed\vspace{3mm} \begin{lem}\label{norm} Let $B$ be an abelian scheme over $S$, where $3$ is prime to the residue characteristics of $S$, $L$ be an ample line bundle on $B$ trivialized along the zero section, $H\subset B_3$ be a finite flat subgroup. Assume that $H$ is isotropic with respect to the symplectic form $e^{L^3}$ on $K(L^3)$ and that $|H|>3$. Then there is a canonical isomorphism of $\operatorname{N}_{H/S}(L|_H)$ with the trivial line bundle on $S$. \end{lem} \noindent {\it Proof} . Note that $L|_H$ is annihilated by $3$ in $\operatorname{Pic}(H)$. Indeed, the cube structure on $L$ gives an isomorphism \begin{equation}\label{triv0} L_{3x}\simeq L_x^3 \otimes\langle x,x\rangle^3 \end{equation} where $\langle\cdot,\cdot\rangle=(\operatorname{id}_B,\phi_L)^*\cal P$ is the symmetric biextension of $B\times B$ associated with $L$ (see \cite{Br}). When $3x=0$ this gives a trivialization of $L_x^3$. Let us consider the finite \'etale covering $c:S'\rightarrow S$ corresponding to a choice of a non-trivial point $\sigma\in H$. The degree of this covering is prime to $3$, so it suffices to prove the triviality of $\operatorname{N}_{H/S}(L|_H)$ after making the corresponding base change. Thus, we can assume that we have a non-trivial $S$-point $\sigma:S\rightarrow H$. To compute $\operatorname{N}_{H/S}(L)$ we decompose the projection $H\rightarrow S$ into the composition $H\rightarrow \overline{H}\rightarrow S$ where $\overline{H}=H/\langle\sigma\rangle$, $\langle\sigma\rangle\subset H$ is the cyclic subgroup in $H$ generated by $\sigma$. Now we claim that \begin{equation}\label{si} \operatorname{N}_{H/\overline{H}}(L)\simeq \pi^*(\sigma^*L\otimes(-\sigma)^*L) \end{equation} where $\pi$ is the projection to $S$. Indeed, to give an isomorphism of line bundles on $\overline{H}$ is the same as to give an isomorphism of their pull-backs to $H$ compatible with the action of $\langle\sigma\rangle\subset H$. Now the cube structure on $L$ gives the following canonical isomorphism \begin{equation}\label{triv1} L_x \otimes L_{x+\sigma}\otimes L_{x-\sigma}\simeq L_x^3\otimes L_{\sigma}\otimes L_{-\sigma}. \end{equation} Composing it with the trivialization of $L_x^3$ obtained above we get an isomorphism \begin{equation}\label{triv} L_x \otimes L_{x+\sigma}\otimes L_{x-\sigma}\simeq L_{\sigma}\otimes L_{-\sigma}. \end{equation} It remains to check that this isomorphism is compatible with the action of ${\Bbb Z}/3{\Bbb Z}$ (the action on the right hand side being trivial), i.~e. that we get the same isomorphism making the cyclic permutation of the left hand side and applying (\ref{triv}) to $x+\sigma$. One can check that the only cause for these isomorphisms to be different is the difference between the two trivializations of $\langle x,\sigma\rangle^3$: the one is obtained using that $3\cdot x=0$ and the other is obtained from $3\cdot\sigma=0$. But this difference is equal to $e^{L^3}(x,\sigma)$ which we assumed to be trivial. Here are more details of this computation. First note that the isomorphism (\ref{triv1}) only uses the cube structure and the fact that $3\cdot\sigma=0$. If we consider the similar isomorphism with $x$ replaced by $x+\sigma$ it will be compatible with the natural isomorphism $\a_{\sigma}:L_{x+\sigma}^3\simeq L_x^3$ which holds for any cube structure and any $\sigma$ such that $3\cdot\sigma=0$. Thus, we have to compute the difference between two trivializations of $L_{x+\sigma}^3$: one which is obtained by directly applying $(\ref{triv0})$ to $x+\sigma$ and then trivializing $\langle x+\sigma,x+\sigma\rangle^3$, and the other which is the composition of $\a_{\sigma}$ with the similar trivialization of $L_x^3$. Note that $\a_{\sigma}$ can be obtained using the isomorphisms $(\ref{triv0})$ and the isomorphism $$\a'_{\sigma}:\langle x+\sigma,x+\sigma\rangle^3\simeq\langle x,x\rangle^3$$ which again only depends on the fact that $3\cdot\sigma=0$. Let us denote by $\b_x:\langle x,x\rangle^3\rightarrow 0$ the natural trivialization for $x\in B_3$. Now our claim follows from the equality $\b_{x+\sigma}=e^{L^3}(x,\sigma)^{\pm 1}\cdot\b_x\circ\a'_{\sigma}.$ Thus, the isomorphism (\ref{triv}) is compatible with the action of ${\Bbb Z}/3{\Bbb Z}$, hence the isomorphism (\ref{si}). But $\sigma^*L\otimes(-\sigma)^*L$ is annihilated by $3$ in $\operatorname{Pic}(S)$ and the degree of the projection $\pi:\overline{H}\rightarrow S$ is divisible by $3$ (here we use the assumption that $|H|>3$). Thus, we obtain $$\operatorname{N}_{H/S}(L)=\operatorname{N}_{\overline{H}/S}\operatorname{N}_{H/\overline{H}}(L)= \operatorname{N}_{\overline{H}/S}(\pi^*\sigma^*L^2)=0$$ as required. \qed\vspace{3mm} \noindent {\it Proof of Theorem \ref{isogmain2}}. Let $H\subset B_{3^k}$ be the preimage of $\ker(\hat{\a})\subset\hat{B}_{3^k}$ under the isomorphism $\phi_L|_{B_{3^k}}:B_{3^k}\rightarrow\hat{B}_{3^k}$. As the proof of Theorem \ref{isogmain} shows we only have to check the triviality of the norm of $M$ restricted to $\ker(\hat{\a})$. Since $\phi_L^*M\simeq L^{-d}$ by Lemma \ref{fur2}, this is equivalent to proving the triviality of $\operatorname{N}_{H/S}(L|_H)$. Consider the subgroup $K=\a^{-1}(H)\subset A$. Then $K\subset A_{3^{2k}}$ and the definition of $H$ implies that $K=K(\a^*L)^{(3)}$. Also since $d$ is prime to $3$, it follows that $\ker(\a)$ is a maximal isotropic subgroup in $K$. Now we claim that after making an \'etale base change of degree prime to $3$ we can find a finite flat subgroup $K_1\subset K$ containing $\ker(\a)$, with the following two properties: \begin{enumerate} \item $K$ is annihilated by $3\cdot |K_1/\ker(\a)|$, \item the quotient $K/K_1$ is annihilated by $3$ and has order $>3$. \end{enumerate} Indeed, since $|K/\ker(\a)|=|K|^{\frac{1}{2}}$, all we need is to find $K_1$ such that $\ker(\a)+3K\subset K_1\subset K$ and $|K/K_1|=9$. To get such $K_1$ we just make an \'etale covering of $S$ (of degree prime to $3$) corresponding to a choice of a subgroup of index $9$ in $K/(\ker(\a)+3K)$. Let us denote $K_1'=K_1/\ker(\a)\subset H\subset B$ and $H'=K/K_1=H/K'_1$. Consider the isogeny $f:B\rightarrow B'=B/K'_1$ and let $L'$ be a line bundle on $B'$ defined by $$L'=\operatorname{N}_{B/B'}(L)\otimes\pi^*\operatorname{N}_{K'_1/S}(L|_{K'_1})^{-1}.$$ Then $L'$ is trivialized along the zero section and we have $$\operatorname{N}_{H/S}(L|_H)=\operatorname{N}_{H'/S}(\operatorname{N}_{B/B'}L|_{H'})= \operatorname{N}_{H'/S}(L'|_{H'})+|H'|\cdot \operatorname{N}_{K'_1/S}(L|_{K'_1}).$$ The latter term is trivial, since $\operatorname{N}_{K'_1/S}(L|_{K'_1})$ is annihilated by $3$ (see the proof of Theorem \ref{isogmain}). Thus, it remains to prove the triviality of $\operatorname{N}_{H'/S}(L'|_{H'})$. Since $|H'|>3$ we can apply Lemma \ref{norm} to $L'$ and $H'$ provided that $H'$ is isotropic with respect to the standard symplectic form on $K(L^{\prime 3})$. This is equivalent to asking that $H$ is isotropic in $K((f^*L')^3)$. But the symplectic structure on the latter group is determined by the the polarization associated with $(f^*L')^3$ (see \cite{MuAb}). Now since $f^*\operatorname{N}_{B/B'}(L)$ is algebraically equivalent to $L^{\deg(f)}$ we obtain that $K((f^*L')^3)=K(L^{3\deg(f)})$ as symplectic groups. Now $H\subset K(L^{3\deg(f)})$ is isotropic iff $K=\a^{-1}(H)\subset K(\a^*L^{3\deg(f)})$ is isotropic. But $K\subset K(\a^*L)$, so this follows from the fact that $K$ is annihilated by $3\deg(f)=3\cdot |K'_1|$ by assumption. \qed\vspace{3mm} \section{The method of Faltings and Chai} \label{meth} In this section we start proving Theorem \ref{main1}. Fix an odd prime number $p$. Following the proof of Faltings and Chai we consider the homomorphism $f_p:{\Bbb Z}\rightarrow\operatorname{Pic}(S)^{(p)}$, such that $f_p(n)$ is the $p$-primary component of $\Delta(L^n)$. This is a "polynomial" function in $n$, which means that $\delta^if_p=0$ for some $i$ where $\delta$ is the difference operator: $\delta\phi(n)=\phi(n+1)-\phi(n)$. As was noticed in \cite{FC} this can be seen by embedding $A$ into the product of projective bundles $\P(\pi_*L^a)\times_S\P(\pi_*L^b)$ for relatively prime $a$ and $b$ (see Lemma \ref{degree} below for a more precise result). This implies immediately that the image of $f_p$ belongs to some finitely generated subgroup of $\operatorname{Pic}(S)^{(p)}$ and that $f_p(n+p^N)=f_p(n)$ for sufficiently large $N$. By Serre duality one has $$f_p(1)+(-1)^g\cdot f_p(-1)=0$$ where $g$ is the relative dimension of $A/S$. Thus, if we find an integer $k\equiv -1\mod(p^N)$ such that $f_p(k)=k^g\cdot f_p(1)$ this would imply that $\Delta(L)^{(p)}=0$. This is always possible when $p\equiv 1\mod(4)$. Indeed, we claim that if $p>3$ then $$f_p(m^2n)=m^{2g}f_p(n)$$ for all $n$ and $m\neq 0$. This follows immediately from Theorem \ref{isogmain} applied to the isogeny $[m]_A:A\rightarrow A$ and the line bundle $L^n$ (we don't have to worry about the factor $12$ since we only consider $p$-primary component of the equality of Lemma \ref{degree} and $p>3$). If $p\equiv 1\mod(4)$ then we can find $k=m^2\equiv -1\mod(4)$, so we are done. Hence, we can assume that $p\equiv -1\mod(4)$. In this case one can always find some integers $n$ and $m$ such that $n^2+m^2\equiv -1\mod(p^N)$. Now let us consider the isogeny $\a:A^2\rightarrow A^2$ given by the matrix $\left( \matrix {[n]_A} & {[m]_A} \\ {[-m]_A} & {[n]_A} \endmatrix\right)$. Then it is easy to see that $\a^*(L\boxtimes L)\simeq L^k\boxtimes L^k$ where $k=n^2+m^2$. Note that possibly changing initial $n$ and $m$ we can achieve that $k$ is prime to $3$. Applying Theorem \ref{isogmain} we find that $$4\cdot\Delta(L^k\boxtimes L^k)= 4\cdot k^{2g}\Delta(L\boxtimes L).$$ Hence, $d\cdot f_p(k)=d\cdot k^g\cdot f_p(1)$. As we noticed above this implies that $d\cdot\Delta(L)^{(p)}=0$. This finishes the first step in the proof of Theorem \ref{main1}. Now let us prove that $\Delta(L)^{(p)}=0$ for $p\ge 2g+1$, $p\neq 3$. The only new ingredient we need is the following lemma. Let us say that a function $\phi:{\Bbb Z}\rightarrow G$, where $G$ is an abelian group, has degree $\le l$ if $\delta^{l+1}\phi=0$ where $\delta\phi(n)=\phi(n+1)-\phi(n)$. \begin{lem}\label{degree} Let $\pi:X\rightarrow S$ be a smooth projective morphism of pure dimension $g$, $L$ be a line bundle on $X$. Then the function $f:{\Bbb Z}\rightarrow\operatorname{Pic}(S)$ defined by $f(n)=\det\pi_*(L^n)$ has degree $\le g+1$. \end{lem} \noindent {\it Proof} . This follows from Elkik's construction (based on ideas of Deligne in \cite{De}), see \cite{Elkik}, IV.1.3. \qed\vspace{3mm} Applying this lemma to our abelian scheme $A/S$ and the line bundle $L$ on it we deduce that $f_p$ has degree $\le g+1$. Now the vanishing of $\Delta(L)^{(p)}$ for $p\ge 2g+1$, $p\neq 3$, is implied by the following lemma. \begin{lem}\label{ar1} Let $p$ be a prime number, such that $p\ge 2g+1$, $p\neq 3$, $\phi:{\Bbb Z}\rightarrow{\Bbb Z}/p^k{\Bbb Z}$ be a function of degree $\le g+1$, such that $\phi(n+p^N)=\phi(n)$ for sufficiently large $N$. Assume that $\phi(m^2n)=m^{2g}\cdot \phi(n)$ for all $n$ and $m$. Then $\phi(n)=n^g\cdot \phi(1)$ for all $n$. \end{lem} \noindent {\it Proof} . Replacing $\phi$ by $\phi(n)-n^g\cdot \phi(1)$ we can assume that $\phi(1)=0$. In this case the assertion of Lemma is that $\phi=0$. An easy induction in $k$ shows that it suffices to prove this for $k=1$. Then we can find a polynomial $\phi'(x)\in{\Bbb Z}/p{\Bbb Z}[x]$ of degree $\le p-1$ such that $\phi'(n)=\phi(n)$ for $n=0,1,\ldots,p-1$. Since $\phi$ is the function of degree $\le p-1$, it is determined uniquely by the set of its $p$ consequtive values. The same is true for $\phi'$ considered as a function ${\Bbb Z}\rightarrow{\Bbb Z}/p{\Bbb Z}$. It follows that $\phi'(n)=\phi(n)$ for all $n$, in particular, $\phi(n)$ depends only on $n\mod(p)$. Let us fix a non-quadratic residue $a$ modulo $p$. We know that $\phi(n)=0$ if $n$ is a square modulo $p$, and that $\phi(n)=a^{-g}\cdot n^g\cdot\phi(a)$ if $n$ is not a square modulo $p$. Hence, for some $\lambda\in{\Bbb Z}/p{\Bbb Z}$ we have $$\phi(n)=\lambda\cdot n^g\cdot(1-n^{\frac{p-1}{2}})$$ for all $n$. Now if $\lambda\neq 0$ then the right hand side is given by a polynomial of degree $g+\frac{p-1}{2}\le p-1$. Therefore, we actually have an identity of polynomials in ${\Bbb Z}/p{\Bbb Z}[x]$ which implies that $\deg(\phi)=g+\frac{p-1}{2}$. But this contradicts to $\deg(\phi)\le g+1$, hence, $\lambda=0$ as required. \qed\vspace{3mm} \begin{rem} The fact that the element $\Delta(L)\in\operatorname{Pic}(S)$ has finite order is proved in \cite{FC} along the same lines. One should consider the function $f_0:{\Bbb Z}\rightarrow\operatorname{Pic}(S)/\operatorname{Pic}(S)^{tors}$ where $\operatorname{Pic}(S)^{tors}$ is the torsion subgroup of $\operatorname{Pic}(S)$, such that $f_0(n)=\Delta(L^n)\mod\operatorname{Pic}(S)^{tors}$. Then $f_0$ is a function of finite degree, hence, its image is a finitely generated free group. Then the identity $f_0(n^2)=n^{2g}\cdot f_0(1)$ for infinitely many $n$ implies that $f_0(n)=n^g\cdot f_0(1)$ for all $n$. Applying this to $n=-1$ and using Serre duality as above we deduce that $f_0=0$. At last, the bound on the 2-primary torsion of $\Delta(L)$ is obtained by considering the isogeny $A^4\rightarrow A^4$ given by a $4\times 4$ matrix of multiplication by a quaternion $n+m\cdot i+p\cdot j+q\cdot k$ such that $n^2+m^2+p^2+q^2\equiv -1(N)$ for sufficiently divisible $N$ (see \cite{FC}). \end{rem} \section{Some arithmetics} \label{arsec} Let $\phi:{\Bbb Z}\rightarrow{\Bbb Z}/p^k{\Bbb Z}$ be a map, $g\ge 1$ be an integer. Let us say that $\phi$ is $g$-{\it special}, if $\phi$ has degree $\le g+1$ and $\phi(m^2n)=m^{2g}\cdot\phi(n)$ for all $n$ and $m$. In particular, since $\phi$ has finite degree it factors through ${\Bbb Z}/p^N{\Bbb Z}$ for sufficiently large $N$. In this terminology Lemma \ref{ar1} says that for $p>3$, $g\le\frac{p-1}{2}$ any $g$-special map has form $\phi(n)=n^g\cdot \phi(1)$. In this section we'll study $g$-special maps for other values of $g$. Our main result is the following theorem, which combined with results of the previous section implies the first part of Theorem \ref{main1}, except for the fact that $\Delta(L)^{(p)}$ is annihilated by $p^2$ if $p=2g-1$. The latter statement will be proved together with the second half of Theorem \ref{main1} in the end of this section. \begin{thm}\label{ar2} If $p>3$ and $g<\frac{3p-1}{2}$, $g\neq\frac{p+1}{2}$, then for any $g$-special map $\phi:{\Bbb Z}\rightarrow{\Bbb Z}/p^k{\Bbb Z}$ one has $p\cdot \phi(n)=p\cdot n^g\cdot \phi(1)$. \end{thm} We will use the condition that the degree of $\phi$ is $\le g+1$ in the following form. Let us consider the generating function $$F(t)=\sum_{n\ge 0}\phi(n)t^n\in{\Bbb Z}/p^k{\Bbb Z}[[t]].$$ Then the condition $\deg(\phi)\le g+1$ implies that $$F(t)\cdot (t-1)^{g+2}=P(t)$$ where $P(t)\in{\Bbb Z}/p^k{\Bbb Z}[t]$ is a polynomial in $t$. In particular, if $\phi(n+p^N)=\phi(n)$ for all $n$ then $$F(t)=Q_{\phi}(t)\cdot (1-t^{p^N})^{-1}$$ where $Q_{\phi}(t)=\sum_{n=0}^{p^N-1}\phi(n)$. Thus, $Q_{\phi}(t)\cdot (t-1)^{g+2}$ is divisible by $t^{p^N}-1$ in ${\Bbb Z}/p^k{\Bbb Z}[t]$. We are particularly interested in the case $k=1$. In this case we obtain that $Q_{\phi}(t)$ is divisible by $(t-1)^{p^N-g-2}$. Let us denote $$S_r(t)=\sum_{n=0}^{p-1}n^r\cdot t^n\in{\Bbb Z}/p{\Bbb Z}[t]$$ for $r\ge 0$ (in case $r=0$ our convention is that $0^0=1$). Note that for $r>0$ one has $S_r(t)=S_{r+p-1}(t)$. For every polynomial $Q\in{\Bbb Z}/p{\Bbb Z}[t]$ we denote by $v_{(t-1)}(Q)$ the maximal power of $(t-1)$ dividing $Q$. \begin{lem}\label{val} One has $$v_{(t-1)}(S_r(t))=p-1-r$$ for $0\le r\le p-1$. \end{lem} \noindent {\it Proof} . For $r=0$ we have an identity $$S_0(t)=(t-1)^{p-1}$$ which follows from the congruence ${p-1 \choose i}\equiv (-1)^i\mod(p)$. Now the identity $S_{r+1}(t)=t\cdot\frac{d}{dt}S_r(t)$ and an easy induction show that $$S_r(t)\equiv (-1)^r\cdot r!\cdot t^r\cdot (t-1)^{p-1-r} \mod((t-1)^{p-r})$$ for $0\le r\le p-1$. \qed\vspace{3mm} The first step in the proof of Theorem \ref{ar2} is the following lemma. \begin{lem}\label{modp} Let $\phi:{\Bbb Z}\rightarrow{\Bbb Z}/p{\Bbb Z}$ be a $g$-special map. If $p>3$ and $\frac{p-1}{2}<g<2p-1$ then $$\phi(n)=\lambda\cdot n^g+\mu\cdot n^{g-\frac{p-1}{2}}\mod(p)$$ for some constants $\lambda,\mu\in{\Bbb Z}/p{\Bbb Z}$. \end{lem} \noindent {\it Proof}. It is easy to see that for any $\lambda$ and $\mu$ the map $$n\mapsto \lambda\cdot n^g+\mu\cdot n^{g-\frac{p-1}{2}}\mod(p)$$ is $g$-special. Hence, if we write $$\phi(n)=\lambda\cdot n^g+\mu\cdot n^{g-\frac{p-1}{2}}+\phi'(n)$$ for some $\lambda$ and $\mu$ then $\phi'$ will also be a $g$-special map. Choosing $\lambda$ and $\mu$ appropriately we can achieve that $\phi'(1)=\phi'(a)=0$ for some $a$ which is a not a square modulo $p$. Replacing $\phi$ by $\phi'$ we can assume that this condition holds for $\phi$. Since $\phi(n+p)=\phi(n)$ for every $n\not\equiv 0\mod(p)$ (this follows from $g$-speciality and the fact that $(n+p)n^{-1}$ is a square in ${\Bbb Z}/p^N{\Bbb Z}$) we deduce that $\phi(n)=0$ for all $n\not\equiv 0\mod(p)$. On the other hand, $\phi(p^2n)=p^{2g}\phi(n)=0$, hence the only non-trivial values of $\phi$ are $\phi(pn)$ for $n\not\equiv 0\mod(p)$. If $\deg\phi<p$ then this implies immediately that $\phi=0$, so we can assume that $g\ge p-1$. For all $n\not\equiv 0\mod(p)$ we have $\phi(pn+p^2)=(1+p\cdot n^{-1})^g \cdot\phi(pn)=\phi(pn)$. In particular, $\phi$ depends only on $p\mod(p^2)$ and $$Q_{\phi}(t)=\sum_{n=0}^{p^2-1}\phi(n)t^n= \sum_{n=1}^{p-1}\phi(pn)t^{pn}.$$ Now $n\mapsto\phi(pn)$ is a $p$-special map depending only on $n\mod(p)$. Hence, $$\phi(pn)=a\cdot n^g+b\cdot n^{g-\frac{p-1}{2}}$$ for some $a,b\in{\Bbb Z}/p{\Bbb Z}$. Therefore, $$Q_{\phi}(t)=a\cdot S_g(t^p)+b\cdot S_{g-\frac{p-1}{2}}(t^p).$$ As we have seen above the fact that $\deg\phi\le g+1$ implies that $v_{(t-1)}(Q_{\phi})\ge p^2-g-2$. Now we claim that for all $g$ such that $p-1\le g\le 2(p-1)$ the valuations of $S_g(t^p)$ and $S_{g-\frac{p-1}{2}}(t^p)$ at $(t-1)$ are less than $p^2-g-2$. This would imply that $a=b=0$, hence $\phi=0$ as required. To prove our claim let us apply Lemma \ref{val} to compute $v_{(t-1)}S_g$ and $v_{(t-1)}S_{g-\frac{p-1}{2}}$. For $g=p-1$ we get $v_{(t-1)}S_g=0$, while for $p-1<g\le 2(p-1)$ we have $v_{(t-1)}S_g(t^p)=p\cdot(2(p-1)-g)<p^2-g-2$. Similarly, for $p-1\le g\le\frac{3(p-1)}{2}$ we get $v_{(t-1)}S_{g-\frac{p-1}{2}}(t^p)= p\cdot(p-1-g+\frac{p-1}{2})<p^2-g-2$, while for $\frac{3(p-1)}{2}<g\le 2(p-1)$ we have $v_{(t-1)}S_{g-\frac{p-1}{2}}(t^p)= p\cdot(2(p-1)-g+\frac{p-1}{2})$. Thus, to finish the proof we need the inequality $$p\cdot(\frac{5(p-1)}{2}-g)<p^2-g-2$$ in this case, but when $p>3$ it follows from $g>\frac{3(p-1)}{2}$. \qed\vspace{3mm} \begin{rem} In fact, for $p>3$ one can prove that the conclusion of the previous lemma remains true when $g$ belongs to one of the following intervals of integers: $[2p,\frac{5}{2}(p-1)]$, $[\frac{5p+1}{2},3(p-1)]$, $[3p,\frac{7}{2}(p-1)]$, $[\frac{7p+1}{2},4(p-1)]$, etc. (for given $p$ the set of such $g$ is finite). \end{rem} Next step is to consider $g$-special maps depending only on $n\mod(p^2)$. \begin{lem}\label{modp2} Let $\phi:{\Bbb Z}\rightarrow{\Bbb Z}/p^2{\Bbb Z}$ be a $g$-special map such that $\phi(n+p^2)=\phi(n)$ for all $n$. Assume that $p>3$ and $\frac{p+1}{2}<g<\frac{3p-1}{2}$. Then $$p\cdot\phi(n)=p\cdot n^g\cdot\phi(1)$$ for all $n$. \end{lem} \noindent {\it Proof} . Replacing $\phi$ by $\phi-n^g\cdot\phi(1)$ we can assume that $\phi(1)=0$. In this case we need to show that $\phi=0\mod(p)$. Applying Lemma \ref{modp} to $\phi\mod(p)$ we obtain \begin{equation}\label{phipsi} \phi(n)=c\cdot (n^g- n^{g-\frac{p-1}{2}})+ p\cdot\psi(n) \end{equation} for some constant $c\in{\Bbb Z}/p^2{\Bbb Z}$ and some map $\psi:{\Bbb Z}\rightarrow{\Bbb Z}/p{\Bbb Z}$. Now for every $n\not\equiv 0\mod(p)$ we have \begin{align*} &\phi(n+p)-\phi(n)=(1+p\cdot n^{-1})^g\cdot\phi(n)-\phi(n)= p\cdot g\cdot n^{-1}\cdot\phi(n)=\\ &p\cdot c\cdot g\cdot (n^{g-1}-n^{g-\frac{p+1}{2}}). \end{align*} On the other hand, subtracting the equation (\ref{phipsi}) for $n$ from that for $n+p$ we get $$\phi(n+p)-\phi(n)= p\cdot c\cdot(g\cdot n^{g-1}- (g-\frac{p-1}{2})\cdot n^{g-\frac{p+1}{2}})+ p\cdot(\psi(n+p)-\psi(n)).$$ It follows that $$\psi(n+p)-\psi(n)=-c\cdot\frac{p-1}{2}\cdot n^{g-\frac{p+1}{2}}=\frac{c}{2}\cdot n^{g-\frac{p+1}{2}}.$$ Hence, for every $m$ one has \begin{equation}\label{psi1} \psi(n+p\cdot m)=\psi(n)+\frac{c}{2}\cdot m\cdot n^{g-\frac{p+1}{2}} \end{equation} provided that $n\not\equiv 0\mod(p)$. Also we claim that \begin{equation}\label{psi2} \psi(p\cdot n)=\lambda\cdot n^g+\mu\cdot n^{g-\frac{p-1}{2}} \end{equation} for some $\lambda,\mu\in{\Bbb Z}/p{\Bbb Z}$. Indeed, since $g>\frac{p+1}{2}$ the equation (\ref{phipsi}) shows that $\phi(p\cdot n)=p\cdot\psi(p\cdot n)$. Hence, the map $n\mapsto\psi(p\cdot n)$ is $g$-special and the assertion follows from Lemma \ref{modp}. Now (\ref{phipsi}) shows that $\psi$ depends only on $n\mod(p^2)$ and has degree $\le g+1$. Hence, the corresponding polynomial $Q_{\psi}(t)=\sum_{n=0}^{p^2-1}\psi(n)t^n$ is divisible by $(t-1)^{p^2-g-2}$. Using (\ref{psi1}) and (\ref{psi2}) we can write \begin{align*} &Q_{\psi}(t)= \sum_{n=1}^{p-1}\sum_{m=0}^{p-1}\psi(n+p\cdot m)t^{n+p\cdot m} +\sum_{n=1}^{p-1}\psi(p\cdot n)=\\ &(\sum_{n=1}^{p-1}\psi(n)t^n)\cdot S_0(t^p)+ \frac{c}{2}\cdot S_{g-\frac{p+1}{2}}(t)\cdot S_1(t^p) +\lambda\cdot S_g(t^p)+\mu\cdot S_{g-\frac{p-1}{2}}(t^p). \end{align*} In the case $g\le p-2$ we have $p^2-g-2\ge p^2-p$, hence, $v_{(t-1)}Q_{\psi}\ge p^2-p$. To prove that $c=0$ in this case it is sufficient to check that $v(g):=v_{(t-1)}S_{g-\frac{p+1}{2}}(t)+v_{(t-1)}S_1(t^p)<p^2-p$ and that $v(g)$ differs from $v_{(t-1)}S_g(t^p)$ and $v_{(t-1)}S_{g-\frac{p-1}{2}}(t^p)$. One can check using Lemma \ref{val} that this is indeed the case. When $g\ge p-1$ we can omit the first term in the above expression for $Q_{\psi}(t)$ when considering $Q_{\psi}(t)\mod(t-1)^{p^2-g-2}$. Hence, to deduce that $c=0$ one should check using Lemma \ref{val} that $v(g)<p^2-g-2$ and $v(g)$ differs from the valuations of two other terms. We omit the details of this simple computation. \qed\vspace{3mm} \noindent {\it Proof of Theorem \ref{ar2}}. The case $g\le\frac{p-1}{2}$ follows from Lemma \ref{ar1} so we only consider $g>\frac{p+1}{2}$. Also as usual we can assume that $\phi(1)=0$. An easy induction in $k$ shows that it suffices to consider the case $k=2$. In the latter case we have $\phi(p^2\cdot n)=0$ for all $n$ and $\phi(n+p^2)=\phi(n)$ for $n\not\equiv 0\mod(p)$. According to Lemma \ref{modp} we can write $$\phi(n)=c\cdot (n^g- n^{g-\frac{p-1}{2}})+ p\cdot\psi(n)$$ for some constant $c\in{\Bbb Z}/p^2{\Bbb Z}$ and some map $\psi:{\Bbb Z}\rightarrow{\Bbb Z}/p{\Bbb Z}$. In particular, $\phi(p\cdot n)=p\cdot\psi(p\cdot n)$ and $n\mapsto\psi(p\cdot n)$ is a $g$-special map. Now Lemma \ref{modp} implies that $\psi(p\cdot(n+p))=\psi(p\cdot n)$, hence $\phi(p\cdot n+p^2)=\phi(p\cdot n)$. Therefore, $\phi(n)$ depends only on $n\mod(p^2)$ and we can apply Lemma \ref{modp2} to finish the proof. \qed\vspace{3mm} Now we are turning to the proof of the second half of Theorem \ref{main1}. Note that for any prime $p$ the function $f_p:n\mapsto\Delta(L^n)^{(p)}$ satisfies the following property: $$f_p(m^2)=m^{2g}\cdot f_p(1)$$ for all $m$ such that $m\not\equiv 0\mod(p)$. Indeed, changing $m$ by $m+p^N$ if necessary we may assume that $m$ is odd and is prime to $3$, hence this follows from Theorem \ref{isogmain}. \begin{prop} Let $f:{\Bbb Z}\rightarrow{\Bbb Z}/p^k{\Bbb Z}$ be a function of degree $\le g+1$, such that $f(m^2)=m^{2g}\cdot f(1)$ for all $m$ such that $m\not\equiv 0\mod(p)$. Then there exists an integer $n(p,g)$ depending only on $p$ and $g$ such that $p^{n(p,g)}\cdot (f(n)-n^g\cdot f(1))=0$ for all $n$. \end{prop} \noindent {\it Proof} . First of all, replacing $f$ by $(g+1)!\cdot f$ we can assume that $$f(n)-n^g\cdot f(1)=a_0+n\cdot a_1+\ldots +n^{g+1}\cdot a_{g+1}$$ for some $a_i\in {\Bbb Z}/p^k{\Bbb Z}$. Let $n_0,\ldots,n_{g+1}$ be the first $g+2$ positive integers of the form $m^2$ with $m\not\equiv 0\mod(p)$. Then $f(n_i)-n_i^g\cdot f(1)=0$ by assumption, hence every coefficient $a_i$ is annihilated by the Vandermonde determinant $\Delta(n_0,\ldots,n_{g+1})=\prod_{i<j}(n_j-n_i)$. It follows that we can take $n(p,g)$ to be $v_p((g+1)!\cdot\Delta(n_0,\ldots,n_{g+1}))$. \qed\vspace{3mm} The proof of the second part of Theorem \ref{main1} follows immediately from this proposition: we can take $N(g)=\prod_{p\le 2g-1}p^{n(p,g)}$. To complete the proof of Theorem \ref{main1} it remains to prove the statement concerning the prime $p=2g-1$. This is the content of the following lemma. \begin{lem} Let $p>3$ be a prime, $f:{\Bbb Z}\rightarrow{\Bbb Z}/p^k{\Bbb Z}$ be a $g$-special map where $g=\frac{p+1}{2}$. Then $p^2\cdot (f(n)-n^g\cdot f(1))=0$. \end{lem} \noindent {\it Proof} . Since $g+1<p$ we have $$f(n)-n^g\cdot f(1)=a_0+n\cdot a_1+\ldots +n^{g+1}\cdot a_{g+1}$$ for some $a_i\in {\Bbb Z}/p^k{\Bbb Z}$. Let $n_0=0$, $n_1,\ldots,n_{g+1}$ be the first $g+1$ positive integers of the form $m^2$ with $m\not\equiv 0\mod(p)$. Then $f(n_i)-n_i^g\cdot f(1)=0$ for all $i=0,\ldots,g+1$ by $g$-speciality, therefore every coefficient $a_i$ is annihilated by the Vandermonde determinant $\Delta=\Delta(n_0,\ldots,n_{g+1})=\prod_{i<j}(n_j-n_i)$. Since $g+1=\frac{p-1}{2}+2$, it follows that $v_p(\Delta)=2$, hence the assertion. \qed\vspace{3mm} \section{Eliminating of primes not dividing characteristic} \label{charsec} In the case when an odd prime $p$ is not among residue characteristics of the base we can evaluate $\Delta(L)^{(p)}$ over $S[\frac{1}{p}]$ using the following result. \begin{thm}\label{bc} Let $L$ be a symmetric, relatively ample line bundle over an abelian scheme $A/S$ trivialized along the zero section. Let $p$ be an odd prime number which is not equal to any of residue characteristics of $S$. Then there exists a finite flat base change $c:S'\rightarrow S$ of degree prime to $p$, an isogeny of abelian $S'$-schemes $\a:A'\rightarrow B$, where $A'$ is obtained from $A$ by this base change, such that $\deg(\a)$ is the power of $p$, and a symmetric line bundle $M$ on $B$ together with a symmetric isomorphism $\a^*M\simeq L'$ (where $L'$ is obtained from $L$ by the base change), such that $\deg\phi_M$ is prime to $p$. \end{thm} Let us deduce Theorem \ref{main2} from this. Choose a base change $c:S'\rightarrow S$ as in Theorem \ref{bc}. Since $\operatorname{Pic}(S)^{(p)}\rightarrow\operatorname{Pic}(S')^{(p)}$ is injective and the construction of $\det\pi_*$ commutes with this base change we can work with $A'$ instead of $A$. It remains to apply Theorem \ref{isogmain} to isogeny $\a$ and Theorem \ref{main1} to $M$ to deduce that $\Delta(L')^{(p)}=0$ if $p\neq 3$. In the case $p=3$ by the same argument we always have $3\cdot\Delta(L')^{(3)}=0$. Now if the $3$-type of the polarization is different from $(1,\ldots,1,3^k)$, then one can see easily from the construction of the isogeny $\a$ below that the conditions of Theorem \ref{isogmain2} are satisfied for $\a$ and $M$. Hence, the triviality of $\Delta(L')^{(3)}$ in this case. \vspace{3mm} \noindent {\it Proof of Theorem \ref{bc}}. We can assume that the base $S$ is connected. Let $K=K(L)^{(p)}$ be the $p$-primary component of the finite flat group scheme $K(L)$ over $S$. Then $K$ is \'etale over $S$ (since $p$ is not among residue characteristics of $S$) and $\phi_L$ induces a skew-symmetric isomorphism $K\simeq \hat{K}$. The fiber of $K$ over a geometric point of $S$ is a discrete group of the form $({\Bbb Z}/p^{n_1}{\Bbb Z})^2\times\ldots\times({\Bbb Z}/p^{n_k}{\Bbb Z})^2$ where the factors $({\Bbb Z}/p^{n_i}{\Bbb Z})^2$ are orthogonal to each other with respect to a symplectic form, ${\Bbb Z}/p^{n_i}{\Bbb Z}\times \{0\}\subset({\Bbb Z}/p^{n_i}{\Bbb Z})^2$ is a lagrangian subgroup for every $i$. Since $S$ is connected the collection $n_1,\ldots,n_k$ doesn't depend on a point. Let $n$ be the maximum of $n_1,\ldots, n_k$, so that $n$ is the minimal number such that $p^n\cdot K=0$. Now let us construct a canonical isotropic subgroup $I_0\subset K$, \'etale over $S$, such that $K_0=I_0^{\perp}/I_0$ is annihilated by $p$. If $n=1$ we can take $I_0=0$ so let's assume that $n\ge 2$. Then $p^{n-1}\cdot K$ is an isotropic subgroup in $K$ so we can consider the reduction $\overline{K}=(p^{n-1}\cdot K)^{\perp}/(p^{n-1}\cdot K)$ with its induced symplectic form. By induction we may assume that we already found an isotropic subgroup $\overline{I}_0\subset K'$ such that $(\overline{I}_0)^{\perp}/\overline{I}_0$ is a $p$-group. Now take $I_0$ to be the preimage of $\overline{I}_0$ in $K$. Our base change $c:S'\rightarrow S$ will be the finite flat covering corresponding to a choice of a lagrangian subgroup in $K_0$. One can construct such a covering in the following way. Let $p^{2r}$ be the order of $K^0$. Start with a subscheme $\widetilde{S}$ in $K_0\times_S\ldots\times_S K_0$ ($2r$ times) corresponding to symplectic bases in $K_0$. Then $\widetilde{S}\rightarrow S$ is a $\Sp_{2r}({\Bbb Z}/p{\Bbb Z})$-torsor. Now let $P\subset\Sp_{2r}({\Bbb Z}/p{\Bbb Z})$ be the subgroup preserving the standard $r$-dimensional lagrangian subgroup in $({\Bbb Z}/p{\Bbb Z})^{2r}$, then we can take $S'=P\backslash \widetilde{S}$. The degree of the covering $S'\rightarrow S$ is equal to the number of lagrangian subgroups in $({\Bbb Z}/p{\Bbb Z})^{2r}$ which is easily seen to be equal to $\prod_{i=1}^r(p^i+1)$ (first compute the number of isotropic flags and then divide by the number of flags in $({\Bbb Z}/p{\Bbb Z})^r$), which is prime to $p$. Let $(A',L',K',I'_0,K'_0)$ be the data obtained from $(A,L,K,I_0,K_0)$ by the base change $S'\rightarrow S$. Then by construction we have a lagrangian subgroup $\overline{I}\subset K'_0$. Taking its preimage by the morphism $(I'_0)^{\perp}/I'_0\rightarrow K'_0$ we obtain a lagrangian subgroup $I\subset K'$. It remains to prove that $L'$ descends to a symmetric bundle on $B=A'/I$. Let $p:G\rightarrow K'$ be the restriction of the Mumford's group of $L'$. It is well-known that a descent of $L'$ to a line bundle on $A'/I$, is equivalent to choosing a splitting of $p$ over $I$. Thus, we are reduced to finding a trivialization of the group extension ${\Bbb G}_m\rightarrow p^{-1}(I)\rightarrow I$, which is compatible with the isomorphism $\tau:G\widetilde{\rightarrow}(-\operatorname{id})^*G$. Note that such a trivialization is necessarily unique since a homomorphism $f:I\rightarrow{\Bbb G}_m$ satsifying $f(-x)=f(x)$ is trivial (recall that the order of $I$ is odd). Hence, it is sufficient to prove local existence of such trivialization. Now locally there exists a homomorphism $\sigma:I\rightarrow p^{-1}(I)$ splitting $p$. Then $x\mapsto \tau^{-1}\sigma(-x)$ is another such splitting, so that $\tau^{-1}\sigma(-x)=\psi(x)\cdot \sigma(x)$ for some homomorphism $\psi:I\rightarrow{\Bbb G}_m$. Since the multiplication by $2$ is invertible on $I$ there exists a homomorphism $\phi:I\rightarrow{\Bbb G}_m$ such that $\psi(x)=\phi(x)^2$. Then $x\mapsto \phi(x)\sigma(x)$ gives the symmetric splitting. \qed\vspace{3mm} \section{Complements}\label{comp} \subsection{Case of elliptic curves} Let us evaluate determinant bundles in the case $g=1$. It is known (see e.g. \cite{Mu}) that in characteristics $\neq 2,3$ the Picard group of the moduli stack of elliptic curves (with one fixed point) is ${\Bbb Z}/12{\Bbb Z}$ and as generator one can take the line bundle $\overline{\omega}$ on this moduli stack that associates to every family of elliptic curves $\pi:E\rightarrow S$ the relative canonical bundle $\overline{\omega}_E\in\operatorname{Pic}(S)$. If $S$ is connected then any symmetric line bundle $L$ on $E$, trivialized along the zero section, is either isomorphic to $L_d(e):=\O(d\cdot e)\otimes\omega_{E/S}^d$ where $e:S\rightarrow E$ is the zero section, or to $L_d(\eta):=\O((d-1)\cdot e+\eta)\otimes\omega_{E/S}^{d-1}$ where $\eta:S\rightarrow E$ is an everywhere non-trivial point of order 2. Note that these line bundles are trivialized along the zero section, since $e^*\O(e)\simeq\overline{\omega}_E^{-1}$. \begin{prop}\label{ell} One has $$\det\pi_*(L_d(e))=(\frac{d(d-1)}{2}+1)\cdot\overline{\omega}_E,$$ $$\det\pi_*(L_d(\eta))=\frac{d(d-1)}{2}\cdot\overline{\omega}_E.$$ In particular, $$\Delta(L_d(e))=(d^2+2)\cdot\overline{\omega}_E,$$ $$\Delta(L_d(\eta))=d^2\cdot\overline{\omega}_E.$$ Furthermore, these equalities are represented by canonical isomorphisms of line bundles. \end{prop} \noindent {\it Proof}. Considering the push-forward of the exact sequence $$0\rightarrow\O((d-1)\cdot e)\rightarrow\O(d\cdot e)\rightarrow e_*e^*\O(d\cdot e)\rightarrow 0$$ we deduce that $$\det\pi_*\O(d\cdot e)-\det\pi_*\O((d-1)\cdot e)= -d\cdot\overline{\omega}_E.$$ Since $\pi_*\O(e)\simeq\O_S$ it follows that $$\det\pi_*\O(d\cdot e)=(1-\frac{d(d+1)}{2})\overline{\omega}_E,$$ hence $\det\pi_*L_d(e)=(\frac{d(d-1)}{2}+1)\overline{\omega}_E$. The case of $\O_d(\eta)$ is considered similarly using the exact sequence $$0\rightarrow\O((d-1)\cdot e+\eta)\rightarrow\O(d\cdot e+\eta)\rightarrow e_*e^*\O(d\cdot e+\eta)\rightarrow 0$$ and the triviality of $\pi_*\O(\eta)$. \qed\vspace{3mm} Note that $\Delta(L_d(e))$ gives a line bundle on the moduli stack of elliptic curves $\AA_1$, while $\Delta(L_d(\eta))$ lives on the stack $\widetilde{\AA}_1$ classifying elliptic curves with a non-trivial point of order 2. Now Proposition \ref{ell} combined with Theorem \ref{main1} implies immediately that the order of $\overline{\omega}$ in $\operatorname{Pic}(\AA_1)$ is 12, while the order of the pull-back of $\overline{\omega}$ to $\widetilde{\AA}_1$ is 4. In particular, $\Delta(L_3(e))=-\overline{\omega}$ in $\operatorname{Pic}(\AA_1)$. \subsection{Linear relations between determinant bundles} Let us first consider the determinant bundles $\det\pi_*(L^n)$ on an abelian scheme $A/S$ of relative dimension $g=2$ or $g=3$, where $L$ is a relatively ample, symmetric line bundle on $A$ trivialized along the zero section. \begin{prop}\label{Ln} If $g=2$ then one has $$\det\pi_*L^n=\frac{4n-n^3}{3}\cdot\det\pi_*L+ \frac{n^3-n}{6}\cdot\det\pi_*L^2+ \frac{n(n-1)(n-2)}{6}\cdot d\cdot\overline{\omega}_A,$$ where $d=\operatorname{rk}\pi_*L$. In particular, $$\Delta(L^n)=\frac{4n-n^3}{3}\cdot\Delta(L)+\frac{n^3-n}{6}\cdot\Delta(L^2).$$ If $g=3$ then one has $$\det\pi_*L^n=\frac{4n^2-n^4}{3}\cdot\det\pi_*L+ \frac{n^4-n^2}{12}\cdot\det\pi_*L^2+ \frac{n^2(n-1)(n-2)}{6}\cdot d\cdot\overline{\omega}_A.$$ In particular, in this case $$\Delta(L^n)=\frac{4n^2-n^4}{3}\cdot\Delta(L)+\frac{n^4-n^2}{12}\cdot \Delta(L^2).$$ \end{prop} \noindent {\it Proof} . This is easily deduced from the fact that the function $f:n\mapsto\det\pi_*L^n$ has degree $\le g+1$. Indeed, by Serre duality the values of this function at $n=-2,-1$ are expressed via those for $n=1,2$. Also $\det\pi_*\O_A=0$, hence, we know values of $f$ at $n\in [-2,2]$ and we can interpolate the rest. \qed\vspace{3mm} \begin{cor} If $g=2$ or $g=3$, then $$8\cdot 9\cdot\Delta(L)=4\cdot 9\cdot\Delta(L^2)=0.$$ \end{cor} \noindent {\it Proof} . This is proved by considering separately $2$-primary and $3$-primary parts of $\Delta(L)$ and $\Delta(L^2)$, using the previous proposition and Theorem \ref{isogmain}. \qed\vspace{3mm} If $g=2$, $gcd(d,3)=1$ and $n$ is odd then we also get from Proposition \ref{Ln} that $\Delta(L^n)=n\cdot\Delta(L)$. If $g=3$, $gcd(d,3)=1$, and the characteristic is zero then according to Kouvidakis one has $\Delta(L^2)=0$, hence in this case for odd $n$ we get $\Delta(L^n)=\Delta(L)$. It would be interesting to find similar dependences between $\Delta(L^n)$ in higher dimensions (see section \ref{tor} for the case of even principal polarization). Recall (see Lemma \ref{degree}) that $n\mapsto\det\pi_*(L^n)$ is a function of degree $\le g+1$. Hence, using Serre's duality, one can express all $\det\pi_*(L^n)$ as linear combinations of $\overline{\omega}$ and $\det\pi_*(L^i)$ where $0\le i\le \frac{g}{2}+1$. However, we expect much more relations between $\Delta(L^n)$. Here are some examples. \begin{enumerate} \item For $g\ge 2$ one has $\det\pi_*\O_A=0$. \item Let $p$ be a prime, $p\equiv -1\mod(4)$. Then for any $n$ such that $(n,p)=1$ one has $$\Delta(L^n)^{(p)}=\left(\frac{n}{p}\right)\cdot n^g\cdot\Delta(L)^{(p)}$$ where $\left(\frac{n}{p}\right)=\pm 1$ is the Legendre symbol. \item Assume that $g\ge 2$. Let $p$ be a prime such that $p\equiv -1\mod(4)$ and $p\ge (g+3)/2$. Then one has $\Delta(L^n)^{(p)}\in{\Bbb Z}\Delta(L)^{(p)}$ for all $n$. \item For odd $n$ one has $\Delta(L^{n+8})^{(2)}=((n+8)/n)^g\cdot\Delta(L^n)^{(2)}$. In particular, if $n$ and $d$ are odd then $\Delta(L^{n+8})^{(2)}=\Delta(L^n)^{(2)}$. \end{enumerate} (1) follows from the fact that $R^i\pi_*\O_A=\bigwedge^i R^1\pi_*\O_A$. For the proof of (2) note that for $(n,p)=1$ Theorem \ref{isogmain} implies that $\Delta(L^n)^{(p)}=n^g\cdot\Delta(L)^{(p)}$ if $n$ is a square modulo $p$, and that $\Delta(L^n)^{(p)}=(-n)^g\cdot\Delta(L^{-1})^{(p)}$ if $-n$ is a square modulo $p$. But Serre's duality implies that $\Delta(L^{-1})=-(-1)^g\cdot\Delta(L)$, hence the assertion. To prove (3) note that for $p\ge (g+3)/2$ all the elements $\Delta(L^n)^{(p)}$ are linear combinations of $\Delta(L^i)^{(p)}$ with $|i|<p$. It remains to apply (1) and (2). At last, (4) follows from Theorem \ref{isogmain} since $\frac{n+8}{n}$ is a square modulo resp. $2^k$. \begin{prop} $d\cdot (\Delta(L^3)^{(2)}+3^g\cdot\Delta(L)^{(2)})=0$. \end{prop} \noindent {\it Proof} . Let us denote $\Delta(L,n)=\det\pi_*(L^n)-n^g\cdot\det\pi_*L$. Then it is easy to see that $$2\Delta(L,n)=\Delta(L^n)-n^g\cdot\Delta(L),$$ in particular, $\Delta(L,n)$ is a torsion element in $\operatorname{Pic}(S)$. Note also that $n\mapsto\Delta(L,n)$ is a polynomial function. Let us choose a sufficiently divisible integer $N>0$ such that both functions $\Delta(L^n)$ and $\Delta(L,n)$ of $n$ are $N$-periodic and are annihilated by $N$, and $N$ is divisible by $6$. Now let $l$ be a prime, such that $N$ is not divisible by $l$ and such that $l\equiv 3\mod(2^m)$ where $m>>0$. Then there exists a solution $(a,b)$ to the congruence $a^2+lb^2\equiv -1\mod(N)$. Consider the isogeny $\a:A^2\rightarrow A^2$ given by the matrix $\left( \matrix a & -lb \\ b & a \endmatrix\right)$. Then it is easy to see that $\a^*(L\boxtimes L^l)\simeq L^k\boxtimes L^{lk}$ where $k=a^2+lb^2$. In particular, $\deg(\a)=k^{2g}$ and applying Theorem \ref{isogmain} we obtain the equality $$\det\pi_*(L^k\boxtimes L^{lk})=k^{2g}\cdot \det\pi_*(L\boxtimes L^l).$$ Now using the equalities $\Delta(L,k)=\Delta(L,-1)$, $\Delta(L,lk)=\Delta(L,-l)$, Serre's duality, the fact that $k\equiv -1\mod(N)$, and the conditions on $N$ one can easily deduce that $$d\cdot(\Delta(L^l)+l^g\cdot\Delta(L))=0.$$ It remains to take the 2-primary part of this equality and replace $l$ by $3$ in the obtained identity (this is justified by the congruence $l\equiv 3\mod(2^m)$ with $m>>0$). \qed\vspace{3mm} \begin{cor} Assume that $n$ and $d$ are odd. Then $$\Delta(L^n)^{(2)}=n^{g-1}\cdot\Delta(L)^{(2)}.$$ In particular, if $d=1$ and $g\ge 2$ then $$\Delta(L^n)=n^{g-1}\cdot\Delta(L)$$ for any odd $n$. \end{cor} \noindent {\it Proof} . Since $\Delta(L^n)^{(2)}$ for odd $n$ depends only on $n\mod(8)$ the first equality follows from the case $n=3$ considered above, Serre's duality, and the vanishing of $4\cdot\Delta(L^n)^{(2)}$ for odd $n$. Now the second statement follows from the fact that for $d=1$ and $g\ge 2$ one has $4\cdot\Delta(L^n)=0$ for any $n$. \qed\vspace{3mm} \subsection{Torsion in the Picard group of moduli}\label{tor} Let $\widetilde{\AA}_g$ be the moduli stack of the data $(A/S,\Theta)$ where $A/S$ is an abelian scheme of relative dimension $g$, $\Theta\subset A$ is an effective (relative) divisor which is symmetric and defines a principal polarization of $A$. One can normalize the line bundle $\O(\Theta)$ over the universal abelian scheme over $\widetilde{\AA}_g$ to obtain the line bundle $L$ which is trivial along the zero section. In particular, we have an element $\Delta(L)\in\operatorname{Pic}(\widetilde{\AA}_g)$. Let $\widetilde{\AA}_g^+$ be the irreducible component of $\widetilde{\AA}_g$ corresponding to even theta divisors, $\widetilde{\AA}_g^+[\frac{1}{2}]$ be the localization of this stack over $\operatorname{Spec}({\Bbb Z}[\frac{1}{2}])$. \begin{thm}\label{torsion} Assume that $g\ge 3$. Then the torsion subgroup in $\operatorname{Pic}(\widetilde{\AA}_g^+[\frac{1}{2}])$ is isomorphic to ${\Bbb Z}/4{\Bbb Z}$ and is generated by $\Delta(L)$. \end{thm} \noindent {\it Proof} . Since $\widetilde{\AA}_g^+$ has smooth geometrically irreducible fibers over $\operatorname{Spec}({\Bbb Z}[\frac{1}{2}])$ (cf. \cite{FC}, IV 7.1) it is sufficient to prove this statement in characteristic zero. Indeed, it is known that the order of $\Delta(L)$ is precisely 4 (cf. \cite{MB2}), hence it would follow that $\Delta(L)$ generates the entire torsion subgroup in the Picard group of the general fiber of $\widetilde{\AA}_g^+$. Now it remains to prove that if some line bundle over $\widetilde{\AA}_g^+$ is trivial over the general fiber then it is trivial everywhere. If $\widetilde{\AA}_g^+$ were represented by a scheme then we could apply the argument from \cite{Mumst}, p. 103, to prove this. Since it is not, we have to replace $\widetilde{\AA}_g^+$ by a $\operatorname{PGL}_N$-torsor over it which is representable (see remark 1 after Theorem \ref{main2}), apply the cited argument, and use the fact that $\operatorname{PGL}_N$ has no non-trivial characters. The corresponding analytic stack is the quotient (in the sense of stacks) of the Siegel's half-space ${\goth H}_g$ by the subgroup $\Gamma_{1,2}\subset\Sp_{2g}({\Bbb Z})$ consisting of matrices whose reduction modulo 2 preserves the standard even quadratic form $\sum_{i=1}^g x_iy_i$. (cf. \cite{MB}, VIII, 3.4). It follows that the torsion in the Picard group of this stack is an abelian group dual to $\Gamma_{1,2}/[\Gamma_{1,2},\Gamma_{1,2}]$ (cf. \cite{Mu}). It remains to prove that the latter group is isomorphic to ${\Bbb Z}/4{\Bbb Z}$. As is shown in \cite{Theta3}, Prop. 8.10, there is a normal subgroup $\Delta\subset\Gamma_{1,2}$ such that $\Gamma_{1,2}/\Delta\simeq{\Bbb Z}/4{\Bbb Z}$. Furthermore, it is shown there that $\Delta$ is generated by the matrices of the form $\left(\matrix A & 0 \\ 0 & \sideset{^t}{^{-1}}{A}\endmatrix\right)$ where $A\in\operatorname{SL}_g({\Bbb Z})$, $\left(\matrix 1 & B \\ 0 & 1\endmatrix\right)$ and $\left(\matrix 1 & 0 \\ B & 1\endmatrix\right)$ where $B$ is integral symmetric $g\times g$ matrix with even diagonal (here we use the standard symplectic basis $e_1,\ldots,e_g,f_1,\ldots f_g$ such that $(e_i,f_j)=\delta_{i,j}$). We claim that $\Delta\subset [\Gamma_{1,2},\Gamma_{1,2}]$. For the proof let us introduce the relevant elementary matrices following the notation of \cite{Cl} 5.3.1. Let $\S_{2g}$ be the set of pairs $(i,j)$ where $1\le i,j\le 2g$ which are not of the form $(2k-1,2k)$ or $(2k,2k-1)$. Then for for every $(i,j)\in \S_{2g}$ we define an elementary matrix $E_{ij}$ as follows: $$E_{2k,2l}= \left(\matrix 1 & 0 \\ \gamma_{k,l} & 1\endmatrix\right),$$ $$E_{2k-1,2l-1}= \left(\matrix 1 & -\gamma_{k,l} \\ 0 & 1\endmatrix\right),$$ $$E_{2k-1,2l}= \left(\matrix e_{kl} & 0 \\ 0 & e_{lk}^{-1}\endmatrix\right),$$ $$E_{2l,2k-1}=E_{2k-1,2l}$$ where $\gamma_{kl}$ has zero $(\a,\b)$-entry unless $(\a,\b)=(k,l)$ or $(\a,\b)=(l,k)$, in the latter case $(\a,\b)$-entry is 1; $e_{kl}$ for $k\neq l$ is the usual elementary matrix with units on the diagonal and at $(k,l)$-entry and zeros elsewhere. Using these matrices one can say that $\Delta$ is generated by $E_{2k-1,l}$ with $k\neq l$, $E_{2k,2l}$ and $E_{2k-1,2l-1}$ with $k\neq l$, and $E_{i,i}^2$ for all $1\le i\le 2g$. It remains to notice that all the matrices $E_{ij}$ with $i\neq j$ belong to $\Gamma_{1,2}$ and use the following relations (cf. \cite{Cl} 9.2.13): \begin{enumerate} \item $[E_{ij},E_{kl}]=E_{il}$, if $(j,k)\not\in\S_{2g}$, $j$ is even, and $i$, $j$, $k$, and $l$ are distinct, \item $[E_{ij},E_{ki}]=E_{ii}^2$, if $(j,k)\not\in\S_{2g}$, $j$ is even, and $i$, $j$, and $k$ are distinct. \end{enumerate} \qed\vspace{3mm} \subsection{Case of principally polarized abelian surfaces} Let $A/S$ be a relative abelian surface, $L$ be a symmetric line bundle trivialized along the zero section. Assume also that $d=1$ that is $L$ gives a principal polarization. Then $L\simeq \O(\Theta)\otimes\pi^*(\pi_*L)$ where $\Theta\subset A$ is theta-divisor. \begin{prop} Assume that $S$ is smooth. Then one has the following equalities in $\operatorname{Pic}(S)$: $$\det\pi_*\O_{\Theta}=\overline{\omega}_A,$$ $$5\cdot\overline{\omega}_A=\delta+\Delta'(L^2),$$ where $\Delta'(L^2)=\det\pi_*(L^2)+2\cdot\overline{\omega}_A$, $\delta$ is the class of the divisor consisting of points $s\in S$ such that $\Theta_s$ is singular, \end{prop} \noindent {\it Proof} . First of all, we note that $\det\pi_*\O_{\Theta}=\det\pi_*\omega_{\Theta}$ by Serre's duality. Now by adjunction we have $$\omega_{\Theta}=\O_{\Theta}(\Theta)\otimes\pi^*\overline{\omega}_A,$$ which implies the first equality due to triviality of $\det\pi_*(\O_A(\Theta))$ and $\det\pi_*(\O_A)$. We also deduce that $$\det\pi_*(\omega_{\Theta}^2)=\det\pi_*\O_{\Theta}(2\Theta)+6\cdot\overline{\omega}_A.$$ Next, since $L^2\simeq\O(2\Theta)\otimes\pi^*(\pi_*L)^2$ we obtain that $$\det\pi_*\O_A(2\Theta)=\det\pi_*L^2+4\cdot\overline{\omega}_A.$$ The exact sequence $$0\rightarrow\O_A(\Theta)\rightarrow\O_A(2\Theta)\rightarrow\O_{\Theta}(2\Theta)\rightarrow 0$$ shows that $\det\pi_*\O_{\Theta}(2\Theta)=\det\pi_*\O_A(2\Theta)$. Combining it with the above equalities we get $$\det\pi_*(\omega_{\Theta}^2)=\det\pi_*L^2+10\cdot\overline{\omega}_A=\Delta'(L^2)+ 8\cdot\overline{\omega}_A.$$ On the other hand, since $\Theta$ is a stable curve over $S$ we have according to Mumford's Theorem 5.10 in \cite{Mumst} $$\det\pi_*(\omega_{\Theta}^2)=13\cdot\det\pi_*\omega_{\Theta}-\delta= 13\cdot\overline{\omega}_A-\delta.$$ Comparing this with the previous expression for $\det\pi_*(\omega_{\Theta}^2)$ we obtain the result. \qed\vspace{3mm} Let $\overline{{\cal M}}_2$ be the moduli stack of stable curves of genus 2, ${\cal M}_2$ be the open substack corresponding to smooth curves, ${\cal M}'_2$ be the substack of $\overline{{\cal M}}_2$ corresponding to curves which are either smooth or reducible. The Picard groups of these stacks can be described as follows (see \cite{Mute}, \cite{Mumst}, \cite{Muen}). $\operatorname{Pic}(\overline{{\cal M}}_2)$ is isomorphic to ${\Bbb Z}^2$ and is generated by the classes $\delta_0$, $\delta_1$ and $\lambda$, where $\delta_0$ (resp. $\delta_1$) is the class of the divisor of singular irreducible curves (resp. reducible curves), $\lambda=\det\pi_*\omega_{{\cal C}}$ where $\pi:{\cal C}\rightarrow\overline{{\cal M}}_2$ is the universal curve, with the only relation \begin{equation}\label{Mure} 10\cdot\lambda=\delta_0+2\cdot\delta_1. \end{equation} It follows that $\operatorname{Pic}({\cal M}'_2)$ is generated by $\lambda$ and $\delta_1$ with the relation $10\cdot\lambda=2\cdot\delta_1$, and $\operatorname{Pic}({\cal M}_2)$ is generated by $\lambda$ with the relation $10\cdot\lambda=0$. Note that the theta divisors $\Theta_s$ are either smooth or reducible, so in the above situation we get a morphism $f:S\rightarrow{\cal M}'_2$ and our computation shows that $\Delta'(L)=f^*(5\cdot\lambda-\delta_1)$. \begin{cor} In the above situation $\Delta(L^{2n})=0$ for any $n$. \end{cor} \noindent {\it Proof} . For $n=1$ this follows from the triviality of $10\cdot\lambda-2\cdot\delta_1$ in $\operatorname{Pic}({\cal M}'_2)$. Now the triviality of $\Delta(L^{2n})$ in general follows from Proposition \ref{Ln}. \qed\vspace{3mm} \begin{rem} Note that $L^2\simeq L_{\phi}:=(\operatorname{id},\phi)^*\cal P$, where $\phi:A\rightarrow\hat{A}$ is the polarization corresponding to $L$. Hence, $\Delta'(L^2)$ is the pull-back of the line bundle $\Delta'(L_{\phi})$ over the moduli stack $\AA_2$. The explicit trivialization of $2\cdot\Delta'(L_{\phi})=10\cdot\overline{\omega}-2\cdot\delta$ in the analytic situation can be found by considering the following modular form of weight 5 on Siegel half-space ${\goth H}_2$ (cf. \cite{Muen}) $$f(Z)=\prod_{a,b\ even}\theta \left[\matrix a\\ b\endmatrix\right](0,Z)$$ where the product is taken over all 10 even theta-characteristics. Then $f$ defines a section of $\overline{\omega}^5$ vanishing precisely on the locus $\Delta\subset{\goth H}_2$ corresponding to products of elliptic curves. It is known that $f$ is a modular form for the group $\Sp_{4}({\Bbb Z})$ with a non-trivial character $\chi_0:\Sp_4({\Bbb Z})\rightarrow\{\pm 1\}$ (such a character is unique, and is obtained from the sign character on $\Sp_4({\Bbb Z}/2{\Bbb Z})\simeq S_6$). Thus, $f^2$ gives the $\Sp_4({\Bbb Z})$-equivariant trivialization of $\overline{\omega}^{10}(-2\Delta)$ which descends to a trivialization over $\AA_2$. This argument also shows that the element $\Delta'(L_{\phi})=5\cdot\overline{\omega}-\delta\in\operatorname{Pic}(\AA_2)$ is non-trivial. In fact, it generates the torsion subgroup of $\operatorname{Pic}(\AA_2)$ (cf. \cite{GH}). Furthermore, one can show that pull-backs of $\Delta'(L_{\phi})$ to either of two irreducible components of $\widetilde{\AA}_2$ are non-trivial. Indeed, it is sufficient to check that the subgroup in $\Sp_4({\Bbb Z}/2{\Bbb Z})\simeq S_6$ preserving a quadratic form $q$ on $({\Bbb Z}/2{\Bbb Z})^4$, such that $q(x+y)+q(x)+q(y)$ is the symplectic form, contains an odd permutation. Recall that the identification of $\Sp_4({\Bbb Z}/2{\Bbb Z})$ with $S_6$ is obtained by considering the action on the set of 6 odd quadratic forms $q$ as above. Using this it is easy to compute that the matrix $E_{14}$ (see the proof of Theorem \ref{torsion}), preserving the standard even form $q_0=x_1y_1+x_2y_2$, corresponds to the product of three transpositions. Similarly, the matrix $E_{11}$ preserves the odd form $x_1y_1+x_2y_2+x_2^2+y_2^2$ and corresponds to a transposition. \end{rem}

9703

alg-geom/9703033

en

https://arxiv.org/abs/alg-geom/9703033

alg-geom/9703033

Wolfgang Eholzer

W. Eholzer, T. Ibukiyama

Rankin-Cohen Type Differential Operators for Siegel Modular Forms

19 pages LaTeX2e using amssym.def

preprint MPI, DAMTP-97-26

Let H_n be the Siegel upper half space and let F and G be automorphic forms on H_n of weights k and l, respectively. We give explicit examples of differential operators D acting on functions on H_n x H_n such that the restriction of D(F(Z_1) G(Z_2)) to Z = Z_1 = Z_2 is again an automorphic form of weight k+l+v on H_n. Since the elliptic case, i.e. n=1, has already been studied some time ago by R. Rankin and H. Cohen we call such differential operators Rankin-Cohen type operators. We also discuss a generalisation of Rankin-Cohen type operators to vector valued differential operators.

alg-geom

\section{Introduction} In this paper we are concerned with the explicit construction of bilinear differential operators for Siegel modular forms mapping $M(\Gamma)_k\times M(\Gamma)_l$ to $M(\Gamma)_{k+l+v}$ for all even non-negative integers $v$ and $\Gamma$ some discrete subgroup of $\mbox{{\rm{Sp}}}(2n,\R)$ with finite co-volume. It has been shown in ref.\ \cite{Ibukiyama} that if the weights $k$ and $l$ are sufficiently large then there is a one-to-one correspondence between such bilinear differential operators and certain invariant pluri-harmonic polynomials (for more details see \S\ref{basics}). Therefore, one possibility for constructing such differential operators is to construct the corresponding invariant pluri-harmonic polynomials. As we will choose exactly this way for our construction one may view our paper also as an attempt to describe certain spaces of invariant pluri-harmonic polynomials explicitly. Some results concerning the explicit construction of bilinear differential operators for Siegel modular forms --which we will call Rankin-Cohen type operators following ref.\ \cite{Zagier}-- are already known in the literature: the genus one case has been considered R.\ Rankin \cite{Rankin} and H.\ Cohen \cite{Cohen} and, more recently, the genus two case by Y.\ Choie and the first author~\cite{ChoieEholzer}. Both of these results are special cases of the construction presented in this paper. We are, however, not able to give closed explicit formulas for all Rankin-Cohen type operators, {\it i.e.} for general values of $n$ and $v$. Instead we derive a system of recursion equations which is indeed very simple to solve for any {\it numerical} values of $n$ and $v$. In several special cases, however, we obtain explicit closed formulas. In addition we show that the image of the Rankin-Cohen type operators is, for $v>0$, contained in the space of cusp forms. Finally, we discuss vector valued generalisations of the Rankin-Cohen type operators, {\it i.e.} bilinear differential operators which map two Siegel modular forms to a vector valued modular Siegel modular form that transforms under a certain representation $\rho$ of $\GL{n}$. This paper is organised as follows. In section \ref{basics}, after reviewing some basic definitions and standard notations, we recall the one-to-one correspondence between invariant pluri-harmonic polynomials and covariant differential operators. In \S\ref{main} we derive our main result in form of a set of recursion relations which allow to determine certain invariant pluri-harmonic polynomials $Q_{n,v}$ (Theorem \ref{vgeneral}). We also show that, for $v>0$, the bilinear differential operators associated to these polynomials map two Siegel modular forms to a Siegel cusp form. Section \ref{proofs} contains the proof of a uniqueness result in \S\ref{basics} as well as the proofs of three lemmas and our main theorem in \S\ref{main} and the result about cusp forms. We then discuss in \S\ref{examples} several special cases of our main result explicitly: the cases $v=2,4$ for general $n$ and the cases $n=1,2$ for general $v$. Finally, we generalise Rankin-Cohen type differential operators to vector valued differential operators in section \ref{vector}. Here the case $n=2$ is treated explicitly and in some detail. We conclude in section \ref{conclusion} with some remarks and open questions. \section{Siegel modular forms, pluri-harmonic polynomials and differential operators} \label{basics} In this section we recall some standard notations and review a result of the second author on the relation between invariant pluri-harmonic polynomials and differential operators (Theorem \ref{Ibuthm}). Let $\H_n$ be the space of complex symmetric $n\times n$ matrices with positive definite imaginary part and define an action of $\mbox{{\rm{Sp}}}(2n,\R)$ on functions $f:\H_n\to\C$ by $$ (f|_M^k)(Z) = f(MZ) \det(CZ+D)^{-k} \qquad (M\in\mbox{{\rm{Sp}}}(2n,\R))$$ where $Z\in\H_n$, $M=\left( \begin{array}{cc} A & B \\ C & D \end{array}\right)$ with $n\times n$ matrices $A,B,C,D$, and, where $MZ = (AZ+B)(CZ+D)^{-1}$. We define Siegel modular forms on some discrete subgroup $\Gamma\subset \mbox{{\rm{Sp}}}(2n,\R)$ of finite co-volume (finite co-volume means that the volume of $\mbox{{\rm{Sp}}}(2n,\R)/\Gamma$ is finite). \begin{df} A holomorphic function $f:\H_n\to\C$ is called a Siegel modular form of non-negative weight $k$ on $\Gamma$ if $$ (f|_M^k)(Z) = f(Z)\qquad {\rm{for}}\ {\rm{all}}\ M\in\Gamma $$ (and $f(Z)$ is bounded at the cusps for $n=1$). \end{df} We denote the space of all Siegel modular forms of weight $k$ on $\Gamma$ by $M(\Gamma)_k$. We review some notations concerning pluri-harmonic polynomials. \begin{df} Let $P$ be a polynomial in the matrix variable $X = (x_{r,s})\in M_{n,d}$ and define $$ \Delta_{i,j}(X) = \sum_{\nu=1}^{d} \frac{\partial^2} {\partial x_{i,\nu} \partial x_{j,\nu}} \qquad (1\le i,j\le n). $$ Then the polynomial $P$ is called harmonic if $\sum_{i=1}^n \Delta_{i,i}(X) P = 0$ and $P$ is called pluri-harmonic if $\Delta_{i,j}(X) P = 0$ for all $1\le i,j\le n$. \end{df} The group $\GL{n}\times O(d)$ acts on such polynomials by $P(X) \to P(A^t X B)$ ($A\in\GL{n}$, $B\in O(d)$). We will be interested in pluri-harmonic polynomials which are invariant under a subgroup of $O(d)$ and transform under a certain representation of $\GL{n}$. A polynomial $P$ of a matrix variable $X \in M_{n,d}$ is called homogeneous of weight $v$ if $P(A^t X) = \det(A)^v\, P(X)$ for all $A\in\GL{n}$. It was already pointed out in ref.\ \cite{KashiwaraVergne} that a polynomial $P(X)$ is pluri-harmonic if and only if $P(A X)$ is harmonic for all $A\in\GL{n}$. Therefore, a homogeneous polynomial of some weight $v$ is pluri-harmonic if and only if it is harmonic. Let $d_i$ ($1\le i\le r$) be natural numbers such that $d_i\ge n$ and $d_1+\dots+d_r = d$. Define an embedding of $K=O(d_1)\times\dots\times O(d_r)$ into $O(d)$ by $$ (B_1,\dots, B_r) \to \left(\begin{array}{ccccc} B_1 & 0 & \dots & \dots & 0 \cr 0 & B_2 & 0 & \dots & 0 \cr \vdots & 0 & \ddots& & \vdots\cr \vdots & \vdots & & \ddots& 0\cr 0 & 0 & & 0 & B_r \end{array} \right). $$ A polynomial $P$ of a matrix variable $X \in M_{n,d}$ which is invariant under the action of $K$ is called $K$-invariant. If we write $X = (X_1,\dots,X_r)$ with $X_i\in M_{n,d_i}$ ($1\le i\le r$) then, by virtue of H. Weyl, for each $K$-invariant polynomial $P$ there exists a polynomial $Q$ such that $P(X) = Q(X_1 X_1^t,\dots,X_r X_r^t)$. (Here we have used the assumption $d_i\ge n$ ($1\le i \le r$).) Following ref.\ \cite{Ibukiyama} we call $Q$ the associated polynomial (map) of $P$. Note that $\GL{n}$ acts on associated polynomials $Q$ by mapping $Q(R_1,\dots,R_r)$ to $Q(A^t R_1 A,\dots,A^t R_r A)$ ($A\in \GL{n}$). Let us introduce some notations for the certain spaces of polynomials. Denote the space of all homogeneous polynomials $P$ of weight $v$ which are $K$-invariant by ${\cal P}_{n,v}(d_1,\dots,d_r)$. Furthermore, let ${\cal Q}_{n,v}(r)$ be the space of all polynomials $Q$ in (the coefficients of) symmetric matrices $R_1,\dots,R_r\in M_{n,n}$ such that $Q(A^t R_1 A, \dots, A^t R_r A) = \det(A)^v Q(R_1,\dots,R_r)$ for all $A\in \GL{n}$. Finally, let ${\cal H}_{n,v}(d_1,\dots,d_r)$ be the subspace of ${\cal Q}_{n,v}(r)$ consisting of elements $Q$ such that $Q(X_1 X_1^t,\dots, X_r X_r^t)$ is pluri-harmonic for $X= (X_1,\dots, X_r)$ where $X_i\in M_{n,d_i}$ ($1\le i\le r$). When $d_i\ge n$ then ${\cal H}_{n,v}(d_1,\dots,d_r)$ is the space of polynomials associated to the invariant pluri-harmonic polynomials in ${\cal P}_{n,v}(d_1,\dots,d_r)$. Before we specialise to the case $r=2$ on which we will concentrate in the following sections let us recall the connection between pluri-harmonic polynomials and differential operators. For $Z=(z_{r,s})\in\H_n$ we define $\partial_{Z}$ to be the $n\times n$ matrix with components $$ \frac{1}{2}(1+\delta_{r,s})\frac{\partial}{\partial z_{r,s}} $$ Let $Q$ be a polynomial in ${\cal Q}_{n}(r)$ and set $D = Q(\partial_{Z_1},\dots,\partial_{Z_r})$. For fixed, even $d_{i} = 2k_{i}$ ($1 \leq i \leq r$) and fixed non-negative $v$, {\it i.e.} $v \in\N_0$, we say that $D$ satisfies the `commutation relation' if the following condition is satisfied. \noindent {\it For all holomorphic functions $F(Z_{1},\ldots,Z_{r}):\H_{n}^{r}\to\C$ and all $\gamma \in \mbox{{\rm{Sp}}}(2n,\R)$ one has} \[ D\left( F(Z_{1},\ldots,Z_{r})\prod_{i=1}^{r}\det(CZ_{i}+D)^{-k_{i}} \right) |_{Z=Z_{1}=\cdots=Z_{r}} = DF(\gamma Z,\ldots,\gamma Z) \det(CZ+D)^{-k-v} \] {\it where $k= k_1+\dots+k_r$ and $\gamma =\left( \begin{array}{cc} A & B \\ C & D \end{array}\right)$ with $n\times n$ matrices $A,B,C$ and $D$.} Using this condition we can now state (a special case of) Theorem 2 of ref. \cite{Ibukiyama}. \begin{thm}\cite{Ibukiyama} \label{Ibuthm} For fixed integers $k_{1},\ldots,k_{r}$ with $2k_i\ge n$ and a non-negative integer $v$ the above commutation relation is satisfied if and only if $Q \in {\cal H}_{n,v}(2k_{1},\ldots,2k_{r})$. In particular, let $F_i:\H_n\to\C$ ($1\le i\le r$) be Siegel modular forms on some discrete subgroup $\Gamma\subset\mbox{{\rm{Sp}}}(2n,\R)$ of finite co-volume with weights $2 k_i \ge n$, respectively. Let $Q$ be in ${\cal H}_{n,v}(2k_1,\dots,2k_r)$ and set $D = Q(\partial_{Z_1},\dots,\partial_{Z_r})$. Then the restriction of $$ D\left( F_1(Z_1)\cdots F_n(Z_r) \right) $$ to $Z = Z_1=\dots=Z_r$ is a Siegel modular form of weight $k_1+\dots+k_r+v$ on $\Gamma$. \end{thm} {}From this theorem it is obvious that every element of ${\cal H}_{n,v}(d_1,\dots,d_r)$ defines a $r$-linear covariant differential operator and vice versa. Here a differential operator $D$ is called covariant if $D$ satisfies the commutation relation above. In the rest of this paper we will be interested in the case of bilinear differential operators, {\it i.e.} the case $r=2$. Finally, let us recall a result on the dimension of ${\cal H}_{n,v}(d_1,d_2)$ which was already stated in ref.\ \cite{Ibukiyama}. \begin{prop}\cite{Ibukiyama} \label{Hdim} For $d_1\ge n$ and $d_2\geq n$ one has $\dim({\cal H}_{n,v}(d_1,d_2)) = 1$ for all non-negative, even $v$. \end{prop} {\it Proof.} The proof for the case $d_1,d_2\ge 2n$ is already contained in ref.\ \cite{Ibukiyama}; we give the complete proof in section \ref{proofs}. \bigskip\noindent In the next section we will, for $d_1, d_2 \geq n$, give a (quite explicit) description of ${\cal H}_{n,v}(d_1,d_2)$. \section{Explicit description of ${\cal H}_{n,v}(d_1,d_2)$} \label{main} In this section we will obtain a description of ${\cal H}_{n,v}(d_1,d_2)$ for $d_1\ge n$ and $d_2 \geq n$. As ${\cal H}_{n,v}(d_1,d_2)$ is one dimensional in this case we only have to find a non-zero element in this space. To find such an element we study the structure of ${\cal Q}_{n}(2)$ first. More precisely, we show that ${\cal Q}_{n}(2)$ is freely generated by certain homogeneous polynomials $P_\alpha$ ($0\le\alpha\le n$) of weight 2 (Lemma \ref{hompoly}). Using this explicit description of ${\cal Q}_{n}(2)$ we can write an element $Q_{n,v}\in {\cal H}_{n,v}(d_1,d_2)$ as a linear combination of monomials in the $P_\alpha$'s. We then calculate the Laplacian of $Q_{n,v}$ and find a set of recursion relations for the coefficients in the linear combination expressing $Q_{n,v}$ in terms of the $P_\alpha$'s (Theorem \ref{vgeneral}). These recursion relations allow to solve uniquely (up to multiplication by a scalar) for the coefficients. In the case of general $n$ and $v$, however, we do not obtain explicit closed formulas for the coefficients; several special cases where we find such formulas are discussed in \S\ref{examples}. In order to present our line of arguments in a transparent way we will postpone the proofs of various lemmas and our main theorem to the next section. \bigskip\noindent We can give ${\cal Q}_{n}(2)$ (and therefore also each ${\cal P}_{n,v}(d_1,d_2)$) explicitly as follows. Firstly, it is clear that ${\cal Q}_{n,v}(2) = 0$ if $v$ is odd. Indeed, any $Q(R,R') \in {\cal Q}_{n,v}(2)$ is determined by its values for $R = {\mathchoice {\rm 1\mskip-4mu l} {\rm 1\mskip-4mu l_{n}$ and diagonal matrices $R'$. Taking $A= (a_{i,j})$ as $a_{1,1}=-1$, $a_{i,i} = 1$ for $i \neq 1$ and $a_{i,j}=0$ for all $i \neq j$, we get $\det(A) = -1$ and $A^t R A = R$, $A^t R' A= R'$ for the above $R$ and $R'$. Hence, if $v$ is odd, then $Q(R,R') = (-1)^v Q(R,R')$ and $Q = 0$. Secondly, we give some typical elements of ${\cal Q}_{n,2}(2)$. For $0\le \alpha\le n$ denote by $P_\alpha$ the polynomial in the matrix variables $R\in M_{n,n}$ and $R'\in M_{n,n}$ defined by $$ \det(R + \lambda R' ) = \sum_{\alpha=0}^{n} P_{\alpha}(R,R')\ \lambda^{\alpha}. $$ Then, each $P_\alpha$ obviously belongs to ${\cal Q}_{n,2}(2)$ and, therefore, all polynomials in the $P_\alpha$ belong to ${\cal Q}_n(2)$. Thirdly, we show that the converse also holds true so that we obtain an explicit description of ${\cal Q}_n(2)$. \begin{lem} \label{hompoly} The ring ${\cal Q}_n(2)$ is generated by the algebraically independent polynomials $P_0,\dots,P_n$. \end{lem} Now let $Q_{n,2v}$ be an element of ${\cal H}_{n,2v}(d_1,d_2)\subset{\cal Q}_{n,2v}(2)$. Then, by the last lemma, $Q_{n,2v}$ can be written in the form $$ Q_{n,2v} = \sum_{a\in I_{n,2v}} C(a)\ \prod_{\alpha=0}^n P_{\alpha}^{a_\alpha}$$ where $I_{n,2v} = \{ a=(a_0,\dots,a_n)\in \N_0^{n+1}\ |\ \sum_{\alpha=0}^n a_\alpha = v \}$. The definition of ${\cal H}_{n,2v}(d_1,d_2)$ in the last section was rather indirect as ${\cal H}_{n,2v}(d_1,d_2)$ was defined of the space of polynomials $Q\in {\cal Q}_{n,2v}(2)$ such that $Q(X_1 X_1^t,X_2 X_2^t)$ is pluri-harmonic for $X= (X_1,X_2)$ with $X_i\in M_{n,d_i}$ ($1\le i\le 2$). One can, however, also give a direct definition as follows. Define two differential operators $L^{(d_1)}_{i,j}$ and ${L'}^{(d_2)}_{i,j}$ acting on elements of ${\cal Q}_{n,v}(2)$ by \begin{eqnarray*} L^{(d_1)}_{i,j} &=& d_{1}(1+\delta_{ij})D_{ij} + 4 R_{i,j} D_{i,i} D_{j,j} +\\ &&\qquad \sum_{m' \neq j, m \neq i} R_{m',m} D_{m',j} D_{m,i} + 2\sum_{m' \neq j} R_{m',i} D_{m',j} D_{i,i} + 2\sum_{m \neq i} R_{m ,j} D_{j ,j} D_{m,i},\\ {L'}^{(d_2)}_{i,j} & = & d_{2}(1+\delta_{ij})D'_{ij} + 4 R'_{ij} D'_{ii} D'_{jj} + \\ &&\qquad \sum_{m' \neq j, m \neq i} R'_{m',m} D'_{m',j} D'_{m,i} + 2\sum_{m'\neq j} R'_{m',i} D'_{m',j} D'_{i,i} + 2\sum_{m \neq i} R'_{m ,j} D'_{ j,j} D'_{m,i} \end{eqnarray*} where we have used $R = (R_{i,j}),R'=(R'_{i,j})\in M_{n,n}$ for the two matrix variables and $D_{i,j}$ and $D'_{i,j}$ for the differential operators $$ D_{i,j} = \frac{\partial}{\partial R_{ij}}, \qquad D'_{i,j} = \frac{\partial}{\partial R'_{ij}} \qquad (1\le i,j\le n). $$ It is now easy to see that the operators $L^{(d_1)}_{i,j}$ and ${L'}^{(d_2)}_{i,j}$ describe the action of $\Delta_{i,j}(X)$ and $\Delta_{i,j}(X')$, respectively on associated polynomials. More precisely, let $Q\in{\cal Q}_{n,v}(2)$ be the associated polynomial of a polynomial $P\in{\cal P} _{n,v}(d_1,d_2)$, {\it i.e.} $Q(X X^t, X' {X'}^t) = P(X,X')$ with $X\in M_{n,d_1}$ and $X'\in M_{n,d_2}$, then one has \begin{eqnarray*} \Delta_{i,j}(X) P(X,X') &=& (L^{(d_1)}_{i,j} Q)(X X^t, X' {X'}^t)\\ \Delta_{i,j}(X') P(X,X') &=& ({L'}^{(d_2)}_{i,j} Q)(X X^t, X' {X'}^t). \end{eqnarray*} By the remarks in the last section a ``homogeneous'' polynomial is pluri-harmonic if and only if it is harmonic. Hence a necessary and sufficient condition for a polynomial $Q_{n,2v}\in {\cal Q}_{n,2v}(2)$ being in ${\cal H}_{n,2v}(d_1,d_2)$ is given by $$ \sum_{i=1}^n (L^{(d_1)}_{i,i} + {L'}^{(d_2)}_{i,i}) Q_{n,2v} = 0.$$ In order to evaluate this equation we have to calculate the action of the Laplacians $L^{(d_1)}_{i,i}$ and ${L'}^{(d_2)}_{i,i}$ on products of $P_{\alpha}$'s. Let us first describe the action of these operators on a product of two polynomials $Q$ and $Q'$ \begin{eqnarray*} L^{(d_1)}_{i,i}(Q Q') & = & (L^{(d_1)}_{i,i}Q) Q' + Q (L^{(d_1)}_{i,i}Q') + 8(Q,Q')_{i,R}, \\ {L'}^{(d_2)}_{i,i}(Q Q') & = & ({L'}^{(d_2)}_{i,i}Q) Q' + Q({L'}^{(d_2)}_{i,i}Q') + 8(Q,Q')_{i,R'} \end{eqnarray*} where \begin{eqnarray*} 4 (Q,Q')_{i,R} &=& R_{i,i} (D_{i,i}Q) (D_{i,i} Q')+ \sum_{1\le l\le n} R_{l,i} (D_{l,i}Q) (D_{i,i} Q') +\\ &&\qquad +\sum_{1\le m\le n} R_{m,i} (D_{i,i}Q) (D_{m,i} Q') +\sum_{1\le l,m\le n} R_{l,m} (D_{l,i}Q) (D_{m,i} Q')\, ,\\ 4 (Q,Q')_{i,R'} &=& R'_{i,i} (D'_{i,i}Q) (D'_{i,i} Q')+ \sum_{1\le l\le n} R'_{l,i} (D'_{l,i}Q) (D'_{i,i} Q') +\\ &&\qquad +\sum_{1\le m\le n} R'_{m,i} (D'_{i,i}Q) (D'_{m,i} Q') +\sum_{1\le l,m\le n} R'_{l,m} (D'_{l,i}Q) (D'_{m,i} Q'). \end{eqnarray*} To state the corresponding formulas for $Q=P_\alpha$ and $Q' = P_\beta$ let us define polynomials $(P_{i_1,\dots,i_g;j_1,\dots,j_g})$ depending on two matrix variables $R,R'\in M_{n,n}$ and the variable $\lambda$ as the determinant of the matrix $R+\lambda R'$ with the rows $i_1,\dots,i_g$ and columns $j_1,\dots,j_g$ removed. Furthermore, we denote by $(P_{i_1,\dots,i_g;j_1,\dots,j_g})_\alpha$ the coefficient of $\lambda^\alpha$ in $(P_{i_1,\dots,i_g;j_1,\dots,j_g})$. Then one has the following result. \begin{lem} \label{deltagrad} With the notations above one has \begin{eqnarray*} L^{(d_1)}_{i,i} P_\alpha &=& 2(d_1+1-n+\alpha)\ (P_{i;i})_\alpha \qquad\ \ (0\le \alpha\le n-1)\, ,\\ {L'}^{(d_2)}_{i,i} P_\alpha &=& 2(d_2+1-\alpha)\qquad\, (P_{i;i})_{\alpha-1} \qquad (1\le \alpha\le n)\, ,\\ L^{(d_1)}_{i,i} P_n &=& {L'}^{(d_2)}_{i,i} P_0 = 0\, ,\\ (P_\alpha, P_\beta)_{i,R} &=& P_{\alpha} (P_{i;i})_{\beta} - P_{\beta+1}(P_{i;i})_{\alpha-1}+ (P_{\alpha-1},P_{\beta+1})_{i,R}\, ,\\ (P_\alpha, P_\beta)_{i,R'} &=& P_{\beta} (P_{i;i})_{\alpha-1} - P_{\alpha-1}(P_{i;i})_{\beta}+ (P_{\alpha-1},P_{\beta+1})_{i,R'} \end{eqnarray*} where in the two last equations $\alpha\le \beta$ and we have set $P_\gamma = (P_{i;i})_\gamma =0$ for $\gamma<0$ or $\gamma>n$. \end{lem} Using these formulas one can now calculate $\sum_{i=1}^n (L^{(d_1)}_{i,i}+{L'}^{(d_2)}_{i,i})$ of any monomial in the $P_\alpha$'s. To be able to extract the equations which the coefficients $C(a)$ have to satisfy if $Q_{n,v}$ is pluri-harmonic we need to know that the polynomials $\sum_{i=1}^n (P_{i;i})_\beta$ are linearly independent over ${\cal Q}_{n}(2)$. \begin{lem} \label{lindep} For any $n \geq 1$, the polynomials $\sum_{i=1}^n (P_{i;i})_\alpha$ ($0\le \alpha \le n-1$) are linearly independent over ${\cal Q}_{n}(2)$. \end{lem} We can now use the last three lemmas to calculate the equations satisfied by the coefficients $C(a)$. \begin{thm} \label{vgeneral} Let $d_1\ge n$ and $d_2\ge n$. With the notations as above the polynomial $$ Q_{n,2v} = \sum_{a\in I_{n,2v}} C(a)\ \prod_{\alpha=0}^n P_{\alpha}^{a_\alpha} $$ is a pluri-harmonic of weight $2v$, {\it i.e.} $Q_{n,2v}$ is in ${\cal H}_{n,2v}(d_1,d_2)$, if the coefficients $C(a)$ satisfy \begin{eqnarray*} (d_1+1-n+i+2(a_i-1)) a_i\ C(a) &&= -(d_2-i) (a_{i+1}+1)\ C(a-e_i+e_{i+1}) \\ && -2\sum_{{i<l\le l'}\atop{l'+l-i\le g}} \tilde a(i,l,l')_l\ ( \tilde a(i,l,l')_{l'}-\delta_{l,l'})\ C( \tilde a(i,l,l') )\\ && +2\sum_{{i<l\le l'}\atop{l'+l-i-1\le g}} \hat a(i,l,l')_l\ (\hat a(i,l,l')_{l'}- \delta_{l,l'})\ C( \hat a(i,l,l') ) \end{eqnarray*} for all $a\in I_{n,2v}$ such that $i:=\mbox{\rm min}\{j | a_j \not=0\} < n$. Here we have used $\tilde a(i,l,l') = a-e_i+e_l+e_{l'}-e_{l+l'-i}$, $\hat a(i,l,l') = a-e_i+e_l+e_{l'}-e_{l+l'-i-1}$ and $e_j\in \{0,1\}^{n+1}$ for the vector with components $(e_j)_l = \delta_{j,l}$. Note that in the above formula we have set $C(a)=0$ if $a_j<0$ for some $0\le j\le n$. \end{thm} Firstly, note that the equations determine the coefficients $C(a)$ uniquely as multiples of $C((0,\dots,0,v))$. To see this define an order for the elements of $I_{n,v}$ by saying that $a < b$ for some $a,b\in I_{n,v}$ if there exists some $j$ with $0\le j\le n$ so that $a_j<b_j$ and $a_i=b_i$ for all $0\le i<j$. Looking at the above recursion equations it is obvious that, for $d_1\ge n$, they can be used to express $C(a)$ in terms of coefficients $C(b)$ with $b<a$. Hence we obtain inductively that once given $C((0,\dots,0,v))$ all other coefficients are uniquely determined (assuming still $d_1\ge n$). This implies that there is exactly one solution to the equations which is indeed easy to calculate for any given {\it numerical} values of $n$ and $v$. It would of course be desirable to give closed explicit formulas for the coefficients $C(a)$ in general but we have not succeeded in doing so; several special cases where we obtain such formulas are discussed in \S\ref{examples}. As, by Proposition \ref{Hdim}, the dimension of ${\cal H}_{n,2v}(d_1,d_2)$ is equal to one for $d_1,d_2\ge n$ Theorem \ref{vgeneral} gives a (more or less explicit) description of ${\cal H}_{n,2v}(d_1,d_2)$. \medskip Finally, the differential operators obtained from non-constant invariant pluri-harmonic polynomials in ${\cal H}_{n,v}(d_1,d_2)$ map two modular forms to a cusp form. \begin{prop} \label{cusp} Let $\Gamma$ be a subgroup of $\mbox{{\rm{Sp}}}(2n,\Qq)$ which is commensurable with $\mbox{{\rm{Sp}}}(2n,\Z)$ and let $F$ and $G$ be Siegel modular forms on $\Gamma$ of weight $k$ and $l$, respectively. Let $D$ be a covariant differential operator obtained from a non-constant invariant pluri-harmonic polynomial in ${\cal H}_{n,v}(2k,2l)$. Then $D(F(Z_{1})G(Z_{2}))|_{Z_{1}=Z_{2}}$ is a cusp form. \end{prop} \medskip In section \ref{proofs} we will give the proofs of the Lemmas \ref{hompoly}, \ref{deltagrad} and \ref{lindep} as well as the proof of Theorem \ref{vgeneral} and Proposition \ref{cusp}. \section{Proofs} \label{proofs} In this section we have collected those proofs which have not been given so far. \subsection{Proof of Proposition \ref{Hdim}} \label{Hdimproof} {\it Proof of Proposition \ref{Hdim}.} Roughly speaking, the irreducible representations of $O(d)$ are parametrised by ``Young diagrams'' $(f_{1},...,f_{k})_{+}$ and $(f_{1},...,f_{k})_{-}$ where $f_{1} \geq f_{2} \geq \cdots \geq f_{k} \geq 0$, $k = d/2$, and, where, $+$ and $-$ coincide if $f_{k} \neq 0$. The space of pluri-harmonic polynomials $P(X,X^{'})$ ($(X,X^{'}) \in M_{n,d_{1}} \times M_{n,d_{2}}$) such that $P(A^t X,A^t X^{'}) = \det(A)^{v}P(X,X^{'})$ gives an irreducible representation of $O(d)$ ($d = d_{1} + d_{2}$) corresponding to the Young diagram $(v,...,v)_{+}$ of depth $n$ ({\it cf.} the notation of M.\ Kashiwara and M.\ Vergne in ref.\ \cite{KashiwaraVergne}). If we take the restriction of this representation to $O(d_{1}) \times O(d_{2})$ then ${\cal H}_{n,v}(d_{1},d_{2})$ is the subspace which corresponds to the trivial representation of $O(d_{1}) \times O(d_{2})$. So, what we should do is to count the multiplicity of the trivial representation in the restriction of $(v,...,v)_{+}$ to $O(d_{1}) \times O(d_{2})$. The irreducible decomposition of a similar restriction has already been worked out by K.\ Koike and I.\ Terada in Theorem 2.5 and Corollary 2.6 on p.\ 115 of ref.\ \cite{KoikeTerada} but here a subtle point is different. For the sake of simplicity let us denote by $R(O(d))$ the ring of those characters of $SO(d)$ which can be obtained as restrictions of characters of irreducible representations of $O(d)$. K.\ Koike and I.\ Terada take $\lambda_{SO(d)} \in R(O(d))$ and give the following formula for the restriction of $\lambda_{SO(d)}$ to $O(d_{1}) \times O(d_{2})$. \[ \lambda_{SO(d)} = \sum_{\beta, \mu, \kappa, \nu} L^{\lambda}_{\beta,\mu}L^{\beta}_{2\kappa,\nu} \pi_{SO(d_{1})}(\mu_{SO})\times \pi_{SO(d_{2})}(\nu_{SO})), \] where $L^{a}_{b,c}$ are the so-called Littlewood-Richardson coefficients, which can be calculated explicitly in principle, where the parameters $\beta$, $\kappa$, $\mu$, $\nu$ are any partitions or ``universal characters'' and, where $\pi_{SO(c)}$ is a ``specialisation'' homomorphism whose image is contained in $R(O(c))$ ({\it cf. loc. cit.}). Then what we should do is as follows. First we calculate the coefficients for those pairs $\mu$ and $\nu$ where $\pi_{SO(d_{1})}(\mu)$ and $\pi_{SO(d_{1})}(\nu)$ are the trivial characters. If $d_{1} > n$ and $d_{2} > n$ it is not difficult to show that this occurs only when $\mu$ and $\nu$ are trivial and that the coefficient is one. This is seen as follows. There are exactly two irreducible representations of $O(c)$ whose restriction to $SO(c)$ is trivial: the trivial representation and the determinant representation. We must exclude the latter possibility. Since we are assuming $d_{1} > n$ and $d_{2} > n$ the fundamental theorem on invariants ({\it cf.} Theorem 2.9A on p.\ 53 of ref.\ \cite{Weyl}) implies that any $SO(d_{i})$ invariant vector is also $O(d_{i})$ invariant. This means in our case that if the restriction to $SO(d_{i})$ is trivial then it comes from the trivial representation of $O(d_{i})$. This proves our assertion for $d_{1} > n$ and $d_{2} > n$. When $d_{1} = n$ or $d_{2} = n$ the proof is more involved. First of all, for $\lambda = (v,\ldots,v)_{+}$, we pick up pairs of $\mu$ and $\nu$ such that $L^{\lambda}_{\beta,\mu}L^{\beta}_{2\kappa,\nu} \neq 0$ and $\pi_{SO(d_{1})}(\mu) \times \pi_{SO(d_{2})}(\nu)$ is trivial. Under the assumption that $d_{1} \geq n$ and $d_{2} \geq n$ these pairs can be described as follows (for the sake of simplicity we set $\rho_{a}=(a,\ldots,a)$ with depth $n$). \\ (1) If $v$ is even and $\mu$ and $\nu$ are trivial then the coefficient is one. \\ (2) If $d_{1} = n$ and $v$ is odd then the coefficient is one for $\mu = \rho_{1}$, $\nu$ trivial, $\beta=\rho_{v-1}$ and $\kappa = \rho_{(v-1)/2}$. \\ (3) If $d_{2} = n$ and $v$ is odd then the coefficient is one for $\mu$ trivial, $\nu = \rho_{1}$, $\beta = \lambda$ and $\kappa=\rho_{(v-1)/2}$. \\ (4) If $d_{1} = d_{2} = n$ and $v$ is even with $v \geq 2$ then the coefficient is one for $\mu=\rho_{1}$, $\nu=\rho_{1}$, $\beta=\rho_{v-1}$ and $\kappa=\rho_{(v-2)/2}$. \\ These possibilities exhaust all cases giving the trivial representation of $SO(d_{1}) \times SO(d_{2})$. We have already shown in \S\ref{main} that $v$ is even in our case so that the cases (2) and (3) do not occur. If $d_{1} > n$ or $d_{2} > n$ then the pluri-harmonic polynomial in question is invariant by $O(d_{1})$ or $O(d_{2})$, respectively. Indeed, if this were not the case then, by the theory of invariants, it is of the form $\det(X^{'})Q(R,R^{'})$ ($d_{2} = n$ and $X^{'} \in M_{n}$) or $\det(X)Q(R,R^{'})$ ($d_{1} = n$, $X \in M_{n}$), respectively, where $Q \in {\cal Q}_{n,v-1}(2)$. But since $Q = 0$ unless $v-1$ is even this cannot be the case. Therefore, the only remaining case is $d_{1} = d_{2} = n$ and we must show that the trivial representation of $O(n) \times O(n)$ occurs exactly once. We prove this by showing that there exists exactly one $\det(g) \times \det(h)$ representation of $O(d_{1}) \times O(d_{2})$ in our restriction. Since, for $d_{1} = d_{2} = n$ and $v$ even, the multiplicity of the trivial representation of $SO(d_{1}) \times SO(d_{2})$ is two this indeed implies that our restriction contains the trivial representation of $O(n) \times O(n)$ exactly once. For even $v$ assume that $P(X,X^{'})$ ($X$, $X^{'} \in M_{n}$) is a pluri-harmonic polynomial such that $P(A^t X,A^t X^{'}) = \det(A)^{v}P(X,X^{'})$. By the above considerations this polynomial is invariant both by $O(d_{1})$ and $O(d_{2})$ or odd invariant for both $O(d_{1})$ and $O(d_{2})$, {\it i.e.} $P(Xg,X^{'}h) = P(X,X^{'})$ or $P(Xg,X^{'}h) = \det(g)\det(h)P(X,X^{'})$, respectively. We show the latter case occurs just for one polynomial in question (up to multiplication by a constant). By the classical theorem of invariants (Weyl, {\it loc.cit.}) we find in the latter case $P(X,X^{'}) = \det(X)\det(X^{'})Q(R,R^{'})$ for some $Q \in {\cal Q}_{n,v-2}(2)$. By applying $\Delta_{11} = \Delta_{11}(X) + \Delta_{11}(X^{'})$ to this polynomial it is easy to see that \[ \Delta_{11}(\det(X)\det(X^{'})Q(R,R^{'})) = (\det(X)\det(X^{'})) (\Delta_{11}Q + 2(\frac{\partial Q}{\partial R_{11}} +\frac{\partial Q}{\partial R^{'}_{11}})). \] Hence the polynomial is pluri-harmonic if and only if $\Delta_{11}Q + 2(\frac{\partial Q}{\partial R_{11}} +\frac{\partial Q}{\partial R^{'}_{11}}) = 0$. The latter action, however, is nothing but the action of $L^{(d_{1}+1)}_{1,1} + L^{'(d_{2}+1)}_{1,1}$. Since $d_{1} + 1 > n$ and $d_{2} + 1 > n$ we have already shown that the kernel of $L^{(d_{1}+1)}_{1,1} + L^{'(d_{2}+1)}_{1,1}$ is one dimensional. \qed \subsection{Proof of Lemmas \ref{hompoly}-\ref{lindep}, Theorem \ref{vgeneral} and Proposition \ref{cusp}} This subsection contains the proofs of the three lemmas stated in section \ref{main} which we then use to prove Theorem \ref{vgeneral}. We also give a proof of Proposition \ref{cusp}. \smallskip\noindent {\it Proof of Lemma \ref{hompoly}.} For any polynomial $Q(R,R') \in {\cal Q}_{n,v}(2)$ the polynomial $Q({\mathchoice {\rm 1\mskip-4mu l} {\rm 1\mskip-4mu l_{n},T)$ is a polynomial of the coefficients of $T$ where $T$ is symmetric and ${\mathchoice {\rm 1\mskip-4mu l} {\rm 1\mskip-4mu l_{n}$ is the unit matrix of size $n$. Since $Q({\mathchoice {\rm 1\mskip-4mu l} {\rm 1\mskip-4mu l_{n},T) = Q({\mathchoice {\rm 1\mskip-4mu l} {\rm 1\mskip-4mu l_{n},O^{-1} T O)$ for any orthogonal matrix $O$ the polynomial $Q({\mathchoice {\rm 1\mskip-4mu l} {\rm 1\mskip-4mu l_{n},T)$ is a polynomial in the functions $\mu_{i}(T)$ ($1 \leq i \leq n$) defined by $\det(t {\mathchoice {\rm 1\mskip-4mu l} {\rm 1\mskip-4mu l_n + T) = \sum_{i=0}^n \mu_i(T) t^i$. Indeed, if $T$ is a diagonal matrix then $Q({\mathchoice {\rm 1\mskip-4mu l} {\rm 1\mskip-4mu l_{n},T)$ is a symmetric function of the diagonal components and hence a function of the $\mu_{i}(T)$. Since $\mu_{i}(T)$ is invariant under conjugation of $T$ we obtain that $Q({\mathchoice {\rm 1\mskip-4mu l} {\rm 1\mskip-4mu l_{n},T)$ is a polynomial in the $\mu_{i}(T)$. Furthermore, for $0 \leq i \leq n$, we find that $\det(R)\mu_i(T) = P_i(R,R')$ where we have set $R= A^t A$ and $T = (A^t)^{-1}R' A^{-1}$ (strictly speaking we are working here over an algebraic extension of our ring $\C[R_{i,j}]$ ($R = (R_{i,j})$) which allows to find such a matrix $A$ with $R=A^t A$). Hence, for any non-negative integers $e_{i}$ ($1\leq i \leq n$) and $l_{0} = \sum_{i=1}^{n}e_{i}$ we get $$\det(R)^{l_{0}}\prod_{i=1}^{n}\mu_{i}(T)^{e_{i}} = \prod_{i=1}^{n}P_{i}^{e_{i}}(R,R'). $$ Now, any linear combination of the above ``monomials'' for fixed $l_{0}$ and various $e_{i}$ ($i \geq 1$) is not divisible by $\det(R)$ since even if some column of $R$ is a zero vector such a linear combination does not vanish. This is seen as follows. Choose $R' = {\mathchoice {\rm 1\mskip-4mu l} {\rm 1\mskip-4mu l_{n}$ and $R$ as a diagonal matrix with $R_{1,1} = 0$. Then, each $P_{i}$ ($ 1 \leq i \leq n-1$) becomes an elementary symmetric function of the $R_{i,i}$ ($2 \leq i \leq n$). This means that the above linear combination does not vanish identically. Furthermore, it also means that, for a fixed $v$, any linear combination of the ``monomials'' $\det(R)^{l}\prod_{i=1}^{n}P_{i}^{e_{i}}(R,R')$ with $l + \sum_{i=1}^{n}e_{i} = v$ where $l$ is an integer and the $e_i$ are non-negative integers is a polynomial if and only if $l$ is positive. Since any element in $Q_{n,v}(2)$ is a linear combination of ``monomials'' of the above type $Q_{n,v}(2)$ is generated by the $P_\alpha$ ($0\le \alpha\le n$). Furthermore, the $P_i$ ($ 1 \leq i \leq n$) are obviously algebraic independent with each other. This completes the proof of the lemma.\qed \bigskip\noindent {\it Proof of Lemma \ref{deltagrad}.} To prove the equations in Lemma \ref{deltagrad} we choose without loss of generality $i=1$. Firstly, note that one has $$D_{l,1}P_{\alpha} = (2-\delta_{1,l})(-1)^{l+1}(P_{1;l})_{\alpha} \qquad (0 \leq \alpha \leq n) $$ (here we regard $(P_{1;j})_{n} = 0$) and $$ D_{l',1} (P_{1;l})_{\alpha} = (-1)^{l'}(P_{1,l';1,l})_{\alpha} \qquad (l'\not=1). $$ This directly implies $$ L^{(d_1)}_{1,1} P_\alpha = 2 d_1 (P_{1;1})_\alpha - 2 \sum_{l,l'\not=1} (-1)^{l+l'} R_{l,l'}\, (P_{1,l';1,l})_{\alpha}. $$ Secondly, we show that $$\sum_{l,l'\not=1} (-1)^{l+l'} R_{l,l'}\, (P_{1,l';1,l})_{\alpha} = (n-1-\alpha) (P_{1;1})_\alpha. $$ This obviously implies the formula stated in the lemma. The last equality can be proven as follows. Multiplying both sides with $\lambda^\alpha$ and summing over $\alpha$ gives, as an equivalent equation, $$ \lambda \frac{d}{d\lambda} P_{1;1} = (n-1)\ P_{1;1} - \sum_{j,k\not=1} (-1)^{j+k}R_{j,k} P_{1,j;1,k}. $$ Instead of proving this equation is suffices to prove the equation with $P_{1;1}$ replaced by $P$ and $n-1$ replaced by $n$, {\it i.e.} to prove $$ \lambda \frac{d}{d\lambda} P = n\, P - \sum_{j,k=1}^n (-1)^{j+k}\ R_{j,k} P_{j;k}. $$ Expanding the determinants on the right hand side gives \begin{eqnarray*} && n\sum_{\sigma\in S_n} (-1)^\sigma \prod_{i=1}^{n} (R_{i,\sigma(i)} + \lambda R'_{i,\sigma(i)})\\ && - \sum_{k=1}^n \sum_{\sigma\in S_n} (-1)^\sigma \prod_{i=1}^{n} (R_{i,\sigma(i)} + \lambda (1-\delta_{i,k}) R'_{i,\sigma(i)}) \\ &=& \sum_{k=1}^n \sum_{\sigma\in S_n} (-1)^\sigma \lambda R'_{k,\sigma(k)} \prod_{i\not=k} (R_{i,\sigma(i)} + \lambda R'_{i,\sigma(i)}) \end{eqnarray*} where $S_n$ is the symmetric group of $n$ elements. Note that the last expression on the r.h.s. is equal to $\lambda \frac{d}{d\lambda} P$ so that we have proven the desired equality. The analogous equation for ${L'}^{(d_2)}_{i,i}$ follows directly from the symmetry $R \leftrightarrow R'$, $\alpha \leftrightarrow n-\alpha$. \bigskip\noindent We now prove the equation equation for $(P_\alpha,P_\beta)_{1,R}$. Using again the equality $D_{l,1}P_{\alpha} = (2-\delta_{1,l})(-1)^{l+1}(P_{1;l})_{\alpha}$ we immediately get $$ (P_{\alpha},P_{\beta})_{1,R} = \sum_{l,m=1}^{n}(-1)^{l+m}R_{l,m}(P_{1;l})_{\alpha}(P_{1;m})_{\beta}. $$ Furthermore, we find $$ \sum_{m=1}^{n}(-1)^{l+m}R_{l,m}(P_{1;m})_{\alpha} = \delta_{1,l}P_{\alpha}- \sum_{m=1}^{n}(-1)^{l+m} R_{l,m}^{'}(P_{1;m})_{\alpha-1} $$ and $$ \sum_{l=1}^{n}(-1)^{l+m}R_{l,m}^{'}(P_{1;l})_{\beta} = \delta_{m,1}P_{\beta+1} - \sum_{l=1}^{n}(-1)^{l+m}R_{l,m}(P_{1;l})_{\beta+1}. $$ Finally, collecting the above formulas gives the desired equation \begin{eqnarray*} (P_{\alpha},P_{\beta})_{1,R} & = & \sum_{l=1}^{n}\delta_{l,1}P_{\alpha}(P_{1;l})_{\beta} -\sum_{l,m=1}^{n}(-1)^{l+m}R^{'}_{l,m}(P_{1;m})_{\alpha-1}(P_{1;l})_{\beta} \\ & = & P_{\alpha}(P_{1;1})_{\beta} \\ && \qquad - \sum_{m=1}^{n}\left(\delta_{m,1}P_{\beta+1}(P_{1;m})_{\alpha-1} -\sum_{l=1}^{n}(-1)^{l+m}R_{l,m}(P_{1;m})_{\alpha-1}(P_{1;l})_{\beta+1}\right) \\ & = & P_{\alpha}(P_{1;1})_{\beta} - P_{\beta+1}(P_{1;1})_{\alpha-1} + \sum_{l,m=1}^{n}(-1)^{l+m}R_{l,m}(P_{1;m})_{\alpha-1}(P_{1;l})_{\beta+1} \\ & = & P_{\alpha}(P_{1;1})_{\beta} - P_{\beta+1}(P_{1;1})_{\alpha-1} + (P_{\alpha-1},P_{\beta+1}). \end{eqnarray*} The corresponding equation for $(P_\alpha, P_\beta)_{1,R'}$ follows, again, directly from the symmetry $R \leftrightarrow R'$, $\alpha\leftrightarrow n-\alpha$. \qed \bigskip\noindent {\it Proof of Lemma \ref{lindep}.} For the sake of simplicity we denote by $(P_{i;i})_{\beta}^{n}$ the differentiation of $(P_{i;i})_{\beta}$ by $\frac{\partial }{\partial R_{nn}}$. (In other words, $(P_{i;i})_{\beta}^{n}$ is obtained from $(P_{i;i})_{\beta}$ by substituting $R_{n,j} = 0$ ($n \neq j$), $R_{n,n}=1$ and $R^{'}_{n,j} = 0$ for all $j$ with $1 \leq j \leq n$.) For $\beta$ with $0 \leq \beta \leq n-1$, we take polynomials $Q_{\beta}$ of $n+1$ variables and assume that they satisfy $$ \sum_{\beta=0}^{n-1}Q_{\beta}(P_{0},...,P_{n}) (\sum_{i=1}^{n}(P_{i;i})_{\beta}) = 0. $$ We show that $Q_{\beta} = 0$ for all $\beta$. We set the last row (and the last column) of $R^{'}$ to 0 and also set $R_{n,j} = R_{j,n} = 0$ for all $j \neq n$. Then $P_{n}$ becomes $0$ and $(P_{n;n})_{\beta}$ is unchanged for any $\beta$. Furthermore, $P_{\alpha}$ becomes equal to $R_{n,n} (P_{n;n})_{\alpha}$ for $\alpha\le n-1$ and $(P_{i;i})_{\alpha}$ becomes equal to $R_{nn} (P_{i,i})_{\alpha}^{n}$ for $i\le n-1$. Hence we have $$ \sum_{\beta=0}^{n-1} Q_{\beta}(R_{n,n} (P_{n;n})_{0},...,R_{n,n}(P_{n;n})_{n-1},0) (\sum_{i=1}^{n-1} R_{n,n} (P_{i;i})_{\beta}^{n} + (P_{n;n})_{\beta}) = 0. $$ Note that $(P_{n;n})_{\beta}$ does not contain $R_{n,n}$ so that by comparing the degree of $R_{n,n}$ on both sides we obtain $$ \sum_{\beta=0}^{n-1} Q_{\beta}(R_{n,n} (P_{n;n})_{0},...,R_{n,n} (P_{n;n})_{n-1},0) (\sum_{i=1}^{n-1} R_{n,n} (P_{i;i})_{\beta}^n) = 0. $$ Now we set $R_{n,n}=1$ and use induction by $n$. Since for $n=2$ the lemma is easily verified we can assume that the lemma holds for $n-1$. Then the above induction step gives nothing but the relation for $n-1$ and we obtain $Q_{\beta}(X_{0},...,X_{n-1},0) = 0.$ Therefore, all $Q_{\beta}$ are divisible by $X_{n}$. By repeating this process we find that $Q_{\beta} = 0$. \qed \bigskip\noindent {\it Proof of Theorem \ref{vgeneral}.} Using Lemma \ref{deltagrad} we can calculate the Laplacian of $Q_{n,v}$ and obtain, since the $\sum_{i=1}^n (P_{i;i})_\alpha$ ($0\le \alpha \le n-1$) are linearly independent over ${\cal Q}_{n}(2)$ by Lemma \ref{lindep}, the following set of equations for the coefficients $C(a)$ \begin{eqnarray*} (d_1+1-g+i)a_i C(a) &+& (d_2-i)(a_{i+1}+1) C(a-e_i+e_{i+1}) =\\ && -2\sum_{l\le l'\le i} \tilde a(i,l,l')_l\ ( \tilde a(i,l,l')_{l'}-\delta_{l,l'})\ C( \tilde a(i,l,l') )\\ && -2\sum_{i\le l\le l'} \hat a(i,l,l')_l\ ( \hat a(i,l,l')_{l'}-\delta_{l,l'})\ C( \hat a(i,l,l') )\\ && +2\sum_{i<l\le l' } \tilde a(i,l,l')_l\ ( \tilde a(i,l,l')_{l'}-\delta_{l,l'})\ C( \tilde a(i,l,l') )\\ &&+2\sum_{l\le l'< i} \hat a(i,l,l')_l\ ( \hat a(i,l,l')_{l'}-\delta_{l,l'})\ C( \hat a(i,l,l') ) \end{eqnarray*} for all $0\le i\le n-1$. Choosing $i=\mbox{\rm min}\{j|a_j\not=0\}$ gives the equations in the formulation of the theorem. (These equations are equivalent to the vanishing of coefficient in front of $(\sum_{k=1}^{n} (P_{k;k})_i) \prod_{j=0,n} P_j^{a_j-\delta_{i,j}}$.) Finally, note that by Lemma \ref{hompoly}, Proposition \ref{Hdim} and the remarks after Theorem \ref{vgeneral} we know that the other equations, {\it i.e.} those with $i\not=\mbox{\rm min}\{j|a_j\not=0\}$, do not contain any further information about the coefficients $C(a)$. \qed \bigskip\noindent {\it Proof of Proposition \ref{cusp}.} Firstly, we show that if $R$ and $R^{'}$ are positive semi-definite symmetric real matrices and if $\det(R+R^{'}) = 0$ then $\det(R+\lambda R^{'})=0$ and hence $P_{\alpha}(R,R^{'}) = 0$ for all $\alpha$ with $0 \leq \alpha \leq n$. In general, if $A$ and $B$ are positive semi-definite symmetric matrices so is $A+B$ and $\det(A+B) \geq \det(A) \geq 0$. Now, let $\lambda$ be any real number between 0 and 1. Then we obtain $0 = \det(R+R^{'}) \geq \det(R+\lambda R^{'}) \geq 0$ so that $\det(R+\lambda R^{'})$ vanishes identically as $\det(R+\lambda R^{'})$ is a polynomial in $\lambda$. Secondly, let $F$ and $G$ be Siegel modular forms of weight $k$ and $l$, respectively. We consider $(F|^{k}_{M})(Z_{1})$ and $(G|^{l}_{M})(Z_{2})$ for $M \in \mbox{{\rm{Sp}}}(2n,\Qq)$. As these are modular forms on $M^{-1}\Gamma M$ there exists some natural number $N$ such that they have a Fourier expansion of the form \begin{eqnarray*} (F|^{k}_{M})(Z_{1}) & = & \sum_{T_{1}}a_{1}(T_{1}) \exp(2\pi i\ {\mbox{\rm tr}}(T_{1}Z_{1}/N)), \\ (G|^{l}_{M})(Z_{2}) & = & \sum_{T_{2}}a_{2}(T_{2}) \exp(2\pi i\ {\mbox{\rm tr}}(T_{2}Z_{2}/N)), \end{eqnarray*} where $T_{1}$ and $T_{2}$ run over all positive semi-definite half-integral matrices. We set $H = (F|^{k}_{M})(Z_{1})(G|^{l}_{M})(Z_{2})$. Assume now that $D$ is obtained from an associated polynomial $Q\in {\cal H}_{n,v}(2k,2l)$. Then we easily obtain \[ (DH)|_{Z_{1}=Z_{2}}= \sum_{T_{1},T_{2}}a_{1}(T_{1})a_{2}(T_{2}) Q(T_{1}/N,T_{2}/N)\exp(2\pi i\ {\mbox{\rm tr}}((T_{1}+T_{2})Z_{2}/N)), \] where, again, $T_{1}$ and $T_{2}$ run over all positive semi-definite half-integral matrices, and $Q(T_{1}/N,T_{2}/N) = 0$ if $\det(T_{1}+T_{2}) = 0$ by the discussion above. By Theorem \ref{Ibuthm} we also know that \[ (D(FG)|_{Z_{1}=Z_{2}=Z})|^{k+l+v}_{M} = D(H)|_{Z_{1}=Z_{2}=Z}. \] This means that $\Phi((D(FG)|_{Z_{1}=Z_{2}=Z})|^{k+l+v}_{M}) = 0$ for all $M\in\mbox{{\rm{Sp}}}(2n,\Qq)$ where $\Phi$ is the Siegel operator and hence $D(FG)|_{Z_{1}=Z_{2}=Z}$ is a cusp form. \qed \section{Some explicit examples} \label{examples} In this section we discuss some special cases of Theorem \ref{vgeneral}. In particular we recover the results of ref.\ \cite{Cohen} for $n=1$ (see also the examples in ref.\ \cite{Ibukiyama}) and ref.\ \cite{ChoieEholzer} for $n=2$. Furthermore, we give closed explicit formulas for $Q_{n,v}$ for $v=2,4$ and general $n$. \subsection{The case $v=2$} By the discussion in section \ref{main} we can write $Q_{n,2}$ as $$ Q_{n,2} = \sum_{\alpha=0}^n C(\alpha) P_\alpha.$$ The recursion equations given in Theorem \ref{vgeneral} simplify in this case to $$ (d_2-\alpha)C(\alpha) = -(d_1+n-\alpha+1) C(\alpha+1) \qquad (0\le \alpha \le n-1). $$ Hence we find that, up to a constant multiple, $Q_{n,2}$ is given by $$ Q_{n,2} = \sum_{\alpha+\beta=n} (-1)^\alpha\, \alpha!\, \beta!\, \bin{d_2-\alpha}{\beta}\, \bin{d_1-\beta}{\alpha}\, P_\alpha. $$ \subsection{The case $v=4$} \label{v4} We consider the case $v=4$ for general $n$. Writing $$ Q_{n,4} = \sum_{i,j=0}^n C_{i,j} P_i P_j $$ with $C_{i,j} = C_{j,i}$ one finds from Theorem \ref{vgeneral} the following recursion equations $$ (d_1+1-n+i + 2\delta_{i,j} ) C_{i,j} + (d_2-i + 2\delta_{i<j} ) C_{i+1,j} = 2\sum_{r=1}^{j-i-1} \left( C_{i+r,j-r} - C_{i+r,j+1-r} \right) $$ where $0\le i\le j \le n$ and $i\le n-1$. One can give an explicit solution to these equations \footnote{This explicit form of the solution is due to D.\ Zagier}. The coefficients $C_{r,s}$ ($0\le r,\,s\le n$) are given by the closed formula $$ C_{r,s}=(-1)^{r-s}\,\frac{(d_1-n+r)!(d_1-n+s)!(d_2-r)!(d_2-s)!} {(d_1-n)!(d_1-n+2)!(d_2-n)!(d_2-n+2)!}\, p_{r-s}\bigl(\kappa_1\bigl(\frac{r+s}2\bigr),\, \kappa_2\bigl(\frac{r+s}2\bigr)\bigr)\,,$$ where $\kappa_1(r) = d_1+2-(n-r)$, $\kappa_2(r)= d_2+2-r$ and $p_e(x,y)$ is the polynomial of degree 4 in $e^2$, $x$ and $y$ given by $$p_e(x,y) = x^2y^2+(e^2-1)\,xy\bigl(x+y+\frac{e^2-6}6\bigr) +\frac{e^2(e^2-4)}{12}\,\bigl(x^2+y^2-\frac14\bigr)\,.$$ The first few values of this polynomial are given by \begin{eqnarray*} p_0(x,y)&=&x(x-1)y(y-1)\,,\\ p_1(x,y)&=&(x+\tfrac12)(x-\tfrac12)(y+\tfrac12)(y-\tfrac12)\,,\\ p_2(x,y)&=&xy(xy+3x+3y-1)\,,\\ p_3(x,y)&=&(x+\tfrac12)(y+\tfrac12) (xy+\tfrac{15}2x+\tfrac{15}2y-\tfrac{15}4)\,,\\ p_4(x,y)&=&x^2y^2+15x^2y+15xy^2+16x^2+25xy+16y^2-4\,. \end{eqnarray*} To check the correctness of the formula, it is convenient to replace the recursion relation above by the simpler 4-term recurrence $$\kappa_1(r+3)\,C_{r,s}+\kappa_2(r+2)\,C_{r+1,s} =\kappa_1(r-2)\,C_{r-1,s+1}+\kappa_2(r-3)\,C_{r,s+1}\,.$$ It is easy to verify that the solution given above indeed satisfies this recursion relation. Note that the denominator in the formula for $C_{r,s}$ is of course just conventional. One could also make the choice $(d_1-n)!^2(d_2-n)!^2$, which gives the simpler expression $$ C_{r,s}=(-1)^{r-s}(n-1-d_1)_r(n-1-d_1)_s(n-1-d_2)_{n-r}(n-1-d_2)_{n-s} p_{r-s}\bigl(\kappa_1\bigl(\frac{r+s}2\bigr), \kappa_2\bigl(\frac{r+s}2\bigr)\bigr)$$ (here we have used $(x)_n$ for $\prod_{i=0}^{n-1} (x-i)$). \subsection{The genus one case} \label{genus1} In this case ${\cal Q}_{1}(2)$ is generated by $P_0$ and $P_1$. Writing $C_{r,s}$ for $C(a)$ with $a=(r,s)$ the recursion equations in Theorem \ref{vgeneral} for $C(a)$ become $$ 2(d_1/2+r-1)r \ C_{r,s} = - d_2 (s+1)\ C_{r-1,s+1} + 2 s (s+1)\ C_{r-1,s+1} = -2 (d_2/2-s)(s+1)\ C_{r-1,s+1} $$ for $1\le r \le v$. Hence we find that, up to a constant multiple, $Q_{1,2v}$ is given by $$ Q_{1,2v} = \sum_{r+s=v} (-1)^r\, \bin{v+d_2/2-1}{r}\, \bin{v+d_1/2-1}{s}\ P_0^r\, P_1^s. $$ The differential operators corresponding to $Q_{1,2v}$ are precisely the operators studied by H.\ Cohen in ref.\ \cite{Cohen}. Note that this case has already been discussed in the context of pluri-harmonic polynomials in ref.\ \cite{Ibukiyama}. \subsection{The genus two case} \label{genus2} To make contact with the results of ref.\ \cite{ChoieEholzer} we use $P^*_0=P_0,P^*_2=P_2$ and $P^*_1=P_0+P_1+P_2$ instead of $P_0, P_1$ and $P_2$ to generate ${\cal Q}_{2}(2)$. Writing $Q_{2,2v}$ as $$ Q_{2,2v} = \sum_{r+s+p=v} C_{r,s,p}\, (P^*_0)^r\, (P^*_2)^s\, (P^*_1)^p $$ and applying Lemma \ref{deltagrad} and Lemma \ref{lindep} we find the following two recursion relations for the coefficients $C_{r,s,p}$ ({\it cf.} the equations on page 13 of {\it loc. cit.}) \begin{eqnarray*} 0 &=& (r+1)((d_1-3)/2 +r+1) C_{r+1,s,p} + (p+1)((d_1+d_2-3)/2 +p+1) C_{r,s,p+1}\,, \\ 0 &=& (s+1)((d_2-3)/2 +s+1) C_{r,s+1,p} + (p+1)((d_1+d_2-3)/2 +p+1) C_{r,s,p+1}. \end{eqnarray*} Hence we find that, up to a constant multiple, $Q_{2,2v}$ is given by \begin{eqnarray*} Q_{2,2v} &=& \sum_{r+s+p=v} \frac{1}{r!s!p!} (d_1/2-3/2+v)_{v-r}\ (d_2/2-3/2+v)_{v-s} \\ &&\qquad \qquad (-((d_1+d_2)/2-3/2+v))_{v-p}\ (P^*_0)^r (P_2^*)^s (P^*_1)^p \end{eqnarray*} where we have, again, used $(x)_n$ for $\prod_{i=0}^{n-1} (x-i)$. Since it is obvious that \begin{eqnarray*} && {\Bbb D}^p( {\Bbb D}^r F(Z)\ {\Bbb D}^s G(Z) ) = \\ && \left( (P^*_1(\partial_{Z_1},\partial_{Z_2}))^p\ (P^*_0(\partial_{Z_1},\partial_{Z_2}))^r\ (P^*_2(\partial_{Z_1},\partial_{Z_2}))^s F(Z_1) G(Z_2) \right) \vert_{Z=Z_1=Z_2} \end{eqnarray*} where ${\Bbb D} = \det(\partial_{Z})$ we obtain exactly the formula in Theorem 1.2 of {\it loc. cit.}. \section{A vector valued generalisation} \label{vector} In this section we describe vector valued differential operators $D$ such that $$ D(F(Z_{1})G(Z_{2}))_{|Z_{1}=Z_{2}} $$ is a vector valued modular form if $F$ and $G$ are modular forms. Let us introduce some notation first. For any representation $\rho$ of $\GL{n}$ we denote by $d(\rho)$ its dimension. Furthermore, for any even positive integers $d_{1}$ and $d_{2}$, we denote by ${\cal H}_{n,\rho}(d_{1},d_{2})$ the space of $d(\rho)$ dimensional vectors of $O(d_{1}) \times O(d_{2})$-invariant pluri-harmonic polynomials $P(X,X^{'})=(P_{i}(X,X^{'}))_{1 \leq i \leq d(\rho)}$ such that $P(A X,A X^{'}) = \rho(A)P(X,X^{'})$. The main result of ref.\ \cite{Ibukiyama} not only includes the case of invariant pluri-harmonic polynomials in ${\cal H}_{n,v}(d_{1},d_{2})$ ({\it c.f.} Theorem \ref{Ibuthm}) but also applies to the case of pluri-harmonic polynomials in ${\cal H}_{n,\rho}(d_{1},d_{2})$. It is shown in {\it loc. cit.} that the vector valued differential operators corresponding to the latter polynomials map two modular forms to a vector valued modular form. We will, therefore, concentrate on giving some examples of pluri-harmonic polynomials in ${\cal H}_{n,\rho}(d_{1},d_{2})$. More precisely, we will consider only the representations of $\GL{n}$ which are of the form $\rho_{m,v}=\det^{v} \mbox{\rm Sym}^{m}$ where $\mbox{\rm Sym}^{m}$ is the symmetric tensor representation of degree $m$. When $n = 2$ these representations exhaust all polynomial representations of $\GL{n}$. For given $m$ and $v$ the Young diagram corresponding to $\rho_{m,v}$ is given by $(v+m,v,...,v)$ where the depth is $n$ if $v \neq 0$ and $1$ if $v = 0$. For the sake of simplicity, we assume from now on that $d_{1} \geq 2n$ and $d_{2} \geq 2n$. Then one can easily see that $\dim({\cal H}_{n,\rho_{m,v}}(d_{1},d_{2})) = 1$ if and only if $m$ and $v$ are even. We want to give explicit bases of (some of) the one dimensional spaces ${\cal H}_{n,\rho_{m,v}}(d_1,d_2)$. Firstly, we consider the simplest case, {\it i.e.} the case of the symmetric representation $\rho_{m,0}$. In this case everything can be reduced to the genus one case which was already discussed in \S\ref{genus1}. Let $u_{1},...,u_{n}$ be $n$ independent variables and, for any multi-index $\nu=(\nu_{1},...,\nu_{n})\in\N_0^n$, write $u^{\nu}$ for $\prod_{i=1}^{n}u_{i}^{\nu_{i}}$ and $|\nu|$ for $\nu_{1} + \cdots + \nu_{n}$. Denote by $I(m)$ the set of all multi-indices with $|\nu|=m$ and set $S = \sum_{i,j=1}^{n}R_{ij}u_{i}u_{j}$ and $S^{'} = \sum_{i,j=1}^{n}R^{'}_{ij}u_{i}u_{j}$. \begin{prop} \label{vectorprop1} Let $m$ be even and let $Q(r,r^{'})$ be a basis of ${\cal H}_{1,m}(d_1,d_2)$. For each multi-index $\nu$ with $|\nu| = m$ we define $Q_{\nu}(R,R^{'})$ by $Q(S,S^{'}) = \sum_{|\nu|=m}Q_{\nu}(R,R^{'})u^{\nu}$. Then the vector $(Q_{\nu}(R,R^{'}))_{\nu \in I(m)}$ gives a basis of the one dimensional space ${\cal H}_{n,\rho_{m,0}}(d_{1},d_{2})$. \end{prop} {\it Proof.} Since it is clear that \[ (Q_{\nu}(ARA^{t},AR^{'}A^{t}))_{\nu \in I(m)} = \rho_{m,0}(A)(Q_{\nu}(R,R^{'}))_{\nu \in I(m)} \] all we should do is to prove that the polynomials $Q_{\nu}(R,R^{'})$ are harmonic. This is easily proved as follows. Using that $Q(r,r^{'})$ is harmonic one obtains \[ \left(2d_{1}\frac{\partial Q(r,r^{'})}{\partial r} + 4r\frac{\partial^{2} Q(r,r^{'})}{\partial r^{2}}\right) +\left(2d_{2}\frac{\partial Q(r,r^{'})}{\partial r^{'}} + 4r^{'}\frac{\partial^{2} Q(r,r^{'})}{\partial r^{'2}}\right) = 0. \] Furthermore, a simple calculation shows that \begin{eqnarray*} L^{(d_{1})}_{ij}(Q(S,S^{'})) & = & u_{i}u_{j} (2d_{1}\frac{\partial Q}{\partial r} + 4S\frac{\partial^{2} Q}{\partial r^{2}})\, , \\ L^{(d_{2})}_{ij}(Q(S,S^{'})) & = & u_{i}u_{j} (2d_{2}\frac{\partial Q}{\partial r^{'}} + 4S^{'}\frac{\partial^{2} Q}{\partial r^{'2}}) \end{eqnarray*} so that the proposition becomes obvious. \qed Finally, we give some examples for the case of representations of mixed type when $n=2$. For $(m+v,v) = (m+2,2)$ with even $m$ an invariant pluri-harmonic polynomial in ${\cal H}_{2,\rho_{m,2}}$ is given by $(Q_{\nu}(R,R^{'}))_{\nu \in I(m)}$ where $Q_{\nu}(R,R^{'})$ is the coefficients of $u^{\nu}$ of the following polynomial $Q(S,S^{'})$. \begin{eqnarray*} Q(S,S^{'}) & = & Q_{2,d_{1},d_{2}}(R_{1},R_{2})F_{m,d_{1}+2,d_{2}+2}(S,S^{'}) \\ &&+\frac{1}{2}((d_{2}-1)P_{0}S-(d_{1}-1)P_{2}S^{'}) (\frac{\partial F_{m.d_{1}+2,d_{2}+2}}{\partial r} -\frac{\partial F_{m.d_{1}+2,d_{2}+2}}{\partial s})(S,S^{'}). \end{eqnarray*} Here we have denoted by $Q_{v,d_{1},d_{2}}$ the non-zero invariant pluri-harmonic polynomial in ${\cal H}_{2,v}(d_{1},d_{2})$ normalised as in section \ref{genus2} and by $F_{m,d_{1},d_{2}}(r,s)$ a non-zero polynomial in ${\cal H}_{1,m}(d_{1},d_{2})$. When $(m+2,2) = (4,2)$, for example, then $Q$ this is given by \begin{eqnarray*} Q(S,S^{'}) & = & \ \ (d_{2}-1)d_{2}(d_{2}+2)P_{0}S - (d_{1}-1)(d_{2}-1)(d_{2}+2)P_{1}S \\ & & +(d_{1}-1)(d_{1}+2)(d_{4}+4)P_{2}S - (d_{1}+4)(d_{2}-1)(d_{2}+2)P_{0}S^{'} \\ & & +(d_{1}-1)(d_{1}+2)(d_{2}-1)P_{1}S^{'} - d_{1}(d_{1}-1)(d_{1}+2)P_{2}S^{'}. \end{eqnarray*} \section{Conclusion} \label{conclusion} In this paper we have described certain spaces of invariant pluri-harmonic polynomials. These polynomials are in one-to-one correspondence with Rankin-Cohen type differential operators which, in the case of non-constant polynomials, map two Siegel modular forms to a cusp form. In particular, we have derived a set of recursion equations (Theorem \ref{vgeneral}) which uniquely (up to multiplication by non-zero elements in $\C^*$) determine an invariant pluri-harmonic polynomial $Q_{n,v}$ in ${\cal H}_{n,v}(d_1,d_2)$ (for $d_1\ge n$ and $d_2\ge n$). Although the recursion equations can easily be solved for any {\it numerical} values of $n$ and $v$ we have not been able to give the closed explicit formulas for the solutions for general $n$ and $v$. However, in several examples we have obtained such formulas for the solutions. In addition, we have discussed certain vector valued bilinear differential operators. Let us conclude with a few remarks and point out some interesting open questions in connection with our results. Firstly, the polynomials $Q_{n,v}\in {\cal H}_{n,v}(d_1,d_2)$ are in one-to-one correspondence with the differential operators $D$ defined in Theorem \ref{Ibuthm} only if $d_1\ge n$ and $d_2\ge n$. Throughout our analysis we have always assumed that this condition is satisfied. For fixed $n$ and $v$ the polynomial $Q_{n,v}$ depends only on $d_1$ and $d_2$ and, with a suitable normalisation, is well defined and non-vanishing even if $d_1< n$ or $d_2< n$. Therefore, one might speculate that the differential operators corresponding to $Q_{n,v}$ in the latter cases also map any two automorphic forms to an automorphic form (for $n=1,2$ this follows from the results in ref.\ \cite{Cohen,ChoieEholzer}). In contrast to the situation for $d_1\ge n$ and $d_2\ge n$ the dimension of ${\cal H}_{n,v}(d_1,d_2)$ can be larger than one if $d_1$ or $d_2$ are `small'. For example if $d_1=d_2=n-1$ and $v=2$ then ${\cal H}_{n,2}(n-1,n-1)$ is spanned by $P_0$ and $P_n$ (note that, in this case, the pluri-harmonic polynomials $P_0(X X^t, X' {X'}^t)$ and $P_n(X X^t, X' {X'}^t)$ are identically zero). In this case the differential operators corresponding to $P_0$ and $P_n$ satisfy the commutation relation ({\it c.f.} \S\ref{basics}). For odd $n$, however, all modular forms of weight $(n-1)/2$ on congruence subgroups are singular so that these differential operators act as zero (this special case is already contained in the results of Kapitel III, \S6 of ref.\ \cite{Freitag} and ref.\ \cite{Resnikoff}). Furthermore, note that if $d_1$ and $d_2$ are such that $d_1+d_2<2n$ then there is no pluri-harmonic polynomial in ${\cal P}_{n,v}(d_1,d_2)$ ({\it c.f.} the discussion in ref. \cite{KashiwaraVergne}). The last remarks show that it would be interesting to compute the dimension of ${\cal H}_{n,v}(d_1,d_2)$ for $d_1+d_2 \ge 2n$ and either $d_1$ or $d_2$ small and to understand the relation between covariant differential operators and invariant pluri-harmonic polynomials in this case. Secondly, the differential operators $D$ give rise to differential operators for Jacobi forms of higher degree. The relation between the Rankin-Cohen type operators for $n=2$ and the corresponding differential operators for Jacobi forms has been discussed in detail in ref.\ \cite{ChoieEholzer}. One would expect that --like in the case $n=2$-- the dimension of the space of covariant differential operators for higher degree Jacobi forms is, for fixed $v$, generically greater than one. It would be interesting if one could compute this dimension and obtain, as in the case of $n=2$, closed explicit formulas. Thirdly, in ref.\ \cite{Zagier} certain algebraic structures --called Rankin-Cohen algebras-- have been defined using only the Rankin-Cohen operators for $n=1$. There is an obvious generalisation of this definition for the case of arbitrary $n$ using the corresponding Rankin-Cohen type differential operators studied in this paper. Therefore, one is naturally led to the question whether one can describe the structure of these generalised Rankin-Cohen algebras in an independent way analogous to the case $n=1$ ({\it cf.} the Theorem in {\it loc. cit.}). Furthermore, it would be interesting to know if other examples of these algebraic structures can be found in mathematical nature. Fourthly, it seems quite natural to look for applications of the Rankin-Cohen type operators constructed in this paper which involve theta series and/or Eisenstein series. Finally, let us mention that there are other very interesting types of pluri-harmonic polynomials. The polynomials considered in this paper correspond to the second case in ref.\ \cite{Ibukiyama}. The first case considered in {\it loc. cit.} is concerned with differential operators $D$ acting on automorphic functions $F$ on $\H_n$ such that the restriction of $D(F)$ to $\H_{n_1}\times\cdots\times\H_{n_r}$ with $n=n_1+\cdots+n_r$ is again an automorphic form and the corresponding pluri-harmonic polynomials \cite{IbukiyamaZagier}. \medskip\noindent {\bf Acknowledgements} We would like to thank D.\ Zagier for many inspiring discussions and, in particular, for the explicit solution of the recursion equation in \S\ref{v4}. W.E. would also like to thank Y.\ Choie for many stimulating discussions. The second author is grateful to the Max-Planck-Institut f\"ur Mathematik in Bonn for its hospitality during his visit when this work was done.

9703

alg-geom/9703022

en

https://arxiv.org/abs/alg-geom/9703022

alg-geom/9703022

Edward Frenkel

E. Frenkel, D. Gaitsgory, D. Kazhdan, K. Vilonen

Geometric Realization of Whittaker Functions and the Langlands Conjecture

33 pages, LATEX2e

We prove the equivalence of two conjectural constructions of unramified cuspidal automorphic functions on the adelic group GL_n(A) associated to an irreducible l-adic local system of rank n on an algebraic curve X over a finite field. The existence of such a function is predicted by the Langlands conjecture. The first construction, which was proposed by Shalika and Piatetski-Shapiro following Weil and Jacquet-Langlands (n=2), is based on considering the Whittaker function. The second construction, which was proposed recently by Laumon following Drinfeld (n=2) and Deligne (n=1), is geometric: the automorphic function is obtained via Grothendieck's ``faisceaux-fonctions'' correspondence from a complex of sheaves on an algebraic stack. Our proof of their equivalence is based on a local result about the spherical Hecke algebra, which we prove for an arbitrary reductive group. We also discuss a geometric interpretation of this result.

alg-geom

\section{Introduction} \subsection{} Let $X$ be a smooth, complete, geometrically connected curve over $\Fq$. Denote by $F$ the field of rational functions on $X$, by ${\mathbb A}$ the ring of adeles of $F$, and by $\on{Gal}(\ol{F}/F)$ the Galois group of $F$. The present paper may be considered as a step towards understanding the geometric Langlands correspondence between $n$--dimensional $\ell$--adic representations of $\on{Gal}(\ol{F}/F)$ and automorphic forms on the group $GL_n({\mathbb A})$. We follow the approach initiated by V.~Drinfeld \cite{Dr}, who applied the theory of $\ell$--adic sheaves to establish this correspondence in the case of $GL_2$. The starting point of Drinfeld's approach is the observation that an unramified automorphic form on the group $GL_n({\mathbb A})$ can be viewed as a function on the set $M_n$ of isomorphism classes of rank $n$ bundles on the curve $X$. The set $M_n$ is the set of $\Fq$--points of ${\mathcal M}_n$, the algebraic stack of rank $n$ bundles on $X$. One may hope to construct the automorphic form associated to a Galois representation as a function corresponding to an $\ell$--adic perverse sheaf on ${\mathcal M}_n$. This is essentially what Drinfeld did in \cite{Dr} in the case of $GL_2$. In abelian class field theory (the case of $GL_1$) this was done previously by P.~Deligne (see \cite{La1}). \subsection{} Let $M'_n$ denote the set of isomorphism classes of pairs $\{L,s\}$, where $L\in M_n$ is a rank $n$ bundle on $X$ and $s$ is a regular non-zero section of $L$. Using a well-known construction due to Weil \cite{W} and Jacquet-Langlands \cite{JL} for $n=2$, and Shalika \cite{Sha} and Piatetski-Shapiro \cite{PS} for general $n$, one can associate to an unramified $n$--dimensional representation $\sigma$ of $\on{Gal}(\ol{F}/F)$, a function $f'_\sigma$ on $M'_n$. The construction of $f'_\sigma$ is obtained from the Whittaker function $W_{\sigma}$, a function canonically attached to $\sigma$. The Langlands conjecture predicts that when $\sigma$ is geometrically irreducible, the function $f'_\sigma$ is constant along the fibers of the projection $p: M'_n\to M_n$. In other words, conjecturally, $f'_\sigma$ is the pull-back of a function $f_{\sigma}$ on $M_n$; the function $f_{\sigma}$ is then the automorphic function corresponding to $\sigma$. \subsection{} Let now $\M'_n$ be the moduli stack of pairs $\{L,s\}$, where $L$ is a rank $n$ bundle on $X$ and $s$ is a regular non-zero section of $L$. We have: $M_n=\M_n(\Fq),M'_n=\M'_n(\Fq)$. Each Galois representation $\sigma$ gives rise to an $\ell$--adic local system $E$ on $X$ of rank $n$. Drinfeld's idea, developed further by G.~Laumon \cite{La1}, can be interpreted as follows. Suppose there exists an irreducible perverse sheaf ${\mathcal S}'\si$ on $\M'_n$, with the property that the function $S'\si$ associated to ${\mathcal S}'\si$ on $M'_n=\M'_n(\Fq)$ equals $f'_\sigma$. Then showing that the function $f'_\sigma$ is constant along the fibers of the projection $p: M'_n\to M_n$ becomes a geometric problem of proving that ${\mathcal S}'\si$ descends to a perverse sheaf ${\mathcal S}\si$ on $\M_n$. In \cite{Dr}, Drinfeld constructed such a sheaf ${\mathcal S}'\si$ in the case of $GL_2$. He started with a geometric realization of the Whittaker function as a perverse sheaf on the symmetric power of the curve $X$. Then he defined the sheaf ${\mathcal S}'\si$ using a geometric version of the Weil-Jacquet-Langlands construction. Drinfeld showed that the sheaf ${\mathcal S}'\si$ is locally constant along the fibers of $p$. Since the fibers of $p$ are projective spaces, hence simply-connected, this implies that ${\mathcal S}'\si$ is constant along the fibers of $p$. One may hope to use a similar argument in the case of $GL_n$. \subsection{} In order to construct ${\mathcal S}'\si$ in general, Laumon \cite{La1} defined a sheaf ${\mathcal L}\si$, which he considered as a geometric analogue of the Whittaker function $W_\sigma$. However, the function $L\si$ corresponding to ${\mathcal L}\si$ and the Whittaker function $W_\sigma$ are defined on different sets and their values are different, see \cite{La1} and Sect. 3.5 below. Using the sheaf ${\mathcal L}\si$, Laumon \cite{La2} constructed a candidate for the sheaf ${\mathcal S}'\si$ on $\M'_n$ (this construction was independently found by one of us; D.G., unpublished). In order to justify this construction, one has to prove that the function $S'\si$ on $M'_n$, corresponding to the sheaf ${\mathcal S}'\si$, coincides with the function $f'_\sigma$. This equality was conjectured by Laumon in \cite{La2} (Conjecture 3.2), and its proof is one of the main goals of this paper. To prove the equality $S'\si=f'_\sigma$, we reduce it to a local statement (see \thmref{local}) which we make for an arbitrary reductive group. We prove \thmref{local} using the Casselman-Shalika formula for the Whittaker function \cite{CS} (actually, \thmref{local} is equivalent to the Casselman-Shalika formula). \thmref{local} can be translated into a geometric statement about intersection cohomology sheaves on the affine Grassmannian (see \conjref{second}). \subsection{} One essential difference between Laumon's approach and our approach is in interpretation of the sheaf ${\mathcal L}\si$ and the function $L\si$ associated to this sheaf. Laumon interprets the local factors of $L\si$ via the Springer sheaves and the Kostka-Foulkes polynomials (see \remref{3}). We interpret the local factors of $L\si$ via the perverse sheaves on the affine Grassmannian and the Hecke algebra (see Sect.~4.2). Our interpretation, which was inspired by \cite{Lu1}, allows us to gain more insight into Laumon's construction. In particular, it helps to explain the apparent discrepancy between $L\si$ and $W_\sigma$: it turns out that $L\si$ is related to $W_\sigma$ by a Fourier transform. Using this result, we demonstrate that the outputs of the two constructions -- $S'\si$ and $f'_\sigma$ -- coincide. \subsection{} Let us now briefly describe the contents of the paper: In Sect.~2 we review some background material concerning the Langlands conjecture and the classical construction of the function $f'_\sigma$ together with its geometric interpretation. We follow closely Sect.~1 of \cite{La1}. In Sect.~3 we describe the construction of the sheaf ${\mathcal S}'\si$ on $\M'_n$ and the function $S'\si$ on $M'_n$. We state the main conjecture (\conjref{princ}) about the geometric Langlands correspondence for $GL_n$ and our main result (\thmref{prin}). In Sect.~4 we give an adelic interpretation of the construction of the function $S'\si$ and reduce \thmref{prin} to a local statement, \propref{Fplus}. In Sect.~5 we prove \thmref{local} for an arbitrary reductive group and derive from it \propref{Fplus}. In Sect.~6 we interpret the results of Sect.~5 from the point of view of the spherical functions. In Sect.~7 we give a geometric interpretation of \thmref{local} and discuss a possible generalization of Laumon's construction to other groups. \subsection{} In this paper we work with algebraic stacks in the smooth topology in the sense of \cite{La:s}. All stacks that we consider have locally the form ${\mathcal Y}/G$ where ${\mathcal Y}$ is a scheme and $G$ is an algebraic group acting on it. We will use the following notion of perverse sheaves on such algebraic stacks: for ${\mathcal V}={\mathcal Y}/G$, a perverse sheaf on ${\mathcal V}$ is just a $G$--equivariant perverse sheaf on ${\mathcal Y}$, approprietly shifted. Throughout this paper, for an $\Fq$--scheme (resp., for an $\Fq$--stack) ${\mathcal V}$ and for an algebra $R$ over $\Fq$, ${\mathcal V}(R)$ will denote the set of $R$--points of ${\mathcal V}$ (resp., the set of isomorphism classes of $R$--points of ${\mathcal V}$). In most cases, schemes and stacks are denoted by script letters and their sets of ${\mathbb F}_q$--points are denoted by the corresponding roman letters (e.g., ${\mathcal V}$ and $V$). We use the same notation for a morphism of stacks and for the corresponding map of sets. If ${\mathcal S}$ is a sheaf or a complex of sheaves on a stack ${\mathcal V}$, then the corresponding set $V$ is endowed with a function of ``alternating sum of traces of the Frobenius'' (as in \cite{De}) which we denote by the corresponding roman letter $S$ (we assume that a square root of $q$ in $\Ql$ is fixed throughout the paper). If ${\mathcal V}$ is a stack over ${\mathbb F}_q$, the set $V$ is endowed with a canonical measure $\mu$, which in the case when ${\mathcal V}$ is a scheme has the property $\mu(v)=1$, $\forall v\in V$. For example, if ${{\mathcal G}}$ is a group, $\mu((pt/{{\mathcal G}})({\mathbb F}_q))=|G|^{-1}$. \subsection{Acknowledgments} We express our gratitude to J.~Bernstein and I.~Mirkovi\'c for valuable discussions and to B.~Gross for useful remarks concerning the Whittaker models. We thank A.~Beilinson and V.~Drinfeld for sharing with us their ideas and unpublished results about the affine Grassmannian, which we used in Sect.~7.2. We are indebted to the referee for valuable comments and suggestions. The research of E.~Frenkel was supported by grants from the Packard and Sloan Foundations, and by the NSF grants DMS 9501414 and DMS 9304580. The research of D.~Gaitsgory was supported by the NSF grant DMS 9304580. D.~Kazhdan was supported by the NSF grant DMS 9622742. K.~Vilonen was supported by the NSF grant DMS 9504299 and by the NSA grant MDA 90495H103. \section{Background and the Shalika-Piatetski-Shapiro construction} \subsection{Langlands conjecture} Let $k={\mathbb F}_q$ be a finite field, and let $X$ be a smooth complete geometrically connected curve over $k$. Denote by $F$ the field of rational functions on $X$. For each closed point $x$ of $X$, denote by $\K_x$ the completion of $F$ at $x$, by $\OO_x$ the ring of integers of $\K_x$, and by $\pi_x$ a generator of the maximal ideal of $\OO_x$. Let $k_x$ be the residue field $\OO_x/\pi_x \OO_x$, and $q_x = q^{\deg x}$ be its cardinality. We denote by $\A = \prod'_{x \in |X|} \K_x$ the ring of adeles of $F$ and by $\OO = \prod'_{x \in |X|} \OO_x$ its maximal compact subring. Consider the set ${\mathfrak G}_n$ of unramified and geometrically irreducible $\ell$--adic representations of the Galois group $\on{Gal}(\overline{F}/F)$ in $GL_n(\Ql)$, where $\ell$ is relatively prime to $q$, as in \cite{La1}, (1.1). Let ${\mathfrak A}_n$ be the set of cuspidal unramified automorphic functions on the group $GL_n(\A)$ -- these are cuspidal functions on the set $GL_n(F)\backslash GL_n(\A)/GL_n(\OO)$, which are eigenfunctions of the Hecke operators. Recall that for each $x \in |X|$ and $i=1,\ldots,n$, one defines the Hecke operator $T^i_x$ by the formula: $$\left( T^i_x \cdot f \right)(g) = \int_{M^i_n(\OO_x)} f(gh) dh,$$ where $$M^i_n(\OO_x) = GL_n(\OO_x) \cdot D_x^i \cdot GL_n(\OO_x) \subset GL_n(\K_x) \subset GL_n(\A),$$ $D_x^i$ is the diagonal matrix whose first $i$ entries equal $\pi_x$, and the remaining $n-i$ entries equal $1$, and $dh$ stands for the Haar measure on $GL_n(\K_x)$ normalized so that $GL_n(\OO_x)$ has measure $1$. Let $B$ be the Borel subgroup of upper triangular matrices and $T N$ be its standard Levi decomposition. Cuspidality condition means that for each proper parabolic subgroup of $GL_n$, whose unipotent radical $V$ is contained in the upper unipotent subgroup $N$, $$\int_{V(F) \backslash V(\A)} f(vg) dv = 0, \quad \quad \forall g \in GL_n(\A).$$ \begin{conj} \label{Lan} For each $\sigma \in {\mathfrak G}_n$, there exists a unique (up to a non-zero constant multiple) function $f_\sigma \in {\mathfrak A}_n$, such that for any $x \in |X|$ $$T^i_x \cdot f_\sigma = q_x^{-i(i-1)/2} \on{Tr}(\Lambda^i \sigma(\on{Fr}_x)) f_\sigma, \quad \quad i=1,\ldots,n,$$ where $T^i_x$ is the $i$th Hecke operator, and $\on{Fr}_x \in \on{Gal}(\overline{F}/F)$ is the geometric Frobenius element. \end{conj} Let $P_1 \subset GL_n$ be the subgroup \begin{equation} \label{pp} \left\{ \begin{pmatrix} g & h \\ 0 & 1 \end{pmatrix} \cond g \in GL_{n-1} \right\}. \end{equation} Following Shalika and Piatetski-Shapiro, we construct a function $f'_\sigma$ on the double-quotient $$P_1(F)\backslash GL_n(\A)/GL_n(\OO),$$ which is cuspidal and which satisfies the Hecke eigenfunction property: \begin{equation} \label{hep} T^i_x \cdot f'_\sigma = q_x^{-i(i-1)/2} \on{Tr}(\Lambda^i \sigma(\on{Fr}_x)) f'_\sigma, \quad \quad i=1,\ldots,n, \forall x\in |X|. \end{equation} The first step in the construction of $f'_\sigma$ is the construction of the Whittaker function. \subsection{Whittaker functions} Introduce the following notation: for a homomorphism $\on{Spec} R \arr X$ and an $\OO_X$--module $M$ we denote by $M_R$ the $R$--module of sections of the pull-back of $M$ to $\on{Spec} R$. Denote by $\Omega$ the canonical bundle over $X$. Let $GL_n^J(R)$ be the group of invertible $n \times n$ matrices $A = (A_{ij})_{0\leq i,j \leq n-1}$, where $A_{ij} \in \Omega_R^{j-i}$. The group $GL_n^J(R)$ is locally (for the Zariski topology) isomorphic to the corresponding non-twisted group $GL_n(R)$. To establish such an isomorphism, one has to choose a non-vanishing regular section $\delta$ of $\Omega$, so the isomorphism is not canonical. It is easy to see that $GL_n^J(R)$ is the group of $R$--points of a group scheme over $X$, but in this paper we will not use this fact. We denote by $N^J(R), T^J(R), P_1^J(R)$, etc., the corresponding subgroups of $GL_n^J(R)$. The twisted forms $GL_n^J(R)$ have the following advantage. Let $u_{i,i+1}$ the $i$th component of the image of $u \in N^J(R)$ in $N^J(R)/[N^J(R),N^J(R)]$ corresponding to the $(i,i+1)$ entry of $u$. Then $u_{i,i+1} \in \Omega_R$. Let us fix once and for all a non-trivial additive character $\psi: k \arr \Ql^\times$. We define the character $\Psi_x$ of $N^J(\K_x)$ by the formula $$\Psi_x(u) = \prod_{i=1}^{n-1} \psi(\on{Tr}_{k_x/k}(\on{Res}_x u_{i,i+1})),$$ and the character $\Psi$ of $N^J(\A)$ by $$\Psi((u_x)) = \prod_{x \in |X|} \Psi_x(u_x).$$ It follows that $\Psi(u)=1$ if $u \in N^J(\OO)$ or $u \in N^J(F)$. For each $x\in |X|$ consider the group $GL_n^J(\K_x)$. Let $\gamma$ be a semi-simple conjugacy class in $GL_n(\Ql)$. The following result is due to Shintani \cite{Shi}, and Casselman and Shalika \cite{CS}. \begin{thm} \label{wgamma} (1) {\em There exists a unique function $W_{\gamma,x}$ on $GL_n^J(\K_x)$ that satisfies the following properties: \begin{itemize} \item $W_{\gamma,x}(gh) = W_{\gamma,x}(g), \forall h \in GL_n^J(\OO_x)$, $W_{\gamma,x}(1)=1$; \item $W_{\gamma,x}(ug) = \Psi_x(u) W_{\gamma,x}(g), \forall u \in N^J(\K_x)$; \item $W_{\gamma,x}$ is an eigenfunction with respect to the local Hecke-operators $T^i_x$, $i=1,\ldots,n$: $$T^i_x \cdot W_{\gamma,x}=q_x^{-i(i-1)/2}\on{Tr}({\Lambda}^i(\gamma))W_{\gamma,x}.$$ \end{itemize}} (2) {\em The function $W_{\gamma,x}$ is given by the following formula. For $\lambda=(\la_1,\ldots,\la_n) \in P^+_n$, the set of dominant weights of $GL_n$ (i.e., such that $\la_1 \geq \la_2 \geq \ldots \geq \la_n$) \begin{equation} \label{cassha} W_{\gamma,x}(\on{diag}(\pi_x^{\la_1},\ldots,\pi_x^{\la_n})) = q_x^{n(\la)} \on{Tr}(\gamma,V(\la)), \end{equation} where $V(\la)$ is the irreducible representation of $GL_n(\Ql)$ of highest weight $\la$, and $n(\la) = \sum_{i=1}^n (i-1) \la_i$. For $\lambda \in \Z^n - P^+_n$, $W_{\gamma,x}(\on{diag}(\pi_x^{\la_1},\ldots,\pi_x^{\la_n})) = 0$.} \end{thm} There is a bijection between the weight lattice of $GL_n$ and the double quotient $N^J(\K_x)\backslash GL_n^J(\K_x)/GL_n^J(\OO_x)$, which maps $(\la_1,\ldots,\la_n)$ to the double coset of \newline $\on{diag}(\pi_x^{\la_1},\ldots,\pi_x^{\la_n})$. This explains the fact that $W_{\gamma,x}$ is uniquely determined by its values at the points $\on{diag}(\pi_x^{\la_1},\ldots,\pi_x^{\la_n})$. \begin{rem} \label{1} The uniqueness of the Whittaker function is connected with the fact that an irreducible smooth representation of a reductive group $G$ over a local non-archimedian field has at most one Whittaker model, see \cite{GK}. There is an explicit formula for the Whittaker function associated to an arbitrary reductive group, due to Casselman and Shalika \cite{CS}, which we will use in Sect.~5.\qed \end{rem} \medskip Now we attach to $\sigma \in {\mathfrak A}_n$ the global Whittaker function $W_\sigma$ on $GL_n^J(\A)$ by the formula \begin{equation} \label{wsigma} W_\sigma((g_x)) = \prod_{x \in |X|} W_{\sigma(\on{Fr}_x),x}(g_x). \end{equation} It satisfies: \begin{itemize} \item $W_\sigma(gh) = W_\sigma(g), \forall h \in GL_n^J(\OO)$, $W_\sigma(1)=1$; \item $W_\sigma(ug) = \Psi(u) W_\sigma(g), \forall u \in N^J(\A)$; in particular, $W_\sigma$ is left invariant with respect to $N^J(F)$; \item For all $i=1,\ldots,n$ and $x\in |X|$ we have: $$T^i_x \cdot W_{\sigma} = q_x^{-i(i-1)/2}\on{Tr}({\Lambda}^i \sigma(Fr_x))W_{\sigma}.$$ \end{itemize} \subsection{The construction of $f'_\sigma$} Let $C^\infty(GL_n^J(\A))^{N^J(\A)}_\Psi$ be the space of $\Ql$--valued smooth (see, e.g., \cite{BZ}) functions $f$ on $GL_n^J(\A)$, such that $f(ug) = \Psi(u) f(g), \forall u \in N^J(\A)$; we call such functions $(N^J(A),\Psi)$--equivariant. Let $C^\infty(GL_n^J(\A))^{P_1^J(F)}_{\on{cusp}}$ be the space of smooth functions $f$ on $GL_n^J(\A)$, which satisfy $f(pg) = f(g), \forall p \in P_1^J(F)$ and are cuspidal, i.e., for each proper parabolic subgroup of $GL_n$, whose unipotent radical $V$ is contained in $N$, $$\int_{V^J(F) \backslash V^J(\A)} f(vg) dv = 0, \quad \quad \forall g \in GL_n^J(\A).$$ The following result is the main step in constructing automorphic functions for $GL_n$. The existence of the subgroup $P_1$ plays a key role in this result, and this makes the case of $GL_n$ special. \begin{thm} \label{shaps} {\em There is a canonical isomorphism $$\phi: C^\infty(GL_n^J(\A))^{N^J(\A)}_\Psi \arr C^\infty(GL_n^J(\A))^{P_1^J(F)}_{\on{cusp}}$$ given by the formula $$(\phi(f))(g) = \sum_{y \in N_{n-1}^J(F)\backslash GL_{n-1}^J(F)} f \left( \begin{pmatrix} y & 0 \\ 0 & 1 \end{pmatrix} g \right).$$ This isomorphism commutes with the right action of $GL_n^J(\A)$ on both spaces.} \end{thm} For the proof, see \cite{Sha} and \cite{PS}. Note that \thmref{shaps} is not stated exactly in this form in \cite{Sha} but its proof can be extracted from the proof of Theorem 5.9 there. We remark that for each $g \in GL_n^J(\A)$ the sum above has finitely many non-zero terms. By construction, $W_\sigma \in C^\infty(GL_n^J(\A))^{N^J(\A)}_\Psi$. Let $f'_\sigma = \phi(W_\sigma)$. The isomorphism of \thmref{shaps} clearly preserves the spaces of right $GL_n^J(\OO)$--invariant functions and commutes with the action of the Hecke operators on them. Therefore $f'_\sigma$ is right $GL_n^J(\OO)$--invariant and satisfies formula \eqref{hep}, i.e., it is an eigenfunction of the Hecke operators with the same eigenvalues as those prescribed by the Langlands conjecture. Furthermore, uniqueness of the Whittaker function $W_\sigma$ implies that the function $f'_\sigma$ is the unique function on $GL_n^J(\A)$ satisfying the above properties (up to a non-zero constant multiple). Thus, the Langlands \conjref{Lan} is equivalent to \begin{conj} \label{precise} For each $\sigma \in {\mathfrak G}_n$, the function $f'_\sigma$ is left $GL_n^J(F)$--invariant. \end{conj} \begin{rem} There is an approach to proving this conjecture using analytic properties of the $L$--function of the Galois representation $\sigma$, see \cite{JL,JPSS,La0}, which will not be discussed here.\qed \end{rem} \subsection{Interpretation in terms of vector bundles} \label{geomint} We begin by fixing notation. Recall that $B$ is the Borel subgroup in $GL_n$ (consisting of upper triangular matrices) and $T \subset B$ is the maximal torus (consisting of diagonal matrices). Let $P$ be the maximal parabolic subgroup of $GL_n$ containing the subgroup $P_1$ of $GL_n$ defined by formula \eqref{pp}. Denote $T^J(\K_x)^+ = T^J(\K_x) \cap \on{Diag}(\OO_x)$, where $\on{Diag}(\OO_x)$ is the set of diagonal $n \times n$ matrices with coefficients in $\OO_x$, $B^J(\K_x)^+ = N^J(\K_x) T^J(\K_x)^+$ and $$P^J(\K_x)^+ = \left\{ \begin{pmatrix} a & b \\ 0 & c \end{pmatrix} \cond a \in GL_{n-1}^J(\K_x), c \in \K_x^\times, |c| \leq 1 \right\}.$$ Let $B^J(\A)^+ = \prod'_{x \in |X|} B^J(\K_x)^+$ and $P^J(\A)^+ = \prod'_{x \in |X|} P^J(\K_x)^+$. Denote by $Q$ the double quotient $N^J(F)\backslash B^J(\A)^+/B^J(\OO)$. Note that since $$B^J(\A)/B^J(\OO) \simeq GL_n^J(\A)/GL_n^J(\OO),$$ $Q$ is naturally a subset of $N^J(F)\backslash GL_n^J(\A)/GL_n^J(\OO)$. Let $M'_n$ denote the double quotient $P_1^J(F)\backslash P^J(\A)^+/P^J(\OO)$. Note that since $$P^J(\A)/P^J(\OO) \simeq GL_n^J(\A)/GL_n^J(\OO),$$ $M'_n$ is naturally a subset of $P_1^J(F)\backslash GL_n^J(\A)/GL_n^J(\OO)$. Finally, set $M_n = GL_n^J(F)\backslash GL_n^J(\A)/GL_n^J(\OO)$. Let us denote by $\nu$ and $p$ the obvious projections $Q \arr M'_n$ and $M'_n \arr M_n$, respectively. \begin{lem} \label{easy} {\em (1)} There is a canonical bijection between the set $Q$ and the set of isomorphism classes of the following data: $\{ L,(F_i),(s_i) \}$, where $L$ is a rank $n$ vector bundle over $X$, $0=F_n \subset F_{n-1} \subset \ldots \subset F_1 \subset F_0= L$ is a full flag of subbundles in $L$, and $s_i: \Omega^i \arr L_i = F_i/F_{i+1}$ is a non-zero $\OO_X$--module homomorphism. {\em (2)} There is a canonical bijection between the set $M'_n$ and the set of isomorphism classes of pairs $\{ L,s \}$, where $L$ is a rank $n$ vector bundle over $X$, and $s: \Omega^{n-1} \arr L$ is a non-zero $\OO_X$--module homomorphism. The natural projection $\nu: Q\arr M'_n$ sends $\{ L,(F_i),(s_i) \} \in Q$ to $\{ L,s_{n-1} \} \in M'_n$. {\em (3)} There is a canonical bijection between the set $M_n$ and the set of isomorphism classes of rank $n$ vector bundles over $X$. The natural map $p: M'_n \arr M_n$ corresponds to forgetting the section $s$. \end{lem} \begin{proof} Recall that for a morphism $\on{Spec} R \arr X$ and an $\OO_X$--module $M$ we denote by $M_R$ the space of sections of the pull-back of $M$ to $\on{Spec} R$. Denote by $J^0$ the vector bundle $\oplus_{i=0}^{n-1} \Omega^i$. Let $Bun$ be the set of data $\{ L,\vf_{\gen},\vf_x \}$, where $L$ is a rank $n$ bundle on $X$, and $\vf_{\gen}: J^0_F \arr L_F$ and $\vf_x: J^0_{\OO_x} \arr L_{\OO_x}, \forall x \in |X|$ are isomorphisms (generic and local ``trivializations'', respectively). We construct a map $b: Bun \arr GL_n^J(\A)$ as follows. After the identifications of $J^0_F \otimes_F \K_x \simeq J^0_{\OO_x} \otimes_{\OO_x} \K_x$ with $J^0_{\K_x}$, and of $L_F \otimes_F \K_x \simeq L_{\OO_x} \otimes_{\OO_x} \K_x$ with $L_{\K_x}$, $\vf_x$ and $\vf_{\gen}$ give rise to homomorphisms $J^0_{\K_x} \arr L_{\K_x}$ which we denote by the same characters. Let $\mu_x = (\vf_x)^{-1} \vf_{\gen}$ be the corresponding automorphism of $J^0_{\K_x}$. To represent the element $\mu_x$ by an $n \times n$ matrix $g_x = (g_{x,ij})$ of the form given in Sect.~2.2, we set $g_{x,ij}$ to be equal to the element of $\Omega^{j-i}_{\K_x}$ corresponding to the map $\Omega^i_{\K_x} \arr J^0_{\K_x} \stackrel{\mu_x}{\longrightarrow} J^0_{\K_x} \arr \Omega^j_{\K_x}$. Thus, $g_x$ is the transpose\footnote{taking the transpose has some advantages, in particular, it agrees with the conventions adopted in \cite{La1}} of the matrix representing the action of $\mu_x$ on $J^0_{\K_x}$. The map $b$ sends $\{ L,\vf_{\gen},\vf_x \}$ to $(g_x)_{x \in |X|} \in GL_n^J(\A)$. It is easy to see that this map is a bijection. Now to prove part (1) of the lemma, let us observe that given a triple $\{ L,(F_i),(s_i) \}$, we can choose $\vf_{\gen}$ and $\vf_x$'s in such a way that for each $j=0,\ldots,n-1$, they map $\oplus_{i=j}^{n-1} \Omega^i_R$ to $F_{j,R} \subset L_R$ and the associated maps $\Omega^j_R \arr F_{j,R}/F_{j+1,R}$ coincide with restrictions of $s_j$ to $\on{Spec} R$ (here $R=F$ or $\OO_x$). With such a choice, $g_x \in B^J(\K_x)^+, \forall x \in |X|$, and the arbitrariness in the choice of $\vf_{\gen}$ (resp., $\vf_x$) corresponds to left (resp., right) multiplication of $(g_x)_{x \in |X|}$ by elements of $N^J(F)$ (resp., $B^J(\OO_x)$). This proves part (1) of the lemma. The proof of parts (2) and (3) is similar. \end{proof} Note that the function $W_\sigma$ (resp., $f'_\sigma$), which is defined on the set \newline $N^J(F)\backslash GL_n^J(\A)/GL_n^J(\OO)$ (resp., $P_1^J(F)\backslash GL_n^J(\A)/GL_n^J(\OO)$) is uniquely determined by its restriction to the subset $Q$ (resp., $M'_n$), since it is an eigenfunction of the Hecke operators $T^n_x$. Now $f'_\sigma$ is a function on $P_1^J(F)\backslash GL_n^J(\A)/GL_n^J(\OO)$. Its restriction to $M'_n$, which we also denote by $f'_\sigma$, equals, by definition, $\nu_{!}(W_\sigma)$, where $\nu_{!}$ denotes the operation of summing up a function along the fibers of the map $\nu$ (note that these fibers are finite). \conjref{precise} can now be stated in the following way. \begin{conj} \label{precise1} The function $f'_\sigma$ is constant along the fibers of the map $p: M'_n \arr M_n$. \end{conj} In the next section we discuss a geometric version of this conjecture. \section{Conjectural geometric construction of an automorphic sheaf} \subsection{Definitions of stacks} Let ${\mathcal M}_n$ be the moduli stack of rank $n$ bundles on $X$. Recall that for an $\Fq$--scheme $S$, $\on{Hom}(S,{\mathcal M}_n)$ is the grouppoid, whose objects are rank $n$ bundles on $X \times S$ and morphisms are isomorphisms of such bundles. Let ${\mathcal M}'_n$ be the moduli stack of pairs $\{ L,s \}$, where $L$ is a rank $n$ bundle on $X$ and $s: \Omega^{n-1} \arr {\mathcal M}'_n$ is an embedding of $\OO_X$--modules. More precisely, $\on{Hom}(S,{\mathcal M}'_n)$ is the grouppoid, whose objects are pairs $\{ L_S,s_S \}$, where $L_S \in \on{Ob}\on{Hom}(S,{\mathcal M}_n)$ and $s_S: \Omega_X^{n-1} \boxtimes \OO_S \to L_S$ is an embedding, such that the quotient $L_S/\on{Im} s_S$ is $S$--flat; morphisms are isomorphisms of such pairs which make the natural diagram commutative. The set $M_n$ (resp. $M'_n$) can be identified with the set of ${\mathbb F}_q$--points of $\M_n$ (resp. $\M'_n$). As was explained in the introduction, we expect that $f'_\sigma$ is the function attached to a complex of $\ell$--adic sheaves ${\mathcal S}'\si$ on $\M'_n$. In this section we present the construction of a candidate for the complex ${\mathcal S}'\si$ following Laumon \cite{La2}. At the level of $\Fq$--points, this construction is actually different from the construction of $f'_\sigma$ given in Sect.~2.3. The reason is the following. It is easy to define a ``naive'' stack ${\mathcal Q}$ classifying triples $\{ L,(F_i),(s_i) \}$ (as in \lemref{easy}) with ${\mathcal Q}(\Fq)=Q$ and a morphism ${\mathcal Q} \arr {\mathcal M}'_n$ corresponding to the map of sets $\nu: Q \arr M'_n$. But this ${\mathcal Q}$ is a disjoint union of connected components labeled by the $n$--tuples $(d_0,\ldots,d_{n-1})$, where $d_i$ is the degree of the divisor of zeros of the map $s_i: \Omega^i \arr F_i/F_{i+1}$. On the other hand, the stack ${\mathcal M}'_n$ is a disjoint union of connected components corresponding to the degree of the divisor of zeros of $s: \Omega^{n-1} \arr L$. Recall that under $\nu$, $s_{n-1}$ becomes $s$. This means that the fibers of $\nu$ are disconnected. Hence one can not obtain an irreducible sheaf on ${\mathcal M}'_n$ as the direct image of a sheaf on ${\mathcal Q}$. In this section we replace the ``naive'' stack ${\mathcal Q}$ by a stack $\wt{\mathcal Q}$, and the Whittaker function $W_\sigma$ by a perverse sheaf $\f\si$ on $\wt{\mathcal Q}$. The pair ($\wt{\mathcal Q}$,$\f\si$) was first constructed by Laumon \cite{La2}. \subsection{The stack $\wt{\mathcal Q}$} The algebraic stack $\wt{\mathcal Q}$ is defined as follows. For an $\Fq$--scheme $S$, $\on{Hom}(S,\wt{\mathcal Q})$ is the grouppoid, whose objects are quintuples $\{L_S,\s_S,J_S,(J_{i,S}),(s_{i,S})\}$, where $L_S$ and $J_S$ are rank $n$ bundles on $X \times S$, $\s_S: J_S \arr L_S$ is an embedding of the corresponding $\OO_{X \times S}$--modules, such that the quotient is $S$--flat, $(J_{i,S})$ is a full flag of subbundles $$0=J_{n,S} \subset J_{n-1,S} \subset \ldots J_{1,S} \subset J_{0,s} =J_S,$$ and $s_{i,S}$ is an isomorphism ${\Omega}^i_X \boxtimes \OO_S \simeq J_{i,S}/J_{i+1,S}, i=0,\ldots,n-1$. Morphisms are isomorphisms of the corresponding $\OO_{X \times S}$--modules making all natural diagrams commutative. We have a natural representable morphism of stacks $\wt{\nu}:\wt{\mathcal Q}\to\M'_n$, which for each $\Fq$--scheme $S$ maps $\{L_S,\s_S,J_S,(J_{i,S}),(s_{i,S})\}$ to the pair $\{L_S,\s_S \circ s_{n-1,S}\}$, where $s_{n-1,S}$ is viewed as an embedding of ${\Omega}^n_X \boxtimes \OO_S$ into $J_S$. Let $\wt{Q} = \wt{\mathcal Q}({\mathbb F}_q)$ be the set of $\Fq$--points of $\wt{\mathcal Q}$ (see Sect.~1.7). By definition, it consists of quintuples $\{L,\s,J,(J_i),(s_i)\}$, where $L$ and $J$ are rank $n$ bundles on $X$, $\s:J \arr L$ is an embedding of the corresponding $\OO_X$--modules, $(J_i)$ is a full flag of subbundles $$0=J_n \subset J_{n-1} \subset \ldots J_1 \subset J_0 =J,$$ and $s_i$ is an isomorphism $s_i: {\Omega}^i \simeq J_i/J_{i+1}, i=0,\ldots,n-1$. There is a natural map of sets $r:\wt{Q}\to Q$ defined as follows. Given an object $\{L,\s,J,(J_i),(s_i)\}$, define $F_i$ to be the maximal locally free submodule of $L$ of rank $n-i$, which contains the image of $J_i \subset J$ under $\mathbf{s}$. Then $(F_i)$ is a full flag of subbundles of $L$. The composition of $s_i: {\Omega}^i \to J_i/J_{i+1}$ with the natural map $J_i/J_{i+1}\to F_i/F_{i+1}$ induced by $\s$ is an $\OO$--module homomorphism $s'_i: \Omega^i \arr F_i/F_{i+1}$ for each $i=0,\ldots,n-1$. Then $\{ L,(F_i),(s'_i) \}$ is a point of $Q$. Thus we obtain a map $r: \wt{Q} \arr Q$. \begin{lem} \label{r1} The composition $\nu\circ r:\wt{Q}\to M'_n$ coincides with the map $\wt{\nu}$. Moreover for every function $f$ on $\wt{Q}$ we have $\nu_{!}(r_{!}(f))=\wt{\nu}_{!}(f)$, the integrations being taken with respect to the canonical measures on each of the three sets. \end{lem} \subsection{The sheaf ${\mathcal L}_E$} In this section we recall Laumon's construction \cite{La1} of the sheaf ${\mathcal L}_E$. Let $\Coh$ be the stack classifying torsion sheaves of finite length on $X$, i.e., for an $\Fq$--scheme $S$, $\on{Hom}(S,\Coh)$ is the grouppoid, whose objects are coherent sheaves ${\mathcal T}_S$ on $X \times S$, which are finite and flat over $S$ (see \cite{La1}\footnote{the notation used in \cite{La1} is $\Coh^0$; we suppress the upper index $0$ to simplify notation}). Let $\Cohn$ be the open substack of $\Coh$ that classifies torsion sheaves that have at most $n$ indecomposable summands supported at each point. The stack $\Cohn$ can be understood as follows. Let $K$ be a field containing $\Fq$, and let ${\mathcal T}\in \Coh(K)$. We have a (non-canonical) isomorphism \begin{equation} \label{torsion} {\mathcal T} \simeq {\mathcal O}_{X_K}/{\mathcal O}_{X_K}(-D_1) \oplus \dots \oplus {\mathcal O}_{X_K}/{\mathcal O}_{X_K}(-D_h), \end{equation} where $X_K = \on{Spec} K \times_{\on{Spec} \Fq} X$, and $D_1\geq D_2 \geq \dots \geq D_h$ is a decreasing sequence (uniquely determined by ${\mathcal T}$) of effective divisors on $X_K$. The torsion sheaf ${\mathcal T}$ belongs to $\Cohn(K)$ precisely when $h\leq n$. Let $S$ be an $\Fq$--scheme. Then a torsion sheaf ${\mathcal T}_S\in\Coh(S)$ belongs to $\Cohn(S)$ if it does so at every closed point of $S$. The stack $\Cohn$ is a disjoint union of connected components $\Cohn=\bigcup_{m\in{\mathbb Z}_+} \Coh_{n,m}$, where the component $\Coh_{n,m}$ classifies torsion sheaves of degree $m$; the degree of the torsion sheaf ${\mathcal T}$ in \eqref{torsion} is $\sum_i \on{deg} (D_i)$. Each $\Coh_{n,m}$ has an open substack $\Coh_{n,m}^{rss}$ classifying regular semi-simple torsion sheaves: an $S$--point ${\mathcal T}_S$ of $\Cohn$ is said to be a point of $\Coh_{n,m}^{rss}$ if for any $\Fqb$--point of $S$ the corresponding sheaf over $X\underset{\on{Spec}\Fq}\times\on{Spec}\Fqb$ is isomorphic to $\oplus \OO_X/\OO_X(-x_i)$, where the points $x_i$ are distinct. Let $X^{(m)}$ denote the $m$th symmetric power of $X$ and let $X^{(m),rss}$ be the complement to the divisor of diagonals in $X^{(m)}$. We have a smooth map $X^{(m),rss}\to \Coh_{n,m}^{rss}$. When we make base change from $\Fq$ to $\Fqb$, $\Coh_{n,m}^{rss}$ can be identified with the quotient of $X^{(m)}$ by $GL_1^m$ with respect to the trivial action. Let us consider the rank $n$ local system $E=E_\sigma$ on $X$ corresponding to the Galois representation $\sigma$. Let us also write $\pi: X^m \arr X^{(m)}$ for the natural projection. Define the $m$th symmetric power $E^{(m)}$ of $E$ as the sheaf of invariants of $\pi_* E^{\boxtimes m}$ under the natural action of the symmetric group $S_m$, i.e., $E^{(m)} = (\pi_* E^{\boxtimes m})^{S_m}$. The restriction $E^{(m)}|_{X^{(m),rss}}$ clearly descends to a local system ${\mathcal L}_{E,m}^0$ on $\Coh_{n,m}^{rss}$, since it does over the algebraic closure of $\Fq$. We define the perverse sheaf ${\mathcal L}_E$ as the sheaf on $\Coh_n$, whose restriction to each $\Coh_{n,m}$ is the Goresky-MacPherson extension of ${\mathcal L}_{E,m}^0$, i.e., ${\mathcal L}_E|_{\Coh_{n,m}} = j_{!*}{\mathcal L}_{E,m}^0$, where $j: \Coh_{n,m}^{rss} \hookrightarrow\Coh_{n,m}$. We will need an explicit description of the function $L_E$ corresponding to ${\mathcal L}_E$, on the set $Coh_{n,m}$. Let $x\in|X|$ be a closed point with residue field $k_x$. Then we can regard $x$ as a $k_x$--rational point of $X$. Recall from Sect.~2.1 that we denote by $q_x$ the cardinality of $k_x$, and that $q_x=q^{\deg x}$. We denote by $\Coh_{n,m}(x)$ the algebraic stack over $k_x$ that classifies torsion sheaves of degree $m$ on $X\underset{\on{Spec}\Fq}\times\on{Spec} k_x$ supported at $x$ that have at most $n$ indecomposable summands. Obviously, $\Coh_{n,m}(x)$ is a locally closed sub-stack of $\Coh_{n,m}\underset{\on{Spec}\Fq}\times\on{Spec} k_x$; we denote by $I_{m,x}$ the corresponding embedding. Let ${\mathcal L}_{E,m,x}$ be the pull-back to $\Coh_{n,m}(x)$ of the sheaf ${\mathcal L}_{E,m}$ under the composition $$\Coh_{n,m}(x)\overset{I_{m,x}}\longrightarrow \Coh_{n,m}\underset{\on{Spec}\Fq}\times\on{Spec} k_x \to\Coh_{n,m}.$$ We will denote the corresponding function by $L_{E,m,x}$. In what follows, we fix, once and for all, a geometric point $\bar x$ over each closed point $x \in |X|$. We denote by $E_x$ the stalk of $E$ at $\bar{x}$. Let $P^{++}_{n,m}$ be the set $\{ \la=(\la_1,\la_2,\ldots,\la_n) \cond \la_i \in \Z, \la_1\geq \la_2\geq \ldots\geq \la_n \geq 0, \sum_{i}\la_i=m \}$. We can consider $P^{++}_{n,m}$ as a subset of the set $P^+_n$ of dominant weights of $GL_n$. For $\lambda\in P^{++}_{n,m}$, we write $E_x(\la)$ for the representation of $GL_n(\Ql)\cong GL(E_x)$ of highest weight $\la$. The stack $\Coh_{n,m}(x)$ has a stratification by locally closed substacks $\Coh_{n,m}^{\la}(x)$ indexed by $\la\in P^{++}_{n,m}$. The stratum corresponding to $\la=(\la_1,\la_2,\ldots,\la_n)\in P^{++}_{n,m}$ parametrizes torsion sheaves of the form ${\mathcal T} \simeq \OO_X/\OO_X(-\la_1 x) \oplus\dots\oplus\OO_X/\OO_X(-\la_n x)$. Let ${\mathcal B}_{\la,x}$ denote the intersection cohomology sheaf associated to the constant sheaf on the stratum $\Coh_{n,m}^{\la}(x)$. Let ${\mathcal T}$ be an $\Fq$--valued point of $\Coh_{n,m}$ and let $x\in|X|$ be a closed point with residue field $k_x$. The pull-back of ${\mathcal T}$ to a sufficiently small Zariski neighborhood of $x$ in $X\underset{\on{Spec}\Fq}\times\on{Spec} k_x$ gives rise to an object ${\mathcal T}_x$ of $\Coh_{n,m_x}(x)$, i.e., to a $k_x$--rational point of $\Coh_{n,m_x}(x)$, for some $m_x$. We have: $\sum_{x\in|X|} m_x\cdot\deg(x) = m$. Moreover, there is a bijection: \begin{equation} \label{bijec} \Cohn(\Fq)=\prod_{x\in|X|} {}\!\!^{'} \; \; \Coh_n(x)(k_x). \end{equation} The explicit description of $L_E$ is given in the following proposition. Note that the shifts in degrees are due to the fact that the stack $\Coh_{n,m}(x)$ has dimension $-m$, whereas the dimension of $\Coh_{n,m}$ is zero. \begin{prop}[\cite{La1},(3.3.8)] \label{descr} {\em(1)} Let ${\mathcal T}$ be an $\Fq$--point of $\Coh_{n,m}$. Then, using the notation above, we have: $$L_{E,m}({\mathcal T})=\prod_{x\in|X|} L_{E,m_x,x}({\mathcal T}_x).$$ {\em(2)} Furthermore, $${\mathcal L}_{E,m,x} \simeq \oplus_{\la\in P^{++}_{n,m}} {\mathcal B}_{\la,x}[m]\left( -n(\la) \right) \otimes E_x(\la),$$ where $n(\la)=\sum_{i=1}^n (i-1) \la_i$. \end{prop} \begin{rem} \label{3} Let $x$ be an $\Fq$-rational point of $X$. Consider now the variety ${\mathcal N}_m\subset \mathfrak{gl}_m$ of nilpotent matrices. The stack $\Coh_{m,m}(x)$ is isomorphic to the stack ${\mathcal N}_m/GL_m$, where $GL_m$ acts on ${\mathcal N}_m$ by conjugation. Let $\pi:\wt{\mathcal N}_m \to {\mathcal N}_m$ denote the Springer resolution and let ${{\mathcal S}p}_m= R\pi_*\Ql$ denote the Springer sheaf on ${\mathcal N}_m$. The sheaf ${{\mathcal S}p}_m$ has a natural action of the symmetric group $S_m$. It is shown in \cite{La1} that $${\mathcal L}_{E,m,x} \simeq \left( {{\mathcal S}p}_m \otimes (E_x)^{\otimes m} \right)^{S_m}|_{\Coh_{n,m}}$$ (note that $\Coh_{n,m}$ is an open substack of $\Coh_{m,m}$). Hence the function $L_E$ associated to ${\mathcal L}_E$ can be expressed via the Kostka-Foulkes polynomials, see \cite{La1}. However, it will be more convenient for us to use another interpretation of the sheaf ${\mathcal L}_{E,m,x}$, via the affine Grassmannian (see Sect.~4.2). This interpretation allows us to express $L_E$ in terms of the Hecke algebra ${\mathcal H}(GL_n(\K),GL_n(\OO))$, see Sect.~5.5. The fact that the two interpretations agree is due to Lusztig \cite{Lu1}.\qed \end{rem} \subsection{The sheaf $\f\si$} Define a morphism of stacks $\alpha: \wt{\mathcal Q}\to \Cohn$ that sends a quintuple $\{L_S,\s_S,J_S,(J_{i,S}),(s_{i,S})\}$ to the sheaf $L_S/\on{Im} \s_S$. Now we define a morphism $\beta: \wt{\mathcal Q} \to \Gaf$, which at the level of $\Fq$--points sends $\{L,\s,J,(J_i),(s_i)\}$ to the sum of $n-1$ classes in $$\Fq \simeq Ext^1(\Omega^i,\Omega^{i-1}) \simeq Ext^1(J_i/J_{i+1},J_{i-1}/J_i)$$ that correspond to the successive extensions $$0\to J_{i-1}/J_i \to J_{i-1}/J_{i+1}\to J_i/J_{i+1}\to 0.$$ Given two coherent sheaves $L$ and $L'$ on $X$, consider the stack ${\mathcal E}xt^1(L',L)$, such that the objects of the grouppoid $\on{Hom}(S,{\mathcal E}xt^1(L',L))$ are coherent sheaves $L''$ on $X \times S$ together with a short exact sequence $$0 \arr L \boxtimes \OO_S \arr L'' \arr L' \boxtimes \OO_S \arr 0,$$ and morphisms are morphisms between such exact sequences inducing isomorphisms at the ends. There is a canonical morphism of stacks ${\mathcal E}xt^1(L',L) \arr Ext^1(L',L)$. We have for each $i=1,\ldots,n-1$, a natural morphism $\beta_i: \wt{\mathcal Q} \arr {\mathcal E}xt^1(\Omega^i,\Omega^{i-1})$, as above. Now $\beta$ is the composition $$\wt{\mathcal Q} \arr \prod_{i=1}^{n-1} {\mathcal E}xt^1(\Omega^i,\Omega^{i-1}) \arr \prod_{i=1}^{n-1} Ext^1(\Omega^i,\Omega^{i-1}) \arr \Gaf^{n-1} \arr \Gaf.$$ Let ${\mathcal I}_\psi$ be the Artin-Schreier sheaf on $\Gaf$ corresponding to the character $\psi$. Recall that the Galois representation $\sigma$ gives rise to a rank $n$ local system $E$ on $X$ and to the sheaf ${\mathcal L}_E$ on $\Cohn$. Define the sheaf $\f\si$ on $\wt{\mathcal Q}$ as $$\f\si:=\alpha^*({\mathcal L}_E)\otimes \beta^*({\mathcal I}_\psi).$$ Note that $\wt{\mathcal Q}$ is an open sub-stack in a vector bundle over the product of $\Cohn$ and a smooth stack that classifies extensions $J$ as above. Hence the map $\alpha$ is smooth, and $\f\si$ is the Goresky-MacPherson extension from its restriction to the open substack $\alpha^{-1}(\Cohn^{rss})$. \subsection{Geometric Langlands conjecture for $GL_n$} Recall that we have a representable morphism of stacks $\wt{\nu}:\wt{\mathcal Q}\to\M'_n$ that associates to an object $\{L,\s,J,(J_i),(s_i)\}$, the pair $\{L,\s \circ s_{n-1}\}$. We define the complex of $\ell$--adic sheaves ${\mathcal S}'\si$ on $\M'_n$ to be the direct image $${\mathcal S}'\si:=\wt{\nu}_{!}(\f\si).$$ The following conjecture of Laumon is a geometric version of \conjref{precise1}. \begin{conj}[\cite{La2}] \label{princ} Let $\sigma$ be in ${\mathfrak G}_n$ and $E$ be the corresponding irreducible $\ell$--adic local system on $X$. Then \begin{itemize} \item The restriction of the complex ${\mathcal S}'\si$ to each connected component of ${\mathcal M}'_n$ is an irreducible perverse sheaf up to a shift in degree. \item ${\mathcal S}'\si\simeq p^*({\mathcal S}\si)$, where $p$ is the natural morphism $\M'_n\to \M_n$, and ${\mathcal S}\si$ is a complex of sheaves on $\M_n$, whose restriction to each connected component of ${\mathcal M}_n$ is an irreducible perverse sheaf up to a shift. \item The sheaf ${\mathcal S}_E$ is an eigensheaf of the Hecke correspondences in the sense of \cite{La1}, (2.1.1). \end{itemize} \end{conj} If this conjecture is true, then the function on $M_n$ associated to the sheaf ${\mathcal S}_E$ is the automorphic function $f_\sigma$ corresponding to $\sigma$. The sheaf ${\mathcal S}_E$ can therefore be called the automorphic sheaf corresponding to $\sigma$. The conjecture means that the sheaf ${\mathcal S}'_E$ is constant along the fibers of the morphism $p$. Thus, it is analogous to \conjref{precise1}. The advantage of dealing with \conjref{princ} as compared to \conjref{precise1} is that while the latter is a global statement, one could use local geometric information about the sheaf ${\mathcal S}'_E$ to tackle \conjref{princ} (as Drinfeld did in the case of $GL_2$ \cite{Dr}). \begin{rem} The above conjecture is obviously false if one does not assume the irreducibility of $\sigma$ (the complex ${\mathcal S}'\si$ must be corrected by the corresponding ``constant terms'' in this case). However one can construct the automorphic sheaves ${\mathcal S}\si$ corresponding to $\sigma$'s, which are direct sums of one-dimensional representations, by means of the geometric Eisenstein series \cite{La:e}.\qed \end{rem} \subsection{Main theorem} Let $S'\si$ denote the function on $M'_n$ associated to ${\mathcal S}'\si$. If \conjref{princ} is true, then this function has the same properties as the function $f'_\sigma$ defined in Sect.~2.3. Recall that these properties uniquely determine $f'_\sigma$ up to a non-zero factor. Therefore \conjref{princ} can be true only if the functions $S'_E$ and $f'_\sigma$ are proportional. This was conjectured by Laumon in \cite{La2} (Conjecture 3.2). One of our motivations was to prove this conjecture. More precisely, we prove the following: \begin{thm} \label{prin} {\em The functions $S'\si$ and $f'_\sigma$ are equal.} \end{thm} \thmref{prin} means that the function $f'_\sigma$ does come from a complex of $\ell$--adic sheaves on ${\mathcal M}'_n$. It also provides a consistency check for \conjref{princ}. Laumon has proved \thmref{prin} in \cite{La1} for $GL_2$ by a method different from the one we use below. We derive \thmref{prin} for $GL_n$ with arbitrary $n$ from the following statement. \begin{prop} \label{r} Let $F\si$ be the function on $\wt{Q}$ corresponding to $\f\si$. The function $r_{!}(F\si)$ coincides with the restriction of the Whittaker function $W_\sigma$ to $Q$. \end{prop} \thmref{prin} immediately follows from \propref{r} because of \lemref{r1} as shown on the diagram below. \setlength{\unitlength}{1mm} \begin{center} \begin{picture}(65,60)(-30,-55) \put(-4,0){\vector(-1,0){20}} \put(4,0){\vector(1,0){20}} \put(0,-3.5){\vector(0,-1){20}} \put(0,-32){\vector(0,-1){20}} \put(2.5,-2.5){\vector(1,-1){8.5}} \put(11,-15.5){\vector(-1,-1){8.5}} \put(-1.5,-1){$\widetilde{Q}$} \put(-30,-1){${\mathbb F}_q$} \put(-2,-29){$M'_n$} \put(-2,-56){$M_n$} \put(12,-14){$Q$} \put(26,-1){${Coh}_n$} \put(-15,2){$\beta$} \put(13,2){$\alpha$} \put(-4,-14){$\widetilde{\nu}$} \put(-4,-43){$p$} \put(8,-6){$r$} \put(8,-21){$\nu$} \end{picture} \end{center} \bigskip The proof of \propref{r} will occupy Sects.~4 and 5 below. \propref{r} can be interpreted in the following way. Let $Coh_n$ (resp., $Coh_{n,m}(x)$) be the set of $\Fq$--points (resp., $k_x$--points) of $\Coh_n$ (resp., $\Coh_{n,m}(x)$). Set $Coh_n(x) = \cup_{m\geq 0} Coh_{n,m}(x)$. Then by formula \eqref{bijec}, $Coh_n = \prod'_{x\in |X|} Coh_n(x)$. We have: $$Coh_n(x) \simeq P^{++}_n = \cup_{m\geq 0} P^{++}_{n,m}.$$ Hence we can identify $Coh_n(x)$ with the set $$\{ \on{diag}(\pi_x^{\la_1},\ldots,\pi_x^{\la_n})|\la_1 \geq \ldots \geq \la_n \geq 0 \}.$$ The Whittaker function $W_\sigma$ can then be restricted to the $Coh_n$, and it is uniquely determined by this restriction, see Sect.~2.2. Thus, both $L_E$ and $W_\sigma$ give rise to functions on $Coh_n$. For each point $t \in Coh_n$, the value of $L_E$ at $t$ is given by taking the alternating sum of traces of the Frobenius on the stalk cohomologies of ${\mathcal L}_E$ at $t$. On the other hand, the value of $W_\sigma$ is given by the trace of the Frobenius on the top stalk cohomology of ${\mathcal L}_E$ at $t$ (see \cite{La1}). Therefore \propref{r} says that the contributions of all stalk cohomologies, other than the top one, are killed by the summation along the fibers of the projection $r$ against the non-trivial character $\Psi$. \begin{rem} As we mentioned in the introduction, Drinfeld has proved a version of \conjref{princ} for $GL_2$. The case of $GL_2$ is special as explained below. Let $\wt{\mathcal M}'_2$ be the open substack of ${\mathcal M}'_2$, which parametrizes $\{ L,s \}$, such that the image of $s: \Omega \arr L$ is a maximal invertible subsheaf of $L$. Due to the Hecke eigenfunction property of $f'_\sigma$ with respect to $T^2_x$, $f'_\sigma$ is uniquely determined by its restriction to $\wt{M}'_2 = \wt{\mathcal M}'_2(\Fq)$. But the map $r$ is a bijection over $\nu^{-1}\left( \wt{M}'_2 \right)$, and hence $\nu^{-1} \left( \wt{M}'_2 \right)$ can be considered as a subset of $\wt{Q}$. Clearly, the map $\alpha$ of Sect.~3.3 sends $\nu^{-1} \left( \wt{M}'_2 \right)$ to $Coh_1 \subset Coh_2$. Furthermore, the restriction of ${\mathcal L}_E$ to the stack $\Coh_1$ is simply a sheaf, i.e., it has stalk cohomology in only one degree. Therefore on $Coh_1$ the function $L_E$ equals the Whittaker function $W_\sigma$. Hence, restricted to $\wt{\mathcal M}'_2$, the geometric construction coincides with the construction described in Sect.~2.3.\qed \end{rem} \section{Reduction to a local statement} \subsection{Adelic interpretation of $\wt{Q}$} Recall that $\wt{Q}$ is the set of isomorphism classes of ${\mathbb F}_q$--points of $\wt{\mathcal Q}$, and in Sect.~3.1 we defined a map $r:\wt{Q} \to Q$. Recall further that $Coh_n(x) \simeq P^{++}_n =\cup_{m\geq 0} P^{++}_{n,m}$, and $Coh_n = \prod'_{x\in |X|} Coh_n(x)$. Denote $GL_n^J(\K_x)^+ = GL_n^J(\K_x) \cap \on{Mat}_n^J(\OO_x)$. Let $GL_n^J(\A)^+$ be the restricted product $\prod'_{x\in |X|} GL_n^J(\K_x)^+$. We have bijections: \begin{equation} \label{iso1} GL_n^J(\OO_x) \backslash GL_n^J(\K_x)^+/GL_n^J(\OO_x) \simeq Coh_n(x). \end{equation} and \begin{equation} \label{iso2} GL_n^J(\OO) \backslash GL_n^J(\A)^+/GL_n^J(\OO) \simeq Coh_n. \end{equation} \begin{prop} \label{adelic} {\em (1)} There is a bijection between the set $\wt{Q}$ and the set $$(N^J(F)\backslash N^J(\A))\underset{N^J(\OO)}\times (GL_n^J(\A)^+/GL_n^J(\OO)).$$ The group $N^J(\OO)$ acts on the product $(N^J(F)\backslash N^J(\A))\times (GL_n^J(\A)^+/GL_n^J(\OO))$ according to the rule $y\cdot (u,g)=(u\cdot y^{-1},y\cdot g)$. {\em (2)} The map $r: \wt{Q} \arr Q$ identifies with the map $$(N^J(F)\backslash N^J(\A))\underset{N^J(\OO)}\times (GL_n^J(\A)^+/GL_n^J(\OO)) \to Q \subset N^J(F)\backslash GL_n^J(\A)/GL_n^J(\OO)$$ given by $(u,g)\to u\cdot g$. The map $\al$ sends $(u,g)$ to the image of $g$ in $GL_n^J(\OO) \backslash GL_n^J(\A)^+/GL_n^J(\OO) \simeq Coh_n$. The map $\beta: \wt{Q} \arr \Fq$ is the composition of the natural map $N^J(F) \backslash N^J(\A)/N^J(\OO) \arr \Fq^{n-1}$ and the summation $\Fq^{n-1} \arr \Fq$. \end{prop} \begin{proof} We will use the notation introduced in the proof of \lemref{easy}. Let $\{ L,\s,J,(J_i),$ $(s_i) \}$ be an element of $\wt{Q}$. Then the triple $\{ J,(J_i),(s_i) \}$ is an element of $Q$. Hence we can associate to it homomorphisms $\vf^J_x: J^0_{\K_x} \arr J_{\K_x}$, $\vf^J_{\gen}: J^0_{\K_x} \arr J_{\K_x}$ and $\mu^J_x = (\vf^J_x)^{-1} \vf^J_{\gen}: J^0_{\K_x} \arr J^0_{\K_x}$, as in the proof of \lemref{easy}. On the other hand, let $\vf_x^L$ be an isomorphism $J^0_{\OO_x} \arr L_{\OO_x}$. We extend it to a homomorphism $J^0_{\K_x} \arr L_{\K_x}$, which we denote by the same symbol. Denote by $\s_x$ the homomorphism $J_{\K_x} \arr L_{\K_x}$ induced by $\s$. Consider the automorphism $\nu_x = (\vf_x^L)^{-1} \s_x \vf_x^J$ of $J^0_{\K_x}$. Now we assign to $\{ L,\s,J,(J_i),(s_i) \}$ the element $((u_x),(g_x))$ of $GL_n^J(\A) \times GL_n^J(\A)$, where $u_x$ is the transpose of the matrix representing $\mu_x$, and $g_x$ is the transpose of the matrix representing $\nu_x$ (see the proof of \lemref{easy}). By construction, $u_x \in N^J(\K_x)$ and $g_x \in GL_n^J(\K_x)^+$. Furthermore, the arbitrariness in the choice of $\vf_{\gen}^J$ corresponds to left multiplication of $u_x$ by elements of $N^J(F)$, the arbitrariness in $\vf_x^L$ corresponds to right multiplication of $g_x$ by elements of $GL_n^J(\OO_x)$, and the arbitrariness in $\vf_x^J$ corresponds to the action of $N^J(\OO_x)$ on $(u_x,g_x)$ according to the rule $y \cdot (u_x,g_x) = (u_x \cdot y^{-1}, y \cdot g_x)$. This proves part (1) of the proposition. The proof of part (2) is now straightforward. \end{proof} Recall that $L_E$ is the function on $Coh_n$ associated to the sheaf ${\mathcal L}_E$. Denote by $L_{E,x}$ the function on $Coh_n(x)$, whose restriction to $\Coh_{n,m}(x)(\Fq)$ is the function associated to ${\mathcal L}_{E,m,x}$. Part (1) of \propref{descr} implies that $$L_E((\la_x)) = \prod_{x \in |X|} L_{E,x}(\la_x),$$ for all $(\la_x) \in \prod'_{x \in |X|} P_n^{++}$. Using the bijection \eqref{iso1} (resp., \eqref{iso2}) we consider $L_{E,x}$ (resp., $L_E$) as a function on $GL_n^J(\K_x)^+$ (resp., $GL_n^J(\A)^+$). Let $\Psi_x: N^J(\K_x)\to\Ql^{\times}$ be the character defined in Sect.~2.2. Note that the function $\Psi_x$ (resp., $L_{E,x}$) is right (resp., left) $N^J(\OO_x)$--invariant. Recall further that $F_E$ is the function on $\wt{Q}$ associated to the sheaf ${\mathcal F}_E$. We conclude: \begin{lem} \label{ide} Under the isomorphism of \propref{adelic}, $$F_E = \prod_{x\in |X|} \Psi_x \times L_{E,x}.$$ \end{lem} Let us extend the function $L_{E,x}$ by zero from $GL_n^J(\K_x)^+$ to $GL_n^J(\K_x)$. Then \propref{adelic} and \lemref{ide} imply: $$(r_! F_E)((g_x)) = \prod_{x \in |X|} \; \; \sum_{u_x \in N^J(\K_x)/N^J(\OO_x)} L_{E,x}(u_x^{-1} \cdot g_x) \Psi_x(u_x),$$ for all $g_x \in GL_n^J(\K_x)^+$ (each sum is actually finite). Let $du_x$ be the Haar measure on $N^J(\K_x)$ normalized so that $\int_{N^J(\OO_x)} du_x = 1$. Using left $GL_n^J(\OO_x)$--invariance of $L_{E,x}$, we can rewrite the last formula as \begin{equation} \label{globalint} (r_! F_E)((g_x)) = \prod_{x \in |X|} \int_{N^J(\K_x)} L_{E,x}(u_x^{-1} \cdot g_x) \Psi_x(u_x) \; du_x. \end{equation} \propref{r} states that $(r_! F_E)((g_x)) = W_\sigma((g_x))|_Q$. According to formulas \eqref{wsigma} and \eqref{globalint}, this is equivalent to the formula \begin{equation} \label{show} \int_{N^J(\K_x)} L_{E,x}(u_x^{-1} \cdot g_x) \Psi_x(u_x) \; du_x = W_{\sigma(\on{Fr}_x)}(g_x), \end{equation} for all $g_x \in GL_n^J(\K_x)^+$. Since both the left and the right hand sides of \eqref{show} are left $(N^J(\K_x),\Psi_x)$--equivariant and right $GL_n^J(\OO)$--invariant, it suffices to check formula \eqref{show} when $g_x = \on{diag}(\pi^{\nu_1},\ldots,\pi^{\nu_n})$, where $\nu = (\nu_1,\ldots,\nu_n) \in P^{++}_n$. Using the explicit formula \eqref{cassha}, we reduce \propref{r} to the following local statement. \begin{prop} \label{Fplus} \begin{equation} \label{oh} \int_{N^J(\K_x)} L_{E,x}(u_x \cdot \on{diag}(\pi_x^{\nu_1},\ldots,\pi_x^{\nu_n})) \Psi^{-1}(u_x) \; du_x = q_x^{n(\nu)} \on{Tr}(\sigma(\on{Fr}_x),E_x(\nu)), \end{equation} where $\nu \in P^{++}_n$. \end{prop} \subsection{Positive part of the affine Grassmannian} Recall that $J^0 = \oplus_{i=0}^{n-1} \Omega^i$. From now on we work in the local setting. Hence we choose once and for all a trivialization of $J^0$ on the formal neighborhood of $x \in |X|$ and identify $GL_n^J(\K_x)$ with $GL_n(\K_x)$. For this reason we suppress the index $J$ in what follows. According to part (2) of \propref{descr}, \begin{equation} \label{le} L_{E,x} = \sum_{\la \in P^{++}_n} \on{Tr}(\sigma(\on{Fr}_x),E_x(\la)) \cdot B_\la, \end{equation} where $B_\la$ is the function on $Coh_n(x)$ associated to the sheaf ${\mathcal B}_{\la,x}[m](-n(\la))$ (see Sect.~3.2). We view $B_\la$ as a $GL_n(\OO_x)$--invariant function on $GL_n(\K_x)/GL_n(\OO_x)$. Let $x$ be a closed point of $X$. To simplify notation, from now on we will assume that $x$ is an $\Fq$--rational point of $X$ (otherwise, we simply make a base change from $\Fq$ to the residue field $k_x$ of $x$). Define the functor which sends an $\Fq$--scheme $S$ to the set of isomorphism classes of pairs $\{ L,t \}$, where $L$ is a rank $n$ bundle on $X \times S$ and $t$ is its trivialization on $(X \times S) - (\{ x \} \times S)$. This functor is representable by an ind--scheme $\G_x$ (see \cite{BL1}), which we call the affine Grassmannian $\G_x$ (for the group $GL_n$). The ind--scheme $\G_x$ splits into a disjoint union of connected components: $\G_x=\cup_{m\in{\mathbb Z}} \G_x^m$ indexed by the degree of $L$ for $\{ L,t \} \in \G_x$. There is a bijection between the set $Gr_x$ of ${\mathbb F}_q$--points of $\G_x$ and the quotient $GL_n(\K_x)/GL_n(\OO_x)$, see \cite{BL}. The analogous quotient over the field of complex numbers is known as the affine, or periodic, Grassmannian. This explains the name that we use. There exists a proalgebraic group $\GG(\OO_x)$ whose set of $\Fq$--points is $GL_n(\OO_x)$. The group $\GG(\OO_x)$ acts on $\G_x$, and its orbits stratify $\G_x$ by locally closed finite-dimensional subschemes $\G_x^\la$ indexed by the set $P^+_n$ of dominant weights $\la$ of $GL_n$. The stratum $\G_x^\la$ is the $\GG(\OO_x)$--orbit of the coset $\on{diag}(\pi_x^{\la_1},\ldots,\pi_x^{\la_n}) \cdot GL_n(\OO_x) \in Gr_x$. Recall that there is an inner product on the set of $GL_n$ weights defined by the formula $(\la,\mu) = \sum_{i=1}^n \la_i \mu_i$, and that $\dim \G_x^\la = 2(\la,\rho)$, where $\rho$ is the half sum of the positive roots of $GL_n$, $\rho = ((n-1)/2,(n-3)/2,\ldots,-(n-1)/2)$. Let $\ol{\mathbb Q}_{\ell,\la}$ be the constant sheaf supported on the stratum $\G_x^\la$. Denote by ${\mathcal A}_{\la,x}$ the intersection cohomology sheaf on the closure of $\G_x^\la$, which is the Goresky-MacPherson extension of the sheaf $\ol{\mathbb Q}_{\ell,\la}[2(\la,\rho)]((\la,\rho))$. Let $A_\la$ be the $GL_n(\OO_x)$--invariant function on $GL_n(\K_x)/GL_n(\OO_x)$, which is the extension by zero of the function associated to ${\mathcal A}_{\la,x}$. Define now the closed subscheme $\G_x^+$ of $\G_x$ which at the level of points corresponds to pairs $\{L,t\}$, for which $t$ extends to an embedding of $\OO_X$--modules $\OO_X^{\oplus n} \arr L$. The scheme $\G_x^+$ also splits into a disjoint union of connected components $\G_x^+=\cup_{m\in \Z^+} \G_x^{m,+}$, where $\G_x^{m,+} = \G_x^+ \cap \G_x^m$. It is clear that $\G_x^{m,+}$ is a union of the strata $\G_x^\la$, for $\la\in P^{++}_{n,m}$ (see Sect.~3.2). The set $\G_x^+(\Fq)$ identifies with the quotient $GL_n(\K_x)^+/GL_n(\OO_x)$. Consider the morphism $q_{m,x}: \G_x^{m,+} \to \Coh_{m,n}(x)$, which sends a pair $\{L,t\}$ to the quotient $\OO_X^{\oplus n}/\on{Im} t^*$, where $t^*: L^* \arr \OO_X^{\oplus n}$. This morphism can be described as follows. There is a natural vector bundle $\wt{\Coh}_{m,n}$ over the stack $\Coh_{m,n}$, whose fiber at ${\mathcal T}$ is $\on{Hom}(\OO_X^{\oplus n},{\mathcal T})$; $\G_x^{m,+}$ is an open substack of the total space of the bundle $\wt{\Coh}_n$, which corresponds to epimorphic elements of $\on{Hom}(\OO_X^{\oplus n},{\mathcal T})$. The map $q_{m,x}$ is simply the projection from this substack to the base. Hence $q_{m,x}$ is a smooth morphism of algebraic stacks. It is clear that it preserves the stratification (compare with \cite{Lu1}). Note that $\dim \G_x^{m,+} = m(n-1)$ and $\dim \Coh_{n,m}(x) = -m$. Therefore \begin{lem} \label{p} \begin{equation} \label{smooth} q_{m,x}^* {\mathcal B}_{\la,x}[m](-n(\la)) = {\mathcal A}_{\la,x}[-m(n-1)](-(\la,\rho)-n(\la)). \end{equation} \end{lem} Recall that $m=|\la|=\sum_{i=1}^n \la_i$. The lemma implies that as functions on \newline $GL_n(\K_x)^+/GL_n(\OO_x)$, \begin{equation} \label{equ} B_\la = (-1)^{|\la|(n-1)} q_x^{(\la,\rho)+n(\la)} A_\la. \end{equation} Hence, according to \eqref{le}, \begin{equation} \label{le1} L_{E,x} = \sum_{\la \in P^{++}_n} (-1)^{|\la|(n-1)} q_x^{(\la,\rho)+n(\la)} \on{Tr}(\sigma(\on{Fr}_x),E_x(\la)) \cdot A_\la, \end{equation} This formula will be used in the next section in the proof of \propref{Fplus}. \section{Proof of the local statement} In this section we will state and prove a general result concerning reductive groups over local non-archimedian field of positive characteristic. In the case of $GL_n$ this result implies \propref{Fplus}. \subsection{General set-up} Let $G=G({\mathcal K})$ be a connected, reductive, split algebraic group over the field ${\mathcal K}=\Fq((\pi))$, $T$ -- its split maximal torus contained in a Borel subgroup $B$, and $N$ -- the unipotent radical of $B$. We again denote by ${\mathcal O}$ the ring of integers of ${\mathcal K}$, by $\pi$ a generator of its maximal ideal, and by $q$ the cardinality of the residue field $k={\mathbb F}_q$. Let $K$ be the compact subgroup $G({\mathcal O})$ of $G$. We fix a Haar measure of $G$, such that $K$ has measure $1$. Let ${\mathcal H}(G,K)$ denote the Hecke algebra of $G$ with respect to $K$, i.e., ${\mathcal H}(G,K)$ is the algebra of $\Ql$--valued compactly supported $K$--bi-invariant functions on $G$ with the convolution product: \begin{equation} \label{conv} (f_1 \cdot f_2)(g) = \int_G f_1(x) f_2(gx^{-1}) \; dx. \end{equation} Let $\GL$ be the Langlands dual group of $G$ (without the Weil group of $\K$), and $\PL$ (resp., $\PL^+$) be the set of weights (resp., dominant weights) of $\GL$. Each $\la$ can be viewed as a one-parameter subgroup of $G(\K)$, and hence $\la(\pi)$ is a well-defined element of $G(\OO_x)$. We denote by $c_\la$ the characteristic function of the double coset $K \la(\pi) K \subset G$. The functions $c_\la$ form a basis of ${\mathcal H}(G,K)$. Let $\on{Rep} \GL(\Ql)$ denote the Grothendieck ring of the category of finite-dimensional representations of $\GL(\Ql)$. We consider it as a $\Ql$--algebra. If $V$ is a finite-dimensional representation of $\GL$, denote by $[V]$ the corresponding element of $\on{Rep} \GL(\Ql)$. In particular, for each $\la \in \PL^+$, let $V(\la)$ be the finite-dimensional representation of $\GL$ with highest weight $\la$. The following statement, often referred to as the Satake isomorphism, is well-known, see \cite{Sa,La,Mac,G}. \begin{thm} \label{Satake} {\em There is a unique isomorphism $S: \on{Rep} \GL(\Ql) \arr {\mathcal H}(G,K)$, which maps $[V(\la)]$ to} $$H_\la = q^{-(\la,\rho)} \left( c_\la + \sum_{\mu \in \PL^+; \mu<\la} a_{\la\mu} c_\mu \right), \quad \quad a_{\la\mu} \in \Z.$$ \end{thm} \begin{rem} \label{character} Each semi-simple conjugacy class $\gamma$ of the group $\GL(\Ql)$ defines a homomorphism $\chi_\ga: \on{Rep} \GL(\Ql) \arr \Ql$, which maps $[V]$ to $\on{Tr}(\gamma,V)$. We denote the corresponding homomorphism ${\mathcal H} \arr \Ql$ by the same symbol $\chi_\ga$. This allows us to identify the spectrum of the commutative algebra ${\mathcal H}$ with the set of semi-simple conjugacy classes of $\GL(\Ql)$. In particular, we have: $\chi_\ga(H_\la) = \on{Tr}(\ga,V(\la))$.\qed \end{rem} \subsection{Hecke algebra and the affine Grassmannian} \label{he} Let again $X$ be as in Sect.~1.1, and $x$ be its $\Fq$--point. We define, in the same way as in Sect.~4.2 for $G=GL_n$, the ind--scheme ${{\mathcal G}r}(G)=\G(G)_x$ that classifies pairs $\{{\mathcal P},t\}$, where ${\mathcal P}$ is a principal $G$--bundle on $X$ and $t$ is its trivialization over $X-x$. The ind--scheme structure on $\G(G)$ is described, e.g., in \cite{LS}. Note that due to the results of \cite{BL,DS}, the global curve $X$ is inessential in the above definition. We could simply take $X = \on{Spec} \OO]$. In particular, there is a bijection between the set of ${\mathbb F}_q$--points of $\G(G)$ and the set $G/K$. There is a proalgebraic group $\GG(\OO)$, whose set of $\Fq$--points is $G(\OO)$. This group acts on $\G(G)$, and its orbits stratify $\G(G)$ by locally closed finite-dimensional subschemes $\G(G)^\la$ indexed by the set $\PL^+$ of dominant weights $\la$ of $\GL$. The stratum $\G(G)^\la$ is the $G(\OO)$--orbit of the coset $\la(\pi) \cdot G(\OO)$, where $\la(\pi) \in T(\K) \subset G(\K)$ is defined above. Denote by $(\la,\rho)$ the pairing between $\la \in \PL$ and the sum of the fundamental coweights $\rho$ of $\GL$. Let $\ol{\mathbb Q}_{\ell,\la}$ be the constant sheaf supported on the stratum $\G(G)^\la$. Denote by ${\mathcal A}_\la={\mathcal A}_{\la,x}$ the intersection cohomology sheaf on the closure of $\G(G)^\la$, which is the Goresky-MacPherson extension of the sheaf $\ol{\mathbb Q}_{\ell,\la}[2(\la,\rho)]((\la,\rho))$ (note that $\dim \G(G)^\la =$ $2(\la,\rho)$). Let $A_\la$ be the function associated to ${\mathcal A}^\la$. We use the same notation for its extension by zero to the whole $\G(G)$. Clearly, the functions $A_\la, \la \in \PL^+$, form a basis in the $\Ql$--vector space $\Ql(G/K)^K$ of $K$--invariant functions on $G/K$ with compact support. We have an isomorphism of vector spaces: ${\mathcal H} \simeq \Ql(G/K)^K$, which commutes with the action of ${\mathcal H}$. Therefore $H_\la \in {\mathcal H}, \la \in \PL$, can also be considered as elements of $\Ql(G/K)^K$. \begin{prop} \label{hla} $H_\la = (-1)^{2(\la,\rho)} A_\la$. \end{prop} This result is due to Lusztig \cite{Lu1,Lu2} and Kato \cite{Ka} (see, e.g., Theorem 1.8, Lemma 2.7, formula (3.5) of \cite{Ka}). It implies that \begin{equation} \label{stalk} H_\la(y) = (-1)^{2(\la,\rho)} \sum_{i \in \Z} \dim \on{H}^i({\mathcal A}_\la)|_y \; q^{i/2}, \end{equation} where $\on{H}^j({\mathcal A}_\la)|_y$ is the $j$th stalk cohomology of ${\mathcal A}_\la$ at $y \in Gr(G)$. \subsection{Fourier transform} Let us denote by $\on{Res}: \K \arr \Fq$ the map defined by the formula $$\on{Res}\left( \sum_{n \in \Z} f_i \pi^i \right) = f_{-1}.$$ We define a character $\Psi$ of $N$ in the following way: $$\Psi(u) = \sum_{i=1}^l \psi\left(\on{Res}(u_i)\right),$$ where $u_i, i=1\ldots,l=\dim N/[N,N]$ are natural coordinates on $N/[N,N]$ corresponding to the simple roots and $\psi: \Fq \arr \Ql^{\times}$ is a fixed non-trivial character. Consider the space $\Ql(G/K)^N_\Psi$ of left $(N,\Psi)$--equivariant and right $K$--invariant functions on $G$ that have a compact support modulo $N$. This space is a module over ${\mathcal H}(G,K)$, with the action defined by formula \eqref{conv}. For $\la \in \PL^+$, let $\phi_\la$ be the function from $\Ql(G/K)^N_\Psi$, which vanishes outside the $N$--orbit of $\la(\pi)$ and equals $q^{-(\la,\rho)}$ at $\la(\pi)$. The elements $\phi_\la, \la \in \PL^+$, provide a $\Ql$--basis for $\Ql(G/K)^N_\Psi$. Define the linear map $\Phi: \Ql(G/K)^K \arr \Ql(G/K)^N_\Psi$ by the formula \begin{equation} \label{check1} (\Phi(f))(g) = \int_{N} f(ug) \Psi^{-1}(u) \; du, \end{equation} where $du$ stands for the Haar measure on $N$ normalized so that $\int_{N({\mathcal O})} \; du = 1$. \begin{lem} \label{F} The map $\Phi$ defines an isomorphism of ${\mathcal H}$--modules $$\Ql(G/K)^K \simeq \Ql(G/K)^N_\Psi.$$ \end{lem} \begin{proof} Let $c_\la \in \Ql(G/K)^K$ be the characteristic function of the $K$--orbit of the coset $\la(\pi) \cdot K \in G/K$. It follows from the definition that $$\Phi(c_\la) = q^{(\la,\rho)} \phi_\la + \sum_{\mu \in \PL^+; \mu<\la} b_{\la,\mu} \phi_\mu.$$ This implies the lemma. \end{proof} It is natural to call $\Phi$ the Fourier transform. We are now ready to state our main local theorem. \begin{thm} \label{local} {\em The map $\Phi$ sends $H_\la$ to $\phi_\la$.} \end{thm} This theorem is equivalent to the formula \begin{equation} \label{formula} \int_{N} H_\la(u\cdot \nu(\pi)) \Psi^{-1}(u) \; du = q^{-(\la,\rho)} \delta_{\la,\nu}. \end{equation} \subsection{Proof of \thmref{local}} \label{proof} Our proof relies on the result of Casselman-Shalika (and Shintani for $G=GL_n$), which describes the values of the Whittaker function at the points $\mu(\pi)$ (cf. Theorem 2.1 and \remref{1}). Let $\gamma$ be a semi-simple conjugacy class in $\GL$ and let $W_{\gamma}$ be a $\Ql$-valued function on $G$ with the following three properties: \begin{itemize} \item $W_{\gamma}(gh) = W_{\gamma}(g), \forall h \in K$, $W_{\gamma}(1)=1$; \item $W_{\gamma}(ug) = \Psi^{-1}(u) W_{\gamma}(g), \forall u \in N$; \item \begin{equation} \label{hecke} \int_G f(x) W_{\gamma}(gx) dx = \chi_\gamma(f) W_{\gamma}(g), \quad \quad \forall g \in G, f \in {\mathcal H} \end{equation} (see \remref{character} for the definition of $\chi_\gamma$). \end{itemize} \begin{thm}[\cite{CS},\cite{Shi}] \label{prime} {\em The function $W_\gamma$ satisfying these properties exists, and it is unique. For $\mu \in \PL$, the value of this function at $\mu(\pi)$ is \begin{equation} \label{css} W_{\gamma}(\mu(\pi))=q^{-(\mu,\rho)} \on{Tr}(\gamma,V(\mu)), \end{equation} if $\mu$ is a dominant weight, and $0$, otherwise.} \end{thm} The function $W_\ga$ is called the Whittaker function corresponding to $\ga$. Now we can prove \thmref{local}. Let $\gamma$ be as above and let $s_{\gamma}$ be a linear functional $\Ql(G/K)^N_\Psi \to \Ql$ given by the formula \begin{equation} \label{sgamma} s_{\gamma}(\phi)=\int_{N\backslash G} W_{\gamma}(g) \phi(g) \; dg, \end{equation} where $dg$ is the measure on $N \backslash G$ induced by the Haar measure on $G$ from Sect.~5.1 and the Haar measure on $N$ from Sect.~5.3. By construction, the function $W_{\gamma}(u) \phi(u)$ is left $N$--invariant. The integral \eqref{sgamma} converges, because, by definition, $\phi$ has compact support modulo $N$. \begin{lem} \label{mapp} The map $s_{\gamma}$ is a homomorphism of ${\mathcal H}(G,K)$--modules $\Ql(G/K)^N_\Psi \to \ol{\mathbb Q}_{\ell,\ga}$, where $\ol{\mathbb Q}_{\ell,\ga}$ is the one-dimensional representation of ${\mathcal H}(G,K)$ corresponding to its character $\chi_\ga$. \end{lem} \begin{proof} Each $f \in {\mathcal H}$ acts on $\Ql(G/K)^N_\Psi$ by mapping $\phi \in \Ql(G/K)^N_\Psi$ to $f \cdot \phi$. By definition of the convolution product (see formula \eqref{conv}), we have: $$(f \cdot \phi)(y) = \int_G f(x) \phi(yx^{-1}) \; dx.$$ Hence $$s_\gamma(f \cdot \phi) = \int_{N\backslash G} W_{\gamma}(g) \left( \int_G f(x) \phi(g x^{-1}) \; dx \right) \; dg.$$ Changing the order of integration and using the invariance of the Haar measure, we obtain $$s_\gamma(f \cdot \phi) = \int_{N\backslash G} \left( \int_G W_\gamma(gx) f(x) \; dx \right) \phi(g) \; dg.$$ By \eqref{hecke}, $$s_\gamma(f \cdot \phi) = \chi_{\gamma}(f) s_\gamma(\phi).$$ \end{proof} By formula \eqref{css} and the definition of the function $\phi_\la$, the function $W_\gamma \cdot \phi_\la$ equals $q^{-2(\la,\rho)} \on{Tr}(\gamma, V(\la))$ times the characteristic function of the double coset $N \la(\pi) K$. Hence $s_\gamma(\phi_\la)=\int_{N \backslash G} W_\gamma(g)\phi_\la(g) dg$ equals $q^{-(\la,\rho)} \on{Tr}(\gamma, V(\la))$ times the measure of the right $K$--orbit $\la(\pi) \cdot K$ in $N \backslash G$. This measure equals $\mu(K/Ad_{\la(\pi)}(N(\OO))) = \mu(N(\OO))/\mu(Ad_{\la(\pi)}(N(\OO)))$ due to our normalization. The latter equals $q^{2(\la,\rho)}$. Therefore $s_{\gamma}(\phi_\la) = \on{Tr}(\gamma, V(\la))$ for each $\la\in \PL^+$. Any $\phi \in \Ql(G/K)^N_\Psi$ can be written as a finite sum $\sum_{\la \in \PL^+} a_\la \phi_\la$, where $a_\la \in \Ql$. We can identify the vector space $\Ql(G/K)^N_\Psi$ with $\on{Rep} \GL(\Ql)$, by mapping $\phi_\la$ to $[V(\la)]$. Let $\ol{\phi}$ be the image of $\phi$ in $\on{Rep} \GL(\Ql)$ under this identification. Then $s_\gamma(\phi) = \sum_{\la \in \PL^+} a_\la \on{Tr}(\gamma,V(\la))$ is simply the value of $\ol{\phi}$ at $\gamma \in \on{Spec} \on{Rep} \GL(\Ql)$. Since the algebra $\on{Rep} \GL(\Ql)$ has no nilpotents, $\phi=\phi'$, if and only if $s_\gamma(\phi) = s_\gamma(\phi')$ for all semi-simple conjugacy classes $\ga$ in $\GL(\Ql)$, Therefore to prove \thmref{local}, it is sufficient to check that for each semi-simple conjugacy class $\ga$ in $\GL(\Ql)$ and $\la\in \PL^+$, $$s_{\gamma}\circ \Phi(H_\la)=\on{Tr}(\gamma,V(\la)).$$ But the composition $s_{\gamma}\circ \Phi: \Ql(G/K)^K \to \ol{\mathbb Q}_{\ell,\ga}$ is a homomorphism of ${\mathcal H}$-modules, by \lemref{F} and \lemref{mapp}. It is easy to check directly that the value of this homomorphism on the element $H_0 = \on{ch}_K \in \Ql(G/K)^K$ equals $1$. Therefore $$s_{\gamma}\circ \Phi(H_\la) = s_\gamma \circ \Phi(H_\la \cdot H_0) = \chi_{\gamma}(H_\la) \cdot s_{\gamma} \circ \Phi(H_0) = \chi_{\gamma}(H_\la) = \on{Tr}(\gamma,V(\la))$$ (see \remref{character}), and \thmref{local} follows. \begin{rem} \label{equi} Our proof shows that \thmref{local} is equivalent to \thmref{prime}.\qed \end{rem} \subsection{Proof of \propref{Fplus}} Note that $(-1)^{2(\la,\rho)} = (-1)^{|\la|(n-1)}$. Hence we obtain from \propref{hla} and formula \eqref{le1}: \begin{equation} \label{lenew} L_{E,x} = \sum_{\la \in P^{++}_n} q_x^{(\la,\rho)+n(\la)} \on{Tr}(\sigma(\on{Fr}_x),E_x(\la)) \cdot H_\la. \end{equation} Therefore we find: $$\int_{N^J(\K_x)} L_{E,x}(u_x \cdot \on{diag}(\pi_x^{\nu_1},\ldots,\pi_x^{\nu_n})) \Psi^{-1}(u_x) \; du_x =$$ $$\sum_{\la \in P^{++}_n} q_x^{(\la,\rho)+n(\la)} \on{Tr}(\sigma(\on{Fr}_x),E_x(\la)) \cdot \int_{N^J(\K_x)} H_\la(u_x \cdot \on{diag}(\pi_x^{\nu_1},\ldots,\pi_x^{\nu_n})) \Psi^{-1}(u_x) \; dx.$$ According to \eqref{formula}, the latter sum equals $q_x^{n(\nu)} \on{Tr}(\sigma(\on{Fr}_x),E_x(\nu))$, which is the right hand side of formula \eqref{oh}. Now \propref{Fplus} is proved, and this finishes the proof of \propref{r} and \thmref{prin}. \section{Whittaker functions and spherical functions} In this section we give an interpretation of \thmref{local} from the point of view of the theory of spherical functions. Throughout this section we will work over the field of complex numbers instead of $\Ql$. In particular, all functions will be $\C$--valued, and ${\mathcal H}$ will be a $\C$--algebra. \subsection{The map $\Th$} Denote by $C^\infty(G/K)^K$ (resp., $C^\infty(G/K)^N_\Psi$) the space of smooth left $K$--invariant (resp., $(N,\Psi)$--equivariant) and right $K$--invariant functions on $G$. We also denote by $\C(G/K)^K$ (resp., $\C(G/K)^N_\Psi$) the subspace of compactly supported (resp., compactly supported modulo $N$) functions. Each element of $C^\infty(G/K)^K$ can be written as an infinite sum $\sum_{\la \in \PL^+} a_\la c_\la$, where $c_\la$ is the characteristic function of the $G(\OO)$--orbit $Gr(G)^\la$. \begin{lem} \label{fini} For each $g \in G$, $(\Phi(c_\la))(g) = 0$ for all but finitely many $\la \in \PL^+$. \end{lem} \begin{proof} It suffices to prove the statement for $g=\mu(\pi)$. In this case, it is easy to see that for all but finitely many $\la$, there exists an element $v \in N$ (depending on $\la$) with $\Psi(v) \neq 1$, such that $\forall u \in N$, $u \cdot \mu(\pi) \in Gr(G)^\la$ if and only if $(vu) \cdot \mu(\pi) \in Gr(G)^\la$. But then $(\Phi(c_\la))(\mu(\pi)) = \Psi(v) (\Phi(c_\la))(\mu(\pi))$, and hence $(\Phi(c_\la))(g) = 0$. \end{proof} Therefore $\Phi$ defines a map $C^\infty(G/K)^K \arr C^\infty(G/K)^N_\Psi$, $f \arr \Phi(f)$, which is equivariant with respect to the action of Hecke operators. Now we define a map $\Th: C^\infty(G/K)^N_\Psi \arr C^\infty(G/K)^K$ by the formula \begin{equation} \label{inverse} (\Th(f))(g) = \int_K f(kg) dk, \end{equation} where $dk$ stands for the Haar measure on $K$ of volume $1$. This map is also equivariant with respect to the action of Hecke operators. We define $a$ as the element of $\C(G/K)^K$ equal to $(\Th \circ \Phi)(\on{ch}_K) = \Th(\phi_0)$. The same argument as in the proof of \lemref{fini} shows that the map $\Th$ sends functions from $\C(G/K)^N_\Psi$ to $\C(G/K)^K$. Hence $a \in \C(G/K)^K = {\mathcal H}$. Introduce the notation $$(a*f)(g) = \int_G a(x) f(gx) \; dx.$$ Then we obtain: \begin{equation} \label{star} (\Th \circ \Phi)(f) = a*f, \quad \quad \forall f \in C^\infty(G/K)^K. \end{equation} In the next section we will use the element $a$ to clarify the connection between Whittaker functions and spherical functions. \subsection{Connection between $a$ and the Plancherel measure} Let $\gamma$ be a semi-simple conjugacy class in the group $\GL(\C)$. Recall \cite{Sa,Mac} that the spherical function $S_\gamma$ is the unique $K$--bi-invariant function on $G$, such that \begin{itemize} \item $f*S_\ga = \chi_\gamma(f) S_\ga, \forall f \in {\mathcal H}$, where $\chi_\ga: {\mathcal H} \arr \C$ is the character corresponding to $\ga$ defined in \remref{character}; \item $S_\ga(1) = 1$. \end{itemize} These properties imply that \begin{equation} \label{vazh} \int_G f(x) S_\ga(x) \; dx = \chi_\gamma(f). \end{equation} Now let $W_\ga$ be the Whittaker function on $G$ as defined in Sect.~5.4 but with the character $\Psi^{-1}$ of $N$ replaced with the character $\Psi$. It is straightforward to check that the function $\Phi(S_\gamma)$ satisfies all the properties of the function $W_\ga$ from \secref{proof}, except for the normalization condition $W_\ga(1)=1$. By \thmref{prime}, $\Phi(S_\gamma)$ is proportional to $W_\gamma$. \begin{lem} \label{whereitgoes} $$\Phi(S_\ga) = \chi_\ga(a) W_\ga.$$ \end{lem} \begin{proof} Introduce $a(\ga)$ by the formula $\Phi(S_\ga) = a(\ga) W_\ga$. Since $\Th(W_\ga) = S_\ga$ by definition, we obtain, using formula \eqref{star} and the properties of $S_\ga$: $a(\ga) S_\ga = (\Th \circ \Phi)(S_\ga) = a*S_\ga = \chi_\ga(a) S_\ga$. \end{proof} According to \cite{Mac}, (1.5.1), there exists a unimodular measure $d\mu(\gamma)$ (Plancherel measure) on the maximal compact subtorus $\TL^u$ of $\TL$, which satisfies \begin{equation} \label{plancherel} \int_G f_1(g) \overline{f_2(g)} dg = \int_{\TL^u} \chi_\ga(f_1) \overline{\chi_\ga(f_2)} \; d\mu(\gamma), \end{equation} for all $f_1, f_2 \in {\mathcal H}$. Setting $f_2=\on{ch}_K$, we obtain: \begin{equation} \label{raz} f(1) = \int_{\TL^u} \chi_\ga(f) \; d\mu(\gamma), \quad \quad \forall f \in {\mathcal H}. \end{equation} By \thmref{local}, $\Phi(H_\la) = \phi_\la$. But it is clear that $(\Th(\phi_\la))(1) = \delta_{\la,0}$. Therefore, using \eqref{star}, we see that $(a*H_\la)(1)=\delta_{\la,0}$. Substituting this into formula \eqref{raz} and using the formula $\chi_\ga(H_\la) = \on{Tr}(\ga,V(\la))$, we obtain: \begin{equation} \label{del} \int_{\TL^u} \on{Tr}(\gamma,V(\la)) a(\gamma) \; d\mu(\gamma) = \delta_{\la,0}. \end{equation} There exists a unique measure $d\wt{\mu}(\gamma)$ on $\TL^u$ (induced by the Haar measure on $\GL^u$), such that \begin{equation} \label{orth} \int_{\TL^u} \on{Tr}(\gamma,V(\nu)) \ol{\on{Tr}(\gamma,V(\la))} \; d\wt{\mu}(\gamma) = \delta_{\la,\nu}. \end{equation} Formula \eqref{del} then implies \begin{prop} \label{aga} $$a(\ga) = \frac{d\wt{\mu}(\gamma)}{d\mu(\gamma)}.$$ \end{prop} Now \lemref{whereitgoes} gives us: \begin{equation} \label{proportion} \Phi(S_\gamma) = \frac{d\wt{\mu}(\gamma)}{d\mu(\gamma)} W_\gamma. \end{equation} \subsection{Another proof of \thmref{local}} In this subsection, which is independent from the previous one, we use spherical functions to give another proof of \thmref{local}. Substituting $f_1=\on{ch}_{KgK}$ into formula \eqref{plancherel} and using formula \eqref{vazh}, we obtain that for any $f \in {\mathcal H}$, \begin{equation} \label{fourier} \ol{f(g)} = \int_{\TL^u} S_{\gamma}(g) \ol{\chi_\gamma(f)} \; d\mu(\gamma). \end{equation} Since $\chi_\ga(H_\la) = \on{Tr}(\gamma,V(\la))$, we have: \begin{equation} \label{alambda} H_\la(g) = \int_{\TL^u} S_\ga(g) \ol{\on{Tr}(\gamma,V(\la))} \; d\mu(\gamma). \end{equation} According to formula \eqref{alambda}, \begin{equation} \label{a0} H_0(g) = \int_{\TL^u} S_\ga(g) \; d\mu(\gamma). \end{equation} Hence $$\Phi(H_0)(g) = \int_{\TL^u} \Phi(S_\ga)(g) \; d\mu(\gamma) = \int_{\TL^u} W_\ga(g) a(\gamma) \; d\mu(\gamma).$$ On the other hand, it is clear from definition that $\Phi(H_0) = \phi_0$. Therefore, substituting $g=\la(\pi)$ and using formula \eqref{css}, we obtain formula \eqref{del}. Repeating the argument with the Haar measure given above, we obtain \eqref{proportion}. Now formulas \eqref{alambda}, \eqref{proportion} and \eqref{css} give: $$(\Phi(H_\la))(\nu(\pi)) = \int_{\TL^u} W_\gamma(\nu(\pi)) \ol{\on{Tr}(\gamma,V(\la))} \; d\wt{\mu}(\gamma) =$$ $$= q^{-(\nu,\rho)} \int_{\TL^u} \on{Tr}(\gamma,V(\nu)) \ol{\on{Tr}(\gamma,V(\la))} \; d\wt{\mu}(\gamma) = q^{-(\la,\rho)} \delta_{\la,\nu}.$$ This proves formula \eqref{formula} and \thmref{local} over the field of complex numbers. Since $H_\la$ takes values in rational numbers and $\Psi$ takes values in the roots of unity, the validity of \eqref{formula} over $\C$ actually implies its validity over $\Ql$. \subsection{The function $L_\ga$} The Whittaker function can be written as a series $$W_\ga = \sum_{\la \in \PL^+} \on{Tr}(\gamma,V(\la)) \cdot \phi_\la.$$ This series obviously makes sense, since the supports of the functions $\phi_\la$ do not intersect. In view of \thmref{local}, it is natural to consider the series \begin{equation} \label{Fgamma} L_\ga = \sum_{\la \in \PL^+} \on{Tr}(\gamma,V(\la)) \cdot H_\la. \end{equation} However, the convergence of this series is not at all automatic, because the supports of functions $H_\la$ do intersect; for instance, each $H_\la$ has a non-zero value at $1$. In this section we study the question of convergence of $L_\ga$. Let us write: $$H_\la = q^{-(\la,\rho)} \sum_{\mu \leq \la} P_{\mu\la}(q^{-1}) \cdot c_\mu,$$ where $q^{-(\la,\rho)} P_{\mu\la}$ is a polynomial in $q^{-1}$ (recall that $c_\la$ is the characteristic function of the $G(\OO)$--orbit $\G(G)^\la$). Formula \eqref{alambda} can be rewritten as follows: \begin{equation} \label{al1} H_\la(g) = \int_{\TL^u} S_\ga(g) a(\ga)^{-1} \ol{\on{Tr}(\gamma,V(\la))} \; d\wt{\mu}(\gamma). \end{equation} Using the defining properties of the spherical function $S_\ga$, we can write it as a series \begin{equation} \label{sga} S_\ga = \sum_{\mu \in \PL^+} s_\ga^\mu(q^{-1}) \cdot c_\mu. \end{equation} where $s_\ga^\mu(q^{-1})$ is a rational function in $q^{-1}$ of the form $Q(q^{-1}) \wt{s}_\ga^\mu(q^{-1})$. Here $$Q(q^{-1}) = \prod_{i=1}^l \frac{1-q^{-m_i-1}}{1-q^{-1}}$$ ($l$ is the rank of $G$, $m_i$'s are the exponents of $G$; note that $Q(q) = \# G/B(\Fq)$), and $\wt{s}_\ga^\mu$ is a polynomial in $q^{\pm 1}$. Its coefficients are finite integral linear combinations of characters of irreducible representations of $\GL$ (for an explicit formula, see \cite{Mac}). It follows from formula \eqref{whereitgoes} that $a(\ga)$ has the same structure as a function of $q^{-1}$. Hence both $s_\ga^\mu(q^{-1})$ and $a(\ga)^{-1}$ can be viewed as formal Laurent power series in $q^{-1}$ and formula \eqref{al1} can be viewed as an identity on such power series. We have: $$s_\ga^\mu = \sum_{m > -M} \on{Tr}(\ga,R^\mu_m) q^{-m},$$ and $$a(\ga)^{-1} = \sum_{m > -M'} \on{Tr}(\ga,U_m) q^{-m},$$ where $R^\mu_m$ and $U_m$ are finite linear combinations of irreducible representations of $\GL$ (the summation is actually only over $m \in \Z_+$). Then formula \eqref{al1} can be written as follows: \begin{equation} \label{f1} q^{-(\la,\rho)} P_{\mu\la}(q^{-1}) = \sum_{N \in \Z} q^{-N} \int_{\TL^u} \on{Tr}(\ga,\oplus_{m \in \Z} R^\mu_m \otimes U_{N-m}) \ol{\on{Tr}(\gamma,V(\la))} \; d\wt{\mu}(\gamma). \end{equation} By formula \eqref{orth}, the $q^{-N}$ coefficient of $P_{\mu\la}(q^{-1})$ equals the multiplicity of $V(\la)$ in $\oplus_{m \in \Z} R^\mu_m \otimes U_{N-m}$. But by construction the latter is a finite linear combination of irreducible representations of $\GL$. Hence we obtain the following \begin{lem} For each $\mu \in \PL^+$ and $N \in \Z_+$ there are only finitely many $\la \in \PL^+$, such that $q^{-(\la,\rho)} P_{\mu\la}$ has a non-zero coefficient in front of $q^{-N}$. \end{lem} \begin{rem} This can also be seen from the explicit formula for $P_{\mu\la}$ obtained in \cite{Lu2,Ka}.\qed \end{rem} Therefore for each $g \in G$, $L_\ga(g)$ given by formula \eqref{Fgamma} makes sense as a formal power series in $q^{-1}$, each coefficient being a finite linear combination of characters. Furthermore, we see, by reversing the argument above that as formal power series in $q^{-1}$, \begin{equation} \label{propor} L_\ga(g) = a(\ga)^{-1} S_\gamma(g), \quad \quad \forall g \in G. \end{equation} In order to estimate the convergence of this series, we have to compute $a(\ga)$ explicitly. According to \cite{Mac}, (5.1.2), $$d\mu(\gamma) = \frac{Q(q^{-1})}{|W|} \frac{\prod_{\al \in \De} (1-\al(\gamma))}{\prod_{\al \in \De} (1-q^{-1} \al(\gamma))} d\gamma,$$ where $|W|$ is the number of elements in the Weyl group, and $\De$ is the set of roots of $G$. The notation $d\ga$ means the Haar measure on $\TL^u$, which gives it volume $1$. On the other hand, $$d\wt{\mu}(\gamma) = \frac{1}{|W|} \prod_{\al \in \De} (1-\al(\gamma)) d\gamma.$$ Now \propref{aga} gives: \begin{equation} \label{propsg} a(\ga) = \frac{\prod_{\al \in \De} (1-q^{-1} \al(\gamma))}{Q(q^{-1})}. \end{equation} Let $^L\!{\mathfrak g}$ be the adjoint representation of $\GL$, and $\Lambda^i(^L\!{\mathfrak g})$ be its $i$th exterior power. Formula \eqref{propsg} means that $a \in {\mathcal H}$ is the image under the Satake isomorphism $S$ of the following element in $\on{Rep} \GL$: $$\prod_{i=1}^l (1-q^{-m_i-1})^{-1} \sum_{i=0}^{\dim ^L\!{\mathfrak g}} \left[ \Lambda^i(^L\!{\mathfrak g}) \right] (-1)^i q^{-i}.$$ Formula \eqref{propor} implies the following result. \begin{prop} \label{converge} If the conjugacy class $\ga$ satisfies: $q^{-1} < |\al(\ga)| < q, \forall \al \in \De_+$, then $L_\ga(g)$ converges absolutely to $$\frac{Q(q^{-1})}{\prod_{\al \in \De} (1-q^{-1} \al(\gamma))} S_\ga(g)$$ for all $g \in G$. \end{prop} Note that when $G=GL_n$, formula \eqref{Fgamma} looks similar to formula \eqref{lenew} for the function $L_{E,x}$. Besides powers of $q_x$, the difference is that in \eqref{lenew} the summation is restricted to the subset $P^{++}_n$ of the set $P^+_n = \PL^+$ of all dominant weights of $GL_n$. However, $L_{E,x}$ is not the restriction of \eqref{Fgamma} to the union of strata $Gr(GL_n)^\la$ with $\la \in P^{++}_n$, because the functions $H_\la$ with $\la \in P^+_n - P^{++}_n$ do not vanish on those strata. While $L_\gamma$ given by \eqref{Fgamma} is manifestly an eigenfunction of the Hecke operators, $L_{E,x}$ is not\footnote{it is actually an eigenfunction of some other operators, similar to the Hecke operators, which were defined by Laumon \cite{La1}}. For general $G$ there is no analogue of the subset $\PL^{++} \subset \PL^+$, and so the function $L_\gamma$ seems to be the closest analogue of $L_{E,x}$ in the general setting. According to \thmref{local} and formula \eqref{css}, $\Phi(L_\gamma)$ equals the Whittaker function $W_\ga$. \begin{rem} \label{lfunct} Let $\ga$ be a semi-simple conjugacy class of $\GL(\Ql)$ and $r: \GL(\Ql) \arr \on{Aut} V$ be a finite-dimensional representation of $\GL(\Ql)$. Recall that the local $L$--function associated to the pair $(\ga,V)$ is defined by the formula \begin{equation} \label{lfun} L(\ga,V;s) = \on{det} \left( 1 - r(\ga) q^{-s} \right)^{-1}. \end{equation} In particular, if $V = ^L\!{\mathfrak g}$ is the adjoint representation, then $$L(\ga,^L\!{\mathfrak g};s) = (1-q^{-s})^{-l} \prod_{\al \in \De} (1 - \al(\ga) q^{-s})^{-1}.$$ Hence $a(\ga)$ can be written as $$a(\ga) = L(\ga,^L\!{\mathfrak g};1) \prod_{i=1}^l (1-q^{-m_i-1}).$$ Thus, we obtain: \begin{equation} \label{vani} \Phi(S_\ga) = L(\ga,^L\!{\mathfrak g};1)^{-1} \prod_{i=1}^l (1-q^{-m_i-1})^{-1} \cdot W_\ga. \end{equation} Using arguments similar to those of Casselman and Shalika \cite{CS}, one can show that the irreducible unramified representation corresponding to $\ga$ has a Whittaker model if and only if $\Phi(S_\ga) \neq 0$. Formula \eqref{vani} means that $\Phi(S_\ga) \neq 0$ if and only if $L(\ga,^L\!{\mathfrak g};s)$ is regular at $s=1$. We conclude that the irreducible unramified representation of $G$ with the Langlands parameter $\ga$ has a Whittaker model if and only if $L(\ga,^L\!{\mathfrak g};s)$ is regular at $s=1$ (in that case the Whittaker model is actually unique). This agrees with a special case of a conjecture of Gross and Prasad \cite{GP} (Conjecture 2.6).\qed \end{rem} \subsection{Identities on $P_{\mu\la}$} According to formula \eqref{propor}, for each $\mu \in \PL^+$ we have the following equality of power series in $q^{-1}$: \begin{equation} \label{id1} \sum_{\la: \la \geq \mu} \on{Tr}(\gamma,V(\la)) q^{-(\la,\rho)} P_{\mu\la}(q^{-1}) = \prod_{i=1}^l \frac{1-q^{-m_i-1}}{1-q^{-1}} \prod_{\al \in \De} (1-q^{-1} \al(\gamma))^{-1} S_\ga(\mu(\pi)). \end{equation} Recall that the coefficients of the polynomial $P_{\mu\la}$ (which can be interpreted as a Kazhdan-Lusztig polynomial for the affine Weyl group \cite{Lu2,Ka}) are given by dimensions of stalk cohomologies of the perverse sheaf ${\mathcal A}_\la$: \begin{equation} P_{\mu\la} = q^{(\la,\rho)} \sum_{i \in \Z} \dim \on{H}^i({\mathcal A}_\la)|_{\mu(\pi)} \; q^{i/2}. \end{equation} Thus, formula \eqref{id1} is an identity which connects these dimensions with the values of the spherical functions. The latter are known explicitly; they can be expressed via the Hall-Littlewood polynomials \cite{Mac}. For example, let us apply formula \eqref{id1} when $G=SL_2$ and $\mu=0$. In this case, $\gamma \in \C^\times$, and the set $\PL^+$ of dominant weights of the dual group $\GL=PGL_2$ can be identified with the set of non-negative even integers. To weight $2m$ corresponds the $2m+1$--dimensional representation $V(2m)$ of $PGL_2$, and $\on{Tr}(\gamma,V(2m)) = \sum_{i=-m}^m \gamma^i$. Formula \eqref{id1} then gives: $$\sum_{m \in \Z_+} \left( \sum_{j=-m}^m \gamma^j \right) \cdot \sum_{i \in \Z} \dim \on{H}^i({\mathcal A}_{2m})|_1 \; q^{i/2} = \frac{1+q^{-1}}{(1-q^{-1} \gamma)(1-q^{-1} \gamma^{-1})}.$$ This is easy to see directly, because ${\mathcal A}_{2m}$ is known to be the constant sheaf on the closure of the stratum $\G(SL_2)^{2m}$ placed in degree $-2m$. For general $G$, formula \eqref{id1} with $\mu=0$ can be interpreted as follows. Let $R(\g)$ be the graded ring of polynomials on $\g$, $J(\g)$ be its subgring of $\GL$--invariants, and $H(\g)$ be the subspace of $\GL$--harmonic polynomials on $\g$. For a graded space $V$, we denote by $V_j$ its $j$th homogeneous component. If each $V_j$ is a representation of $\GL$, we denote by $\on{Ch}(\ga,V)$ the graded character of $V$: $$\on{Ch}(\ga,V) = \sum_{j=0}^\infty \on{Tr}(\ga,V_j) q^{-j}.$$ Note that the graded character of $R(\g)$ equals \begin{equation} \label{tozh1} \on{Ch}(\ga,R(\g)) = \prod_{i=1}^l (1-q^{-1})^{-l} \prod_{\al \in \De} (1-q^{-1} \al(\gamma))^{-1} \end{equation} (compare with \remref{lfunct}), while $$\on{Ch}(\ga,J(\g)) = \prod_{i=1}^l (1-q^{-m_i-1})^{-1}.$$ Now let $H(\g)$ be the (graded) space of $\GL$--harmonic polynomials on $\g$. According to Theorem 0.2 of B.~Kostant \cite{Ko}, $R(\g) = J(\g) \otimes H(\g)$. Hence $$\on{Ch}(\ga,H(\g)) = \prod_{i=1}^l \frac{1-q^{-m_i-1}}{1-q^{-1}} \prod_{\al \in \De} (1-q^{-1} \al(\gamma))^{-1} = a(\ga)^{-1}.$$ Thus, we obtain another interpretation of $a(\ga)^{-1}$: it is equal to the graded character of the space of $\GL$--harmonic polynomials. Note that it also equals the graded character of the space of regular functions on the nilpotent cone ${\mathcal N}$ in $\g$. Formula \eqref{id1} together with this interpretation give us the following result. \begin{prop} \begin{equation} \label{new} P_{0,\la}(q^{-1}) = q^{(\la,\rho)} \sum_{j=0}^\infty q^{-j} \on{mult}(V(\la),H(\g)_j), \end{equation} where $\on{mult}(V(\la),H(\g)_j)$ is the multiplicity of $V(\la)$ in $H(\g)_j$. \end{prop} A complete description of these multiplicities has been given by Kostant in \cite{Ko}. In fact, applying Theorem 0.11 of \cite{Ko} to formula \eqref{new}, we obtain: \begin{equation} \label{tozh2} P_{0,\la} = q^{(\la,\rho)} \sum_{i=1}^{l_\la} q^{-m_i(\la)}, \end{equation} where $m_i(\la)$ are the generalized exponents associated to the representation $V(\la)$, defined in \cite{Ko}. In the special case when $V(\la)$ is the adjoint representation $^L\!{\mathfrak g}$, these are just the exponents of $\GL$, and formula \eqref{tozh2} specializes to Lusztig's formula (see \cite{Lu2}, p.~226) $$P_{0,\la_{\on{adj}}} = \sum_{i=1}^l q^{m_i-1}.$$ Formula \eqref{tozh2} is not new: R.~Brylinski \cite{Br} observed that it immediately follows if one compares the Lusztig-Kato formula \cite{Lu2,Ka} for $P_{0,\la}$ and the Hesselink-Peterson formula \cite{He} for the right hand side of \eqref{tozh2}. Note that in contrast to her argument, our proof of formula \eqref{tozh2} does not use the Lusztig-Kato formula. Using \eqref{f1} it is easy to derive a formula analogous to \eqref{new} for $P_{\mu\la}$ in terms of Hall-Littlewood polynomials. \section{Geometric analogue of \thmref{local} and some open problems} \subsection{} In this subsection we will formulate a geometric analogue of \thmref{local}. Recall the Grassmannian $\G(G)$ of section ~5.2. This is a strict ind--scheme over $\Fq$, i.e., an inductive system of $\Fq$--schemes $\G(G)_k, k\geq 0$, where all maps $i_{k,m}: \G(G)_k \rightarrow \G(G)_m, k<m$, are closed embeddings. For more details, see, e.g., \cite{LS,MV}. By a $\Ql$--sheaf on $\G(G)$ we will understand a system of $\Ql$--sheaves $\cf_k$ on each $\G(G)_k$ and a compatible system of isomorphisms $\cf_k \simeq i_{k,m}^* \cf_m$ for all $k<m$. There exists an ind--group scheme ${\mathcal N}(\K)$, whose set of $\Fq$--points is $N(\K)$. This ind-group scheme acts on the Grassmannian $\G(G)$, and its orbits stratify $\G(G)$ by ind-schemes $S^\nu, \nu \in \PL$. The stratum $S^\nu$ is the $\NK$--orbit of $\nu(\pi) \in Gr(G)$. Denote by $j^\nu$ the embedding $S^\nu \hookrightarrow \G(G)$. We choose a generic additive character $\wt\Psi: \NK \to {\mathbb G}_{a,\mathcal K}$ and define a homomorphism $\Psi: \NK \to \Gaf$ by the formula $$ \Psi \ = \ \operatorname{Res} \circ \,\wt\Psi: \NK \to \Gaf\,, $$ where $\operatorname{Res}$ is the geometric analogue of the residue map of Sect.~5.3. As before, let $\psi: \Fq \to \Ql^\times$ denote a non-trivial character and let $\mathcal I_\psi$ denote the corresponding Artin-Schreier sheaf on $\Gaf$. Consider the category $\on{P}_{\GG(\OO)}(\G(G))$ of ${\GG(\OO)}$--equivariant perverse sheaves on $\G(G)$ with finite-dimensional support. This category is a geometric analogue of the Hecke algebra ${\mathcal H}=\Ql(G/K)^K$ (see \remref{cate} below). We define an abelian category $\on{Sh}^\Psi_{\NK}(\G(G))$ (a geometric analogue of $\Ql(G/K)^N_\Psi$) and a collection of cohomology functors $\W^i: \on{P}_{\GG(\OO)}(\G(G)) \arr \on{Sh}^\Psi_{\NK}(\G(G))$, which are a geometric analogue of the map $\W$ of Sect.~5.3. The objects of the category $\on{Sh}^\Psi_{\NK}(\G(G))$ are $\Ql$--sheaves ${\mathcal E}$ on $\G(G)$ which satisfy the following conditions: (1) ${j^\nu}^* {\mathcal E} = 0$ except for finitely many $\nu\in \PL$; (2) $t_\nu^* {j^\nu}^* {\mathcal E} \otimes\varphi^*{\mathcal I}_{\psi^{-1}}$ are trivial local systems of finite rank for all $\nu$, where $t_\nu: \NK \to S^\nu$ is the map $u \mapsto u \cdot\nu(t)$. Morphisms in this category are defined in an obvious way. \begin{lem} \label{first} If $\nu\in\PL$ is not dominant, then for every ${\mathcal E} \in\on{Ob} (\on{Sh}^\Psi_{\NK}\G(G))$, ${j^\nu}^*({\mathcal E})=0$. \end{lem} The proof is analogous to the proof of the corresponding statement for functions. Thus, ${\mathcal E}$ not only satisfies property (1) above, but also satisfies the stronger property (1') ${j^\nu}^* {\mathcal E} = 0$ except for finitely many $\nu\in \PL^+$. Now we construct the functors $\W^i:\on{P}_{\GG(\OO)}(\G(G))\to \on{Sh}^\Psi_{\NK}(\G(G))$. Consider the sequence of maps: $$ \begin{CD} \G @<a<< \NK\times\G @>q>> \NK/\NO\times\G @>p>> \G\,, \end{CD} $$ where $a$ is given by acting with $\NK$ on $\G$ and $p,q$ are projections. For $\cf\in \on{P}_{\GG(\OO)}(\G(G))$ we then set: $${\W}^i(\cf) = R^i p_!(\wt \cf \otimes \varphi^*{\mathcal I}_\psi), \qquad \text{with} \ \ q^*\wt\cf = a^*\cf\,. $$ To formulate the geometric analogue of \thmref{local}, recall that for $\la\in {^LP^+}$ we denote by ${\mathcal A}_\lambda$ the Goresky-MacPherson extension of the sheaf $\ol{\mathbb Q}_{\ell,\la}[2(\la,\rho)]((\la,\rho))$ associated to the $\GG(\OO)$--orbit $\G(G)^\la$. For each $\nu \in \PL^+$, denote by $\NK_\nu$ the isotropy group of $\nu(\pi)$. Since $\nu$ is dominant, the restriction of $\Psi$ to $\NK_\nu$ equals $0$. Therefore the map $\Psi$ restricts to a map on $S^\nu$, which we continue to denote by the same letter $\Psi: S^\nu\to\Gaf$. With this notation, the sheaf $\Psi^*{\mathcal I}_\psi$ is a sheaf on $S^\nu$. Recall that $j^\nu$ denotes the embedding $S^\nu \hookrightarrow \G(G)$. Now we are ready to state the geometric analogue of \thmref{local}. \begin{conj} \label{general2} $${\W}^i({\mathcal A}_\lambda) = \left\{\aligned &j^\la_! \Psi^*{\mathcal I}_\psi((\la,\rho)) \qquad\text{if} \ \ i=2(\lambda,\rho) \\ &\ 0 \qquad\text{if} \ \ i\neq 2(\lambda,\rho) \,.\endaligned\right. $$ \end{conj} We will now formulate \conjref{second} describing the stalk cohomologies of the sheaves ${\W}^i({\mathcal A}_\lambda)$. The statement of \conjref{second} does not explicitly involve the category $\on{Sh}^\Psi_{\NK}(\G(G))$ and the functors $\Phi^i$. However, \conjref{general2} can be derived from \conjref{second}. Note that the support of the restriction of ${\mathcal A}_\la$ to $S^\nu$, i.e., $\ol{\G(G)^\la} \cap S^\nu$, is finite-dimensional. To work in a geometric setting, let us extend the base field from $\Fq$ to $\ol{\mathbb F}_q$ and use Weil sheaves. Denote by $\Psi_{\la,\nu}$ the restriction of $\Psi$ to $\ol{\G(G)^\la} \cap S^\nu$. \begin{conj} \label{second} For $\lambda$ dominant $$\operatorname{H}_c^k(\ol{\G(G)^\la} \cap S^\nu, {\mathcal A}_\lambda\otimes \Psi_{\la,\nu}^*{\mathcal I}_\psi) = \left\{\aligned &\Ql(-(\la,\rho)) \ \ \text{if} \ \nu=\lambda \ \text{and} \ \ k=2(\la,\rho) \\ &\ 0\qquad\qquad\ \ \text{otherwise}\,.\endaligned \right. $$ \end{conj} Proving \conjref{second} would yield an alternative proof of \thmref{local}, and hence of the Casselman-Shalika formula \eqref{css} (see \remref{equi}). One sees readily that \conjref{second} holds when $\la \leq \nu$. Let us, then, assume that $\nu$ is dominant and $\nu<\lambda$. The projection formula implies that $$ \operatorname {H}_c^*(S^\nu, {\mathcal A}_\lambda\otimes \Psi^*{\mathcal I}_\psi) = \operatorname {H}_c^*(\Gafb, R\Psi_!{\mathcal A}_\lambda\otimes {\mathcal I}_\psi). $$ Therefore \conjref{second} follows from the following \begin{conj} \label{third} For $\lambda, \nu$ dominant and $\nu\neq\lambda$ the sheaves $ R^k\Psi_!{\mathcal A}_\lambda$ are constant on $\Gafb$. \end{conj} By theorem 4.3a of \cite{MV} we see that if \conjref{third} holds then $R^k\Psi_!{\mathcal A}_\lambda = 0$ unless $k = 2((\la,\rho)-1)$. Here we are using the fact that the results of \cite{MV}, stated there over $\mathbb C$, extend to our current context. \subsection{} In this subsection we speculate about what could be the analogue of Laumon's sheaf ${\mathcal L}_E$ in the case of an arbitrary reductive group. Recall from Sect.~3.2 that ${\mathcal L}_E$ is a sheaf on the stack $\Coh_n$ canonically attached to a rank $n$ local system $E$ on $X$. This sheaf is used as the starting point of the conjectural construction of the automorphic sheaf on ${\mathcal M}_n$ associated to $E$, see \cite{La2} and Sect.~3 above. First we revisit the case of $GL_n$ and introduce a scheme $\G^+_{X^{(\infty)}}$ with a smooth morphism $q: \G^+_{X^{(\infty)}} \arr \Coh_n$, and take the pull-back of ${\mathcal L}_E$ to $\G^+_{X^{(\infty)}}$. The scheme $\G^+_{X^{(\infty)}}$ classifies pairs $\{ L,t \}$, where $L$ is a rank $n$ bundle on $X$ and $t: \OO_X^{\oplus n} \arr L$ is an embedding of $\OO_X$--modules. The scheme $\G^+_{X^{(\infty)}}$ is a disjoint union of the smooth schemes $\G^{+,m}_{X^{(\infty)}}, m\geq 0$, corresponding to bundles of degree $m$. The scheme $\G^{+,m}_{X^{(\infty)}}$ is isomorphic to the Grothendieck Quot--scheme $\on{Quot}^m_{\OO^{\oplus n}_X/X/\Fq}$. Recall \cite{Gr} that $\on{Quot}^m_{\OO^{\oplus n}_X/X/k}$ classifies the quotients of $\OO^{\oplus n}_X$ that are torsion sheaves of length $m$; at the level of points, $\{ L,t \}$ corresponds to the quotient of $\OO^{\oplus n}_X$ by the image of $L^*$ under the transpose homomorphism $t^*: L^* \arr \OO^{\oplus n}_X$. The morphism $q: \G^+_{X^{(\infty)}} \arr \Coh_n$ sends $\{ L,t \}$ to $\OO^{\oplus n}_X/\on{Im} t^*$. In the same way as in Sect.~4.2, one can show that $q$ is smooth. We denote by the same character ${\mathcal L}_E$ the pull-back of ${\mathcal L}_E$ by $q$. It is the pair $(\G^+_{X^{(\infty)}},{\mathcal L}_E)$ that we would like to generalize to other groups. Let us describe the basic structure of $\G^+_{X^{(\infty)}}$. For each $k\geq 1$, we introduce the scheme $\G^+_{X^{(k)}}$ over $X^{(k),rss}$ (see Sect.~3.2), which parametrizes the objects $\{ (x_1,\ldots,x_k),$ $L,t \}$, where $(x_1,\ldots,x_k)$ is a set of $k$ non-ordered distinct points of $X$, $L$ is a rank $n$ bundle over $X$, and $t$ is its trivialization over $X - \{ x_1,\ldots,x_k \}$, which extends to an embedding of $\OO_X$--modules $\OO_X^{\oplus n} \arr L$. The fiber of this scheme over $(x_1,\ldots,x_k) \in X^{(k),rss}$ is the product of the $\G^+_{x_i}$. It is easy to describe the pull-back ${{\mathcal L}}_{X^{(k)}}^{E,+}$ of ${\mathcal L}_E$ under the natural morphism $\delta_k: \G^+_{X^{(k)}} \arr \G^+_{X^{(\infty)}}$. In particular, the restriction of ${{\mathcal L}}_{X^{(k)}}^{E,+}$ to the fiber of $\G^+_{X^{(k)}}$ over $(x_1,\ldots,x_k)$ is $\boxtimes_{i=1}^k {{\mathcal L}}_{x_i}^{E,+}$, where \begin{equation} \label{box1} {{\mathcal L}}_x^{E,+} = \sum_{\la \in P^{++}_n} {\mathcal A}_{\la,x}[|\la|(n-1)](|\la|(n-1)/2) \otimes E_x(\la). \end{equation} The set of $\Fq$--points of $\G^+_{X^{(\infty)}}$ is isomorphic to the quotient $GL_n(\A)^+/GL_n(\OO)$. For groups other than $GL_n$ we do not have analogues of the subset $GL_n(\A)^+ \subset GL_n(\A)$, the subset $P^{++}_n \subset \PL^+$, and the subscheme $\G^+$ of the affine Grassmannian. For this reason, we can not avoid considering a substantially larger object in place of $\G^+_{X^{(\infty)}}$. Thus, for a reductive group $G$, we consider the set $G(\A)/G(\OO)$. This set, which we denote by $Gr(G)_{X^{(\infty)}}$, is isomorphic to the set of isomorphism classes of pairs $\{ {\mathcal P},t \}$, where ${\mathcal P}$ is a principal $G$--bundle over $X$ and $t$ is an isomorphism between ${\mathcal P}$ and the trivial bundle on a Zariski open subset of $X$. It is not difficult to define a functor $\G(G)_{X^{(\infty)}}$ from the category of $\Fq$--schemes to the category of sets, whose set of $\Fq$--points is $Gr(G)_{X^{(\infty)}}$. It would be desirable to have a notion of perverse sheaf on $\G(G)_{X^{(\infty)}}$. A $\GL$--local system $E$ on $X$ should give rise to a perverse sheaf ${\mathcal L}_E$ on $\G(G)_{X^{(\infty)}}$, analogous to the sheaf ${\mathcal L}_E$ in the case of $GL_n$; this sheaf should be irreducible if $E$ is irreducible. Although we do not know how to define such an object, we describe below what its pull-backs should be under certain natural morphisms. For each $k\geq 1$, following Beilinson and Drinfeld, we introduce the ind--scheme $\G(G)_{X^{(k)}}$ over $X^{(k),rss}$, which parametrizes the objects $\{ (x_1,\ldots,x_k),{\mathcal P},t \}$, where $(x_1,\ldots,x_k)$ is a set of non-ordered distinct points of $X$, ${\mathcal P}$ is a principal $G$--bundle over $X$, and $t$ is its isomorphism with the trivial bundle over $X - \{ x_1,\ldots,x_k \}$. The fiber of this scheme over $(x_1,\ldots,x_k) \in X^{(k),rss}$ is the product of the $\G_{x_i}$ (see \cite{MV}, Sect.~3). We have an obvious set-theoretic map $\delta_k: Gr(G)_{X^{(k)}} \arr G(G)_{X^{(\infty)}}$, which can also be defined on the level of functors: schemes $\arr$ sets. The pull-back of ${\mathcal L}_E$ to $\G(G)_{X^{(k)}}$ should be the sheaf ${{\mathcal L}}_{X^{(k)}}^E$ on $\G(G)_{X^{(k)}}$ (inductive limit of perverse sheaves), such that its restriction to the fiber $\prod_{i=1}^k \G(G)_{x_i}$ over $(x_1,\ldots,x_k)$ is $\boxtimes_{i=1}^k {{\mathcal L}}_{x_i}^E$, where \begin{equation} \label{rx} {{\mathcal L}}_x^E = \oplus_{\la \in \PL^+} {\mathcal A}_{\la,x} \otimes E_x(\la)^* \end{equation} (up to shifts in degree). Here $E_x(\la)$ has the same meaning as in the case of $GL_n$. The sheaves ${{\mathcal L}}_{X^{(k)}}^E$ have been previously considered by Beilinson and Drinfeld in the context of the geometric Langlands correspondence. Formula \eqref{rx} is analogous to formula \eqref{box1}. The main difference is that in \eqref{box1} the summation is restricted to the subset $P^{++}_n$ of the set $P^+_n = \PL^+$ of all dominant weights of $GL_n$, which does not have an analogue for general $G$ (compare with Sect.~6.4). \begin{rem} \label{cate} Let $\on{P}_{\GG(\OO_x)}(\G(G)_x)$ be the category of $\GG(\OO_x)$--equivariant perverse sheaves on $\G(G)_x$ (we consider objects of $\on{P}_{\GG(\OO_x)}(\G(G)_x)$ as pure of weight $0$). This category is a tensor category, and as such, it is equivalent to the tensor category ${{\mathcal R}ep} \GL$ of finite-dimensional representations of $\GL$. To be precise, this result has been proved in \cite{Gi,MV} over the ground field $\C$ (in this setting, this isomorphism was conjectured by V.~Drinfeld; see also \cite{Lu2}). But the proof outlined in \cite{MV} can be generalized to the $\Fq$--case, so here we assume the result to be true over $\Fq$ as well. Note that there is a small error in \cite{MV}. The tensor structure (or, more precisely, the commutativity constraint), which is given by the convolution product, should be altered slightly. This alteration does not affect the structure of $\on{P}_{\GG(\OO_x)}(\G(G)_x)$ as a monoidal category. We simply replace the perverse sheaf ${\mathcal A}_{\la,x}$ with $(-1)^{2(\la,\rho)}{\mathcal A}_{\la,x}$, where the sign $(-1)^{2(\la,\rho)}$ is to be viewed as a formal symbol. The symbol $(-1)^{2(\la,\rho)}$ has the effect of making the cohomology of $(-1)^{2(\la,\rho)}{\mathcal A}_{\la,x}$ lie in even degrees only. Then the equivalence of categories ${{\mathcal R}ep} \GL \arr \on{P}_{\GG(\OO_x)}(\G(G)_x)$ takes the irreducible representation $V(\la)$ to the perverse sheaf $(-1)^{2(\la,\rho)}{\mathcal A}_{\la,x}$. With this adjustment the sign in \propref{hla} disappears and the equivalence above can be viewed as a categorical version of the Satake isomorphism $\on{Rep} \GL \arr {\mathcal H}$ (see \thmref{Satake}). Indeed, an equivalence of two categories induces an isomorphism of their Grothendieck rings. But the Grothendieck ring of $\on{P}_{\GG(\OO_x)}(\G(G)_x)$ is canonically isomorphic to the Hecke algebra ${\mathcal H}$ via the ``faisceaux--fonctions'' correspondence. Now consider the left regular representation of $\GL$, $$\oplus_{\la \in \PL^+} V(\la) \otimes V(\la)^*$$ as an ind--object of the category ${{\mathcal R}ep} \GL$. The corresponding ind--object of the category $\on{P}_{\GG(\OO_x)}(\G(G)_x)$ is ${{\mathcal L}}^E_x$ given by formula \eqref{rx}, adjusted for the formal signs, i.e., \begin{equation} \label{rxx} {{\mathcal L}}_x^E = \oplus_{\la \in \PL^+} (-1)^{2(\la,\rho)}{\mathcal A}_{\la,x} \otimes E_x(\la)^*. \end{equation} \qed \end{rem} \begin{rem} Let $L_X^E$ be the function associated to the sheaf ${{\mathcal L}}_X^E$, and $L_x^E$ be the restriction of $L_X^E$ to $Gr(G)_x \subset Gr(G)_X$. Using \propref{hla} we obtain: \begin{equation} \label{series} L_x^E = \sum_{\la \in \PL^+} \on{Tr}(\gamma_x,V(\la)^*) \cdot H_{\la,x}, \end{equation} where $\gamma_x = \sigma(\on{Fr}_x)$. Hence the function $L_x^E$ coincides with the function $L_{\gamma_x^{-1}}$ given by formula \eqref{Fgamma}. According to \propref{converge}, the series \eqref{series} converges absolutely, if $q_x^{-1} < |\al(\gamma_x)| < q_x, \forall \al \in \De_+$, and is proportional to the spherical function $S_{\gamma_x^{-1}}$.\qed \end{rem}

9703

alg-geom/9703024

en

https://arxiv.org/abs/alg-geom/9703024

alg-geom/9703024

Ludwig Balke

Ludwig Balke

A note on P-resolutions of cyclic quotient singularities

8 pages, 4 Postscript figures

P-resolutions of a cyclic quotient singularity are known to be in one-to-one correspondence with the components of the base space of its semi-universal deformation. Stevens and Christophersen have shown that P-resolutions are parametrized by so-called chains representing zero or, equivalently, certain subdivisions of polygons. I give here a purely combinatorial proof of the correspondence between subdivisions of polygons and P-resolutions.

alg-geom

\section{Introduction} The discovery of Koll\'ar and Shepherd--Barron in \cite{Kollar-Shepherd-Barron} that the reduced components of the versal base space of a quotient singularity are in one-to-one correspondence with the P-resolutions of this singularity provided a more conceptual understanding of the deformation theory of cyclic quotient singularities. Christophersen \cite{Christophersen} and Stevens \cite{Stevens} found a beautiful description of these components in terms of so-called chains respresenting zero or subdivisions of polygons. Nevertheless, their proofs were very involved and used algebro-geometric methods. Altmann \cite{Altmann} gave a description of P-resolutions and their correpondence to chains representing zero in terms of toric varieties. In this note, I will emphasize that P-resolutions on the one hand (cf.~Lemma~\ref{lem_Presolution}) and subdivisions of polygons on the other hand are purely combinatorial objects. Hence it is interesting to establish the correspondence between both classes by purely combinatorial methods. Therefore, I will refer to cyclic quotient singularities and P-resolutions only in so far as it is necessary to define the objects to be investigated here. The reader can find in \cite{Stevens} more information on cyclic quotient singularities and their deformation theory. \section{Sequences and subdivisions} Let $\cW$ denote the monoid of sequences $(a_1,\dots,a_k)$ with $a_i\in \mathbb{N}} \newcommand{\cW}{\mathcal{W}\setminus\{0\}$ and $\cW_2$ the submonoid of such sequences with $a_i\geq 2$. The empty sequence $()$ is denoted by $\varepsilon} \newtheorem{lemma}{Lemma$. Furthermore, let $M$ denote the free monoid with generators $\alpha,\beta$ and $M_2$ the free monoid with generators $\alpha, \gamma$. Both monoids act on $\cW$ respectively $\cW_2$ in the following manner from the lefthand side or from the righthand side, respectively : \begin{alignat*}{2} \alpha \varepsilon} \newtheorem{lemma}{Lemma & = \varepsilon} \newtheorem{lemma}{Lemma & \varepsilon} \newtheorem{lemma}{Lemma\alpha & = \varepsilon} \newtheorem{lemma}{Lemma\\ \alpha(a_1,\dots,a_k)& =(a_1,\dots,a_k +1) & \qquad (a_1,\dots,a_k)\alpha& =(a_1+1,\dots,a_k) \\ \beta a & = a(1) & a\beta & = (1)a\\ \gamma a & = \alpha \beta a = a(2) & a\gamma & = a \beta\alpha = (2)a \end{alignat*} Here $a$ is an arbitrary element of $\cW$ respectively $\cW_2$ and the composition in the respective monoids is denoted by juxtaposition. The following lemma is easy to observe: \begin{lemma} For each $a\in\cW$ there exists a unique element $\rho\in M$ such that $a=\rho \varepsilon} \newtheorem{lemma}{Lemma$ and $\rho=1$ or $\rho=\rho'\beta$ for a suitable $\rho'$. An analogous statement holds for $\cW_2$ and $M_2$. The inversion of a sequence in $\cW$ is defined by $\overleftarrow{(a_1,\dots,a_k)}:=(a_k,\dots,a_1)$. Analogous inversions are defined for the free monoids $M$ and $M_2$. Then we have for $a\in\cW, \rho \in M$ or $a\in\cW_2,\rho\in M_2$, respectively. \[ \rho a = \overleftarrow{a}\overleftarrow{\rho}\] \end{lemma} We define a map $R:\cW\to\cW$ inductively as follows. On $\cW_2$ we have $R(\varepsilon} \newtheorem{lemma}{Lemma)=\varepsilon} \newtheorem{lemma}{Lemma$ and \[R(\alpha a) =\gamma R(a)\quad\text{and}\quad R(\gamma a) = \alpha R(a)\] for $a\in\cW_2, a\neq\varepsilon} \newtheorem{lemma}{Lemma$. For arbitrary elements $a\in\cW$, the map $R$ is defined by the rule \[ R(a(1)a') = R(a)(1)R(a').\] A straightforward induction shows: \begin{lemma} $R$ is an involution, i.e.~$R^2=\mbox{id}_{\cW}$, and \[ R(\overleftarrow{a})=\overleftarrow{R(a)}. \]\label{lem_involution} \end{lemma} \begin{lemma} For $a',a''\in\cW$ and $e,f\geq 2$: \[ R(a' (e+f-1) a'') = R(a'(e))R ((f)a'').\] \label{lem_split} \end{lemma} A nonempty sequence $(a_1,\dots,a_k)\in\cW_2$ defines a continued fraction \[ [a_1,\dots,a_k] =a_1 - \cfrac{1}{a_2 - \cfrac{1}{\dots - \cfrac{1}{a_k }}} \] It is an easy consequence from Lemma~\ref{lem_involution} that $[R(a)]=\frac{n}{n-q}$ if $[a] =\frac{n}{q}$. Hence the operator $R$ generalizes Riemenschneider's point diagram rule to $\cW$ (cf.~\cite{Riemenschneider}, \cite{Stevens}, 1.2). For later use, we need a relation between the positions in the sequence $a$ and the positions in the sequence $R(a)$. This relation will be inductively defined. Let $\ell$ be the length of $a$ and $r$ the length of $R(a)$. The $(\ell+1)$-th position of $\gamma a$ is associated with the $r$-th position of $R(\gamma a)=\alpha R(a)$ and the $\ell$-th postion of $\alpha a$ is associated with the $(r+1)$-th position of $R(\alpha a) = \gamma R(a)$. Furthermore, the $(\ell +1)$-th position of $\beta a$ is associated with the $(r+1)$-th position of $R(\beta a)=\beta R(a)$. Obviously, the $i$-th position of $a$ is associated with the $j$-th position of $R(a)$ if and only if the $j$-th position of $R(a)$ is associated with the $i$-th position of $R(R(a))=a$. We fix a countable set $\mathcal{V}} \newcommand{\cS}{\mathcal{S}$ together with a linear ordering $\leq$ of $V$. Elements of $V$ will be called vertices and if $V$ is a finite subset of $r$ vertices then $v_i$ with $1\leq i\leq r$ is defined by the equation $V=\{v_0< \dots <v_r\}$. A {\em subdivision of an polygon with distinguished vertex}, or shortly subdivision, is a pair $(V,\Delta)$ such that $V$ is a nonempty finite subset of $\mathcal{V}} \newcommand{\cS}{\mathcal{S}$ and $\Delta$ is a set of subsets of $V$ with the following property. There exists a subdivision of a plane polygon into triangles and a bijection between the vertices of the polygon and the elements of $V$ such that $v_i$ and $v_{i+1}$ correspond to consecutive vertices and each set of vertices of a triangle is identified with an element of $\Delta$ and vice versa. We accept a digon as plane polygon, hence $(V,\emptyset)$ is a subdivision if $\# V =2$. An {\em isomorphism} between subdivisions $(V,\Delta)$ and $(V',\Delta')$ is a bijection from $V$ to $V'$ which maps $\Delta$ bijectively onto $\Delta'$. Hence this bijection either preserves or reverses the ordering $\leq$. Obviously, the notion of subdivision can be defined in a purely combinatorial way without reference to plane polygons. But we do not bother the reader with an axiomatic approach since this would not enhace the understanding of this concept. We define two laws of compositions on the set of subdivisions. Let $(V_1,\Delta_1)$ and $(V_2,\Delta_2)$ be subdivisions with \begin{eqnarray*} v^0&:=&\min V_1 = \min V_2\\ v^1&:=&\max V_1 < v^2:=\min V_2\setminus \{\min V_2\}. \end{eqnarray*} Then we can define a new subdivision $(V_1,\Delta_1)(V_2,\Delta_2):=(V,\Delta)$ with \[V:=V_1\cup V_2\quad\text{and}\quad \Delta := \Delta_1\cup \Delta_2\cup \{\{v^0,v^1,v^2\}\}\] A subdivision is called {\em irreducible} if it is not the product of two subdivisions and it is called {\em primary} if there is atmost one triangle with $\min V$ as vertex. Obviously, a primary subdivision is also a irreducible one, but, in general, the converse is not true. Furthermore, each subdivision has a unique decomposition into irreducible factors. Assume now $v^1=v^2$, then $(V_1,\Delta_1)*(V_2,\Delta_2):=(V,\Delta)$ with \[V:= V_1\cup V_2\quad\text{and}\quad \Delta:=\Delta_1\cup\Delta_2\] Each irreducible subdivision can by uniquley decomposed into primary factors with respect to the law of composition $*$. Let $(V,\Delta)$ be a subdivison of an $n$-gon and $a=(a_1,\dots,a_n)\in\cW$. The sequence $a$ can be understood as a labelling of the vertices $v$ of $V$ and we define the {\em degree} $\deg v=\deg_a v$ of a vertex with respect to this labelling in such a way that it emphasizes the special role of vertices with label $1$. Consider triangles of $\Delta$ as equivalent if they have an edge in common and one of the triangles has a vertex with label $1$. Then the degree of a vertex $v$ is the number of equivalence classes of triangles having $v$ as one of its vertices. In particular, a vertex with label $1$ has always degree $1$. We call $a$ {\em admissible} for $(V,\Delta)$, if we always have \[\deg_a v_i \leq a_i.\] The difference $\operatorname{def}_a v_i := a_i - \deg_a v_i$ is called the {\em defect} of the vertex $v_i$. The set of triples $(V,\Delta,a)$ with $(V,\Delta)$ a subdivision and $a$ admissible for $(V,\Delta)$ is the set of {\em labelled subdivisions} and is denoted by $\cS$. A vertex $v$ of $(V,\Delta)$ is called a {\em splitting vertex} if it is the maximal or the second vertex of an irreducible factor of $(V,\Delta)$. A vertex $v$ is called {\em saturated} if its defect is $0$, or if it is a splitting vertex and both neighbouring vertices have a label different from $1$, or if this vertex together with its both neighbours forms a triangle. A triangle $t\in\Delta$ is called an {\em interior triangle} if and only if $t=\{v_i,v_j,v_k\}$ with $i+1<j<k-1$ and $0<i$. A nonempty sequence $a\in\cW$ can be associated with a labelled graph $\Gamma (a)$ as follows. If $a=(a_1,\dots,a_k)$ then $\Gamma (a)$ is a chain with $k$ nodes which are labelled consecutively with the numbers $-a_1,\dots,-a_k$. For $a\in\cW_2$ with $[a]=\frac{n}{q}$, the graph $\Gamma (R(a))$ is the graph of the minimal resolution of the cyclic quotient singularity $X(n,q)$ (cf. \cite{Riemenschneider}). \begin{figure} \epsfysize3cm \hspace*{\fill}\epsffile{tsub.eps}\hspace*{\fill} \caption{Inductive definition of T-subdivisions} \label{fig_tsubdivisions} \end{figure} \section{P-resolutions and subdivisions} We recall the inductive definition of graphs of type T using our terminology (cf.~\cite{Stevens}). The graph $\Gamma (R(2,a,2))$ for $a\geq 2$ is a graph of type T of the first kind. If $\Gamma (b)$ is a graph of type T , then $\Gamma (\alpha b \gamma)$ and $\Gamma (\gamma b \alpha)$ are graphs of type T of the second kind. We define a subset $T$ of subdivions inductively as depicted in Fig.\ref{fig_tsubdivisions}. If $(V,\Delta)\in T$, then there exists exactly one vertex $v=v_i\in V$ such that $\{v_{i-1},v_i,v_{i+1}\}\in\Delta$. This vertex is called the {\em central vertex} and $i$ the {\em central index} of $(V,\Delta)$. By induction, one easily obtains: \begin{lemma} Let $(V,\Delta)\in T$ and $(V,\Delta,a)\in \cS$ such that all vertices different from the central one have defect $0$. Then $\Gamma (R(a))$ is a graph of type T. On the other hand, assume that $\Gamma (R(a))$ is a graph of type T. Then there exists a subdivision $(V,\Delta)\in T$ such that $(V,\Delta,a)\in\cS$ and all vertices different from the central one have defect $0$. For each set $V$ of vertices with the appropriate cardinality there exists exactly one $\Delta$ with these properties. \label{lem_tsubdivision} \end{lemma} \begin{lemma} Let $(V,\Delta,a)\in \cS$ with $(V,\Delta)\in T$ such that all vertices $v>v_0$ are saturated. Then there exist unique numbers $\ell,r\geq 0$ and a unique $a'\in\cW_2$ such that $\Gamma (R(a'))$ is a graph of type T and $a=\alpha^\ell a' \alpha^r$. \label{lem_characterizeT} \end{lemma} A P-resolution of a quotient singularity $X$ is a partial resolution $Y\longrightarrow X$ such that $K_Y\cdot E_i$ for all exceptional divisors, and all singularities of $Y$ have a resolution graph of type T or $A_k$ (cf. \cite{Stevens}, 3.1). An easy calculation with the adjunction formula shows \begin{lemma} Let $Y$ be a P-resolution of the cyclic quotient singularity $X$ and $G$ the graph of its minimal resolution $\Tilde{Y}$. Let $J$ be the subset of nodes which blow down to singularities of type T in $Y$. Then each connected component of $J$ is a subgraph of $G$ being of type $T$. The $-1$-nodes in $G$ are adjacent to two nodes in $J$ and at least one of this node belongs to a component of $J$ which is of the second kind. The subset $K$ of nodes of $G$ which blow down to $A_k$--singularities can be characterized as follows: A $-2$-node belongs to $K$ if and only if none of its neighbours belongs to $J$. \label{lem_Presolution} On the other hand: If $G$ is a resolution graph of the quotient singularity $X$ and $J$ a subset of nodes of $G$ satisfying the above properties, then blowing down the nodes in $J$ and those in $K$ which is defined as above yields a P-resolution of $X$. \end{lemma} Assume $G=\Gamma (b)$ for a graph $G$ of the minimal resolution of a P-resolution. Let $I_1,\dots,I_r$ be the maximal intervals of $J$, the set defined in lemma \ref{lem_Presolution}. Each interval $I_j$ is associated with a interval $H_j$ of positions of $a=R(b)$. These intervals need not longer to be disjoint, they may have extremal positions in common. Let $\{M_1,\dots,M_m\}$ be the set of all sets of positions of $R(b)$ which are either equal to one of the $H_j$ or a maximal interval in the complement of the union of all $H_j$. We define inductively a subdivision $(V,\Delta)$ with $(V,\Delta,a)\in\cS$. If $M_i$ is one of the $H_j$ then $(V_i,\Delta_i)$ is chosen according to Lemma~\ref{lem_tsubdivision} for the subgraph of $G$ determined by the interval $I_j$. For all other $M_i$ the subdivision $(V_i,\Delta_i)$ is chosen such that all triangles have the minimal vertex as vertex. Then we set \[(V,\Delta)=(V_1,\Delta)x \dots x(V_m,\Delta_m),\] where $x$ denotes $\cdot$ or $*$, respectively, depending on whether $M_i\cap M_{i+1} = \emptyset$ or not. The set of all these labelled subdivisions is the set $\cP$. Obviously, we can reconstruct the resolution graph $G$ and the subset $J$ of nodes of $G$ from an element of $\cP$ which is associated with $G$ by our construction, i.e.~there exists a canonical bijection between P-resolutions and isomorphism classes of $\cP$. On the other hand, we have the set $\mathcal{M}} \newcommand{\then}{\Longrightarrow$ of all labelled subdivisions $(V,\Delta,a)$ with $a\in \cW_2$. The isomorphism classes of this set are in a canonical one-to-one correspondence with graphs of minimal resolutions of cyclic quotient singularities via the map $\Gamma\circ R$ (cf.~\cite{Riemenschneider}). The remaining part of this paper will provide a proof of: \begin{theorem} There exists a canonical one-to-one correspondence between isomorphism classes of $\mathcal{M}} \newcommand{\then}{\Longrightarrow$ and isomorphism classes of $\cP$ with the following property. If $(V,\Delta,a)\in \cP$ and $(V',\Delta',a') \in \mathcal{M}} \newcommand{\then}{\Longrightarrow$ correspond to each other, then $\Gamma R(a')$ can be obtained from $\Gamma R(a)$ by successively blowing down $-1$-nodes. \end{theorem} \begin{proof} Blowing down and blowing up yields operations on the labels of subdivisions via the operator $R$, which is an involution. In order to find the right correspondence between $\mathcal{M}} \newcommand{\then}{\Longrightarrow$ and $\cP$, it is necessary to find suitable positions for blowing up and to define suitable modifications of the underlying subdivision. As we will see later, the subdivision itself gives a hint where to blow up. First of all, we prove a useful characterization of the set $\cP$: \begin{lemma} A subdivision $(V,\Delta,a)$ is an element of $\cP$ if and only if \begin{enumerate} \item There exist no interior triangles.\label{lem_char1} \item All vertices are saturated.\label{lem_char2} \item If an irreducible component has more than two vertices then its label is in $\cW_2$.\label{lem_char3} \item A vertex with label $1$ has two neighbouring vertices which belong to primary factors being elements of $T$. One of this factor has more than four vertices.\label{lem_char4} \end{enumerate}\label{lem_characterize_cP} \end{lemma} \begin{proof} The necessity of these conditions follows easily from Lemma~\ref{lem_Presolution}. Let us assume that all these conditions hold. Condition~\ref{lem_char1}--\ref{lem_char3} imply with Lemma~\ref{lem_characterizeT} that a primary component $(V',\Delta')$ with more than two vertices is an element of $T$ and its label is $\alpha^\ell a' \alpha^r$ with $\Gamma R(a')$ a graph of type T and $\ell,r\geq 0$. If the next vertex to the lefthand or the righthand side of this component has label $1$ then $\ell = 0$ or $r=0$, respectively, due to the definition of a saturated vertex. The conditons~\ref{lem_char2} and \ref{lem_char4} guarantee that the $-1$--nodes of $\Gamma (R(a))$ have the properties of lemma \ref{lem_Presolution}. Hence $\Gamma (R(a))$ is the graph of the minimal resolution of a P-resolution. \end{proof} This lemma shows that one should blow up in such a way that interior triangles and non-saturated vertices disappear. In Fig.~\ref{fig_blowup1} and Fig.~\ref{fig_blowup2}, one can see three operations on subdivisions. We call the first one an operation of type B1, and the second and third one operations of type B2a or B2b , respectively. All of them are operations of type B. \begin{figure} \epsfysize4cm \hspace*{\fill}\epsffile{blowup1.eps}\hspace*{\fill} \caption{An operation of type B1} \label{fig_blowup1} \end{figure} \begin{figure} \epsfysize7.5cm \hspace*{\fill}\epsffile{blowup2.eps}\hspace*{\fill} \caption{Operations of type B2a and B2b} \label{fig_blowup2} \end{figure} An operation of type B changes the label of the subdivision as follows. From $a=a'(e+f-1)a''$ we proceed to $b=a'(e,2,1,2,f)a''$. According to Lemma~\ref{lem_split}, we have $R(a)=R(a'(e))R((f)a'')$. Therefore, $R(b) = R(a'(e+1))(1)R((f+1)a'')$, i.e.~$\Gamma (R(b))$ is obtained by blowing up from $\Gamma (R(a))$. An analogous computation shows that for operations of type B2a and B2b, the graph $\Gamma (R(B))$ is obtained from the graph $\Gamma (R(a))$ by blowing up between a $-1$-node and another node. We will define a map $F:\cS \longrightarrow \cS$ which consists of applying a suitable operation of type $B$, if appropriate, and show that $F^k(V,\Delta,a)\in\cP$ for $(V,\Delta,a)\in\mathcal{M}} \newcommand{\then}{\Longrightarrow$ and $k$ sufficiently large. On the other hand, a map $G$ will be defined with the property $F(G(V,\Delta,a))=(V,\Delta,a) =G(F(V,\Delta,a))$ if $F(V,\Delta,a) \neq (V,\Delta,a)$ and $G(V,\Delta,a)\neq (V,\Delta,a)$. Furthermore, $G^k (V,\Delta,a)\in\mathcal{M}} \newcommand{\then}{\Longrightarrow$ for $(V,\Delta,a)\in\cP$ and $k$ sufficiently large. Before we come to the definition of $F$ we need some technical definitions. A configuration as depicted in Fig.~\ref{fig_hexafan} is called a {\em hexagonal fan}. The vertex which is incident to all four triangles is the {\em root} of the fan and the vertex opposite to the root is the {\em apex} of the fan. Furthermore, we require that the two triangles incident to the apex are not interior ones. \begin{figure} \epsfysize2.5cm \hspace*{\fill}\epsffile{hexafan.eps}\hspace*{\fill} \caption{A hexagonal fan} \label{fig_hexafan} \end{figure} Assume that we are given a subdivision $(V,\Delta)$ and a subset $W$ of $V$ with $v_0\notin W$. The {\em height} of $W$ is the number i+(k+1-j) where $v_i$ is the minimal element of $W$, $v_j$ the maximal element of $W$ and $k+1=\# V$. Let $(V,\Delta,a)\in\cS$. If conditions~\ref{lem_char1}--\ref{lem_char3} hold, then $F(V,\Delta,a):=(V,\Delta,a)$. Otherwise, there is a vertex $v_j$ such that one of the following conditions hold. \begin{enumerate} \item There exists an interior triangle $\{v_i,v_j,v_k\}\in\Delta$ with $i<j<k$.\label{case1} \item The vertex $v_j$ is not saturated.\label{case2} \item We have $a_j=1$ and $\{v_0,v_j\}$ is not an irreducible component of $(V,\Delta,a)$.\label{case3} \end{enumerate} Choose $j$ minimal with this property. Note that case~\ref{case1} and case~\ref{case2} may occur simultaneously. If condition~\ref{case1} holds with $a_j\geq 3$ and all involved vertices have a label different from $1$, then we apply an operation of type B1 to the triangle $\{v_i,v_j,v_k\}$, otherwise we set $F(V,\Delta,a) = (V,\Delta,a)$. For this, we have to specify numbers $e,f$ with $a_j = e + f -1$. Choose $e$ such that it is the degree of the vertex whose label it is in the new subdivision. After having applied this operation either the number of interior triangles has become smaller or the height of the first such triangle has become smaller. Furthermore, we have obtained a hexagonal fan whose root is its minimal or maximal vertex, its apex has label $1$ and all other vertices of this fan have label different from $1$. Furthermore, no interior triangle has this apex as vertex. In the remaining cases condition~\ref{case2} holds in, we apply again a operation of type B1. First we specify the triangle the operation is applied to. Two cases can occur. First, that there exists a triangle $\{v_{j-1},v_j,v_k\}$ with $k>j$, secondly, that there exists a triangle $\{v_j, v_{j+1},v_k\}$ with $k<j$. In the first case, the number $f$ is chosen such that it is the degree of the vertex whose label it is in the new subdivision. In the second case, the number $e$ is chosen in such manner. We observe, that the number of non-saturated vertices is reduced by this operation, since the vertex with label $e$ or $f$, respectively, will be a vertex of a triangle with three consecutive vertices, due to the construction. We now deal with the case that condition~\ref{case3} holds. We assume that there is a hexagonal fan with $v_j$ as apex such that its root is the minimal or maximal vertex of this fan and all vertices of the fan being different from $v_j$ have a label $\geq 2$. In all other cases, we define $F(V,\Delta,a)=(V,\Delta,a)$. If the root is minimal, we can apply operation B2a, otherwise we apply operation B2b. Having applied this operation, we have again a hexagonal fan as above. If the root is not the minimal vertex $v_0$ then its height is smaller than the height of the fan above. If $(V,\Delta,a)$ is obtained from an element of $\mathcal{M}} \newcommand{\then}{\Longrightarrow$ by succesive application of $F$, then proceeeding by induction, one can easily verify the following facts: \begin{itemize} \item If $v_j$ is as above, then all vertices with label $1$ are smaller than this vertex. \item Each vertex with label $1$ is apex of a hexagonal fan whose root is its minimal or maximal vertex. If $v_0$ is vertex of the hexagonal fan then it is its root. \item A vertex with label $1$ has two neighbouring vertices which belong to primary factors being elements of $T$. One of these factors has more than four vertices. \end{itemize} Let us call a labelled subdivision with these properties a good one. Hence, if $F(V,\Delta,a)=(V,\Delta,a)$ and $(V,\Delta,a)$ is good, then there exists no vertex $v_j$ as above and hence $(V,\Delta,a)\in\cP$, as desired. The considerations above show that this happens after finitely many applications of $F$, since in each step we diminish one of the following numbers: the number of interior triangles, the number of non-saturated vertices, the number of vertices $v_j$ such that $a_j=1$ and $\{v_0,v_j\}$ is not an irreducible component, the height of the first interior triangle, the height of the first hexafan whose apex has label $1$. Furthermore, the inverse map $G$ can be obtained as follows. Apply on good labelled subdivisions, if possible, the inverse of one of the operations of type $B$ which yield a blowing down of the rightmost $-1$-node in $\Gamma ( R (a))$. Otherwise, $G$ maps a labelled subdivision onto itself. The desired properties of $G$ are now easily to observe. \end{proof}

9703

alg-geom/9703012

en

https://arxiv.org/abs/alg-geom/9703012

alg-geom/9703012

Nitin Nitsure

Nitin Nitsure

Moduli of regular holonomic D-modules with normal crossing singularities

LaTeX, 41 pages, 124 KB

This paper solves the global moduli problem for regular holonomic D-modules with normal crossing singularities on a nonsingular complex projective variety. This is done by introducing a level structure (which gives rise to ``pre-D-modules''), and then introducing a notion of (semi-)stability and applying Geometric Invariant Theory to construct a coarse moduli scheme for semistable pre-D-modules. A moduli is constructed also for the corresponding perverse sheaves, and the Riemann-Hilbert correspondence is represented by an analytic morphism between these moduli spaces.

alg-geom

\subsection*{\hbox{}\hfill{\normalsize\sl #1}\hfill\hbox{}}} \textheight 23truecm \textwidth 15truecm \addtolength{\oddsidemargin}{-1.05truecm} \addtolength{\topmargin}{-1.5truecm} \makeatletter \def\l@section{\@dottedtocline{1}{0em}{1.2em}} \makeatother \begin{document} \title{Moduli of regular holonomic ${\cal D}$-modules \\ with normal crossing singularities } \author{Nitin Nitsure} \date{09-III-1997} \maketitle \centerline{Tata Institute of Fundamental Research, Mumbai 400 005, India.} \centerline{e-mail: [email protected]} \centerline{Mathematics Subject Classification: 14D20, 14F10, 32G81, 32C38} \begin{abstract} This paper solves the global moduli problem for regular holonomic $\cal D$-modules with normal crossing singularities on a nonsingular complex projective variety. This is done by introducing a level structure (which gives rise to ``pre-$\cal D$-modules''), and then introducing a notion of (semi-)stability and applying Geometric Invariant Theory to construct a coarse moduli scheme for semistable pre-$\cal D$-modules. A moduli is constructed also for the corresponding perverse sheaves, and the Riemann-Hilbert correspondence is represented by an analytic morphism between these moduli spaces. \end{abstract} \vfill \newpage \tableofcontents \vfill \newpage \section{Introduction} The moduli problem for regular holonomic ${\cal D}$ modules on a non-singular complex projective variety $X$ has the following history. Around 1989, Carlos Simpson solved the problem in the case when the ${\cal D}$-modules are ${\cal O}$-coherent, which first appeared in a preliminary version of his famous paper [S]. In this case, a ${\cal D}$-module $M$ on $X$ is the same as a vector bundle together with an integrable connection. The next case, that of meromorphic connections with regular singularity along a fixed normal crossing divisor $Y\subset X$ was solved in [N]. As explained there, one has to consider a level structure in the form of logarithmic lattices for the meromorphic connections in order to have a good moduli problem (or an Artin algebraic stack), and secondly, a notion of semi-stability has to be introduced in order to be able to apply Geometric Invariant Theory. In collaboration with Claude Sabbah, a more general case was treated in [N-S], where the divisor $Y\subset X$ is required to be smooth, but the only restriction on the regular holonomic module $M$ is that its characteristic variety should be contained in $X\cup N^*_{S,X}$ (this is more general than being a non-singular or meromorphic connection). Here, we introduced the notion of pre-${\cal D}$-modules, which play the same role for these regular holonomic modules that logarithmic connections play for regular meromorphic connections. A notion of semistability was introduced for the pre-${\cal D}$-modules, and a moduli was constructed. We also constructed a moduli for the corresponding perverse sheaves, and showed that the Riemann-Hilbert correspondence defines an analytic morphism from the first moduli to the second, and has various good properties. This is already the most general case if $X$ is $1$ dimensional. The present paper solves the moduli problem in the general case where we have a divisor with {\sl normal crossings}, and the caracteristic variety of the regular holonomic ${\cal D}$-modules is allowed to be any subset of the union of the conormal bundles of the nonsingular strata of the divisor. This is done by extending the notion of pre-${\cal D}$-modules to this more general case, defining semistability, and constructing a moduli for these using GIT methods and Simpson's construction in [S] of moduli for semistable $\Lambda$-modules. Also, a moduli is constructed for the corresponding perverse sheaves, and the Riemann-Hilbert correspondence is represented by an analytic morphism having various good properties. We now give a quick overview of the contents of this paper. Let $X$ be a nonsingular variety of dimension $d$, and let $Y\subset X$ be a divisor with normal crossings (the irreducible components of $Y$ can be singular). Let $S_d=X$, $S_{d-1}=Y$, and let $S_i$ be the singular locus of $S_{i+1}$ for $i< d-1$. This defines a filtration of $X$ by closed reduced subschemes $$S_d\supset S_{d-1}\supset \ldots \supset S_0$$ where each $S_i$ is either empty or of pure dimension $i$. Let $S'_d=X-Y$ and for $i<d$ let $S'_i =S_i-S_{i-1}$. The stratification $X= \cup _iS'_i$ is called the {\bf singularity stratification} of $X$ induced by $Y$. Let $T^*X$ be the total space of the cotangent bundle of $X$, and for $i\le d$ let $N^*_i\subset T^*X$ be the locally closed subset which is the conormal bundle of the closed submanifold $S'_i$ of $X-S_{i-1}$. In particular, $N^*_d$ is the zero section $X\subset T^*X$. Let $N^*(Y)\subset T^*X$ be defined to be the union $$N^*(Y) = N^*_d \cup N^*_{d-1} \cup N^*_{d-2} \cup \ldots$$ of all the $N^*_i$ for $i\le d$. Note that $N^*(Y)$ is a closed lagrangian subset of $T^*X$, and any irreducible component of $N^*(Y)$ is contained in the closure $\ov{N^*_i}$ for some $i$. In this paper we consider those regular holonomic ${\cal D}$ modules on $X$ whose characteristic variety is contained in $N^*(Y)$. Equivalently (under the Riemann-Hilbert correspondence), we consider perverse sheaves on $X$ which are cohomologically constructible with respect to the singularity stratification $$X=\cup_{0\le i\le d}S'_i$$ Such regular holonomic ${\cal D}$ modules (equivalently, such perverse sheaves) form an abelian category, in which each object is of finite length. These will be called regular holonomic ${\cal D}$-modules (or perverse sheaves) on $(X,Y)$. In section 2, we introduce the basic notation involving various morphisms which arise out of the singularity strata, their normalizations, and their \'etale coverings. One of the problems we had to overcome was to give a convenient ${\cal O}$-coherent description of regular holonomic ${\cal D}$-modules on $(X,Y)$. This is done in section 3, by extending the notion of a pre-${\cal D}$-modules from [N-S] to this more general setup. In section 4, we explain the functorial passage from pre-${\cal D}$-modules to ${\cal D}$-modules. In the case of a smooth divisor, this was only indirectly done, via Malgrange's presentation of ${\cal D}$-modules, in [N-S]. Here we do it more explicitly in our more general situation. In section 5, we define a notion of (semi-)stability, and construct a moduli scheme for (semi-)stable pre-${\cal D}$-modules on $(X,Y)$ with prescribed numerical data (in the form of Hilbert polynomials). We have given here an improved quotient construction, which allows us to give a much more simplified treatment of stability when compared with [N-S]. In section 6, we represent perverse sheaves on $(X,Y)$ with prescribed numerical data in {\sl finite terms}, via a notion of `Verdier objects', generalizing the notion of Verdier objects in [N-S]. We then construct a moduli scheme for these Verdier objects. This is a quotient of an affine scheme by a reductive group, so does not need any GIT. Though a more general moduli construction due to Gelfand, MacPherson and Vilonen exists in literature (see [G-M-V]), the construction here is particularly suited for the study the Riemann-Hilbert morphism (in section 7), which represents the de Rham functor. In section 7 we show that the Riemann-Hilbert correspondence defines an analytic morphism from (an open set of) the moduli of pre-${\cal D}$-modules to the moduli of perverse sheaves. We extend the rigidity results in [N] and [N-S] to this more general situation, which in particular means that this morphism is a local isomorphism at points with {\bf good residual eigenvalues} as defined later. {\footnotesize {\bf Erratum} I point out here some mistakes which have remained in [N] and [N-S], and their corrections. (1) The lemma 2.9 of [N] is false as stated, and needs the additional hypothesis that $A$ is reduced. This additional hypothesis of reducedness is satisfied in the part of proposition 2.8 of [N] where this lemma is employed. (2) On page 58 of [N-S], in the construction of a local universal family, we take the action of $G_i=PGL(p_i(N))$ on $Q_i$. This should read $G_i=SL(p_i(N))$ and not $PGL(p_i(N))$. (3) In Theorem 4.19.3 on page 63 of [N-S], the statement ``The S-equivalence class of a semistable reduced module ${\bf E} $ equals its isomorphism class if and only if ${\bf E} $ is stable'' should be corrected to read ``The S-equivalence class of a semistable reduced module ${\bf E}$ equals its isomorphism class if ${\bf E}$ is stable''. The ``... and only if...'' should be removed. (The proof only proves the ``if'' part, ignores the ``only if'' part, and the ``only if'' statement is in fact trivially false.) {\bf Acknowledgement} I thank the International Center for Theoretical Physics, \\ Trieste, for its hospitality while part of this work was done. I thank H\'el\`ene Esnault for providing the interesting example \ref{Esnault} below, in answer to a question. \section{Preliminaries on normal crossing divisors} In section 2.1 we define various objects naturally associated with a normal crossing divisor. The notation introduced here is summarized in section 2.2 for easy reference, and is used without further comment in the rest of the paper. \subsection{Basic definitions} Let $X$ be a nonsingular variety of dimension $d$, $Y$ a normal crossing divisor, and let closed subsets $S_i\subset X$ for $i\le d$ be defined as in the introduction ($S_d=X$, $S_{d-1}=Y$, and by descending induction, $S_{i-1}$ is the singularity locus of $S_i$). Let $m$ be the smallest integer such that $S_m$ is nonempty. We allow the possibility that $m$ is any integer from $0$ to $d$ (for example, if $m=d$ then $Y$ is empty, and if $m=d-1$ when $Y$ is smooth). For $i\ge m$, let $X_i$ be the normalization of $S_i$, with projection $p_i:X_i\to S_i$. This is a reduced, nonsingular, $i$-dimensional scheme of finite type over $C\!\!\!\!I$ for $i\ge m$. This may not be connected (same as irreducible), and we denote its components by $X_{i,a}$, as $a$ varies over the indexing set $\pi_0(X_i)$. Note that the fiber of $p_i:X_i\to S_i$ over a point $y\in S_i$ is the set of all branches of $S_i$ which pass through $y$ (where by definition a branch means a component in the completion of the local ring). Now let $I$ be a nonempty subset of $\{\, m,\ldots ,d-1\,\}$, and for any such $I$, let $m(I)$ denote the smallest element of $I$. To any such $I$, we now associate a scheme $X_I$ and a morphism $p_I:X_I\to S_{m(I)}$ as follows. By definition, $X_I$ is the finite scheme over $S_{m(I)}$ whose fiber over any point $y$ of $S_{m(I)}$ consists of all nested sequences $(x_j)$ of branches of $S_j$ at the point $y$, for $j$ varying over $I$. `Nested' means for any two $j,\,k\in I$ with $j \le k$, $x_k\in X_k$ is a branch of $S_k$ containing the branch $x_j\in X_j$ of $S_j$. In particular when $I=\{\, i\,\}$, $p_{\{ i\}}:X_{\{ i\} }\to S_i$ is just the normalization $p_i:X_i\to S_i$ of $S_i$. For nonempty subsets $I\subset J\subset \{\, m,\ldots ,d-1\,\}$, we have a canonical forgetful map $p_{I,J}:X_J\to X_I$. If $I\subset J\subset K$, then by definition we have the equality $$p_{I,K}=p_{I,J}\circ p_{J,K}:X_K\to X_I$$ For $m\le i\le d-1$ we denote by $Y^*_i$ the $d-i$ sheeted finite \'etale cover $p_{\{ i\},\{ i+1 \} }: X_{\{ i,i+1 \} }\to X_i$ of $X_i$, which splits the branches of $S_{i+1}$ which meet along $S_i$. Similarly, we denote by $Z_i$ the $C^{d-i}_2$ sheeted finite etale cover $p_{\{ i\} ,\{ i+2\} }: X_{\{i,i+2\} } \to X_i$ of $X_i$, which splits the branches of $S_{i+2}$ which meet along $S_i$. We denote by $Z^*_i$ the $2$ sheeted finite \'etale cover $p_{\{ i, i+2\} ,\{ i,i+1,i+2\} }: X_{\{i,i+1,i+2 \} } \to X_{\{ i,i+2 \} }$ of $Z_i$. Let $f_i:X_i\to X$ be the composite $X_i\stackrel{p_i}{\to}S_i\to X$. Let $N_i$ be the normal bundle of $X_i$ in $X$. This is defined by means of the exact sequence $$ 0\to T_{X_i} \to f_i^*T_X \to N_i\to 0 $$ Let $T(Y)\subset T_X$ be the closed subset of $T_X$ consisting of vectors tangent to branches of $Y$. This gives a closed subset $f_i^*T(Y)\subset f_i^*T_X$. Let $F_i$ be the closed subset of $N_i$ which is the image of $f^*T(Y)$ under the morphism $f^*T_X\to N_i$ of geometric vector bundles. Similarly, let $N_{i,i+1}$ be the normal bundle to $Y^*_i=X_{i,i+1}$ in $X$, defined with respect to the composite morphism $Y^*_i\to X_i\to X$, and let $F_{i,i+1}\subset N_{i,i+1}$ be the normal crossing divisor in the total space of $N_{i,i+1}$, defined by vectors tangent to branches of $Y$. Let $Y^*=Y^*_{d-1} =X_{d-1}$ be the normalization of $Y$. Let $h_i:Y^*_i\to Y^*$ be the canonical map, which associates to a point $(x_i, x_{i+1})\in X_{\{i,i+1\} }= Y^*_i$ the unique branch $y_{d-1}\in Y^*$ of $Y$ at $p_i(x_i)\in S_i$ such that $$x_i = x_{i+1}\cap y_{d-1}$$ This defines a vector hyper subbundle $H_{i,i+1} \subset N_{i,i+1}$ (hyper subbundle means rank is less by $1$) of $N_{i,i+1}$, which is given by vectors tangent to the branch of $Y$ given by $h_i:Y^*_i\to Y^*$. Note that $H_{i,i+1}$ is a nonsingular irreducible component of $F_{i,i+1}$. We now define some sheaves ${\cal D}_i$ of algebras differential operators on $X_i$ and ${\cal D}^*_i$ on $Y^*_i$. (These are `split almost polynomial algebras' of differential operators in the terminology of Simpson [S] as explained in later in section 5.1.) For this we need the following general remark: \refstepcounter{theorem}\paragraph{Remark \thetheorem} Let $V$ be nonsingular, $M\subset V$ a divisor with normal crossing, and $M'\subset M$ a nonsingular component of $M$, with inclusion $f:M'\hookrightarrow V$. Let $\Lambda = {\cal D}_V[\log M]$ be the subring of ${\cal D}_V$ which preserves the ideal sheaf $I_M\subset {\cal O}_V$, and let $\Lambda'= {\cal O}_{M'}\otimes_{{\cal O}_V}\Lambda$ (we will write $\Lambda' =f^*(\Lambda)$ or $\Lambda'=\Lambda|M'$ for brevity, though this abbreviated notation conceals that we have tensored with ${\cal O}_{M'}$ on the {\sl left}, for it does not matter on what side we tensor). Then $\Lambda'$ is naturally a split almost polynomial algebra of differential operators on $M'$ (see section 5.1), and the category of $\Lambda$ modules on $V$ which are schematically supported on $M'$ is naturally equivalent to the category of $\Lambda'$-modules on $M'$. This equivalence follows from the fact that $\Lambda$ necessarily preserves the ideal sheaf of $M'$ in $V$. Now we come back to our given set up, where we apply the above remark with $N_i$ (or $N_{i,i+1}$) as $V$, $F_i$ (or $F_{i,i+1}$) as $M$, and the zero section $X_i$ of $N_i$ (or the zero section $Y^*_i$ of $N_{i,i+1}$) as $M'$. As usual, let ${\cal D}_{N_i}[\log F_i]$ denote the subring of ${\cal D}_{N_i}$ consisting of all operators which preserve the ideal sheaf of $F_i$ in ${\cal O}_{N_i}$. Then we define ${\cal D}_i$ to be the restriction (see the above remark) of ${\cal D}_{N_i}[\log F_i]$ to the zero section $X_i\subset N_i$, which is canonically isomorphic to $f_i^*{\cal D}_X[\log Y]$. Similarly, we define ${\cal D}^*_i$ to be the restriction of ${\cal D}_{N_{i,i+1}}[\log F_{i,i+1}]$ to the zero section $Y^*_i$. If $H\subset M$ is a nonsingular irreducible component of a normal crossing divisor $M$ in a nosingular variety $V$, then the {\bf Euler operator along $H$ in ${\cal D}_V[\log M]$} is the element $\theta_H \in ({\cal D}_V[\log M]|H)$ which has the usual definition: if $V$ has local analytic coordinates $(x_1,\ldots x_d)$ with $M$ locally defined by $x_1\cdots x_{d-m}=0$ and $H$ by $x_1=0$, then the operator $\theta_H$ is given by the action of $x_1(\partial/\partial x_1)$. For each $i\le d-1$ we define the section $\theta_i$ of ${\cal D}^*_i$ as the restriction to $Y^*_i$ of the Euler operator along $H_{i,i+1}$ in ${\cal D}_{N_{i,i+1}}[\log F_{i,i+1}]$. The above definitions work equally well in the analytic category. For the remaining basic definitions, we restrict to the analytic category, with euclidean topology (in particular, if $T$ was earlier a finite type, reduced scheme over $C\!\!\!\!I$ then now the same notation $T$ will denote the corresponding analytic space). Vector bundles will denote their respective total analytic spaces. Let $U_i$ be the open subset $U_i =N_i-F_i$ of $N_i$, and let $R_i$ be the open subset $N_{i,i+1}-F_{i,i+1}$ of $N_{i,i+1}$. Let $N_{i,i+2}$ be the normal bundle to $Z_i$ in $X$, and let $W_i$ be the open subset of $N_{i,i+2}$ which is the complement in $N_{i,i+2}$ of vectors tangent to branches of $Y$. Similarly, let $N_{i,i+1,i+2}$ be the normal bundle to $Z^*_i$ in $X$. and let $W^*_i$ be the open subset of $N_{i,i+1,i+2}$ which is the complement in $N_{i,i+1,i+2}$ of vectors tangent to branches of $Y$. Finally, for $i\le d-1$ we define some central elements $\tau_i(c)$ of certain fundamental groups, which are the topological counterparts of the operators $\theta_i$. Let $Y^*_i(c)$, where $c$ varies over $\pi_0(Y^*_i)$, be the connected components of $Y^*_i$, and let $N_{i,i+1}(c)$, $F_{i,i+1}(c)$, $H_{i,i+1}(c)$, and $R_i(c)$ be the restrictions of the corresponding objects to $Y^*_i(c)$. Let $\tau_i(c)$ be the element in the center of $\pi_1(R_i(c))$, (with respect to any base point) which is represented by a positive loop around $H_{i,i+1}(c)$. The fact that $\tau_i(c)$ is central, and is unambigously defined, follows from the following lemma in the topological category. \begin{lemma} Let $S$ be a connected topological manifold, and $p:N\to S$ a complex vector bundle on $S$ of rank $r$. Let $F\subset N$ be a closed subset such that locally over $S^1$, the subset $F$ is the union of $r$ vector subbundles of $N$, each of rank $r-1$, in general position. Let $H\subset N$ be a vector subbundle of rank $r-1$ such that $H\subset F$. Let $U=N-F$ with projection $p:U\to S$, which is a locally trivial fibration with fiber $(C\!\!\!\!I\,^*)\,^r$. The fundamental group of any fiber is $Z\!\!\!Z^r$, with a basis given by positive loops around the various hyperplanes. Let $u_0\in U$ be a base point, and let $p(u_0)=s_0\in S$. Let $\tau_H\in \pi_1(U, u_0)$ be represented by the positive loop around $H\cap p^{-1}(s_0)$ in the fiber $U_{s_0}$. Then we have the following: (1) The element $\tau_H$ is central in $\pi_1(U,u_0)$. (2) Let $u_1\in U$ be another base point, and let $\tau'_H\in \pi_1(U,u_1)$ be similarly defined. Let $\sigma:[0,1]\to U$ be a path joining $u_0$ to $u_1$ , and let $\sigma^*: \pi_1(U,u_1)\to \pi_1(U,u_0)$ be the resulting isomorphism. Then $\sigma^*(\tau'_H)=\tau_H$. \end{lemma} \paragraph{Proof} (Sketch) Let $S^1$ be the unit circle with base point $1$, and let $\gamma: S^1\to U : 1\mapsto u_0$ be another loop in $U$, based at $u_0$. By pulling back the bundle $N$ under the base chang $p\circ \gamma :S^1\to S$, we can reduce the statement (1) to the case that $S=S^1$. If base is $S^1$, then as all complex vector bundles become trivial, the space $U$ becomes a product $C\!\!\!\!I\,^*\times U'$ for some $U'$, and $\tau_H$ is the positive generator of the fundamental group of $C\!\!\!\!I\,^*$, so is central in this product. Similarly, (2) follows from a base change to the unit interval $[0,1]$. \subsection{Summary of basic notation} \begin{tabular}{ll} $X$ & $=$ a nonsingular variety. \\ $d$ & $=$ the dimension of $X$. \\ $Y$ & $=$ a divisor with normal crossing in $X$.\\ $S_d$ & $=$ $X$ \\ $S_{d-1}$ & $=$ $Y$ \\ & \\ & By decreasing induction we define starting with $i=d-2$,\\ $S_i$ & $=$ the singular locus of $S_{i+1}$ for $i\le d-2$. \\ $m$ & $=$ the smallest $i$ for which $S_i$ is nonempty. \\ $I$ & $=$ any nonempty subset of $\{\, m,\ldots ,d-1\,\}$\\ $m(I)$ & $=$ the smallest element of $I$. \\ $p_I:X_I\to S_{m(I)}$ & $=$ the finite scheme over $S_{m(I)}$ whose fiber over \\ & point of $S_{m(I)}$ consists of all nested sequences of branches \\ & of $S_j$ at that point, for $j$ varying over $I$. \\ & In particular when $I=\{\, i\,\}$, we have \\ $p_i:X_i\to S_i$ & $=$ the normalization of $S_i$. \\ & For nonempty subsets $J\subset I\subset \{\, m,\ldots ,d-1\,\}$, \\ $X_{i,a}$ & $=$ connected components of $X_i$, \\ & as $a$ varies over the indexing set $\pi_0(X_i)$ \\ $p_{J,I}:X_I\to X_J$ & $=$ the canonical map. (All these commute.) \\ \end{tabular} \bigskip \bigskip \hfill {\footnotesize (Continued on next page) } \vfill \pagebreak \begin{tabular}{ll} & For $m\le i\le d-1$ we put \\ $Y^*_i$ & $=$ $X_{i,i+1}$, the $d-i$ sheeted finite \'etale cover of $X_i$ \\ & which splits the branches of $S_{i+1}$ which meet along $S_i$ \\ & For $m\le i\le d-2$ we put \\ $Z_i$ & $=$ $X_{i,i+2}$, the $C^{d-i}_2$ sheeted finite \'etale cover of $X_i$ \\ & which splits the branches of $S_{i+2}$ which meet along $S_i$ \\ $Z^*_i$ & $=$ $X_{i,i+1,i+2}$, the $2$ sheeted finite \'etale cover of $Z_i$ \\ $f_i:X_i\to X$ & $=$ the composite $X_i\stackrel{p_i}{\to}S_i\to X$ \\ $N_i$ & $=$ the normal bundle to $X_i$ in $X$ under $f_i$, defined by \\ & the exact sequence $0\to T_{X_i}\to f_i^*T_X\to N_i\to 0$\\ $T(Y)\subset T_X$ & $=$ the closed subset of $T_X$ consisting of \\ & vectors tangent to branches of $Y$ \\ $F_i$ & $=$ the closed subset of $N_i$ which is the image of $f^*T(Y)$ \\ & under the morphism $f^*T_X\to N_i$ of geometric vector bundles. \\ $U_i$ & $=$ $N_i-F_i$ \\ ${\cal D}_i$ & $=$ the restriction of ${\cal D}_{N_i}[\log F_i]$ to the zero section $X_i\subset N_i$,\\ & which is canonically isomorphic to $f_i^*{\cal D}_X[\log Y]$ \\ $Y^*$ & $=$ $Y^*_{d-1}$ $=$ $X_{d-1}$, the normalization of $Y$ \\ $N_{i,i+1}$ & $=$ the normal bundle to $Y^*_i=X_{i,i+1}$ in $X$, \\ & defined with respect to the morphism $Y^*_i\to X_i\to X$ \\ $F_{i,i+1}$ & $=$ the normal crossing divisor in the total space of $N_{i,i+1}$, \\ & defined by vectors tangent to branches of $Y$. \\ $h_i:Y^*_i\to Y^*$ & $=$ the canonical map, sending $(x_i,x_{i+1})$ to the branch \\ & $y_{d-1}$ of $Y^*$ which intersects the given branch $x_{i+1}$ of $S_{i+1}$ \\ & along given the branch $x_i$ of $S_i$. \\ $H_{i,i+1}$ & $=$ the hypersubbundle of $N_{i,i+1}$ contained in $F_{i,i+1}$, \\ & defined by vectors tangent to the branch of $Y$ \\ & given by $h_i:Y^*_i\to Y^*$ \\ ${\cal D}^*_i$ & $=$ the restriction of ${\cal D}_{N_{i,i+1}}[\log F_{i,i+1}]$ \\ & to the zero section $Y^*_i\subset N_{i,i+1}$ \\ $\theta_i$ & $=$ the Euler operator along $H_{i,i+1}$ in ${\cal D}^*_i$. \\ $R_i$ & $=$ the open subset of $N_{i,i+1}-F_{i,i+1}$ of $N_{i,i+1}$ which is the \\ & complement in $N_{i,i+1}$ of vectors tangent to branches of $Y$\\ $R_i(c)$ & $=$ connected components of $R_i$ as $c$ varies over $\pi_0(Y^*_i)$.\\ $\tau_i(c)$ & $=$ the element in the center of $\pi_1(R_i(c))$, which is \\ & given by a positive loop around $H_{i,i+1}(c)$\\ $N_{i,i+2}$ & $=$ the normal bundle to $Z_i$ in $X$. \\ $W_i$ & $=$ the open subset of $N_{i,i+2}$ which is the complement \\ & in $N_{i,i+1,i+2}$ of vectors tangent to branches of $Y$\\ $N_{i,i+1,i+2}$ & $=$ the normal bundle to $Z^*_i$ in $X$. \\ $W^*_i$ & $=$ the open subset of $N_{i,i+1,i+2}$ which is the complement \\ & in $N_{i,i+1,i+2}$ of vectors tangent to branches of $Y$\\ \end{tabular} \vfill \section{Pre-${\cal D}$-modules on $(X,Y)$} In this section, we first define the notion of a pre-${\cal D}$-module. Then we consider the special case when $X$ is a polydisk. Finally, we give some historical motivation. \subsection{Global definition} Let $X$ is any smooth variety and $Y$ a normal crossing divisor. We follow the notation introduced in section 2. The following definitnion works equally well in the algebraic or the analytic categories. \begin{definition}\rm A {\bf pre-${\cal D}$-module} ${\bf E} = (E_i,t_i,s_i)$ on $(X,Y)$ consists of the following. (1) For each $m\le i\le d$, $E_i$ is a vector bundle on $X_i$ (of not necessarily constant rank) together with a structure of ${\cal D}_i$-module, (Note that by (1), for each $m\le i\le d-1$, the pullbacks $E_{i+1}| Y^*_i$ and $E_i| Y^*_i$ under the respective maps $Y^*_i\to X_{i+1}$ and $Y^*_i\to X_i$ have a natural structure of a ${\cal D}^*_i$-module.) (2) For each $m\le i\le d-1$, $t_i:(E_{i+1}| Y^*_i) \to (E_i| Y^*_i)$ and $s_i:(E_i |Y^*_i) \to (E_{i+1} | Y^*_i)$ are ${\cal D}^*_i$-linear maps, such that \begin{eqnarray*} s_it_i &=& \theta_i {\mbox{~{\rm on}~}} E_{i+1}|Y^*_i \\ t_is_i &=& \theta_i {\mbox{~{\rm on}~}} E_i| Y^*_i \\ \end{eqnarray*} (3) Let $m\le i \le d-2$. Let $\pi:Z^*_i\to Z_i$ be the projection $$p_{\{\,i,i+2\,\} ,\{\,i,i+1,i+2\,\} }: X_{\{\,i,i+1,i+2\,\} }\to X_{\{\,i,i+2\,\} }$$ Let $E_{i+2}|Z_i$ and $E_i|Z_i$ be the pullbacks of $E_{i+2}$ and $E_i$ under respectively the composites $Z_i=X_{\{\,i,i+2\,\} }\to X_{i+2}$ and $Z_i=X_{\{\,i,i+2\,\} }\to X_i$. {\rm (Note that there is no object called $E_{i+1}|Z_i$.)} We will denote the pullback of $E_{i+1}$ under $Z^*_i=X_{\{\,i,i+1,i+2\,\} } \to X_{i+1}$ by $E_{i+1}|Z^*_i$. We will denote $\pi^*(E_{i+2}|Z_i)$ by $E_{i+2}|Z^*_i$ and $\pi^*(E_i|Z_i)$ by $E_i|Z^*_i$. Let $$a_{i+2}:E_{i+2}|Z_i \to \pi_*\pi^*(E_{i+2}|Z_i) = \pi_*(E_{i+2}|Z^*_i)$$ $$a_i:E_i|Z_i \to \pi_*\pi^*(E_i|Z_i) = \pi_*(E_i|Z^*_i)$$ be adjunction maps, and let the cokernels of these maps be denoted by $$q_{i+2}:\pi_*(E_{i+2}|Z_i)\to Q_{i+2}$$ $$q_i:\pi_*(E_i|Z_i)\to Q_i$$ Then we impose the requirement that the composite map $$E_{i+2}|Z_i\stackrel{a_{i+2}}{\to} \pi_*(E_{i+2}|Z^*_i) \stackrel{\pi_*(t_{i+1}|Z^*_i)}{\to} \pi_*(E_{i+1}\vert Z^*_i) \stackrel{\pi_*(t_i|Z^*_i)}{\to} \pi_*(E_i\vert Z^*_i) \stackrel{q_i}{\to} Q_i $$ is zero. (4) Similarly, we demand that for all $m\le i\le d-2$ the composite map $$Q_{i+2}\stackrel{q_{i+2}}{\leftarrow} \pi_*(E_{i+2}\vert Z^*_i) \stackrel{\pi_*(s_{i+1}|Z^*_i)}{\leftarrow} \pi_*(E_{i+1}\vert Z^*_i) \stackrel{\pi_*(s_i|Z^*_i)}{\leftarrow} \pi_*(E_i\vert Z^*_i) \stackrel{a_i}{\leftarrow} E_i|Z_i$$ is zero. (5) Note that as $\pi:Z^*_i\to Z_i$ is a 2-sheeted cover, for any sheaf ${\cal F}$ on $Z_i$ the new sheaf $\pi_*\pi^*({\cal F})$ on $Z_i$ has a canonical involution coming from the deck transformation for $Z^*_i\to Z_i$ which transposes the two points over any base point. In particular, the bundles $\pi_*(E_{i+2}|Z^*_i)=\pi_*\pi^*(E_{i+2}|Z_i)$ and $\pi_*(E_i|Z^*_i)=\pi_*\pi^*(E_i|Z_i))$ have canonical involutions, which we denote by $\nu$. We demand that the following diagram should commute. Diagram III. $$\begin{array}{ccccc} \pi_*(E_{i+1} | Z^*_i) & \stackrel{\pi_*(s_{i+1}|Z^*_i)}{\to} & \pi_*(E_{i+2}|Z^*_i) & \stackrel{\nu}{\to} & \pi_*(E_{i+2}|Z^*_i) \\ {\scriptstyle \pi_*(t_i|Z^*_i)}\downarrow & & & & \downarrow {\scriptstyle \pi_*(t_i|Z^*_i)}\\ \pi_*(E_i|Z^*_i) & \stackrel{\nu}{\to} & \pi_*(E_i|Z^*_i) & \stackrel{\pi_*(s_i|Z^*_i)}{\to} & \pi_*(E_{i+1} | Z^*) \\ \end{array}$$ A {\bf homomorphism} $\varphi :{\bf E} \to {\bf E}'$ of pre-${\cal D}$-modules consists of a collection $\varphi_i:E_i\to E'_i$ of ${\cal D}_i$-linear homomorphisms which make the obvious diagrams commute. \end{definition} \refstepcounter{theorem}\paragraph{Remark \thetheorem} As the adjunction maps are injective (in particular as $a_i$ is injective), the condition (3) is equivalent to demanding the existence of a unique $f$ which makes the following diagram commute. Diagram I. $$\begin{array}{ccccc} E_{i+2}|Z_i & & \stackrel{f}{\longrightarrow} & & E_i|Z_i \\ a_{i+2}\downarrow & & & & \downarrow a_i\\ \pi_*(E_{i+2}\vert Z^*_i) & \stackrel{\pi_*(t_{i+1}|Z^*_i)}{\to} & \pi_*(E_{i+1}\vert Z^*_i) & \stackrel{\pi_*(t_i|Z^*_i)}{\to} & \pi_*(E_i\vert Z^*_i)\\ \end{array}$$ Similarly, the condition (4) is equivalent to the existence of a unique homomorphism $g$ which makes the following diagram commute. Diagram II. $$\begin{array}{ccccc} E_{i+2}|Z_i & & \stackrel{g}{\longleftarrow} & & E_i|Z_i \\ a_{i+2}\downarrow & & & & \downarrow a_i\\ \pi_*(E_{i+2}\vert Z^*_i) & \stackrel{\pi_*(s_{i+1}|Z^*_i)}{\leftarrow} & \pi_*(E_{i+1}\vert Z^*_i) & \stackrel{\pi_*(s_i|Z^*_i)}{\leftarrow} & \pi_*(E_i\vert Z^*_i)\\ \end{array}$$ \refstepcounter{theorem}\paragraph{Remark \thetheorem}\label{local} The above definition is local, in the sense that (1) if $U_{\alpha}$ is an open covering of $X$ in the algebraic or analytic category, and ${\bf E}_{\alpha}$ is a collection of pre-${\cal D}$-modules on $(U_{\alpha},\,Y\cap U_{\alpha})$ together with isomorphisms $\varphi_{\alpha,\beta}:({\bf E}_{\beta}|U_{\alpha, \beta}) \to ({\bf E}_{\alpha}|U_{\alpha, \beta})$ which form a $1$-cocycle, then there exists a unique (upto unique isomorphism) pre-${\cal D}$-module ${\bf E}$ on $(X,Y)$ obtained by gluing. (2) If ${\bf E}$ and ${\bf E}'$ are two pre-${\cal D}$-modules on $(X,Y)$, then a homomorphism $\varphi :{\bf E}\to {\bf E}'$ is uniquely defined by a collection of homomorphisms over $U_{\alpha}$ which match in $U_{\alpha, \beta}$. \begin{definition}\label{goodresieigen}\rm Consider the action of $\theta_i$ on $E_{i+1}|Y^*_i$, which is ${\cal O}_{Y^*_i}$-linear. Note that in the global algebraic case, compactness of $Y^*_i$ implies that the characteristic polynomial of the resulting endomorphism of $E_{i+1}|Y^*_i$ is constant on each component of $Y^*_i$. We say that a {\bf pre-${\cal D}$-module ${\bf E}=(E_i,s_i,t_i)$ has good residual eigenvalues} if for each $i \le d-1$ no two eigenvalues of $\theta_i\in End(E_{i+1}|Y^*_i)$, on any two components of $Y^*_i$ which map down to intersecting subsets of $X$, differ by a non-zero integer. \end{definition} \refstepcounter{theorem}\paragraph{Remark \thetheorem} Note that the above definition does not prohibit two eigenvalues of $\theta_i$ on $E_i|Y^*_i$ from differing by non-zero integers. Also, note that the definition can involve more than one component of $Y^*_i$ at a time: it is stronger than requiring that on each component of $Y^*_i$ no two eigenvalues should differ by nonzero integes. \subsection{Restriction to a polydisk} There exists an open covering of $X$, where each open subset is a polydisk in $C\!\!\!\!I^d$ with coordinates $x_i$, defined by $|x_i|< 1$, whose intersection with $Y$ is defined by $\prod_{i\le r}x_i=0$ for a variable integer $r\le d-m$. It is possible globally that the irreducible components of $Y$ are singular. Moreover, it is possible that various branches of $Y$ meeting at a point get interchanged as one moves around. This does not happen in a polydisk of the above kind, so the definition of a pre-${\cal D}$-module becomes much simpler. We give it in detail in view of remark \ref{local} above. Let $X$ be a polydisk in $C\!\!\!\!I^d$ around the origin, with coordinates $x_1,\ldots, x_d$, and let $m$ be some fixed integer with $0\le m\le d$. If $m\le d-1$, let $Y\subset X$ be the normal crossing divisor defined by $\prod _{1\le i\le d-m}x_i=0$. If $m=d$, we take $Y$ to be empty. We will follow the notation summarised in section 2. We have a filtration $Y=S_{d-1}\supset \ldots \supset S_m$ where each $S_i$ is the singularity set of $S_{i+1}$ for $m\le i\le d-2$, and $S_m$ is nonsingular. Note that the irreducible components of $S_i$ are as follows. For any subset $A\subset \{\,1,\ldots,d-m\,\}$ of cardinality $d-i$, we have a component $S_A$ of $S_i$ defined by the ideal generated by all $x_j$ for $j\in A$ (total $C^{d-m}_{d-i}$ components). Then the normalization $X_i$ of $S_i$ is simply the disjoint union of all the $S_A$. Therefore $X_{i,A} = S_A$, which are polydisks of dimension $d- |A|$, are the components of $X_i$. Whenever $k\in A$, we have an inclusion $X_A\to X_{A-\{ k\}}$. This is identified in our earlier notation with a component of $Y^*_{i-1} \to X_i$. It follows from its general definition that a pre-${\cal D}$-module ${\bf E}$ on $(X,Y)$ consists of the following data. (1) For each $A\subset \{\, 1,\ldots , d-m\,\}$, we are given a vector bundle $E_A$ on $S_A$, together with the structure of a ${\cal D}_X[\log Y]$-module. The $E_A$ with $|A|=d-i$ are the restrictions of $E_i$ to the components $S_A$ of $X_i$. (2) For any $k \in A\subset \{\, 1,\ldots , d-m\,\}$, we have ${\cal D}_X[\log Y]$-linear homomorphisms $t^k_A:E_{A-k}|X_A \to E_A$ and $s^k_A:E_A \to E_{A-k}|X_A$ such that $$s^k_At^k_A = x_k\partial/\partial x_k {\mbox{~{\rm on}~}} E_{A-k}|X_A$$ $$t^k_As^k_A = x_k\partial/\partial x_k {\mbox{~{\rm on}~}} E_A$$ In terms of earlier notation, the $t^k_A$ (respectively, the the $s^k_A$) with $|A|=d-i$ make up the $t_i$ (respectively the $s_i$). (3) Let $k\ne \ell$ such that $k,\, \ell \in A\subset \{\, 1,\ldots , d-m\,\}$. Then we must have \begin{eqnarray*} t^k_At^{\ell}_{A-k} = t^{\ell}_At^k_{A-{\ell}}\\ s^k_{A-{\ell}}s^{\ell}_A = s^{\ell}_{A-k}s^k_A \\ t^k_{A-{\ell}}s^{\ell}_{A-k} = s^{\ell}_A t^k_A \\ \end{eqnarray*} The above three equations respectively embody the conditions that diagrams I, II, and III in the definition of a pre-${\cal D}$-module must commute. \subsection{Motivation for the definition} The definition of a pre-${\cal D}$-module may be regarded as another step in the programme of giving concrete representations of regular holonomic ${\cal D}$ modules and perverse sheaves. The earlier steps relevant to us are the following. {\bf (1)} Deligne's description (1982) of a perverse sheaf on a disk with singularity at the origin, in terms of pairs of vector spaces and linear maps and Malgrange's description of corresponding regular holonomic ${\cal D}$-modules. {\bf (2)} Verdier's functor of specialization (Asterisque 101-102), and his description of extension of a perverse sheaf across a closed subspace (Asterisque 130). {\bf (3)} Similar construction by Malgrange for regular holonomic ${\cal D}$-modules in place of perverse sheaves. {\bf (4)} Verdier's description of a perverse sheaf on the total space of a line bundle $L$ on a smooth variety $S$, in terms of two local systems on $L-S$ ($=$ the com\-ple\-ment of the zero section) and maps between them (Asterisque 130, 1985). {\bf (5)} Definition of a pre-${\cal D}$-module on $(X,Y)$ when $Y$ is nonsingular, which can be obtained by choosing compatible logarithmic lattices in a combination of Step 3 and Step 4 (see [N-S]). {\bf (6)} Description by Galligo, Granger, Maisonobe of perverse sheaves on a polydisk with coordinates $(z_1,\ldots,z_n)$ with respect to the smoothening stratification induced by the normal crossing divisor $z_1\cdots z_n=0$, in terms of a hypercube of vector spaces and linear maps (1985). The description (6) for a polydisk with a normal cross\-ing divisor is local and coordinate dependent like the description (1) for the disk with a point. One first makes it coordinate free and globalizes it in order to have the equivalent of (4) (which gives us finite descriptions of perverse sheaves described in section 6 below), and then puts level structures generalizing (5) to arrive at the above definition of a pre-${\cal D}$-module. One of the problems in globalizing the local hypercube description is that one can not unambiguously label the branches of $Y$ which meet at a point, because of twistedness of the divisor. This is taken care of by normalizing the closed strata $S_i$ and going to the coverings $\pi :Z^*_i\to Z_i$. The requirement that the various composites of $s$ and $t$ should give endomorphisms expressible in terms of Euler vector fields is present in the local hypercube description in much the same form. The three commutative diagrams I, II, and III in the definition respectively embody the globalizations of the conditions in the hypercube description that I: the two canonical maps $C_x$ and $C_y$ should commute, II: the two variation maps $V_x$ and $V_y$ should commute, and III: we should have $C_xV_y=V_yC_x$. \bigskip \section{From pre-${\cal D}$-modules to ${\cal D}$-modules} In section 4.1, we directly describe the ${\cal D}$-module associated to a pre-${\cal D}$-module in the special case where $Y$ is nonsingular. In section 4.2, we will associate a ${\cal D}$-module to a pre-${\cal D}$-module when $Y$ is normal crossing. This is done by first doing it on polydisks, and then patching up. Finallly, in section 4.3 we show how to find a pre-${\cal D}$-module with good residual eigenvalues over a given ${\cal D}$-module, proving that the functor from pre-${\cal D}$-modules on $(X,Y)$ with good residual eigenvalues to regular holonomic ${\cal D}$-modules on $X$ whose caracteristic variety is contained in $N^*(Y)$ is essentially surjective. \subsection{The case when $Y$ is smooth} First we treat the case where the divisor $Y$ is nonsingular. To a pre-${\cal D}$-module $(E,F,t,s)$ on $(X,Y)$, where $E=E_d$ is a logarithmic connection on $(X,Y)$, $F=E_{d-1}$ is a vector bundle on $Y$ with structure of a ${\cal D}_X[\log Y]$-module, and $t:E|Y\to F$ and $s:F\to E|Y$ are ${\cal D}_X[\log Y]$-linear maps with $st = \theta_Y$ on $E|Y$, and $ts=\theta_Y$ on $F$, we will directly associate the following ${\cal D}$-module $M$ on $X$. (This was indirectly described in [N-S]). Let $M_0=E$, and let $E\oplus_sF$ denote the subsheaf of $E\oplus F$ consisting of sections $(e,f)$ such that $(e|Y)=s(f)$. Let ${\cal O}_X(Y)$ be the line bundle on $X$ defined by the divisor $Y$ as usual, and let $M_1={\cal O}_X(Y)\otimes (E\oplus_sF)$. Let $M_0\hookrightarrow M_1$ be the inclusion defined by sending a local section $e$ of $M_0=E$ to the local section $(1/x)\otimes (xe,0)$, where $x$ is a local generator for the ideal of $Y$ in $X$ (this can be readily seen to be independent of the choice of $x$). We make $E\oplus_sF$ is a ${\cal D}_X[\log Y]$-module by putting for any local section $\xi$ of $T_X[\log Y]$ and $(e,f)$ of $E\oplus_sF$, $$\xi (e,f) = (\xi(e) , \, t(e|Y) + \xi(f))$$ The right hand side may again be checked to be in $E\oplus_sF$, using the relation $st=\theta_Y$ on $E|Y$. As ${\cal O}_X(Y)$ is naturally a ${\cal D}_X[\log Y]$-module, this now gives the structure of a (left) ${\cal D}_X[\log Y]$-module on the tensor product $M_1={\cal O}_X(Y)\otimes_{{\cal O}_X}(E\oplus_sF)$. Moreover, the inclusion $M_0\hookrightarrow M_1$ defined above is ${\cal D}_X[\log Y]$-linear. We now define a connection $\nabla : M_0 \to \Omega^1_X\otimes M_1$ by putting, for any local sections $\eta$ of $T_X$ and $e$ of $M_0$, $$\eta (e) = (1/x)\otimes ((x\eta)(e),\, \eta(x) t(e|Y))$$ where $x$ is any local generator of the ideal of $Y$. The right hand side makes sense because $x\eta$ is a section of $T_X[\log Y]$, and so $(x\eta)(e)$ is defined by the logarithmic connection on $E$. It can be checked that the above formula is independent of the choice of $x$, is ${\cal O}_X$-linear in the variable $\eta$, and the resulting map $\nabla: M_0 \to \Omega^1_X\otimes M_1$ satisfies the Leibniz rule. Moreover, the following diagram commutes, where the maps $M_i\to \Omega^1_X[\log Y]\otimes M_i$ (for $i=0$ and for $i=1$) are given by the ${\cal D}_X[\log Y]$-module structure on $M_i$. $$\begin{array}{ccc} M_0 & \to & \Omega^1_X\otimes M_1 \\ \downarrow & & \downarrow \\ M_1 & \to & \Omega^1_X[\log Y]\otimes M_1 \\ \end{array}$$ Now let $M$ be the ${\cal D}_X$-module which is the quotient of ${\cal D}_X\otimes_{{\cal D}_X[\log Y]}M_1$ by the submodule generated by elements of the type $\eta\otimes e - 1\otimes \eta(e)$ where $e$ is a local section of $E=M_0\subset M_1$ and $\eta$ is a local section of $T_X$. Then $M$ is the ${\cal D}_X$-module that we associate to the pre-${\cal D}$-module $(E,F,t,s)$. \centerline{\sl Relation with $V$-filtration} We now assume that the generalized eigenvalues of $\theta_Y$ on $E|Y$ do not differ by nonzero integers (it is actually enough to assume this along each connected component of $Y$, but for simplicity we will assume that $Y$ is connected). Let $\mu$ the only possible integral eigenvalue (when there are more components in $Y$, there can be a possibly different $\mu$ along each component). Under this assumption, We now construct a $V$-filtration on $M$ along the divisor $Y$. Put $V^{\mu}(M)$ to be the image of $M_0$ and $V^{\mu +1}(M)$ to be the image of $M_1$ in $M$. For $k\ge 1$ put $V^{\mu -k}(M)$ to be the image of $I_Y^kM_0 \subset M_0$ and $V^{\mu +k}(M)$ to be the image of ${\cal O}_X((k-1)Y)\otimes M_1$ in $M$. Then by definition each $V^kM$ for $k\in Z\!\!\!Z$ is an ${\cal O}_X$-coherent ${\cal D}_X[\log Y]$-module, with $\eta (V^k(M))\subset V^{k+1}(M)$ and $I_YV^{k+1}(M) \subset V^k(M)$ for all $k$. Conversely, let $M$ be a regular holonomic ${\cal D}$-module on $X$ with $car(M)\subset N^*(Y)$. Let $V^k(M)$ be a $V$-filtration with $\mu$ the only integer in the fundamental domain chosen for the exponential map $C\!\!\!\!I \to C\!\!\!\!I\,^*$). Then put $E=V^{\mu}(M)$, $F = N^*_{Y,X}\otimes (V^{{\mu}+1}(M)/V^{\mu}(M))$, $t:(E|Y)\to F$ is defineded by putting $$t(e|Y) = x\otimes (\partial/\partial x)e$$ where $x$ is a local generator of $I_Y$ (which is independent of the choice of $x$), and $s:F\to E|Y$ defined by simply the multiplication $I_Y\times V^{\mu +1}M\to V^{\mu}M$ (note for this that $N^*_{Y,X}=I_Y/I_Y^2$). Then we get a pre-${\cal D}$-module $(E,F,t,s)$. Given a choice of a fundamental domain for the exponential map, the above two processes are inverses of each other. \subsection{General case of a normal crossing $Y$} Let ${\bf E} =(E_i,t_i,s_i)$ be a pre-${\cal D}$-module on $(X,Y)$, where we now allow $Y$ to have normal crossings. Let the sheaf $F$ on $X$ be the subsheaf of $\oplus (p_i)_*E_i$ whose local sections consist of all tuples $(e_i)$ where $e_i\in (p_i)_*E_i$, such that $s_i(e_i|Y^*_i)=e_{i+1}|Y^*_i$. This is a ${\cal D}_X[\log Y]$-submodule of $\oplus (p_i)_*E_i$ as may be seen. Let $G={\cal O}_X(Y)\otimes F$. As ${\cal O}_X(Y)$ is naturally a ${\cal D}_X[\log Y]$-module, $G$ has a natural structure of a left ${\cal D}_X[\log Y]$-module. The ${\cal D}_X$-module $M$ that we are going to associate to the pre-${\cal D}$-module ${\bf E}$ is going to be a particular quotient of the ${\cal D}_X$-module ${\cal D}_X\otimes_{{\cal D}_X[\log Y]}G$. \medskip \centerline{\sl The case of a polydisk} Let $x_1,\ldots,x_d$ be coordinates on the polydisk, let $0\le r\le d$ and let $Y$ be defined by the polynomial $P(x)=\prod_{1\le k\le r}x_k$ (in particular $P=1$ and therefore $Y$ is empty if $r=0$). A pre-${\cal D}$-module ${\bf E}$ on $X$ has been described already in section 3.1 above. For each $k$ such that $1\le k\le r$, we define a subsheaf $F_k\subset F$ (where $F$ is the submodule of $\oplus (p_A)_*E_A$ defined above using the $s_A$) as follows. $$F_k = x_kF\subset F$$ This defines an ${\cal O}_X$-coherent module, which is in fact a ${\cal D}_X[\log Y]$-submodule of $F$ as may be checked. We now put $G_k={\cal O}_X(Y)\otimes F_k \subset {\cal O}_X(Y)\otimes F = G$. This is therefore a ${\cal D}_X[\log Y]$-submodule of $G$. We now define the operator $$\partial/\partial x_k:G_k \to G$$ as follows. If $e=(1/P)\otimes (e_A)$ is a section of $G_k$, we put $\partial_k(e) = (1/P)\otimes (f_B)$ where $$f_B = (x_k\partial/\partial x_k)(e_B) + t^k_B(e_{B-k})$$ where by convention $t^k_B=0$ whenever $k$ does not belong to $B$. Let $K$ be the ${\cal D}_X$-submodule of ${\cal D}_X\otimes_{{\cal D}_X[\log Y]}G$ generated by elements of the form $$\partial_k\otimes e -1\otimes \partial_k(e)$$ where $e\in G_k$. \begin{lemma} The submodule $K$ is independent of the choice of local coordinates, and restricts to the corresponding submodule on a smaller polydisk. \end{lemma} \paragraph{Proof} This is a local coordinate calculation, using the chain rule of partial differentiation under a change of coordinates. We omit the details. \medskip \centerline{\sl Back to the global case} By the above lemma applied to an open covering of $X$ by polydisks, we get a globally defined submodule $K$ of ${\cal D}_X\otimes G$ (where the later is already defined globally). We now put $M$ to be the quotient of ${\cal D}_X\otimes G$ by $K$. This is our desired ${\cal D}_X$-module. \subsection{The $V$ filtration for a polydisk} Let $X$ be a polydisk with coordinates $x_1,\ldots,x_d$ and $Y$ be defined by $\prod_{k\in\Lambda}x_k$ for some initial subset ${\Lambda}=\{\,1,\ldots,r\,\}\subset \{\,1,\ldots,d\,\}$. Let $M$ be a regular holonomic ${\cal D}_X$-module on $(X,Y)$. We now describe a pre-${\cal D}$-module ${\bf E}$ such that $M$ is associated to it. This is done via a $V$-filtration of $M$. We assume that we have fixed some fundamental domain $\Sigma$ for $exp:C\!\!\!\!I\to C\!\!\!\!I\,^*:z\mapsto e^{2\pi iz}$, in order to define the $V$-filtration. Let $\mu\in \Sigma$ be the only integer. Note that the filtration will be by sub ${\cal D}_X[\log Y]$-modules $V^{(n_1,\ldots,n_r)}$ where $n_k\in Z\!\!\!Z$, partially ordered as follows. If $(n_1,\ldots,n_r)\le (m_1,\ldots,m_r)$ (which means $n_k\le m_k$ for each $k\in {\Lambda}$), then $V^{(n_1,\ldots,n_r)} \subset V^{(m_1,\ldots,m_r)}$. In particular, there is a portion of the filtration indexed by the power set of ${\Lambda}$, where for any subset $A$ of ${\Lambda}$, we put $V^A = V^{(n_1,\ldots,n_r)}$ where $n_k=\mu +1$ if $k\in A$ and $n_k=\mu$ otherwise. By this convention, note that $V^{\phi} =V^{(\mu ,\ldots,\mu )}$ for the empty set $\phi$, and $V^{\Lambda}=V^{(\mu+1,\ldots,\mu +1)}$. Note that if $k\in A$ then multiplication by $x_k$ defines a map $$x_k:V^AM\to V^{A-k}M$$ and differentiation by $\partial/\partial x_k$ defines a map $$\partial_k:V^{A-k}M \to V^AM$$ Let for any nonempty $A$, $$gr^AM = {V^AM \over \sum_B V^BM}$$ where $B$ varies over all proper subsets of $A$. All eigenvalues of $x_k\partial/\partial x_k$ on $gr^AM$ lie in $\Sigma$. The above defines maps $$x_k: gr^AM\to gr^{A-k}M$$ and $$\partial_k : gr^{A-k}M\to gr^AM$$ Then we can define a pre-${\cal D}$-module ${\bf E}=(E_A,t^k_A,s^k_A)$ as follows: The ${\cal D}_X[\log Y]$-modules $E_A$ are defined by $$E_A = {\cal O}_X(-\sum_{k\in A} Y^k)\otimes_{{\cal O}_X}gr^AM$$ where $Y^k\subset X$ is the divisor defined by $x_k=0$. Let $k\in A$. To define $t^k_A$ and $s^k_A$, note that the differentiation $\partial_k: gr^{A-k}M\to gr^AM$ induces a map ${\cal O}_X(Y^k)\otimes gr^{A-k}M\to gr^AM$, as $\partial_k$ can be canonically identified with the section $1/x_k$ of ${\cal O}_X(Y^k)$. Tensoring this by the identity map on ${\cal O}_X(-\sum_{\ell\in A}Y^{\ell})$, and restricting to $X_A$, we get $t^k_A: (E_{A-k}|X_A)\to E_A$. Also, the map $x_k: gr^AM\to gr^{A-k}M$ defines a homomorphism ${\cal O}_X(-Y^k)\otimes gr^AM\to gr^{A-k}M$ which after tensoring by the identity map on ${\cal O}_X(-\sum_{\ell \in A-k}Y^{\ell})$ gives $s^k_A: E_A\to (E_{A-k}|X_A)$. Then it can be checked that we indeed get a pre-${\cal D}$-module ${\bf E}$, such that it has good residual eigenvalues, lying in $\Sigma$, such that $M$ is the ${\cal D}$-module associated to it. It can again be checked that the above proceedure over a polydisk is coordinate independent, so glues up to give such a correspondence globally over $(X,Y)$. \refstepcounter{theorem}\paragraph{Remark \thetheorem} If we began with the ${\cal D}$-module $M$ associated to a pre-${\cal D}$-module ${\bf E}$ with good residual eigenvalues lying in $\Sigma$, then the above will give back ${\bf E}$, as then $V^{\phi}M = E_d$ and for $A\subset \Lambda$ we will get $$V^AM = (\prod_{\ell \in {\Lambda}-A}x_{\ell}) G \subset G$$ where $G={\cal O}_X(Y)\otimes F$ where $F\subset \oplus_BE_B$ as above. \refstepcounter{theorem}\paragraph{Remark \thetheorem} (The [G-G-M]-hypercube for a polydisk) : Let $W_A$ be the fiber of $E_A$ at the origin of the polydisk. For $k\in A$ let $t^k_A : W_{A-k}\to W_A$ again denote the restriction of $t^k_A :E_{A-k} \to E_A$ to the fiber at origin, and let $v^k_A: W_A\to W_{A-k}$ be defined by the following formula. $$v^k_A = {\exp(2\pi i\theta_k)-1\over \theta_k} s^k_A$$ Then $(W_A,t^k_A,v^k_A)$ is the hypercube description of $M$ as given by Galligo, Granger, Maisonobe in [G-G-M]. \bigskip \section{Moduli for semistable pre-${\cal D}$-modules} In this section we define the concepts of semistability and stability for pre-${\cal D}$-modules, and construct a coarse moduli. The main result is Theorem \ref{maintheorem} below. \subsection{Preliminaries on $\Lambda$-modules} Simpson has introduced a notion of modules over rings of differential operators which we first recall (see section 2 of [S]). Let $X$ be a complex scheme, of finite type over $C\!\!\!\!I$, and let $\Lambda$ be a sheaf of ${\cal O}_X$-algebras (not necessarily non-commutative), together with a filtration by subsheaves of abelian groups $\Lambda_0\subset \Lambda_1 \subset \ldots \Lambda$ which satisfies the following properties. (1) $\Lambda = \cup \Lambda_i$ and $\Lambda_i\cdot \Lambda_j \subset \Lambda_{i+j}$. (In particular, $\Lambda_0$ is a subring, and each $\Lambda_i$ is a $\Lambda_0$-bimodule.) (2) The image of the homomorphism ${\cal O}_X\to \Lambda$ is equal to $\Lambda_0$. (In particular, each $\Lambda_i$ is an ${\cal O}_X$-bimodule). (3) Under the composite map $C\!\!\!\!I_X\hookrightarrow {\cal O}_X \to \Lambda$, the image of the constant sheaf $C\!\!\!\!I_X$ is contained in the center of $\Lambda$. (4) The left and right ${\cal O}_X$-module structures on the $i$th graded piece $Gr_i(\Lambda) = \Lambda_i/\Lambda_{i-1}$ are equal. (5) The sheaves of ${\cal O}_X$-modules $Gr_i(\Lambda)$ are coherent. (In particular, each $\Lambda_i$ is bi-coherent as a bi-module over ${\cal O}_X$, and their union $\Lambda$ is bi-quasicoherent.) (6) The associated graded ${\cal O}_X$-algebra $Gr(\Lambda)$ is generated (as an algebra) by the piece $Gr_1(\Lambda)$. (7) (`Split almost polynomial' condition) : The homomorphism ${\cal O}_X\to \Lambda_0$ is an isomorphism, the ${\cal O}_X$-module $Gr_1(\Lambda)$ is locally free, the graded ring $Gr(\Lambda)$ is the symmetric algebra over $Gr_1(\Lambda)$, and we are given a left-${\cal O}_X$-linear splitting $\xi :Gr_1(\Lambda)\to \Lambda_1$ for the left-${\cal O}_X$-linear projection $\Lambda_1\to Gr_1(\Lambda)$. \refstepcounter{theorem}\paragraph{Remark \thetheorem} The condition (7) is not necessary for the moduli construction, but allows a simple description (see lemma 2.13 of [S]) of the structure of $\Lambda$-module, just as the structure of a ${\cal D}_X$-module on an ${\cal O}_X$-module can be described in terms of the action of $T_X$. The pair $(\Lambda, \xi)$ is called a {\bf split almost polynomial algebra of differential operators} on $X$. Simpson also defines this in the relative situation $X\to S$, and treats basic concepts such as base change, which we will assume. A {\bf $\Lambda$-module} will always mean a left $\Lambda$-module unless otherwise indicated. For any complex scheme $T$, a {\bf family} $E_T$ of $\Lambda$-modules parametrized by $T$ has an obvious definition (see [S]). The following basic lemma is necessary to parametrize families of pre-${\cal D}$-modules. \begin{lemma}\label{nnreplem} ({\bf Coherence and representability of integrable direct images}) Let $\Lambda$ be an algebra of differential operators on $X$. Let $E_T$ and $F_T$ be a families of $\Lambda$-modules on $X$ parametrised by a scheme $T$ (which is locally noetherian and of finite type over the field of complex numbers). (i) The sheaf $(\pi _T)_*\underline{Hom}_{\Lambda_T}(E_T,F_T)$ is a coherent sheaf of ${\cal O}_T$ modules. (ii) Consider the contravarient functor from schemes over $T$ to the category of abelian groups, which associates to ${T'}\longrightarrow T$ the abelian group $Hom_{\Lambda_{T'}}(E_{T'}),F_{T'})$ where $E_{T'}$ and $F_{T'}$ are the pullbacks under $X\times {T'}\to X\times T$. Then there exists a linear scheme $V\longrightarrow T$ which represents this functor. \end{lemma} \paragraph{Proof} The above lemma is a stronger version of lemma 2.7 in [N]. The first step in the proof is the following lemma, which is an application of the Grothendieck complex of semi-\-continuity theory. \begin{lemma}\label{ega} {\rm (EGA III 7.7.8 and 7.7.9)} Let $Z\to T$ be a projective morphism where $T$ is noetherian, and let ${\cal F}$ and ${\cal G}$ be coherent sheaves on $Z$ such that ${\cal G}$ is flat over $T$. Consider the contra functor $\varphi$ from the category of schemes over $T$ to the category of abelian groups, which associates to any $T'\to T$ the abelian group of all ${\cal O}_{Z\times_TT'}$-linear homomorphisms from ${\cal F}_{T'}$ to ${\cal G}_{T'}$. Then $\varphi$ is representable by a linear scheme $W$ over $T$. \end{lemma} Now we prove lemma \ref{nnreplem}. Forgetting the structure of $\Lambda$-modules on $E_T$ and $F_T$ and treating them just as ${\cal O}$-modules, let $W\to T$ be the linear scheme given by the above lemma \ref{ega}. Then $W$ parametrizes a universal family of ${\cal O}_{X\times W}$-linear morphism $u:E_W\to F_W$. The condition of $\Lambda_W$-linearity on $u$ defines a closed linear subscheme $V$ of $W$. By its construction, for any base change $T'\to T$, we have canonical isomorphism $Mor_T(T,V) = Hom_{\Lambda_{T'}}(E_{T'},F_{T'})$ which proves the lemma \ref{nnreplem}. \subsection{Families of pre-${\cal D}$-modules} We now come back to $(X,Y)$ as before, and our earlier notation. Recall that $X_i$ is the normalization of $S_i$, which is $i$-dimensional, nonsingular if non-empty. Let $X_{i,a}$, as $a$ varies over the indexing set $\pi_0(X_i)$, be the connected components of $X_i$. We denote by ${\cal D}_{i,a}$ the restriction of ${\cal D}_i$ to the component $X_{i,a}$ of $X_i$. \begin{lemma}\label{lambda} Each ${\cal D}_{i,a}$ satisfies the above properties (1) to (7) (where in (7), we take $\xi : ({\cal D}_{i,a})_1\to {\cal O}_{X_{i,a}}$ to be induced by the splitting $({\cal D}_V)_1 = {\cal O}_V \oplus T_V$ for any non-singular variety $V$), so is a split almost polynomial algebra of differential operators on $X_{i,a}$. \end{lemma} \begin{definition}\rm For any complex scheme $T$, a {\bf family} $${\bf E}_T =(E_{i,T},\,s_{i,T},\,t_{i,T})$$ of pre-${\cal D}$-modules parametrized by $T$ is defined as follows. The $E_{i,T}$ are vector bundles on $X_i\times T$ with structure of ${\cal D}_{i,T}$ modules, where ${\cal D}_{i,T}$ are the relative versions of the ${\cal D}_i$. The morphisms $s_{i,T}$ and $t_{i,T}$ are the relative versions of the morphisms $s_i$ and $t_i$ in the definition of a pre-${\cal D}$-module. \end{definition} Given any morphism $f:T'\to T$ of complex schemes and a family ${\bf E}_T$ of pre-${\cal D}$-modules parametrized by $T$, the pullback family $f^*{\bf E}_T$ on $T'$ has again the obvious definition. This therefore defines a fibered category over the base category of complex schemes. When we put the restriction that all morphisms in each fiber category be isomorphisms, we get a fibered category ${\cal P}\!{\cal D}$ of groupoids over $Schemes_{C\!\!\!\!I}$. \begin{proposition} The fibered category of groupoid ${\cal P}\!{\cal D}$ of pre-${\cal D}$-modules on $(X,Y)$ is an algebraic stack in the sense of Artin. \end{proposition} \paragraph{Proof} (Sketch) We refer the reader to the notes of Laumon [L] for basic concepts and constructions involving algebraic stacks. As fpqc descent and fpqc effective descent is obviously satisfied by ${\cal P}\!{\cal D}$, it follows that ${\cal P}\!{\cal D}$ is a stack. It remains to show that this stack is algebraic in the sense of Artin. For this, first note that if $\Lambda$ is a split almost polynomial algebra of differential operators, then ${\cal O}$-coherent $\Lambda$-modules form an algebraic stack, for the forgetful functor (1-morphism of stacks) from $\Lambda$-modules to ${\cal O}$-modules is representable (as follows from the alternative description of the structure of a $\Lambda$-module given in lemma 2.13 of [S]), and coherent ${\cal O}$-modules form an algebraic stack in the sense of Artin. Now from the lemma \ref{nnreplem} on coherence and representability of the integrable direcrt image functor it can be seen that the forgetful functor (1-morphism of stacks) from pre-${\cal D}$-modules to the product of the stacks of its underlying ${\cal D}_i$-modules is representable (in fact, the details of this occur below in our construction of a local universal family for pre-${\cal D}$-modules). Hence the result follows. \subsection{Filtrations and 1-parameter deformations} We first recall some standard deformation theory, for convinience of reference. Recall that in this paper a vector bundle means a locally free module (but not necessarily of constant rank), and a subbundle of a vector bundle will mean a locally free submodule such that the quotient is also locally free. Let $E$ be a vector bundle on a complex scheme $X$ together with an exhaustive increasing filtration $E_p$ by vector subbundles, indexed by $Z\!\!\!Z$. (The phrase {\bf exhaustive} means that $E_p=0$ for $p\ll 0$ and $E_p=E$ for $p\gg 0$.) Let $A^1 = \mathop{\rm Spec}\nolimits C\!\!\!\!I[\tau]$ be the affine line, and let $U =\mathop{\rm Spec}\nolimits C\!\!\!\!I[\tau, \tau^{-1}]$ be the complement of the origin with inclusion $j:U\hookrightarrow A^1$. Let $\pi_X:X\times A^1 \to X$ be the projection. Consider the quasi-coherent sheaf $(1_X\times j)_*(\pi_X^*E|X\times U)$ on $X\times A^1$, which is usually denoted by $E\otimes C\!\!\!\!I[\tau,\tau^{-1}]$. This has a subsheaf $\ov{E}$ generated by all local sections of the type $\tau^p v_p$ where $v_p$ is a local section of $\pi^*E_p$. It is common to write $$\ov{E} = \sum_{p\inZ\!\!\!Z}E_p\tau^p\subset E\otimes C\!\!\!\!I[\tau,\tau_{}^{-1}]$$ Then we have the following basic fact: \refstepcounter{theorem}\paragraph{Remark \thetheorem}\label{fact1} It can be seen that $\ov{E}$ is an ${\cal O}_{X\times A^1}$-coherent submodule, which is in fact locally free, and $\ov{E}|X\times U$ is just $\pi_X^*E$ where $\pi_X:X\times U\to X$ is the projection. On the other hand, the specialization of $\ov{E}$ at $\tau =0$ is canonically isomorphic to the graded object $E'=\oplus(E_p/E_{p-1})$ associated with $E$. So, the 1-parameter family $E_{\tau}$ is a deformation of $E$ to its graded object $E'$. Now let $F$ vector bundle on $X$ together with filtration $F_p$, and let $f : E\to F$ be an ${\cal O}_X$-homomorphism. We have an induced ${\cal O}_{X\times A^1}$-homomorphism $$\pi_X^*f\,:\,E\otimes C\!\!\!\!I[\tau,\tau^{-1} ]\to F\otimes C\!\!\!\!I[\tau,\tau^{-1} ]$$ Then we have the following basic fact : \refstepcounter{theorem}\paragraph{Remark \thetheorem}\label{fact2} The homomorphism $f:E\to F$ is filtered, that is, $f$ maps each $E_p$ into $F_p$, if and only if the above homomorphism $\pi^*f$ carries $\ov{E}$ into $\ov{F}$. In that case, the induced map at $\tau=0$ is the associated graded map $gr(f) : E'\to F'$. As a consequence, we get the following : \refstepcounter{theorem}\paragraph{Remark \thetheorem}\label{fact3} Let $E$ and $F$ be vector bundles with exhaustive filtrations $E_p$ and $F_p$ indexed by $Z\!\!\!Z$, and let $\ov{E}$ and $\ov{F}$ be the corresponding deformations parametrized by $A^1$. Let $f:E\to F$ be an ${\cal O}_X$-homomorphisms, and let $g:\ov{E}\to \ov{F}$ be an ${\cal O}_{X\times A^1}$-homomorphism. Suppose that the restriction of $g$ to $X\times U$ (where $U=A^1-\{ 0\}$) is equal to the pullback $\pi_X^*f$ of $f$ under $\pi_X:X\times U\to X$. Then $f$ preserves the filtrations, and $g_0:E'\to F'$ is the associated graded homomorphism $f':E'\to F'$ (where $E'$ and $F'$ are the graded objects) \begin{definition}\rm A {\bf sub pre-${\cal D}$-module} $F$ of of a pre-${\cal D}$-module $E=(E_i,\,s_i,\,t_i)$ consists of the following data : For each $i$ we are given an ${\cal O}_{X_i}$-coherent ${\cal D}_i$-submodule $F_i\subset E_i$ such that for each $i$, (i) the ${\cal O}_{X_i}$-modules $F_i$ and $E_i/F_i$ are locally free (but not necessarily of constant ranks over $X_i$). In other words, for each $(i,a)$, we are given a vector subbundle $F_{i,a}\subset E_{i,a}$ which is a sub ${\cal D}_{i,a}$-module. (ii) the maps $s_i$ and $t_i$ preserve $F_i$, that is $s_i$ maps $F_i|Y^*_i$ into $F_{i+1}|Y^*_i$ and $t_i$ maps $F_{i+1}|Y^*_i$ into $F_i|Y^*_i$. Note that consequently, $F=(F_i,\,s^F_i,\,t^F_i)$ is also a pre-${\cal D}$ module, where $s^F$ and $t^F$ denote the restrictions of $s$ and $t$. Also, the quotients $E_i/F_i$ naturally form a pre-${\cal D}$-module $E/F$ which we call as the corresponding {\bf quotient pre-${\cal D}$-module}. \end{definition} \refstepcounter{theorem}\paragraph{Remark \thetheorem} Given a pre-${\cal D}$-module ${\bf E}$ and a collection ${\bf F}$ of subbundles $F_i\subset E_i$ which are ${\cal D}_i$-submodules, the job of checking whether these subbundles are preserved by the $s_i$ and $t_i$ is made easier by the following: it is enough to check this in the fiber of a point of each of the connected components of $Y^*_i-p^{-1}(S_{i-1})$ where $p:Y^*_i\to S_i$ is the projection. This is because the $s_i$ and $t_i$ are `integrable' in a suitable sense, and if an integrable section $\sigma$ of a vector bundle with an integrable connection has a value $\sigma(P)$ in the fibre at $P$ of a subbundle preserved by the connection, then it is a section of this subbundle. \begin{definition}\rm An {\bf exhaustive filtration on a pre-${\cal D}$-module} ${\bf E}=(E_i,\,s_j,\,t_j)$ consists of an increasing sequence ${\bf E}_p$ of sub pre-${\cal D}$-modules of ${\bf E}$ indexed by $Z\!\!\!Z$ such that ${\bf E}_p=0$ for $p\ll 0$ and ${\bf E}_p={\bf E}$ for $p\gg 0$.) A filtration is {\bf nontrivial} if ${\bf E}_p$ is a nonzero proper sub pre-${\cal D}$-module of ${\bf E}$ for some $p$. \end{definition} For a filtered pre-${\cal D}$-module, each step ${\bf E}_p=((E_i)_p,\,(s_i)_p,\,(t_i)_p)$ of the filtration, as well as the associated graded object ${\bf E}'=(E'_i,\,s'_i,\,t'_i)$ are pre-${\cal D}$-modules. Applying the remark \ref{fact2} to filtrations of pre-${\cal D}$-modules we get the following. \refstepcounter{theorem}\paragraph{Remark \thetheorem}\label{deform} Let ${\bf E}$ be a pre-${\cal D}$-module, together with an exhaustive filtration ${\bf E}_p$. Then there exists a family $({\bf E}_\tau)_{\tau\in A^1}^{}$ of pre-${\cal D}$-modules parametrized by the affine line $A^1=\specC\!\!\!\!I[\tau]$, for which the specialization at $\tau=0$ is the graded object ${\bf E}'$ while the family over $\tau_0\neq0$ is the constant family made from the original pre-${\cal D}$-module ${\bf E}$ defined as follows: put ${\bf E}_i=\sum_{p\inZ\!\!\!Z}(E_i)_p\tau^p\subset E_i\otimes C\!\!\!\!I[\tau,\tau^{-1}]$. \subsection{Quot scheme and group action on total family} Let $X$ be a projective scheme over a base $S$ and ${\cal V}$ a coherent ${\cal O}_X$-module. Let $$G =Aut_{{\cal V}}: {\rm Schemes}/S \to {\rm Groups}$$ be the contrafunctor which associates to any $T\to S$ the group of all ${\cal O}_{X_T}$-linear automorphisms of the pullback ${\cal V}_T$ of ${\cal V}$ under $X_T=X\times_ST\to X$. Note that then $G$ is in fact an affine group scheme over $S$, but this will not be relevant to us. Let $Q=Quot_{{\cal V}/X/S}$ be the relative quot scheme of quotients of ${\cal V}$ on fibers of $X\to S$. A $T$-valued point $y:T\to Q$ is represented by a surjective ${\cal O}_{X_T}$-linear homomorphism $q:{\cal V}_T\to F$ where $F$ is a coherent sheaf on $X_T$ which is flat over $T$. Two such surjections $q_1:{\cal V}_T\to F_1$ and $q_2:{\cal V}_T\to F_2$ represent the same point $y\in Q(T)$ if and only if either of the following two equivalent conditions is satisfied: (i) there exists an isomorphism $f:F_1\to F_2$ such that $q_2=f\circ q_1$, or (ii) the kernels of $q_1$ and $q_2$ are identical as a subsheaf of ${\cal V}_T$. Therefore, a canonical way to represent the point $y\in Q(T)$ is the quotient ${\cal V}_T\to {\cal V}_T/K_y$ where $K_y=ker(q)$ depends only on $y$. A natural group action $Q\times G\to Q$ over $S$ is defined as follows: in terms of valued points $y\in Q(T)$ represented by $q:{\cal V}_T\to F$, and $(g:{\cal V}_T\to {\cal V}_T) \in G(T)$, the point $yg\in Q(T)$ is represented by $q\circ g:{\cal V}_T\to F$. In other words, if $y$ is canonically represented by ${\cal V}_T\to {\cal V}_T/K_y$ then $yg$ is canonically represented by ${\cal V}_T\to {\cal V}_T/g^{-1}(K_y)$. This means $K_{yg}= g^{-1}(K_y)$. Let $q: {\cal V}_Q\to {\cal U}$ be the universal quotient family on $X_Q$. The action $Q\times G\to Q$ over $S$ when pulled back under $X\to S$ gives an action $Q_X \times G_X \to Q_X$ over $X$ (where $G_X=G\times _SX$ and $Q_X=Q\times_SX=X_Q$). The action $Q_X\times G_X\to Q_X$ has a natural lift to the sheaf ${\cal U}$ on $Q_X$ as follows. For $y\in Q(T)$ and $g\in G(T)$, the pull backs ${\cal U}_y$ and ${\cal U}_{yg}$ of the universal quotient sheaf under $y$ and $yg$ are canonically isomorphic to ${\cal V}_T/K_y$ and ${\cal V}_T/g^{-1}(K_y)$ respectively, so $g:{\cal V}_T\to {\cal V}_T$ induces a canonical isomorphism $\varphi_g^y: {\cal U}_{yg}\to {\cal U}_g$. Hence we get an isomorphism $\varphi : g^*({\cal U})\to {\cal U}$ over $X_Q$. Since $\varphi_{gh}^y = \varphi_g^y\circ \varphi_h^{yg}$ for any $g,\,h\in G(T)$ and $t\in Q(T)$, we get $$\varphi_{gh} = \varphi_g\circ g^*(\varphi_h)$$ Thus, $\varphi$ is a `factor of automorphy', and so defines the required lift. \refstepcounter{theorem}\paragraph{Remark \thetheorem}\label{liftformula} By definition, if $y\in Q(T)$ is represented by the surjection $q:{\cal V}_T\to F$, then we get a canonical identification of $F$ with ${\cal U}_y={\cal V}_T/K_y$. For $g\in G(T)$ the point $yg\in Q(T)$ is represented by $q\circ g:{\cal V}_T\to F$, so we get another canonical identification of $F$ with ${\cal U}_{gy}={\cal V}_T/g^{-1}(K_y)$. Under these identifications, the isomorphism $\varphi_g^y:{\cal U}_{yg}\to U_y$ simply becomes the identity map $1_F :F\to F$. (This is so simple that it can sometimes cause confusion.) Hence if $\sigma$ is a local section of ${\cal U}_T$ represented by $(q,\, s)$, where $q:{\cal V}_T\to F$ and $s$ is a local section of $F$, then the action of $g\in G(T)$ can be written as $$(q,\,s)\cdot g = (q\circ g,\, s)$$ \refstepcounter{theorem}\paragraph{Remark \thetheorem}\label{scalarmulti} The central subgroup scheme ${\bf G}_m\subset G$ (where by definition $\lambda\in {\bf G}_m(T)=\Gamma(T,{\cal O}_T^{\times})$ acts by scalar multiplication on ${\cal V}_T$) acts trivially on $Q$ but its action on the universal family ${\cal U}$ is again by scalar multiplication so is non-trivial. In terms of the above notation have the equality $$(q,\,s)\cdot \lambda = (q\circ \lambda, \,s) = (q, \,\lambda s)$$ Hence the induced action of $PG=G/{\bf G}_m$ on $Q$ does not lift to ${\cal U}$, which is the basic reason why a Poincar\'e bundle does not in general exist in the kind of moduli problems we are interested in. \refstepcounter{theorem}\paragraph{Remark \thetheorem}\label{glnaction} In the applications below, $X$ will be in general a projective scheme over $C\!\!\!\!I$ and the sheaf ${\cal V}$ will be of the type ${\cal O}_X^{\oplus N}\otimes_{{\cal O}_X}{\cal W} = C\!\!\!\!I^N\otimes_{C\!\!\!\!I}{\cal W}$ where ${\cal W}$ is some coherent sheaf over $X$. Then $GL(N)$ is naturally a subgroup scheme of $G=Aut_{{\cal V}}$, and we will only be interested in the resulting action of $GL(N)$. \subsection{Semistability and moduli for $\Lambda$-modules} We now recall the moduli construction of Simpson for $\Lambda$-modules (see section 2, 3 and 4 of [S] for details). Let $X$ be projective, with ample line bundle ${\cal O}_X(1)$. Let $E$ be an ${\cal O}_X$-coherent $\Lambda$-module on $X$. Then Simpson defines $E$ to be a {\bf semistable $\Lambda$ module} if (i) the ${\cal O}_X$-module $E$ is pure dimensional, and (ii) for each non-zero ${\cal O}_X$-coherent $\Lambda$-submodule $F\subset E$, the inequality $$\dim\, H^0(X,\,F(n))/{\rm rank}\,(F) \le {\rm dim}\,H^0(X,\,E(n))/{\rm rank}\,(E) $$ holds for $n$ sufficiently large (where ${\rm rank}\,(F)$ for any coherent sheaf on $(X,\,{\cal O}_X(1))$ is by definition the leading coefficient of the Hilbert polynomial of $F$). If the $\Lambda$-module $E$ is nonzero and moreover we can always have strict inequality in the above for $0\ne F\ne E$, then $E$ is called {\bf stable}. An {\bf S-filtration} of a semistable $\Lambda$-module $E$ is a filtration $0=E_0\subset E_1\subset \ldots E_{\ell}=E$ by ${\cal O}_X$-coherent $\Lambda$-submodules, such that each graded piece $E_i/E_{i-1}$ is a semistable $\Lambda$-module, with the same normalized Hilbert polynomial as that of $E$ if non-zero (where `normalized Hilbert polynomial' means Hilbert polynomial divided by its leading coefficient). It can be seen that an S-filtration on a non-zero $E$ is maximal (that is, can not be further refined) if and only if each graded piece $E_i/E_{i-1}$ is stable. The associated graded object $\oplus_{1\le i\le \ell}(E_i/E_{i-1})$ to a maximal S-filtration, after forgetting the gradation, is independent (upto isomorphism) of the choice of an S- filtration, and two non-zero semistable $\Lambda$-modules are called {\bf S-equivalent} if they have S-filtrations with isomorphic graded objects (after forgetting the gradation). The zero module is defined to be S-equivalent to itself. By a standard argument using Quot schemes (originally due to Narasimhan and Ramanathan), it can be seen that semistability is a Zariski open condition on the parameter scheme of any family of $\Lambda$-modules. In order to construct a moduli for semi-stable $\Lambda$-modules, Simpson first shows that if we fix the Hilbert polynomial $P$, then all semi-stable $\Lambda$-modules whose Hilbert polynomial is $P$ form a bounded set, and then shows the following: \begin{proposition}\label{simplocuni} (Simpson [S]) : Let $(X,\,{\cal O}_X(1))$ be a projective scheme, $P$ a fixed Hilbert polynomial, and $\Lambda$ a sheaf of differential operators on $X$. Then there exists a quasi-projective scheme $C$ together with an action of $PGL(N)$ (for some large $N$), and a family $E_C$ of semistable $\Lambda$-modules on $X$ with Hilbert polynomial $P$ para\-met\-rized by $C$ such that (1) the family $E_C$ is a local universal family for semistable $\Lambda$-modules with Hilbert polynomial $P$, (2) two morphisms $f_1,f_2:T\stackrel{\to}{\to}C$ give isomorphic families $f_1^*(E_C)$ and $f_2^*(E_C)$ if and only if there exists a Zariski open cover $T'\to T$ (that is, $T'$ is a disjoint union of finitely many open subsets of $T$ whose union is $T$) and a $T'$-valued point $g:T'\to PGL(N)$ of $PGL(N)$ which carries $f_1|T'$ to $f_2|T'$. (3) a good quotient (`good' in the technical sense of GIT) $C\to C/\!/PGL(N)$ exists, and is (as a consequence of (1) and (2)) the coarse moduli scheme for $S$-equivalence classes of semistable $\Lambda$-modules with Hilbert polynomial $P$. (4) {\bf Limit points of orbits }: Let $\lambda :GL(1)\to GL(N)$ be a 1-parameter subgroup, and let $q\in C$ have limit $q_0\in C$ under $\lambda$, that is, $$q_0 =\lim_{\tau\to 0}\, q\cdot \lambda(\tau)$$ Let $E_{\tau}$ be the pullback of $E_C$ to $X\times A^1$ under the resulting morphism $\lambda :A^1\to C$ on the affine line $A^1$. Then there exists an exhaustive filtration $F_p$ of $E$ (where $E$ is the module associated to $\tau =1$) by $\Lambda$-submodules such that each $F_p/F_{p-1}$ is semistable, and when non-zero it has the same reduced Hilbert polynomial as $E$, and the family $E_{\tau}$ is isomorphic to the family $\ov{E}=\sum_{p\in Z\!\!\!Z} F_p \tau^p \,\subset \, E \otimes C\!\!\!\!I[\tau,\, \tau^{-1}]$ occuring in remark \ref{fact1} above. In particular, the limit $E_0$ is the graded object corresponding to $F_p$, and so the closed points of the quotient are in bijection with the set of $S$-equivalence classes. \end{proposition} \refstepcounter{theorem}\paragraph{Remark \thetheorem} The following feature of the family $E_C$ and the action of $GL(N)$ will be very important: $C$ is a certain locally closed subscheme of the Quot scheme $Quot_{C\!\!\!\!I^N\otimes {\cal W}/X/C\!\!\!\!I}$ of quotients $$q :C\!\!\!\!I^N\otimes {\cal W} \to E$$ where $W$ is some fixed coherent sheaf on $X$, such that $C$ is invariant under $GL(N)$, and $E_C$ is the restriction to $C$ of the universal family ${\cal U}$ on the Quot scheme, The action of $GL(N)$ on the Quot scheme and its lift to ${\cal U}$ is as explained in remark \ref{glnaction} above. \vfill \pagebreak \subsection{Strong local freeness for semistable $\Lambda$-modules} \begin{definition}\label{defstrong}\rm We will say that a semistable $\Lambda$-module $E$ on $X$ is {\bf strongly locally free} if for every S-filtration $0=E_0\subset E_1\subset \ldots E_{\ell}=E$, the associated graded object $\oplus_{1\le i\le \ell}(E_i/E_{i-1})$ is a locally free ${\cal O}_X$-module. (In particular, the zero module is strongly locally free.) \end{definition} \refstepcounter{theorem}\paragraph{Remarks \thetheorem} (1) A nonzero semistable $\Lambda$-module $E$ is strongly locally free if and only if there exists some maximal S-filtration such that the graded pieces are are locally free ${\cal O}_X$-modules. (2) A strongly locally free semistable $\Lambda$-module is necessarily locally free. However, a locally free semistable $\Lambda$-module is not necessarily strongly locally free. \begin{proposition}\label{strong} (1) Let $E_C$ be the family of semistable $\Lambda$-modules parametrized by $C$. Then the condition that the associated graded object to any S-filtration $F_k$ of $E_q$ should be locally free defines a $GL(N)$-invariant open subset $C^o$ of $C$ which is closed under limits of orbits, and hence has a good quotient under $GL(N)$, which is an open subscheme of the moduli of semistable $\Lambda$-modules. (2) For any family $E_T$ of semistable $\Lambda$-modules parametrized by a scheme $T$, the condition that $E_t$ is strongly locally free is a Zariski open condition on $T$. \end{proposition} \paragraph{Proof} Let $U\subset C$ be the open subscheme defined by the condition that $E_q$ is locally free for $q\in U$ (this is indeed open as a consequence of the flatness of $E_C$ over $C$). Note that $U$ is $G=GL(N)$-invariant. Let $\pi :C\to C/\!/G$ be the good quotient. Then as $F=C-U$ is a $G$-invariant closed subset, $\pi(F)$ is closed in $C/\!/G$ by properties of good quotient. Let $C^o = \pi^{-1} (C/\!/G -\pi(F))$. Then $C^o$ is the desired open subscheme of $C$, which proves the statement (1). It follows from the relation between closure of orbits and S-filtrations that points of $C/\!/G$ correspond to S-equivalence classes. Therefore the statement (2) now follows from (1) by the universal property of the moduli $C/\!/G$. \subsection{Semistable pre-${\cal D}$-modules --- definition} We now come back to $(X,Y)$ as before, and our earlier notation. We choose an ample line bundle on each $X_{i,a}$, and fix the resulting Hilbert polynomials $p_{i,a}(n)$ of the sheaf $E_{i,a}$ on $X_{i,a}$. By lemma \ref{lambda} the theory of semistability and moduli for $\Lambda$-modules can be applied to ${\cal D}_{i,a}$-modules. \begin{definition}\rm We say that a pre-${\cal D}$-module is {\bf semistable} if the following two conditions are satisfied. (i) Each of the ${\cal D}_{i,a}$-modules $E_{i,a}$ is semistable in the sense of Simpson. (ii) Each $E_{i,a}$ is strongly locally free, that is, the associated graded object to $E_{i,a}$ under any S-filtration is again a locally free ${\cal O}_{X_{i,a}}$-module (see definition \ref{defstrong} and its subsequent remarks). \end{definition} \refstepcounter{theorem}\paragraph{Remark \thetheorem} The notion of stability is treated later. \begin{proposition} A vector bundle $E_{i,a}$ on $X_{i,a}$ with the structure of a semistable ${\cal D}_{i,a}$-module is strongly locally free if it has good residual eigenvalues as defined in definition \ref{goodresieigen}. As a consequence, a pre-${\cal D}$-module $E$ such that each of the ${\cal D}_{i,a}$-modules $E_{i,a}$ is semistable and has good residual eigenvalues is a semistable pre-${\cal D}$-module. \end{proposition} \paragraph{Proof} Let $Z$ be the polydisk in $C\!\!\!\!I^n$ defined by $|z_i|< 1$, and $W\subset Z$ the divisor with normal crossings defined by $\prod_{k\le r}z_k =0$ (if $r=0$ then $W$ is empty). Let there be given a set theoretic section (fundamental domain) for the exponential map $C\!\!\!\!I\to C\!\!\!\!I\,^* \,:\,z\mapsto \exp(2\pi iz)$. Then the Deligne construction, which associates to a local system ${\cal L}$ on $Z-W$ an integrable logarithmic connection $E({\cal L})$ on $(Z,W)$ with residual eigenvalues in the given fundamental domain, has the following property: if ${\cal K}\subset{\cal L}$ are two local systems, then $E({\cal K})$ is a subbundle of $E({\cal L})$. Now suppose $F$ is an ${\cal O}_Z$-coherent ${\cal D}_Z[\log W]$-submodule of $E({\cal L})$. Let ${\cal K}$ be the local system on $Z-W$ defined by $F$, and let $E({\cal K})$ be its associated logarithmic connection given by Deligne's construction. Then $F|(Z-W)=E({\cal K})|(Z-W)$. As $E({\cal K})$ is a vector subbundle of the vector bundle $E({\cal L})$, it follows that $E({\cal K})$ is just the ${\cal O}_Z$-saturation of $F$. Hence if $E$ has good residual eigenvalues and if $F\subset E$ is an ${\cal O}_Z$-coherent and saturated ${\cal D}_Z[\log W]$-submodule, then $F$ is a vector subbundle. Now we apply this to each $E_{i,a}$ as follows. Let $E=E_{i,a}$ be ${\cal D}_{i,a}$-semistable, and let $F\subset E$ be a step in an S-filtration of $E$. Hence we must have $$ {p_F\over rank(F)} = {p_E\over rank(E)}$$ where $p_F$ denotes the Hilbert polynomial of a sheaf $F$. Note that the ${\cal O}_{X_{i,a}}$-saturation $F'$ of $F$ in $E$ is again an ${\cal O}$-coherent ${\cal D}_{i,a}$-submodule, which restricts to $F$ on a dense open subset of $X_{i,a}$, in particular, $F'$ has the same rank as $F$. Hence with respect to any ample line bundle on $X_{i,a}$, the normalized Hilbert polynomials of $F$ and $F'$ satisfy the relation $$ {p_F\over rank(F)} \le {p_{F'}\over rank(F')}$$ with equality only if $F=F'$. As $E=E_{i,a}$ is semistable, we have $${p_{F'}\over rank(F')} \le {p_{E}\over rank(E)}$$ and hence $F=F'$. Hence we can assume that any step $F$ in an S-filtration of $E=E_{i,a}$ is ${\cal O}$-saturated. Hence the proposition would follow if we show that if for each $i\ge m+1$, $E_{i,a}$ has good residual eigenvalues under $\theta_{i-1}$ (see definition \ref{goodresieigen}), then any ${\cal O}$-coherent and saturated ${\cal D}_{i,a}$-submodule $F$ is a vector subbundle. This is a purely local question on $X$ in the euclidean topology, so we may assume that $X$ is a polydisk with local coordinates $z_i$ and $Y$ is defined by a monomial in the $z_i$. Then $X_{i,a}$ is just some coordinate $i$-plane $Z$ contained in $Y$, and $W=X_{i,a}\cap S_{i-1}$ is a normal crossing divisor in the polydisk $Z$, defined by a monomial in the coordinates. Using the local coordinates $z_k$, we get a structure of ${\cal D}_Z [\log W]$-module on the vector bundle $E=E_{i,a}$ on $Z$ with good residual eigenvalues, and $F$ becomes an ${\cal O}_Z$-coherent and saturated ${\cal D}_Z[\log W]$-submodule. (Note that these ${\cal D}_Z[\log W]$-structures very much depend on the choice of local coordinates $z_k$). As $E$ has good residual eigenvalues by assumption, it follows from the first part of our argument (involving Deligne constructions over polydisks) that $F$ is a vector subbundle of $E_{i,a}$. This proves the proposition. \refstepcounter{theorem}\paragraph{Remark \thetheorem}\label{goodresi2} If $E_{i,a}$ has good residual eigenvalues under $\theta_{i-1}$, and if $F$ is a vector subbundle which is a ${\cal D}_i$-submodule, then it can be seen that the associated graded ${\cal D}_i$-module $F\oplus (E_{i,a}/F)$ also has good residual eigenvalues under $\theta_{i-1}$. The following example shows that if $E$ is a vector bundle with an integrable logarithmic connection, and $F\subset E$ is an ${\cal O}$-coherent sub connection such that $E/F$ is torsion free, then it can still happen that $E/F$ is not locally free if $E$ has bad residual eigenvalues. \begin{example}\label{Esnault}\rm (Due to H\'el\`ene Esnault) Let $X$ be a polydisk in $C\!\!\!\!I^2$, with divisor $Y$ defined by $xy=0$. Let $E={\cal O}_X^{\oplus 2}$ with basis $v_1=(1,0)$ and $v_2=(0,1)$. On $E$ we define an integrable logarithmic connection $\nabla :E\to \Omega^1_X[\log Y]\otimes E$ by putting $\nabla(v_1) = (dx/x)\otimes v_1 $ and $\nabla(v_2)=(dy/y)\otimes v_2$ (this has curvature zero, as $(-\log x)v_1$ and $(-\log y)v_2$ form a flat basis of $E|X-Y$). Note that both $0$ and $1$ are eigenvalues of the residue of $(E,\nabla)$, along any branch of $Y$. Let ${\bf m}\subset {\cal O}_X$ be the ideal sheaf generated by $x$ and $y$. This is a sub ${\cal D}_X[\log Y]$-module of ${\cal O}_X$, which is torsion free but not locally free. Now define a surjective homomorphism $\varphi : E\to {\bf m}$ sending $v_1\mapsto x$ and $v_2\mapsto y$, which can be checked to be ${\cal D}_X[\log Y]$-linear, and put $F=ker(\varphi)$. Then $F$ is an ${\cal O}_X$-saturated subconnection (which is in fact locally free), but $E/F={\bf m}$ is not locally free. \end{example} \subsection{Semistable pre-${\cal D}$-modules --- local universal family} We now construct a local universal family for semistable pre-${\cal D}$-modules with given Hilbert polynomials. Let $C_{i,a}$ be the scheme for $({\cal D}_{i,a},\,p_{i,a})$ given by the proposition \ref{simplocuni}, with the action of $PGL(p_{i,a}(N_{i,a}))$ as in the proposition \ref{simplocuni}. Let $C^o_{i,a}\subset C_{i,a}$ be the open subset where $E_{i,a}$ is strongly locally free. By proposition \ref{strong}, $C^o_{i,a}$ is an open subset of $C_{i,a}$ which is $PGL(p_{i,a}(N_{i,a}))$-invariant and admits a good quotient for the action of $PGL(p_{i,a}(N_{i,a}))$, which is an open subscheme of $C_{i,a}/\!/PGL(p_{i,a}(N_{i,a}))$. Let $C^o_i= \prod_a C^o_{i,a}$ and let $C=\prod_i C^o_i=\prod_{i,a} C^o_{i,a}$. Let $E_{i,a}$ again denote the pullback of $E_{i,a}$ to $X_{i,a}\times C$ under the projection $C\to C^o_{i,a}$. Let $E_i|Y^*_i$ and $E_{i+1}|Y^*_i$ be regarded as families of ${\cal D}^*_i$-modules parametrized by $C$. These are again flat over $C$ as the $E_i$ are locally free on $X_i$. Hence by applying lemma \ref{nnreplem} to the pair $E_{i+1}|Y^*_i$ and $E_i|Y^*_i$ of ${\cal D}^*_i$-modules parametrized by $C$, we get linear schemes $A_i$ and $B_i$ over $C$ which parametrize ${\cal D}^*_i$-linear maps $t_i$ and $s_i$ in either direction between the specializations of these two families. Let $H_i\subset A_i\times_CB_i$ be the closed subscheme defined by the conditions on $t_i$ and $s_i$ imposed by the definition of a pre-${\cal D}$-module (it can be seen that the conditions indeed define a closed subscheme $H_i$). Finally, let $H$ be the fibered product over $C$ of all the $H_i$. By its construction, $H$ parametrizes a natural family of pre-${\cal D}$-modules on $(X,Y)$. Let $G_i =\prod_a G_{i,a}$ and let $G = \prod_i G_i=\prod_{i,a} G_{i,a}$. Note that this is a reductive group. We define an action of $G$ on $H$ as follows. Any point $\underline{q}_{i,a}$ of $C^o_{i,a}$ is represented by a quotient $q_{i,a} :{\cal O}_{X_{i,a}}(-N_{i,a})^{p_{i,a}(N_{i,a})} \to E_{i,a}$ (which satisfies some additional properties) and a point of $H$ over a point $(\underline{q}_{i,a})\in C=\prod_{i,a}C^o_{i,a}$ is given by the additional data $s_j:(E_j|Y^*_j)\to (E_{j+1}|Y^*_j)$ and $t_j:(E_{j+1}|Y^*_j)\to (E_j|Y^*_j)$, and so a point of $H$ is represented by the data $(q_i,s_j,t_j)$. \refstepcounter{theorem}\paragraph{Remark \thetheorem}\label{eq} Note that two such tuples $(q_i,s_j,t_j)$ and $(q'_i,s'_j,t'_j)$ represent the same point of $H$ if there exists an isomorphism $\phi: E \to E'$ of pre-${\cal D}$-modules $E=(E_i,s_it_i)$ and $E' =(E'_i,s'_i,t'_i)$ such that $q'_i = \phi_i\circ q_i$ for each $i$. \begin{definition}\label{defaction} \rm (Right action of the group $G$ on the scheme $H$.) In terms of valued points, we define this as follows. For any point $h$ of $H$ represented by $(q_i,s_j,t_j)$, and an element $g=(g_i)\in G = \prod_i G_i$, put $$ (q_i,\,s_j,\,t_j)\cdot g = (q_i\circ g_i,\,s_j,\,t_j )$$ Note that this is well defined with respect to the equivalence given by remark \ref{eq}, and indeed defines an action of $G$ on $H$ lifting its action on $C$, as follows from remark \ref{liftformula}. \end{definition} It is clear from the definitions of $H$ and this action that two points of $H$ parametrise isomorphic pre-${\cal D}$-modules if and only if they lie in the same $G$ orbit. The morphism $H\to C$ is an affine morphism which is $G$-equivariant, where $G$ acts on $C$ via $\prod_{i,a} G_{i,a}$. As seen before, the action of each $G_{i,a}$ on $C^o_{i,a}$ admits a good quotient in the sense of geometric invariant theory, and hence the action of $G$ on $C$ admits a good quotient $C/\!/G$. A well known lemma of Ramanathan (see Proposition 3.12 in [Ne]) asserts that if $G$ is a reductive group acting on two schemes $U$ and $V$ such that $V$ admits a good quotient $V/\!/G$, and if there exists an affine, $G$-equivariant morphism $U\to V$, then there exists a good quotient $U/\!/G$. Applying this to the $G$-equivariant affine morphism $H\to C$, a good quotient $H/\!/G$ exists, which by construction and universal properties of good quotients is the coarse moduli scheme of semistable pre-${\cal D}$-modules with given Hilbert polynomials. By construction this is a separated scheme of finite type over $C\!\!\!\!I$, and is, in fact, quasiprojective. Note that under a good quotient in the sense of geometric invariant theory, two different orbits can in some cases get mapped to the same point (get identified in the quotient). In the rest of this section, we determine what are the closed points of the quotient $H/\!/G$. \subsection{Stability and points of the moduli} Let $T_{i,a}\subset G_{i,a}$ be its center. Let $T_i=\prod_aT_{i,a}$, and $T= \prod_iT_i=(\prod_{i,a}T_{i,a})$. Note that $T \subset G$ as a closed normal subgroup (which is a torus), which acts trivially on $C$. By definition of $T$ we have a canonical identification $$T = \prod_i \Gamma(X_i,{\cal O}^{\times}_{X_i})$$ By definition \ref{defaction} and remark \ref{scalarmulti}, the action of $\lambda =(\lambda_i)\in T$ on $h=(q_i,s_i,t_i)\in H$ is given by $$h\cdot g =( \lambda_iq_i,\,s_i,\,t_i) = (q_i,\, {\lambda_i\over\lambda_{i+1}}s_i,\, {\lambda_{i+1}\over\lambda_i}t_i)$$ In [N-S],the construction of the quotient $H/\!/G$ was made in a complicated way in two steps: by Ramanathan's lemma, we can first have the quotient $R=H/\!/T$, and then as the second step we have the quotient $H/\!/G = R/\!/(G/T)$. However, we now do it in a much simplified way, which gives a simplification also of [N-S]. \begin{definition}\rm A sub pre-${\cal D}$-module ${\bf F}$ of a semistable pre-${\cal D}$-module ${\bf E}$ will be called an {\bf S-submodule} if each non-zero $F_{i,a}$ has the same normalized Hilbert polynomial $p_{i,a}$ as that of $E_{i,a}$. A filtration ${\bf E}_p$ on a semistable pre-${\cal D}$-module ${\bf E}$ is an {\bf S-filtration of the pre-${\cal D}$-module} if each ${\bf E}_p$ is an S-submodule, equivalently the given filtration on each $E_{i,a}$ is an S-filtration. \end{definition} \refstepcounter{theorem}\paragraph{Remark \thetheorem} Given a semistable pre-${\cal D}$-module ${\bf E}$, an S-submodule, the corresponding quotient module, and the graded pre-${\cal D}$-module ${\bf E}'$ associated with an S-filtration are again semistable pre-${\cal D}$-modules. Moreover, ${\bf E}'$ has the same Hilbert polynomials $p_{i,a}$ as ${\bf E}$. We will say that ${\bf E}$ is (primitively) S-equivalent to ${\bf E}'$. \begin{definition}\rm The equivalence relation on the set of isomorphism classes of all semistable pre-${\cal D}$-modules generated by the above relation, under which the graded module ${\bf E} '$ associated to an S-filtration of ${\bf E}$ is taken to be equivalent to ${\bf E}$, will be called {\bf S-equivalence for pre-${\cal D}$-modules}. \end{definition} \begin{definition}\label{defstable}\rm We say that a semistable pre-${\cal D}$-module is {\bf stable} if it nonzero and does not admit any nonzero proper S-submodule. \end{definition} \begin{proposition}\label{1parafamily} Let ${\bf E}_H$ denote the tautological family of pre-${\cal D}$-modules para\-met\-rized by $H$. Let $\lambda :GL(1)\to G$ be a 1-parameter subgroup of $G=\prod G_i$, and let $h=(q_i, s_j, t_j)\in H$ be a point such that the limit $\lim_{\tau \to 0}\,h\lambda(\tau)$ exists in $H$. Let $\ov{\lambda}:A^1\to H$ be the resulting morphism. Then there exists an S-filtration $({\bf E}_h)p$ of the pre-${\cal D}$-module $({\bf E}_h)_p$ such that the pullback of ${\bf E}_H$ to $A^1$ under $\ov{\lambda}:A^1\to H$ is isomorphic to the family constructed in remark \ref{deform} \end{proposition} \paragraph{Proof} By the definition of the action of $G$ on $H$, the family ${\bf E}_{\tau}$ satisfies the following properties: (i) The families $(E_{i,a})_{\tau}$ are of the necessary type by proposition \ref{simplocuni}.(4). (ii) Outside $\tau=0$, the homomorphisms $(s_j)_{\tau}$ and $(t_j)_{\tau}$ are pull backs of $s_j$ and $t_j$. Therefore now the proposition follows from remark \ref{fact3}. The following lemma, whose proof is obvious, is necessary to show that stability in an open condition on the parameter scheme $T$ of a family ${\bf E}_T$ of pre-${\cal D}$-modules. \begin{lemma} Let ${\bf E}_T$ be a family of pre-${\cal D}$-modules parametrized by $T$. Let there be given a family ${\bf F}_T$ of sub vectorbundles $F_{i,T}\subset E_{i,T}$ which are ${\cal D}_{i,T}$-submodules. Then there exists a closed subscheme $T_o\subset T$ with the following universal property. Given any base change $T'\to T$, the pullback ${\bf F}_{T'}$ is a sub pre-${\cal D}$-module of ${\bf E}_{T'}$ if and only if $T'\to T$ factors through $T_o$. \end{lemma} Using the above lemma, the `quot scheme argument for openness of stability' can now be applied to a family of pre-${\cal D}$-modules, to give \begin{proposition}\label{open} Stability is a Zariski open condition on the parameter scheme $T$ of any family ${\bf E}_T$ of semistable pre-${\cal D}$-modules. \end{proposition} \paragraph{Proof} Let $\pi:P\to T$ be the projective scheme, which is closed subscheme of a fibered product over $T$ of relative quot schemes of the $E_{i,T}$, which parametrizes families of sub vector bundles $F_i$ which are ${\cal D}_i$-submodules with the same reduced Hilbert polynomials as those of ${\bf E}$. (Actually, the Quot scheme parametrizes coherent quotients flat over the base, and the condition of ${\cal D}_i$-linearity gives a closed subscheme. Now by the assumption of strong local freeness on the $E_{i,a}$, it follows that the quotients are locally free). Now by the above lemma, $P$ has a closed subscheme $P_o$ where ${\bf F}_{P_o}$ is a family of sub pre-${\cal D}$-modules with the same normalized Hilbert polynomials, and every such sub pre-${\cal D}$-module of a ${\bf E}_t$ for $t\in T$ occures among these. Hence $T-\pi(P_o)$ is the desired opoen subset of $T$. \refstepcounter{theorem}\paragraph{Remark \thetheorem} As semistability is itself a Zariski open condition on any family of pre-$D$-modules, it now follows that stability is a Zariski open condition on the parameter scheme $T$ of any family ${\bf E}_T$ of pre-${\cal D}$-modules. Now all the ingredients are in place for the following main theorem, generalizing theorem 4.19 in [N-S]. \begin{theorem}\label{maintheorem} Let $X$ be a non-singular variety with a normal crossing divisor $Y$. Let a numerical polynomial $p_{i,a}$ and an ample line bundle on $X_{i,a}$ be chosen for each $X_{i,a}$. Then we have the following. (1) There exists a coarse moduli scheme ${\cal M}$ for semistable pre-${\cal D}$-modules ${\bf E}$ on $(X,Y)$ where $E_{i,a}$ has Hilbert polynomial $p_{i,a}$. The scheme ${\cal M}$ is quasiprojective, in particular, separated and of finite type over $C\!\!\!\!I$. (2) The points of ${\cal M}$ are S-equivalence classes of semistable pre-${\cal D}$-modules. (3) The S-equivalence class of a stable pre-${\cal D}$-module equals its isomorphism class. (4) ${\cal M}$ has an open subscheme ${\cal M} ^s$ whose points are the isomorphism classes of all stable pre-${\cal D}$-modules. This is a coarse moduli for (isomorphism classes of) stable pre-${\cal D}$-modules. \end{theorem} \paragraph{Proof} The statement (1) is by the construction of ${\cal M}=H/\!/G$ and properties of a good quotient. The statement (2) follows from remark \ref{deform} and propositon \ref{1parafamily}. Let $x\in H$ and let $x_0 \in H$ be a limit point of the orbit $Gx$. Then by properties of GIT quotients, there exists a $1$-parameter subgroup $\lambda:GL(1)\to G$ such that $x_0=\lim_{\tau\to 0}\,x\cdot\lambda(\tau)$. Any such limit is of the type given by proposition \ref{1parafamily}, made from an S-filtration of the corresponding pre-${\cal D}$-module ${\bf E}_x$. If $x\in H$ be stable (means corresponds to a stable pre-${\cal D}$-module), then it has no non-trivial S-filtration, so the orbit of $x$ is closed. As stability is an open condition on $H$ by proposition \ref{open}, if the orbit of a point $y$ in $H$ has a limit $x$ which is stable, then the point $y$ must itself be stable. So by the above, the orbit of $y$ must be closed, so $x\in Gy$. Hence a stable point $x$ is not the limit point of any other orbit. Hence (3) follows. Finally, the statement (4) follows from (1), (2), (3), and proposition \ref{open}. This completes the proof of the theorem. {\footnotesize Note: the statement (3) in theorem 4.19 of [N-S] has a mistake - the `if and only if' should be changed to `if', removing the `only if' part.) \bigskip \section{Perverse Sheaves on $(X,Y)$} In this section, we give a finite description (in terms of a finite quiver of finite dimensional vector spaces and linear maaps) of perverse sheaves on $(X,Y)$, that is, perverse sheaves on $X$ that are cohomologically constructible with respect to the stratification $X=\cup_i\, (S_i-S_{i-1})$, which closely parallels our definition of pre-${\cal D}$-modules. This enables us to describe the perverse sheaf associated to the ${\cal D}$-module associated to a pre-${\cal D}$-module directly in terms of the pre-${\cal D}$-module. In turn, this allows us to deduce properties of the analytic morphism from the moduli of pre-${\cal D}$-modules to the moduli of such quivers (which is the moduli of perverse sheaves with given kind of singularities). More general finite descriptions of perverse sheaves in terms of quivers exist in literature (see for example MacPherson and Vilonen [M-V]), where the requirement of normal crossing singularities is not needed, and the resulting moduli space has been constructed by Brylinski, MacPherson, and Vilonen [B-M-F]). We cannot use their moduli directly, as it is too general for the specific purpose of describing the Riemann-Hilbert morphism by a useable formula (where, for example, we can see the differential of the map). \subsection{The specialization functor} In this section, a vector bundle will usually mean a geometric vector bundle in the analytic category. That is, if $E$ is a locally free sheaf (but not necessarily of constant rank) on a reduced scheme $X$ of finite type over $C\!\!\!\!I$ , then when we refer to the {\bf vector bundle $E$}, what we mean is the analytic space (with euclidean topology) associated to the scheme $\mathop{\rm Spec}\nolimits_X Sym^{\cdot}(E^*)$. The replacement by Verdier (see [V1] and [V2]) of the not so canonical operation of {\sl restriction to a tubular neighbourhood} by {\sl specialization to normal cone (or bundle)} is used below in a somewhat more general set up as follows. Let $M$ be a complex manifold, $T \subset M$ be a divisor with normal crossings, and for some integer $k$ let $T_k$ be a union of components of the $k$-dimensional singularity stratum of $T$. Let $C \to T_k$ be a union of components of the normalization of $T_k$, and $f: C\to M$ the composite map $C\to T_k\to M$. Let $N_f$ be the normal bundle to $f:C\to M$, and let $U\subset N_f$ be the open subset which is the complement of the normal crossing divisor $F_f$ in $N_f$ defined by vectors tangent to branches of $T$. Note that in particular $F_f$ contains the zero section of $N_f$. Then we have a functor from local systems on $M-T$ to local systems on $U$ defined as follow. For each $x\in C$, there exists an open neighbourhood $V_x$ of $f(x)$ in $M$ such that the restricted map $f_x:C_x \to M$ where $C_x=f^{-1}(V_x)$ and $f_x=f|C_x$ is is a closed imbedding of the manifold $C_x$ into $V_x$. Then note that the normal bundle $N_{f_x}$ of $f_x:C_x\to V_x$ is the restriction of $N_f$ to $C_x\subset C$. Let $F_{f_x}=F_f\cap N_{f_x}$, and $U_x=N_{f_x}-F_{f_x}$. The usual functor of specialization (see [V1] and [V2]) now associates a local system on $U_x$ to a local system on $V_x-T$. These glue together to define our desired functor. Given a local system ${\cal E}$ on $M_T$, we denote its specialization by ${\cal E}|\!|U$, which is a local system on $U$. More generally, the above method gives a definition of a specialization functor \\ between the derived categories of cohomologically bounded constructible complexes of sheaves of complex vector spaces on $M$ and $N_f$. If the complex ${\cal F}^{\cdot}$ is cohomologically constructible with respect to the singularity stratification of $(M, T)$, then its specialization ${\cal F}^{\cdot}|\!|N_f$ to $N_f$ is cohomologically constructible with respect to the singularity stratification of $(N_f,\,F_f)$. This functor carries perverse sheaves to perverse sheaves. \refstepcounter{theorem}\paragraph{Remark \thetheorem}\label{composite} For a topological manifold $M$ which is possibly disconnected, choose a base point in each component $M_a$, and let $\Gamma^M_a$ denote the indexed set of the fundamental groups of the components of $M$ with respect to the chosen base points, indexed by $a\in \pi_0(M)$. We have an equivalence categories between local systems on $T$ and an indexed collection of group representations $\rho_a: \Gamma^T_a \to GL(n_a)$. In the above situation, we have the groups $\Gamma^U_a$ and $\Gamma^{M-T}_b$. Then there exists a map $\gamma: \pi_0(U)\to \pi_0(M-T)$ and a group homomorphism $$\psi_a:\Gamma^U_a\to \Gamma^{M-T}_{\gamma(a)}$$ for each $a\in \pi_0(U)$, such that the above functor of specialization from local systems on $M-T$ to local systems on $U$ is given by associating to a collection of representations $\rho_b :\Gamma^{M-T}_b \to GL(n_b)$ where $b$ varies over $\pi_0(M-T)$ the collection composite representations $\rho_{\gamma(a)}\circ \psi_a$. In this sense, specialization is like pullback. \subsection{Finite representation} We now return to $(X,Y)$ and use our standard notation (see section 2). We will apply the above specialization functor to the following cases. Case(1): $M$ is $X$, $T$ is $Y$, and $k=d-1$, so $C=X_{d-1}=Y^*$ is the normalization of $Y$. In this case, starting from a local system ${\cal E}_d$ on $X-Y$ we get a local system on $U_{d-1}$, which we denote by ${\cal E}_d|\!|U_{d-1}$. Case(2): $M$ is $N_{i+1}$ ($=$ the normal bundle to $f_i:X_i\to X$), $T$ is the divisor $F_{i+1}$ in $N_{i+1}$ defined by vectors tangent to branches of $Y$, $k=i$, $T$ the inverse image of $S_i$ under the map $p_{i+1}:X_{i+1}\to S_{i+1}$, and $C=Y^*_i$. In this case, starting from a local system ${\cal E}_{i+1}$ on $U_{i+1}$, we get a local system on $R_i$ which we denote by ${\cal E}_{i+1}|\!|R_i$. Case(3): Apply this with $M=N_{i+2}$, $T=F_{i+2}$, $C=Z_i$ and $C\to M$ the composite $$Z_i\stackrel{p_{\{i+2 \},\{ i,i+2\} }}{\to} X_{i+2} \hookrightarrow N_{i+2}$$ In this case, starting from a local system ${\cal E}_{i+2}$ on $U_{i+2}$, we get a local system on $W_i$ which we denote by ${\cal E}_{i+2}|\!|W_i$. Case(4): Apply this with $M=N_{i+1}$, $T=F_{i+1}$, $C=Z^*_i$ and $C\to M$ the composite $$Z^*_i\stackrel{p_{\{i+1 \},\{ i,i+1, i+2\} }}{\to} X_{i+1} \hookrightarrow N_{i+1}$$ In this case, starting from a local system ${\cal E}_{i+1}$ on $U_{i+1}$, we get a local system on $W^*_i$ which we denote by ${\cal E}_{i+1}|\!|W^*_i$. Case(5): Apply this with $M=N_{i+1,i+2 }$, $T=F_{i+1,i+2}$, $C = Z^*_i$, and $C\to M$ the composite $$Z^*_i\stackrel{p_{\{i+1, i+2 \},\{ i,i+1, i+2\} }}{\to} Y^*_{i+1} \hookrightarrow N_{i+1, i+2}$$ In this case, starting from a local system ${\cal F}_{i+1}$ on $R_{i+1}$, we get a local system on $W^*_i$ which we denote by ${\cal F}_{i+1}|\!|W^*_i$. On the other hand, note that the derivative of the covering projection $p_{\{i \},\{ i,i+1\} } : Y^*_i\to X_i$ is a map $dp : N_{i,i+1} \to N_i$ under which $F_{i,i+1}\subset N_{i,i+1}$ is the inverse image of $F_i\subset N_i$. Hence $dp$ induces a map $R_i\to U_i$. If ${\cal E}_i$ is a local system on $U_i$, then we denote its pullback under this map by ${\cal E}_i|R_i$, which is a local system on $R_i$. Similarly, for the covering projection $p_{\{i \},\{ i,i+2\} } :Z_i\to X_i$ the derivative induces a map $W_i\to U_i$. If ${\cal E}_i$ is a local system on $U_i$ then we denote its pullback under this map by ${\cal E}_i|W_i$ which is a local systems on $W_i$. Note that the derivative of the $2$-sheeted covering projection $X_{\{ i,i+1,i+2 \} }\to X_{\{ i,i+2\} }$ induces a $2$ sheeted covering projection $\pi : W^*_i\to W_i$. We will denote the pullbacks of ${\cal E}_i|W_i$ and ${\cal E}_{i+2}|\!|W_i$ under $\pi :W^*_i\to W_i$ by ${\cal E}_i|W^*_i$ and ${\cal E}_{i+2}|\!|W^*_i$ respectively. Note that the same ${\cal E}_{i+2}|\!| W^*_i$ could have been directly defined by specializing, similar to case 4 above. In summary, for a collection of local systems ${\cal E}_i$ on $U_i$, we have various pullbacks or specializations associated with it follows: (i) On $R_i$ we have local systems ${\cal E}_i |R_i$ and ${\cal E}_{i+1} |\!| R_i$, for $i\le d-1$ (ii) On $W_i$ we have local systems ${\cal E}_i |W_i$ and ${\cal E}_{i+2}|\!|W_i$, for $i\le d-2$ (iii) On $W^*_i$ we have local systems ${\cal E}_i|W^*_i$, ${\cal E}_{i+1}|\!|W^*_i$, and ${\cal E}_{i+2}|\!|W^*_i$, for $i\le d-2$ . \refstepcounter{theorem}\paragraph{Remark \thetheorem} On $W^*_i$, we have a canonical identification between the sheaf ${\cal E}_{i+1}|\!| W^*_i$ (defined as in case (4) above) and the sheaf $({\cal E}_{i+1}|\!|R_{i+1})|\!|W^*_i$ (defined as in case (5) above). \refstepcounter{theorem}\paragraph{Remark \thetheorem} We have earlier defined central elements $\tau_i(c)$ in the fundamental group of each connected component $R_i(c)$ of $R_i$ (see section 2). Given a linear system ${\cal F}$ on $R_i$, we denote by $\tau_i$ the automorphism of ${\cal F}$ induced by the monodromy action of the central element $\tau_i(c)$ on $R_i(c)$. \begin{definition}\rm A {\bf Verdier object} $({\cal E}_i,C_i,V_i)$ on $(X,Y)$ consists of the following. (1) For each $m\le i\le d$, ${\cal E}_i$ is a local system on $U_i$ (the ranks of the local systems are not necessarily constant.) (2) For each $m\le i\le d-1$, $C_i:({\cal E}_{i+1}|\!| R_i) \to ({\cal E}_i| R_i)$ and $V_i:({\cal E}_i |R_i) \to ({\cal E}_{i+1} |\!| R_i)$ are homomorphisms of local systems, such that \begin{eqnarray*} V_iC_i &=& 1-\tau_i {\mbox{~{\rm on}~}} {\cal E}_{i+1}|\!| R_i \\ C_iV_i &=& 1-\tau_i {\mbox{~{\rm on}~}} {\cal E}_i| R_i \\ \end{eqnarray*} (3) Let $m\le i \le d-2$. Let $\pi: W^*_i\to W_i$ be the covering projection induced by $\pi:Z^*_i\to Z_i$. Let $$a_{i+2}:{\cal E}_{i+2}|\!|W_i \to \pi_*\pi^*({\cal E}_{i+2}|\!|W_i) = \pi_*({\cal E}_{i+2}|\!|W^*_i)$$ $$a_i:{\cal E}_i|W_i \to \pi_*\pi^*({\cal E}_i|W_i) = \pi_*({\cal E}_i|W^*_i)$$ be adjunction maps, and let the cokernels of these maps be denoted by $$q_{i+2}:\pi_*({\cal E}_{i+2}|\!|W_i)\to Q_{i+2}$$ $$q_i:\pi_*({\cal E}_i|W_i)\to Q_i$$ Then we impose the requirement that the composite map $${\cal E}_{i+2}|\!|W_i\stackrel{a_{i+2}}{\to} \pi_*({\cal E}_{i+2}|\!|W^*_i) \stackrel{\pi_*(C_{i+1}|\!|W^*_i)}{\to} \pi_*({\cal E}_{i+1}|\!| W^*_i) \stackrel{\pi_*(t_i|W^*_i)}{\to} \pi_*({\cal E}_i | W^*_i) \stackrel{q_i}{\to} Q_i $$ is zero. (4) Similarly, we demand that for all $m\le i\le d-2$ the composite map $$Q_{i+2}\stackrel{q_{i+2}}{\leftarrow} \pi_*({\cal E}_{i+2}|\!| W^*_i) \stackrel{\pi_*(V_{i+1}|\!|W^*_i)}{\leftarrow} \pi_*({\cal E}_{i+1}|\!| W^*_i) \stackrel{\pi_*(V_i|W^*_i)}{\leftarrow} \pi_*({\cal E}_i | W^*_i) \stackrel{a_i}{\leftarrow} {\cal E}_i|W_i$$ is zero. (5) Note that as $\pi:W^*_i\to W_i$ is a 2-sheeted cover, for any sheaf ${\cal F}$ on $W_i$ the new sheaf $\pi_*\pi^*({\cal F})$ on $W_i$ has a canonical involution coming from the deck transformation for $W^*_i\to W_i$ which transposes the two points over any base point. In particular, the local systems $\pi_*({\cal E}_{i+2}|\!|W^*_i)=\pi_*\pi^*({\cal E}_{i+2}|\!|W_i)$ and $\pi_*({\cal E}_i|W^*_i)=\pi_*\pi^*({\cal E}_i|W_i))$ have canonical involutions, which we denote by $\nu$. We demand that the following diagram should commute. Diagram III. $$\begin{array}{ccccc} \pi_*({\cal E}_{i+1} |\!| W^*_i) & \stackrel{\pi_*(V_{i+1}|\!|W^*_i)}{\to} & \pi_*({\cal E}_{i+2}|\!|W^*_i) & \stackrel{\nu}{\to} & \pi_*({\cal E}_{i+2}|\!|W^*_i) \\ {\scriptstyle \pi_*(C_{i+1}|\!|W^*_i)}\downarrow & & & & \downarrow {\scriptstyle \pi_*(C_i|W^*_i)}\\ \pi_*({\cal E}_i|W^*_i) & \stackrel{\nu}{\to} & \pi_*({\cal E}_i|W^*_i) & \stackrel{\pi_*(V_i|W^*_i)}{\to} & \pi_*({\cal E}_{i+1} |\!| W^*_i) \\ \end{array}$$ \end{definition} \refstepcounter{theorem}\paragraph{Remark \thetheorem} As the adjunction maps are injective (in particular as $a_i$ is injective), the condition (3) is equivalent to demanding the existence of a unique $f$ which makes the following diagram commute. Diagram I. $$\begin{array}{ccccc} {\cal E}_{i+2}|\!|W_i & & \stackrel{f}{\longrightarrow} & & {\cal E}_i|W_i \\ a_{i+2}\downarrow & & & & \downarrow a_i\\ \pi_*({\cal E}_{i+2}|\!| W^*_i) & \stackrel{\pi_*(C_{i+1}|\!|W^*_i)}{\to} & \pi_*({\cal E}_{i+1}|\!| W^*_i) & \stackrel{\pi_*(C_i|W^*_i)}{\to} & \pi_*({\cal E}_i | W^*_i)\\ \end{array}$$ Similarly, the condition (4) is equivalent to the following: there must exist a unique homomorphism $g$ which makes the following diagram commute. Diagram II. $$\begin{array}{ccccc} {\cal E}_{i+2}|\!|W_i & & \stackrel{g}{\longleftarrow} & & {\cal E}_i|W_i \\ a_{i+2}\downarrow & & & & \downarrow a_i\\ \pi_*({\cal E}_{i+2}|\!| W^*_i) & \stackrel{\pi_*(V_{i+1}|\!|W^*_i)}{\leftarrow} & \pi_*({\cal E}_{i+1}|\!| W^*_i) & \stackrel{\pi_*(V_i|W^*_i)}{\leftarrow} & \pi_*({\cal E}_i | W^*_i)\\ \end{array}$$ It can be seen that the above definition of a Verdier object on $(X,Y)$ reduces in the case of a polydisk to the hypercube description of perverse sheaf on a polydisk. As Verdier objects, perverse sheaves, and the specialization functors are all local in nature, we get the following by gluing up. \begin{proposition} There is an equivalence of categories between the category of \\ Verdier objects and the category of perverse sheaves on $(X,Y)$. \end{proposition} \subsection{Moduli for perverse sheaves} As the various fundamental groups are finitely generated, the definition of a Verdier object has an immediate translation in terms of {\bf quivers}, that is, diagrams of finite dimensional vector spaces and linear maps, by means of remark \ref{composite}. We define a {\bf family of Verdier objects} parametrized by some space $T$ as a family of such quivers over $T$, in which vector spaces are replaced by vector bundles over $T$ and linear maps (or group representations) are replaced by endomorphisms of the bundles. These obviously form an algebraic stack in the sense of Artin, if we work in the category of schemes over $C\!\!\!\!I$. When we fix the ranks $n_{i,a}$ of the restrictions of local systems ${\cal E}_i$ on connected components $X_{i,a}$ of $X_i$, and go modulo the conjugate actions of the various $GL(n_a)$ (this is exactly as in the section 6 of [N-S] so we omit the details), we get an affine scheme of finite type over $C\!\!\!\!I$ as the moduli of Verdier objects with given ranks. The points of this moduli space are Jordan-Holder classes (that is, semisimplifications) of Verdier objects. The above definition of families and construction of moduli is independent (upto isomorphism) of the choices of base points and generators for the various fundamental groups. We define an {\bf algebraic} (or {\bf holomorphic}) {\bf family of perverse sheaves on $(X,Y)$} to be an algebraic (or holomorphic) family of Verdier objects, parametrized by a complex scheme (or a complex analytic space) $T$. Therefore, we have \begin{proposition}\label{points} There exists a coarse moduli scheme $\cal P$ for perverse sheaves on $(X,Y)$ of fixed numerical type. The scheme ${\cal P}$ is an affine scheme of finite type over $C\!\!\!\!I$, and points of ${\cal P}$ correspond to Jordan-Holder classes of perverse sheaves. \end{proposition} \section{The Riemann-Hilbert morphism} In section 7.1 we define an analytic morphism ${\cal R\!\!H}$ from the stack (or moduli) of pre-${\cal D}$-modules to the stack (or moduli) of Verdier objects, which reperesents the de Rham functor. Note that even if both sides are algebraic, the map is only analytic in general, as it involves integration in order to associate to a connection its monodromy. Next (in section 7.2) we prove some properties of the above Riemann-Hilbert morphism ${\cal R\!\!H}$, in particular that it is a local isomorphism at points representing pre-${\cal D}$-modules which have good residual eigenvalues. This generalizes the rigidity results in [N] and [N-S]. \subsection{Definition of the Riemann-Hilbert morphism} The following allows us to go from pre-${\cal D}$-modules to perverse sheaves. \begin{proposition}\label{map} Let $X$ be a nonsingular variety, $Y\subset X$ a divisor with normal crossing, and let $Y^*\to Y$ the normalization of $Y$, with $f:Y^*\to X$ the composite map. Let $N$ be the normal bundle to $f: Y^* \to X$, and let $F\subset N$ be the closed subset of the total space of $N$ defined by vestors tangent to branches of $Y$ (in particular, this includes the zero section $Y^*$ of $N$). Let $\pi : N_f\to Y$ be the bundle projection. Then we have (1) If $E$ is a vector bundle on $Y^*$ together with the structure of a ${\cal D}_N[\log F]|Y^*$-module, then $\pi^*F$ is naturally a ${\cal D}_N[\log F]$-module. (2) Let $E$ be a vector bundle on $X$ together with the structure of a ${\cal D}_X[\log Y]$-module, and let $E|Y^*$ be its pullback to $Y^*$, which is naturally a module over $f^*{\cal D}_X[\log Y]$. Then $\pi^*(E|Y^*)$ is naturally a ${\cal D}_N[\log F]$-module. (3) If $E$ is as in (2) above and if the residual eigenvalues of $E$ do not differ by non-zero integers on any component of $Y^*$, then the local system $(\pi^*(E|Y^*))^{\nabla}$ on $N-F$ of integrable sections of $\pi^*(E|Y^*)$ is canonically isomorphic to the specialization of the local system $(E|X-Y)^{\nabla}$ on $X-Y$ of integrable sections of $E$. (4) If $E_T$ is a holomorphic family of vector bundles with integrable logarithmic connections on $(X,Y)$ parametrized by a complex analytic space $T$, such that each $E_t$ has good residual eigenvalues, then the corresponding local systems on $N-F$ given by (3) form a holomorphic family of local systems on $N_F$ parametrized by $T$. \end{proposition} \paragraph{Proof} The statement (1) is a special case of the following more general statement. Let $S$ be any nonsingular variety, $\pi:N\to S$ any geometric vector bundle on $S$, and $F\subset N$ a normal crossing divisor in the total space of $N$ such that analytic (or \'etale) locally $F$ is the union of $r$ vector subbundles of $N$ of rank $r-1$ where $r$ is the rank of $N_S$. Then for any vector bundle $E$ on $S$ together with the structure of a ${\cal D}_N[\log F]$-module, the vector bundle $\pi^*(E)$ on $N$ has a natural structure of a ${\cal D}_N[\log F]$-module. This can be seen by choosing analytic local coordinates $(x_1,\ldots,x_m, y_1, \ldots, y_r)$ on $N$ where $(x_i)$ are local coordinates on $S$ and $y_i$ are linear coordinates on the fibers such that $F$ is locally defined by $\prod_iy_i=0$, and defining a logarithmic connection on $\pi^*(E)$ in terms of the actions of $\partial/\partial x_i$ and $y_i\partial/\partial y_i$ given by \begin{eqnarray*} \nabla_{\partial/\partial x_i}(g(y)\otimes_{{\cal O}_S} e) & = & g(y)\otimes_{{\cal O}_S} \nabla_{\partial/\partial x_i} e \\ \nabla_{y_i\partial/\partial y_i}(g(y)\otimes_{{\cal O}_S} e) & = & (y_i(\partial/\partial y_i)g(y))\otimes_{{\cal O}_S} e \end{eqnarray*} The statement (2) follows from the canonical isomorphism between $f^*({\cal D}_X[\log Y])$ and ${\cal D}_N[\log f]|Y^*$. The statements (3) and (4) follows over polydisks from the relation between V-filtrations and specializations, and we have the global statements by gluing. \refstepcounter{theorem}\paragraph{Remark \thetheorem}\label{notalg} As we have to integrate in order to associate its monodromy representation to an integrable connection, even if $T$ was associated to an algebraic variety and the original family $(E_T, \, \nabla_T)$ was algebraic, the associated family of local systems will in general only be an analytic family. \refstepcounter{theorem}\paragraph{Remark \thetheorem} It is sometimes erroneously believed that if $E$ is a a locally free logarithmic connection on $(X,Y)$, and $Y$ is nonsingular, then the restriction $E|Y$ has a natural connection on it. This is false in general, and is correct in some special case if the exact sequencence $0\to {\cal O}_Y\to T_X[\log Y]|Y \to T_Y$ has a natural splitting in some special case under consideration. The above proposition allows us to directly associate a Verdier object $({\cal E}_i, C_i, V_i)$ to a pre-${\cal D}$-module $(E_i, s_i, t_i)$ which has good residual eigenvalues in the sense of definition \ref{goodresieigen}, as follows. \begin{definition}\label{defrh}\rm We put ${\cal E}_d$ to be the local system on $X-Y$ given by $E_d$, and for $i\le d-1$ we define ${\cal E}_i$ to be the local system on $U_i$ associated to the logarithmic connection on $(N_i,F_i)$ associated to $E_i$ by proposition \ref{map}(1). For $m\le j\le d-1$ we put $C_j$ to be the map induced (using the statements (2) and (3) in proposition \ref{map} above) by $\pi_j^*(t_j)$ where $\pi_j:N_{j,j+1}\to Y^*_j$ is the bundle projection, and define $V_j$ by the formula $$V_j = {\exp(2\pi i\theta_j)-1\over \theta_j} \pi^*(s_j)$$ \end{definition} It is immediate from the definitions that this gives a Verdier object starting from a pre-${\cal D}$-module. By proposition \ref{map}(4), the above association is well behaved for analytic families, and gives rise to an analytic family of Verdier objects when we apply it to an analytic family of pre-${\cal D}$-modules. This gives the {\bf Riemann-Hilbert morphism ${\cal R\!\!H}$} at the level of analytic stacks. \refstepcounter{theorem}\paragraph{Remark \thetheorem} As a consequence of remark \ref{notalg}, the above morphism ${\cal R\!\!H}$ of stacks is holomorphic but not algebraic. \refstepcounter{theorem}\paragraph{Remark \thetheorem} By its definition, the Verdier object ${\cal E}=({\cal E}_i,C_i,V_i)$ associated by definition \ref{defrh} to a pre-${\cal D}$-module ${\bf E} =(E_i,t_i,s_i)$ with good residual eigenvalues defines the perverse sheaf associated by the de Rham functor to the ${\cal D}$-module $M$ associated to ${\bf E}$ in section 4.2 above, as follows locally from the hypercube descriptions of these objects restricted to polydisks. If a semistable pre-${\cal D}$-module has good residual eigenvalues, then the graded object associated to any S-filtration again has good residual eigenvalues by remark \ref{goodresi2}. It follows that the condition that the residual eigenvalues be good defines an analytic open subset ${\cal M}_o$ of the moduli space ${\cal M}$ by theorem \ref{maintheorem}(2). It can be proved as in Lemma 7.3 of [N-S] and its following discussion, using analytic properties of a GIT quotient proved by Simpson in [S], that our association of a Verdier object to a pre-${\cal D}$-module with good residual eigenvalues now descends to an {\bf analytic} morphism ${\cal R\!\!H} :{\cal M}_o\to P$ from ${\cal M}_o$ to the moduli ${\cal P}$ of Verdier objects. This is the Riemann-Hilbert morphism at the level of the moduli spaces. \subsection{Properties of the Riemann-Hilbert morphism} This section contains material which is a straightforward generalization of [N-S], so we omit the details. In the reverse direction, can construct a pre-${\cal D}$-module with good residual eigenvalues over a given Verdier object by using repeatedly the {\bf Deligne construction}. This gives the {\bf surjectivity} of the Riemann-Hilbert morphism. The exponential map $M(n,C\!\!\!\!I)\to GL(n,C\!\!\!\!I)$ is a submersion at points where the eigenvalues do not differ by $2\pi i$ times a nonzero integer. Using this, we can extend the Deligne construction to families of local systems parametrized by Artin local rings (e.g., $C\!\!\!\!I[\epsilon ]/(\epsilon ^2)$) to get families of logarithmic connections with good residual eigenvalues. From this it follows that the Riemann-Hilbert morphism is {\bf surjective at tangent level} at points above having good residual eigenvalues. Proposition 5.3 of [N] shows that for a meromorphic connection $M$ on $X$ with regular singularities on a normal crossing divisor $Y$, any locally free logarithmic lattice whose residual eigenvalues do not differ by positive integers is infinitesimally rigid. In Proposition 8.6 of [N-S], this is extended to pre-${\cal D}$-modules on $(X,Y)$ when $Y$ is smooth, by analyzing the derivative of a map of the form $$(s,\,t)\mapsto (s,\,t{e^{st}-1\over st})$$ for matrices $s$ and $t$ (lemma 3.10 of [N-S]). By a similar proof applied to the formula given in definition \ref{defrh}, we have the following when $Y$ is normal crossing. \begin{proposition} {\rm ({\bf Infinitesimal rigidity}):} A pre-${\cal D}$-module on $(X,Y)$ with good residual eigenvalues does not admit any nontrivial infinitesimal deformations such that the associated ${\cal D}$ module (or perverse sheaf) on $(X,Y)$ is constant. \end{proposition} This shows that the Riemann-Hilbert morphism is a {\bf tangent level isomorphism} of stacks at points above with good residual eigenvalues. The above properties, which are for the Riemann-Hilbet morphism as a morphism of analytic stacks, are valid by \ref{maintheorem} and \ref{points} for the morphism ${\cal R\!\!H} : {\cal M}_o \to {\cal P}$ at the level of the two moduli spaces at stable points of ${\cal M}_o$ which go to simple Verdier objects. \section*{References} \addcontentsline{toc}{section}{References} [G-G-M] Galligo, Granger, Maisonobe : ${\cal D}$-modules et faisceaux pervers dont le support singulier est un croissement normal. Ann. Inst. Fourier, Grenoble 35 (1985) 1-48. [G-M-V] Gelfand, MacPherson, Vilonen : Perverse sheaves and quivers. Duke Math. J. 83 (1996) 621-643. [L] Laumon, G. : Champs alg\'ebriques. Preprint no. 88-33, Universit\'e Paris Sud, 1988. [M-V] MacPherson and Vilonen : Elementary construction of perverse sheaves. Invnt. Math. 84 (1986), 403-435. [Mal] Malgrange, B. : Extension of holonomic ${\cal D}$-modules, in Algebraic Analysis (dedicated to M. Sato), M. Kashiwara and T. Kawai eds., Academic Press, 1988. [Ne] Newstead, P.E. : {\sl Introduction to moduli problems and orbit spaces}, TIFR lecture notes, Bombay (1978). [N] Nitsure, N. : Moduli of semistable logarithmic connections. J. Amer. Math. Soc. 6 (1993) 597-609. [N-S] Nitsure, N. and Sabbah, C. : Moduli of pre-${\cal D}$-modules, perverse sheaves, and the Riemann-Hilbert morphism -I, Math. Annaln. 306 (1996) 47-73. [S] Simpson, C. : Moduli of representations of the fundamental group of a smooth projective variety - I, Publ. Math. I.H.E.S. 79 (1994) 47-129. [V1] Verdier, J.-L. : Extension of a perverse sheaf across a closed subspace, \\ Ast\'erisque 130 (1985) 210-217. [V2] Verdier, J.-L. : Prolongements de faisceaux pervers monodromiques, \\ Ast\'erisque 130 (1985) 218-236. \bigskip Address: School of Mathematics, Tata Institute of Fundamental Research, Homi Bhabha Road, Mumbai 400 005, India. e-mail: [email protected] \bigskip \centerline{09-III-1997} \end{document}

9703

alg-geom/9703016

en

https://arxiv.org/abs/alg-geom/9703016

alg-geom/9703016

Misha S. Verbitsky

Misha Verbitsky

Hypercomplex Varieties

40 pages LaTeX 2e

Comm. Anal. Geom. 7 (1999), no. 2, 355--396.

We give a number of equivalent definitions of hypercomplex varieties and construct a twistor space for a hypercomplex variety. We prove that our definition of a hypercomplex variety (used, e. g., in alg-geom/9612013) is equivalent to a definition proposed by Deligne and Simpson, who used twistor spaces. This gives a way to define hypercomplex spaces (to allow nilpotents in the structure sheaf). We give a self-contained proof of desingularization theorem for hypercomplex varieties: a normalization of a hypercomplex variety is smooth and hypercomplex.

alg-geom

\section{Introduction} Hypercomplex varieties are a very natural class of objects. This notion allows one to speak uniformly of a number of disparate examples coming from hyperk\"ahler geometry and the theory of moduli spaces (\ref{_triana_hyperco_Remark_}, Subsection \ref{_hyperholomo_Subsection_}). A main property of hypercomplex varieties is that they can be desingularized in a functorial way, and this desingularization is a hypercomplex manifold. Moreover, this desingularization is achieved by taking normalization. This is a very useful result -- see \cite{_Arapura_}, \cite{_Verbitsky:New_} for some of its uses. This is why it is important to dwell on the definition of hypercomplex varieties. We give a number of equivalent definitions, in order to produce conditions which are easy to check in every situation. It is now possible to prove that something is {\bf not} hypercomplex. In \ref{_quotie_not_hyperco_Proposition_}, we show that for a hypercomplex variety $M$, and a finite group $G$ acting on $M$ by automorphisms, the quotient $M/G$ is {\bf not} hypercomplex, unless $G$ acts on $M$ freely. More precisely, we show that, for $M_0\subset M$ the part of $M$ where $G$ acts freely, the natural hypercomplex structure on $M_0/G$ cannot be extended to $M/G$. \subsection{An overview} The definitions of a hypercomplex variety, given in \cite{_Verbitsky:Desingu_} and \cite{_Verbitsky:DesinguII_}, on one hand, and in \cite{_Verbitsky:Deforma_}, \cite{_Verbitsky:Hyperholo_bundles_} on the other hand, are not identical. We show that these two definitions are in fact equivalent. We give the weakest form of the definition of hypercomplex varieties (one used in \cite{_Verbitsky:Deforma_}, \cite{_Verbitsky:Hyperholo_bundles_}), and give the proof of the desingularization theorem under these assumptions. This proof is essentially identical to the proof used in \cite{_Verbitsky:Desingu_} and \cite{_Verbitsky:DesinguII_}, though our assumptions are weaker, and the arguments are more rigorous. Then we obtain the stronger version of the definition (used in \cite{_Verbitsky:Desingu_} and \cite{_Verbitsky:DesinguII_}) from the desingularization and a recent result of Kaledin \cite{_Kaledin_}. An almost hypercomplex variety (\ref{_almost_hyperco_Definition_}) is a real analytic variety $M$ with a quaternionic action on its sheaf of real analytic differentials. For each $L\in \Bbb H$, $L^2 = -1$, $L$ gives an almost complex structure on $M$. This almost complex structure is called {\bf induced by the quaternionic action}. We say that $M$ is {\bf hypercomplex} (\ref{_hyperco_Definition_}) if there are induced almost complex structures $I, J \in \Bbb H$, such that $I\neq \pm J$, and $I$, $J$ are integrable (\ref{_integra_almost_comple_Definition_}). We show that then {\bf all} induced complex structures are integrable. This was {\em a definition} of the hypercomplex variety which we used in \cite{_Verbitsky:Desingu_} and \cite{_Verbitsky:DesinguII_}. For almost hypercomplex manifolds (i. e., smooth almost hypercomplex varieties), D. Kaledin \cite{_Kaledin_} proved that integrability of two $I, J \in \Bbb H$, $I\neq \pm J$ implies integrability of all induced complex structures. We prove the desingularization theorem (\ref{_desingu_Theorem_}) for hypercomplex varieties, and then apply Kaledin's result to obtain integrability of all induced complex structures (\ref{_all_indu_comple_integra_Theorem_}). The last part of this paper deals with several other versions of the definition of hypercomplex variety, which are, as we show, equivalent. We define the {\bf twistor space} of a hypercomplex variety (\ref{_twistor_Definition_}), constructed as an almost complex variety. Using Kaledin's theorem and desingularization, we prove that the almost complex structure on the twistors is integrable, i. e., the twistor space is a complex variety. We give a description of the hypercomplex structure in terms of the twistor space, following \cite{_HKLR_}, \cite{_Simpson:hyperka-defi_} and \cite{_Deligne:defi_}. The twistor space $\operatorname{Tw}$ is a complex variety equipped with a holomorphic map $\pi:\; \operatorname{Tw} {\:\longrightarrow\:} {\Bbb C} P^1$ and an anticomplex involution $\iota:\; \operatorname{Tw} {\:\longrightarrow\:} \operatorname{Tw}$. The original hypercomplex variety is identified with a fiber $\pi^{-1}(I)$, $I\in {\Bbb C} P^1$, and the data $(\operatorname{Tw}, \pi, \iota)$ are essentially sufficient to recover the hypercomplex structure on $M = \pi^{-1}(I)$ (see \eqref{_twistor_data_Equation_} for details and a precise statement). It is possible to define a hypercomplex variety in terms of $(\operatorname{Tw}, \pi, \iota)$. This definition was proposed by Deligne and Simpson (\cite{_Deligne:defi_}, \cite{_Simpson:hyperka-defi_}). We show that their definition is equivalent to ours. Their definition has the significant advantage that it does not assume that $M$ is reduced, and indeed might be used to define hypercomplex spaces, i. e., to allow nilpotents in the structure sheaf. \subsection{Twistor spaces: an introduction} The twistor space of a hypercomplex variety is the following object. Let $M$ be a hypercomplex variety. Then the quaternion algebra $\Bbb H$ acts in $\Omega^1M$ in such a way that for all $L\in \Bbb H$, $L^2 =-1$, the corresponding operator is an almost complex structure. Identifying the set $\{ L\in \Bbb H \;\; |\;\; L^2=-1\}$ with ${\Bbb C} P^1$, we obtain an almost complex structure $\c I$ on ${\Bbb C} P^1 \times M$ acting on $T _{L,m} {\Bbb C} P^1 \times M = T_L {\Bbb C} P^1 \oplus T_m M$ as $I_{{\Bbb C} P^1} \oplus L$, where $L\in {\Bbb C} P^1$ acts on $T_m M$ as the corresponding quaternion, and $I_{{\Bbb C} P^1}$ is the complex structure operator on ${\Bbb C} P^1$. This almost complex structure is integrable (\ref{_twi_integra_Claim_}; essentially, this is proven by D. Kaledin \cite{_Kaledin_}). The corresponding variety is called {\bf the twistor space of $M$}, denoted by $\operatorname{Tw}$ (\ref{_twistor_Definition_}). Consider the holomorphic projection map $\pi:\; \operatorname{Tw}{\:\longrightarrow\:} {\Bbb C} P^1$. Let $\iota_0:\; {\Bbb C} P^1 {\:\longrightarrow\:} {\Bbb C} P^1$ be the anticomplex involution with no fixed points given by the central symmetry of $S^2 \subset {\Bbb R}^3$, and $\iota:\; \operatorname{Tw} {\:\longrightarrow\:} \operatorname{Tw}$ map $(s, m)$ to $(\iota_0(s), m)$. Clearly, $\iota$ is also an anticomplex involution. The set of sections of the projection $\pi$ is called {\bf the space of twistor lines}, denoted by $\operatorname{Sec}$. This space is equipped with complex structure, by Douady (\cite{_Douady_}). Consider a twistor line $I \stackrel {s_m} {\:\longrightarrow\:} (I \times m)\in {\Bbb C} P^1 \times M = \operatorname{Tw}$, where $m\in M$. Then $s_m$ is called {\bf a horisontal twistor line}. The variety of horisontal twistor lines is denoted by $\operatorname{Hor}$. Clearly, the line $\operatorname{im}(s_m)\subset \operatorname{Tw}$ is fixed by the involution $\iota$. One can show that a space of horisontal twistor lines is a connected component of the space $\operatorname{Sec}^\iota$ of all twistor lines fixed by $\iota$. Clearly, the real analytic variety $\operatorname{Hor}$ is identified with $M$, and the induced complex structures on $M$ come from the natural isomorphisms between $\operatorname{Hor}$ and fibers of $\pi$. This shows how to recover the hypercomplex variety $M$ from $\operatorname{Tw}$, $\iota$ and $\pi$. \hfill According to Deligne and Simpson (\cite{_Deligne:defi_}, \cite{_Simpson:hyperka-defi_}), singular hyperk\"ahler varieties should be defined in terms of the following data (see \eqref{_twistor_data_Equation_} for details). \begin{description} \item[(i)] $\operatorname{Tw}, \pi, \iota$ \item[(ii)] a component $\operatorname{Hor}$ of the variety of $\iota$-invariant holomorphic sections of $\pi$. \end{description} satisfying the following conditions (see \eqref{_twistor_properties_Equation_} for details). \begin{description} \item[(a)] Through every point $x\in \operatorname{Tw}$ passes a unique line $s\in \operatorname{Hor}$ \item[(b)] Let $s\in Hor$. Consider the points $x$, $y$ situated in a small neighbourhood $U$ of $\operatorname{im} s \subset \operatorname{Tw}$, $\pi(x)\neq \pi(y)$. Assume that $x$ and $y$ belong to the same irreducible component of $U$. Then there exist a unique twistor line $s_{xy}\in \operatorname{Sec}$ passing through $x$, $y$. \end{description} The advantage of the definition of Deligne--Simpson (see \ref{_Deligne_Simpson_Definition_}) is that it is easy to adopt for the scheme situation: one may allow nilpotents in structure sheaf. For the definition of a {\bf hypercomplex space}, see \ref{_hyperco_spaces_Definition_}. We show that Deligne--Simpson's definition of a hypercomplex variety is equivalent to ours (\ref{_hype_vari_and_twi_equiva_Theorem_}). The condition (b) in the above listing is very difficult to check. We replace it by an equivalent condition, which should be thought of as its linearization (see \ref{_twi_hyperco_type_Definition_} for details). \begin{description} \item[(b$'$)] For each $s\in \operatorname{Hor}$, the conormal sheaf \[ \ker \bigg (\Omega^1(\operatorname{Tw})\restrict{\operatorname{im} s} {\:\longrightarrow\:} \Omega^1(\operatorname{im} s)\bigg) \] is isomorphic to $\oplus({\cal O}(-1))$. \end{description} \ref{_Deli_Simpsi_equi_infinite_Theorem_} shows that the above conditions (b) and (b$'$) are equivalent. The condition (b$'$) is very easy to check, and hence is extremely useful. In Subsection \ref{_hyperholomo_Subsection_}, we use the condition (b) to give a new proof that a deformation space of a stable holomorphic bundle $B$ over a hyperk\"ahler manifold is a hyperk\"ahler variety, provided that the Chern classes of $B$ are $SU(2)$-invariant. \subsection{Desingularization of hypercomplex varieties} Let $M$ be a hypercomplex variety (\ref{_hyperco_Definition_}), $I$ an integrable induced complex structure. Consider the complex variety $(M,I)$, which is $M$ with the complex structure defined by $I$. The Desingularization Theorem (\ref{_desingu_Theorem_}) says that the normalization $\widetilde{(M,I)}$ of $(M,I)$ is smooth and hypercomplex. This desingularization is canonical and functorial: the hypercomplex manifold $\widetilde{(M,I)}$ is independent from the choice of induced complex structure $I$. The proof of the Desingularization Theorem goes along the following lines. We define spaces with locally homogeneous singularities (\ref{_SLHS_Definition_}). A space with locally homogeneous singularities (SLHS) is an analytic space $X$ such that for all $x\in X$, the $x$-completion of a local ring ${\cal O}_xX$ is isomorphic to an $x$-completion of the associated graded ring $({\cal O}_xX)_{gr}$. We show that hypercomplex varieties are always SLHS (\ref{_hyperco_SLHS_Theorem_}). This is proven using an elementary argument from commutative algebra. We work with a complete local Noetherian ring $A$ over ${\Bbb C}$, with a residual field ${\Bbb C}$. By definition, an automorphism $e:\; A {\:\longrightarrow\:} A$ is called {\bf homogenizing} (\ref{_homogeni_automo_Definition_}) if its differential acts as a dilatation on the Zariski tangent space of $A$, with dilatation coefficient $|\lambda|<1$. As usual, by the Zariski tangent space we understand the space $(\mathfrak m_A /\mathfrak m_A^2)^*$, where $\mathfrak m_A$ is a maximal ideal of $A$. For a ring $A$ equipped with a homogenizing automorphism $e:\; A {\:\longrightarrow\:} A$, we show that $A$ has locally homogeneous singularities. We complete the proof that all hypercomplex varieties are SLHS by constructing an explicit homogenizing automorphism in a local ring of germs of holomorphic functions on a complex variety underlying a given hypercomplex variety (\ref{_homogenizing_Proposition_}). The proof of Desingularization Theorem proceeds then with a study of a tangent cone of a hypercomplex variety. For every point of a hypercomplex variety, the corresponding Zariski tangent space $T_xM$ is equipped with a quaternionic action. This makes $T_zM$ into a hyperk\"ahler manifold, with appropriately chosen metric. We show that the tangent cone $Z_xM$ of $M$, considered as a subvariety of $T_xM$, is trianalytic, i. e. analytic with respect to all induced complex structures (\ref{_tange_cone_underly_Proposition_}). It was proven in \cite{_Verbitsky:Deforma_} (see also \ref{_triana_comple_geo_Proposition_}), that trianalytic subvarieties are {\bf totally geodesic}, i. e. all geodesics in such a subvariety remain geodesics in the ambient manifold. Since $T_xM$ is flat, its totally geodesic subvariety $Z_xM$ is a union of planes. Now from the local homogeneity of singularities of $M$ it follows that $M$ looks locally as its tangent cone, i. e. as a union of non-singular hypercomplex varieties. This finishes the proof of Desingularization Theorem. \subsection{Contents} The paper is organized as follows. \begin{itemize} \item The Introduction is independent from the rest of this paper. \item In Section \ref{_real_ana_Section_}, we recall some standard definitions and results from the theory of real analytic spaces. We define an almost complex real analytic space and show that the complex structure on a complex variety can be recovered from the corresponding almost complex structure on the underlying real analytic variety. \item Section \ref{_hyperka_Section_} contains some well-known results and definitions from the hyperk\"ahler geometry. We define trianalytic subvarieties of hyperk\"ahler manifolds and show that trianalytic subvarieties naturally apppear in the complex geometry of hyperk\"ahler manifolds. \item In Section \ref{_hypercomple_Section_}, we define a hypercomplex variety. Examples of hypercomplex varieties include trianalytic subvarieties of hyperk\"ahler manifolds (\ref{_trianalytic_Definition_}) and moduli spaces of certain kinds of stable bundles over hyperk\"ahler manifolds (Subsection \ref{_hyperholomo_Subsection_}). We study the tangent cone $Z_xM\subset T_xM$ of a hypercomplex variety, and show that it is a union of linear subspaces of the Zariski tanhgent space $T_xM$. \item Section \ref{_LHS_Section_} deals with spaces having locally homogeneous singularities (SLHS). Roughly speaking, these are analytic spaces $M$ for which every point $x\in M$ has a system of coordinates $z_1, ..., z_n$ such that the corresponding epimorphism of formal completions \[ {\Bbb C} [[z_1, ..., z_n]] {\:\longrightarrow\:} \hat{\cal O}_x M \] has a homogeneous kernel (\ref{_SLHS_Definition_}, \ref{_locally_homo_coord_Claim_}). We show that a space has LHS if it is endowed with a system of infinitesimal automorphisms acting as dilatation on the tangent spaces (\ref{_homogeni_LHS_Proposition_}). We show that every hypercomplex variety is naturally equipped with such a system of automorphisms, thus proving that it is SLHS (\ref{_hyperco_SLHS_Theorem_}). \item In Section \ref{_desingu_Section_}, we prove the desingularization theorem for hypercomplex varieties: a normalization of a hypercomplex variety is smooth and hypercomplex (\ref{_desingu_Theorem_}). This result is used to show that all induced complex structures on a hypercomplex variety are integrable (\ref{_all_indu_comple_integra_Theorem_}). \item In Section \ref{_twistors_Section_}, we define a twistor space of a hypercomplex variety. We show how to reconstruct a hypercomplex variety from its twistor space. We axiomatize this situation, giving two sets of conditions which are satisfied for twistor spaces of hypercomplex varieties: we define {\bf twistor spaces of hypercomplex type} (\ref{_twi_hyperco_type_Definition_}) and {\bf twistor spaces of Deligne-Simpson type} (\ref{_Deligne_Simpson_Definition_}). \item In Section \ref{_Deli_Si_equi_hyperco_Section_}, we prove that these sets of conditions are equivalent: a variety is a twistor space of hypercomplex type if and only if it is a twistor space of Deligne--Simpson type. \item In Section \ref{_hype_type_equi_hype_Section_}, we show how to construct a hypercomplex variety starting from an arbitrary twistor space of hypercomplex (or, equivalently, Deligne-Simpson) type. This proves that a functor associating to each hypercomplex variety a twistor space of hypercomplex type is an equivalence of categories. \item In Section \ref{_twi_applications_Section_}, we give some applications of the equivalence of categories constructed in Section \ref{_hype_type_equi_hype_Section_}. We define hypercomplex spaces (\ref{_hyperco_spaces_Definition_}), thus generalizing the definition of hypercomplex varieties to spaces with nilpotents. We show that a quotient of a hypercomplex variety by an action of a finite group $G$ cannot be hypercomplex, unless $G$ acts freely. Finally, we give another proof that the space of stable bundles over a compact hyperk\"ahler manifold is hypercomplex, assuming its Chern classes are ``suitable'' (for the precise definition of suitability and the full statement, see Subsection \ref{_hyperholomo_Subsection_}; see also \cite{_Verbitsky:Hyperholo_bundles_}). \end{itemize} \section{Real analytic varieties and spaces} \label{_real_ana_Section_} In this section, we follow \cite{_GMT_}. \subsection{Real analytic varieties and spaces: reduction, differentials} Let $I$ be an ideal sheaf in the ring of real analytic functions in an open ball $B$ in ${\Bbb R}^n$. The set of common zeroes of $I$ is equipped with a structure of ringed space, with ${\cal O}(B)/I$ as the structure sheaf. We denote this ringed space by $Spec({\cal O}(B)/I)$. \hfill \definition By a {\bf weak real analytic space} we understand a ringed space which is locally isomorphic to $Spec({\cal O}(B)/I)$, for some ideal $I \subset {\cal O}(B)$. A {\bf real analytic space} is a weak real analytic space for which the structure sheaf is coherent (i. e., locally finitely generated and presentable). \hfill For every real analytic variety $X$, there is a natural sheaf morphism of evaluation, ${\cal O}(X) \stackrel{ev} {\:\longrightarrow\:} C(X)$, where $C(X)$ is the sheaf of real analytic functions on $X$. \hfill \definition A {\bf real analytic variety} is a weak real analytic space for which the natural sheaf morphism ${\cal O}(X) {\:\longrightarrow\:} C(X)$ is injective. \hfill Let $(X, {\cal O}(X))$ be a real analytic space and $N(X)\subset {\cal O}(X)$ be the kernel of the natural sheaf morphism ${\cal O}(X) {\:\longrightarrow\:} C(X)$. Clearly, the ringed space $(X, {\cal O}(X)/N(X))$ is a real analytic variety. This variety is called {\bf a reduction of $X$}, denoted $X_r$. The structure sheaf of $X_r$ is not necessarily coherent, for examples see \cite{_GMT_}, III.2.15. \hfill For an ideal $I\subset {\cal O}(B)$ we define the module of real analytic differentials on ${\cal O}(B)/I$ by \[ \Omega^1({\cal O}(B)/I) = \Omega^1({\cal O}(B))\bigg/\bigg ( I \cdot \Omega^1({\cal O}(B))+ d I \bigg), \] where $B$ is an open ball in ${\Bbb R}^n$, and $\Omega^1({\cal O}(B))\cong {\Bbb R}^n \otimes {\cal O}(B)$ is the module of real analytic differentials on $B$. Patching this construction, we define the sheaf of real analytic differentials on any real analytic space. Likewise, one defines sheaves of analytic differentials for complex varieties and in other similar situations. \subsection{Real analytic spaces underlying complex analytic varieties} Let $X$ be a complex analytic variety. The {\bf real analytic space underlying $X$} (denoted by $X_{\Bbb R}$) is the following object. By definition, $X_{\Bbb R}$ is a ringed space with the same topology as $X$, but with a different structure sheaf, denoted by ${\cal O}_{X_{\Bbb R}}$. Let $i:\; U \hookrightarrow B^n$ be a closed complex analytic embedding of an open subset $U\subset X$ to an open ball $B^n\subset{\Bbb C}^n$, and $I$ be an ideal defining $i(U)$. Then \[ {\cal O}_{X_{\Bbb R}}\restrict U:=\; {\cal O}_{B^n_{\Bbb R}}/Re(I)\] is a quotient sheaf of the sheaf of real analytic functions on $B^n$ by the ideal $Re(I)$ generated by the real parts of the functions $f\in I$. Note that the real analytic space underlying $X$ needs not be reduced, though it has no nilpotents in the structure sheaf. Consider the sheaf ${\cal O}_X$ of holomorphic functions on $X$ as a subsheaf of the sheaf $C(X,{\Bbb C})$ of continuous ${\Bbb C}$-valued functions on $X$. The sheaf $C(X,{\Bbb C})$ has a natural automorphism $f{\:\longrightarrow\:} \bar f$, where $\bar f$ is complex conjugation. By definition, the section $f$ of $C(X,{\Bbb C})$ is called {\bf antiholomorphic} if $\bar f$ is holomorphic. Let ${\cal O}_X$ be the sheaf of holomorphic functions, and $\bar {\cal O}_X$ be the sheaf of antiholomorphic functions on $X$. Let ${\cal O}_X \otimes_{\Bbb C} \bar{\cal O}_X \stackrel i{\:\longrightarrow\:} {\cal O}_{X_{\Bbb R}}\otimes {\Bbb C}$ be the natural multiplication map. \hfill \claim \label{_comple_real_ana_produ_Claim_} Let $X$ be a complex variety, $X_{\Bbb R}$ the underlying real analytic space. Then the natural sheaf homomorphism $i:\; {\cal O}_X \otimes_{\Bbb C} \bar{\cal O}_X {\:\longrightarrow\:} {\cal O}_{X_{\Bbb R}}\otimes {\Bbb C}$ is injective. For each point $x\in X$, $i$ induces an isomorphism on $x$-completions of ${\cal O}_X \otimes_{\Bbb C} \bar{\cal O}_X$ and ${\cal O}_{X_{\Bbb R}}\otimes {\Bbb C}$. {\bf Proof:} Clear from the definition. $\;\;\hbox{\vrule width 4pt height 4pt depth 0pt}$ \hfill In the assumptions of \ref{_comple_real_ana_produ_Claim_}, let \[ \Omega^1({\cal O}_{X_{\Bbb R}}), \ \ \Omega^1({\cal O}_X \otimes_{\Bbb C} \bar{\cal O}_X),\ \ \Omega^1({\cal O}_{X_{\Bbb R}}\otimes {\Bbb C}) \] be the sheaves of real analytic differentials associated with the corresponding sheaves of rings. There is a natural sheaf map \begin{equation} \label{_Omega_X_R_and_Omega_X_Equation_} \Omega^1({\cal O}_{X_{\Bbb R}})\otimes {\Bbb C} = \Omega^1({\cal O}_{X_{\Bbb R}}\otimes {\Bbb C}){\:\longrightarrow\:} \Omega^1({\cal O}_X\otimes_{\Bbb C} \bar {\cal O}_X), \end{equation} correspoding to the monomorphism \[ {\cal O}_X\otimes_{\Bbb C} \bar {\cal O}_X\hookrightarrow{\cal O}_{X_{\Bbb R}}\otimes {\Bbb C}. \] \hfill \claim \label{_differe_real_ana_and_co_ana_Claim_} Tensoring both sides of \eqref{_Omega_X_R_and_Omega_X_Equation_} by ${\cal O}_{X_{\Bbb R}}\otimes {\Bbb C}$ produces an isomorphism \[ \Omega^1({\cal O}_X\otimes_{\Bbb C} \bar {\cal O}_X) \bigotimes_{{\cal O}_X\otimes_{\Bbb C} \bar {\cal O}_X}\bigg({\cal O}_{X_{\Bbb R}}\otimes {\Bbb C}\bigg) =\Omega^1({\cal O}_{X_{\Bbb R}}\otimes {\Bbb C}). \] {\bf Proof:} Clear. $\;\;\hbox{\vrule width 4pt height 4pt depth 0pt}$ \hfill According to the general results about differentials (see, for example, \cite{_Hartshorne:Alg_Geom_}, Chapter II, Ex. 8.3), the sheaf $\Omega^1({\cal O}_X\otimes_{\Bbb C} \bar {\cal O}_X)$ admits a canonical decomposition: \[ \Omega^1({\cal O}_X\otimes_{\Bbb C} \bar {\cal O}_X) = \Omega^1({\cal O}_X)\otimes_{\Bbb C} \bar {\cal O}_X \oplus{\cal O}_X\otimes_{\Bbb C}\Omega^1(\bar {\cal O}_X). \] Let $\tilde I$ be an endomorphism of $\Omega^1({\cal O}_X\otimes_{\Bbb C} \bar {\cal O}_X)$ which acts as a multiplication by $\sqrt{-1}\:$ on \[ \Omega^1({\cal O}_X)\otimes_{\Bbb C} \bar {\cal O}_X \subset \Omega^1({\cal O}_X\otimes_{\Bbb C} \bar {\cal O}_X) \] and as a multiplication by $-\sqrt{-1}\:$ on \[ {\cal O}_X\otimes_{\Bbb C}\Omega^1(\bar {\cal O}_X) \subset \Omega^1({\cal O}_X\otimes_{\Bbb C} \bar {\cal O}_X). \] Let $\underline I$ be the corresponding ${\cal O}_{X_{\Bbb R}}\otimes {\Bbb C}$-linear endomorphism of \[ \Omega^1({\cal O}_{X_{\Bbb R}})\otimes {\Bbb C} = \Omega^1({\cal O}_X\otimes_{\Bbb C} \bar {\cal O}_X) \otimes_{{\cal O}_X\otimes_{\Bbb C} \bar {\cal O}_X} \bigg({\cal O}_{X_{\Bbb R}}\otimes {\Bbb C}\bigg). \] A quick check shows that $\underline I$ is {\it real}, that is, comes from the ${\cal O}_{X_{\Bbb R}}$-linear endomorphism of $\Omega^1({\cal O}_{X_{\Bbb R}})$. Denote this ${\cal O}_{X_{\Bbb R}}$-linear endomorphism by \[ I:\; \Omega^1({\cal O}_{X_{\Bbb R}}){\:\longrightarrow\:} \Omega^1({\cal O}_{X_{\Bbb R}}), \] $I^2=-1$. The endomorphism $I$ is called {\bf the complex structure operator on the underlying real analytic space}. In the case when $X$ is smooth, $I$ coinsides with the usual complex structure operator on the cotangent bundle. \hfill \definition Let $M$ be a weak real analytic space, and \[ I:\; \Omega^1({\cal O}_M){\:\longrightarrow\:}\Omega^1({\cal O}_M) \] be an endomorphism satisfying $I^2=-1$. Then $I$ is called {\bf an almost complex structure on $M$}. \subsection{Real analytic varieties and almost complex structures} Let $B$ be an open ball in ${\Bbb C}^n$, and $X\subset B$ a closed complex subvariety defined by an ideal $I \subset {\cal O}_B$. Let $X_{\Bbb R}\subset B_{\Bbb R}$ be the underlying real analytic space, and $X_{\Bbb R}^r\subset B_{\Bbb R}$ the underlying real analytic variety, with the respective ideal sheaves denoted by $I_{\Bbb R}$, $I^r_{\Bbb R}$. Consider the ideal $I_{\Bbb R}^r \otimes {\Bbb C} \subset {\cal O}_{B_{\Bbb R}}\otimes{\Bbb C}$. \hfill \lemma\label{_I_R^r_generators_Lemma_} In the above assumptions, the ideal $I_{\Bbb R}^r\otimes {\Bbb C}$ is generated by elements $f\cdot \bar g$, where $f, g\in {\cal O}_B$ are holomorphic functions on $B$ satisfying $fg \in I$ {\bf Proof:} Clear. \blacksquare \hfill Let $F$ be a finite generated sheaf over a real analytic variety $Z$. For all $z\in Z$, let ${\mathfrak m}_z\subset {\cal O}_Z$ be the ideal of all functions vanishing in $z$. The vector space $ \cdot F / {\mathfrak m}_z \cdot F$ is called {\bf the fiber of $F$ in $z$}, denoted by $F\restrict z$. Likewise we define the fiber $f\restrict z\in F\restrict z$ for a section $f$ of $F$. \hfill \lemma\label{_all_fibers_zero=>Fzero_Lemma_} Let $F$ be a finite generated ${\cal O}_Z$-sheaf over a real analytic variety $Z$, and $f$ its section. Assume that $f\restrict z=0$ for all $z\in Z$. Then $f=0$. \hfill {\bf Proof:} Going to a closed subvariety if necessary, we may assume that $Sup(f)=Z$. Let $Z_0\subset Z$ be an open subset such that $F\restrict {Z_0}$ is free. Clearly, it suffices to show that when $f\restrict{Z_0}$ is zero. Since $F\restrict{Z_0}$ is free, $f\restrict{Z_0}$ is an $n$-tuple of functions $(f_1, ... f_n)$, and $f\restrict z=0$ if and only if all $f_i$ take value $0$ at $z$. Applying the definition of real analytic variety, we obtain $f=0$. \blacksquare \hfill \proposition \label{_redu_differe_Lemma_} Let $X_{\Bbb R}$ be a real analytic space underlying a complex variety $X$, $X_{\Bbb R}^r$ be its reduction, and $\Omega^1(X)$, $\Omega^1(X_{\Bbb R}^r)$ the corresponding sheaves of real analytic differentials. Consider the natural map \begin{equation} \label{_diffe_multipli_redu_arrow_Equation_} \Omega^1(X_{\Bbb R})\otimes_{{\cal O}_X}{\cal O}_{X_{\Bbb R}^r} \stackrel{\phi_r} {\:\longrightarrow\:} \Omega^1(X_{\Bbb R}^r). \end{equation} Then $\phi_r$ is an isomorphism. \hfill {\bf Proof:} We work in notation introduced earlier in this section. Consider the closed embedding $X_{\Bbb R}^r \hookrightarrow X_{\Bbb R}$. Let $N\subset {\cal O}_{X_{\Bbb R}}$ be the ideal defining $X_{\Bbb R}^r$, \[ N = \ker \bigg({\cal O}_{X_{\Bbb R}} \stackrel{ev} {\:\longrightarrow\:} C(X) \bigg).\] Clearly from the definitions, \[ \Omega^1 (X_{\Bbb R}^r) = \Omega^1(X_{\Bbb R}) \otimes {\cal O}_{X_{\Bbb R}^r} \bigg/ d N \otimes{\cal O}_{X_{\Bbb R}^r}. \] To show that \eqref{_diffe_multipli_redu_arrow_Equation_} is an isomorphism, it suffices to prove that the subsheaf $dN \subset \Omega^1(X_{\Bbb R})$, tensored by ${\cal O}_{X_{\Bbb R}^r}$, gives zero. By \ref{_all_fibers_zero=>Fzero_Lemma_}, it suffices to show that every section of \[ d N \otimes{\cal O}_{X_{\Bbb R}^r}\subset \Omega^1(X_{\Bbb R}) \otimes {\cal O}_{X_{\Bbb R}^r}, \] has zero fibers in $x$, for all points $x\in X$. The fiber of $\Omega^1 X_{\Bbb R}$ in $x\in X$ is ${\mathfrak m}_x/{\mathfrak m}_x^2$, where ${\mathfrak m}_x\subset {\cal O}_{X_{\Bbb R}}$ is the ideal generated by all functions vanishing in $x$. For all $f\in {\cal O}_{X_{\Bbb R}}$ such that $f(x)=0$, the fiber $df\restrict x$ of $df\in\Omega^1 X_{\Bbb R}$ in $x$ coinsides with the class of $f$ in ${\mathfrak m}_x/{\mathfrak m}_x^2$. By \ref{_I_R^r_generators_Lemma_}, $N\otimes {\Bbb C}$ is generated by $f\bar g$, $f, g \in {\cal O}_B$, $fg\in I$. Therefore, $N\subset {\mathfrak m}_x^2\otimes {\Bbb C}$ and by the above, the fiber $dN\restrict x$ iz zero. This proves \ref{_redu_differe_Lemma_}. \blacksquare \hfill {}From \ref{_redu_differe_Lemma_}, it follows that a a real analytic variety underlying a given complex variety is equipped with a natural almost complex structure. The corresponding operator is called {\bf the complex structure operator in the underlying real analytic variety.} \subsection{Integrability of almost complex structures} \definition\label{_commu_w_comple_str_Definition_} Let $X$, $Y$ be complex analytic varieties, and \[ f:\; X_{\Bbb R}{\:\longrightarrow\:} Y_{\Bbb R}\] be a morphism of underlying real analytic varieties. Let $f^* \Omega^1_{Y_{\Bbb R}} \stackrel P{\:\longrightarrow\:} \Omega^1_{X_{\Bbb R}}$ be the natural map of sheaves of differentials associated with $f$. Let \[ I_X:\; \Omega^1_{X_{\Bbb R}}{\:\longrightarrow\:} \Omega^1_{X_{\Bbb R}}, \;\;\; I_Y:\; \Omega^1_{Y_{\Bbb R}}{\:\longrightarrow\:} \Omega^1_{Y_{\Bbb R}} \] be the complex structure operators, and \[ f^* I_Y:\; f^*\Omega^1_{Y_{\Bbb R}}{\:\longrightarrow\:} f^*\Omega^1_{Y_{\Bbb R}} \] be ${\cal O}_{X_{\Bbb R}}$-linear automorphism of $f^*\Omega^1_{Y_{\Bbb R}}$ defined as a pullback of $I_Y$. We say that $f$ {\bf commutes with the complex structure} if \begin{equation}\label{_commu_w_comle_Equation_} P\circ f^* I_Y = I_X \circ P. \end{equation} \hfill \theorem \label{_commu_w_comple_str_Theorem_} Let $X$, $Y$ be complex analytic varieties, and \[ f_{\Bbb R}:\; X_{\Bbb R}{\:\longrightarrow\:} Y_{\Bbb R}\] be a morphism of underlying real analytic varieties which commutes with the complex structure. Then there exist a morphism $f:\; X{\:\longrightarrow\:} Y$ of complex analytic varieties, such that $f_{\Bbb R}$ is its underlying morphism. \hfill {\bf Proof:} By Corollary 9.4, \cite{_Verbitsky:Deforma_}, the map $f$, defined on the sets of points of $X$ and $Y$, is meromorphic; to prove \ref{_commu_w_comple_str_Theorem_}, we need to show it is holomorphic. Let $\Gamma \subset X \times Y$ be the graph of $f$. Since $f$ is meromorphic, $\Gamma$ is a complex subvariety of $X\times Y$. It will suffice to show that the natural projections $\pi_1:\; \Gamma {\:\longrightarrow\:} X$, $\pi_2:\; \Gamma {\:\longrightarrow\:} Y$ are isomorphisms. By \cite{_Verbitsky:Deforma_}, Lemma 9.12, the morphisms $\pi_i$ are flat. Since $f_{\Bbb R}$ induces isomorphism of Zariski tangent spaces, same is true of $\pi_i$. Thus, $\pi_i$ are unramified. Therefore, the maps $\pi_i$ are etale. Since they are one-to-one on points, $\pi_i$ etale implies $\pi_i$ is an isomorphism. $\;\;\hbox{\vrule width 4pt height 4pt depth 0pt}$ \hfill \hfill \definition \label{_integra_almost_comple_Definition_} Let $M$ be a real analytic variety, and \[ I:\; \Omega^1({\cal O}_M){\:\longrightarrow\:}\Omega^1({\cal O}_M) \] be an endomorphism satisfying $I^2=-1$. Then $I$ is called {\bf an almost complex structure on $M$}. If there exist a structure $\mathfrak C$ of complex variety on $M$ such that $I$ appears as the complex structure operator associated with $\mathfrak C$, we say that $I$ is {\bf integrable}. \ref{_commu_w_comple_str_Theorem_} implies that this complex structure is unique if it exists. \section{Hyperk\"ahler manifolds} \label{_hyperka_Section_} \subsection{Definitions} This subsection contains a compression of the basic definitions from hyperk\"ahler geometry, found, for instance, in \cite{_Besse:Einst_Manifo_} or in \cite{_Beauville_}. \hfill \definition \label{_hyperkahler_manifold_Definition_} (\cite{_Besse:Einst_Manifo_}) A {\bf hyperk\"ahler manifold} is a Riemannian manifold $M$ endowed with three complex structures $I$, $J$ and $K$, such that the following holds. \begin{description} \item[(i)] the metric on $M$ is K\"ahler with respect to these complex structures and \item[(ii)] $I$, $J$ and $K$, considered as endomorphisms of a real tangent bundle, satisfy the relation $I\circ J=-J\circ I = K$. \end{description} \hfill The notion of a hyperk\"ahler manifold was introduced by E. Calabi (\cite{_Calabi_}). \hfill Clearly, a hyperk\"ahler manifold has the natural action of the quaternion algebra ${\Bbb H}$ on its real tangent bundle $TM$. Therefore its complex dimension is even. For each quaternion $L\in \Bbb H$, $L^2=-1$, the corresponding automorphism of $TM$ is an almost complex structure. It is easy to check that this almost complex structure is integrable (\cite{_Besse:Einst_Manifo_}). \hfill \definition \label{_indu_comple_str_Definition_} Let $M$ be a hyperk\"ahler manifold, $L$ a quaternion satisfying $L^2=-1$. The corresponding complex structure on $M$ is called {\bf an induced complex structure}. The $M$ considered as a complex manifold is denoted by $(M, L)$. \hfill Let $M$ be a hyperk\"ahler manifold. We identify the group $SU(2)$ with the group of unitary quaternions. This gives a canonical action of $SU(2)$ on the tangent bundle, and all its tensor powers. In particular, we obtain a natural action of $SU(2)$ on the bundle of differential forms. \hfill \lemma \label{_SU(2)_commu_Laplace_Lemma_} The action of $SU(2)$ on differential forms commutes with the Laplacian. {\bf Proof:} This is Proposition 1.1 of \cite{_Verbitsky:Hyperholo_bundles_}. \blacksquare Thus, for compact $M$, we may speak of the natural action of $SU(2)$ in cohomology. \subsection{Trianalytic subvarieties in compact hyperk\"ahler manifolds.} In this subsection, we give a definition and a few basic properties of trianalytic subvarieties of hyperk\"ahler manifolds. We follow \cite{_Verbitsky:Symplectic_II_}. \hfill Let $M$ be a compact hyperk\"ahler manifold, $\dim_{\Bbb R} M =2m$. \hfill \definition\label{_trianalytic_Definition_} Let $N\subset M$ be a closed subset of $M$. Then $N$ is called {\bf trianalytic} if $N$ is a complex analytic subset of $(M,L)$ for any induced complex structure $L$. \hfill Let $I$ be an induced complex structure on $M$, and $N\subset(M,I)$ be a closed analytic subvariety of $(M,I)$, $dim_{\Bbb C} N= n$. Denote by $[N]\in H_{2n}(M)$ the homology class represented by $N$. Let $\inangles N\in H^{2m-2n}(M)$ denote the Poincare dual cohomology class. Recall that the hyperk\"ahler structure induces the action of the group $SU(2)$ on the space $H^{2m-2n}(M)$. \hfill \theorem\label{_G_M_invariant_implies_trianalytic_Theorem_} Assume that $\inangles N\in H^{2m-2n}(M)$ is invariant with respect to the action of $SU(2)$ on $H^{2m-2n}(M)$. Then $N$ is trianalytic. {\bf Proof:} This is Theorem 4.1 of \cite{_Verbitsky:Symplectic_II_}. \blacksquare \hfill \remark \label{_triana_dim_div_4_Remark_} Trianalytic subvarieties have an action of quaternion algebra in the tangent bundle. In particular, the real dimension of such subvarieties is divisible by 4. \hfill Let $M$ be a hyperk\"ahler manifold, $\c R$ the set of induced complex structures. The following result implies that for generic $I\in \c R$, all complex subvarieties of $(M, I)$ are trianalytic in $M$. \hfill \definition Let $M$ be a compact hyperk\"ahler manifold, $\c R$ the set of induced complex structures. The complex structure $I\subset \c R$ is called {\bf of general type with respect to the hyperk\"ahler structure} if all rational $(p,p)$-classes \[ \omega \in \bigoplus\limits_p H^{p,p}(M)\cap H^{2p}(M,{\Bbb Z})\subset H^*(M) \] are $SU(2)$-invariant. \hfill \proposition Let $M$ be a compact hyperk\"ahler manifold, $\c R$ the set of induced complex structures. Then for all $I\in \c R$ except a countable subset, $I$ is of general type. {\bf Proof:} This is Proposition 2.2 from \cite{_Verbitsky:Symplectic_II_}. \blacksquare \subsection{Totally geodesic submanifolds.} \nopagebreak \hspace{5mm} \proposition \label{_comple_geodesi_basi_Proposition_ Let $X \stackrel \phi\hookrightarrow M$ be an embedding of Riemannian manifolds (not necessarily compact) compatible with the Riemannian structure. Then the following conditions are equivalent. \begin{description} \item[(i)] Every geodesic line in $X$ is geodesic in $M$. \item[(ii)] Consider the Levi-Civita connection $\nabla$ on $TM$, and restriction of $\nabla$ to $TM \restrict{X}$. Consider the orthogonal decomposition \begin{equation} \label{TM_decompo_Equation_} TM\restrict{X} = TX \oplus TX^\bot. \end{equation} Then, this decomposition is preserved by the connection $\nabla$. \end{description} {\bf Proof:} Well known; see, for instance, \cite{_Besse:Einst_Manifo_}. \hbox{\vrule width 4pt height 4pt depth 0pt} \hfill \proposition \label{_triana_comple_geo_Proposition_} Let $X\subset M$ be a trianalytic submanifold of a hyperk\"ahler manifold $M$, where $M$ is not necessarily compact. Then $X$ is totally geodesic. {\bf Proof:} This is \cite{_Verbitsky:Deforma_}, Corollary 5.4. \blacksquare \section{Hypercomplex varieties} \label{_hypercomple_Section_} \subsection{Definition and examples} \definition\label{_almost_hyperco_Definition_} Let $M$ be a real analytic variety equipped with almost complex structures $I$, $J$ and $K$, such that $I\circ J = -J \circ I = K$. Then $M$ is called {\bf an almost hypercomplex variety.} \hfill An almost hypercomplex variety is equipped with an action of quaternion algebra in its differential sheaf. Each quaternion $L\in \Bbb H$, $L^2=-1$ defines an almost complex structure on $M$. Such an almost complex structure is called {\bf induced by the hypercomplex structure}. \hfill \definition\label{_hyperco_Definition_} Let $M$ be an almost hypercomplex variety. We say that $M$ is {\bf hypercomplex} if there exist a pair of induced complex structures $I_1, I_2\in \Bbb H$, $I_1\neq \pm I_2$, such that $I_1$ and $I_2$ are integrable. \hfill {\bf Caution:} Not everything which looks hypercomplex satisfies the conditions of \ref{_hyperco_Definition_}. Take a quotient $M/G$ of a hypercomplex manifold by an action of a finite group $G$, acting compatible with the hyperk\"ahler structure. Then $M/G$ is {\it not} hypercomplex, unless $G$ acts freely (\ref{_quotie_not_hyperco_Proposition_}). \hfill \noindent \claim \label{_hyperka_hyperco_Claim_} Let $M$ be a hyperk\"ahler manifold. Then $M$ is hypercomplex. {\bf Proof:} Let $I$, $J$ be induced complex structures. We need to identify $(M, I)_{\Bbb R}$ and $(M,J)_{\Bbb R}$ in a natural way. These varieties are canonically identified as $C^\infty$-manifolds; we need only to show that this identification is real analytic. This is \cite{_Verbitsky:Deforma_}, Proposition 6.5. \blacksquare \hfill \remark\label{_triana_hyperco_Remark_} Trianalytic subvarieties of hyperk\"ahler manifolds are obviously hypercomplex. Define trianalytic subvarieties of hypercomplex varieties as subvarieties which are complex analytic with respect to all integrable induced complex structures. Clearly, trianalytic subvarieties of hypercomplex varieties are equipped with a natural hypercomplex structure. Another example of a hypercomplex variety is given in Subsection \ref{_hyperholomo_Subsection_}. For additional examples, see \cite{_Verbitsky:Deforma_}. \subsection{Tangent cone of a hypercomplex variety.} Let $M$ be a hypercomplex variety, $I$ an integrable induced complex structure, $\tilde Z_x(M,I)$ be the Zariski tangent cone to $(M,I)$ in $x\in M$ and $Z_x(M,I)$ its reduction. Consider $Z_x(M,I)$ as a closed subvariety in the Zariski tangent space $T_xM$. The space $T_xM$ has a natural quaternionic structure and admits a compatible metric. This makes $T_xM$ into a hyperk\"ahler manifold, isomorphic to ${\Bbb H}^n$. \hfill \noindent\theorem \label{_cone_hype_Theorem_} Under these assumptions, the following assertions hold: \begin{description} \item[(i)] The subvariety $Z_x(M,I)\subset T_x M$ is independent from the choice of integrable induced complex structure $I$. \item [(ii)] Moreover, $Z_x(M,I)$ is a trianalytic subvariety of $T_x M$. \end{description} {\bf Proof:} Clearly, \ref{_cone_hype_Theorem_} (ii) follows from \ref{_cone_hype_Theorem_} (i). \ref{_cone_hype_Theorem_} (i) is directly implied by the following general result. \hfill \proposition\label{_tange_cone_underly_Proposition_} Let $M$ be a complex variety, $x\in X$ a point, and $Z_xM\subset T_xM$ be the reduction of the Zariski tangent cone to $M$ in $x$, considered as a closed subvariety of the Zariski tangent space $T_xM$. Let $Z_x M_{\Bbb R} \subset T_x M_{\Bbb R}$ be the Zariski tangent cone for the underlying real analytic space $M_{\Bbb R}$. Then $(Z_x M)_{\Bbb R} \subset (T_x M)_{\Bbb R} = T_x M_{\Bbb R}$ coinsides with $Z_x M_{\Bbb R}$. {\bf Proof:} For each $v\in T_x M$, the point $v$ belongs to $Z_x M$ if and only if there exist a real analytic path $\gamma:\; [0, 1] {\:\longrightarrow\:} M$, $\gamma(0)=x$ satisfying $\frac{d\gamma}{dt}=v$. The same holds true for $Z_x M_{\Bbb R}$. Thus, $v\in Z_x M$ if and only if $v\in Z_x M_{\Bbb R}$. \blacksquare \hfill The following theorem shows that the Zariski tangent cone $Z_xM\subset T_x M$ is a union of planes $L_i\subset T_x M$. \hfill \theorem \label{_cone_flat_Theorem_} Let $M$ be a hypercomplex variety, $I$ an induced complex structure and $x\in M$ a point. Consider the reduction of the Zariski tangent cone (denoted by $Z_x M$) as a subvariety of the quaternionic space $T_x M$. Let $Z_x(M, I)= \cup L_i$ be the irreducible decomposition of the complex variety $Z_x(M,I)$. Then \begin{description} \item[(i)] The decomposition $Z_x(M, I)= \cup L_i$ is independent from the choice of induced complex structure $I$. \item[(ii)] For every $i$, the variety $L_i$ is a linear subspace of $T_x M$, invariant under quaternion action. \end{description} {\bf Proof:} Let $L_i$ be an irreducible component of $Z_x(M, I)$, $Z_x^{ns}(M,I)$ be the non-singular part of $Z_x(M,I)$, and $L_i^{ns}:=Z_x^{ns}(M,I) \cap L_i$. Then $L_i$ is a closure of $L_i^{ns}$ in $T_xM$. Clearly from \ref{_cone_hype_Theorem_}, $L_i^{ns}(M)$ is a hyperk\"ahler submanifold in $T_xM$. By \ref{_triana_comple_geo_Proposition_}, $L_i^{ns}$ is totally geodesic. A totally geodesic submanifold of a flat manifold is again flat. Therefore, $L_i^{ns}$ is an open subset of a linear subspace $\tilde L_i\subset T_xM$. Since $L_i^{ns}$ is a hyperk\"ahler submanifold, $\tilde L_i$ is invariant with respect to quaternions. The closure $L_i$ of $L_i^{ns}$ is a complex analytic subvariety of $T_x(M,I)$. Therefore, $\tilde L_i = L_i$. This proves \ref{_cone_flat_Theorem_} (ii). From the above argument, it is clear that $Z_x^{ns}(M,I)= \coprod L_i^{ns}$ (disconnected sum). Taking connected components of $Z_x^{ns}M$ for each induced complex structure, we obtain the same decomposition $Z_x(M, I)= \cup L_i$, with $L_i$ being closured of connected components. This proves \ref{_cone_flat_Theorem_} (ii). \blacksquare \section[Hypercomplex varieties have locally homogeneous singularities]{Hypercomplex varieties \\have locally homogeneous singularities} \label{_LHS_Section_} This section follows \cite{_Verbitsky:DesinguII_}. \subsection{Spaces with locally homogeneous singularities.} \noindent \definition (local rings with LHS) Let $A$ be a local ring. Denote by $\mathfrak m$ its maximal ideal. Let $A_{gr}$ be the corresponding associated graded ring for the $\mathfrak m$-adic filtration. Let $\hat A$, $\widehat{A_{gr}}$ be the $\mathfrak m$-adic completion of $A$, $A_{gr}$. Let $(\hat A)_{gr}$, $(\widehat{A_{gr}})_{gr}$ be the associated graded rings, which are naturally isomorphic to $A_{gr}$. We say that $A$ {\bf has locally homogeneous singularities} (LHS) if there exists an isomorphism $\rho:\; \hat A {\:\longrightarrow\:} \widehat{A_{gr}}$ which induces the standard isomorphism $i:\; (\hat A)_{gr}{\:\longrightarrow\:} (\widehat{A_{gr}})_{gr}$ on associated graded rings. \hfill \definition\label{_SLHS_Definition_} (SLHS) Let $X$ be a complex or real analytic space. Then $X$ is called {\bf a space with locally homogeneous singularities} (SLHS) if for each $x\in X$, the local ring ${\cal O}_x X$ has locally homogeneous singularities. \hfill The following claim might shed a light on the origin of the term ``locally homogeneous singularities''. \hfill \claim \label{_locally_homo_coord_Claim_} Let $A$ be a complete local Noetherian ring over ${\Bbb C}$, with a residual field ${\Bbb C}$. Then the following statements are equivalent \begin{description} \item[(i)] $A$ has locally homogeneous singularities \item[(ii)] There exist a surjective ring homomorphism $\rho:\; {\Bbb C}[[x_1, ... , x_n]] {\:\longrightarrow\:} A$, where ${\Bbb C}[[x_1, ... , x_n]]$ is the ring of power series, and the ideal $\ker \rho$ is homogeneous in ${\Bbb C}[[x_1, ... , x_n]]$. \end{description} {\bf Proof:} Clear. \blacksquare \hfill \definition Let $M$ be a hypercomplex variety. Then $M$ is called a space with locally homogeneous singularities (SLHS) if for all integrable induced complex structures $I$ the $(M, I)$ is SLHS. \subsection{Complete rings with automorphisms} \definition \label{_homogeni_automo_Definition_} Let $A$ be a local Noetherian ring over ${\Bbb C}$, with a residual field ${\Bbb C}$, equipped with an automorphism $e:\; A {\:\longrightarrow\:} A$. Let $\mathfrak m$ be a maximal ideal of $A$. Assume that $e$ acts on $\mathfrak m /\mathfrak m^2$ as a multiplication by $\lambda\in {\Bbb C}$, $|\lambda|< 1$. Then $e$ is called {\bf a homogenizing automorphism of $A$}. \hfill The aim of the present subsection is to prove the following statement. \hfill \proposition \label{_homogeni_LHS_Proposition_} Let $A$ be a complete Noetherian ring over ${\Bbb C}$, with a residual field ${\Bbb C}$, equipped with a homogenizing authomorphism $e:\; A {\:\longrightarrow\:} A$. Then there exist a surjective ring homomorphism $\rho:\; {\Bbb C}[[x_1, ... , x_n]] {\:\longrightarrow\:} A$, such that the ideal $\ker \rho$ is homogeneous in ${\Bbb C}[[x_1, ... , x_n]]$. In particular, $A$ has locally homogeneous singularities.\footnote{See \ref{_locally_homo_coord_Claim_} for LHS property in terms of coordinate systems.} \hfill This statement is well known. A reader who knows its proof should skip the rest of this section. \hfill \proposition \label{_homogeni_auto_then_basis_Proposition_} Let $A$ be a complete Noetherian ring over ${\Bbb C}$, with a residual field ${\Bbb C}$, equipped with a homogenizing authomorphism $e:\; A {\:\longrightarrow\:} A$. Then there exist a system of ring elements \[ f_1 , ..., f_m \in \mathfrak m, \ \ m = \dim_{\Bbb C}\mathfrak m /\mathfrak m^2, \] which generate $\mathfrak m /\mathfrak m^2$, and such that $e(f_i) = \lambda f_i$. \hfill {\bf Proof:} Let $\underline a\in\mathfrak m /\mathfrak m^2$. Let $a\in \mathfrak m$ be a representative of $\underline a$ in $\mathfrak m$. To prove \ref{_homogeni_auto_then_basis_Proposition_} it suffices to find $c \in \mathfrak m^2$, such that $e(a-c) = \lambda a -\lambda c$. Thus, we need to solve an equation \begin{equation}\label{_a_through_a_Equation_} \lambda c - e(c) = e(a) - \lambda(a). \end{equation} Let $r:= e(a)-\lambda a$. Clearly, $r\in \mathfrak m ^2$. A solution of \eqref{_a_through_a_Equation_} is provided by the following lemma. \hfill \lemma \label{_e-lambda_invertible_Lemma_} In assumptions of \ref{_homogeni_auto_then_basis_Proposition_}, let $r\in \mathfrak m^2$. Then, the equation \begin{equation}\label{_finding_eigen_Equation_} e(c) - \lambda c = r \end{equation} has a unique solution $c \in \mathfrak m^2$. \hfill {\bf Proof:} We need to show that the operator $P:= (e-\lambda)\restrict{\mathfrak m^2}$ is invertible. Consider the $\mathfrak m$-adic filtration $\mathfrak m^2 \supset \mathfrak m^3 \supset ...$ on $\mathfrak m^2$. Clearly, $P$ preserves this filtration. Since $\mathfrak m^2$ is complete with respect to the adic filtration, it suffices to show that $P$ is invertible on the successive quotients. The quotient $\mathfrak m^2/\mathfrak m^i$ is finite-dimensional, so to show that $P$ is invertible it suffices to calculate the eigenvalues. Since $e$ is an automorphism, restriction of $e$ to $\mathfrak m^i/\mathfrak m^{i-1}$ is a multiplication by $\lambda^i$. Thus, the eigenvalues of $e$ on $\mathfrak m^2/\mathfrak m^i$ range from $\lambda^2$ to $\lambda^{i-1}$. Since $|\lambda|>|\lambda|^2$, all eigenvalues of $P\restrict{\mathfrak m^2/\mathfrak m^i}$ are non-zero and the restriction of $P$ to $\mathfrak m^2/\mathfrak m^i$ is invertible. This proves \ref{_e-lambda_invertible_Lemma_}. $\hbox{\vrule width 4pt height 4pt depth 0pt}$ \hfill {\bf The proof of \ref{_homogeni_LHS_Proposition_}.} Consider the map \[ \rho:\; {\Bbb C}[[x_1, ... x_m]] {\:\longrightarrow\:} A,\ \ \rho(x_i) = f_i,\] where $f_1, ... , f_m$ is the system of functions constructed in \ref{_homogeni_auto_then_basis_Proposition_}. By Nakayama's lemma, $\rho$ is surjective. Let $e_\lambda:\; {\Bbb C}[[x_1, ... x_m]] {\:\longrightarrow\:}{\Bbb C}[[x_1, ... x_m]] $ be the automorphism mapping $x_i $ to $\lambda x_i$. Then, the diagram \[\begin{CD} {\Bbb C}[[x_1, ... x_m]] @>{\rho}>> A \\ @V{e_\lambda}VV @VV{e}V\\ {\Bbb C}[[x_1, ... x_m]] @>{\rho}>> A \end{CD} \] is by construction commutative. Therefore, the ideal $I= \ker \rho$ is preserved by $e_\lambda$. Clearly, every $e_\lambda$-preserved ideal $I\subset {\Bbb C}[[x_1, ... x_m]]$ is homogeneous. \ref{_homogeni_LHS_Proposition_} is proven. \blacksquare \subsection{Automorphisms of local rings of hypercomplex varieties} Let $M$ be a hypercomplex variety, $x\in M$ a point, $I$ an integrable induced complex structure. Let $A_I:= \hat {\cal O}_x(M,I)$ be the adic completion of the local ring ${\cal O}_x(M,I)$ of $x$-germs of holomorphic functions on the complex variety $(M,I)$. Clearly, the sheaf ring of the antiholomorphic functions on $(M,I)$ coinsides with ${\cal O}_x(M,-I)$. Thus, the corresponding completion ring is $A_{-I}$. The isomorphism of \ref{_comple_real_ana_produ_Claim_} produces a natural epimorphism \begin{equation}\label{_co_ana_and_rea_isom_Equation_} \widehat{A_I \otimes_{\Bbb C} A_{-I}} {\:\longrightarrow\:} A_{\Bbb R}, \end{equation} where \[ A_{\Bbb R} := \widehat{{\cal O}_x(M_{\Bbb R})\otimes_{\Bbb R} {\Bbb C}}\] is the $x$-completion of the ring of germs of real analytic complex-valued functions on $M$. Consider the natural quotient map \[ p:\;A_{-I}{\:\longrightarrow\:} {\Bbb C}.\] Consider the natural epimorphism of complete rings \begin{equation}\label{_epi_from_pro_to_A_I_Equation_} \widehat{A_I \otimes_{\Bbb C} A_{-I}} {\:\longrightarrow\:} A_I,\ \ a\otimes b \mapsto a\otimes p(b), \end{equation} where $a\in A_I$, $b\in A_{-I}$, and \[ a\otimes b\in{A_I \otimes_{\Bbb C} A_{-I}}.\] \hfill \lemma\label{_p_zero_on_kernel_of_multi_Lemma_} The kernel of \eqref{_epi_from_pro_to_A_I_Equation_} contains the kernel of \eqref{_co_ana_and_rea_isom_Equation_}. \hfill {\bf Proof:} Consider an epimorphism $\phi:\; B_x {\:\longrightarrow\:} A_I$ where \[ B_x = {\Bbb C} [[z_1, ... , z_n]].\] Let ${\mathfrak I}\subset B_x$ be the kernel of $\phi$. By \ref{_I_R^r_generators_Lemma_}, the ring $A_{\Bbb R}$ is naturally isomorphic to \[ (B_x)_{\Bbb R} = {\Bbb C} [[z_1, ... , z_n, \bar z_1, ... \bar z_n]]/ {\mathfrak I}_{\Bbb R}, \] where ${\mathfrak I}_{\Bbb R}$ is an ideal generated by all the products \[ f(z_1, ... z_n)\cdot\bar g(\bar z_z , ... \bar z_n),\] such that $fg \in \mathfrak I$. Likewise, $\widehat{A_I \otimes_{\Bbb C} A_{-I}}$ is a quotient of \[ (B_x)_{\Bbb R} = {\Bbb C} [[z_1, ... , z_n, \bar z_1, ... \bar z_n]] \] by the ideal \[ {\mathfrak I} \cdot {\Bbb C} [[\bar z_1, ... \bar z_n]] + {\Bbb C} [[z_1, ... , z_n]\cdot \bar {\mathfrak I}. \] Slightly abusing the notation, we denote the corresponding quotient map by \[ \phi:\; {\Bbb C} [[z_1, ... , z_n, \bar z_1, ... \bar z_n]] {\:\longrightarrow\:} \widehat{A_I \otimes_{\Bbb C} A_{-I}}. \] Let $a\in \widehat{A_I \otimes_{\Bbb C} A_{-I}}$ be an element which is mapped to zero by the map \eqref{_co_ana_and_rea_isom_Equation_}. Then $a$ is a linear combination of $\phi(f_i\bar g_i)$, for $f_i, g_i\in {\Bbb C} [[z_1, ... , z_n]]$, $fg\in \mathfrak I$. Therefore, it suffices to show that $a$ lies in the kernel of \eqref{_epi_from_pro_to_A_I_Equation_} for $a = \phi(f \bar g)$. Either $g$ is invertible and $f\in \mathfrak I$, or $g(0, ... 0)=0$. In the first case, $f\bar g\in {\mathfrak I}\otimes {\Bbb C} [[\bar z_1, ... \bar z_n]]$, so $\phi(f\bar g)=0$. In the second case, $p(\bar g) =0$, so $1\otimes p(a)=0$ and $a$ lies in the kernel of the map \eqref{_epi_from_pro_to_A_I_Equation_}. This proves \ref{_p_zero_on_kernel_of_multi_Lemma_}. \blacksquare \hfill Consider the diagram \[\begin{CD} \widehat{A_I \otimes_{\Bbb C} A_{-I}} @>>> A_I\\ \searrow \\ \ \ \ \ \ \ \ \ \ A_{\Bbb R} \end{CD} \] formed from the arrows of \eqref{_epi_from_pro_to_A_I_Equation_} and \eqref{_co_ana_and_rea_isom_Equation_}. By \ref{_p_zero_on_kernel_of_multi_Lemma_}, there exists an epimorphism $e_I:\; A_{\Bbb R} {\:\longrightarrow\:} A_I$, making this diagram commutative. Let $i_I:\; A_I \hookrightarrow A_{\Bbb R}$ be the natural embedding \[ a \mapsto a\otimes 1\in\widehat{A_I \otimes_{\Bbb C} A_{-I}}.\] For an integrable induced complex structure $J$, we define $A_J$, $A_{-J}$, $i_J$, $e_J$ likewise. Let $\Psi_{I,J}:\; A_I {\:\longrightarrow\:} A_I$ be the composition \[ A_I \stackrel {i_I}{\:\longrightarrow\:} A_{\Bbb R}\stackrel {e_J}{\:\longrightarrow\:} A_J \stackrel {i_J}{\:\longrightarrow\:} A_{\Bbb R}\stackrel {e_I}{\:\longrightarrow\:} A_I. \] Clearly, for $I=J$, the ring morphism $\Psi_{I,J}$ is identity, and for $I=-J$, $\Psi_{I,J}$ is an augmentation map. \hfill \proposition \label{_homogenizing_Proposition_} Let $M$ be a hypercomplex variety, $x\in M$ a point, and $I$, $J$ induced complex structures, such that $I\neq J$ and $I\neq -J$. Consider the map $\Psi_{I,J}:\; A_I {\:\longrightarrow\:} A_I$ defined as above. Then $\Psi_{I,J}$ is a homogenizing automorphism of $A_I$.\footnote{For the definition of a homogenizing automorphism, see \ref{_homogeni_automo_Definition_}.} \hfill {\bf Proof:} Let $d\Psi$ be the differential of $\Psi_{I,J}$, that is, the restriction of $\Psi_{I,J}$ to $*\mathfrak m/\mathfrak m^2)^*$, where $\mathfrak m$ is the maximal ideal of $A_I$. By Nakayama's lemma, to prove that $\Psi_{I,J}$ is an automorphism it suffices to show that $d\Psi$ is invertible. To prove that $\Psi_{I,J}$ is homogenizing, we have to show that $d\Psi$ is a multiplication by a complex number $\lambda$, $|\lambda|<1$. As usually, we denote the real analytic variety underlying $M$ by $M_{\Bbb R}$. Let $T_I$, $T_J$, $\underline {T}_{\Bbb R}$ be the Zariski tangent spaces to $(M,I)$, $(M,J)$ and $M_{\Bbb R}$, respectively, in $x\in M$. Consider the complexification $T_{\Bbb R}:= \underline {T}_{\Bbb R}\otimes {\Bbb C}$, which is a Zariski tangent space to the local ring $A_{\Bbb R}$. To compute $d\Psi:\; T_I {\:\longrightarrow\:} T_I$, we need to compute the differentials of $e_I$, $e_J$, $i_I$, $i_J$, i. e., the restrictions of the homomorphisms $e_I$, $e_J$, $i_I$, $i_J$ to the Zariski tangent spaces $T_I$, $T_J$, $T_{\Bbb R}$. Denote these differentials by $de_I$, $de_J$, $di_I$, $di_J$. \hfill \lemma \label{_i_e_through_Hodge_Lemma_} Let $M$ be a hypercomplex variety, $M_{\Bbb R}$ the associated real analytic variety, $x\in M$ a point. Consider the space $T_{\Bbb R} := T_x (M_{\Bbb R})\otimes {\Bbb C}$. For an induced complex structure $I$, consider the Hodge decomposition $T_{\Bbb R}= T^{1,0}_I \oplus T^{0,1}_I$. In our previous notation, $T_I^{1,0}$ is $T_I$. Then, $di_I$ is the natural embedding of $T_I = T_I^{1,0}$ to $T_{\Bbb R}$, and $de_I$ is the natural projection of $T_{\Bbb R}= T^{1,0}_I \oplus T^{0,1}_I$ to $T_I^{1,0}=T_I$. {\bf Proof:} Clear. \blacksquare \hfill We are able now to describe the map $d\Psi:\; T_I {\:\longrightarrow\:} T_I$ in terms of the quaternion action. Recall that the space $T_I$ is equipped with a natural ${\Bbb R}$-linear quaternionic action. For each quaternionic linear space $\underline V$ and each quaternion $I$, $I^2=-1$, $I$ defines a complex structure in $\underline V$. Such a complex structure is called {\bf induced by the quaternionic structure}. \hfill \lemma \label{_Psi_through_quate_Lemma_} Let $\underline V$ be a space with quaternion action, and $V:= \underline V \otimes {\Bbb C}$ its complexification. For each induced complex structure $I\in {\Bbb H}$, consider the Hodge decomposition $V:= V_I^{1,0} \oplus V_I^{0,1}$. For induced complex structures $I, J\in \Bbb H$, let $\Phi_{I,J}(V)$ be the composition of the natural embeddings and projections \[ V_I^{1,0} {\:\longrightarrow\:} V {\:\longrightarrow\:} V_J^{1,0} {\:\longrightarrow\:} V {\:\longrightarrow\:} V_I^{1,0}. \] Using the natural identification $\underline V \cong V_I^{1,0}$, we consider $\Phi_{I,J}(V)$ as an ${\Bbb R}$-linear automorphism of the space $\underline V$. Then, applying the operator $\Phi_{I,J}(V)$ to the quaternionic space $T_I$, we obtain the operator $d\Psi$ defined above. {\bf Proof:} Follows from \ref{_i_e_through_Hodge_Lemma_} \blacksquare \hfill As we have seen, to prove \ref{_homogenizing_Proposition_} it suffices to show that $d\Psi$ is a multiplication by a non-zero complex number $\lambda$, $|\lambda| < 1$. Thus, the proof of \ref{_homogenizing_Proposition_} is finished with the following lemma. \hfill \lemma\label{_compu_of_Psi_for_qua_Lemma_} In assumptions of \ref{_Psi_through_quate_Lemma_}, consider the map \[ \Phi_{I,J}(V):\; V_I^{1,0} {\:\longrightarrow\:} V_I^{1,0}.\] Then $\Phi_{I,J}(V)$ is a multiplication by a complex number $\lambda$. Moreover, $\lambda$ is a non-zero number unless $I=-J$, and $|\lambda|< 1$ unless $I=J$. \hfill {\bf Proof:} Let $\underline V= \oplus \underline V_i$ be a decomposition of $V$ into a direct sum of $\Bbb H$-linear spaces. Then, the operator $\Phi_{I,J}(V)$ can also be decomposed: $\Phi_{I,J}(V) = \oplus \Phi_{I,J}(V_i)$. Thus, to prove \ref{_compu_of_Psi_for_qua_Lemma_} it suffices to assume that $\dim_{\Bbb H} \underline V=1$. Therefore, we may identify $\underline V$ with the space $\Bbb H$, equipped with the right action of quaternion algebra on itself. Consider the left action of $\Bbb H$ on $\underline V = \Bbb H$. This action commutes with the right action of $\Bbb H$ on $\underline V$. Consider the corresponding action \[ \rho:\; SU(2) {\:\longrightarrow\:} \operatorname{End}(\underline V) \] of the group of unitary quaternions ${\Bbb H}^{un}=SU(2)\subset \Bbb H$ on $\underline V$. Since $\rho$ commutes with the quaternion action, $\rho$ preserves $V^{1,0}_I \subset V$, for every induced complex structure $I$. In the same way, for each $g\in SU(2)$, the endomorphism $\rho(g)\in \operatorname{End}(V^{1,0}_I)$ commutes with $\Phi_{I,J}(V)$. Consider the 2-dimensional ${\Bbb C}$-vector space $V^{1,0}_I$ as a representation of $SU(2)$. Clearly, $V^{1,0}_I$ is an irreducible representation. Thus, by Schur's lemma, the automorphism $\Phi_{I,J}(V)\in \operatorname{End}(V^{1,0}_I))$ is a multiplication by a complex constant $\lambda$. The bounds $0< |\lambda| < 1$ are implied by the following elementary argument. The composition $i_I \circ e_J$ applied to a vector $v\in V_I^{1,0}$ is a projection of $v$ to $V_J^{1,0}$ along $V_J^{0,1}$. Consider the natural Euclidean metric on $V = \Bbb H$. Clearly, the decomposition $V = V_J^{1,0}\oplus V_J^{0,1}$ is orthogonal. Thus, the composition $i_I \circ e_J$ is an orthogonal projection of $v\in V_I^{1,0}$ to $V_J^{1,0}$. Similarly, the composition $i_J \circ e_I$ is an orthogonal projection of $v\in V_J^{1,0}$ to $V_I^{1,0}$. Thus, the map $\Phi_{I,J}(V)$ is an orthogonal projection from $V_I^{1,0}$ to $V_J^{1,0}$ and back to $V_I^{1,0}$. Such a composition always decreases a length of vectors, unless $V_I^{1,0}$ coincides with $V_J^{1,0}$, in which case $I=J$. Also, unless $V_I^{1,0} = V_J^{0,1}$, $\Phi_{I,J}(V)$ is non-zero; in the later case, $I = -J$. \ref{_homogenizing_Proposition_} is proven. \blacksquare \hfill {}From \ref{_homogenizing_Proposition_} and \ref{_homogeni_LHS_Proposition_}, we obtain the following theorem. \hfill \theorem\label{_hyperco_SLHS_Theorem_} (the main result of this section) Let $M$ be a hypercomplex variety. Then $M$ is a space with locally homogeneous singularities (SLHS). \blacksquare \section{Desingularization of hypercomplex varieties} \label{_desingu_Section_} \subsection{The proof of desingularization theorem} \proposition\label{_normali_smooth_Corollary_} Let $M$ be a hypercomplex variety, and $I$ an integrable induced complex structure. Then the normalization of $(M,I)$ is smooth. \hfill {\bf Proof:} The normalization of $Z_xM$ is smooth by \ref{_cone_flat_Theorem_}. The normalization is compatible with the adic completions (\cite{_Matsumura:Commu_Alge_}, Chapter 9, Proposition 24.E). Therefore, the integral closure of the completion of ${\cal O}_{Z_xM}$ is a regular ring. {}From \ref{_hyperco_SLHS_Theorem_} it follows that the integral closure of $\hat {\cal O}_xM$ is also a regular ring, where $\hat {\cal O}_xM$ is an adic completion of the local ring of holomorphic functions on $(M, I)$ in a neighbourhood of $x$. Applying \cite{_Matsumura:Commu_Alge_}, Chapter 9, Proposition 24.E again, we obtain that the integral closure of ${\cal O}_x M$ is regular. This proves \ref{_normali_smooth_Corollary_} \blacksquare \hfill \theorem \label{_desingu_Theorem_} (Desingularization theorem) Let $M$ be a hypercomplex variety $I$ an integrable induced complex structure. Let \[ \widetilde{(M, I)}\stackrel n{\:\longrightarrow\:} (M,I)\] be the normalization of $(M,I)$. Then $\widetilde{(M, I)}$ is smooth and has a natural hypercomplex structure $\c H$, such that the associated map $n:\; \widetilde{(M, I)} {\:\longrightarrow\:} (M,I)$ agrees with $\c H$. Moreover, the hypercomplex manifold $\tilde M:= \widetilde{(M, I)}$ is independent from the choice of induced complex structure $I$. \hfill {\bf Proof:} The variety $\widetilde{(M, I)}$ is smooth by \ref{_normali_smooth_Corollary_}. Let $x\in M$, and $U\subset M$ be a neighbourhood of $x$. Let ${\mathfrak R}_x(U)$ be the set of irreducible components of $U$ which contain $x$. There is a natural map $\tau: {\mathfrak R}_x(U) {\:\longrightarrow\:} Irr(Spec \hat{\cal O}_xM)$, where $Irr(Spec \hat{\cal O}_xM)$ is a set of irreducible components of $Spec \hat {\cal O}_xM$, where $\hat {\cal O}_xM$ is a completion of ${\cal O}_xM$ in $x$. Since ${\cal O}_x M$ is Henselian (\cite{_Raynaud_}, VII.4), there exist a neighbourhood $U$ of $x$ such that $\tau: {\mathfrak R}_x(U) {\:\longrightarrow\:} Irr(Spec \hat {\cal O}_xM)$ is a bijection. Fix such an $U$. Since $M$ is a space with locally homogeneous singularities, the irreducible decomposition of $U$ coinsides with the irreducible decomposition of the tangent cone $Z_x M$. Let $\coprod U_i \stackrel u {\:\longrightarrow\:} U$ be the morphism mapping a disjoint union of irreducible components of $U$ to $U$. By \ref{_cone_flat_Theorem_}, the $x$-completion of ${\cal O}_{U_i}$ is regular. Shrinking $U_i$ if necessary, we may assume that $U_i$ is smooth. Then, the morphism $u$ coinsides with the normalization of $U$. For each variety $X$, we denote by $X^{ns}\subset X$ the set of non-singular points of $X$. Clearly, $u(U_i) \cap U^{ns}$ is a connected component of $U^{ns}$. Therefore, $u(U_i)$ is trianalytic in $U$. By \ref{_triana_hyperco_Remark_}, $U_i$ has a natural hypercomplex structure, which agrees with the map $u$. This gives a hypercomplex structure on the normalization $\tilde U := \coprod U_i$. Gluing these hypercomplex structures, we obtain a hypercomplex structure $\c H$ on the smooth manifold$\widetilde{(M, I)}$. Consider the normalization map $n:\; \widetilde{(M, I)} {\:\longrightarrow\:} M$, and let $\tilde M^{n}:= n^{-1}(M^{ns})$. Then, $n\restrict{\tilde M^{n}} \tilde M^{n}{\:\longrightarrow\:} M^{ns}$ is an etale finite covering which is compatible with the hypercomplex structure. Thus, $\c H\restrict{\tilde M^{n}}$ can be obtained as a pullback from $M$. Clearly, a hypercomplex structure on a manifold is uniquely defined by its restriction to an open dense subset. We obtain that $\c H$ is independent from the choice of $I$. \blacksquare \subsection{Integrability of induced complex structures} \theorem\label{_all_indu_comple_integra_Theorem_} Let $M$ be a hypercomplex variety, $I$ an induced complex structure. Then $I$ is integrable. \hfill {\bf Proof:} Kaledin has proven \ref{_all_indu_comple_integra_Theorem_} for smooth $M$ (\cite{_Kaledin_}). Let $\tilde M$ be a desingularization of $M$, which is hypercomplex. Then $I$ induces an integrable almost complex structure on $\tilde M$. From the local structure of the singularities of $M$, is is clear that $M$ is obtained from $\tilde M$ by gluing pairwise certain trianalytic subvarieties $X_i\subset \tilde M$. Since $I$ induces an integrable complex structure on $\tilde M$, the $X_i$ are complex subvarieties of $(\tilde M, I)$, and the identification procedures are complex analytic with respect to $I$. Performing these identification morphisms on $(\tilde M, I)$, we obtain a complex variety $M'$ such that $(M,I)$ is the underlying almost complex variety. This proves \ref{_all_indu_comple_integra_Theorem_}. \blacksquare \section{Twistor spaces of hypercomplex varieties} \label{_twistors_Section_} Let $M$ be a hypercomplex variety, $M_{\Bbb R}$ the underlying real analytic variety. Consider the variety $\operatorname{Tw}_{\Bbb R}:= M_{\Bbb R} \times S^2$, where $S^2$ is the 2-dimensional sphere identified with the real variety underlying ${\Bbb C} P^1$. We endow $\operatorname{Tw}_{\Bbb R}$ with an almost complex structure as follows. We have a decomposition \[ \Omega^1 \operatorname{Tw}_{\Bbb R} = \pi^* \Omega^1 S^2 \oplus \sigma^* \Omega^1 M_{\Bbb R}, \] where $\pi:\; \operatorname{Tw}_{\Bbb R} {\:\longrightarrow\:} S^2$, $\sigma:\; \operatorname{Tw}_{\Bbb R} {\:\longrightarrow\:} M$ are the natural projection maps. Let $C$ be the natural map $\Omega^1 M_{\Bbb R}\otimes \Bbb H\stackrel C {\:\longrightarrow\:} \Omega^1 M_{\Bbb R}$ arising from the quaternionic structure. We identify the points of $S^2$ with induced complex structures on $M$, as usually, which are quaternions $L\in \Bbb H$, $L^2=-1$. This gives a natural real analytic map $i:\; S^2 {\:\longrightarrow\:} \Bbb H$. A composition of $C$ and $i$ gives an endomorphism $\c I_0:\; \sigma^* \Omega^1 M_{\Bbb R} {\:\longrightarrow\:} \sigma^* \Omega^1 M_{\Bbb R}$. In terms of the fibers, the endomorphism $\c I_0$ can be described as follows. For $(s,m) \in S^2 \times M_{\Bbb R} = \operatorname{Tw}_{\Bbb R}$, $\c I_0$ acts on $\sigma^* \Omega^1 M_{\Bbb R}\restrict{(s,m)}=\Omega^1 M_{\Bbb R}\restrict m$ by $I_s$, where $I_s= i(s) \in \Bbb H$ is the induced complex structure corresponding to $s\in S^2$. Since $S^2$ is identified with ${\Bbb C} P^1$, this space has a natural complex structure $I_{{\Bbb C} P^1}:\; \Omega^1 S^2{\:\longrightarrow\:} \Omega^1 S^2$. Let $\c I$ be an almost complex structure $\c I:\; \Omega^1 \operatorname{Tw}_{\Bbb R} {\:\longrightarrow\:} \Omega^1 \operatorname{Tw}_{\Bbb R}$ acting as $\pi^* I_{{\Bbb C} P^1}$ on $\pi^* \Omega^1 S^2$ and as $\c I_0$ on $\sigma^* \Omega^1 M_{\Bbb R}$. \hfill \claim\label{_twi_integra_Claim_} The constructed above almost complex structure on $\operatorname{Tw}_{\Bbb R}$ is integrable. {\bf Proof:} For $M$ non-singular, this is proven by D. Kaledin \cite{_Kaledin_}. For $M$ singular, the proof essentially repeats the proof of \ref{_all_indu_comple_integra_Theorem_}: we apply the desingularization theorem (\ref{_desingu_Theorem_}), and then Kaledin's result. \blacksquare \hfill \definition\label{_twistor_Definition_} Let $M$ be a hypercomplex variety. Consider the complex variety $(\operatorname{Tw}, \c I)$ obtained in \ref{_twi_integra_Claim_}. Then $\operatorname{Tw}$ is called {\bf a twistor space} of $M$. \hfill It is possible to characterize the hypercomplex varieties in terms of the twistor spaces. This characterization is the main purpose of the present paper. Consider the unique anticomplex involution $\iota_0:\; {\Bbb C} P^1 {\:\longrightarrow\:} {\Bbb C} P^1$ with no fixed points. This involution is obtained by central symmetry with center in $0$ if we identify ${\Bbb C} P^1$ with a unit sphere in ${\Bbb R}^3$. Let $\iota:\; \operatorname{Tw}{\:\longrightarrow\:} \operatorname{Tw}$ be an involution of the twistor space mapping $(s,m)\in S^2 \times M = \operatorname{Tw}$ to $(\iota_0(s), m)$. Clearly, $\iota$ is anticomplex. \hfill \definition Let $s:\; {\Bbb C} P^1 {\:\longrightarrow\:} \operatorname{Tw}$ be a section of the natural holomorphic projection $\pi:\; \operatorname{Tw} {\:\longrightarrow\:} {\Bbb C} P^1$, $s\circ \pi = Id\restrict{{\Bbb C} P^1}$. Then $s$ is called {\bf the twistor line}. The space $\operatorname{Sec}$ of twistor lines is finite-dimensional and equipped with a natural complex structure, as follows from deformation theory (\cite{_Douady_}). \hfill Let $\operatorname{Sec}^\iota$ be the space of all lines $s\in \operatorname{Sec}$ which are fixed by $\iota$. The space $\operatorname{Sec}^\iota$ is equipped with a structure of a real analytic space. We have a natural map $\tau:\; M_{\Bbb R} {\:\longrightarrow\:} \operatorname{Sec}^\iota$ associating to $m\in M$ the line $s:\; {\Bbb C} P^1 {\:\longrightarrow\:} \operatorname{Tw}$, $s(x) = (x,m) \in S^2 \times M = \operatorname{Tw}$. Such twistor lines are called {\bf horizontal twistor lines.} Denote the set of horizontal twistor lines by $\operatorname{Hor}\subset \operatorname{Sec}$. \hfill \lemma\label{_hori-compo-inSec^iota_Claim_} Let $M$ be a hypercomplex variety, $\operatorname{Tw}$ its twistor space and $\tau:\; M_{\Bbb R} {\:\longrightarrow\:} \operatorname{Sec}^\iota$ the real analytic map constructed above. Then $\tau$ is a closed embedding identifying $M$ with one of connected components of $\operatorname{Sec}^\iota$. \hfill {\bf Proof:} By the Desingularization Theorem (\ref{_desingu_Theorem_}), we may assume that $M$ is smooth. For smooth $M$, \ref{_hori-compo-inSec^iota_Claim_} is a well-known statement which can be easily deduced from the deformation theory. For details, the reader is referred to \cite{_HKLR_}. \blacksquare \hfill The following data suffice to recover the hypercomplex variety $M$: \begin{equation}\label{_twistor_data_Equation_} \begin{minipage}[m]{0.8\linewidth} \begin{itemize} \item A complex analytic variety $\operatorname{Tw}$, equipped with a morphism $\pi:\; \operatorname{Tw} {\:\longrightarrow\:} {\Bbb C} P^1$. \item An anticomplex involution $\iota:\; \operatorname{Tw} {\:\longrightarrow\:} \operatorname{Tw}$ such that $\iota\circ \pi = \pi\circ \iota_0$ \item A choice of connected component $\operatorname{Hor} $ of $\operatorname{Sec}^\iota\subset \operatorname{Sec}$. \end{itemize} \end{minipage} \end{equation} The hypercomplex variety $M$ is reconstructed as follows. The real analytic structure on $M = \operatorname{Hor} $ comes from $\operatorname{Sec}^\iota$. For $I \in {\Bbb C} P^1$, consider the map $p_I:\; \operatorname{Hor} {\:\longrightarrow\:} \pi^{-1}(I) \subset \operatorname{Tw}$, $s\in \operatorname{Hor} {\:\longrightarrow\:} s(I) \in \operatorname{Tw}$. This identifies $\operatorname{Hor} $ with $\pi^{-1}(I) \subset \operatorname{Tw}$. We obtained a complex structure $I$ on $M$, for each $I\in {\Bbb C} P^1$. Identifying ${\Bbb C} P^1$ with a subset of quaternions, we recover the original quaternion action on $\Omega^1 M$. The data \eqref{_twistor_data_Equation_} satisfies the following properties (condition (ii) is implicit in the quaternionic action). \begin{equation}\label{_twistor_properties_Equation_} \begin{minipage}[m]{0.8\linewidth} \begin{description} \item[(i)] For each point $x\in \operatorname{Tw}$, there is a unique line $s\in \operatorname{Hor} \subset \operatorname{Sec}^\iota$, passing through $x$. \\[2mm] \begin{minipage}[m]{0.8\linewidth} {\it For every $s\in \operatorname{Sec}$, we identify the image of $s$, $\operatorname{im} s \subset \operatorname{Tw}$, with ${\Bbb C} P^1$.} \end{minipage} \item[(ii)] For every line $s\in \operatorname{Hor}\subset \operatorname{Sec}^\iota$, the conormal sheaf \[ N^*_s = \ker\left( \Omega^1 \operatorname{Tw}\restrict {\operatorname{im} s} \stackrel {s^*}{\:\longrightarrow\:} \Omega^1 (\operatorname{im} s)\right) \] of $\operatorname{im} s$ is isomorphic to ${\cal O}(-1) \oplus {\cal O}(-1) \oplus ... \oplus {\cal O}(-1)$. \end{description} \end{minipage} \end{equation} \hfill \definition \label{_twi_hyperco_type_Definition_} The data \eqref{_twistor_data_Equation_} satisfying the conditions (i), (ii) of \eqref{_twistor_properties_Equation_} are called {\bf a twistor space of hypercomplex type}. We have shown how to associate a twistor space of hypercomplex type to every hypercomplex variety. Denote the corresponding functor by $\c F$. \hfill Condition (ii) of \eqref{_twistor_properties_Equation_} can be replaced by the following condition. \begin{equation}\tag{\ref{_twistor_properties_Equation_}} \begin{minipage}[m]{0.8\linewidth} \begin{description} \item[(ii$'$)] For every line $s\in \operatorname{Hor}\subset \operatorname{Sec}^\iota$, there exists an open neighbourhood $U\subset \operatorname{Tw}$ of $\operatorname{im} s$, such that for every $x, y \in U$, $\pi(x) \neq \pi(y)$, there exists a unique twistor line $s_{x,y}$ passing through $x$ and $y$, provided that $x$ and $y$ belong to the same irreducible component of $U$. \end{description} \end{minipage} \end{equation} Condition (ii) of \eqref{_twistor_properties_Equation_} should be thought of as a linearization of \eqref{_twistor_properties_Equation_} (ii$'$). In the subsequent section, we shall see that these conditions are equivalent. \hfill \definition\label{_Deligne_Simpson_Definition_} The data \eqref{_twistor_data_Equation_} satisfying the conditions (i), (ii$'$) of \eqref{_twistor_properties_Equation_} are called {\bf a twistor space of Deligne-Simpson type}. These conditions were proposed by Deligne and Simpson (\cite{_Simpson:hyperka-defi_}, \cite{_Deligne:defi_}) in order to define singular hyperk\"ahler manifolds. \section{Twistor spaces of Deligne-Simpson type} \label{_Deli_Si_equi_hyperco_Section_} The main result of this section is the following theorem. \hfill \theorem \label{_Deli_Simpsi_equi_infinite_Theorem_} Let $(\operatorname{Tw}, \pi, \iota, \operatorname{Hor})$ be the data of \eqref{_twistor_data_Equation_} satisfying condition (i) of \eqref{_twistor_properties_Equation_}. Then the conditions (ii) and (ii$'$) are equivalent. In other words, $(\operatorname{Tw}, \pi, \iota, \operatorname{Hor})$ is a twistor space of hypercomplex type if and only if $(\operatorname{Tw}, \pi, \iota, \operatorname{Hor})$ is a twistor space of Deligne--Simpson type. \hfill The proof of \ref{_Deli_Simpsi_equi_infinite_Theorem_} takes the rest of this section. \hfill Under assumptions of \eqref{_twistor_data_Equation_}, \eqref{_twistor_properties_Equation_} (i) consider the map $\sigma:\; \operatorname{Tw} {\:\longrightarrow\:} \operatorname{Hor}$ associating to a point $x\in \operatorname{Tw}$ the unique horisontal line passing through this point. This map is continuous, and induces a homeomorphism $\sigma \times \pi:\; \operatorname{Tw} {\:\longrightarrow\:} \operatorname{Hor} \times {\Bbb C} P^1$. \hfill \lemma\label{_Deli_Si_local_Lemma_} Let $(\operatorname{Tw}, \pi, \iota, \operatorname{Hor})$ be the data of \eqref{_twistor_data_Equation_} satisfying condition (i) of \eqref{_twistor_properties_Equation_}, and $U\subset \operatorname{Hor}$ an arbitrary open subset. Then $\sigma^{-1}(U)$ is preserved by $\iota$. {\bf Proof:} Let $s \in U \subset \operatorname{Hor}$. Then $\iota$ preserves a line $\operatorname{im} s \subset \operatorname{Tw}$, and thus, $\iota(\operatorname{im} s) \subset \sigma^{-1}(s) \subset \sigma^{-1}(U)$. This proves \ref{_Deli_Si_local_Lemma_}. \blacksquare \hfill We prove the implication ``$(\operatorname{Tw}, \pi, \iota, \operatorname{Hor})$ of hypercomplex type'' $\Rightarrow$ ``$(\operatorname{Tw}, \pi, \iota, \operatorname{Hor})$ of Deligne--Simpson type''. The statement of \ref{_Deli_Simpsi_equi_infinite_Theorem_} is local in $\operatorname{Hor}$, as follows from \ref{_Deli_Si_local_Lemma_}. Consider the evaluation maps $p_I:\; \operatorname{Hor} {\:\longrightarrow\:} \pi^{-1}(I)$, $s {\:\longrightarrow\:} s(I)$, defined for all $I\in {\Bbb C} P^1$. The map $\sigma$ gives a homeomorphism $\pi^{-1}(I) \stackrel \sigma {\:\longrightarrow\:} \operatorname{Hor}$. A homeomorphism preserves the dimension of the variety. Thus, for a smooth point $x\in \operatorname{Tw}$, the point $\sigma(x)\in \operatorname{Hor}$ is equidimensional \footnote{Equidimensional point of $X$ is a point where all irreducible components of $X$ have the same dimension.} in $\operatorname{Hor}$. Using the homeomorphism $\sigma \times \pi:\; \operatorname{Tw} {\:\longrightarrow\:} \operatorname{Hor} \times {\Bbb C} P^1$, we find that for a smooth point $x\in \operatorname{Tw}$ and $y\in \operatorname{Tw}$ satisfying $\sigma(y)=\sigma(x)$, the point $y$ is equidimensional in $\operatorname{Tw}$. The real dimensions of the varieties $\operatorname{Hor}$ and $\pi^{-1}(I)$ are equal. On the other hand, for every $m\in \operatorname{Hor}$, the dimension of the tangent space $T_{p_I(m)}\pi^{-1}(I)$ is equal to the dimension of $N^*_m\restrict{p_I(m)}$. Since $N^*_m\restrict{p_I(m)}$ is a bundle, the dimension of $T_{p_I(m)}\pi^{-1}(I)$ is the same for all $I\in {\Bbb C} P^1$. The local ring is regular if and only if the dimension of its tangent space is equal to the dimension of the ring. Dimensions of the varieties $\pi^{-1}(J)$ and $\pi^{-1}(I)$, and the dimensions of the corresponding tangent spaces are equal. Thus, for a smooth point $m\in \pi^{-1}(I)$, and every $J\in {\Bbb C} P^1$, the points $p_J(\sigma (m))\in \pi^{-1}(J)$ are smooth in $\pi^{-1}(J)$. We obtained the following result. \hfill \claim \label{_hori_smooth_Claim_} Let $(\operatorname{Tw}, \pi, \iota, \operatorname{Hor})$ be a twistor space of hypercomplex type, and $m\in \operatorname{Tw}$ a smooth point. Let $\sigma$ denote the natural continuous map $\sigma:\; \operatorname{Tw} {\:\longrightarrow\:} \operatorname{Hor}$. Then, for every point $m'\in \operatorname{Tw}$ such that $\sigma(m) = \sigma(m')$, $m'$ is smooth. \blacksquare \hfill \lemma\label{_hype+smoo=>Deli-Simps_Lemma_} Let $(\operatorname{Tw}, \pi, \iota, \operatorname{Hor})$ be a twistor space of hypercomplex type, and $m\in \operatorname{Tw}$ a smooth point. Then \begin{description} \item[(i)] $\sigma(m)$ is a smooth point of $\operatorname{Hor}$. \item[(ii)] Moreover, for a smooth neighbourhood $U$ of $m\in Tw$, $(\sigma^{-1}(\sigma(U))\subset \operatorname{Tw}, \pi, \iota, \sigma(U))$ is a twistor space of Deligne-Simpson type. \end{description} {\bf Proof:} Let $s= \operatorname{im} m\subset \operatorname{Tw}$ be the horisontal twistor line corresponding to $m$. From the deformation theory we know that the deformations of a smooth curve $s$ are classified by the sections of the normal bundle $\Gamma(Ns)$, with obstructions corresponding to $H^1(Ns)$. The cohomology space $H^1(Ns)$ vanishes, because $Ns=\bigoplus {\cal O}(1)$ is ample. Thus, $\Gamma(Ns)$ has a dimension $2 (\dim \operatorname{Tw}-1)$. For a small deformations $s'$ of $s$, $\Gamma(Ns')= \bigoplus {\cal O}(1)$, since $\bigoplus {\cal O}(1)$ is semistable. Thus, $T_{s'}\operatorname{Sec}$ is constant in a neighbourhood of $s$, where $\operatorname{Sec}$ is the space of sections of $\pi:\; \operatorname{Tw}{\:\longrightarrow\:} {\Bbb C} P^1$. This implies that $s$ is a smooth point of $\operatorname{Sec}$, and hence, $s$ is a smooth point of $\operatorname{Hor}$. \ref{_hype+smoo=>Deli-Simps_Lemma_} (i) is proven. To prove (ii), let $I$, $J$ be distinct points in ${\Bbb C} P^1$ and consider the map $p_{IJ}:\; \operatorname{Sec} {\:\longrightarrow\:} \pi^{-1}(I) \times \pi^{-1}(J)$, $\gamma {\:\longrightarrow\:} (\gamma(I), \gamma(J))$. We have to prove that $p_{IJ}$ has invertible differential in $s$. The tangent space $T_s \operatorname{Sec}$ is, as we have seen, $\Gamma(Ns)$. The differential of the map $p_I:\; \operatorname{Sec}{\:\longrightarrow\:} \pi^{-1}(I)$, $\gamma {\:\longrightarrow\:} \gamma(I)$ coinsides with the restriction map $r_I:\;\Gamma(Ns) {\:\longrightarrow\:} Ns\restrict{I}$. Since $Ns$ is $\bigoplus {\cal O}(1)$, the differential $dp_{IJ} = r_I\times r_J$ is an isomorphism (a section of $\bigoplus {\cal O}(1)$ is uniquely determined by its value in two distinct points). This proves \ref{_hype+smoo=>Deli-Simps_Lemma_} (ii). \blacksquare \hfill We return to the proof of an implication ``$(\operatorname{Tw}, \pi, \iota, \operatorname{Hor})$ of hypercomplex type'' $\Rightarrow$ ``$(\operatorname{Tw}, \pi, \iota, \operatorname{Hor})$ of Deligne--Simpson type''. Let $\operatorname{Tw}^{ns}$ be the set of non-singular points of $\operatorname{Tw}$, $I$, $J$ be two distinct points of ${\Bbb C} P^1$. Let $W\subset \operatorname{Sec}$ be an open neighbourhood of a horisontal line $s\in \operatorname{Sec}$, such that its closure $\bar W$ is compact and $\bar V_{IJ}$ the set of all triples $(x,y, s_{xy}) \in \pi^{-1}(I)\times \pi^{-1}(J)\times \operatorname{Sec}$ such that $s_{xy}\in \bar W$, and $s_{xy}$ is a twistor line passing through $x$ and $y$. Let $V_{IJ}\subset \bar V_{IJ}$ be the set of the triples $(x,y, s_{xy})\in \bar V_{IJ}$, for which the corresponding twistor line $s_{xy}$ belongs to the non-singular part of $\operatorname{Tw}$. \hfill \lemma \label{_closure_of_lines_through_ns_Lemma_} In the above assumptions, consider the forgetful map $p:\; \bar V_{IJ} {\:\longrightarrow\:} \pi^{-1}(I)\times \pi^{-1}(J)$. Then $p\left(\bar V_{IJ}\right)$ is a closure of $p\left( V_{IJ}\right)$. \hfill {\bf Proof:} The space $\bar V_{IJ}$ is compact, and hence its image is compact and thus closed. It remains to show that $V_{IJ}$ is dense in $\bar V_{IJ}$. By \ref{_hori_smooth_Claim_}, for each smooth point $m\in \operatorname{Tw}$, a neighbourhoor of $\sigma(m)\in \operatorname{Sec}$ belings to $V_{IJ}$. Since the set of smooth points of $\operatorname{Tw}$ is dense in $\operatorname{Tw}$, for $\bar V_{IJ}$ sufficiently small, $V_{IJ}$ is dense in $\bar V_{IJ}$. \blacksquare \hfill Let $U$ be an open subset of $\operatorname{Tw}$, and $X_U\subset \pi^{-1}(I)\times \pi^{-1}(J)$ be the set of all $(x,y) \in \pi^{-1}(I)\times \pi^{-1}(J)$ belonging to the same irreducible component of $U\subset \operatorname{Tw}$. Consider the forgetful map $p:\; \bar V_{IJ} {\:\longrightarrow\:} \pi^{-1}(I)\times \pi^{-1}(J)$. Clearly, the image of $p$ intersected with $U\times U$ lies in $X_U$, so we may assume that $p$ maps $\bar V_{IJ}$ to $X_U$. Computing the differential of $p$ as in the proof of \ref{_hype+smoo=>Deli-Simps_Lemma_}, we find that $dp$ is locally injective for $s_{xy}$ in a neighbourhood of $\operatorname{Hor}$. To prove the condition of Deligne and Simpson, it remains to show that $p$ is locally a surjection onto $X_U$, for sufficiently small $U$. By \ref{_hype+smoo=>Deli-Simps_Lemma_}, the image of $p\restrict{V_{IJ}}$ is dense in $X_U\cap \operatorname{Tw}^{ns}\times \operatorname{Tw}^{ns}$, for $U$ sufficiently small. On the other hand, the closure of $\operatorname{im} p\restrict{V_{IJ}}$ is $\operatorname{im} p\restrict{\bar V_{IJ}}$ by \ref{_closure_of_lines_through_ns_Lemma_}, so $p$ is locally a surjection. We proved that the Deligne-Simpson condition holds for all twistor spaces of hypercomplex type. Assume now that $(\operatorname{Tw}, \pi, \iota, \operatorname{Hor})$ is a twistor space of Deligne--Simpson type. Let $s\in \operatorname{Hor}$. Consider the evaluation map $p_{IJ}:\; \operatorname{Sec} {\:\longrightarrow\:} \pi^{-1}(I)\times \pi^{-1}(J)$, $I\neq J\in {\Bbb C} P^1$. By Deligne-Simpson's condition, $p_{IJ}$ induces an isomorphism \begin{equation}\label{_diffe_eva_Tsec_Equation_} dp_{IJ}:\; T_s \operatorname{Sec} {\:\longrightarrow\:} T_{s(I)}\pi^{-1}(I)\times T_{s(I)}\pi^{-1}(I) \end{equation} of the tangent spaces. Thus, dimension of $T_{s(I)} \pi^{-1}(I)$ is independent from the choice of $I$. We obtain that the conormal sheaf $N^*s$ is a bundle, and it makes sense to speak of the normal bundle $Ns$. Through each point in a neighbourhood of $s$ passes a deformation of $s$. Thus, $T_s \operatorname{Sec}= \Gamma(Ns)$; there is no first order obstructions to the deformation. Since $T_s \operatorname{Sec}= Ns$, the map \eqref{_diffe_eva_Tsec_Equation_} can be interpreted as \[ dp_{IJ}:\;\Gamma(Ns) {\:\longrightarrow\:} Ns\restrict{I} \times Ns\restrict{J} \] Since \eqref{_diffe_eva_Tsec_Equation_} is an isomorphism, $Ns$ is a bundle which is isomorphic to $\bigoplus {\cal O}(1)$. \ref{_Deli_Simpsi_equi_infinite_Theorem_} is proven. \blacksquare \section{Hypercomplex varieties and twistor spaces of hypercomplex type.} \label{_hype_type_equi_hype_Section_} The main result of this paper is the following theorem. \hfill \theorem \label{_hype_vari_and_twi_equiva_Theorem_} Consider the functor $\c F$ of \ref{_twi_hyperco_type_Definition_}, associating to a hypercomplex variety the corresponding twistor space of hypercomplex type. Then $\c F$ is equivalence of categories. \hfill {\bf Proof:} We have shown how to recover the hypercomplex structure from the twistor space. This proves that $\c F$ is full and faithful. It remains to show that each twistor space of hypercomplex type is obtained from a hypercomplex variety. Thus, \ref{_hype_vari_and_twi_equiva_Theorem_} is implied by the following statement. \hfill {\bf \ref{_hype_vari_and_twi_equiva_Theorem_}$'$} Let \[ (\operatorname{Tw}, \pi, \iota, \operatorname{Hor})\] be a twistor space of hypercomplex type. Then $\operatorname{Hor}$ admits a hypercomplex structure $\c H$, such that $(\operatorname{Tw}, \pi, \iota, \operatorname{Hor})$ is a twistor space of $(\operatorname{Hor}, \c H)$. \hfill The rest of this section is taken by the proof of \ref{_hype_vari_and_twi_equiva_Theorem_}$'$. \hfill \lemma\label{_irre_compo_twi_hype_type_Lemma_} Let $\operatorname{Hor} = \bigcup H_i$ be an irreducible decomposition of the variety $\operatorname{Hor}$. Let $\operatorname{Tw}_i := \sigma^{-1}(H_i)$, where $\sigma:\; \operatorname{Tw}{\:\longrightarrow\:} \operatorname{Hor}$ is the standard continuous map. Then $\operatorname{Tw}_i$ is preserved by $\iota$, and \[ \left( Tw_i, \pi\restrict{\operatorname{Tw}_i}, \iota\restrict{\operatorname{Tw}_i}, H_i\right) \] is a twistor space of hypercomplex type. \hfill {\bf Proof:} It is clear from the definition that $\operatorname{Tw}_i$ is preserved by $\iota$ and that for every $m\in \operatorname{Tw}_i$ there is a unique horisontal line $s\in H_i$ passing through $m$. It remains to prove the condition (ii) of \eqref{_twistor_properties_Equation_}, or, equivalently, condition (ii)$'$ of \eqref{_twistor_properties_Equation_}. But, since $\operatorname{Tw}$ satisfies the condition (ii)$'$ of \eqref{_twistor_properties_Equation_}, $\operatorname{Tw}_i$ satisfies this condition automatically. \blacksquare \hfill \claim \label{_homeomo_maos_irre_compo_Claim_} Let $f:\; X {\:\longrightarrow\:} Y$ be a homeomorphism of complex analytic varieties. Then $f$ maps irreducible components of $X$ to irreducible components of $Y$. {\bf Proof:} Clear. \blacksquare \hfill \lemma \label{_p_I_iso_Lemma_} Let $(\operatorname{Tw}, \pi, \iota, \operatorname{Hor})$ be a twistor space of hypercomplex type, and $I\in {\Bbb C} P^1$. Consider the map $p_I:\; \operatorname{Hor} {\:\longrightarrow\:} \pi^{-1}(I)$, $s {\:\longrightarrow\:} s(I)$. Let $p:\; \operatorname{Hor} \times {\Bbb C} P^1{\:\longrightarrow\:} \operatorname{Tw}$ map $(s, I)$ to $s(I)$. Then $p$ and $p_I$ induce isomorphisms of corresponding real analytic varieties. \hfill {\bf Proof:} Using \ref{_homeomo_maos_irre_compo_Claim_}, \ref{_irre_compo_twi_hype_type_Lemma_} and \ref{_Deli_Si_local_Lemma_}, we may assume that the fiber $\pi^{-1}(I)$ is locally irreducible and thus, equidimensional. The real dimensions of the varieties $\operatorname{Hor}$, $\pi^{-1}(I)$ are clearly equal. Thus, to show that $p$, $p_I$ induce isomorphisms, it suffices to show that corresponging maps of local rings are surjective. By Nakayama, for this we need to show that $p$, $p_I$ induce surjection on the Zariski tangent spaces. The differential of the evaluation map $ev_{I}:\; \operatorname{Sec} {\:\longrightarrow\:} \pi^{-1}(I)$, $s{\:\longrightarrow\:} s(I)$ is the standard restriction map $r_I:\; \gamma {\:\longrightarrow\:} \gamma\restrict{I}$, where $\gamma\in T_s \operatorname{Sec} = \Gamma(Ns)$, and $ \gamma\restrict{I}\in Ns\restrict{I} = T_{s(I)}\pi^{-1}(I)$. Thus, $dp_I$ is a composition of $r_I$ and the embedding $T_s \operatorname{Hor} \hookrightarrow T_s\operatorname{Sec}$. The image of $T_s \operatorname{Hor} \hookrightarrow T_s\operatorname{Sec}$ coinsides with the set of fixed points of the involution $d\iota:\; T_s\operatorname{Sec} {\:\longrightarrow\:} T_s \operatorname{Sec}$. Thus, $dp_I$ is an isomorphism by the following trivial result of linear algebra. \hfill \lemma \label{_tota_real_iso_linear_alge_Lemma_} Let $V$ be a vector space of complex dimension $2n$, $W$ a vector space of complex dimension $n$, and $\phi:\; V {\:\longrightarrow\:} W$ an epimorphism. Let $V'\subset V$ be a totally real subspace of real dimension $2n$. Then $\phi\restrict{V'}:\; V' {\:\longrightarrow\:} W$ is an isomorphism. \blacksquare A similar argument proves that $p$ is also an isomorphism. \ref{_p_I_iso_Lemma_} is proven. \blacksquare \hfill We obtained that the real analytic variety $\operatorname{Hor}$ is isomorphic to one underlying $\pi^{-1}(I)$, for all $I\in {\Bbb C} P^1$. This gives a set of integrable almost complex structures on $\operatorname{Hor}$, parametrized by ${\Bbb C} P^1$. The following linear algebraic argument shows that these complex structures satisfy quaternionic relations. This finishes the proof of \ref{_hype_vari_and_twi_equiva_Theorem_}. \hfill Let $F$ be an $n$-dimensional holomorphic vector bundle on ${\Bbb C} P^1$, $F \cong \oplus {\cal O}(1)$. Consider the restriction maps $r_I:\; \Gamma(F) {\:\longrightarrow\:} F\restrict{I}$, defined for each $I\in {\Bbb C} P^1$. Let $W\subset \Gamma(F)$ be a totally real subspace of real dimension $2n$. By \ref{_tota_real_iso_linear_alge_Lemma_}, $r_I$ induces an isomorphism between $F\restrict{I}$ and $W$. This gives a set of complex structures on $W$, parametrized by ${\Bbb C} P^1$. \hfill \claim These complex structures satisfy quaternionic relations. {\bf Proof:} Clear. \blacksquare \section{Some applications} \label{_twi_applications_Section_} \subsection{Hypercomplex spaces} Using \ref{_hype_vari_and_twi_equiva_Theorem_}, it is possible to generalize the definition of hypercomplex varieties, allowing nilpotents, in such a way that a reduction of a hypercomplex space is a hypercomplex variety. Consider the anticomplex involution $\iota_0:\; {\Bbb C} P^1 {\:\longrightarrow\:} {\Bbb C} P^1$ defined in \ref{_twistors_Section_}. \hfill \definition\label{_hyperco_spaces_Definition_} (Hypercomplex spaces) Let $\operatorname{Tw}$ be a complex analytic space, $\pi:\; \operatorname{Tw}{\:\longrightarrow\:} {\Bbb C} P^1$ a holomorphic map, and $\iota:\; \operatorname{Tw} {\:\longrightarrow\:} \operatorname{Tw}$ an anticomplex automorphism, such that $\iota\circ\pi = \pi\circ \iota_0$. Let $\operatorname{Sec}$ be the space of sections of $\pi$ equipped with a structure of a complex analytic space, and $\operatorname{Sec}^\iota$ be the real analytic space of sections $s$ of $\pi$ satisfying $s\circ \iota_0 = \iota\circ s$. Let $\operatorname{Hor}$ be a connected component of $\operatorname{Sec}^\iota$. Then $(\operatorname{Tw}, \pi,\iota,\operatorname{Hor})$ is called {\bf a hypercomplex space} if \begin{description} \item[(i)] For each point $x\in \operatorname{Tw}^r$, there exist a unique line $s\in \operatorname{Hor}^r$ passing through $x$, where $\operatorname{Tw}^r$, $\operatorname{Hor}^r$ is a reduction of $\operatorname{Tw}$, $\operatorname{Hor}$. \item[(ii)] Let $s\in \operatorname{Hor}$, and $U\subset \operatorname{Tw}$ be a neighbourhood of $s$ such that an irreducible decomposition of $U$ coinsides with the irreducible decomposition of $\operatorname{Tw}$ in a neighbourhood of $s\subset \operatorname{Tw}^r$. Let \[ \bar X:= \pi^{-1}(I)\times \pi^{-1}(J)\cap U\times U, \] where $I$, $J$ distinct points of ${\Bbb C} P^1$. Let $p_{IJ}:\; U {\:\longrightarrow\:} \bar X\subset \pi^{-1}(I)\times \pi^{-1}(J)$ be the evaluation map, $s{\:\longrightarrow\:} (s(I), s(J))$. Then there exist a closed subspace $X\subset \bar X$, obtained as a union of some of irreductible components of $\bar X$, and an open neighbourhood $V\subset \operatorname{Sec}$ of $s\in \operatorname{Sec}$, such that $p_{IJ}$ is an open embedding of $V$ to $X$. \end{description} \subsection{Stable bundles over hyperk\"ahler manifolds} \label{_hyperholomo_Subsection_} Let $M$ be a compact hyperk\"ahler manifold, $I$ an induced complex structure and $B$ a stable holomorphic bundle over $(M, I)$, such that the first two Chern classes $c_1(B)$, $c_2(B)$ are $SU(2)$-invariant, with respect to the natural action of the group $SU(2)$ on the cohomology of $M$ (\ref{_SU(2)_commu_Laplace_Lemma_}). Recall that $SU(2)$ acts on the space of differential forms on $M$. This allows us to speak of $SU(2)$-invariant differential forms, for instance of connections with $SU(2)$-invariant curvature. In \cite{_Verbitsky:Hyperholo_bundles_}, the following theorem was proven. \hfill \theorem \label{_hyperholomo_conne_Theorem_} There exist a unique Hermitian connection $\nabla$ on $B$ such that its curvature $\Theta$ is $SU(2)$-invariant. Conversely, if such connection exitst on a holomorphic bundle $B$ over $(M,I)$, then $B$ is a direct sum of stable bundles with $SU(2)$-invariant Chern classes. \blacksquare \hfill We show that the moduli space $\operatorname{Def}(B)$ of deformations of $B$ is hypercomplex. Consider the twistor space $\operatorname{Tw}$ of $M$, and a standard real analytic map $\sigma:\; \operatorname{Tw} {\:\longrightarrow\:} M$. Let $\sigma^* B$ be the pullback of $B$ equipped with the connection which is trivial along the fibers of $\sigma$. {}From twistor transform (\cite{_NHYM_}, Section 5) is clear that $\sigma^*B$ is holomorphic. Restricting $\sigma^*B$ to the fibers of $\pi:\; \operatorname{Tw} {\:\longrightarrow\:} {\Bbb C} P^1$, we obtain holomorphic bundles $B_J$ on $\pi^{-1}(J) = (M, J)$. By \ref{_hyperholomo_conne_Theorem_}, $B_J$ is stable. Let $\hat{\operatorname{Tw}}$ be the moduli space of sheaves of type $i^J_* F$, where $i^J:\; (M, J)=\pi^{-1}(J)\hookrightarrow \operatorname{Tw}$ is the natural embedding, and $F$ a stable holomorphic bundle which is a deformation of $B_J$ (i. e. belongs in the same deformation class). Then $\hat{\operatorname{Tw}}$ is equipped with a holomorphic fibration $\hat \pi:\; \hat{\operatorname{Tw}}{\:\longrightarrow\:} {\Bbb C} P^1$, $i^J_* F{\:\longrightarrow\:} J$. Mapping $F$ to $\iota^* F$, we obtain an anticomplex involution $\hat \iota$ of $\hat{\operatorname{Tw}}$. The operation $(B,J){\:\longrightarrow\:} B_J$ gives an $\hat\iota$-invariant section of $\hat\pi$, parametrized by $\operatorname{Def}(B)$. To show that thus obtained quadruple $(\hat{\operatorname{Tw}}, \hat \pi, \hat\iota, \operatorname{Def}(B))$ is hypercomplex, it suffices to prove the condition (ii) of \eqref{_twistor_properties_Equation_}. Equivalently, we may prove (ii$'$) of \ref{_Deligne_Simpson_Definition_}. On the other hand, Proposition 2.19 of \cite{_NHYM_} implies (ii$'$). This gives another proof that the space of stable deformations of $B$ is hypercomplex, in addition to that given in \cite{_Verbitsky:Hyperholo_bundles_}. \subsection{Quotients of hypercomplex varieties by an action of a finite group} Let $M$ be a hypercomplex variety and $G$ a finite group acting on $M$, generically free. Assume that $G$ preserves the hypercomplex structure, and acts freely outside of nonempty finite set of fixed points, denoted by $\Gamma$. Clearly, $\bigg(M\backslash \{\Gamma\}\bigg)/G$ has a natural structure of a hypercomplex variety. \hfill \proposition \label{_quotie_not_hyperco_Proposition_} The hypercomplex structure on $\bigg(M\backslash \{\Gamma\}\bigg)/G\subset M/G$ cannot be extended to $M/G$. \hfill {\bf Proof:} Consider the space $\operatorname{Tw}/G$ fibered over ${\Bbb C} P^1$, with corresponding action of $\iota$. Then $\operatorname{Hor}/G$ gives an open subset in the space of $\iota$-invariant sections of $\pi:\; \operatorname{Tw}/G {\:\longrightarrow\:} {\Bbb C} P^1$. Let $p:\; M {\:\longrightarrow\:} M/G$, $p:\; \operatorname{Tw} {\:\longrightarrow\:} \operatorname{Tw}/G$ be the natural quotient maps. If the hypercomplex structure on $M\backslash \{\Gamma\}/G\subset M/G$ were extended to $M/G$, the space of horisontal sections would have been $p(\operatorname{Hor})$. Applying \ref{_hype_vari_and_twi_equiva_Theorem_}, we obtain the following assertion. \hfill \claim The hypercomplex structure on $M\backslash \{\Gamma\}$ is extended to $M/G$ if an only if for all $s:\; {\Bbb C} P^1 {\:\longrightarrow\:} \operatorname{Tw}/G$, $s \in p(\operatorname{Hor})$, the conormal sheaf of $\operatorname{im} s$ in $\operatorname{Tw}/G$ is $\oplus {\cal O}(-1)$. \blacksquare \hfill Let $s\in \operatorname{Hor}$ be a horisontal twistor line in $\operatorname{Tw}$ which passes through fixed point of $G$-action. Consider its formal neighbourhood in $\operatorname{Tw}$. Let ${\cal O}(s)$ be the corresponding complete ring over ${\Bbb C} P^1$ and ${\cal O}(s)_{gr}$ be the associated graded ring. Then the ring ${\cal O}(s)_{gr}$ is isomorphic to $\oplus S^*({\cal O}(-1))/D$, where $D$ is a graded ideal lying in \[ \bigoplus\limits_{k=2}^{\infty} S^k({\cal O}(-1)) \] Let ${\cal O}(\hat s)_{gr}\subset{\cal O}(s)_{gr} $ be the sheaf of $G$-invariant sections of $\oplus S^*({\cal O}(-1))/D$. Clearly, ${\cal O}(\hat s)_{gr}$ is the graded ring of the formal neighbourhood of $\hat s \subset \operatorname{Tw}/G$. Since $x$ is an isolated fixed point of $G$ action, the group acts on the Zariski tangent space $T_xM$ without invariants. Therefore, ${\cal O}(\hat s)_{gr}$ lies in \[ \bigoplus\limits_{k=2}^{\infty} S^k({\cal O}(-1))/D \subset \oplus S^*({\cal O}(-1))/D = {\cal O}(s)_{gr}. \] Thus, the conormal sheaf of ${\cal O}(\hat s)_{gr}$ is isomorphic to ${\cal O}(i_1) \oplus {\cal O}(i_2) \oplus ...$, where $i_1, ... , i_k<-2$. Therefore, $M/G$ cannot be hypercomplex. \ref{_quotie_not_hyperco_Proposition_} is proven. \blacksquare \hfill {\bf Acknowledegments:} D. Kaledin and T. Pantev explained me the Deligne and Simpson's definition of a hyperk\"ahler variety. They were also very helpful in correcting the errors of the manuscript. P. Deligne kindly pointed out persistent errors in the presentation of \cite{_Verbitsky:Desingu_}, trying to explain to me the theory of real analytic spaces. I am grateful to these and also to A. Beilinson, R. Bezrukavnikov, M. Entov, D. Kazhdan, M. Kontsevich, A. Todorov and S.-T. Yau for valuable discussions.

9703

alg-geom/9703026

en

https://arxiv.org/abs/alg-geom/9703026

alg-geom/9703026

Bill Oxbury

William Oxbury, Christian Pauly

Heisenberg invariant quartics and SU_C(2) for a curve of genus four

LaTeX, 36 pages, 2 figures

If C is a curve of genus 4 without vanishing theta-nulls then there exists a unique (irreducible) Heisenberg-invariant quartic Q_C in |2\Theta| = P^{15} such that Sing Q_C contains the image of SU_C(2), the moduli space of rank 2 vector bundles with trivial determinant. Moreover, in each eigen-P^7 of the Heisenberg action on |2\Theta|, Q_C restricts to the classical Coble quartic of the corresponding Prym-Kummer variety. We compare Q_C with the hypersurface G_3 in |2\Theta| of divisors containing a translate of C in J(C), and show that in the eigen-P^7s G_3 recovers Beauville--Debarre's quadrisecant planes of the Prym-Kummers (this works for any genus). Using the Recillas construction this enables us to deduce, contrary to the analogous result for genus 3, that Q_C and G_3 are distinct.

alg-geom

\section{\@startsection {section}{1}{{\bf Z}@}{-3.5ex plus -1ex minus -.2ex}{1.5ex plus .2ex}{\large\bf} \def\subsection{\@startsection{subsection}{2}{{\bf Z}@}{-3.25ex plus -1ex minus -.2ex}{1.5ex plus .2ex}{\normalsize\it}} \let\emppsubsection\subsection \newcommand{\numberequationsassubsubsections} \newtheorem{prop}{Proposition}[section] \newtheorem{lemm}[prop]{Lemma} \newtheorem{theo}[prop]{Theorem} \newtheorem{cor}[prop]{Corollary} \newtheorem{conj}[prop]{Conjecture} \newtheorem{rem}[prop]{\it Remark} \newtheorem{rems}[prop]{\it Remarks} \newtheorem{ex}[prop]{Example} \newtheorem{exs}[prop]{Examples} \begin{document} \title{Heisenberg invariant quartics and ${\cal SU}_C(2)$ for a curve of genus four} \author{William Oxbury and Christian Pauly} \date{} \maketitle \bigskip The projective moduli variety ${\cal SU}_C(2)$ of semistable rank 2 vector bundles with trivial determinant on a smooth projective curve $C$ comes with a natural morphism $\phi$ to the linear series $|2\Theta|$ where $\Theta$ is the theta divisor on the Jacobian of $C$. Well-known results of Narasimhan and Ramanan say that $\phi$ is an isomorphism to ${\bf P}^3$ if $C$ has genus 2 \cite{NR1}, and when $C$ is nonhyperelliptic of genus 3 it is an isomorphism to a special Heisenberg-invariant quartic $Q_C\subset{\bf P}^7$ \cite{NR2}. The present paper is an attempt to extend these results to higher genus. In the nonhyperelliptic genus 3 case the so-called {\it Coble quartic} $Q_C \subset |2\Theta| = {\bf P}^7$ is characterised by either of two properties: \medskip \par\indent \hangindent2\parindent \textindent{(i)} $Q_C$ is the unique Heisenberg-invariant quartic containing the Kummer variety, i.e. the image of ${\rm Kum}: J_C \rightarrow |2\Theta|$, $x\mapsto \Theta_x +\Theta_{-x}$, in its singular locus; and \par\indent \hangindent2\parindent \textindent{(ii)} $Q_C$ is precisely the set of $2\Theta$-divisors containing some translate of the curve $W_1 \subset J_C^1$. \medskip \noindent We shall examine, for a curve of genus 4, the analogue of each of these properties, and our first main result, analogous to (i), is: \begin{theo} \label{theorem1} If $C$ is a curve of genus 4 without vanishing theta-nulls then there exists a unique (irreducible) Heisenberg-invariant quartic $Q_C\subset |2\Theta| = {\bf P}^{15}$ containing $\phi({\cal SU}_C(2))$ in its singular locus. \end{theo} We prove this in sections \ref{invariantqs} and \ref{cubicnorm} (see corollary \ref{specialq}). The main work involved is first to show cubic normality for $\phi({\cal SU}_C(2))$ (theorem \ref{cubicnormality}). We then use the Verlinde formula to deduce that its ideal contains exactly sixteen independent cubics; by symmetry considerations these cubics are the partial derivatives, with respect to the homogeneous coordinates, of a single quartic $Q_C$. The argument here is identical to that of Coble \cite{C} for the genus 3 case. We conjecture that $\phi({\cal SU}_C(2)) = {\rm Sing}\ Q_C$, or equivalently that the ideal of $\phi({\cal SU}_C(2))$ is generated by cubics. We cannot prove this, but in the rest of the paper we examine the relationship of this problem with property (ii) above. For any curve one may construct a sequence of irreducible, Heisenberg-invariant subvarieties, for $1\leq d\leq g-1$, $$ G_{d} = \{D\ |\ \hbox{$x+W_{g-d} \subset {\rm supp}\ D$ for some $x\in J_C^{d-1}$}\} \subset |2\Theta|. $$ In particular, $G_1$ is the Kummer variety, while $G_{g-1}$ is a hypersurface containing $\phi({\cal SU}_C(2))$ and which coincides with the Coble quartic in the case $g=3$ (and with the Kummer quartic surface in the case $g=2$). When $g=4$, however, $G_3$ turns out to be distinct from $Q_C$---quite contrary to our original expectation. We see this by restricting to the eigen-${\bf P}^7$s of the action on $|2\Theta| = {\bf P}^{15}$ of the group $J_C[2]$ of 2-torsion points. For any nonzero element $\eta \in J_C[2]$ we have an associated double cover $\pi:{\widetilde C} \rightarrow C$ with respect to which ${\rm ker \ } {\rm Nm} = P_{\eta} \cup P_{\eta}^-$, where $(P_{\eta},\Xi)$ is the principally polarised Prym variety. The fixed-point set of the $\eta$-action on $|2\Theta|$ is a pair of ${\bf P}^7$s either of which can be naturally identified with $|2\Xi|$; this fixed-point set therefore contains the Kummer image of $P_{\eta} \cup P_{\eta}^-$, and this is precisely the intersection with $\phi({\cal SU}_C(2))$. Beauville and Debarre \cite{BD} have shown that a $|2\Xi|$-embedded Prym-Kummer variety admits a 4-parameter family of quadrisecant planes analogous to the trisecant lines of a Jacobian Kummer. We prove: \begin{theo} \label{theorem2} Let $C$ be a curve of genus 4 without vanishing theta-nulls; and for any nonzero $\eta \in J_C[2]$ identify $|2\Xi|\hookrightarrow |2\Theta|$ as the component of the fixed-point set of $\eta$ containing the Kummer image of $P_{\eta}$. Then: \begin{enumerate} \item $Q_C\subset {\bf P}^{15}$ restricts on $|2\Xi|$ to the Coble quartic of ${\rm Kum}(P_{\eta})$. \item $G_3\subset {\bf P}^{15}$ restricts on $|2\Xi|$ to the hypersurface ruled by the quadrisecant planes of ${\rm Kum}(P_{\eta})$; and this is distinct from the Coble quartic. \end{enumerate} \end{theo} Part 1 is proved in section \ref{cubicnorm}. Note that $P_{\eta}$ is necessarily the Jacobian $J_X$ of some curve $X$ of genus 3, which can be constructed explicitly (given a choice of trigonal pencil on $C$) via the Recillas correspondence (see section \ref{pryms}). In section \ref{cobleqs} we obtain necessary and sufficient conditions for a secant line of ${\rm Kum}(J_X)$ to lie on $\phi({\cal SU}_X(2))$; and in section \ref{quadrisecants} this is used to show that a generic quadrisecant plane of the family does not lie on $\phi({\cal SU}_X(2))$---hence the final remark in part 2 of the theorem. We introduce the filtration of $|2\Theta|$ by the subvarieties $G_{d}$ in section \ref{abeljacobi}. These varieties are ruled, as $x\in J_C^{d-1}$ varies, by the subseries $N(x) \subset |2\Theta|$ of divisors containing $x+W_{g-d}$. Moreover, for any theta characteristic $\kappa$ on the curve the rulings of $G_d$ and $G_{g+1-d}$ are polar in the sense that $$ N(x) \perp N(\kappa x^{-1}) $$ with respect to the bilinear form on $|2\Theta|$ induced by $\kappa$ (symmetric or skew-symmetric accordingly as $\kappa$ is even or odd). Our results here are somewhat incomplete when $g>4$, and depend on the vanishing, which we are unable to prove, of certain cohomology groups on the symmetric products $S^d C$. This problem and the associated computations are discussed in the appendix. For genus 4, however, we are able to prove what we need, and we arrive at a configuration: $$ {\rm Kum}(J) \subset G_2 \subset \phi({\cal SU}_C(2)) \subset G_3 \subset {\bf P}^{15} $$ where $G_2$ is a divisor in $\phi({\cal SU}_C(2))$ ruled by 4-planes, and $G_3$ is ruled by their polar 10-planes (with respect to any theta characteristic). $G_2$ contains the trisecants of the Jacobian Kummer variety (this is true for any genus, incidentally, and is proved in \cite{OPP}); while the ruling of $G_3$ cuts out in $|2\Xi|$, for each $\eta \in J_C[2]$, precisely the Beauville--Debarre quadrisecant planes. Again this is true for any genus, and is proved in section \ref{prymG3}. \medskip {\it Acknowledgments.} The authors are grateful to Arnaud Beauville and Bert van Geemen for some helpful discussions; and to Miles Reid and Warwick MRC for their hospitality during the early part of 1996, when much of this work was carried out. The second author thanks the University of Durham for its hospitality during 1996, and European networks Europroj and AGE for partial financial support. \vfill\eject \section{Preliminaries} \label{prelims} Given a smooth projective curve $C$ of genus $g\geq 2$, let $J= J^0(C)$ be its Jacobian and $\Theta = W_{g-1} \subset J^{g-1}(C)$ its canonical theta divisor. Let $\vartheta(C) \subset J^{g-1}(C)$ be the set of its theta characteristics. Then the line bundle ${\cal L} = {\cal O}(2\Theta_{\kappa})\in {\rm Pic}(J)$ is independent of $\kappa \in \vartheta(C)$ (where $\Theta_{\kappa} = \Theta - \kappa$); and there is a canonical duality (up to scalar) $$ H^0(J, {\cal L}) = H^0(J^{g-1}, 2\Theta)^{\vee}. $$ Let ${\cal SU}_C(2)$ and ${\cal SU}_C(2,K)$ be the projective moduli varieties of semistable rank 2 vector bundles on $C$ with determinant ${\cal O}_C$ and $K=K_C$ respectively; and let ${\cal L}$, ${\cal L}_K$ denote the respective ample generators of their Picard groups. Then there are canonical identifications: \begin{equation} \label{2theta} |{\cal L}|_J = |2\Theta|^{\vee} = |{\cal L}|_{{\cal SU}_C(2)} \qquad {\rm and} \qquad |{\cal L}|_J^{\vee} = |2\Theta| = |{\cal L}_K|_{{\cal SU}_C(2,K)}. \end{equation} It should not cause any great confusion that we denote by ${\cal L}$ both the line bundle on ${\cal SU}_C(2)$ and its pull-back via the semistable boundary $J\rightarrow {\cal SU}_C(2)$, $L\mapsto L\oplus L^{-1}$. These spaces give us certain maps which are all identified by (\ref{2theta}): \begin{equation} \begin{array}{rcl} \phi: {\cal SU}_C(2) &\rightarrow& |2\Theta| \\ E &\mapsto & D_E = \{ L\in J^{g-1}\ | \ h^0(C,L\otimes E) >0\ \};\\ &&\\ \lambda_{|{\cal L}|} : {\cal SU}_C(2) &\rightarrow& |{\cal L}|^{\vee};\\ &&\\ \psi: {\cal SU}_C(2) &\rightarrow& |{\cal L}_K|\\ E &\mapsto & \{F\in {\cal SU}_C(2,K)\ |\ h^0(C,E\otimes F) >0\ \}_{\rm red}.\\ \end{array} \end{equation} Note that in the definition of $\psi$, the bundle $E\otimes F$ carries a $K$-valued orthogonal structure, so by the Atiyah-Mumford lemma the condition $H^0(C,E\otimes F) \not=0$ determines naturally a divisor with multiplicity 2. Note also that $\phi(E) = D_E$ is the restriction of $\psi(E)$ to the semistable boundary $J^{g-1} \rightarrow {\cal SU}_C(2,K)$, $L\mapsto L\oplus KL^{-1}$. There are likewise (naturally identified) maps: \begin{equation} \begin{array}{rcl} \lambda_{|{\cal L}_K|} : {\cal SU}_C(2,K) &\rightarrow& |{\cal L}_K|^{\vee};\\ &&\\ \phi_K: {\cal SU}_C(2,K) &\rightarrow& |{\cal L}|_J\\ F &\mapsto & \{L\in J\ |\ h^0(C,L\otimes F) >0\ \}.\\ \end{array} \end{equation} Note that any choice of theta characteristic $\kappa \in \vartheta (C)$ sets up a commutative diagram \begin{equation} \label{thetachoice} \begin{array}{rcccl} J&\rightarrow&{\cal SU}_C(2)&\map{\phi}&|2\Theta| = |{\cal L}|^{\vee}\\ &&&&\\ \downarrow&&\otimes \kappa \downarrow&\searrow&\downarrow\\ &&&&\\ J^{g-1}&\rightarrow&{\cal SU}_C(2,K)&\map{\phi_K}&|{\cal L}|,\\ \end{array} \end{equation} in which all the vertical arrows are isomorphisms: the last $|2\Theta | \mathbin{\hbox{$\widetilde\rightarrow$}} |{\cal L}|_J$ is just translation $D\mapsto D-\kappa$; let us briefly recall how the isomorphism $|{\cal L}| \cong |{\cal L}|^{\vee}$ (or $|2\Theta |\cong|2\Theta |^{\vee}$) is induced by the theta characteristic. We shall write $V = H^0(J,{\cal L})$, and let $\theta_{\kappa} \in V$ denote a section cutting out the divisor $\Theta_{\kappa}$. We consider the map $$ \begin{array}{rcl} m: J\times J & \rightarrow & J\times J\\ (u,v) &\mapsto & (u+v,u-v)\\ \end{array} $$ and denote by $\xi_{\kappa} \in \bigotimes^2 V$ the element corresponding to $m^*(pr_1^*\theta_{\kappa} \otimes pr_2^*\theta_{\kappa})$ under the K\"unneth isomorphism $$ \textstyle \bigotimes^2 V \cong H^0(J\times J, pr_1^*{\cal L} \otimes pr_2^*{\cal L}). $$ Then each $\xi_{\kappa}$ is a nondegenerate pairing (and we obtain bases $\{ \xi_{\kappa}\}_{\kappa \in \vartheta^+(C)}$ of $S^2 V$ and $\{ \xi_{\kappa}\}_{\kappa \in \vartheta^-(C)}$ of $\bigwedge^2 V$); this gives the isomorphism $V \cong V^{\vee}$ on the right in diagram (\ref{thetachoice}). Restricted to the Kummer variety this pairing can be written \begin{equation} \label{riemannxi} \xi_{\kappa} (u,v) = \theta_{\kappa}(u+v)\theta_{\kappa}(u-v) \qquad \hbox{for $u,v \in J$.} \end{equation} Finally, note that the whole diagram (\ref{thetachoice}) is acted on by the subgroup $J[2]$ of 2-torsion points in the Jacobian, acting on vector bundles by tensor product and on $|2\Theta|$ by translation. The horizontal maps are all equivariant and the south and south-east arrows are permuted by the action of $J[2]$ on $\vartheta(C)$. \section{Prym varieties} \label{pryms} For each nonzero half-period $\eta \in J[2] \backslash \{{\cal O}\}$ we have an associated unramified double cover $$\pi : {\widetilde C} \rightarrow C.$$ We shall denote by $\sigma$ the involution of ${\widetilde C}$ given by sheet-interchange over~$C$; and by abuse of notation it will denote also the induced involution of ${\rm Pic}({\widetilde C})$. The kernel of the norm map on divisors has two isomorphic connected components: $$ {\rm ker \ } {\rm Nm} = P_{\eta} \cup P_{\eta}^-, $$ where $P_{\eta} = (1 - \sigma)J^0({\widetilde C})$ and $P_{\eta}^- = (1 - \sigma)J^1({\widetilde C})$. We shall refer to the composite $$ S^d {\widetilde C} \rightarrow J^d({\widetilde C}) \rightarrow P_{\eta} \cup P_{\eta}^- $$ as the Abel-Prym map, and for $D\in S^d {\widetilde C}$ write $[D] = D-\sigma D \in P_{\eta}$ for $d$ even, $\in P_{\eta}^-$ for $d$ odd. If we choose $\zeta \in J$ such that $\zeta^2 = \eta$ then we can define a map $$ P_{\eta} \cup P_{\eta}^- \rightarrow {\cal SU}_C(2) \map{\phi} |2\Theta| $$ by $x \mapsto \zeta \otimes \pi_* x \in {\cal SU}_C(2)$. The image is independent of the choice of $\zeta$ and is precisely the fixed-point set of the involution $\otimes \eta$ of ${\cal SU}_C(2)$. Moreover, the linear span of this image can be naturally identified with the linear series $|2\Xi|$ where $\Xi$ is the canonical theta divisor on the dual abelian variety, and represents the principal polarisation on $P_{\eta}$ induced from that on $J({\widetilde C})$. Thus there is a commutative diagram \begin{equation} \label{prymkum} \begin{array}{ccc} P_{\eta} & \rightarrow & {\cal SU}_C(2) \\ &&\\ \downarrow &&\downarrow\\ &&\\ |2\Xi| &\subset & |2\Theta|.\\ \end{array} \end{equation} Finally, we recall that by a construction of S. Recillas \cite{Rec}, if $C$ is a trigonal curve then $P_{\eta} = P_{\eta}(C)$ is isomorphic as a ppav to a tetragonal Jacobian: \begin{prop} There is generically a bijection between the following two sets of data, under which there is a canonical isomorphism of ppavs $(J(X),\Theta) \cong (P_{\eta}(C), \Xi)$: \begin{enumerate} \item $(X,g^1_4)$ where $X$ is a smooth curve of genus $g$, and $g^1_4$ is a tetragonal pencil; \item $(C,\eta,g^1_3)$ where $C$ is a smooth curve of genus $g+1$, $\eta \in J[2]$ is a nonzero 2-torsion point and $g^1_3$ is a trigonal pencil. \end{enumerate} \end{prop} For the details of this construction we refer to \cite{D}---note in particular that the word `generically' means that on each side one must restrict to pencils having smooth ramification behaviour, in order to obtain a smooth curve on the other side. Accordingly, the correspondence can be compactified, but this will not concern us. Here we need only to note that by definition the double cover defined by $\eta$ is the relative symmetric square ${\widetilde C} = S^2_{{\bf P}^1}X$ with respect to the 4 to 1 cover $X\rightarrow {\bf P}^1$ determined by $g^1_4$; and then $C = {\widetilde C} /\sigma$ where $\sigma $ is the obvious involution on $S^2_{{\bf P}^1}X$. We then have a commutative diagram \begin{equation} \label{recdiagram} \begin{array}{ccc} {\widetilde C} & \hookrightarrow & S^2 X\\ &&\\ \downarrow && \downarrow \hbox{Abel-Jacobi} \hidewidth\\ &&\\ P_{\eta}^- & \mathbin{\hbox{$\widetilde\rightarrow$}} & J^2(X).\\ \end{array} \end{equation} We shall be particularly interested in the Recillas correspondence for the case $g=3$. \section{Heisenberg invariant quartics} \label{invariantqs} Let $V= H^0(J,{\cal L}) = H^0({\cal SU}_C(2), {\cal L})$; so from section \ref{prelims} we have maps: $$ J\rightarrow {\cal SU}_C(2) \map{\phi} {\bf P} V^{\vee} = |2\Theta|; $$ and these identify with the maps given by the complete linear series $|{\cal L}|$ on $J$ and ${\cal SU}_C(2)$ respectively. Let us fix a theta structure for the line bundle ${\cal L}$; this allows us to view the vector space $V$ as an irreducible representation of the Heisenberg group $$ H_g = {\bf C}^* \times {\bf F}_2^g \times {\rm Hom}({\bf F}_2^g, {\bf C}^*), $$$$ (s,a,\chi)(t,b,\gamma) = (st\gamma(a), a+b,\chi\gamma). $$ This in turn gives us a canonical basis $\{X_{\sigma}\}_{\sigma \in {\bf F}_2^g}$ for $V$ (i.e. homogeneous coordinates on $|2\Theta| = {\bf P}^{2^g -1}$) such that: $$ (s,a, \chi) : X_{b} \mapsto s\chi(a + b) X_{a + b}, \qquad (s,a, \chi) \in H_g. $$ In particular our theta structure fixes an isomorphism $$ J[2] \cong {\bf F}_2^g \times {\rm Hom}({\bf F}_2^g, {\bf C}^*) = \widehat K \times K $$ where we denote by $K$ and $\widehat K$ the maximal level subgroups $$ \begin{array}{rcl} K = K_g &=& \{(1,0,\chi)| \chi \in {\rm Hom}({\bf F}_2^g, {\bf C}^*)\},\\ \widehat K = \widehat K_g &=& \{(1,a,1)| a \in {\bf F}_2^g\}.\\ \end{array} $$ The Heisenberg group acts also on the spaces of higher degree forms $S^nV$. This action is related to differentiation of polynomials by the following lemma, which is easy to check. \begin{lemm} \label{diffn} For $x=(s,a,\chi)\in H_g$ and $P\in S^n V$ we have: $$ {\partial \over \partial X_{b}}(x\cdot P)= \chi(b)\ x\cdot {\partial P\over \partial X_{a+b}}. $$ \end{lemm} We shall be concerned, in particular, with the spaces $S^3 V$ and $S^4 V$ of cubics and quartics on $|2\Theta|$ respectively, and in particular with the subspace $S^4_0 V \subset S^4 V$ of invariant quartics. This subspace has a basis consisting of: $$ \begin{array}{rclcl} Q_0 &=& \sum_{\sigma \in \widehat K} X_{\sigma}^4,&&\\ Q_{\lambda} &=& \sum_{\sigma \in \widehat K} X_{\sigma}^2 X_{\sigma+\lambda}^2&&\lambda\in {\bf P}(\widehat K),\\ Q_{\Lambda} &=& \sum_{\sigma \in \widehat K} X_{\sigma} X_{\sigma+\lambda} X_{\sigma+\mu} X_{\sigma+\nu}&& \Lambda = \{\lambda,\mu,\nu\}\in \grs{2}{\widehat K},\\ \end{array} $$ and in particular has dimension \begin{equation} \label{dimS40} \dim S^4_0 V = {1\over 3}(2^g +1)(2^{g-1}+1). \end{equation} \begin{prop} \label{partials} {\rm (\cite{vG}, proposition 2)} With respect to the homogeneous coordinates $\{X_{\sigma}\}_{\sigma \in \widehat K}$ we have: \begin{enumerate} \item $\displaystyle {\partial \over \partial X_0} : S^4_0 V \mathbin{\hbox{$\widetilde\rightarrow$}} (S^3 V )^{K}$ is an isomorphism; \item For $\sigma \in \widehat K$ and $Q \in S^4_0 V$ we have $\displaystyle{\partial Q\over \partial X_{\sigma}}= \sigma \cdot {\partial Q\over \partial X_0}$. \end{enumerate} \end{prop} The second part follows at once from lemma \ref{diffn}; the proposition says that an invariant quartic is determined by any of its partial derivatives with respect to the homogeneous coordinates; and these are all $K$-invariant and are permuted by the action of $\widehat K$. \begin{prop} \label{coble} Suppose that $M\subset |2\Theta|$ is a $J[2]$-invariant subvariety and that the restriction map $S^3V \rightarrow H^0(M,{\cal O}(3))$ has kernel of dimension $2^g$. Then there exists a unique invariant quartic $Q\in S^4_0 V$ with $M\subset {\rm Sing}\, Q$. \end{prop} {\it Proof.}\ Since the restriction map is $J[2] = K \times \widehat K$-equivariant, its $K$-invariant part is $$ {\rm ker \ }\{ (S^3 V)^K \rightarrow H^0(M,{\cal O}(3))^K\}, $$ which is therefore 1-dimensional, by \cite{M} proposition 3. By proposition \ref{partials}, therefore, we have a unique invariant quartic whose partial derivatives all vanish along $M$. {\unskip\nobreak\hfill\hbox{ $\Box$}\medskip\par} We shall apply this result to the images in $|2\Theta|$ of ${\cal SU}_C(2)$ and the Kummer variety. \begin{ex} $\bf g=2.$ \rm Here $\phi$ is an isomorphism ${\cal SU}_C(2) \mathbin{\hbox{$\widetilde\rightarrow$}} {\bf P}^3$; see \cite{NR1}. The Jacobian maps to the classical Kummer quartic surface, given by the 1-dimensional kernel of the surjective restriction map $S^4_0 V\rightarrow H^0_0(J,8\Theta)$. \end{ex} \begin{ex} \label{exg=3} $\bf g=3.$ \rm Here $\dim S^3 V = 120$, $\dim H^0_+(J, 6\Theta) = 112$ and for nonhyperelliptic $C$---equivalently for ppav $J_C$ without vanishing theta-nulls---the restriction map $S^3 V \rightarrow H^0_+(J, 6\Theta)$ is surjective with 8-dimensional kernel. So by proposition \ref{coble} there is a unique invariant quartic $Q_C\in S^4_0 V$ containing the Kummer variety in its singular locus---this is the original case of the proposition (see \cite{C} page 104). On the other hand, Narasimhan and Ramanan \cite{NR2} showed that if $C$ is nonhyperelliptic then $\phi:{\cal SU}_C(2)\rightarrow |2\Theta|$ is an embedding whose image is an invariant quartic; in particular this quartic is singular along the Kummer and therefore coincides with $Q_C$. It is called the {\it Coble quartic} of the curve. \end{ex} Alternatively, van Geemen and Previato \cite{vGP1} have shown that for curves without vanishing theta-nulls $\phi({\cal SU}_C(2))$ is projectively normal in degree 4; in particular there is a surjective restriction map \begin{equation} \label{quarticrestriction} S^4_0 V \rightarrow H^0_0({\cal SU}_C(2),{\cal L}^4) \end{equation} where $H^0_0({\cal SU}_C(2),{\cal L}^4)\subset H^0({\cal SU}_C(2),{\cal L}^4)$ again denotes the subspace of $J_2$-invariants, with dimension (see \cite{OP}) $ \dim H^0_0({\cal SU}_C(2),{\cal L}^4) = (3^g +1)/2 $. In the case $g=3$, therefore---where `no vanishing theta-nulls' means nonhyperelliptic---we see that $Q_C$ is the 1-dimensional kernel of~(\ref{quarticrestriction}). \section{Cubic normality for genus 4} \label{cubicnorm} We shall now apply proposition \ref{coble} in the case of genus 4. Our main result is: \begin{theo} \label{cubicnormality} For any curve $C$ of genus 4 without vanishing thetanulls the multiplication map $ S^3 H^0({\cal SU}_C(2),{\cal L}) \rightarrow H^0({\cal SU}_C(2), {\cal L}^3) $ is surjective. \end{theo} Using the Verlinde formula (see \cite{vGP1} or \cite{OW}) one observes that the respective dimensions of these spaces are: $$ \dim S^3 H^0({\cal SU}_C(2),{\cal L}) = 816, \qquad \dim H^0({\cal SU}_C(2), {\cal L}^3) = 800. $$ (Note that the number $816 = 2^4 \times 51$ also comes from (\ref{dimS40}) and proposition \ref{partials}.) So from proposition \ref{coble} we deduce: \begin{cor} \label{specialq} For any curve $C$ of genus 4 without vanishing theta-nulls there exists a unique invariant quartic $Q_C \subset {\bf P} V ^{\vee} \cong {\bf P}^{15}$ with the property that $\phi({\cal SU}_C(2)) \subset {\rm Sing}\, Q_C$. \end{cor} \begin{rems}\rm \par\indent \hangindent2\parindent \textindent{(i)} It is interesting to note what happens when the curve $C$ has a vanishing theta-null: assuming $C$ is nonhyperelliptic this vanishing theta-null is unique, i.e. $C$ has a unique semi-canonical pencil $g^1_3$, so that by \cite{B} proposition 2.6 the image $\phi({\cal SU}_C(2)) \subset |2\Theta| $ lies on a unique Heisenberg-invariant {\it quadric} $G$. In this case, therefore, the non-reduced quartic $Q_C = 2 G$ has the properties stated in the corollary; though we do not know that it is unique. \par\indent \hangindent2\parindent \textindent{(ii)} When $C$ has no vanishing theta-nulls the same result \cite{B} proposition 2.6 tells us that $\phi({\cal SU}_C(2)) \subset |2\Theta| $ does not lie on any quadric, and it follows easily from this that the quartic in corollary \ref{specialq} is irreducible. \end{rems} Before proving theorem \ref{cubicnormality} we shall need some notation. Fix a theta structure and maximal level subgroups $K_g,\widehat K_g$ as in section \ref{invariantqs}. A 2-torsion point $\eta \in K_g\subset J[2]$ acts linearly on $V$, and we shall denote by $V_{\eta}$ (resp. $V_{\eta}^-$) the $+1$- (resp. $-1$-)eigenspace. Since $K_g$ is an isotropic subgroup for the skew-symmetric Weil form on $J[2]$, and since the linear actions of two orthogonal 2-torsion points commute, the restriction map to the eigenspace $V_{\eta}$ maps $K_g$-invariant cubics to $K_g$-invariant cubics. Given any $\eta \in K_g$ we can choose an isomorphism $$ K_g/\<\eta \> \mathbin{\hbox{$\widetilde\rightarrow$}} K_{g-1}, $$ and hence obtain a linear map $$ {\rm res}_{\eta} : (S^3 V)^{K_g} \rightarrow (S^3 V_{\eta})^{K_{g-1}}. $$ Furthermore, by the general theory of Prym varieties, once we choose a theta structure on the associated Prym $P_{\eta}$ the space $V_{\eta}$ becomes an irreducible $H_{g-1}$-module to which we may apply proposition \ref{partials}. \begin{lemm} \label{injlem} For $g\geq 3$ the map obtained by restriction to all the eigenspaces $V_{\eta}$ is injective: $$ \sum {\rm res}_{\eta} : (S^3 V)^{K_g} \hookrightarrow \bigoplus_{\eta \in K_g\backslash\{0\}} (S^3 V_{\eta})^{K_{g-1}}. $$ \end{lemm} \begin{rem}\rm Although we will not need the fact, one may note that the restriction map of $K_g$-invariant cubics to the $-1$-eigenspace $V_{\eta}^-$ is the zero map. \end{rem} {\it Proof.}\ Let $\eta = (1,0,\chi) \in K_g\backslash\{0\}$. By definition, the eigenspace $V_{\eta}$ is spanned by the vectors $X_{\sigma}$ such that $\chi(\sigma) =1$. These $\sigma$s form a subgroup of $\widehat K_g$ which is isomorphic to $\widehat K_{g-1}$; and $\{X_{\sigma}|\chi(\sigma) =1\}$ is a canonical basis for the action of $H_{g-1}$ on $V_{\eta}$. This means that the elements $$ \begin{array}{rclcl} X_{0}^3&&\\ X_{0}^2 X_{0+\lambda}&&{\rm with}\ \chi(\lambda)=1,\\ X_{\lambda} X_{\mu} X_{\lambda+\mu}&&{\rm with}\ \chi(\lambda)=\chi(\mu)=1,\\ \end{array} $$ of $(S^3 V)^{K_{g}}$ map bijectively to a basis of $(S^3 V_{\eta})^{K_{g-1}}$. To prove the lemma it is therefore sufficient to observe that for any $\lambda,\mu \in \widehat K_g \backslash \{0\}$ there exists $\chi \in K_g \backslash \{0\}$ such that $\chi(\lambda) = \chi(\mu) = 1$. For $g\geq 3$ this is obvious. {\unskip\nobreak\hfill\hbox{ $\Box$}\medskip\par} {\it Proof of theorem \ref{cubicnormality}.} Choose a theta structure, so that the Heisenberg group $H_4$ acts on both spaces. Since the multiplication map is Heisenberg-equivariant it is enough to show surjectivity for $K$-invariant elements: $$ m^K: (S^3 V)^K \rightarrow H^0({\cal SU}_C(2), {\cal L}^3)^K. $$ We have already observed that these spaces have dimensions 51 and 50 respectively, and so we have to show that the kernel is 1-dimensional. Consider a cubic $F\in {\rm ker \ } m^K$, and restrict it to the eigenspace ${\bf P} V_{\eta}^{\vee}$. Since the intersection of this eigenspace with $\phi({\cal SU}_C(2))$ is the Kummer image of the Prym variety $P_{\eta}$ (see \cite{NR3}), the restricted cubic ${\rm res}_{\eta}(F)$ is an element of $$ U_{\eta} = {\rm ker \ } \{ (S^3 V_{\eta})^{K_3} \rightarrow H^0_+(P_{\eta}, 6\Xi)\}. $$ By hypothesis $J_C$ has no vanishing theta-nulls, and it follows from the Schottky-Jung identities that $P_{\eta}$ has no vanishing theta-nulls. By example \ref{exg=3}, therefore, $\dim U_{\eta} =1$. It is therefore sufficient to show that for any $\eta$ the map $$ {\rm res}_{\eta}|_{{\rm ker \ } m^K } : {\rm ker \ } m^K \rightarrow U_{\eta} $$ is injective. Suppose that $F\in {\rm ker \ } m^K \cap {\rm ker \ } {\rm res}_{\eta}$. Choose a nonzero $\zeta \in J[2]$ orthogonal to $\eta$ with respect to the Weil pairing; we shall show that $F\in {\rm ker \ } {\rm res}_{\zeta}$. Weil orthogonality implies that the intersection $V_{\eta,\zeta} = V_{\eta}\cap V_{\zeta}$ is 4-dimensional (see, for example, \cite{vGP1}). By hypothesis $F$ vanishes in $V_{\eta}$ and hence in $V_{\eta,\zeta}$. On the other hand, ${\rm res}_{\zeta}(F) \in U_{\zeta}$; if this element is nonzero then it spans $U_{\zeta}$, and in particular its $\widehat K_3$-orbit spans all cubics in ${\bf P} V_{\zeta}^{\vee}$ vanishing on ${\rm Kum}(P_{\zeta})$. We conclude that if ${\rm res}_{\zeta}(F) \not= 0$ then the singular locus of the Coble quartic of $P_{\zeta}$ contains ${\bf P} V_{\eta,\zeta}^{\vee} \cong {\bf P}^3$, a contradiction. By repeating this argument we deduce that if $F\in {\rm ker \ } m^K \cap {\rm ker \ } {\rm res}_{\eta}$ for {\it some} nonzero $\eta \in J[2]$ then the same is true for {\it any} $\eta \in J[2]$; by lemma \ref{injlem} this implies that $F=0$. {\unskip\nobreak\hfill\hbox{ $\Box$}\medskip\par} {\it Proof of theorem \ref{theorem2}(1).} Under the isomorphism of proposition \ref{partials}(1) the quartic $Q_C$ corresponds to a cubic $F$ whose restriction to each ${\bf P} V_{\eta}^{\vee}$ is nonzero, by the proof of theorem \ref{cubicnormality}. Therefore since $K_g /\<\eta\> \cong K_{g-1}$ we see that $Q_C$ restricts to a nonzero invariant quartic in ${\bf P} V_{\eta}^{\vee} \cong {\bf P}^7$ which is singular along ${\rm Kum} (P_{\eta}) \subset \phi({\cal SU}_C(2))$. By example \ref{exg=3} (and since $P_{\eta}$ has no vanishing theta-nulls) this restriction is just the Coble quartic. {\unskip\nobreak\hfill\hbox{ $\Box$}\medskip\par} \section{Lines on the Coble quartic} \label{cobleqs} \begin{prop} \label{seconcob} Let $X$ be a nonhyperelliptic curve of genus 3. For $a,b \in J(X)$ the secant line $\overline{ab} \subset {\bf P}^7$ of the Kummer variety lies on the Coble quartic ${\cal SU}_X(2)\subset {\bf P}^7$ if and only if $a+b \in X-X$ or $a-b \in X-X$. \end{prop} {\it Proof.}\ We make use of the ruling of ${\cal SU}_X(2)\subset {\bf P}^7$ by 3-planes (see \cite{OPP} or \cite{NR2}). First of all, if $a-b \in X-X$ then $a(p) = b(q) = x\in J^1(X)$ for some points $p,q,\in X$, and then the line $\overline{ab} \subset {\bf P}^7$ is precisely the secant line $\overline{pq}$ of the image of the curve in the extension space ${\bf P}(x) \subset {\cal SU}_X(2)$ (see \cite{OPP} for notation). If $a+b \in X-X$ the argument is similar. So now suppose that $\overline{ab} \subset {\cal SU}_X(2)$. If $\overline{ab} \subset {\bf P}(x)$ for some $x\in J^1(X)$, then (since ${\bf P}(x)$ meets the Kummer in the image of $X\hookrightarrow {\bf P}(x)$) $\overline{ab}$ is a secant line of the image curve, and one infers easily that $a+b \in X-X$ or $a-b \in X-X$. We may therefore suppose that $\overline{ab}$ does not lie on any such 3-plane ${\bf P}(x)$. Pick any stable bundle $E$ on the line $\overline{ab}$; then (since $g(X) = 3$) $E \in {\bf P}(x)$ for some $x\in J^1(X)$. Under the embedding $\phi: {\cal SU}_X(2) \hookrightarrow |2\Theta|$ the linear subspace ${\bf P}(x)$ is the linear system of divisors $D\in |2\Theta|$ containing the curve $x+ W_1 \cong X \subset J^2(X)$. We shall view the line $\overline{ab}$ as the pencil of $2\Theta$-divisors spanned by $\Theta_a + \Theta_{-a}$ and $\Theta_b + \Theta_{-b}$, and restrict it to $x+W_1$. Since there is a member of this pencil vanishing identically on the curve we see that there is an equality of effective divisors: \begin{equation} \label{divisor} (\Theta_a + \Theta_{-a})|_{x+W_1} = (\Theta_b + \Theta_{-b})|_{x+W_1}. \end{equation} We note that $\Theta_a$ restricts to the line bundle $Kx^{-1}a$ on $x+W_1 \cong X$, {\it and moreover that $h^0(X,Kx^{-1}a) =1$}. To see this, observe that since $\deg Kx^{-1}a = 3$, $h^0(X,Kx^{-1}a) >1$ only if $Kx^{-1}a = K(-p) $ for some point $p\in X$. So $x = a(p)$ and we deduce that $x+W_1 \subset \Theta_a$. But then the Kummer image of $a$, i.e. the divisor $\Theta_a + \Theta_{-a}$, lies in the space ${\bf P}(x)$ and therefore so does the line $\overline{ab}$, contrary to hypothesis. We may thus write (\ref{divisor}) in the form: $$ p_1+p_2+p_3 + q_1+q_2+q_3 = r_1+r_2+r_3 + s_1+s_2+s_3 $$ where $\{p_1+p_2+p_3\} = |Kx^{-1}a|$, $\{q_1+q_2+q_3\}= |Kx^{-1}a^{-1}|$ etc. If $p_1+p_2+p_3 = r_1+r_2+r_3$ or $s_1+s_2+s_3$ then we deduce that $a=\pm b$. Otherwise, we can find an equation of the form $$ p_1+p_2+p_3 = r_1+r_2+ s_i $$ for some $i=1,2,3$. But this means that $$ a-b = (p_1+p_2+p_3 ) - (r_1+r_2+ r_3) = s_i - r_3 \in X-X. $$ Similarly, each such equation leads either to $a-b\in X-X$ or to $a+b\in X-X$. {\unskip\nobreak\hfill\hbox{ $\Box$}\medskip\par} \begin{cor} The secant lines of a 3-dimensional Kummer variety in ${\bf P}^7$ cover ${\bf P}^7$. \end{cor} {\it Proof.}\ The secant variety of the Kummer is irreducible and contains ${\cal SU}_X(2)$, since the latter is swept out by 3-planes meeting the Kummer in a curve. But the preceding proposition shows that the inclusion is proper; since ${\cal SU}_X(2) \subset {\bf P}^7$ has codimension one, therefore, the result follows. {\unskip\nobreak\hfill\hbox{ $\Box$}\medskip\par} \section{Quadrisecant planes} \label{quadrisecants} In this section we shall recall the result of Beauville--Debarre \cite{BD} which says that the Kummer variety of a Prym: $$ {\rm Kum} : P_{\eta} = {\rm Prym}(C,\eta) \rightarrow |2 \Xi | = {\bf P}^{2^{g} -1} $$ (where we shall take $C$ to have genus $g+1$) possesses a 4-parameter family of quadrisecant ${\bf P}^2$s, analogous to the trisecant lines of a Jacobian Kummer. The base ${\cal B}$ of the family is the fibre product: \begin{equation} \label{BDfamily} \begin{array}{rcl} {\cal B} &\rightarrow & S^4 {\widetilde C}\\ &&\\ \downarrow &&\downarrow \hbox{Abel-Prym}\\ &&\\ P_{\eta}^- & \map{sq} & P_{\eta}\\ \end{array} \end{equation} where $sq$ is the squaring map $a\mapsto a^{2}$. An element of ${\cal B}$, in other words, is a pair $(a,\Gamma) \in P_{\eta}^-\times S^4 {\widetilde C}$ such that $a^{2} = [\Gamma]$ (see section \ref{pryms}). For any points $p,\ldots,q\in {\widetilde C}$ and $a\in P_{\eta}^-$, let us write: $$ \begin{array}{rcl} \<p\>_a &=& {\rm Kum}([p] -a)\in |2 \Xi | = {\bf P}^{2^{g} -1},\\ \<p,\ldots ,q\>_a &=& \hbox{linear span of $\<p\>_a,\ldots ,\<q\>_a$}.\\ \end{array} $$ The following fact then results from \cite{BD} propositions 1 and 2. \begin{prop} \label{linepairs} For each $(a,p+q+r+s) \in {\cal B}$, the four points $ \<p\>_a,\<q\>_a,\<r\>_a,\<s\>_a \in {\rm Kum}\ P_{\eta} \subset {\bf P}^{2^{g} -1} $ are coplanar. \end{prop} We now make the following observation which relates this family to the parameter space ${\cal F}$ of Fay trisecants (see \cite{OPP}, section 2). First, ${\cal F} \subset J^2(C)\times S^4 C$ consists of pairs $(\lambda, D)$ such that $\lambda ^2 = {\cal O}(D) \in J^4(C)$; and we consider the 16:1 cover $$ \begin{array}{rcc} {\cal G} \cup {\cal G}^- = \widetilde{\cal F} &\subset & J^2(C) \times S^4 {\widetilde C}\\ &&\\ 16:1 \downarrow &&\downarrow {\rm id}\times {\rm Nm}_{\pi}\\ &&\\ {\cal F} & \subset& J^2(C) \times S^4 C.\\ \end{array} $$ There is a map $\widetilde{\cal F} \rightarrow P_{\eta} \cup P_{\eta}^-$ defined by $(\lambda,\Gamma) \mapsto \pi^*\lambda^{-1} \otimes {\cal O}(\Gamma)$, and replacing $\Gamma = p+q+r+s$ by $\sigma(p) +q+r+s$, i.e. transposing one point of $\Gamma$ by sheet-interchange over $C$, switches the image from $P_{\eta}$ to $P_{\eta}^-$ and vice versa. This shows that $\widetilde{\cal F}$ has two connected components, and accordingly we have written $\widetilde{\cal F} = {\cal G} \cup {\cal G}^-$. Thus, by definition $$ {\cal G}^- = \{ (\lambda ,\Gamma) \in J^2(C) \times S^4 {\widetilde C} \,|\, \lambda^2 = {\rm Nm}_{\pi} \Gamma;\ \pi^*\lambda^{-1} \otimes {\cal O}(\Gamma)\in P_{\eta}^-\}. $$ If we write $\Gamma = p+q+r+s$ again then the sixteen points of $P_{\eta} \cup P_{\eta}^-$ coming from the fibre of ${\cal G} \cup {\cal G}^-$ over $(\lambda , {\rm Nm}_{\pi} \Gamma) \in {\cal F}$ are (modulo $\sigma$): \begin{equation} \label{FBDtable} \begin{array}{|l|l|} \hline &\\ \qquad P_{\eta}^-&\qquad P_{\eta}\\ \hline &\\ \pi^* \lambda^{-1} (p+q+r+s)=a & \pi^* \lambda^{-1}(\sigma p+q+r+s) = [p] -a\\ \pi^* \lambda^{-1} (\sigma p+\sigma q+r+s) & \pi^* \lambda^{-1}( p+\sigma q+r+s) = [q] -a\\ \pi^* \lambda^{-1} (\sigma p+q+\sigma r+s) & \pi^* \lambda^{-1}( p+q+\sigma r+s) = [r] -a\\ \pi^* \lambda^{-1} (\sigma p+q+r+\sigma s)& \pi^* \lambda^{-1}( p+q+r+\sigma s) = [s] -a\\ &\\ \hline \end{array} \end{equation} Note that we define $a = \pi^* \lambda^{-1} (\Gamma)$ here, and that $-a = \sigma a = \pi^* \lambda^{-1} (\sigma \Gamma)$. Observe also that $a^2 = [\Gamma]$ so that we have a map $g: {\cal G}^- \rightarrow {\cal B}$ sending $(\lambda, \Gamma) \mapsto (a ,\Gamma)$. In addition, we have a map $$ \begin{array}{rcl} f: {\cal F} &\rightarrow & S^4 {\rm Kum}(P_{\eta}) \\ x & \mapsto & \{\hbox{fibre of ${\cal G}$ over $x$}\}/\sigma.\\ \end{array} $$ In other words, $f$ maps $x = (\lambda , {\rm Nm}_{\pi} \Gamma)$ to the four points in the right-hand column of (\ref{FBDtable}). In conclusion, the identifications in the right-hand column of the table show that the following diagram commutes: \begin{equation} \label{FBDdiagram} \begin{array}{rcl} {\cal G}^- & \map{g} & {\cal B}\\ &&\\ 8:1 \downarrow &&\downarrow {\hbox{Beauville-Debarre}\atop\hbox{quadrisecant planes}}\\ &&\\ {\cal F} & \map{f}& S^4 {\rm Kum}(P_{\eta}) \hookrightarrow {\rm Gr}_3 H^0(2\Xi). \end{array} \end{equation} \begin{rems}\rm \par\indent \hangindent2\parindent \textindent{(i)} It is easy to see that ${\cal G}^-$ is \'etale on both ${\cal F}$ and ${\cal B}$ (with degree 2 in the latter case); in particular both spaces have the same image in $S^4 {\rm Kum}(P_{\eta})$. \par\indent \hangindent2\parindent \textindent{(ii)} We shall make use of diagram (\ref{FBDdiagram}) in section \ref{prymG3}. The fact that ${\cal F}$ parametrises both the trisecants of the Jacobian Kummer and the quadrisecants of the Prym Kummers is rather striking, but will not play any role in this paper. \end{rems} We now choose a $g^1_3$ on $C$. The Recillas correspondence then determines a tetragonal curve $(X,g^1_4)$ and identifies $J(X) \cong P_{\eta}(C)$. In particular, we can now view ${\widetilde C} \subset S^2 X$ as in diagram (\ref{recdiagram}); we shall denote this inclusion map by $p\mapsto p_1 + p_2$ for $p\in {\widetilde C}$. Restricting to the case $g=3$, we shall view $X\subset {\bf P}^2$ in its canonical embedding, and consider lines $\overline{p} = \overline{p_1 p_2} \subset {\bf P}^2$, for each $p\in {\widetilde C}$. \begin{prop} Let $g=3$ and ${\cal SU}_X(2) \subset {\bf P}^7$ be the Coble quartic with singular locus ${\rm Kum}\ P_{\eta}(C)$. Then for $(a, p+q+r+s) \in {\cal B}$ the following are equivalent: \begin{enumerate} \item $\<p,q\>_a \subset {\cal SU}_X(2)$; \item $\<r,s\>_a \subset {\cal SU}_X(2)$; \item $\overline{p}\cap \overline{q} \in X\subset {\bf P}^2$ or $\overline{r}\cap \overline{s} \in X\subset {\bf P}^2$. \end{enumerate} \end{prop} {\it Proof.}\ We consider the line $\<p,q\>_a$. By proposition \ref{seconcob} this lies on ${\cal SU}_X(2)$ if and only if the sum $[p]+[q]-2a$ or the difference $[p]-[q]$ of the points $[p]-a, [q]-a \in P_{\eta}(C) = J(X)$ lies on the surface $X-X$. In view of the diagram (\ref{recdiagram}) we have \begin{equation} \label{one} [p]-[q] = p_1 +p_2 - q_1-q_2 \in J(X); \end{equation} whilst by definition $2a = [p+q+r+s] \in P_{\eta}$ is identified with the point $p_1+p_2 + \cdots + s_1+s_2 - 2K_X \in J(X)$, so that \begin{equation} \label{two} [p]+[q] - 2a = K_X -r_1-r_2 -s_1-s_2 \in J(X). \end{equation} By (\ref{one}), $[p]-[q]\in X-X$ if and only if $$ p_1+p_2 +v \sim q_1+q_2+ u $$ for some $u,v \in X$. If these divisors are equal then $p_i = q_j$ for some $i,j\in \{1,2\}$; in which case $\overline{p} \cap \overline{q}$ lies on $X\subset {\bf P}^2$ trivially. Otherwise the two divisors span a $g^1_3 = |K_X(-s)|$ for some $s\in X$; and so in this case $\overline{p} \cap \overline{q}$ is the point $s\in X$. Conversely, if $\overline{p} \cap \overline{q}\in X$ then the same argument shows that $[p]-[q] \in X-X$. By (\ref{two}), on the other hand, $[p]+[q] -2a \in X-X$ if and only if $$ r'_1+r'_2 +v \sim s_1+s_2+ u $$ for $u,v \in X$, where $r_1+r_2+r'_1+r'_2 \in |K_X|$. And now, since $\overline{r'_1 r'_2} = \overline{r_1 r_2} = \overline{r}$, the same argument as before shows that $[p]+[q] -2a \in X-X$ if and only if $\overline{r} \cap \overline{s}\in X$. We conclude that parts 1 and 3 are equivalent to each other and hence also to part 2. {\unskip\nobreak\hfill\hbox{ $\Box$}\medskip\par} From the preceding proposition we can deduce necessary and sufficient conditions for the quadrisecant plane $\<p,q,r,s\>_a \subset {\bf P}^7$ to lie on ${\cal SU}_X(2)$: since ${\cal SU}_X(2)\subset {\bf P}^7$ has degree 4 this is equivalent to ${\cal SU}_X(2)$ containing all of the six lines $\<p,q\>_a, \ldots, \<r,s\>_a$. Let $$ \Delta(p) = \overline{q} \cup \overline{r} \cup\overline{s}\subset {\bf P}^2, $$ with $\Delta(q),\Delta(r),\Delta(s)$ defined similarly. Then from proposition \ref{linepairs} we see that ${\cal SU}_X(2)$ contains the six lines if and only if {\it either} one of the triangles $\Delta(p),\ldots , \Delta(s)$ has its vertices on $X\subset {\bf P}^2$ {\it or} one of the lines, $\overline{p}$ say, meets the remaining three in points of $X\subset {\bf P}^2$. {\it To summarise:} $\<p,q,r,s\>_a \subset {\cal SU}_X(2)$ if and only if the four lines $\overline{p},\overline{q},\overline{r},\overline{s} \subset {\bf P}^2$ form a configuration with respect to the canonical curve of one of the two forms below: $$ \epsfxsize=1.5in\epsfbox{triangle.eps} \qquad\qquad\qquad \epsfxsize=2in\epsfbox{line.eps} $$ \begin{cor} The generic quadrisecant plane $\<p,q,r,s\>_a$ is not contained in ${\cal SU}_X(2)$. \end{cor} {\it Proof.}\ By the Recillas construction $X$ comes equipped with a $g^1_4$, and each of the divisors $p_1 +p_2, \ldots , s_1 + s_2$ is contained in a divisor of this pencil. If the $g^1_4$ is canonical then it is obtained by projection of $X$ away from a point of the plane off the curve; in this case we see that the configurations above can never occur, i.e. that {\it no} quadrisecant plane $\<p,q,r,s\>_a$ is contained in ${\cal SU}_X(2)$. More generally, we can count parameters. There can be at most a 2-parameter family of triangles on the left: one parameter for any side, and one for the opposite vertex. There is a 1-parameter choice of the remaining line; and hence at most a 3-dimensional family of quadrisecants can give rise to the configuration on the left. There is also at most a 3-dimensional family of quadrisecants giving rise to the right-hand configuration. {\unskip\nobreak\hfill\hbox{ $\Box$}\medskip\par} \section{Abel-Jacobi stratification of $|2\Theta|$} \label{abeljacobi} Let $W_d \subset J^d$ be the subscheme of special line bundles of degree $d$, $0\leq d \leq g-1$, i.e. the Abel-Jacobi image of $S^d C$. We introduce the following sets of divisors on the Jacobian: \begin{equation} \label{ajsets} \begin{array}{rcl} G_{d+1} &=& \{\ D\ | \ x+W_{g-1-d} \subset {\rm supp}\ D\ \hbox{for some $x\in J^d$}\ \} \subset |2\Theta|,\\ &&\\ G_{g-d}^{\vee} &=& \{\ D\ | \ -x+W_d \subset {\rm supp}\ D\ \hbox{for some $x\in J^d$}\ \} \subset |{\cal L}|.\\ \end{array} \end{equation} It will often be convenient to identify $|{\cal L}| \mathbin{\hbox{$\widetilde\rightarrow$}} |2\Theta|$ via a choice of theta characteristic as in diagram (\ref{thetachoice}) of section \ref{prelims}. It is easy to see that this induces identifications $G_i^{\vee} \mathbin{\hbox{$\widetilde\rightarrow$}} G_i$ for each $i=1,\ldots, g$; note also that this is unaffected by the choice of theta characteristic as all the sets (\ref{ajsets}) are $J[2]$-invariant. Obviously $G_g = |2\Theta|$ and $G_0$ is empty, whilst by the theorem of the square $G_1 = {\rm Kum}(J)$ where ${\rm Kum}$ is the Kummer map $$ \begin{array}{rcl} {\rm Kum}:J &\rightarrow & |2\Theta|\\ x &\mapsto & \Theta_x + \Theta_{-x}.\\ \end{array} $$ We thus have (isomorphic) filtrations: \begin{equation} \label{AJstrat} \begin{array}{rcl} {\rm Kum}(J) &= G_1 \subset G_2 \subset \cdots \subset G_{g-1}& \subset |2\Theta|,\\ &&\\ {\rm Kum}(J^{g-1}) &= G_1^{\vee} \subset G_2^{\vee} \subset \cdots \subset G_{g-1}^{\vee}& \subset |{\cal L}| = |2\Theta|^{\vee}.\\ \end{array} \end{equation} To interpret these sets scheme-theoretically we consider maps $$ \begin{array}{ccc} J^d \times S^d C&\map{\alpha_d}&J\\ &&\\ \downarrow \pi&&\\ &&\\ J^d&&\\ \end{array} $$ where $\alpha_d: (x,D) \mapsto x^{-1} \otimes {\cal O} (D)$ and $\pi$ is projection to the first factor; and set $Q_d = \pi_* \alpha_d^* {\cal L}.$ In the notation of the appendix and proposition \ref{alphabeta}, $$ Q_d(x) = H^0(S^d C, {\cal L}_x) = H^0({\bf P} (x), {\cal I}_C^{d-1}(d)) $$ at each $x\in J^d$. {\it In the remainder of this section we shall assume that $g\leq 4$.} More generally (see proposition \ref{conjOK}) we shall assume the validity of conjecture \ref{400}.) It then follows that $Q_d$ is locally free and the restriction map $\alpha_d^*:{\cal O}_{J^d} \otimes H^0(J,{\cal L})\rightarrow Q_d$ is surjective. We therefore obtain dual short exact sequences of vector bundles on $J^d$ (in which $N_d = {\rm ker \ } \alpha_d^*$ by definition): \begin{equation} \label{NQsequence} \begin{array}{rcl} 0\rightarrow N_d \rightarrow &{\cal O}_{J^d} \otimes H^0(J,{\cal L})& \map{\displaystyle \alpha_d^*} Q_d \rightarrow 0\\ &&\\ 0\rightarrow Q_d^{\vee} \rightarrow &{\cal O}_{J^d} \otimes H^0(J^{g-1},2\Theta)& \rightarrow N_d^{\vee} \rightarrow 0.\\ \end{array} \end{equation} {\it We define $G_{g-d}^{\vee}$ to be the ruled variety ${\rm im} \{\, {\bf P} N_d \subset J^d \times |{\cal L}| \rightarrow |{\cal L}|\, \}$, and $G_{d+1}\subset |2\Theta|$ to be the ruled variety ${\rm im} \{\, {\bf P} Q_d^{\vee}\subset J^d \times |2\Theta|\rightarrow |2\Theta|\, \}$.} Thus, for $x\in J^d$ the subset ${\bf P} N_d(x)\subset |{\cal L}|$ consists of divisors containing $-x+W_d$. In a moment we shall verify the less obvious fact that \begin{equation} \label{Qdxdual} {\bf P} Q_d (x)^{\vee} = \{\ D\in |2\Theta|\ | \ x+W_{g-1-d} \subset {\rm supp}\ D\ \ \} \end{equation} so that this definition is compatible with (\ref{ajsets}). \begin{rem}\rm \label{dualdims} By \ref{400}(1) we have ${\rm rank \ } Q_d = \sum_{i=0}^d {g\choose i}$. It follows from this and the fact that $h^0(J,{\cal L}) = 2^g$ that $$ {\rm rank \ } N_d = \sum_{i=d+1}^g {g\choose i} = {\rm rank \ } Q_{g-1-d}. $$ Note in particular that $\dim {\bf P} N_1 = \dim {\bf P} Q_{g-2}^{\vee} = 2^g -2$, and we can therefore expect that $G_{g-1}\subset |2\Theta|$ is a hypersurface. We shall be particularly interested in this case in what follows. \end{rem} Recall also that a choice of $\kappa \in \vartheta (C)$ gives rise to an isomorphism $|2\Theta| \cong |2\Theta|^{\vee} = |{\cal L}|$ (see section \ref{prelims}). Given this choice we can prove the following polarity relation: \begin{prop} \label{perp} Fix $x\in J^d$ and theta characteristic $\kappa \in \vartheta (C)$. Then the induced isomorphism $H^0(J^{g-1}, 2\Theta) \mathbin{\hbox{$\widetilde\rightarrow$}} H^0(J,{\cal L}) =H^0(J^{g-1}, 2\Theta) ^{\vee} $ (up to scalar) restricts to $$ Q_d(x) ^{\vee}\mathbin{\hbox{$\widetilde\rightarrow$}} N_{g-1-d}(\kappa x^{-1}) = N_d(x)^{\perp}. $$ \end{prop} {\it Proof.}\ By definition, the annihilator of ${\bf P} N_d(x)\subset |{\cal L}|$ in $|{\cal L}|^{\vee} = |2\Theta|$ is the linear span of $ {\rm Kum}(W_d - x)\subset |2\Theta|. $ Likewise ${\bf P} N_{g-1-d}(\kappa x^{-1})$ has annihilator spanned by $ {\rm Kum}(W_{g-1-d} -\kappa +x) \subset |2\Theta|. $ Now, under the map $m:J\times J \rightarrow J\times J$, $(u,v)\mapsto (u+v,u-v)$ of section \ref{prelims} we note that $$ (W_d - x)\times (W_{g-1-d} -\kappa +x) \map{m} \Theta_{\kappa} \times J $$ and hence by (\ref{riemannxi}) the image is killed by the form $\xi_{\kappa}$ defining the isomorphism $|{\cal L}| \cong |{\cal L}|^{\vee}$. It follows that $$ N_d(x)\perp N_{g-1-d}(\kappa x^{-1}) $$ under this pairing. But from remark \ref{dualdims} we see that $$ \dim N_d(x) + \dim N_{g-1-d}(\kappa x^{-1}) = 2^g, $$ and so $N_{g-1-d}(\kappa x^{-1}) = N_d(x)^{\perp}$. On the other hand, let $V=H^0(J,{\cal L})$ as usual. Since $N_d = N_{g-1-d}^{\perp}$ there is an isomorphism $Q_d = ({\cal O}_J\otimes V) /N_d \mathbin{\hbox{$\widetilde\rightarrow$}} ({\cal O}_J\otimes V) /N_{g-1-d}^{\perp} = N_{g-1-d}^{\vee}$. {\unskip\nobreak\hfill\hbox{ $\Box$}\medskip\par} In the following corollary, let $V_{\eta}^{\pm}\subset V$ be the eigenspaces of the action of $\eta \in J[2]$ on $V$ (or more precisely a lift of $\eta$ to the Heisenberg group); and note that under the pairing induced by any theta characteristic as above, $V_{\eta}^+ \perp V_{\eta}^-$. It is then easy to check the following fact using remark \ref{dualdims} and proposition \ref{perp}. \begin{cor} \label{intdims} For each $x\in J^d$ and $\kappa \in \vartheta(C)$, $\eta\in J[2]\backslash \{0\}$, we have $$ \dim(N_d(x) \cap V_{\eta}^+) - \dim(N_{g-1-d}(\kappa x^{-1}) \cap V_{\eta}^-) = 2^{g-1} - \sum_{i=0}^d {g\choose i}. $$ \end{cor} The first step in the direction of calculating the degrees of the varieties $G_{d+1} \subset |2\Theta|$ is the observation that the self-intersection of the hyperplane class in ${\bf P} N_d$ is the top Segre class of the bundle, and that in view of the exact sequence (\ref{NQsequence}) (since the central term is trivial) this in turn is the top Chern class of $Q_d$: $$ {\cal O}_{{\bf P} N_d}(1)^{\dim {\bf P} N_d} = s_g(N_d) = c_g(Q_d) \in H^{2g}(J^d,{\bf Z}) \cong {\bf Z}. $$ In other words, we have \begin{equation} \deg G_{g-d}\ \times \ \Bigl( {\hbox{no. of translates of $W_d$} \atop \hbox{in a general $D\in G_{g-d}$}} \Bigr) = c_g(Q_d) \in {\bf Z}. \end{equation} We shall restrict attention to the case $d=1$; the Chern classes of $Q_1$ are easily computed by comparison with the direct image sheaf $$ Q' = \pi_* ({\cal P}^2 \otimes pr_C^*K) $$ where ${\cal P} \rightarrow J^1 \times C$ is a Poincar\'e bundle. By proposition \ref{100}(1) we have \begin{equation} \label{QNQ'} Q_1 = {\cal N} \otimes Q' \end{equation} for some line bundle ${\cal N} \in {\rm Pic}(J^1)$. To determine ${\cal N}$, note that if we fix a base-point $p_0 \in C$ the Poincar\'e bundle ${\cal P}$ is uniquely determined by requiring that ${\cal P}|_{J^1 \times \{p_0\}}$ be trivial. This will be satisfied, with \begin{equation} \label{computingN1} {\cal P}^2 \otimes pr_C^*K = \pi^*{\cal N}^{-1} \otimes \alpha_1 ^* {\cal L}, \end{equation} if ${\cal N}$ is chosen so that $\pi^*{\cal N}^{-1} \otimes \alpha_1 ^* {\cal L}$ is trivial on $J^1 \times \{p_0\}$, i.e. if we let \begin{equation} \label{computingN2} {\cal N} = \alpha_{p_0}^* {\cal L} = t_{-p_0}^* {\cal L} \end{equation} where $t_{-p_0} : J^1 \rightarrow J$ denotes translation by $p_0$. Applying $\pi_*$ to both sides of (\ref{computingN1}) we obtain (\ref{QNQ'}). The Chern character of $Q'$ can be computed by Grothendieck-Riemann-Roch (see \cite{ACGH} chapter VIII) and is $g+1 - 4\theta$, where $\theta$ denotes the fundamental class of $\Theta$ in the Jacobian. We therefore conclude from (\ref{QNQ'}) and (\ref{computingN2}) that \begin{equation} \label{chernQ} ch(Q_1) = (g+1-4\theta) e^{2\theta}. \end{equation} Expanding the exponential and applying Newton's formula (\cite{F} page 56) we deduce: \begin{prop} The Chern classes $c_n = c_n(Q_1)$ are given recursively by $$ c_n ={1\over n} (c_{n-1}p_1 - c_{n-2} p_2 + \cdots + (-1)^{n-2} c_1 p_{n-1} + (-1)^{n-1}p_n) $$ where $p_n = (g+1 - 2n) (2\theta)^n$. \end{prop} Although it is not easy to obtain a closed formula for $c_g(Q_1)$ it is readily calculated on a computer, using the above proposition, and the first few values are: $$ \begin{array}{r|l} g & c_g(Q_1) \\ &\\ \hline &\\ 3&32\\ 4&384\\ 5&4096\\ 6&56320\\ 7&872448\\ 8&15368192\\ \end{array} $$ We conjecture that at least for $g=4$, $\deg G_{g-1} = 4$. One might hope to use theorem \ref{theorem2}(2) to verify this. \section{Segre stratification of ${\cal SU}_C(2)$} \label{segrestrat} We shall show in this section that the Abel-Jacobi stratification (\ref{AJstrat}) induces, via $\phi$, the {\it Segre stratification} of ${\cal SU}_C(2)$, i.e. the stratification by maximal degree of a line subbundle. (See \cite{LN}.) For any vector bundle $E$ on $C$ of rank 2 and degree 0 let $$ n(E) = {\rm min}\{\, n\in {\bf Z}\, | \,\hbox{$\exists$ line subbundle $L\subset E$ with $\deg L = -n$}\,\}. $$ This function is nonnegative on semistable bundles; it is also lower semicontinuous on families and determines a filtration of moduli space by closed subvarieties: \begin{equation} \label{Segrestrat} J/{\pm} \cong {\cal SU}_C(2)_0 \subset {\cal SU}_C(2)_1 \subset \cdots \subset {\cal SU}_C(2)_{[g/2]} = {\cal SU}_C(2) \end{equation} where ${\cal SU}_C(2)_d = \{E\in {\cal SU}_C(2)|n(E) \leq d\}$. The right-hand equality in (\ref{Segrestrat}) follows from Nagata's theorem \cite{N}, \cite{L}: every ruled surface of genus $g$ admits a section with self-intersection at most $g$. \begin{prop} \label{strattheorem} For $E\in {\cal SU}_C(2)$, $n(E)\leq d$ if and only if $\phi(E) \in G_{d+1}$. In other words, ${\cal SU}_C(2)_d = \phi^{-1} G_{d+1}$ for $0\leq d \leq [g/2]$. \end{prop} By definition $n(E)\leq d$ if and only if there exists $\xi \in J^d$ such that $h^0(C,\xi \otimes E)>0$; and that this is equivalent to $\phi(E) \in G_{d+1}$ can be restated in the following form, proved in \cite{OPP}, section 4. For any $E\in {\cal SU}_C(2)$ and $\xi \in J^d$: \begin{equation} \label{killingD} h^0(C,\xi \otimes E)>0 \quad \Longleftrightarrow \quad {h^0(C,\xi(D) \otimes E)>0\atop \forall\,\, D\in S^{g-1-d}C.} \end{equation} Note that the case $d=0$ of proposition \ref{strattheorem} just says that ${\rm Kum}(J)\subset |2\Theta|$ comes from the semistable boundary of ${\cal SU}_C(2)$; at the other end we see that $$ \phi({\cal SU}_C(2)) \subset G_{[g/2]+1}. $$ \begin{rems}\rm \par\indent \hangindent2\parindent \textindent{(i)} In a moment (see corollary \ref{G2}) we shall show that $G_2 \subset \phi({\cal SU}_C(2))$. By remark \ref{dualdims} $G_2$ is ruled by ${\bf P}^g$s; this is the `$g$-plane ruling' of \cite{OPP} \S1. \par\indent \hangindent2\parindent \textindent{(ii)} For $g=2$ we know $\phi({\cal SU}_C(2)) = G_2 = |2\Theta|$; while for each $g\geq 3$ we have $\phi({\cal SU}_C(2)) \subset G_{g-1}$. In the case $g=3$ equality holds and $G_2\subset {\bf P}^7$ is the Coble quartic of the curve. \end{rems} By Serre duality the projective space ${\bf P} (x) := |Kx^2|^{\vee}$, for $x\in J^d$, can be identified with the space ${\bf P} H^1(C, x^{-2})$ of nontrivial extensions $$ \ses{x^{-1}}{E}{x} $$ up to isomorphism of $E$. It therefore has a rational map to ${\cal SU}_C(2)$ (we shall see in a moment that the generic extension is semistable) which has been described in detail by Bertram, Lange--Narasimhan and others \cite{Bert}, \cite{LN}. \medskip\noindent {\it Notation:} For each $x\in J^d$ we shall denote by $\varepsilon : {\bf P} (x) \rightarrow {\cal SU}_C(2)$ the rational moduli map, and write $E=\varepsilon(e)$ for the bundle corresponding to a point $e\in {\bf P} (x)$. We shall denote by ${\rm Sec}^n C \subset {\bf P}(x)$ the variety of $n$-secants $\overline D \cong {\bf P} ^{n-1}$, for $D\in S^n C$, of $\lambda_{|K x^2|}:C\hookrightarrow {\bf P}(x)$. Finally, in the lemma below we shall consider the blow-up of ${\bf P}(x)$ along these secant varieties: we shall denote by ${\widetilde \secant}^n C$ the proper transform of any ${\rm Sec}^n C$ in a blow-up with lower dimensional centre, and by ${\cal S}_n$ the exceptional divisor of the blow-up along ${\rm Sec}^n C$ or ${\widetilde \secant}^n C$. \medskip It is easy to show (see for example \cite{LN} proposition 1.1) that \begin{equation} \label{dbar} e\in \overline D \quad \Longleftrightarrow \quad x(-D) \subset E; \end{equation} and in particular that for all $|n| \leq d-1$ \begin{equation} \label{seccondition} n(E)\leq n \quad \Longleftrightarrow \quad e\in {\rm Sec}^{d+n}C \end{equation} Note that the requirement $n\leq d-1$ arises here because $n(E) \leq d$ for all extension classes in ${\bf P} (x)$ since $x^{-1}$ is always a line subbundle. Applying (\ref{dbar}) when $\deg D =d$ says that ${\rm Sec}^d C$ maps---away from ${\rm Sec}^{d-1} C$---to the semistable boundary $J/\pm$, each $d$-secant plane $\overline D$ contracting down to the S-equivalence class of $E= x(-D) \oplus x^{-1}(D)$. Applying (\ref{dbar}) when $\deg D <d$---and note that by (\ref{seccondition}) ${\rm Sec}^{d-1}C$ is the exceptional locus of the rational map $\varepsilon$---we see that in this case $\overline D$ transforms to the image $\varepsilon {\bf P} (x(-D))$. The picture is clarified by the following results of Bertram \cite{Bert}. \begin{lemm} \label{bertram} For $x\in J^d$ the rational map $\varepsilon : {\bf P} (x) \rightarrow {\cal SU}_C(2)$ has the following properties. \begin{enumerate} \item $\varepsilon^*: H^0({\cal SU}_C(2),{\cal L}) \rightarrow H^0({\bf P}(x),{\cal I}_C^{d-1}(d))$; \item $\varepsilon$ resolves to a morphism ${\widetilde \varepsilon}$ of the $(d-1)$-st blow-up: $$ \begin{array}{r} {\bf P} (x) \leftarrow {\rm Bl}_C \leftarrow {\rm Bl}_{{\widetilde \secant}^2 C} \leftarrow \cdots \leftarrow {\rm Bl}_{{\widetilde \secant}^{d-1}C} = \widetilde{\bf P} (x)\\ \\ \downarrow {\widetilde \varepsilon}\\ \\ {\cal SU}_C(2)\\ \end{array} $$ \item For $1\leq n\leq d-1$ we have $\displaystyle {\widetilde \varepsilon}({\cal S}_n) = \bigcup_{D\in S^n C} {\widetilde \varepsilon}\, \widetilde{\bf P} (x(-D))$. \end{enumerate} \end{lemm} Combining these observations with proposition \ref{strattheorem}, the situation can be further summarised in the following diagram: $$ \begin{array}{rcccccl} {\widetilde \secant}^d C&\subset \cdots\subset &{\widetilde \secant}^{2d-1} C&\subset &\widetilde {\bf P} (x)&&\\ &&&&&&\\ \downarrow&&\downarrow&&\downarrow {\widetilde \varepsilon}&&\\ &&&&&&\\ J/\pm \cong {\cal SU}_C(2)_0&\subset \cdots \subset &{\cal SU}_C(2)_{d-1}& \subset& {\cal SU}_C(2)_d& \subset \cdots \subset &{\cal SU}_C(2)\\ &&&&&& \\ \downarrow&&\downarrow && \downarrow &&\downarrow \phi\\ &&&&&&\\ {\rm Kum}(J) = G_1 &\subset \cdots \subset& G_d & \subset& G_{d+1}&\subset \cdots \subset& G_{[g/2]+1}\subset |2\Theta|\\ \end{array} $$ \medskip \begin{rem} \rm Since $\dim {\bf P} (x) = g + 2d -2$ we see that ${\rm Sec}^{d+n} C = {\bf P} (x)$ as soon as $n\geq (g-1)/2$, i.e. $n\geq [g/2]$. This means that for such $n$ the top row of the diagram terminates, which is consistent with the fact that ${\cal SU}_C(2) = {\cal SU}_C(2)_{[g/2]}$. \end{rem} Now consider the linear pull-back map $\varepsilon^* : |{\cal L}| \rightarrow |{\cal I}_C^{d-1}(d)|$ and its dual $(\varepsilon^*)^{\vee}: |{\cal I}_C^{d-1}(d)|^{\vee}\rightarrow |2\Theta|$. We will check that it fits into the following commutative diagram, where $\alpha_x$ and $\beta_x$ are as defined in the appendix ((\ref{alphax}) and (\ref{betax}) respectively): \begin{equation} \label{abdiagram} \begin{array}{ccc} S^d C& \map{\displaystyle\beta_x} & |{\cal I}_C^{d-1}(d)|^{\vee} \\ &&\\ \alpha_x\downarrow& &\downarrow(\varepsilon^*)^{\vee}\\ &&\\ J& \map{\displaystyle{\rm Kum}}& |2\Theta|\\ \end{array} \end{equation} To see this, first note that $D\in S^d C$ maps under ${\rm Kum} \circ \alpha_x$ to the split divisor $\Theta_{x(-D)}+\Theta_{x^{-1}(D)}$, i.e. to the image of the semistable equivalence class of $x(-D)\oplus x^{-1}(D)$ under the morphism $\phi:{\cal SU}_C(2) \rightarrow |2\Theta|$. On the other hand, $\beta_x(D)$ can be viewed as the set of divisors in $|{\cal I}_C^{d-1}(d)|$ containing the linear span $\overline D \subset {\bf P}(x)$. But we have seen above that this linear span contracts under $\varepsilon$ to the point $x(-D)\oplus x^{-1}(D)$; thus $(\varepsilon^*)^{\vee}\circ \beta_x(D)$ coincides with $\phi(x(-D)\oplus x^{-1}(D))$, and the diagram commutes. \begin{cor} \label{completeness} For $x\in J^d$, suppose that $\alpha_x^*$ is surjective (see \ref{400}(2)). Then the pull-back $\varepsilon^* :H^0({\cal SU}_C(2),{\cal L}) \rightarrow H^0({\bf P}(x), {\cal I}_C^{d-1}(d))$ is surjective; in other words, the rational map $\varepsilon: {\bf P}(x) \rightarrow {\cal SU}_C(2) \hookrightarrow |2\Theta|$ cuts out the complete linear system $|{\cal I}_C^{d-1}(d)|$. \end{cor} {\it Proof.}\ In view of the identification $H^0({\cal SU}_C(2),{\cal L}) = H^0(J,{\cal L})$, the commutativity of (\ref{abdiagram}) implies commutativity of $$ \begin{array}{rcl} H^0(J,{\cal L}) & \map{\displaystyle \varepsilon^*} & H^0({\bf P}(x), {\cal I}_C^{d-1}(d))\\ &&\\ &\alpha_x^* \searrow& \downarrow \beta_x^*\\ &&\\ &&H^0(S^d C, {\cal L}_x).\\ \end{array} $$ Surjectivity of $\varepsilon^*$ now follows from that of $\alpha_x^*$ and from proposition \ref{alphabeta}. {\unskip\nobreak\hfill\hbox{ $\Box$}\medskip\par} \begin{cor} \label{Qdxdualagain} If $d=1$ or 2, then for each $x\in J^d$ we have (compare with (\ref{Qdxdual}) in section \ref{abeljacobi} above): $$ {\bf P} Q_d(x)^{\vee} = {\rm Span}({\rm Kum}(W_d -x)) = {\rm Span}(\phi \circ{\widetilde \varepsilon} \ \widetilde{{\bf P}}(x)) \subset |2\Theta|. $$ \end{cor} {\it Proof.}\ We have ${\bf P} Q_d(x)^{\vee} = {\rm Span}({\rm Kum}(W_d -x))$ by construction; the second equality then follows from diagram (\ref{abdiagram}), proposition \ref{alphabeta} and corollary \ref{completeness}. Surjectivity of $\alpha_x^*$, when $d=1$ or 2, follows from proposition \ref{conjOK}. {\unskip\nobreak\hfill\hbox{ $\Box$}\medskip\par} In the case $d=1$ we have $\widetilde{\bf P} (x) = {\bf P} (x)$, and this extension space embeds linearly under $\phi \circ \varepsilon$. It follows from this that: \begin{cor} \label{G2} $G_2 \subset \phi({\cal SU}_C(2))$. \end{cor} \section{Prym quadrisecant planes are cut out by $G_3$} \label{prymG3} In \cite{OPP} it was shown that the ruled subvariety $G_2 \subset \phi({\cal SU}_C(2))$ contains the Fay trisecant lines of the Kummer variety in the planes of its ruling. In this section we shall show that in a curiously analogous way the ruling of $G_3 \subset |2\Theta|$ cuts out the Beauville--Debarre quadrisecant planes of each Prym Kummer variety (see section \ref{quadrisecants}). For $x\in J^2_C$ let us write $$ W(x) = {\bf P} Q_2 (x)^{\vee} = \{\ D\in |2\Theta|\ | \ x+W_{g-3} \subset {\rm supp}\ D\ \ \}; $$ equivalent descriptions being given by corollary \ref{Qdxdualagain}. Then set-theoretically $ G_3 = \bigcup_{x\in J_C^2} W(x) $ (see (\ref{NQsequence}) and (\ref{Qdxdual})). For $\eta\in J_C[2]\backslash\{0\}$ we fix $\zeta \in J_C$ such that $\zeta^2 = \eta$; with respect to this $\zeta$ the Prym-Kummer map is then described in diagram (\ref{prymkum}) in section \ref{pryms}. The description of the quadrisecant planes of the image that we shall use is that given by the map $f:{\cal F} \rightarrow S^4 {\rm Kum}(P_{\eta})$ (see diagram (\ref{FBDdiagram})) where ${\cal F} = \{(\lambda, D)\in J^2 \times S^4 C\,|\, D\in |\lambda^2|\}$. \begin{prop} \label{WcapPrym} For each $x\in J_C^2$ we have $$ W(x) \cap {\rm Kum}(P_{\eta}) = \bigcup_{D\in |x^2 \eta|} f(x\zeta, D). $$ \end{prop} {\it Proof.}\ By construction, if $u\in P_{\eta}$ then ${\rm Kum}(u) \in W(x)$ if and only if $ h^0(C,x(D) \otimes \zeta \otimes \pi_* u) >0$ for all $D\in S^{g-3} C$. By (\ref{killingD}) this is equivalent to $$ 0< h^0(C,x \otimes \zeta \otimes \pi_* u) = h^0({\widetilde C}, u\otimes \pi^*(x \zeta)), $$ and since $\deg u\otimes \pi_*(x \zeta) = 4$ we conclude that ${\rm Kum}(u) \in W(x)$ if and only if $u = \pi^*(x^{-1} \zeta^{-1})\otimes {\cal O}(\Gamma)$ for some $\Gamma \in S^4 {\widetilde C}$. By definition of $P_{\eta} \cup P_{\eta}^-$ we have ${\rm Nm}_{\pi}(u) = {\cal O}_C$ and so ${\rm Nm}_{\pi} (\Gamma) = D$ for some $D\in |(x \zeta)^2 | = |x^2 \eta|$. Conversely, for each such effective divisor $D$ we have sixteen possibilities for $\Gamma$, giving the sixteen points of $P_{\eta} \cup P_{\eta}^-$ in table (\ref{FBDtable}) (with $\lambda = x\zeta$). {\unskip\nobreak\hfill\hbox{ $\Box$}\medskip\par} Finally, let us restrict attention to a nonhyperelliptic curve $C$ of genus 4. We arrive at a configuration $$ G_2 \subset \phi({\cal SU}_C(2)) \subset G_3 \subset |2\Theta| = {\bf P}^{15}. $$ Here $G_2$ is ruled by ${\bf P} N_2 \cong {\bf P} Q_1^{\vee} \rightarrow J^1$ with 4-dimensional projective fibres which we shall denote by $$ {\bf P}^4(y) = \{D\in |2\Theta|\,|\,y+W_2 \subset {\rm supp}\,D\} \qquad \hbox{for $y\in J^1$;} $$ and $G_3$ is ruled by ${\bf P} N_1 \cong {\bf P} Q_2^{\vee} \rightarrow J^2$ with 10-dimensional fibres $$ W(x) = {\bf P}^{10}(x) = \{D\in |2\Theta|\,|\,x+W_1 \subset {\rm supp}\,D\} \qquad \hbox{for $x\in J^2$.} $$ With respect to any theta characteristic $\kappa \in \vartheta(C)$ the two rulings are polar; equivalently they are given by the same grassmannian map (where both vertical maps are isomorphisms): $$ \begin{array}{ccc} J^1 & \map{}& {\rm Gr}_{5} {H^0(2\Theta)}\\ &&\\ \downarrow \kappa &&\downarrow {\rm ann}\\ &&\\ J^2 &\map{}& {\rm Gr}_{11} {H^0(2\Theta)}.\\ \end{array} $$ We wish to consider the restriction of these rulings to the fixed-point set $|2\Theta|^{\eta} \subset |2\Theta|$ under the action of a 2-torsion point $\eta\in J[2]\backslash \{0\}$. We recall from section \ref{pryms} that this set has two components $$ |2\Theta|^{\eta} = {\bf P} (V^{\vee})^+_{\eta} \cup {\bf P} (V^{\vee})^-_{\eta} $$ into which $P_{\eta} \cup P_{\eta}^-$ map (given a choice of $\zeta = {1\over 2}\eta$) as Kummer varieties. We shall identify ${\bf P} (V^{\vee})^{\pm}_{\eta} = |2\Xi|^{\pm}$ and view $|2\Xi|^+ \cup |2\Xi|^- \subset |2\Theta|$. \begin{lemm} \label{intplane} For $x\in J^2$ we have $$ \dim {\bf P}^{10}(x) \cap |2\Xi|^{\pm} = \cases{2& if $h^0(x^{2}\eta) =1$\cr 3& if $h^0(x^{2}\eta) =2$.\cr} $$ \end{lemm} {\it Proof.}\ By corollary \ref{intdims} and Serre duality this is equivalent, for any choice of $\kappa \in \vartheta(C)$, to $$ {\bf P}^{4}(y) \cap |2\Xi|^{\pm} = \cases{\emptyset& if $h^0(y^{2}\eta) =0$\cr {\rm point}& if $h^0(y^{2}\eta) >0$\cr} $$ where $y = \kappa x^{-1}$. Here ${\bf P}^4(y)\hookrightarrow {\cal SU}_C(2)$ is the set of extensions $\ses{y^{-1}}{E}{y}$, and $E\in {\bf P}^4(y)$ is in $|2\Theta|^{\eta}$ if and only if $E\in {\bf P}^4(y) \cap {\bf P}^4(\eta y)$. We now refer to \cite{OPP}, proposition 1.2. This says, if we translate from ${\cal SU}_C(2,K)$ to ${\cal SU}_C(2)$, that ${\bf P}^4(y) \cap {\bf P}^4(y')$ is nonempty only if either $y\otimes y' = {\cal O}(p+q)$ or $y^{-1}\otimes y' = {\cal O}(p-q)$ for some points $p,q \in C$. When $y' = y \eta$ the second case is impossible for nonhyperelliptic $C$; so we see that the intersection is nonempty only in the first case, i.e. if and only if $h^0(y^{2}\eta) >0$. In this case ${\bf P}^4(y) \cap {\bf P}^4(y \eta)$ is a line on which the involution $\eta$ acts with two fixed points, which are the respective intersections ${\bf P}^{4}(y) \cap |2\Xi|^{\pm}$. {\unskip\nobreak\hfill\hbox{ $\Box$}\medskip\par} We conclude from \ref{WcapPrym} and \ref{intplane} that the 10-planes ${\bf P}^{10}(x) \subset |2\Theta|$ cut out in $|2\Xi|$ {\it precisely the quadrisecant planes of the embedded Prym-Kummer variety}---and this concludes the proof of theorem \ref{theorem2}. \section{Appendix: symmetric products of a curve} \label{symm} We shall gather together here some results concerning the symmetric products $S^d C$. We denote by $ p: C^d = C\times \cdots \times C \rightarrow S^d C $ the quotient map. For any $L \in {\rm Pic}(C)$ one can associate a line bundle $S^d L \in {\rm Pic}\ S^d C$ satisfying $$ p^* S^d L = \bigoplus_{i=1}^d pr_i^* L. $$ Let $\Delta $ be the union of the diagonals in $C^d$, i.e. the ramification divisor of $p$; and $\overline \Delta$ the diagonal divisor in $S^d C$; so we have \begin{equation} \label{pbdiag} 2\Delta = p^* \overline \Delta. \end{equation} Noting that $K_{C^d} = p^* S^d K_C$, the ramification formula for the map $p$ tells us that $$ {\cal O}(\Delta) = p^*(S^d K_C \otimes K^{-1}_{S^d C}). $$ In other words, although the divisor $\Delta$ does not descend to the quotient, the line bundle ${\cal O}(\Delta)$ does descend. We next introduce some notation for the cohomology ring of $S^d C$ (see \cite{Mac}). Let $\beta \in H^2(C,{\bf Z})$ be the fundamental class of a point; and let $\{\alpha_i\}_{1\leq i \leq 2g}$ be a symplectic basis of $H^1(C, {\bf Z})$. Then it is well-known that the cohomology ring of $S^d C$ is generated over ${\bf Z}$ by the following elements (see \cite{Mac} (3.1) and (6.3); and note that we identify $H^*(S^d C, {\bf Z})$ with invariant cohomology on $C^d$ under the action of the symmetric group): $$ \begin{array}{rcll} \xi_i &=& \displaystyle \sum_{j=1}^d 1\otimes \cdots \underbrace{\otimes \alpha_i \otimes}_{\hbox{\sixrm j-th positions}} \cdots \otimes 1 &\in H^1(S^d C, {\bf Z}) \qquad \hbox{for $i = 1,\ldots , 2g$;}\\ \eta &=&\displaystyle \sum_{j=1}^d 1\otimes \cdots \overbrace{\otimes \beta \otimes} \cdots \otimes 1 &\in H^2(S^d C,{\bf Z}).\\ \end{array} $$ The $\xi_i$s anticommute with each other and commute with $\eta$. In fact $\eta$ is the fundamental class of the divisor $S^{d-1} C \hookrightarrow S^d C$, $D\mapsto D+p$, where $p\in C$ is any point of the curve. More generally one has \begin{equation} \label{c1SdL} c_1(S^d L) = (\deg L) \eta \qquad \hbox{for any $L\in {\rm Pic}(C)$;} \end{equation} while for the inclusion $S^{d-1} C \hookrightarrow S^d C$ above, ${\cal O}_{S^d C}(S^{d-1} C) = S^d {\cal O}_C(p)$. We shall sometimes (notably in proposition \ref{100} below) simply write $p$ for this line bundle. Next, we define $$ \sigma_i = \xi_i \xi_{i+g} \in H^2(S^d C, {\bf Z}) \qquad \hbox{for $i = 1,\ldots , g$.} $$ Then we have (\cite{Mac} 5.4): \begin{equation} \label{50} \begin{array}{rcll} \sigma_i ^2 &=& 0,&\\ \sigma_i \sigma_j &=& \sigma_j \sigma_i &\hbox{for $i\not= j$,}\\ \sigma_{i_1} \cdots \sigma_{i_a} \eta^b &=& \eta^{a+b}& \hbox{for $b>0$ and distinct ${i_1}, \ldots ,{i_a}$;}\\ \end{array} \end{equation} and the Chern class of the diagonal is: \begin{equation} \label{c1delbar} c_1(\overline \Delta) = 2(d+g-1)\eta -2(\sigma_1 + \cdots + \sigma_g). \end{equation} We now consider, for $x\in J^d$, the map \begin{equation} \label{alphax} \begin{array}{rcl} \alpha_x : S^d C &\rightarrow& J\\ D &\mapsto& x^{-1} \otimes {\cal O}(D).\\ \end{array} \end{equation} We shall write ${\cal L}_x = \alpha_x ^* {\cal L} \in {\rm Pic}\ S^d C$. \begin{prop} \label{100} \begin{enumerate} \item ${\cal L}_x \cong S^d (K x^2) \otimes {\cal O}(-\overline \Delta)$; \item $c_1({\cal L}_x) = 2(\sigma_1 + \cdots + \sigma_g)$; \item $\chi (S^d C, {\cal L}_x) = \sum_{i=0}^d {g \choose i}$; \item $\chi (S^d C, {\cal L}_x(-p)) = {g \choose d}$ for any $p\in C$. \end{enumerate} \end{prop} {\it Proof.}\ 1. The case $d=1$ is well-known; see for example \cite{NR2}. For $d>1$ let $y=x(-p_1-\cdots -p_{d-1})$ for some arbitrary points $p_1,\ldots ,p_{d-1}\in C$. Then, from the case $d=1$, we see that $$ p^*{\cal L}_x|_{\{(p_1,\ldots ,p_{d-1})\}\times C} = Kx^2 (-2p_1-\cdots -2p_{d-1}), $$ and hence $p^* {\cal L}_x = p^*S^d(Kx^2) \otimes {\cal O}(-2\overline \Delta)$. So the result follows from (\ref{pbdiag}) and injectivity of $p^*$ on line bundles. 2. This is immediate from part 1, (\ref{c1SdL}) and (\ref{c1delbar}). 3. One should be able to extract this from \cite{Mac}, but as there appears to be an error in the proof of (14.9) of that paper we shall give the Riemann-Roch calculation here. By \cite{Mac}(14.5) the total Chern class is $ c(S^d C) = (1+\eta)^{d-2g+1} \prod_{i=1}^g (1+\eta -\sigma_i), $ and the Todd class is therefore $$ {\rm td}\ (S^d C) = \Bigl({\eta \over 1-e^{-\eta}}\Bigr)^{d-2g+1} \prod_{i=1}^g \Bigl({\eta -\sigma_i \over 1-e^{-\eta +\sigma_i }}\Bigr). $$ Making use of (\ref{50}), the expression in the product can be rewritten: $$ {\eta -\sigma_i \over 1-e^{-\eta +\sigma_i }} = \Bigl({\eta \over 1-e^{-\eta}}\Bigr)(1+\sigma_i \tau) \qquad {\rm where} \ \tau = {\eta e^{-\eta} + e^{-\eta} - 1 \over \eta (1-e^{-\eta})}, $$ and so $$ {\rm td}\ (S^d C) = \Bigl({\eta \over 1-e^{-\eta}}\Bigr)^{d-g+1} \prod_{i=1}^g (1+\sigma_i \tau). $$ On the other hand ${\rm ch}({\cal L}_x) = e^{2(\sigma_1 + \cdots + \sigma_g)} = \prod_{i=1}^g (1+2 \sigma_i)$; so by Hirzebruch-Riemann-Roch: $$ \chi({\cal L}_x) = \deg \bigg\lbrace \Bigl({\eta \over 1-e^{-\eta}}\Bigr)^{d-g+1} \prod_{i=1}^g (1+\sigma_i (2+\tau)) \bigg\rbrace _d $$ Using \cite{Mac} (5.4) this expression may be computed by setting $\sigma_1 =\cdots = \sigma_g = \eta$ and evaluating the coefficient of $\eta^d$: $$ \begin{array}{rcl} \chi({\cal L}_x) &=& \bigg\lbrace \Bigl({\displaystyle\eta \over \displaystyle 1-e^{-\eta}}\Bigr)^{d-g+1}(1+\eta (2+\tau))^g \bigg\rbrace _{\hbox{coefficient of $\eta^d$}} \\ &=& \bigg\lbrace \Bigl({\displaystyle\eta \over \displaystyle 1-e^{-\eta}}\Bigr)^{d+1}(2-e^{-\eta})^g \bigg\rbrace _{\hbox{coefficient of $\eta^d$}}\\ &&\\ &=&\displaystyle {\rm Res}_{\eta =0}\ {(2-e^{-\eta})^g d\eta\over (1-e^{-\eta})^{d+1}} \\ &&\\ &=&\displaystyle {\rm Res}_{\zeta =0}\ {(1+\zeta)^g d\zeta\over \zeta^{d+1}(1-\zeta)}\\ &=&\displaystyle \sum_{i=0}^d {g \choose i},\\ \end{array} $$ where in the penultimate step we have made a substitution $\zeta = 1 - e^{-\eta}$. The computation for part 4 is entirely similar. {\unskip\nobreak\hfill\hbox{ $\Box$}\medskip\par} For $x\in J^d$ ($d\geq 1$), let ${\bf P} (x) \cong {\bf P} ^{g+2d-2}$ be the extension space introduced in section \ref{segrestrat} into which $C$ maps by the complete linear series $|Kx^2|$: \begin{equation} \lambda_{|Kx^2|}: C \hookrightarrow {\bf P} (x) = |Kx^2|^{\vee}. \end{equation} We shall denote by ${\cal I}_C$ the ideal sheaf of the image, and consider the linear series $|{\cal I}_C^{d-1}(d)|$ on ${\bf P}(x)$. This series contracts each $d$-secant plane $\overline D \cong {\bf P}^{d-1} \subset {\bf P}(x)$ (where $D\in S^d C$) to a point, and so induces a rational map \begin{equation} \label{betax} \beta_x : S^d C \rightarrow |{\cal I}_C^{d-1}(d)|^{\vee}. \end{equation} We observe that by proposition \ref{100}(1) a section of ${\cal L}_x$ can be identified with a symmetric multilinear form $F: \bigotimes^d H^0(C, Kx^2)^{\vee} \rightarrow {\bf C}$ with the property that $F(p_1,\ldots , p_d) = 0$ whenever $p_i = p_j$, $i\not= j$; or equivalently $F \in H^0({\bf P}(x), {\cal I}_C^{d-1}(d))$. From this one may check the following (and we are grateful to Beauville for pointing this out to us): \begin{prop} \label{alphabeta} $\beta_x^* {\cal O}(1) = {\cal L}_x$; and there is an induced isomorphism $$ \beta^*_x : H^0({\bf P} (x), {\cal I}_C^{d-1}(d)) \mathbin{\hbox{$\widetilde\rightarrow$}} H^0(S^d C, {\cal L}_x). $$ \end{prop} One would like higher cohomology of ${\cal L}_x$ to vanish, so that parts 3 and 4 of proposition \ref{100} compute $h^0$. \begin{conj} \label{400} Let $1\leq d\leq g$ and $x\in J_C^d$. Then: \begin{enumerate} \item $h^0(S^d C, {\cal L}_x) = \sum_{i=0}^d {g\choose i}$; \item $\alpha_x^* : H^0(J,{\cal L}) \rightarrow H^0(S^d C, {\cal L}_x)$ is surjective. \end{enumerate} \end{conj} We can verify this at once in certain cases. When $d=1$, for example, ${\cal L}_x = K x^2$ and \ref{400} follows from \cite{NR2} lemma 4.1. In the case $d=2$, part 1 is proved in \cite{BV} proposition 4.9. Part~2 can also be shown---for $g>4$ this will appear elsewhere, but in the case $g= 4$ we can give a simple ad hoc argument as follows. In the notation of section \ref{abeljacobi}, $N_2(x) \perp N_1(\kappa x^{-1})$ where $\dim N_1(\kappa x^{-1}) = 11$. Hence $\dim N_2(x) \leq 5$; but (by \cite{BV} proposition 4.9) $\dim Q_2(x) = 11$, and therefore $\alpha_x^*$ is surjective. In the case $d=g$ the conjecture is also easy to see, since $\alpha_x : S^g C \rightarrow J_C$ is a birational morphism with connected fibres. Indeed, this suggests using a descending induction for $d\leq g$ using the inclusions $$ C\subset S^2 C \subset \cdots \subset S^g C $$ given by some choice of point $p\in C$. At each stage restriction yields an exact sequence on $S^{d}C$ (where the last term is supported on $S^{d-1} C$): $$ \ses{{\cal L}_{x}(-p)}{{\cal L}_{x}}{{\cal L}_{x(-p)}}. $$ Suppose for a moment that \begin{equation} \label{vanishing} H^i(S^{d} C , {\cal L}_{x} (-p))=0 \qquad \hbox{for $i>0$.} \end{equation} This would imply that the restriction map $ H^0(S^{d} C , {\cal L}_{x} ) \rightarrow H^0(S^{d-1} C , {\cal L}_{x(-p)} ) $ is surjective, and hence surjectivity of $\alpha_{x}^*$ would imply that of $\alpha_{x(-p)}^*$; while \ref{400}(1) also follows inductively since $h^0(S^{d} C , {\cal L}_{x} (-p)) = {g\choose d}$ by proposition \ref{100}(4). Unfortunately we cannot prove (\ref{vanishing})---except when $d=g$. For here we have a diagram $$ \begin{array}{rcl} S^{g-1} C &\subset&S^{g} C\\ \downarrow &&\downarrow \\ \Theta &\subset& J\\ \end{array} $$ where the vertical arrows are birational morphisms with connected fibres and so preserve cohomology under pull-back. Hence $h^i(S^g C,{\cal L}_{x} (-p)) = h^i(J,{\cal L}(-\Theta)) = h^i(J,\Theta) = 0$. To summarise, we have shown: \begin{prop} \label{conjOK} Conjecture \ref{400} holds for $d=1,2,g-1,g$; and in particular it holds when~$g\leq 4$. \end{prop} Finally, we remark that the difficulty in proving (\ref{vanishing}) rests principally in the fact that the self-intersection $$ c_1({\cal L}_x K^{-1})^d = {g!\over (g-d)!} + (d+1)^d - g^d $$ is negative in all the cases not covered by proposition \ref{conjOK}, and one can show that the line bundle ${\cal L}_x K^{-1}$ fails to be nef, big or semi-ample. One is therefore unable to use the standard vanishing theorems.

9703

alg-geom/9703031

en

https://arxiv.org/abs/alg-geom/9703031

alg-geom/9703031

Elizabeth Gasparim

Elizabeth Gasparim

On the topology of holomorphic bundles

In this work we study the topology of holomorphic rank two bundles over complex surfaces. We consider bundles that are constructed by glueing and show that under certain conditions the topology of the bundle does not depend on the glueing. We present a simple classification of bundles on blown-up surfaces.

alg-geom

\section{Introduction} Let $X$ be a complex manifold and let $A$ and $B$ be open sets that cover $X.$ Given holomorphic bundles $E_A$ and $E_B$ defined over the subsets $A$ and $B,$ we construct holomorphic bundles over $X$ by glueing the bundles $E_A$ and $E_B$ over $A \cap B.$ We then compare the topology of bundles given by different glueings. Our main motivation is the study of bundles over blown-up surfaces. Therefore we will focus our attention on the cases where the intersection $A \cap B$ is biholomorphic to ${\bf C}^m - \{0\}.$ We then present simple holomorphic and topological classifications of bundles on blown-up surfaces. \section{Glueing bundles over a manifold} In this section we construct bundles over a complex manifold by glueing bundles defined on open subsets and then we compare their topology .\\ \noindent Consider the following data: \vspace{ 2 mm} \noindent i) a complex manifold $X$ with $dim _{\bf C}X = m \ge 2$ and open subsets $A,B,$ and $C$ of $X$ satisfying $X = A \cup B,\, A \cap B = C$ with $C$ biholomorphic to ${\bf C}^m - \{0 \}$ \vspace{2 mm} \noindent ii) holomorphic vector bundles $\pi_A: E_A \rightarrow A$ and $\pi_B : E_B \rightarrow B$ which are trivial when restricted to $C$ \vspace{2 mm} \noindent iii) a trivialization $F: E_A \rightarrow C \times {\bf C}^n$ of $E_A|_C$ and trivializations $G_i: E_B \rightarrow C \times {\bf C}^n, \, \, i = 0,1$ of $E_B|_C.$\\ Given the above data, we define bundles $E_i$ over $X$ by the formula $$E_i = E_A \bigcup _{F = G_i} E_B = (E_A \bigsqcup E_B)/ \sim$$ where for $x \in E_A$ and $y \in E_B$ we define $x \sim y$ if $F(x) = G_i(y).$ \noindent We have the following result about the bundles $E_i.$ \begin{guess}\label{topequ} The topology of the bundle $E_i$ is independent of the glueing, that is $E_0$ and $E_1$ are topologically equivalent vector bundles. \end{guess} \noindent To prove this we present some preliminary lemmas. \begin{lemma}\label{homtri}If the trivializations $G_0$ and $G_1$ are homotopic, then $E_0$ and $E_1$ are topologically equivalent vector bundles. \end{lemma} \noindent{\bf Proof}: By $G_0$ homotopic to $G_1$ we mean that there is a one parameter family of trivializations $G_t$ with $ t\in [0,1]$ taking $G_0$ to $G_1.$ To prove the lemma we consider the product bundles $E_A \times I$ and $E_B \times I$ over $A \times I$ and $B \times I$ respectively, together with given trivializations $\widetilde{F}$ and $\widetilde{G}$ of $E_A \times I |_{C \times I}$ and $E_B \times I |_{C \times I}$ defined by $\widetilde{F}(a,t) = (F(a),t),$ for $(a,t) \in A \times I$ and $\widetilde{G}(b,t) = (G(b),t),$ for $(b,t) \in B \times I.$ We have that $\widetilde{G}(y,0) = G_0(y)$ and $\widetilde{G}(y,1) = G_1(y).$ Defining the bundle $E$ over $X \times I$ by $E = (E_A \times I) \cup_{\widetilde{F} = \widetilde{G}} (E_B \times I)$ it immediately follows that $E|_{X \times \{0\}} \simeq E_0$ and $E|_{X \times \{1\}} \simeq E_1$ and consequently $E_0$ and $E_1$ are topologically equivalent. \hfill\vrule height 3mm width 3mm \begin{lemma}\label{tribun} Consider the trivial vector bundle $D = C \times {\bf C}^n$ over a complex space $C.$ Let $G_0$ and $G_1$ be two trivializations of $D$ over $C$ and let $\Phi:C \rightarrow GL(n,{\bf C})$ be the corresponding transition matrix, i.e. $\Phi(c) G_0(c) = G_1(c).$ Then $G_0$ and $G_1$ are homotopic if and only if $\Phi$ is nullhomotopic. \end{lemma} \noindent The proof is straightforward. \begin{lemma}\label{nulhom} Any holomorphic map $f: {\bf C}^m - \{0\} \rightarrow GL(n,{\bf C})$ for $m\ge 2$ is nullhomotopic. \end{lemma} \noindent {\bf Proof}: By Hartog's Theorem, $f$ extends to a holomorphic function $\widetilde{f}: {\bf C}^m \rightarrow M(n,{\bf C}),$ where $M(n,{\bf C})$ denotes the space of all $n \times n$ matrices with complex coefficients. We claim that $\widetilde{f}(0) \in GL(n,{\bf C}).$ In fact, if $\widetilde{f}(0) \notin GL(n,{\bf C})$ then $det(\widetilde{f}(0)) = 0.$ But then $(det \circ \widetilde{f})^{-1} (0) = \{ 0\} \subset {\bf C}^m$ which is a contradiction, because $(det \circ \widetilde{f}): {\bf C}^m \rightarrow {\bf C}$ is holomorphic and the pre-image of a point by a holomorphic function in ${\bf C}^m$ is either empty or has codimension 1. Which proves the claim. Thus, we can write $\widetilde{f} : {\bf C}^m \rightarrow GL(n,{\bf C}).$ Hence $f$ factors through ${\bf C}^m,$ i.e. $f = \widetilde{f} \circ i$ where $i$ is the inclusion $i: {\bf C}^m -\{0\} \rightarrow {\bf C}^m.$ As ${\bf C}^m $ is contractible it follows that $f$ is nullhomotopic. \vspace {3mm} \noindent Alternative proof of Lemma \ref{nulhom}: Consider the holomorphic function $g = det \circ f : {\bf C}^m - \{0 \} \rightarrow {\bf C}.$ By Hartog's theorem we have that both $g$ and $1/g$ extend to holomorphic functions defined on ${\bf C}^m.$ Let $\widetilde {g} $ be the extension of $g$ to ${\bf C}^m. $ Then, because $1/g$ also extends as a holomorphic function to ${\bf C}^m,$ it follows that $\widetilde {g}(0) \neq 0.$ We have $g = i \circ \widetilde {g}$ where $i$ is the inclusion $i : {\bf C}^m - \{ 0\} \rightarrow {\bf C}^m$ and since ${\bf C}^m$ is contractible it follows that $g$ is nullhomotopic. \hfill\vrule height 3mm width 3mm \vspace{5 mm} \noindent {\bf Proof of Proposition \ref{topequ}}: Lemmas \ref{tribun} and \ref{nulhom} imply that any two holomorphic trivializations of the trivial bundle $({\bf C}^m - \{0 \}) \times {\bf C}^n$ over ${\bf C}^m - \{0 \}$ are homotopic and then Lemma \ref{homtri} implies that the bundles obtained using these trivializations are topologically equivalent.\hfill\vrule height 3mm width 3mm \section{ Bundles on Blown-up Surfaces } In this section we apply Proposition \ref{topequ} to give a simple topological description of bundles on some blown-up surfaces. Let us consider the following case: \vspace{2 mm} \noindent i) $X = \widetilde {S} $ is the blow-up of a complex surface $S$ at a point $P$ and $\ell$ is the exceptional divisor \vspace{2 mm} \noindent ii) the open subsets $A = N_{\ell}$ and $B = \widetilde {S} - \ell \simeq S - \{P\}$ are respectively a neighborhood of the exceptional divisor $\ell$ and the complement of the exceptional divisor. \vspace{2 mm} \noindent iii) $A \cap B \simeq {\bf C}^2 - \{0\} .$\\ Some elementary examples of such surfaces are the blow up of the projective plane ${\bf P}^2$ at a point or the blow-up of a Hirzebruch surface $S_n = {\bf P}( {\cal O}(n) \oplus {\cal O})$ at a point. Studying successive blow-ups on these basic surfaces leads to a similar topological classification of bundles on any rational surface. This just follows from the classification of rational surfaces, see Griffths and Harris [3]. To state our topological classification we first quote some results from previous papers. We write $\widetilde {\bf C}^2 = U \cup V$, where $U = {\bf C}^2 =\{(z,u)\},$\,\, $V = {\bf C}^2 = \{(\xi,v)\},$\,\, $U \cap V = ({\bf C} - \{0\}) \times {\bf C}$ with the change of coordinates $ (\xi,v) = (z^{-1 },zu).$ \begin{theorem}\label{matk=1}{\rm [1, Thm. 2.1]} Let $E$ be a holomorphic rank two vector bundle on $ \widetilde{\bf C}^2 $ with vanishing first Chern class and let $j$ be the non-negative integer that satisfies $E_{\ell} \simeq {\cal O}(j) \oplus {\cal O}(-j). $ Then $E$ has a transition matrix of the form $$\left(\matrix {z^j & p \cr 0 & z^{-j} \cr }\right)$$ from $U$ to $V,$ where $p$ is a polynomial given by $$p = \sum_{i = 1}^{2j-2} \sum_{l = i-j+1}^{j-1}p_{il}z^lu^i.$$ \end{theorem} \begin{theorem}\label{matgen}{\rm [2, Thm. 3.3]} Let $E$ be a holomorphic rank two vector bundle on ${\cal O}(-k)$ whose restriction to the zero section is $E_{\ell} \simeq {\cal O}(j_1) \oplus {\cal O}(j_2),$ with $j_1 \ge j_2.$ Then $E$ has a transition matrix of the form $$\left(\matrix {z^{j_1} & p \cr 0 & z^{j_2} \cr }\right)$$ from $U$ to $V,$ where the polynomial $p$ is given by $$p = \sum_{i = 1}^{ \left[(j_1 - j_2 -2)/k\right]} \sum_{l = ki+j_2+1}^{j_1-1}p_{il}z^lu^i$$ and $p = 0$ if $j_1< j_2 +2.$ \end{theorem} \begin{theorem}\label{triout}{\rm [2, Cor. 4.2]} Holomorphic bundles on the blow up of a surface are trivial on a neighborhood of the exceptional divisor minus the exceptional divisor. \end{theorem} As a consequence of Theorems \ref{matk=1} and \ref{triout} we have the following holomorphic classification of bundles on $\widetilde{S}.$ \begin{corollary}\label{holzer} Every holomorphic rank two vector bundle over $\widetilde{S}$ with vanishing first Chern class is completely determined (up to isomorphism) by a 4-tuple $(E,j,p,\Phi)$ where $E$ is a holomorphic rank two bundle on $S$ with vanishing first Chern class, $j$ is a non-negative integer, $p$ is a polynomial, and $\Phi : {\bf C}^2 -\{0\} \rightarrow GL(2,{\bf C})$ is a holomorphic map. \end{corollary} \noindent {\bf Proof}: The essential ingredient here is that by theorem \ref{triout} every holomorphic bundle on $\widetilde{S}$ is trivial on $N_\ell -\ell$ for some neighborhood $N_\ell$ of the exceptional divisor. It follows that outside $\ell$ we may take a pull-back bundle $\pi^*(E|_{S-p})$ of a holomorphic rank two bundle $E$ on $S$ with vanishing first Chern class and glue it to a bundle on $N_\ell$ using the function $\Phi.$ Now we use Theorem \ref{matk=1} to see that a bundle on $N_\ell$ is determined by a non-negative integer $j$ and a polynomial $p$ whose form is explicitly known. \hfill\vrule height 3mm width 3mm \vspace{5 mm} The corresponding classification for nonvanishing first Chern class is the following. \begin{corollary}\label{holnon} Every holomorphic rank two vector bundle over $\widetilde{S}$ is completely determined (up to isomorphism) by a 5-tuple $(E,j_1,j_2,p,\Phi)$ where $E$ is a holomorphic rank two bundle on $S,$ $j_1$ and $j_2$ are integers, $p$ is a polynomial, and $\Phi : {\bf C}^2 -\{0\} \rightarrow GL(2,{\bf C})$ is a holomorphic map. \end{corollary} \noindent The proof is analogous to the one for Corollary \ref{holzer}. \section {Topology of bundles on $\widetilde{S}$} We now deduce the topological counterparts of Corollaries 3.4 and 3.5. \begin{corollary} \label{topzer}Every holomorphic rank two vector bundle over $\widetilde{S}$ with vanishing first Chern class is topologically determined by a triple $(E,j,p)$ where $E$ is a holomorphic rank two bundle on $S$ with vanishing first Chern class, $j$ is a non-negative integer, and $p$ is a polynomial. \end{corollary} \noindent {\bf Proof}: By Corollary \ref{holzer} we know that such a bundle is holomorphically determined by a 4-tuple $(E,j,p, \Phi)$ and Proposition \ref{topequ} shows that topologically the choice of the map $\Phi$ is irrelevant.\hfill\vrule height 3mm width 3mm \vspace{5 mm} A straightforward generalization of Corollary \ref{topzer} for the case of nonvanishing first Chern class is the following result. \begin{corollary}\label{topnon} Every holomorphic rank two vector bundle over $\widetilde{S}$ is topologically determined by a 4-tuple $(E,j_1,j_2,p)$ where $E$ is a holomorphic rank two bundle on $S,$ $j_1$ and $j_2$ are integers, and $p$ is a polynomial. \end{corollary} \noindent The proof is analogous to the proof of \ref{topzer}. \vspace {5 mm} \noindent {\bf Examples}: Let us write down some examples to clarify the statement of Corollary \ref{topzer}. First we fix a holomorphic rank two bundle $E$ with vanishing first Chern class over the surface $S.$ Then we would like to see which are the possible bundles $\widetilde {E}$ over $\widetilde {S}$ that are a pull-back of $E$ outside the exceptional divisor. According to Corollary \ref{topzer} such bundles are topologically given by a choice of an integer $j$ and a polynomial $p$ whose form is given in Theorem \ref{matk=1} as $p = \sum_{i = 1}^{2j-2} \sum_{l = i-j+1}^{j-1}p_{il}z^lu^i.$ If $j = 0,$ then $p = 0$ and it follows that $\widetilde {E} =\pi^* E$ is globally a pull-back bundle. That is, applying Corollary \ref{holzer}. we verify the well known fact that bundles over a blown-up surface that are trivial when restricted to the exceptional divisor are pull-backs. If $j = 1,$ then also $p = 0.$ In this case we see that on a neighborhood $N_\ell$ of $\ell$ the bundle is ${\cal O}(1) \oplus {\cal O}(-1).$ If follows that all holomorphic bundles $\widetilde {E}$ over $\widetilde {S}$ whose restriction to the exceptional is ${\cal O}(1) \oplus {\cal O}(-1).$ are topologically equivalent. However, clearly these bundles are not pull-backs and are not topologically equivalent to any of the bundles we obtained in the previous case. If $j = 2,$ then $p = (p_{10} + p_{11}z)u + p_{21} zu^2$ depends on three complex parameters. However these are not effective parameters in the sense that some different choices of the polynomial $p$ will give isomorphic bundles over $N_\ell$ (holomorphically and hence also topologically) and therefore will also lead to globally isomorphic bundles over $\widetilde {S}.$ This leads us immediately to the question of determining the ``local moduli space'' structure. That is, to see for a fixed value of $j$ what polynomials define isomorphic bundles over $N_\ell.$ We call ${\cal M}_j$ the moduli space of isomorphism classes of bundles on $N_\ell$ whose restriction to $\ell$ equals ${\cal O}(j) \oplus {\cal O}(-j).$ The answer to the local moduli question is given by the following results. \begin{theorem}\label{modj=2}{\rm [1, Thm. 3.4]} The moduli space ${\cal M}_2$ is homeomorphic to the union ${\bf P}^1 \cup \{q_1,q_2\},$ of a complex projective plane ${\bf P}^1$ and two points, with a basis of open sets given by $${\cal U} \cup \{q_1,U : U \in {\cal U} - \phi \} \cup \{q_1,q_2,U : U \in {\cal U} - \phi \} $$ where ${\cal U}$ is a basis for the standard topology of ${\bf P}^1.$ \end{theorem} \begin{theorem}\label{modgen}{\rm [1, Thm.3.5]} The generic set of the moduli space ${\cal M}_j$ is a complex projective space of dimension $2j-3.$ \end{theorem} \noindent{ \bf Examples}: Let us continue the analysis of the case $j = 2$ started in the preceding example. We have seen that for $j = 2$ the polynomial $p$ is given by three complex parameters. However, by Theorem \ref{modj=2} we see that nonequivalent choices of $p$ are parametrized by a non-Hausdorff space formed by a projective line ${\bf P}^1$ with two extra points. Therefore, it follows from Corollary \ref{holzer} that for each fixed choice of glueing $\Phi,$ isomorphism classes of bundles are parametrized by ${\bf P}^1$ (with the standard topology) plus two extra points. However it is simple to see that any such choices will produce topologically equivalent bundles. For each value of the integer $j$ we can reproduce an analysis similar to the ones in the previous examples. For a chosen bundle $E$ over $S$ and a particular choice of glueing $\Phi$ we have generically a projective space $P^{2j-3}$ parametrizing nonisomorphic bundles $\widetilde {E}.$ Details for the topology as well as explicit calculations of Chern classes for these bundles will appear in a subsequent paper. Let us represent holomorphic bundles on the blown-up surface $\widetilde{S}$ by $(E,j,p,\Phi),$ according to Corollary \ref{holzer}. Then we have just proved the following. \begin{theorem} Let $E$ be a holomorphic rank two bundle with vanishing first Chern class. For fixed $\Phi$ and $j$ the family of isomorphism classes of holomorphic bundles on $\widetilde {S}$ which are a pull-back of $E$ outside the exceptional divisor is generically parametrized by ${\bf P}^{2j-3}.$ \end{theorem} \centerline{\bf Acknowledgments} I am happy to thank Pedro Ontaneda for all the help.

9703

alg-geom/9703014

en

https://arxiv.org/abs/alg-geom/9703014

alg-geom/9703014

Gerd Mueller

Antonio Campillo, Janusz Grabowski, Gerd M\"uller

Derivation algebras of toric varieties

LaTeX, 14 pages

1997/1, Mainz University

Normal affine algebraic varieties in characteristic 0 are uniquely determined (up to isomorphism) by the Lie algebra of derivations of their coordinate ring. This is not true without the hypothesis of normality. But, we show that (in general, non-normal) toric varieties defined by simplicial affine semigroups are uniquely determined by their Lie algebra if they are supposed to be Cohen-Macaulay of dimension at least 2 or Gorenstein of dimension 1. Moreover, every automorphism of the Lie algebra is induced from a unique automorphism of the variety and every derivation of the Lie algebra is inner, i.e., the first cohomology of the Lie algebra with coefficients in the adjoint representation vanishes.

alg-geom

\section{Introduction} Normal affine algebraic varieties in characteristic 0 are uniquely determined (up to isomorphism) by the Lie algebra of derivations of their coordinate ring. This was shown by Siebert \cite{Si} and, independently, by Hauser and the third author \cite{HM}. In both papers the assumption of normality is essential. There are non-isomorphic non-normal varieties with isomorphic Lie algebras. The third author \cite{M} treated certain non-normal varieties defined in combinatorial terms by showing that closed simplicial complexes can be reconstructed from the Lie algebra of their Stanley-Reisner ring. Here we study this problem for (in general, non-normal) toric varieties defined by simplicial affine semigroups. \\[1ex] We show that such toric varieties are uniquely determined by their Lie algebra if they are supposed to be Cohen-Macaulay of dimension $\ge2$. The corresponding statement is false in dimension 1. For toric curves we need the stronger hypothesis that they are Gorenstein. In fact, we can reconstruct from the Lie algebra the semigroup defining the variety. Our result should be compared with a recent one of Gubeladze \cite{Gu} saying that an affine semigroup is uniquely determined by the toric variety it defines (more precisely, by its coordinate ring as an augmented algebra). \\[1ex] The main tool in our proofs is a root space decomposition of the Lie algebra of derivations of a Buchsbaum semigroup ring. The set of roots is closely related to the underlying semigroup. This structural description will be used to prove two more results. We show, in the Cohen-Macaulay case, that every automorphism of the Lie algebra is induced from a unique automorphism of the variety. And we establish an infinitesimal analogue of the last statement: Every derivation of the Lie algebra is inner, i.~e., the first cohomology of the Lie algebra with coefficients in the adjoint representation vanishes. \\[1ex] Our results were obtained during visits at the Mathematics Departments of the Universities in Valladolid, Warszawa, and Mainz. We thank these institutions (as well as the Spanish-German Acciones Integradas) for financial support and their members for their hospitality. \section{The root space decomposition} Let $S$ be an affine semigroup, i.~e., a finitely generated subsemigroup of some $\N^n$. We stress that, in this paper, semigroup always means semigroup with zero element. Denote by $G=G(S)$ the subgroup of $\Z^n$ generated by $S$ and by $r={\rm rk}\: S={\rm rk}\: G(S)$ its rank. Let $C_S$ be the convex polyhedral cone spanned by $S$ in ${\Bbb Q}^n$. We shall suppose throughout that $S$ is {\em simplicial}, i.~e., that the convex cone $C_S$ can be spanned by $r$ elements of $S$. For an algebraically closed field $k$ of characteristic 0 let $k[S]\subseteq k[t]=k[t_1,\ldots,t_n]$ denote the corresponding semigroup ring. We need to recall how the property of $k[S]$ being Cohen-Macaulay or Buchsbaum can be described in terms of $S$. For this purpose, let $F_1,\ldots,F_m$ be the $(r-1)$-dimensional faces of $C_S$. Set \[S'_i=\{\l\in G,\; \l+s\in S\;\mbox{for some}\;s\in S\cap F_i\} \] for $i=1,\ldots,m$, and $S'=\bigcap S'_i$. \begin{trivlist} \item[] \bf Proposition 1. \it For a simplicial affine semigroup $S$ the semigroup ring $k[S]$ is Cohen-Macaulay (resp.\ Buchsbaum) if and only if $S'=S$ (resp.\ $S'+(S\setminus\{0\})\subseteq S$). \end{trivlist} For the proof see \cite{GSW}, \cite[Theorem 6.4]{St}, \cite[section 4]{TH}, and \cite[section 6]{SS}. The semigroup $S'$ is called the {\em Cohen-Macaulayfication} of $S$. Let \[\bar{S}=\{s\in G,\; ms\in S \;\mbox{for some}\; m\in{\Bbb N},m\not= 0\}. \] It is known \cite[section 1]{Ho} that $k[\bar{S}]$ is the normalization of $k[S]$. An affine semigroup $S$ is called {\em standard} if \begin{itemize} \item[(i)] $\bar{S}=G(S)\cap\N^n$. \item[(ii)] For all $i$ the image of $S$ under the the projection $\pi_i$ on the $i$-th component is a numerical semigroup, i.~e., the complement ${\Bbb N}\setminus\pi_i(S)$ is finite. \item[(iii)] The semigroups $S\cap \ker\:\pi_i$, $i=1,\ldots,n$, are distinct of rank equal to ${\rm rk}\: S-1$. \end{itemize} It was shown by Hochster \cite[section 2]{Ho} that every affine semigroup is isomorphic to a standard one. Hence we shall assume throughout that $S$ is standard. In that case the cone $C_S$ has exactly $n$ faces of dimension $r-1$, namely the convex cones spanned by the $S\cap\ker\:\pi_i$. Hence \[S'_i=\{\l\in\N^n,\;\l+s\in S\;\mbox{for some}\;s\in S\;\mbox{with}\; s_i=0\} \] for $i=1,\ldots,n$. A standard affine semigroup $S$ is simplicial if and only if $S$ has elements on every coordinate axis. In fact, the cone of a simplicial affine semigroup of rank $r$ has only $r$ faces of dimension $r-1$. Standardness gives $r=n$. Then the edges of $C_S$ are the intersections of $C_S$ with the coordinate axes, see \cite[section 1]{SS}. The reversed implication is obvious. Let $a_i\in{\Bbb N}$, $a_i\not=0$, be the minimal number such that $\a^i=(0,\ldots,0,a_i,0,\ldots,0)\in S$, where the nonzero entry is at the $i$-th place. \begin{trivlist} \item[] {\bf Proposition 2.} \it Every $k$-linear derivation $D$ of $k[S]$ extends uniquely to a derivation of the polynomial ring $k[t]$. \item[] Proof. \rm As $S\subseteq\N^n$ is standard and simplicial it has rank $n$ and $k[S]$ has dimension $n$. Hence the rational function field $k(t)$ is a separable finite extension of the quotient field $k(S)$ of $k[S]$. Therefore $D$ extends uniquely to a derivation $D$ of $k(t)$. Write $D=\sum f_i \partial_i$ with $f_i\in k(t)$, say $f_i=g_i/h_i$ with coprime $g_i,h_i\in k[t]$. With the semigroup elements $\a^i$ introduced above we have \[a_it_i^{a_i-1}f_i=D(t^{\a^i})\in k[S] \subseteq k[t] \] and $h_i$ divides $t_i^{a_i-1}$. As $\pi_i(S)$ is a numerical semigroup there is $s\in G$ with the $i$-th component $s_i=1$. Using simpliciality we may assume that $s\in\N^n$, hence $s\in\bar{S}$. It was shown by Seidenberg \cite{Se} that $D$ maps the normalization $k[\bar{S}]$ of $k[S]$ into itself. Then \[\sum s_jt^sf_j/t_j=D(t^s)\in k[\bar{S}]\subseteq k[t] \] implies $\prod_{j\not= i}t_j^{a_j-1}t^sf_i/t_i\in k[t]$. Hence $h_i$ divides $\prod_{j\not= i}t_j^{a_j-1}t^s/t_i$. But $t_i$ does not divide this product since $s_i=1$. Thus $h_i\in k$ and $f_i\in k[t]$. This means that $D$ restricts to a derivation of $k[t]$. \hfill $\Box$ \end{trivlist} By Proposition 2 the Lie algebra $\T(S)={\rm Der}\: k[S]$ of $k$-linear derivations of the semigroup ring may be viewed as a subalgebra of ${\Bbb D}={\rm Der}\: k[t]$. Let us first describe the latter Lie algebra. The derivations $D_i=t_i\partial_i$ span an Abelian subalgebra $H$. For a linear form $\l\in H^*$ let \[\D_\l=\{D\in{\Bbb D},\;[h,D]=\l(h)\cdot D\;\mbox{for all}\; h\in H\}. \] Then ${\Bbb D}$ admits a root space decomposition \[{\Bbb D}=\bigoplus_{\l\in H^*}\D_\l. \] Given the basis $D_1,\ldots,D_n$ of $H$ one may identify $H^*$ with $k^n$ by identifying the form $\l$ with the vector $(\l(D_1),\ldots,\l(D_n))$. Then the set of $\l\in H^*$ with $\D_\l\not=0$ equals \[\N^n\cup\{\l\in\Z^n,\;\l_i=-1\;\mbox{for exactly one}\;i\; \mbox{and}\;\l_j\ge0\;\mbox{for all}\;j\not= i\}. \] In fact, for $\l\in\N^n$ the root space $\D_\l$ is spanned by all $D_{\l j}=t^\l t_j\partial_j$, $j=1,\ldots,n$. In particular, ${\Bbb D}_0=H$. And if $\l\in\Z^n$ with $\l_i=-1$ and $\l_j\ge0$ for $j\not= i$ then $\D_\l$ is spanned by the single element $D_{\l i}=t^\l t_i\partial_i$. All these statements follow from the commutator relation \[[D_i,D_{\l j}]=\l_i\cdotD_{\l j}. \] In order to describe the subalgebra $\T(S)$ we need some more notation. Let \begin{eqnarray*} \L_i & = & \{\l\in\Z^n,\; \l+s\in S\;\mbox{for all}\; s\in S \;\mbox{with}\; s_i\not= 0\},\;i=1,\ldots,n \\ \L & = & \L(S) = \bigcup \L_i \\ \tilde{S} & = & \{\l\in\N^n,\; \l+(S\setminus\{0\})\subseteq S\}. \end{eqnarray*} \begin{trivlist} \item[] {\it Remarks.} (i) Let $n=1$. Then $k[S]$ is always Cohen-Macaulay, and the cardinality of $\L\setminus S$ equals the Cohen-Macaulay type of $k[S]$, see \cite{HK}. For $S={\Bbb N}$ one has $\tilde{S}={\Bbb N}$ and $\L=\tilde{S}\cup\{-1\}$. Otherwise $1\notin S$. Then our assumption that ${\Bbb N}\setminus S$ is finite implies $\L\subseteq{\Bbb N}$ and $\L=\tilde{S}$. \item[] (ii) Let $n\ge2$. From $\l+\a^i\in S$ for $\l\in\tilde{S}$ and two indices $i$ one sees $\tilde{S}\subseteq S'$. Hence $\tilde{S}=S'$ in the Buchsbaum case and $\tilde{S}=S$ in the Cohen-Macaulay case. \end{trivlist} \begin{trivlist} \item[] \bf Proposition 3. {\rm (i)} \it The Lie algebra $\T(S)$ admits a root space decomposition \[\T(S)=\bigoplus_{\l\in H^*}\T_\l. \] with $\T_\l=\T(S)\cap\D_\l$. \item[] {\rm(ii)} Suppose that $k[S]$ is Buchsbaum. Then the set of $\l\in H^*$ with $\T_\l\not=0$ equals $\L(S)$. If $\l\in\tilde{S}$ then $\T_\l$ is spanned by $D_{\lambda 1},\ldots,D_{\lambda n}$. And if $\l\in E_i=\L_i\setminus\tilde{S}$ then $\T_\l$ is spanned by the single element $D_{\l i}$. In particular, $\L(S)=\tilde{S}\cup\bigcup E_i$ is a disjoint union. \item[] \rm The elements of $\tilde{S}$ (resp.\ $E_i$) will be called {\em ordinary} (resp.\ {\em i-exceptional}) {\em roots}. \item[] {\it Proof.} (i) For $D_\l=\sum_ib_{\l i}D_{\l i}\in\D_\l$ one has $D_\l t^s=\sum_ib_{\l i}s_i\cdot t^{\l+s}$. Hence $\sum_\l D_\l\in\T(S)$ if and only if $\l+s\in S$ for all $s\in S$ and all occuring $\l$ with $\sum_i b_{\l i}s_i\not=0$ if and only if $D_\l\in\T(S)$ for all occuring $\l$. \item[] (ii) Consider $\l\in\tilde{S}$. Then $D_{\lambda 1},\ldots,D_{\lambda n}$ are defined and contained in $\T(S)$. Next consider $\l\in \L_i$. From $\l+\a^i\in S$ we see $\l_j\ge0$ for all $j\not= i$. Moreover, $\l_i\in\L(\pi_i(S))$ and Remark (i) above yields $\l_i\ge-1$. Hence $D_{\l i}$ is defined and contained in $\T(S)$. Conversely, if $D_{\l i}\in\T(S)$ then $\l\in \L_i$. The proof is completed by the following claim: If $\T_\l$ contains a linear combination of the $D_{\l i}$ with at least two non-vanishing coefficients then $\l\in\tilde{S}$. In fact, if $\sum_i b_iD_{\l i}\in\T(S)$ with $b_1,b_2\not=0$ then $\l+\a^1$ and $\l+\a^2$ are contained in $S$. This gives $\l\in S'\subseteq\tilde{S}$ as $k[S]$ is Buchsbaum. \hfill $\Box$ \end{trivlist} \begin{trivlist} \item[] \it Examples. \rm (i) (\cite[Remark 1.3]{MT}) Let $S\subseteq{\Bbb N}^2$ be generated by (0,10),(3,7),(7,3), (8,2),(10,0) and let $\l=(9,11)$. Then $\l+(3,7)\notin S$ but $\l+s\in S$ for the remaining generators $s$. Hence $\l\in S'\setminus\tilde{S}$ and $k[S]$ is not Buchsbaum. Moreover, $\l\notin\L(S)$ but $\T_\l\not=0$. In fact, $7D_{\l 1}-3D_{\l 2}\in\T_\l$. \item[] (ii) Let $S\subseteq{\Bbb N}^2$ correspond to the affine cone over the $d$-uple embedding of $\P^1$ in $\P^d$, $d\ge2$, i.~e., $S$ is generated by $(0,d),(1,d-1),\ldots,(d-1,1),(d,0)$. Then $k[S]$ is normal and Cohen-Macaulay. The exceptional roots are $(-1,1)+m(0,d)$ and $(1,-1)+m(d,0)$ with $m\in{\Bbb N}$. \item[] (iii) Let $S\subseteq{\Bbb N}^2$ correspond to the product of a cusp with a line, i.~e., $S$ is generated by $(2,0)$, $(3,0)$ and $(0,1)$. Then $k[S]$ is Cohen-Macaulay. The 1-exceptional roots are $(1,0)+m(0,1)$ with $m\in{\Bbb N}$. The 2-exceptional roots are $(0,-1)+m(2,0)$ and $(3,-1)+m(2,0)$ with $m\in{\Bbb N}$. \end{trivlist} Examples (ii) and (iii) illustrate the second part of the next result. \begin{trivlist} \item[] \bf Proposition 4. \it {\rm (i)} $\tilde{S}$ is a finitely generated subsemigroup of $\N^n$. \item[] {\rm (ii)} Suppose that $k[S]$ is Buchsbaum and $n\ge2$. For fixed $i$ let $A_i$ be the semigroup generated by all $\a^j$ with $j\not= i$. Then the set $E_i$ of $i$-exceptional roots is a finitely generated $A_i$-module. \item[] Proof. \rm (i) Clearly $\tilde{S}$ is a subsemigroup of $\N^n$. Let $A$ be the semigroup generated by $\a^1,\ldots,\a^n$. We show more generally that every subsemigroup $T\subseteq\N^n$ containing $A$ is finitely generated. Let $a_i$ be the nonzero entry of $\a^i$. For $\b\in\N^n$ with $\b_i<a_i$ for all $i$ let $T_\b=(\b+A)\cap T$. By Dickson's Lemma each $T_\b$ is a finitely generated $A$-module (or empty). Since $T=\bigcup T_\b$ is a finite union, $T$ is finitely generated as an $A$-module and hence as a semigroup. \item[] (ii) We may assume $i=1$. If $\l\in E_1=\L_1\setminus\tilde{S}$ then clearly $\l+\a^2\in\L_1$. Moreover, $\l+\a^1\in S$ so that $\l\in S'_i$ for $i\ge2$. If $\l+\a^2\in\tilde{S}$ then $\l+2\a^2\in S$, hence $\l\in S'_1$ and $\l\in S'=\tilde{S}$, contradiction. Thus $\l+\a^2\in E_1$. This proves that $E_1$ is an $A_1$-module. It remains to show that it is finitely generated. For $\gamma\in{\Bbb N}\times\{0\}\subseteq\N^n$ and $\b\in\{0\}\times{\Bbb N}^{n-1}\subseteq \N^n$ with $\b_i<a_i$ for all $i$ let $E_{\gamma\b}=(\gamma+\b+A_1)\cap E_1$. As above this is a finitely generated $A_1$-module (or empty). If $E_{\gamma\b}\not=\emptyset$ and $\gamma'=\gamma+m\a^1$ for some $m\in{\Bbb N}$, $m\not=0$ then $E_{\gamma'\b}=\emptyset$. Otherwise, there is $\l\in A_1$ with $\gamma+\b+\l,\gamma'+\b+\l\in E_1$, contradicting $\gamma'+\b+\l=\gamma+\b+\l+m\a^1\in S\subseteq\tilde{S}$. Since there are only finitely many congruence classes of ${\Bbb N}$ modulo $\a^1$ the Proposition is proven. \hfill $\Box$ \end{trivlist} \section{Reconstruction of the semigroup} Before we explain how to reconstruct the semigroup $S$ from its Lie algebra $\T(S)$ we make a remark concerning the reconstruction of $S$ from its semigroup ring $k[S]$ discussed by Gubeladze \cite{Gu}. Consider the augmentation $k[S]\to k$ defined by $t^s\mapsto 0$ for all $s\in S\setminus\{0\}$. Gubeladze \cite[Theorem 2.1]{Gu} proved that affine semigroups $S_1$ and $S_2$ are isomorphic if $k[S_1]$ and $k[S_2]$ are isomorphic as augmented algebras. Moreover \cite[Lemma 2.8]{Gu}, if $k[S_1]$ and $k[S_2]$ are normal and isomorphic just as algebras then they are isomorphic as augmented algebras. We shall extend this result (for simplicial semigroups) to the Buchsbaum case. \\[1ex] Let us say that $S$ corresponds to a {\em product along a line} if, after permutation of coordinates, $S={\Bbb N}\oplus M$ for some semigroup $M\subseteq{\Bbb N}^{n-1}$. We shall see that this property only depends on the algebra $k[S]$ and even on the Lie algebra $\T(S)$. Let $L=[\T(S),\T(S)]$ be the derived algebra. \begin{trivlist} \item[] \bf Proposition 5. \it Suppose that $k[S]$ is Buchsbaum. Then the following are equivalent: \begin{itemize} \item[{\rm (a)}] The semigroup $S$ corresponds to a product along a line. \item[{\rm (b)}] There is $\l\in\L(S)$ with $|\l|<0$. \item[{\rm (c)}] $L=\T(S)$. \end{itemize} \item[] Proof. \rm (a) $\Leftrightarrow$ (b) If $(-1,0,\ldots,0)$ is a root then $(1,0,\ldots,0)\in S$ and $S={\Bbb N}\oplus M$ with $M=S\cap\ker\:\pi_1$. The converse is clear. \item[] (b) $\Rightarrow$ (c) Here and later we use the commutator relation \[[D_{\l i},D_{\m j}]=\m_iD_{\l+\m,j}-\l_jD_{\l+\m,i}. \] It shows $\bigoplus_{\l\not=0}\T_\l\subseteq L$. Let $\l=(-1,0,\ldots,0)\in\L$ so that $\mu=(1,0,\ldots,0)\in S\subseteq\tilde{S}$. Then $L$ contains $2D_1=[D_{\l 1},D_{\mu 1}]$ and $D_j=[D_{\l 1},D_{\mu j}]$ for $j\ge2$. Thus $\Theta_0=H\subseteq L$. \item[] (c) $\Rightarrow$ (b) Assume that $|\l|\ge0$ for all roots $\l$. Then $\eta^1+\eta^2=0$ for roots $\eta^1,\eta^2\not=0$ is possible only if (after permutation of coordinates) $\eta^1=(-1,1,0,\ldots,0)$, $\eta^2=(1,-1,0,\ldots,0)$. In this case $[D_{\eta^1,1},D_{\eta^2,2}]=D_2-D_1$. Since $\Theta_0$ is Abelian we obtain \[L\subseteq\bigoplus_{\l\not=0}\T_\l\oplus<D_n-D_1,\ldots,D_2-D_1> \] and $\Theta_0\not\subseteq L$. \hfill $\Box$ \end{trivlist} \begin{trivlist} \item[] \bf Proposition 6. \it Suppose that $k[S_1]$ and $k[S_2]$ are Buchsbaum. \item[] {\rm (i)} If $k[S_1]$ and $k[S_2]$ are isomorphic as algebras then they are isomorphic as augmented algebras. \item[] {\rm (ii)} If $S_1$ and $S_2$ do not correspond to products along a line then every algebra isomorphism $\phi:k[S_1]\to k[S_2]$ is augmented. \item[] Proof. \rm Let $I\subseteq k[S_2]$ be a proper differential ideal, i.~e., $D(I)\subseteq I$ for every $D\in\Theta(S_2)$. We claim that $I$ is generated by some monomials $t^s$, $s\in S_2$. In particular, $I$ is contained in the augmentation ideal generated by all $t^s$, $s\in S_2\setminus\{0\}$. Given $f=\sum b_st^s\in I$ fix any $s$ with $b_s\not=0$. Take any of the remaining $\l\in S_2$ with $b_\l\not=0$ and choose $j$ with $\l_j\not= s_j$. Then $\sum_\mu(\l_j-\mu_j)b_\mu t^\mu=\l_j f-D_j(f)\in I$ contains less monomials than $f$ but still the monomial $t^s$. Repeated application yields $t^s\in I$, proving the claim. \\[1ex] Now assume $S_1={\Bbb N}^m\oplus M$ for some $M\subseteq {\Bbb N}^{n-m}$ which does not correspond to a product along a line. Let $J$ be the ideal of $k[S_1]$ generated by all $t^\mu$, $\mu\in M\setminus\{0\}$. We claim that $J$ is differential. Consider any $\l\in\L_i$, $i=1,\ldots,n$. In order to show $D_{\l i}(t^\mu)=\m_i t^{\l+\mu}\in J$ we may assume $\m_i\not=0$. Then $\l+\mu\in S_1$. From $|\mu|\ge2$ we conclude $\l+\mu=\nu+\mu'$ with $\nu\in{\Bbb N}^m$ and $\mu'\in M\setminus\{0\}$. Hence $t^{\l+\mu}=t^{\nu+\mu'}\in J$. \\[1ex] Let $\phi:k[S_1]\to k[S_2]$ be an algebra isomorphism. It induces a Lie algebra isomorphism $\phi^\sharp:\Theta(S_1)\to\Theta(S_2)$ by $D\mapsto\phi\circ D\circ\phi^{-1}$. Since $J$ is differential its image in $k[S_2]$ is differential and hence contained in the augmentation ideal of $k[S_2]$. We have $k[S_1]=k[M][t_1,\ldots,t_m]$. For $i=1,\ldots,m$ let $c_i$ be the constant term of $\phi(t_i)$. Define the $k[M]$-automorphism $\psi$ of $k[S_1]$ by $\psi(t_i)=t_i-c_i$, $i=1,\ldots,m$. Then the $\phi\circ\psi(t_i)$ have no constant term. Since the augmentation ideal of $k[S_1]$ is generated by $t_1,\ldots,t_m$ and $J$ this means that $\phi\circ\psi$ is augmented. Assertion (ii) now also is clear because in that case $J$ equals the augmentation ideal. \hfill $\Box$ \end{trivlist} \begin{trivlist} \item[] \bf Theorem 1. \it Let $S_1,S_2$ be simplical affine semigroups such that $k[S_1],k[S_2]$ are Buchsbaum. Suppose that the Lie algebras $\Theta(S_1),\Theta(S_2)$ are isomorphic. Then $S_1,S_2$ have the same rank and the semigroups $\tilde{S_1},\tilde{S_2}$ are isomorphic. \item[] Proof. \rm If $\Theta(S_1)$ equals its derived algebra then $S_1$ and $S_2$ correspond to products along a line. By a result of Skryabin \cite[Theorem 2]{Sk} the semigroup rings $k[S_1],k[S_2]$ are isomorphic. Then \cite[Theorem 2.1]{Gu} and Proposition 6 imply that the semigroups $S_1,S_2$ themselves are isomorphic. Now suppose that the derived algebra is strictly smaller than $\Theta(S_1)$. Then $|\l|\ge 0$ for all $\l\in\L(S_1)$. As $[\T_\l,\T_\m]\subseteq\T_{\l+\m}$ for all roots $\l,\mu$ the subspaces $I_d=\bigoplus_{|\l|\ge d}\T_\l$ are ideals of $\Theta(S_1)$ with finite dimensional quotients $\Theta(S_1)/I_d$ and $\bigcap_{d\in{\Bbb N}}I_d=0$. Given an isomorphism $\Theta(S_1)\simeq\Theta(S_2)$ we obtain an Abelian subalgebra $H_2$ of $\Theta(S_1)$ and another root space decomposition $\Theta(S_1)=\bigoplus_{\mu\in H_2^*}\T_\m'$. Every finite dimensional subspace of $\Theta(S_1)$ is mapped isomorphically onto its image in $\Theta(S_1)/I_d$ if $d$ is sufficiently large. Thus, for $d\gg 0$, $H_2$ embeds into $Q=\Theta(S_1)/I_d$. For $\mu\in H_2^*$ consider the root spaces \[Q_{\mu}'=\{D\in Q,\; [h,D]=\mu(h)\cdot D\;\mbox{for all}\;h\in H_2\}. \] Their sum is direct. Since each $\T_\m'$ is mapped into $Q_{\mu}'$ and the images of the $\T_\m'$ span $Q$ we see $Q=\bigoplus_{\mu\in H_2^*}Q_{\mu}'$ and that each $\T_\m'$ is mapped onto $Q_{\mu}'$. In particular, $Q_0'=H_2$. It follows that $H_2$ equals its normalizer in $Q$ and hence is a Cartan subalgebra of $Q$. Using Proposition 3, Remark (i) preceding it, and Proposition 4 we may assume that the subsemigroup of $H_2^*$ generated by all $\mu$ with $\dim\:Q_{\mu}'=\dim\:H_2={\rm rk}\: S_2$ equals $\tilde{S_2}$. Analogous statements hold true for $H_1$ and $d\gg 0$. Since $Q$ is finite dimensional there is an automorphism of $Q$ mapping the Cartan subalgebra $H_1$ onto the second Cartan subalgebra $H_2$, \cite[section 16]{Hu}. Its dual induces an isomorphism between the semigroups $\tilde{S_1}$ and $\tilde{S_2}$. \hfill $\Box$ \end{trivlist} Using Remark (ii) preceding Proposition 3 we conclude \begin{trivlist} \item[] \bf Corollary 1. \it Simplicial affine semigroups $S$ of rank $\ge 2$ with $k[S]$ Cohen-Macaulay are uniquely determined by their Lie algebra $\T(S)$. \end{trivlist} Look again at Gubeladze's Theorem that $S$ is uniquely determined by the augmented algebra $k[S]$. In the above proof we applied this only in case $S$ does correspond to a product along a line. Therefore, using the Lie algebra $\T(S)$ as an intermediate step, we have reproven Gubeladze's Theorem in the special case that $S$ is simplicial, does not correspond to a product along a line, and $k[S]$ is Cohen-Macaulay of dimension $\ge 2$. But $\T(S)$ cannot distinguish between semigroups with the same Cohen-Macaulayfication: \begin{trivlist} \item[] \it Examples. \rm (i) Fix $d,l\in{\Bbb N}$, both $\ge2$. Let $S$ consist of all $s\in{\Bbb N}^2$ with $|s|=md$, $m\ge l$. Then $k[S]$ is Buchsbaum and the Cohen-Macaulayfication $S'$ is generated by $(0,d),(1,d-1),\ldots,(d-1,1),(d,0)$. Both $S$ and $S'$ have the same exceptional roots, see Example (ii) after Proposition 3. Hence $\T(S)=\Theta(S')$, independently of $l$. \item[] (ii) Let $S_1$ (resp.\ $S_2$) be generated by all $\l\in{\Bbb N}^2$ with $|\l|=6$ except $\l=(3,3)$ (resp.\ $\l=(2,4)$). They have a Buchsbaum semigroup ring and the same Cohen-Macaulayfication generated by all $\l\in{\Bbb N}^2$ with $|\l|=6$. In both cases the exceptional roots are $(-1,7)+m(0,6)$ and $(7,-1)+m(6,0)$ with $m\in{\Bbb N}$. Hence $\Theta(S_1)=\Theta(S_2)$. But $S_1,S_2$ are not isomorphic. In fact, any isomorphism would map the set of extremal elements $\{(6,0),(0,6)\}$ onto itself, hence $(6,6)$ onto $(6,6)$. This contradicts $(6,6)=2(3,3)$ in $S_2$ but $(6,6)\not=2s$ for all $s\in S_1$. Observe that both semigroups correspond to affine cones over smooth projective curves in $\P^5$. \end{trivlist} In the rank 1 case the situation is different. Although the semigroup ring always is Cohen-Macaulay the semigroup is, in general, not determined by the Lie algebra: \begin{trivlist} \item[] \it Examples. \rm (i) The numerical semigroups generated by 2 and 3 (resp.\ 3, 4 and 5) have the same $\tilde{S}={\Bbb N}$, hence the same Lie algebra. Observe that the semigroup ring is Gorenstein in the first case whereas it has Cohen-Macaulay type 2 in the second, see Remark (i) preceding Proposition 3. \item[] (ii) The numerical semigroups generated by 3, 7 and 8 (resp.\ 4, 5 and 7) have the same $\tilde{S}$ generated by 3, 4 and 5, hence the same Lie algebra. Observe that the Cohen-Macaulay type is 2 in both cases. \end{trivlist} \begin{trivlist} \item[] \bf Corollary 2. \it Numerical semigroups $S$ with $k[S]$ Gorenstein are uniquely determined by $\T(S)$ and even by the finite dimensional Lie algebra $\T(S)/[L,L]$. \item[] Proof. \rm If $L=\T(S)$ then $S={\Bbb N}$. So suppose $L\not=\T(S)$. Then $\tilde{S}$ is the set of roots and $L=\bigoplus_{\l\not=0}\T_\l$. This implies $\T_\l\cap[L,L]=0$ for $\l$ in the minimal generator system of $\tilde{S}$ and $\T_\l\subseteq[L,L]$ for every $\l$ which can be decomposed as $\l=\mu+\nu$ with two different $\mu,\nu\in\tilde{S}$. We see that $\T(S)/[L,L]$ is finite dimensional and that we can use the intrinsically defined ideal $[L,L]$ instead of $I_d$ in the proof of Theorem 1. It remains to show that $S$ is uniquely determined by $\tilde{S}$ in the Gorenstein case. By \cite[Satz 1.9, Proposition 2.21]{HK} we know $\tilde{S}=S\cup\{c-1\}$ with the conductor $c$ of $S$. Consider first the case $\tilde{S}={\Bbb N}$. Then $S$ must be the semigroup ${\Bbb N}\setminus\{1\}$, generated by 2 and 3. Now let $\tilde{S}\not={\Bbb N}$. Let $a$ be the smallest element of $S$ different from $0$. As $S$ is a symmetric semigroup we see $c-2,\ldots,c-a\in S$ but $c-a-1\notin S$. Thus $\tilde{S}$ has conductor $c-a$. Then $c-a\in S\setminus\{0\}$ implies $c-1> c-a\ge a$. Hence $a$ is the smallest element of $\tilde{S}$ different from $0$. Therefore, $S=\tilde{S}\setminus\{c-1\}$ is determined via $c-a$ and $a$ by $\tilde{S}$. \hspace*{\fill} $\Box$ \end{trivlist} \section{Automorphisms of the Lie algebra} Every automorphism $\phi$ of $k[S]$ induces a Lie algebra automorphism \[\phi^{\sharp}:\T(S)\to\T(S):D\mapsto\phi\circ D\circ\phi^{-1}. \] The purpose of this section is to show \begin{trivlist} \item[] \bf Theorem 2. \it Let $S$ be a simplicial affine semigroup such that $k[S]$ is Cohen-Macaulay. For every automorphism $\Phi$ of $\T(S)$ there is a unique automorphism $\phi$ of $k[S]$ such that $\Phi=\phi^{\sharp}$. \item[] Proof. \rm If $\Phi=\phi^{\sharp}$ then $\Phi(f\cdot\Phi^{-1}(D))=\phi(f)\cdot D$ for all $f\in k[S]$ and $D\in\T(S)$. This shows uniqueness. Now take an arbitrary automorphism $\Phi$ of $\T(S)$. If $S$ corresponds to a product along a line the assertion follows from \cite[Theorem 2]{Sk}. By Proposition 5 we may assume $|\l|\ge 0$ for all $\l\in\L$. The Lie algebra $\T(S)$ is graded by $\Theta^d=\bigoplus_{|\l|=d}\T_\l$. Note that $\Theta^0$ consists of the linear vector fields in $\T(S)$, i.~e., those $\sum f_i\partial_i\in\T(S)$ where the $f_i$ are linear forms in the variables $t_i$. The homogeneous component of smallest degree of $D\in\T(S)$, $D\not=0$, will be called the leading form of $D$. We claim that every $h'=\Phi(h)\in\Phi(H)$, $h'\not=0$, has leading form of degree zero. In fact, choose $\l\in\L$ with $\l(h)\not=0$ and $Y\in\Phi(\T_\l)$, $Y\not=0$. Comparison of leading forms in $[h',Y]=\l(h)\cdot Y$ yields the claim. Hence the leading forms of the vector fields $Y_i=\Phi(D_i)$, $i=1,\ldots,n$, are linear vector fields and linearly independent. We can thus find a point $p$ in affine space ${\Bbb A}^n$ such that the tangent vectors $Y_1(p),\ldots,Y_n(p)$ are linearly independent. Now consider the polynomial ring $k[t]$ as a subring of the ring ${\cal F}=k[[t-p]]$ of formal power series centered at $p$ and ${\rm Der}\: k[t]$ as a subalgebra of the Lie algebra ${\rm Der}\: {\cal F}$. By Proposition 7 below there are formal coordinates $s_1,\ldots,s_n$ at $p$, i.~e., elements of ${\cal F}$ vanishing at $p$ with $k[[s]]={\cal F}$, such that $Y_i=\d_{s_i}$ in ${\rm Der}\: {\cal F}$. Let $x_i=\exp\; s_i$ for $i=1,\ldots,n$. If $\l\in\Z^n$ then \[Y_i(x^\l)=\l_i\cdot x^\l \quad\mbox{for}\quad i=1,\ldots,n \] and, up to multiplication with a constant, $x^\l$ is the unique element of ${\cal F}$ with this property. This implies that for $\l\in\Z^n$ the root space \[\T_\l'=\{D\in{\rm Der}\:{\cal F},\;[Y_i,D]=\l_i\cdot D\;\mbox{for all}\;i\} \] is spanned by the $Y_{\l i}=x^\l Y_i$, $i=1,\ldots,n$. We conclude that $\Phi(\T_\l)=\T_\l'$ for ordinary roots $\l\in\tilde{S}$ and $\Phi(\T_\l)\subseteq\T_\l'$ for exceptional roots $\l\in\bigcup E_i$. Next we claim \[\Phi(D_{\l i})=b_{\l i}Y_{\l i} \quad\mbox{for all $\l$ and $i$} \] with suitable constants $b_{\l i}\not=0$. To prove this, note that $[D_{\l i},D_{\m j}]=-\l_j D_{\l+\m,i}$ if $\m_i=0$ and thus $Y=\Phi(D_{\l i})$ has the following property: For all $\mu\in\tilde{S}$ with $\m_i=0$ the image of ${\rm ad}\: Y: \T_\m'\to\T_{\l+\m}'$ has dimension $\le 1$. Hence it is enough to show that, up to multiplication with a constant, $Y_{\l i}$ is the unique element of $\T_\l'$ with this property. In fact, for $Y=\sum_k c_kY_{\l k}$ the matrix of coefficients of $([Y,Y_{\mu j}])_j$ with respect to the basis $(Y_{\l+\mu,k})_k$ has determinant equal to the value at $\sum c_k\mu_k$ of the characteristic polynomial of the matrix $(\l_j c_k)_{j,k}$. This value does not vanish for a suitable choice of $\mu\in\tilde{S}$ with $\m_i=0$ if $c_k\not=0$ for some $k\not= i$, and the claim is proven. \\[1ex] For fixed $\l$ choose $\mu\in\tilde{S}$ with $\mu_1\not=\l_1$ and $\m_i\not=0$ for $i\not=1$. Then the usual commutator relation implies $b_{\l i}b_{\mu 1}=b_{\l+\mu,1}$ for all $i$. Hence the $b_{\l i}$ are independent of $i$, say $b_{\l i}=b_\l$. We have \[\Phi(D_{\l i})=b_\lY_{\l i} \quad\mbox{for all $\l$ and $i$}. \] Denote by $\Gamma$ the subgroup of $\Z^n$ generated by $\L$. As $b_\l b_\mu=b_{\l+\mu}$ for all $\l,\mu\in\L$ with $\l+\mu\in\L$ the map $\l\mapsto b_\l$ can be extended to a homomorphism $\Gamma\to k^*$. The group $\Gamma$ is free of rank $n$, say generated by $\gamma_1,\ldots,\gamma_n$. There is a rational matrix $Q=(q_{ij})_{i,j}$ such that $l=Q\cdot\l$ if $\l=\sum l_i\gamma_i\in\Gamma\subseteq\Z^n$ and $l=(l_1,\ldots,l_n)\in\Z^n$. Write $q_{ij}=r_{ij}/s$ with integers $r_{ij},s$, choose $\zeta_i\in k$ such that $b_{\gamma_i}=\zeta_i^s$, and let $c_j=\prod_i \zeta_i^{r_{ij}}$. Then $b_\l=c^\l$ for all $\l\in\Gamma$. Thus, if we replace the $x_j$ by $c_jx_j$ we obtain for the new $Y_{\l i}=x^\l Y_i$ the equations \[\Phi(D_{\l i})=Y_{\l i}\quad\mbox{for all $\l$ and $i$}. \] We have seen above that $\T(S)$ is spanned by all $x^\l Y_i$ with $\l\in\L_i$ and $i=1,\ldots,n$. Using $|\l|\ge0$ for all $\l$ it is easy to show that each $\L_i$ is an $\tilde{S}$-module. Hence $\T(S)$ is a module over the subalgebra of ${\cal F}$ generated by all $x^s$, $s\in\tilde{S}$. Fix $s\in\tilde{S}$. From $x^st_i\d_i=x^sD_i\in\T(S)\subseteq{\rm Der}\: k[t]$ we conclude that the element $x^st_i$ of ${\cal F}$ actually is contained in the subalgebra $k[t]$. Since the same is true for $x^{2s}t_i$ we obtain $x^s\in k[t]$. Even more: $x^sD_i\in\T(S)$ for $i=1,\ldots,n$ shows $x^s\in k[\tilde{S}]$. The $x_i$ are algebraically independent. Therefore, an algebraic relation between finitely many $t^{s_1},\ldots,t^{s_m}$ holds if and only if the same relation holds between $x^{s_1},\ldots,x^{s_m}$. This means that we can define an injective homomorphism $\phi:k[\tilde{S}]\to k[\tilde{S}]$ by $\phi(t^s)=x^s$. The equations $\Phi(t^\l D_i)=x^\l Y_i$ established above translate into $\Phi(D)\circ\phi=\phi\circ D$ for all $D\in\T(S)$. Using $\Phi^{-1}$ instead of $\Phi$ we get an injective endomorphism $\psi$ of $k[\tilde{S}]$ with $D\circ\psi=\psi\circ\Phi(D)$ for all $D$. Then $D_i\circ\psi\circ\phi=\psi\circ\phi\circ D_i$ for all $i$. Using this information one shows that $\psi\circ\phi$ maps each one-dimensional subspace of $k[\tilde{S}]$ spanned by some $t^s$ into itself. Hence injectivity of $\phi$ and $\psi$ implies surjectivity of both. In case $n\ge2$ we are done because then $S=\tilde{S}$ by our hypothesis on $k[S]$ and $\phi$ is an automorphism of $k[S]=k[\tilde{S}]$ with $\Phi(D)=\phi\circ D\circ\phi^{-1}$ for all $D\in \T(S)$, i.~e., $\Phi=\phi^\sharp$. \\[1ex] Finally, consider the case $n=1$. Then $x$ is a single element of ${\cal F}$ with $x^s\in k[t]$ for all $s\in\tilde{S}$. Since $\tilde{S}$ is a numerical semigroup $x$ must be contained in $k(t)$ and, being integral over $k[t]$, even in $k[t]$. As $\phi:t^s\mapsto x^s$ defines an automorphism of $k[\tilde{S}]$ the polynomial $x$ has degree 1, say $x=a+bt$. We had $Y(x)=x$ for $Y=\Phi(t\d_t)\in\T(S)$. This implies $Y=(a/b+t)\d_t$ and $a=0$ because $S\not={\Bbb N}$, i.~e., $-1$ is not a root. Therefore, $\phi$ restricts to an automorphism of $k[S]$ with $\Phi=\phi^\sharp$. \hspace*{\fill} $\Box$ \end{trivlist} It remains to show \begin{trivlist} \item[] \bf Proposition 7. \it Let ${\cal F}=k[[t_1,\ldots,t_n]]$. Suppose that $Y_1,\ldots,Y_n\in{\rm Der}\: {\cal F}$ satisfy $[Y_i,Y_j]=0$ for all $i,j$ and that $Y_1(0),\ldots,Y_n(0)$ are linearly independent. Then there are formal coordinates $s_1,\ldots,s_n$ such that $Y_i=\d_{s_i}$ for all $i$. \item[] Proof. \rm Write $Y_i=\sum_jf^i_j\d_{t_j}$. By hypothesis the matrix $F=(f^i_j)_{i,j}$ is invertible over ${\cal F}$, say with inverse $G=(g^j_k)_{j,k}$. Application of $Y_m$ to \[\sum_jf^i_jg^j_k=\delta^i_k \] yields \renewcommand{\theequation}{\fnsymbol{equation}} \begin{equation} \sum_{l,j}f^m_lf^i_j(\d_{t_l}g^j_k) =-\sum_{l,j}f^m_l(\d_{t_l}f^i_j)g^j_k \end{equation} The hypothesis $[Y_m,Y_i]=0$ means \[\sum_lf^m_l(\d_{t_l}f^i_j)=\sum_lf^i_l(\d_{t_l}f^m_j) \] for all $j$. Hence we may interchange $i$ and $m$ in the right hand side and, therefore, in the left hand side of (*). After renaming the summation indices we obtain \[\sum_{l,j}f^m_lf^i_j(\d_{t_l}g^j_k-\d_{t_j}g^l_k)=0. \] Invertibility of $F$ implies \[\d_{t_l}g^j_k=\d_{t_j}g^l_k \] for all $l$, $j$ and $k$. This condition is equivalent (over a field of characteristic 0) to the existence of $s_1,\ldots,s_n\in{\cal F}$ vanishing at 0 with \[g^j_k=\d_{t_j}s_k \] for all $j$ and $k$. These $s_k$ form a system of coordinates because $G$ is invertible. And clearly $Y_is_k=\delta^i_k$ for all $i,k$. \hfill $\Box$ \renewcommand{\theequation}{\arabic{equation}} \setcounter{equation}{0} \end{trivlist} \begin{trivlist} \item[] \it Remark. \rm Proposition 7 is the special case $r=n$ of a more general statement involving an arbitrary number $r\le n$ of vector fields. The latter usually is stated for differentiable or analytic vector fields over the fields of real or complex numbers and appears in the literature in connection with Frobenius' Theorem. It is surely known to hold for formal power series vector fields over arbitrary fields of characteristic 0. But lacking an explicit reference we have chosen to provide the very simple proof above. \end{trivlist} \section{Derivations of the Lie algebra} In this section we show \begin{trivlist} \item[] \bf Theorem 3. \it Let $S\subseteq\N^n$ be a simplicial affine semigroup such that $k[S]$ is Buchsbaum. Then every derivation $\Delta$ of $\T(S)$ is inner: $\Delta={\rm ad}\: D$ for some $D\in\T(S)$. \item[] Proof. \rm The cochain complex of the Lie algebra $\T(S)$ with coefficients in the adjoint representation has a $\Z^n$-grading given by the root space decomposition. By \cite[Theorem 1.5.2b]{F} it is acyclic in degrees different from zero. Hence we may assume that the given $\Delta$ has degree $0$, i.~e.\ $\Delta(\T_\l)\subseteq\T_\l$ for all $\l$. For each root $\l$ denote by $M(\l)$ the set of $i$ such that $D_{\l i}\in\T(S)$. Thus $M(\l)=\{1,\ldots,n\}$ for ordinary roots and $M(\l)=\{i\}$ for $i$-exceptional roots. We have \begin{equation} \Delta(D_{\l i})=\sum_{m\in M(\l)}b_{\l im}D_{\l m} \quad\mbox{for}\quad i\in M(\l) \end{equation} with suitable constants $b_{\l im}\in k$. The brackets of the generators are given by \begin{equation} [D_{\l i},D_{\m j}]=\m_i D_{\l+\m,j}-\l_j D_{\l+\m,i} \end{equation} Inserting (1) and (2) into the cocycle condition \[\Delta([D_{\l i},D_{\m j}])=[\Delta(D_{\l i}),D_{\m j}]+[D_{\l i},\Delta(D_{\m j})] \] gives \begin{eqnarray*} \lefteqn{ \sum_m(\m_i\cdot b_{\l+\mu,j,m}-\l_j\cdot b_{\l+\mu,i,m})D_{\l+\mu,m}} \hspace{2cm} \\ & = & \sum_m(\m_i\cdot b_{\mu jm}-\l_j\cdot b_{\l im}) D_{\l+\mu,m} \\ & & +(\sum_m\mu_m\cdot b_{\l im})D_{\l+\m,j} -(\sum_m\l_m\cdot b_{\mu jm})D_{\l+\m,i}. \end{eqnarray*} By comparing the coefficients one obtains \begin{equation} \m_i\cdot b_{\l+\mu,j,m}-\l_j\cdot b_{\l+\mu,i,m}= \m_i\cdot b_{\mu jm}-\l_j\cdot b_{\l im} \quad \mbox{for} \quad m\not=i,j \end{equation} \begin{equation} \m_i\cdot b_{\l+\mu,j,j}-\l_j\cdot b_{\l+\mu,i,j}= \m_i\cdot b_{\mu jj}-\l_j\cdot b_{\l ij} +\sum_m\mu_m\cdot b_{\l im} \quad \mbox{for} \quad j\not=i \end{equation} \begin{equation} (\m_i-\l_i)b_{\l+\mu,i,i}=\m_i\cdot b_{\mu ii}-\l_i\cdot b_{\l ii} +\sum_m\mu_m\cdot b_{\l im}-\sum_m\l_m\cdot b_{\mu im} \end{equation} Equation (4) with $\l=\mu=\a^j$ yields \begin{equation} b_{2\a^j,i,j}=0\quad \mbox{for} \quad i\not=j \end{equation} Let us show that $b_{\l ij}=0$ for all $\l\in\tilde{S}$ and all $i,j\in M(\l)$ with $i\not= j$. Set $\mu=2\a^j$. In case $\l_i=0$ the claim follows from (5) and (6). If $\l_i\not=0$ use (3) with $j=i$ and $m$ replaced by $j$ to show $b_{\l+\mu,i,j}=b_{\l ij}$. Then (4) gives the claim. \\[1ex] Now we have \[\Delta(D_{\l i})=b_{\l i}D_{\l i}\quad\mbox{for}\quad i\in M(\l) \] with suitable $b_{\l i}\in k$. Equations (4) and (5) reduce to \begin{equation} \m_i\cdot b_{\l+\mu,j}=\m_i\cdot b_{\mu j}+\m_i\cdot b_{\l i} \quad\mbox{for}\quad j\not=i \end{equation} \begin{equation} (\mu_j-\l_j)b_{\l+\mu,j}=(\mu_j-\l_j)(b_{\l j}+b_{\mu j}) \end{equation} For fixed $\l\in\tilde{S}$ the coefficients $b_{\l i}$ are independent of $i\in M(\l)$. In fact, for $j\not=i$ apply (7) and (8) where $\mu$ is any element of $\tilde{S}$ with $\m_i\not=0$ and $\mu_j\not=\l_j$. Thus we may write $b_\l$ instead of $b_{\l i}$. \\[1ex] Consider first the case $n\ge2$. Then (7) implies $b_{\l+\mu}=b_\l+b_\mu$ for $\l,\mu\in\tilde{S}$. Let $c_i=b_{\a^i}/a_i$ where $a_i$ denotes the nonzero entry of $\a^i$. Using the fact that $\tilde{S}$ is torsion modulo the semigroup generated by the $\a^i$ one shows $b_\l=\sum_ic_i\l_i$ for all $\l\in\tilde{S}$. The same is seen to hold for $\l\in\L_i$ by applying (7) with some $\mu\in S$, $\m_i\not=0$. We have proven \[[\sum_ic_i D_i,D_{\l j}]=\sum_ic_i\liD_{\l j}=b_\lD_{\l j}=\Delta(D_{\l j}) \] for all $\l\in\L$ and $j\in M(\l)$. This means $\Delta={\rm ad}\: D$ for $D=\sum_ic_iD_i$. \\[1ex] In the case $n=1$ only equation (8) is available. Then $b_{5\l}=b_{3\l}+b_{2\l}=2b_{2\l}+b_\l$ and $b_{5\l}=b_{4\l}+b_\l=b_{3\l}+2b_\l=b_{2\l}+3b_\l$, hence $b_{2\l}=2b_\l$ and then $b_{m\l}=m b_\l$ for all $m\in{\Bbb N},\l\in\tilde{S}$ with $m,\l>0$. This shows that the ratio $b_\l/\l$ is independent of $\l$, say $b_\l/\l=c$. Hence $b_\l=c\l$ for all positive roots. Since the same clearly holds for $\l=0$ (and $\l=-1$ in the special case $S={\Bbb N}$) we have again shown that $\Delta$ is inner. \hfill $\Box$ \end{trivlist} \begin{trivlist} \item[] \it Remark. \rm In the special case $S=\N^n$ Theorem 2 was proven by Heinze \cite[Kap.~II, Satz 2.8]{He}. More generally, for semigroups corresponding to a product along a line it follows from work of Skryabin \cite[Theorem 3]{Sk}. \end{trivlist}

9506

alg-geom/9506007

en

https://arxiv.org/abs/alg-geom/9506007

alg-geom/9506007

Lisa Jeffrey

Lisa C. Jeffrey, Frances C. Kirwan

On localization and Riemann-Roch numbers for symplectic quotients

19 pages; September 1994, revised March 1995, LaTeX v. 2.09

Quart. J. Math. 47 (1996) 165-185

Suppose $(M,\omega)$ is a compact symplectic manifold acted on by a compact Lie group $K$ in a Hamiltonian fashion, with moment map $\mu: M \to \Lie(K)^*$ and Marsden-Weinstein reduction $M_{red} = \mu^{-1}(0)/K$. In this paper, we assume that $M$ has a $K$-invariant K\"ahler structure. In an earlier paper, we proved a formula (the residue formula) for $\eta_0 e^{\omega_0}[M_{red}]$ for any $\eta_0 \in H^*(M_{red})$, where $\omega_0$ is the induced symplectic form on $M_{red}$. Here we apply the residue formula in the special case $\eta_0 = Td(M_{red})$; when $K$ acts freely on $\mu^{-1}(0)$ this yields a formula for the Riemann-Roch number $RR (L_{red})$ of a holomorphic line bundle $L_{red}$ on $M_{red}$ that descends from a holomorphic line bundle $L$ on $M$ for which $c_1(L) = \omega$. Using the holomorphic Lefschetz formula we similarly obtain a formula for the $K$-invariant Riemann-Roch number $RR^K(L) $ of $L$. In the case when the maximal torus $T$ of $K$ has dimension one (except in a few special circumstances), we show the two formulas are the same. Thus in this special case the residue formula is equivalent to the result of Guillemin and Sternberg that $RR(L_{red}) = RR^K(L)$. (The residue formula was proved under the assumption that 0 is a regular value of $\mu$, and was given in terms of the restrictions of classes in the equivariant cohomology $H^*_T(M) $ of $M$ to the

alg-geom

\section{Introduction} Let $M$ be a compact symplectic manifold of (real) dimension $2m$, acted on in a Hamiltonian fashion by a compact connected Lie group $K$ with maximal torus $T$, and let $\liek$ and $\liet$ denote the Lie algebras of $K$ and $T$. Let $\mu: M \to \lieks$ be a moment map for this action. The reduced space $\mred$ is defined as $$ \mred = \mu^{-1}(0)/K. $$ We shall assume throughout this paper that $0$ is a regular value of $\mu$, so that $\mred$ is a symplectic orbifold; it has at worst finite quotient singularities and the symplectic form $\omega$ on $M$ induces a symplectic $\omega_0$ on $\mred$. There is a natural surjective ring homomorphism \cite{Ki1} $\kappa_0: \hk(M) \to H^*(\mred)$, where $\hk(M)$ is the $K$-equivariant cohomology of $M$. The main result of \cite{JK1} was the residue formula (Theorem 8.1), which for any $\eta_0 \in H^*(\mred)$ gives a formula for the evaluation of $\eta_0 e^{ \omega_0}$ on the fundamental class of $\mred$. This is given in terms of the restrictions $i_F^* \eta $ to components $F$ of the fixed point set of $T$ in $\symm$, for any class $\eta \in \hk(M)$ which maps to $\eta_0$ under $\kappa_0$. The residue formula is an application of the localization theorem of Berline and Vergne, a result on the equivariant cohomology of torus actions \cite{BV1}, for which a topological proof was later given by Atiyah and Bott \cite{abmm}. The residue formula is related to a result of Witten (the nonabelian localization theorem \cite{tdg}): like the residue formula, Witten's theorem expresses $\eta_0 e^{\omega_0}[\mred] $ in terms of appropriate data on $\symm$. In this paper we assume also that there exists a line bundle ${\mbox{$\cal L$}}$ on $M$ for which $c_1({\mbox{$\cal L$}}) = \omega$, with the action of $K$ on $M$ lifting to an action on the total space of ${\mbox{$\cal L$}}$ (such a lift exists because the action of $K$ on $M$ is Hamiltonian; we choose the lift to be compatible with the chosen moment map). Under the assumption that $K$ acts freely on $\mu^{-1} (0)$, we get a line bundle $\lred$ on $\mred$ whose first Chern class is $\omega_0$, where $\omega_0$ is the induced symplectic form on $\mred$. In the more general case, $\lred$ is only an orbifold bundle. Suppose also that there exists a $K$-invariant K\"ahler structure on $\symm$; more precisely a complex structure compatible with $\omega $ and preserved by the action of $K$. The bundle ${\mbox{$\cal L$}} $ then acquires a holomorphic structure in a standard manner, and we define the {\em quantizations} $\quant$ and $\quantr$ to be the virtual vector spaces \begin{equation} \label{2.1} \quant = \oplus_{j \ge 0 } (-1)^j H^j(\symm, {\mbox{$\cal L$}}) \end{equation} and \begin{equation} \label{2.1b} \quantr = \oplus_{j \ge 0 } (-1)^j H^j(\mred, \lred). \end{equation} The space $\quant$ is a virtual representation of $K$. The {\rm Riemann-Roch numbers} $\rrk({\mbox{$\cal L$}})$ and $\rr (\lred) $ are defined by \begin{equation} \label{2.2} \rrk ({\mbox{$\cal L$}}) = \sum_{j \ge 0 } (-1)^j \dim H^j (\symm, {\mbox{$\cal L$}})^K \end{equation} \begin{equation} \label{2.2b} \rr (\lred) = \sum_{j \ge 0 } (-1)^j \dim H^j (\mred, \lred). \end{equation} The main result of this paper is that when the dimension of the maximal torus $T$ is one, except in a few special circumstances, a particular case of our residue formula is equivalent to the statement that these two Riemann-Roch numbers are equal: \begin{equation} \label{2.02} \rr (\lred) = \rrk({\mbox{$\cal L$}}). \end{equation} This statement was proved by Guillemin and Sternberg [14] under some additional positivity hypotheses on ${\mbox{$\cal L$}}$, and was conjectured by them to hold more generally. It has been called the quantization conjecture: that quantization commutes with reduction. In this paper we show the following: \noindent{\bf Theorem 6.2:} {\em Suppose $K$ is a compact connected group of rank one. Let $K$ act in a Hamiltonian fashion on the symplectic manifold $\symm$, with a moment map $\mu$ for the action of $K$ such that $0$ is a regular value of $\mu$. If $K = SO(3)$, suppose also that there exists a component $F$ of the fixed point set $M^T$ of the maximal torus $T$ such that the constant value taken by the $T$-moment map $\mu_T$ on $F$satisfies $|\mf| > 1$, and if $K=SU(2)$ suppose that there is an $F$ for which $|\mf|>2$ and that there is no $F$ with $\mf=\pm1=n_{F,\pm}$ where $n_{F,\pm}$ is the sum of the positive (respectively negative) weights for the action of $T$ on the normal to $F$ in $M$. Then $\rrk ({\mbox{$\cal L$}}) = \rr (\lred).$ } Our original motivation for considering Riemann-Roch numbers was to provide a link between the residue we had defined and more standard definitions of residues in algebraic geometry (such as the Grothendieck residue \cite{harts}). We had defined the residue as the evaluation at $0$ (suitably interpreted) of the Fourier transform of a particular function on $\liet$: in the case when $T$ has rank one, this may be identified with the residue at $0$ of a meromorphic function on ${\Bbb C }$ whose poles occur only at $0$. Moreover, when $T$ has rank one, the special case of our residue that arises in computing $\rr (\lred)$ may be recast as the residue of a meromorphic 1-form on the Riemann sphere $\hatc$ at one of its poles. This same expression arises when one computes $\rrk({\mbox{$\cal L$}})$ by using the holomorphic Lefschetz formula to give a formula for the character $\charr(k)$ of the action of an element $k$ of $K$ on $\quant$ and then integrating $\charr(k)$ over the group $K$ to get the dimension of the $K$-invariant subspace $\quant^K$. Since writing this paper we have found that it is possible to extend its methods to treat the case when $K$ has higher rank (see \cite{JKQ}); however the arguments become more involved. Since we first began considering the application of the residue formula to Riemann-Roch numbers, several papers have appeared which extend the Guillemin-Sternberg result to a wider class of situations, and in which the main tool is localization in equivariant cohomology. There are two approaches, one due to Guillemin \cite{G}, the other due to Vergne \cite{V}. Guillemin's proof uses the residue formula to reduce the verification of (1.5) to a combinatorial identity involving counting lattice points in polyhedra. Guillemin then observes that this identity is known when $K$ is a torus acting in a quasi-free manner. Meinrenken \cite{mein} has subsequently extended this proof to torus actions which need not be quasi-free. As has been pointed out by Guillemin (\cite{G}, Section 3), the application of the residue formula to yield a formula for $\rr(\lred)$ requires only that there exist an {\em almost complex} structure on $\symm$ compatible with the action of $K$: such a structure enables one to define a spin-${\Bbb C }$ Dirac operator which can be used to define the virtual vector space $\quant$. Guillemin and Sternberg's original proof \cite{gsgq}, on the other hand, depends on the existence of a K\"ahler structure on $\symm$ and on some positivity hypotheses that are not necessary in the approaches based on equivariant cohomology. Thus the use of localization in equivariant cohomology extends Guillemin and Sternberg's original result to a more general situation. The observation that it suffices to assume the existence of a $K$-invariant compatible almost complex structure on $\symm$ applies likewise to the proof we shall present. Vergne [23] has given a different proof of the Guillemin-Sternberg conjecture when $K$ is a torus, also using ideas based on localization in equivariant cohomology. Her proof likewise does not require positivity hypotheses or the existence of a $K$-invariant K\"ahler structure on $\symm$. In later work, Meinrenken \cite{mein2} has proved the Guillemin-Sternberg formula for general compact nonabelian groups $K$; the only hypothesis he imposes is that the symmetric quadratic form on the tangent space to $\symm$ given by the symplectic form and the almost complex structure should be a metric (i.e., that it should be positive definite). Although many features of the rank one case are quite special, and although the proofs of Vergne and Guillemin-Meinrenken described above apply in much greater generality, we felt nevertheless that it was instructive to give a written account of our approach to this case since it is simple and self-contained. The layout of this paper is as follows. In Section 2 we review some basic facts about equivariant cohomology, and find an equivariant cohomology class $\eta$ on $M$ mapping to the Todd class of $\mred$ under the natural surjection $\kappa_0:\hk(M) \to H^*(\mred)$, so that $$\eta_0 = \td(\mred).$$ By the Riemann-Roch formula we have \begin{equation} \label{3.1} RR(\lred) = \eta_0 e^{\omega_0}[\mred], \end{equation} provided $\mred$ is a manifold, which is true if $K$ acts freely on $\mu^{-1}(0)$; in the more general case $RR(\lred)$ is given by Kawasaki's Riemann-Roch theorem for orbifolds (Theorem 6.1). In Section 3 we apply the residue formula to give a formula for the right hand side of (\ref{3.1} as a sum over the components of the fixed point set $M^T$ of $T$. In Section 4, we apply the holomorphic Lefschetz formula to obtain a similar fixed point sum for $RR^K(\lred)$; finally in Sections 5 and 6 we identify the two expressions. \noindent{\em Acknowledgement:} We thank Miles Reid, who originally proposed that we should investigate the possibility of trying to reformulate the residue in \cite{JK1} in more algebro-geometric terms, and suggested that there should be a relation between our residue formula and the Riemann-Roch theorem. { \setcounter{equation}{0} } \section{Preliminaries} If $M$ is a compact oriented manifold acted on by a compact connected Lie group $K$, the $K$-equivariant cohomology $\hk(M)$ of $M$ may be identified with the cohomology of the following chain complex (see Chapter 7 of \cite{BGV}): \begin{equation} \Omega^*_K(M) = \Bigl ( S(\lieks) \otimes \Omega^*(M) \Bigr )^K \end{equation} equipped with the differential\footnote{This (nonstandard) definition of the equivariant cohomology differential is different from that used in \cite{JK1} but consistent with that used in \cite{tdg}. We have found it convenient to introduce this definition to obtain consistency with the formulas in Section 4.} \begin{equation} \label{3.001} D = d - \isq \iota_{X_M} \end{equation} where $X_M$ is the vector field on $M $ generated by the action of $X \in \liek$. The natural map \begin{equation} \label{3.002} \tau_M: \hk(\symm) \to H^*_T (\symm) \end{equation} corresponds to the restriction map $$ \Bigl ( S(\lieks) \otimes \Omega^*(M) \Bigr )^K \to S(\liets) \otimes \Omega^*(M). $$ We shall make use of equivariant characteristic classes on $\symm$: for their definition see Section 7.1 of \cite{BGV}. \begin{prop} \label{p2.1} The equivariant Chern character of the line bundle ${\mbox{$\cal L$}}$ is $$ \chk({\mbox{$\cal L$}}) (X) = \,{\rm Tr}\, (e^{ \omega + \isq (\mu,X) } ) \in \hat{\Omega}^*_K(\symm). $$ \end{prop} Here, $X$ is a parameter in $\liek$, and $\hat{\Omega}^*_K(\symm) $ is the formal completion of the space ${\Omega}^*_K(\symm)$ of equivariant differential forms on $\symm$. Suppose $F$ is a component of the fixed point set of $T$ in $\symm$. We may (formally) decompose the normal bundle $\nu_F$ to $F$ (using the splitting principle if necessary) as a sum of line bundles $\nu_F = \sum_{j = 1}^N \nfj $, in such a way that $T$ acts on $\nfj$ with weight $\bfj \in \liets$.\footnote{Throughout this paper we shall use the convention that weights $\bfj \in \liets$ send the integer lattice $\intlat = \,{\rm Ker} \,(\exp: \liet \to T) $ to ${\Bbb Z }$.} The $T$-equivariant Euler class $e_F$ of the normal bundle $\nu_F$ is then defined for $X \in \liet$ by \begin{equation} \label{3.021} e_F(X) = \prod_{j = 1}^N \bigl (c_1 (\nfj) + \isq \bfj(X) \bigr). \end{equation} Recall that the Todd class of a vector bundle $V$ is given in terms of the Chern roots $x_l$ by $$\td (V) = \prod_l \frac{x_l}{1 - e^{- x_l} } = \sum_{j \ge 0 } \td_j(V), $$ where $\td_j $ is a homogeneous polynomial of degree $j$ in the $x_l$. If the Todd class is given in terms of the Chern roots by $$\td = \tau (x_1, \dots, x_N) $$ then the $T$-equivariant Todd class of the normal bundle $\nf $ is given for $X \in \liet$ by \begin{equation} \label{4.9} \td_T (\nu_F ) (X) = \tau \Bigl ( c_1 (\nu_{F,1}) + \isq \beta_{F,1}(X) , \dots, c_1 (\nu_{F,N}) + \isq \beta_{F,N}(X) \Bigr ) . \end{equation} We may also define the $K$-equivariant Todd class $\tdk(V)$ of any $K$-equivariant vector bundle $V$ on $M$, and in particular the equivariant Todd class $\tdk (\symm) $ $= \tdk(T\symm)$ of $M$. We have $ \tau_M (\tdk(V) ) = \tdt(V)$ and $\tau_M (\chk({\mbox{$\cal L$}}) ) = \ch_T ({\mbox{$\cal L$}}) $, where $\tau_M$ is the natural map introduced at (\ref{3.002}). Moreover one may define the inverse equivariant Todd class $$\tdki(V) = \sum_{j = 0 }^\infty (\tdki)_j (V) $$ as the equivariant extension of the class $\td^{-1}(V)$ given in terms of the Chern roots by $$\td^{-1}(V) = \prod_l \frac{1 - e^{- x_l} } {x_l}. $$ The surjective ring homomorphism $\kappa_0:\hk(\symm) \to H^*(\mred)$ mentioned in the introduction is the composition of the restriction map from $\hk(M)$ to $\hk(\mu^{-1}(0))$ and the natural isomorphism from $\hk(\mu^{-1}(0))$ to $H^*(\mred)$ which exists since $K$ acts locally freely on $\mu^{-1}(0)$ and we are working with cohomology with complex coefficients. This surjection is zero on $H^j_K(M)$ for any $j> dim_{\bf R}(\mred)$, and so it makes sense to apply $\kappa_0$ to formal equivariant cohomology classes such as the equivariant characteristic classes we have been considering. \begin{prop} \label{p2.2} We have $$ \kappa_0 \Bigl ({\tdk(\symm)}{\tdki (\lieka \oplus \lieksa) } \Bigr ) = \td (\mred), $$ where $\kappa_0 $ is the natural surjective ring homomorphism $\hk(\symm) \to H^*(\mred) $. Here, $\lieka $ denotes the product bundle $\symm \times \liek$ where $\liek$ is equipped with the adjoint action of $K$, and $\lieksa $ denotes the product bundle $\symm \times \lieks$ where $\lieks$ is equipped with the coadjoint action of $K$. \end{prop} \Proof The normal bundle $\nu(\zloc) $ to $\mu^{-1}(0)$ (which is a submanifold of $\symm$ since $0$ is a regular value of $\mu$) is isomorphic as an equivariant bundle to $\lieksa$ (since $\mu: \symm \to \lieks$ is an equivariant map). Moreover, when $K$ acts freely on $\mu^{-1} (0)$, we have the following decomposition of $T\symm$ in terms of $K$-equivariant bundles: $$ T \symm|_{\mu^{-1}(0)} = T (\mu^{-1} (0) ) \oplus \lieksa $$ and $ T (\mu^{-1} (0) ) = \proj^* T \mred \oplus \lieka $ where $\proj: \zloc \to \mred$ is the natural projection.$\square$ The following is an immediate consequence of (\ref{4.9}): \begin{lemma} For $X \in \liet$ , the $T$-equivariant Todd class of $\lieka \oplus \lieksa$ is given by \begin{equation} \label{4.10} \tdt(\lieka \oplus \lieksa)(X) = \prod_{\gamma > 0 } \frac{ \hg(X) ^2}{(1 - e^{ \isq \hg(X) } )(1 - e^{ - \isq \hg(X) } )}, \end{equation} where the product is over the positive roots, and we have introduced\footnote{The extra factors of $1/(2 \pi)$ are introduced because of the convention explained in Footnote 2 that weights $\beta \in \liets$ satisfy $ \beta \in \,{\rm Hom}\,(\intlat, {\Bbb Z }) $ rather than $\beta \in \,{\rm Hom}\,(\intlat, 2 \pi {\Bbb Z })$. Our definition of roots is as in Lemma 3.1 of \cite{JK1}: thus the roots $\gamma$ satisfy $\gamma(\intlat) \subset 2 \pi {\Bbb Z }$. In the terminology of \cite{BD} (p. 185), the quantities $i \gamma: \liet \to i {\Bbb R }$ are the {\em infinitesimal} roots whereas the corresponding weights $\hg$ are the {\em real} roots.} $\hg = \gamma/(2\pi)$. \end{lemma} When $K$ is abelian the bundle $\tdk(\lieka \oplus \lieksa) $ is equivariantly trivial as well as trivial, and so we have in this case \begin{equation} \kappa_0 (\tdk(\symm) ) = \td (\mred). \end{equation} { \setcounter{equation}{0} } \section{Review of the residue formula} We now recall the main result (the residue formula, Theorem 8.1) of \cite{JK1}: \begin{theorem} \label{t4.1}{[\cite{JK1}]} Let $\eta \in \hk(\symm) $ induce $\eta_0 \in H^*(\mred)$. Then we have \begin{equation} \label{4.1} \eta_0 e^{{}\omega_0} [\mred] = {\stabo C^K} \res \Biggl ( \nusym^2 (X) \sum_{F \in {\mbox{$\cal F$}}} \rfe(X) [d X] \Biggr ), \end{equation} where $\stabo$ is the order of the subgroup of $K$ that acts trivially on $\zloc$, and the constant $C^K$ is defined by \begin{equation} \label{4.001} C^K = \frac{\isq^l}{(2 \pi)^{s-l} |W| \,{\rm vol}\, (T)}. \end{equation} We have introduced $s = \dim K$ and $l = \dim T$. Also, ${\mbox{$\cal F$}}$ denotes the set of components of the fixed point set of $T$, and if $F$ is one of these components then the meromorphic function $\rfe$ on $\liet \otimes {\Bbb C }$ is defined by \begin{equation} \label{4.01} \phantom{a} \rfe(X ) = e^{i \mu_T(F) (X ) } \int_F \frac{i_F^* (\eta(X ) e^{{} \omega} ) }{e_F(X) } . \end{equation} Here, $i_F: F \to \symm$ is the inclusion and $e_F$ is the $T$-equivariant Euler class of the normal bundle to $F$ in $\symm$, which was defined at (\ref{3.021}). The polynomial $\nusym: \liet \to {\Bbb R }$ is defined by $\nusym(X) = \prod_{\gamma > 0 } \gamma(X) $, where $\gamma$ runs over the positive roots of $K$. \end{theorem} The general definition of the residue $\res$ was given in Section 8 of \cite{JK1}. Here we shall treat the case where $K$ has rank $1$, for which the results are as follows. See Footnotes 2 and 3 for our conventions on roots and weights. \begin{corollary} \label{c4.2} {(\bf \cite{JK1}; \cite{Kalk}, \cite{wu})} In the situation of Theorem \ref{t4.1}, let $K = U(1)$. Then $$ \eta_0 e^{{}\omega_0} [\mred] = {\isq\stabo} \res_0 \Bigl ( \sum_{F \in {\mbox{$\cal F$}}_+} \rfe(X) d \lx \Bigr ) . $$ Here, the meromorphic function $\rfe $ on $ {\Bbb C }$ was defined by (\ref{4.01}), and $\res_0 $ denotes the coefficient of the meromorphic 1-form $d\lx/\lx$ on $\liek \otimes {\Bbb C }$, where $X \in \liek$ and $\lambda$ is the generator of the weight lattice of $U(1)$. The set ${\mbox{$\cal F$}}_+ $ is defined by ${\mbox{$\cal F$}}_+ = \{ F \in {\mbox{$\cal F$}}: \mu_T(F) > 0 \}. $ The integer $\stabo$ is as in Theorem \ref{t4.1}. \end{corollary} \begin{corollary} \label{c4.3}{\bf(cf. \cite{JK1}, Corollary 8.2)} In the situation of Theorem \ref{t4.1}, let $K = SU(2)$ or $K = SO(3) $. Then $$ \eta_0 e^{{}\omega_0} [\mred] = \frac{\isq \stabo }{2} \res_0 \Bigl ( \hg(X)^2 \sum_{F \in {\mbox{$\cal F$}}_+} \rfe(X) d\lambda(X) \Bigr ) . $$ Here, $\res_0$, $\rfe$ and ${\mbox{$\cal F$}}_+$ are as in Corollary \ref{c4.2}, and $\lambda = \lambda_K \in \liets$ is the generator of the weight lattice of $K$. We have $\lambda_{SO(3)} = \hg$ and $\lambda_{SU(2)} = \hg/2$, where $\hg = \gamma/( 2 \pi)$ was defined in terms of the positive root $\gamma$. The integer $\stabo$ is as in Theorem \ref{t4.1}. \end{corollary} We now specialize to the case $\eta_0 = \td (\mred)$. Assume that $T$ acts at the fixed point $F$ with weights $\bfj \in \liets$. From now on we assume that the action of $K$ on $\zloc$ is effective, so that $\stabo = 1$ in Theorem \ref{t4.1}. \begin{prop} \label{p4.4} We have \begin{equation} \chtd = {C^K} \res \Biggl ( \sum_{F \in {\mbox{$\cal F$}}} {\nusym^2(X)e^{\isq \mu_T(F)(X)} }{\tdti(\lieka \oplus \lieksa) (X) } \end{equation} $$\times \int_F \frac{e^{{}\omega} \tdt(\nu_F)(X) \td(F) }{e_F(X) } \Biggr ). $$ This is equal to $\rr (\lred)$ provided $K$ acts freely on $\zloc$. \end{prop} Here, the constant $C^K$ was defined at (\ref{4.001}). We have used the definitions of equivariant characteristic classes given in Section 2. We have also decomposed the restriction to $F$ of the $T$-equivariant Todd class of $\symm$ as \begin{equation} \td_T (M) (X) = \td_T (\nu_F)(X) \td (TF). \end{equation} Here, we have used the multiplicativity of the Todd class and the fact that the action of $T$ on $TF$ is trivial. Then the Proposition follows immediately from Theorem \ref{t4.1}. The special case of Proposition \ref{p4.4} when $K = U(1)$ is: \begin{prop} \label{p4.5} If $K = U(1)$, we have \begin{equation} \label{4.010} \chtd = {\isq} \res_0 \Bigl ( \sum_{F \in {\mbox{$\cal F$}}_+} \expmf \int_F \frac{e^{{}\omega} \tdt(\nu_F)(X) \td(F) }{e_F(X) } d \lx \Bigr ) \end{equation} This is equal to $\rr (\lred)$ provided $K$ acts freely on $\zloc$. Here, $X \in \liet$ and $\res_0$ denotes the coefficient of the meromorphic 1-form $d \lx/\lx$ on $\liet \otimes {\Bbb C }$, where the element $\lambda$ $ \in \liets$ is the generator of the weight lattice of $\liet$. \end{prop} The corresponding result for $K = SU(2)$ or $K = SO(3) $ is \begin{prop} \label{p4.6} Let $K = SU(2)$ or $K = SO(3)$. Then we have \begin{equation} \label{4.11} \chtd = \frac{\isq}{2} \res_0 \Bigl ( (1 - e^{\isq \hg(X) } ) (1 - e^{- \isq \hg(X) } ) \sum_{F \in {\mbox{$\cal F$}}_+} \expmf\end{equation} $$ \times \int_F \frac{e^{{}\omega} \tdt(\nu_F )(X) \td(F) }{e_F(X) } d \lambda(X) \Bigr ). $$ This is equal to $\rr (\lred)$ provided $K$ acts freely on $\zloc$. Here, $ X \in \liet$, and $\res_0$ denotes the coefficient of the meromorphic 1-form $d \lx/\lx$ on $\liet \otimes {\Bbb C }$, where the element $\lambda = \lambda_K\in \liets$ is the generator of the weight lattice of $K$; also, $ \lambda_{SO(3) } = $ $\hg = \gamma/(2 \pi)$ where $\gamma$ is the positive root of $SO(3)$, and $\lambda_{SU(2)} = \hg/2$, as in the statement of Corollary \ref{c4.3}. \end{prop} Notice that it is valid to apply the residue formula for groups of rank one to formal equivariant cohomology classes in this way, because both sides of the formula send to zero all elements of $H^j_K(M)$ when $j>dim_{\bf R}(\mred)$. { \setcounter{equation}{0} } \section{The holomorphic Lefschetz formula} We now describe the application of the holomorphic Lefschetz theorem in our situation. The theorem is proved by Atiyah and Singer (\cite{AS3}, Theorem 4.6), and is based on results of Atiyah and Segal \cite{AS2}: an exposition of the general result from which the theorem follows is given in Theorem 6.16 of \cite{BGV}. A more general equivariant index theorem involving equivariant cohomology is proved by Berline and Vergne in \cite{BV2}. The following statement is in a form that will be convenient for us. We introduce the notation that if $X \in \Lie(T)$ then $t = \exp (X) \in T$. For any weight $\beta$, we define $t^\beta$ as $\exp ( 2 \pi i \beta(X) ) $ $ \in U(1) \subset {\Bbb C }^\times$, where the weights $\beta$ have been chosen to send the integer lattice $\intlat $ in $\liet$ to $ {\Bbb Z } \subset {\Bbb R }$. \begin{theorem} \label{t5.0} {\bf (Holomorphic Lefschetz formula)} Let $ t \in T $ be such that the fixed point set of $t$ in $\symm$ is the same as the fixed point set $\cup_{F \in {\mbox{$\cal F$}}} F$ of $T$ in $M$; then the character $\charr(t)$ of the virtual representation of $t$ on $\quant$ is given by $$ \charr(t) = \sum_{F \in {\mbox{$\cal F$}}} \charr_F (t), $$ where \begin{equation} \label{5.1} \charr_F (t) = \int_F\frac{ i_F^* \cht({\mbox{$\cal L$}}) (t) \td(F) } {\prod_j (1 - t^{- \bfj} e^{ - c_1(\nfj) }) } \end{equation} $$ \phantom{a} = t^{\mf} \int_F \frac{e^{{}\omega} \td(F) } {\prod_j (1 - t^{- \bfj} e^{ - c_1(\nfj) }) } $$ Here, the $\bfj \in \,{\rm Hom}\,(T, U(1)) \subset \liets$ are the weights of the action of $T$ on the normal bundle $\nu_F$ of $F$ in $\symm$, and the $T$-moment map $\mu_T$ is the composition of $\mu$ with restriction from $\lieks$ to $\liets$. \end{theorem} \Proof Equation (\ref{5.1}) follows immediately from the statement given in Theorem 4.6 of \cite{AS3}. We need only observe that the action of $t$ on the fibre of ${\mbox{$\cal L$}}$ above any point in $F$ is given by multiplication by $t^{\mf}$. Thus $i_F^* \ch_T ({\mbox{$\cal L$}}) (t) = e^{{}\omega} t^{\mf}. $$\square$ When the $T$ action has isolated fixed points, (\ref{5.1}) reduces to \begin{equation} \label{5.2} \charr_F (t) = \frac{t^{\mf} } {\prod_j (1 - t^{- \bfj} ) } \end{equation} In the general case, the structure of the right hand side of (\ref{5.1}) is given as follows: \begin{lemma} \label{l5.0} The expression $$ \frac{1} {\prod_j (1 - t^{- \bfj} e^{ - c_1(\nfj) }) } $$ appearing in (\ref{5.1}) is given by \begin{equation} \label{5.003} \prod_{j} \sum_{r_j \ge 0 } \frac{t^{- r_j \bfj} (e^{ - c_1(\nfj) } - 1 )^{r_j} } { (1 - t^{ - \bfj} )^{r_j + 1} }. \end{equation} In particular the only poles occur when $t^{\bfj} = 1. $ \end{lemma} \Proof This follows by examining for each $j$ $$ \frac{1}{ 1 - t^{ - \bfj} e^{ - c_1 (\nfj) } } = \frac{1}{1 - y (1 +u) } = \frac{1}{1 - y } \sum_{ r \ge 0 } \frac{y^r u^r }{ (1 - y)^r} $$ where $y = t^{ - \bfj} $ and $u = e^{ - c_1 (\nfj)} - 1 $ is nilpotent. $\square$ We restrict from now on to the case $T = U(1)$, which is regarded as embedded in $\hatc$ in the standard way. We identify the weights with integers by writing them as multiples of the generator $\lambda$ of the weight lattice of $U(1)$. \begin{prop} \label{p5.1} The character $\charr(t )$ extends to a holomorphic function on ${\Bbb C }^\times = \hatc - \{ 0, \infty \}$. \end{prop} \Proof This follows since $\charr$ is the character of a finite dimensional (virtual) representation of $U(1)$, so it is of the form $\charr(t) = \sum_{m\in {\Bbb Z }} c_m t^m $ for some integer coefficients $c_m$, finitely many of which are nonzero. $\square$ The following is immediate: \begin{prop} \label{p5.2} The expression $\charr_F$ given in (\ref{5.1}) defines a meromorphic function on $\hatc$ such that $\sum_{F \in {\mbox{$\cal F$}}} \charr_F(t) $ agrees with $\charr(t)$ on the open subset of $ U(1)$ consisting of those $t$ whose action does not fix any point of $\symm - \symm^T$. Hence, by analyticity, $\charr(t) = \sum_{F \in {\mbox{$\cal F$}}} \charr_F(t)$ on an open set in $\hatc$ containing $ {\Bbb C }^\times - U(1)$. \end{prop} \begin{prop} \label{p5.3} The virtual dimension of the $T$-invariant subspace of $\quant$ is given by \begin{equation} \label{5.3} \dim \quant^T = \frac{1}{2 \pi \isq} \int_{|t| \in \Gamma} \frac{dt}{t} \sum_{F \in {\mbox{$\cal F$}}} \charr_F(t), \end{equation} where $\charr_F$ was defined after (\ref{5.1}). Here, for any $\epsilon > 0 $, $\Gamma = \{ t \in \hatc: |t| = 1 + \epsilon \} \subset \Omega $ is a cycle in $\hatc $ on which the $\charr_F$ have no poles. \end{prop} \Proof This follows since $$ \dim \quant^T = \frac{1}{2 \pi \isq} \int_{|t| = 1} \frac{dt}{t} \charr(t) $$ $$ \phantom{bbbbb} = \frac{1}{2 \pi \isq} \int_{|t| \in \Gamma} \frac{dt}{t} \charr(t), $$ and by applying Proposition \ref{p5.2} to identify $\charr $ with $\sum_{F \in {\mbox{$\cal F$}} } \charr_F $ on $\Gamma$. $\square$ \noindent{\em Remark:} One obtains an equivalent formula by defining $\Gamma = \{ t \in {\Bbb C }: |t| = 1 - \epsilon \} $ for $0 < \epsilon < 1$. Let us now regard \begin{equation} \label{5.03} h_F = \charr_F (t) \frac{dt}{t} = \frac{dt}{t} t^{\mf} \int_F \frac{ e^{{}\omega} \td(F) }{\prod_j (1 - t^{ - \bfj} e^{- c_1( \nfj) } ) } \end{equation} as a meromorphic 1-form on $\hatc$, whose poles may occur only at $0$, $\infty $ and $\sss \in \wf$, where we define \begin{equation} \label{5.06} \wf = \{ \sss \in U(1): \sss^\bfj = 1 \mbox{ for some } \bfj \}. \end{equation} (This is true by inspection of (\ref{5.2}) when the fixed point set of the action of $T$ consists of isolated points. In the general case it follows from Lemma \ref{l5.0}.) The integral (\ref{5.3}) then yields \begin{equation} \label{5.4} \dim \quant^T = - \sum_{F \in {\mbox{$\cal F$}}} \res_\infty h_F. \end{equation} Let us examine the poles of $h_F $ on $\hatc$. We have \begin{lemma} \label{l5.4} For a given $F$, let $n_{F,\pm}$ be $\sum_{j: \pm \bfj > 0 } |\bfj|$. If $\mf > - \nplf$ then $\res_0 h_F = 0 $, while if $\mf < \nminf$ then $\res_\infty h_F = 0 $. \end{lemma} \Proof To study the residue at $0$, we assume $|t| < 1 $, so that $ (1 - t)^{-1} = \sum_{ n \ge 0 } t^n $ and $(1 - t^{-1} )^{-1} = - t \sum_{n \ge 0 } t^n $. For $r \ge 1 $ we examine $$ \frac{ t^{ \mf} }{ \prod_j ( 1 - t^{ - \bfj} )^r } \frac{dt}{t} $$ \begin{equation} \label{5.5} = t^{ \mf} (-1)^{l_+} t^{r \nplf} \Bigl ( \prod_{j } \sum_{n_j \ge 0 } t^{|\bfj| n_j} \Bigr )^r \frac{dt}{t}, \end{equation} where $l_+$ is the number of $\bfj$ that are positive. It follows that if $\nplf + \mf > 0 $ then the residue at $0$ is zero. A similar calculation yields the result for the residue at $\infty$. $\square$ Recall that the action of $T$ on $M$ is said to be {\em quasi-free} if it is free on the complement of the fixed point set of $T$ in $M$. The following is shown in \cite{DGP}: \begin{lemma} \label{l5.3} The action of $T= U(1)$ on $\symm$ is quasi-free if and only if the weights are $\bfj = \pm 1$. \end{lemma} \begin{prop} \label{p5.5} If the action of $T$ is quasi-free, then we have \begin{equation} \label{5.7} \res_\infty \sum_{F \in {\mbox{$\cal F$}}} h_F = - \sum_{F \in {\mbox{$\cal F$}}_+ } \res_1 h_F . \end{equation} Here, ${\mbox{$\cal F$}}_+ = \{ F \in {\mbox{$\cal F$}}: \mf > 0 \}$. More generally the result is true if $\res_1 h_F $ is replaced by $\sum_{\sss \in \wf} \res_\sss h_F$, where the set $\wf$ was defined at (\ref{5.06}). \end{prop} \Proof Assume for simplicity that the action of $T$ is quasi-free: the proof of the general case is almost identical. Lemma \ref{l5.4} establishes that $$\res_\infty \sum_{F \in {\mbox{$\cal F$}}} h_F = \sum_{F \in {\mbox{$\cal F$}}_+} \res_\infty h_F. $$ Also, if $F \in {\mbox{$\cal F$}}_+$ then $\mf > - n_+$ so $\res_0 h_F = 0 $; hence (\ref{5.7}) follows because the meromorphic 1-form $h_F$ has poles only at $0, 1 $ and $\infty$ and their residues must sum to zero, so $\res_1 h_F = - \res_\infty h_F $ when $F \in {\mbox{$\cal F$}}_+$.$\square$ \noindent{\em Remark:} Recall that $\mu_T(F)$ is never zero. The following is an immediate consequence of combining Proposition \ref{p5.5} with Proposition \ref{p5.3}: \begin{corollary} \label{c5.7} If the action of $T = U(1)$ on $\mred$ is quasi-free, we have $\rrt({\mbox{$\cal L$}}) = \sum_{F \in {\mbox{$\cal F$}}_+} \res_1 h_F$. More generally we have $\rrt ({\mbox{$\cal L$}}) = \sum_{F \in {\mbox{$\cal F$}}_+} \sum_{\sss \in \wf} \res_\sss h_F, $ where $\wf$ was defined by (\ref{5.06}). \end{corollary} We now treat the cases $K = SU(2)$ and $K = SO(3)$. We shall first need the following \begin{lemma} \label{l5.10} There is no component $F$ of the fixed point set of $T$ on $\symm$ for which $\mf = 0 $. \end{lemma} \Proof Because the $K$ moment map is equivariant, $\mu(F)$ is fixed by the action of $T$ on $\lieks$ for every $F \in {\mbox{$\cal F$}}$; thus $\mu(F) \subset \liet$ (identifying $\liek$ with $\lieks$ and $\liet$ with $\liets$ by the choice of an inner product), so that $\mu(F) = \mf$. Thus $\mf = 0 $ implies $\mu(F) = 0 $. However, because $K$ acts locally freely on $\mu^{-1}(0)$, no $F$ may intersect $\mu^{-1}(0)$.$\square$ We shall prove the following result: \begin{prop} \label{p5.9} \noindent{\bf (a)} Suppose $M$ is connected, and suppose $K = SO(3)$ acts on $\symm$ in such a way that the action of $T$ is quasi-free. Suppose also that there exists $F$ for which $|\mu_T(F)| > 1$. Then $$ \rrk({\mbox{$\cal L$}}) = {\frac{1}{2} } \sum_{F \in {\mbox{$\cal F$}}_+} \res_1 (2 - t - t^{-1}) h_F$$ where the meromorphic 1-form $h_F $ on $\hatc$ was defined by (\ref{5.03}). More generally we have when $K = SO(3)$ (provided there exists $F$ for which $|\mu_T(F)| > 1$) that $$ \rrk({\mbox{$\cal L$}}) = {\frac{1}{2} } \sum_{F \in {\mbox{$\cal F$}}_+} \sum_{\sss \in \wf} \res_\sss (2 - t - t^{-1}) h_F,$$ where $\wf$ was defined by (\ref{5.06}). \noindent{\bf (b)} Let $K = SU(2)$, and suppose that there is an $F$ for which $|\mu_T(F) | > 2, $ and also that there is no $F$ with either $\mu_T(F) = 1$ and $\nplf = 1$ or $\mu_T(F) = - 1 $ and $\nminf = 1$. Then we have that $$ \rrk({\mbox{$\cal L$}}) = {\frac{1}{2} } \sum_{F \in {\mbox{$\cal F$}}_+} \sum_{\sss \in \wf} \res_\sss (2 - t^2 - t^{-2}) h_F.$$ \end{prop} \Proof {\bf (a)}:If $K = SO(3)$, we have by the Weyl integral formula for Lie groups that \begin{equation} \label{5.07}\dim \quant^K = \frac{1}{ \,{\rm vol}\, K} \int_{k \in K} dk \charr(k) = \frac{1}{|W|} \frac{1}{2 \pi \isq} \int_{t \in T} \frac{dt}{t} (1 - t) (1 - t^{-1}) \charr(t) \end{equation} $$ \phantom{bbbbb} = {\frac{1}{2} } \frac{1}{2 \pi \isq} \int_{t \in \Gamma} \frac{dt}{t} (2 - t - t^{-1}) \charr(t) $$ $$ \phantom{bbbbb} = {\frac{1}{2} } \frac{1}{2 \pi \isq} \int_{t \in \Gamma} \frac{dt}{t} (2 - t - t^{-1}) \sum_{F \in {\mbox{$\cal F$}}} t^{\mf} \int_F \frac{ e^{{}\omega} \td(F) }{\prod_j (1 - t^{ - \bfj} e^{- c_1( \nfj)} ) }. $$ (In (\ref{5.07}), the factor $(1 - t) (1 - t^{-1})$ is the volume of the conjugacy class of $K$ containing $t$, in a normalization where $ \,{\rm vol}\, K = 1$: see for instance \cite{BD}, (IV.1.11).) By the previous argument (Lemmas \ref{l5.0} to \ref{l5.4}), this is equal to \begin{equation} \label{5.8} \dim \quant^K = - {\frac{1}{2} } \sum_{F \in {\mbox{$\cal F$}}} \res_\infty (2- t - t^{-1}) h_F \end{equation} where the meromorphic 1-form $h_F $ was defined at (\ref{5.03}). By the proof of Lemma \ref{l5.4} this becomes \begin{equation} \label{5.9} {\frac{1}{2} } \bigl ( 2 \res_1 \sum_{F \in {\mbox{$\cal F$}}_+} h_F - \res_1 \sum_{F \in {\mbox{$\cal F$}}, \mf + 1 \ge \nminf} t h_F - \res_1 \sum_{F \in {\mbox{$\cal F$}}, \mf - 1 \ge \nminf} t^{-1} h_F \bigr ). \end{equation} By Lemma \ref{l5.10}, $\mf \ne 0 $ for any $F$; thus it suffices to check that \begin{equation} \label{5.91} \sum_{F \in {\mbox{$\cal F$}}: \phantom{a} \mu_T(F) + 1 \ge \nminf} \res_1 (t h_F) = \sum_{F \in {\mbox{$\cal F$}}: \phantom{a} \mu_T(F) > 0 } \res_1 (t h_F) \end{equation} and likewise that \begin{equation} \label{5.92} \sum_{F \in {\mbox{$\cal F$}}: \phantom{a} \mu_T(F) - 1 \ge \nminf} \res_1 (t^{-1} h_F) = \sum_{F \in {\mbox{$\cal F$}}: \phantom{a} \mu_T(F) > 0 } \res_1 (t^{-1} h_F). \end{equation} Equation (\ref{5.91}) follows by the proof of Lemma \ref{l5.4}, unless $\nminf = 0 $ and $\mf = - 1$: we apply the proof, replacing $\mf$ by $\mf + 1 $ and using the fact that for any $r \in {\Bbb Z }$, \begin{equation} \label{5.94} \res_0(t^r h_F) = \res_\infty (t^r h_F) = 0 ~~~\mbox{if}~~~\mf+ r \in [ - \nplf + 1, \nminf - 1]. \end{equation} Likewise, equation (\ref{5.92}) follows unless $\nplf = 0$ and $\mf = 1$. However if $(\mf,\nminf) = (-1,0)$ then $F$ gives a local minimum of $\mu_T$. Since $M$ is connected and $\mu_T$ is a perfect Morse function (\cite{Ki1}, (5.8)), the local minimum must be a global minimum, contradicting the assumption that there exists an $F'$ for which $|\mu_T(F')| > 1$.\footnote{Recall that by Weyl symmetry there exists $F'$ for which $\mu_T (F') > 1 $ if and only if there exists $F'$ for which $\mu_T (F') < - 1 $.} Similarly the case $(\mf, \nminf) = (1,0) $ gives a maximum of $\mu_T$ and hence cannot occur. \noindent{\bf (b)} If $K = SU(2)$ we obtain instead of (\ref{5.9}) \begin{equation} \label{5.93} \dim \quant^K = {\frac{1}{2} } \bigl ( 2 \res_1 \sum_{F \in {\mbox{$\cal F$}}_+} h_F - \res_1 \sum_{F \in {\mbox{$\cal F$}}, \mf + 2\ge \nminf} t^2 h_F - \res_1 \sum_{F \in {\mbox{$\cal F$}}, \mf - 2 \ge \nminf} t^{-2} h_F \bigr ). \end{equation} Using (\ref{5.94}) we find that the second sum in (\ref{5.93}) is equal to $- \res_1 \sum_{F \in {\mbox{$\cal F$}}, \mf > 0 } t^2 h_F $ except when $\mf,\nminf)$ is $(-2,0)$, $(-1,0)$ or $(-1,1)$. Likewise the third sum is equal to $- \res_1 \sum_{F \in {\mbox{$\cal F$}}, \mf > 0 } t^{-2} h_F $ except when $(\mf, \nplf)$ is $ (2,0)$, $(1,0)$ or $(1,1)$. The first, second, fourth and fifth of these six cases are excluded if we assume that there is some $F$ with $|\mf| > 2$. $\square$ \noindent{\em Remark:} The technical hypothesis in Proposition \ref{p5.9}(a) that there should exist $F$ for which $|\mf| \ne 1$ can be satisfied by replacing ${\mbox{$\cal L$}}$ by ${\mbox{$\cal L$}}^k$ with $k \ge 2$. Similarly the technical hypotheses in \ref{p5.9}(b) can be satisfied by taking $k \ge 3.$ \noindent { \setcounter{equation}{0} } \section{Identification with the residue formula} In this section we shall assume the weights are $\bfj = \pm 1$, so that the action of $T$ on $M$ is quasi-free. In order to treat the general case one needs to use Kawasaki's Riemann-Roch theorem for orbifolds \cite{Kaw}: we do this in the next section. Let us examine the residue $ \res_1 h_F$ in the case $K = U(1)$. We denote a generator of the weight lattice of $\liet$ by $\lambda$, and replace the parameter $t$ (in a small neighbourhood of $1 \in \hatc$) by \begin{equation} \label{6.001} t = e^{i \lambda(X) } \end{equation} (where $X \in \liet \otimes {\Bbb C }$ is in a small neighbourhood of $0$ in $\liet \otimes {\Bbb C }$), so that $$ \frac{ dt}{t} = i { d \lx} $$ defines a meromorphic 1-form on $\liet \otimes {\Bbb C }$. (The substitution (\ref{6.001}) differs from the substitution used in Section 4, where we set $t = e^{2 \pi i \lambda(X)}$: however the value of the residue obviously is independent of which of these substitutions is used, and the substitution (\ref{6.001}) yields the formulas in Section 3.) We then find that \begin{equation} \res_1 h_F = \isq \res_0 \Bigl ( \expmf \int_F \frac{e^{{}\omega} \td(F)} {\prod_{j } (1 - e^{- \isq \bfj( X) - c_1 (\nfj) } ) } d \lx \Bigr ) \end{equation} \begin{equation} \label{6.1} \phantom{bbbbb} = \isq \res_0 \Bigl ( \expmf \int_F \frac{ e^\omega \tdt (\nu_F )(X)\td(F) } { \prod_j (\isq \bfj(X) + c_1 (\nfj) ) } d \lx \Bigr ),\end{equation} \begin{equation} \label{6.2} \phantom{bbbbb} = \isq \res_0 \Bigl ( e^{ \isq \mf(X) } \int_F \frac{ e^{{} \omega} \tdt (\nu_F)(X) \td(F) }{e_F(X) } d \lx \Bigr ) \end{equation} where $\res_0 $ denotes the coefficient of $\srlx$. Combining (\ref{6.2}) with Proposition \ref{p4.5}, one obtains \begin{prop} \label{p6.1} We have \begin{equation} \chtd = \sum_{F \in {\mbox{$\cal F$}}_+} \res_1 h_F. \end{equation} This equals $\rr(\lred) $ provided $K = U(1) $ acts freely on $\zloc$. \end{prop} Comparing Proposition \ref{p6.1} with Corollary \ref{c5.7}, we have \begin{prop} \label{p6.1p} Let the action of $K = U(1) $ on $\symm$ be quasi-free (which implies $K$ acts freely on $\mu^{-1}(0) $). Then $\rrk({\mbox{$\cal L$}}) = \rr(\lred)$. \end{prop} To treat $K = SO(3)$ and $K=SU(2)$, using the substitution (\ref{6.001}) in $\res_1 (2 - t - t^{-1}) h_F$, we recover the right hand side of Proposition \ref{p4.6}: \begin{prop} \label{p6.04} Let $K = SO(3)$ or $SU(2)$ act on $\symm$. Then $$\chtd = {\frac{1}{2} } \sumpl \res_1 (2 - t - t^{-1}) h_F. $$ This equals $\rr(\lred) $ provided $K$ acts freely on $\zloc$. \end{prop} Combining Proposition \ref{p6.04} with Proposition \ref{p5.9} we get\footnote{We do not treat $K = SU(2)$, since in this case the action of $T$ can only be quasi-free if all the $F$ are in the fixed point set of $K$. For unless $F$ is fixed by all of $K$, the orthocomplement $\liet^\perp$ of $\liet$ (equipped with the adjoint action) injects into the normal bundle $\nf$ under the action of $K$. There is thus a subbundle of $\nf$ on which $T$ acts with weight $2$ or $-2$, and so the action of $T$ cannot be quasi-free by Lemma \ref{l5.3}.} \begin{prop} \label{p6.04p} Let $K = SO(3) $ act on $\symm$ in such a way that the action of $T$ is quasi-free (which implies $K$ acts freely on $\mu^{-1}(0) $). Then $\rrk({\mbox{$\cal L$}}) = \rr(\lred)$. \end{prop} Thus we have \begin{theorem} \label{t6.2} Suppose $T = U(1) $ and either $K = U(1) $ or $K = SO(3)$. Suppose $K$ acts in a Hamiltonian fashion on the K\"ahler manifold\footnote{{\rm As described in the Introduction, it actually suffices to assume that $M$ is equipped with a $K$-invariant {\em almost} complex structure compatible with the symplectic structure. This applies likewise to Theorem \ref{t7.2} below.}} $\symm$, in such a way that the action of $T$ is quasi-free. If $K = SO(3)$, suppose also that there exists $F$ for which $|\mf| > 1$. We assume a moment map $\mu$ for the action of $K$ has been chosen in such a way that $0$ is a regular value of $\mu$. Then $\rrk({\mbox{$\cal L$}}) = \rr(\lred).$ \end{theorem} \section{Kawasaki's Riemann-Roch theorem} In this final section we sketch the proof of the Guillemin-Sternberg result $\rr(\lred) = \rrk({\mbox{$\cal L$}})$ when $K$ has rank one, without the assumption that the action of $T$ is quasi-free. In this more general case, $\mred$ is an orbifold and $\lred$ an orbifold bundle. The Riemann-Roch number of $\lred$ is then given by applying Kawasaki's Riemann-Roch theorem for orbifolds. We state Kawasaki's result only as it applies in our particular situation: the special case when $K=T$ in fact appears in earlier work of Atiyah. \begin{theorem} \label{t7.1} {(\bf Atiyah \cite{A:ell}; Kawasaki \cite{Kaw})} The Riemann-Roch number of the orbifold bundle $\lred$ is given by \begin{equation} \label{7.1} \rr(\lred) = \chtd + \sum_{ 1\neq\sss \in {\mbox{$\cal S$}}} \sum_{a \in {\mbox{$\cal A$}}_s} \frac{1}{n_{s,a} } \int_{\msoab} \isa. \end{equation} Here, ${\mbox{$\cal S$}}$ is a set of representatives $s \in T $ for the conjugacy classes in $K$ of elements whose fixed point set $\ms$ is strictly larger than the fixed point set of any subgroup of $K$ of dimension at least one. The components of $\ms$ are denoted $\msa$, where $a \in {\mbox{$\cal A$}}_s$; we introduce $\msoa = \msa \cap \zloc$, and $\msoab = \msoa/K_s$ where $K_s$ is the centralizer of $s$ in $K$. The positive integer $n_{s,a} $ is the order of the stabilizer of the action of $K_s$ at a generic point of $\msa$. The class $\isa $ $ \in H^*(\msoab) $ is defined by \begin{equation} \label{7.01} \isa = \frac{\ch({{\mbox{$\cal L$}}}^a_{s,red}) s^{\mu_a} \td(\msoab) } {\prod_{k \in \kappind} \bigl ( 1 - s^{ - \betak} e^{- \cnus} \bigr ) }. \end{equation} Here, $\mu_a$ is the weight of the action of $s$ on the fibre of ${\mbox{$\cal L$}}$ over any point in $\msa$ and ${{\mbox{$\cal L$}}}^a_{s,red}$ is the induced orbifold bundle on $\msoab$. If $\nu(\msoab)$ denotes the orbifold bundle which is the pullback to $\msoab$ of the normal to the image of the natural map from $\msoa$ to $\zloc$, we decompose $\nu(\msoab)$ as a formal sum of line bundles \begin{equation} \nu(\msa) = \oplus_{k \in \kappind} \nusak , \end{equation} and denote by $\betak \in {\Bbb Z }$ the weight of the action of $s$ on the formal line subbundle of the normal bundle to $\msoa$ in $\zloc$ corresponding to $\nusak$. \end{theorem} We can use this Theorem to prove Guillemin and Sternberg's result for groups of rank one, by identifying the additional terms on the right hand side of (\ref{7.1}) with the additional residues at the points $1 \ne $ $\sss \in\wf$ that appear in the statement of Proposition \ref{p5.5} when the action of $T$ is not quasi-free. Meinrenken uses Kawasaki's theorem in a different way to eliminate the quasi-free action hypothesis from the proof given by Guillemin in \cite{G}: see \cite{mein}, Remark 1 following Theorem 2.1. The proofs of Corollary \ref{c5.7} and Proposition 4.11 give when $K=T$ \begin{equation} \label{7.2} \rrk({\mbox{$\cal L$}}) = \sum_{s \in {\mbox{$\cal S$}}} \sum_{a \in {\mbox{$\cal A$}}_s} \bigl ( \sumfms \res_s h_F \bigr ), \end{equation} and when $K=SO(3)$ or $SU(2)$ \begin{equation} \label{7.25} \rrk({\mbox{$\cal L$}}) = \sum_{s \in W{\mbox{$\cal S$}}} \sum_{a \in {\mbox{$\cal A$}}_s} \bigl( \sumfms \res_s (2 - t - t^{-1}) h_F/2 \bigr ), \end{equation} where the meromorphic 1-form $h_F$ on $\hatc$ was defined at (\ref{5.03}) and $W$ is the Weyl group of $K$. The terms in the second sum indexed by different elements $s$ of the same Weyl group orbit are equal, so the sum can be rewritten as a sum over ${\mbox{$\cal S$}}$ instead of $W{\mbox{$\cal S$}}$. We know from Proposition \ref{p6.1} (a consequence of applying the residue formula to the class $\ch(\lred) \td(\mred)$ on $\mred$) that the term in each of these sums indexed by $s=1$ is \begin{equation} \label{7.3} \chtd . \end{equation} To deal with the other terms we apply the residue formula (Theorem \ref{t4.1}) to the action of $K_s$ on the symplectic manifold $\msa$: in the notation of that theorem, we introduce an appropriate equivariant cohomology class $\eta e^{\bom } = $ ${{\mbox{$\cal I$}}}^{s,a}_{K_s} \in H^*_{K_s}(\msa)$ which descends on the symplectic quotient $\msoab$ to $\isa$ $ = \eta_0 e^{\omega_0} $. When $K=T$ the class $\isat$ is given by \begin{equation} \label{7.03} \isat = {\cht(L) s^{\mu_a} \tdt(\msa) } \Bigl ( \prod_{k \in \kappind} \bigl ( 1 - s^{ - \betak} e^{- \cnust} \bigr )^{-1} \Bigr ) \end{equation} where $\cnust$ is the $T$-equivariant first Chern class of the virtual line bundle $\nusak$. In the other cases ${{\mbox{$\cal I$}}}^{s,a}_{K_s}$ is defined similarly, using Proposition 2.2 applied to $K_s$. This yields for each $s \in {\mbox{$\cal S$}} $ and $a \in {\mbox{$\cal A$}}_s$ that the term in the right hand side of (\ref{7.2}) or (\ref{7.25}) indexed by $s$ and $a$ is\begin{equation} \label{7.4} \frac{1}{n_{s,a}} \int_{\msoab} \isa. \end{equation} (Here, the factor $n_{s,a}$ is the order of the subgroup of $K_s$ that acts trivially on $\msa$: see the statement of Theorem \ref{t4.1}.) Substituting (\ref{7.4}) in (\ref{7.2}) or (\ref{7.25}) we recover the right hand side of (\ref{7.1}). Thus we obtain the Guillemin-Sternberg result in the special case when $K $ has rank one: \begin{theorem} \label{t7.2} Suppose that a compact group $K$ with maximal torus $T = U(1) $ acts in a Hamiltonian fashion on the K\"ahler manifold $\symm$, in such a way that $0$ is a regular value of $\mu$. Then if the hypotheses of Proposition 4.11(a) and (b) are satisfied $$\rrk({\mbox{$\cal L$}}) = \rr(\lred).$$ \end{theorem} \noindent{\bf Remark:} It has been pointed out to us by M. Vergne that there are examples (such as the action of $SU(2)$ on the complex projective line) to show that this result is not true without some hypotheses such as those of Proposition 4.11.

9506

alg-geom/9506023

en

https://arxiv.org/abs/alg-geom/9506023

alg-geom/9506023

Kai Behrend

K. Behrend and Yu. Manin

Stacks of Stable Maps and Gromov-Witten Invariants

Postscript file available at http://www.math.ubc.ca/people/faculty/behrend/gwi.ps , AMSLaTeX

We correct some errors in the earlier version of this paper. Most importantly, the definition of isogeny of marked stable graphs has changed.

alg-geom

\section{Introduction} Let $V$ be a projective algebraic manifold. In \cite{KM}, Sec.\ 2, Gromov-Witten invariants of $V$ were described axiomatically as a collection of linear maps \[I^{V}_{g,n,\beta}:\ H^{\ast}(V)^{\otimes n}\longrightarrow H^{\ast}(\overline{M}_{g,n},{\Bbb Q}),\quad \beta\in H_2(V,{\Bbb Z})\] satisfying certain axioms, and a program to construct them by algebro-geometric (as opposed to symplectic) techniques was suggested. The program is based upon Kontsevich's notion of a stable map $(C,x_1,\dots ,x_n,f)$, $f:C\rightarrow V$. This data consists of an algebraic curve $C$ with $n$ labeled points on it and a map $f$ such that if an irreducible component of $C$ is contracted by $f$ to a point, then this component together with its special points is Deligne-Mumford stable. For more details, see \cite{K} and below. \smallskip The construction consists of three major steps. \medskip A. Construct an orbispace (or rather a stack) of stable maps $\overline{M}_{g,n}(V,\beta)$ such that $g=\text{genus of }C$, $f_{\ast}{([C])}=\beta$, and its two morphisms to $V^n$ and $\overline{M}_{g,n}$. On the level of points, these morphisms are given respectively by \begin{eqnarray*} p:(C,x_1,\dots ,x_n,f) & \longmapsto & (f(x_1),\dots ,f(x_n)), \\ q:(C,x_1,\dots ,x_n,f) & \longmapsto & [(C,x_1,\dots ,x_n)]^{\mbox{\tiny stab}}, \end{eqnarray*} where the last expression means the stabilization of $(C,x_1,\dots ,x_n)$. \medskip B. Construct a ``virtual fundamental class'' $[\overline{M}_{g,n}(V,\beta)]_{\mbox{\tiny virt}}$, or ``orientation'' (see Definition~\ref{domb} below) and use it to define a correspondence in the Chow ring $C^V_{g,n,\beta}\in A(V^n\times \overline{M}_{g,n})$. \smallskip This step suggested in \cite{K} is quite subtle and has not been spelled out in full detail. It can be bypassed for $g=0$ and $V=G/P$ (generalized flag spaces) where the virtual class coincides with the usual one (see \cite{KM}). \smallskip In general, it involves a definition of a new ${\Bbb Z}$-graded supercommutative structure sheaf on $\overline{M}_{g,n}(V,\beta)$. The virtual class is obtained as a product of the class of this sheaf and the inverse Todd class of an appropriate tangent complex. Geometrically, it serves as a general position argument furnishing the Dimension Axiom of \cite{KM} and replacing the deformation of the complex structure used in the symplectic context. \medskip C. Use $C^V_{g,n,\beta}$ in order to construct the induced maps $I^V_{g,n,\beta}$ on any cohomology satisfying some version of the standard properties making it functorial on the category of correspondences. \medskip In this approach, the main features of $I^V_{g,n,\beta}$ axiomatized in \cite{KM} reflect functorial properties of $\overline{M}_{g,n}(V,\beta)$ and the cotangent complex with respect to degenerations of stable maps. In particular, the key ``Splitting Axiom'' (or Associativity Equations for $g=0$) expresses the compatibility between the divisors at infinity of $\overline{M}_{g,n}(V,\beta)$ and $\overline{M}_{g,n}.$ \smallskip A neat way to organize this information is to introduce the category of marked stable modular graphs indexing degeneration types of stable maps and to treat various modular stacks $\overline{M}_{g,n}(V,\beta)$ as values of this modular functor on the simplest one-vertex graphs. Then the check of the axioms in \cite{KM} essentially boils down to a calculation of this functor on a family of generating morphisms and objects in the graph category. \smallskip The degeneration type of $(C,x_1,\dots ,x_n, f)$ is described by the graph whose vertices are the irreducible components of $C$, edges are singular points of $C$, and tails (``one-vertex edges'') are $x_1,\dots ,x_n.$ In addition, each vertex is marked by the homology class in $V$ which is the $f$-image of the fundamental class of the respective component of $C$ and by the genus of the normalization of this component. The description of morphisms is somewhat more delicate, cf.\ Sec.\ 1 below. \smallskip This philosophy is an extension of the operadic picture which already gained considerable importance from various viewpoints. In turn, it leads to a new notion of a $\Gamma$-operad as a monoidal functor on an appropriate category $\Gamma$ of graphs, and an algebra over an operad as a morphism of such functors. This approach will be developed elsewhere (see \cite{KapMan}). It clarifies the origin of the proliferation of the types of operads considered recently (May's, Markl's, modular, cyclic, ...) \smallskip In Part~I of the present paper we treat in this way Step~A, stressing the functoriality not only with respect to the degeneration types with fixed $V$ but also with respect to $V$, expressed by the change of the marking semigroup of abstract non-negative homology classes. We hope also that our approach will help to introduce quantum cohomology with coefficients and to understand better the K\"unneth formula for quantum cohomology from \cite{KM2}. \smallskip Part~II is devoted to Steps~B and~C for $g=0$ and convex manifolds $V$. The formalism of orientation classes is introduced axiomatically, but we did not attempt to justify the relevant claims of \cite{K} in general. \smallskip A word of warning and apology is due. The reader will meet several different categories of marked graphs in this paper of which the most important are ${\frak G}_s$ (cf.\ Definition~\ref{dmmsg}), $\tilde{{\frak G}}_s(A)$ (cf.\ Definition~\ref{ecisg} and the preceding discussion) and $\tilde{{\frak G}}_s(V)_{{\mbox{\tiny cart}}}$ (cf.\ Definition~\ref{ceicv}). They differ mainly by their classes of morphisms. Specifically, certain elementary arrows which are combinatorially ``the same'', run in opposite directions in different categories, which affects the whole structure of the morphism semigroups. The reason is that functorial properties of moduli stacks of maps considered {\em by themselves } are different form the functorial properties of their virtual fundamental classes treated {\em as correspondences}. Since graphs are used mainly as a bookkeeping device, their categorical properties must reflect this distinction. \medskip \subsection{Acknowledgements} The authors are grateful to the Max-Planck-Institut f\"ur Mathematik in Bonn where this work was done for support and stimulating atmosphere. \vfill\eject \part{Stacks of Stable Maps} \section{Graphs} \label{graphs} \begin{defn} \label{dog} A {\em graph }$\tau$ is a quadruple $(F\t,V\t,j\t,\partial\t)$, where $F\t$ and $V\t$ are finite sets, $\partial\t:F\t\rightarrow V\t$ is a map and $j\t:F\t\rightarrow F\t$ an involution. We call $F\t$ the set of {\em flags}, $V\t$ the set of {\em vertices}, $S\t=\{f\in F\t\mathrel{\mid} j\t f=f\}$ the set of {\em tails } and $E\t=\{\{f_1,f_2\}\subset F\t\mathrel{\mid}\text{$f_2=j\t f_1$}\}$ the set of edges of $\tau$. For $v\in V\t$ let $F\t(v)=\partial^{-1}\t(v)$ and $|v|=\# F\t(v)$, the {\em valence }of $v$. \end{defn} \begin{defn} \label{geomr} Let $\tau$ be a graph. We define the {\em geometric realization $|\tau|$ of $\tau$} as follows. We start with the disjoint union of closed intervals and singletons \[\coprod_{f\in F\t}[0,{\textstyle{1\over2}}]\amalg\coprod_{v\in V\t}\{|v|\}.\] We denote the real number $x\in [0,{1\over2}]$ in the component indexed by $f\in F\t$ by $x_f$. Then for every $v\in V\t$ we identify all elements of $\{0_f\mathrel{\mid} f\in F\t(v)\}$ with $|v|$ and for every edge $\{f_1,f_2\}$ of $\tau$, we identify ${1\over2}_{f_1}$ and ${{1\over2}}_{f_2}$. Finally, we remove for every tail $f\in S\t$ the point ${1\over2}_f$. We consider $|\tau|$ as a topological space with base points given by $\{|v|\mathrel{\mid} v\in V\t\}$, the {\em vertices } of $|\tau|$. It should always be clear from the context whether $|v|$ denotes the geometric realization of a vertex or it's valence. \end{defn} \begin{defn} \label{doc} Let $\tau$ and $\sigma$ be graphs. A {\em contraction }$\phi:\tau\rightarrow\sigma$ is a pair of maps $\phi^F:F_{\sigma}\rightarrow F\t$ and $\phi_V:V\t\rightarrow V_{\sigma}$ such that the following conditions are satisfied. \begin{enumerate} \item $\phi^F$ is injective and $\phi_V$ is surjective. \item The diagram \[\begin{array}{ccc} F\t & \stackrel{\partial\t}{\longrightarrow} & V\t \\ \ldiagup{\phi^F} & & \rdiag{\phi_V} \\ F_{\sigma} & \stackrel{\partial_{\sigma}}{\longrightarrow} & V_{\sigma} \end{array}\] commutes. \item $\phi^F\mathbin{{\scriptstyle\circ}} j_{\sigma} =j\t\mathbin{{\scriptstyle\circ}} \phi^F$, so that $\phi$ induces injections $\phi^S:S_{\sigma}\rightarrow S\t$ and $\phi^E:E_{\sigma}\rightarrow E\t$ on tails and edges. \item $\phi^S$ is a bijection, so $F\t-\phi^F(F_{\sigma})$ consists entirely of edges, the edges being contracted. \item Call two vertices $v,w\in V\t$ {\em equivalent}, if there exists an $f\in F\t-\phi^F(F_{\sigma})$ such that $f\in F\t(v)$ and $j\t f\in F\t(w)$. Then pass to the associated equivalence relation on $V\t$. The map $\phi_V:V\t\rightarrow V_{\sigma}$ induces a bijection $V\t/\mathop{\sim}\rightarrow V_{\sigma}$. \end{enumerate} For a vertex $v\in V_{\sigma}$ the graph whose set of flags is \[\{f\in F\t\mathrel{\mid} \rtext{$f\not\in\phi^F(F_{\sigma})$ and $\phi_V(\partial\t f)=v$}\},\] whose set of vertices is $\phi_V^{-1}(v)$ and whose $j$ and $\partial$ are obtained from $j\t$ and $\partial\t$ by restriction, is called the {\em graph being contracted onto $v$}. If the graphs being contracted have together exactly one edge, we call $\phi$ an {\em elementary contraction}. \end{defn} \begin{numrmks} \label{rmc} \begin{enumerate} \item It is clear how to compose contractions, and that the composition of contractions is a contraction. \item If $\phi:\tau\rightarrow\sigma$ and $\phi':\tau\rightarrow\sigma'$ are contractions with the same set of edges being contracted, then there exists a unique isomorphism $\psi:\sigma\rightarrow\sigma'$ such that $\psi\mathbin{{\scriptstyle\circ}}\phi=\phi'$. \item Every contraction is a composition of elementary contractions. \item \label{rmcone} To carry out a construction for contractions of graphs, which is compatible with composition of contractions, it suffices to perform this construction for elementary contractions and check that the construction is independent of the order in which it is realized for two elementary contractions. \end{enumerate} \end{numrmks} \begin{defn} \label{domg} A {\em modular graph } is a graph $\tau$ endowed with a map $g\t:V\t\rightarrow{\Bbb Z}_{\geq0};v\mapsto g(v)$. The number $g(v)$ is called the {\em genus }of the vertex $v$. \end{defn} We say that a semigroup $A$ has {\em indecomposable zero}, if $a+b=0$ implies $a=0$ and $b=0$, for any two elements $a,b\in A$. \begin{defn} \label{doasmg} Let $\tau$ be a modular graph and $A$ a semigroup with indecomposable zero. An {\em $A$-structure } on $\tau$ is a map $\alpha:V_{\tau}\rightarrow A$. The element $\alpha(v)$ is called the {\em class } of the vertex $v$. The pair $(\tau,\alpha)$ is called a {\em modular graph with $A$-structure }(or {\em $A$-graph}, by abuse of language). A {\em marked graph } is a pair $(A,\tau)$, where $A$ is a semigroup with indecomposable zero and $\tau$ an $A$-graph. \end{defn} \begin{defn} \label{commor} Let $(\sigma,\alpha)$ and $(\tau,\beta)$ be $A$-graphs. A {\em combinatorial morphism $a:(\sigma,\alpha)\rightarrow(\tau,\beta)$} is a pair of maps $a_F:F_{\sigma}\rightarrow F\t$ and $a_V:V_{\sigma}\rightarrow V\t$, satisfying the following conditions. \begin{enumerate} \item \label{commor1} The diagram \[\comdia{F_{\sigma}}{\partial_{\sigma}}{V_{\sigma}}{a_F}{}{a_V}{F\t}{\partial\t}{V\t}\] commutes. In particular, for every $v\in V_{\sigma}$, letting $w=a_V(v)$, we get an induced map $a_{V,v}:F_{\sigma}(v)\rightarrow F\t(w)$. \item With the notation of (\ref{commor1}), for every $v\in V_{\sigma}$ the map $a_{V,v}:F_{\sigma}(v)\rightarrow F\t(w)$ is injective. \item \label{commor3} Let $f\in F_{\sigma}$ and $\overline{f}=j_{\sigma}(f)$. If $f\not=\overline{f}$, there exists an $n\geq1$ and $2n$ (not necessarily distinct) flags $f_1,\ldots,f_n,\overline{f}_1,\ldots,\overline{f}_n\in F\t$ such that \begin{enumerate} \item $f_1=a_F(f)$ and $\overline{f}_n=a_F(\overline{f})$, \item $j\t(f_i)=\overline{f}_i$, for all $i=1,\ldots,n$, \item $\partial\t(\overline{f}_i)=\partial\t(f_{i+1})$ for all $i=1,\ldots,n-1$, \item for all $i=1,\ldots,n-1$ we have \[\overline{f}_i\not= f_{i+1}\Longrightarrow\text{$g({v_i})=0$ and $\beta(v_i)=0$},\] where $v_i=\partial(\overline{f}_i)=\partial(f_{i+1})$, \end{enumerate} \item for every $v\in V_{\sigma}$ we have $\alpha(v)=\beta(a_V(v))$, \item for every $v\in V_{\sigma}$ we have $g({v})=g({a_V(v)})$. \end{enumerate} A {\em combinatorial morphism of marked graphs }$(B,\sigma,\beta)\rightarrow(A,\tau,\alpha)$ is a pair $(\xi,a)$, where $\xi:A\rightarrow B$ is a homomorphism of semigroups and $a:(\sigma,\beta)\rightarrow(\tau,\xi\mathbin{{\scriptstyle\circ}}\alpha)$ is a combinatorial morphism of $B$-graphs. Usually, we will suppress the subscripts of $a$. \end{defn} \begin{rmks} \begin{enumerate} \item The composition of two combinatorial morphisms is again a combinatorial morphism. \item We say that a combinatorial morphism $a:\sigma\rightarrow\tau$ is {\em complete}, if for every $v\in V_{\sigma}$ the map $a_{V,v}:F_{\sigma}(v)\rightarrow F\t(a(v))$ is bijective. Examples of complete combinatorial morphism are \begin{enumerate} \item the inclusion of a connected component, \item the morphism $\sigma\rightarrow\tau$, where $\sigma$ is obtained from $\tau$ by {\em cutting an edge}, i.e.\ changing $j$ in such a way as to turn a two element orbit into two one element orbits. \end{enumerate} \item Let $\tau$ be an $A$-graph and $f\in S\t$ a tail of $\tau$. Let $F_{\sigma}=F\t-\{f\}$, $V_{\sigma}=V\t$ and define $\partial_{\sigma}$ and $j_{\sigma}$ by restricting $\partial\t$ and $j\t$. Then $\sigma$ is naturally an $A$-graph called {\em obtained from $\tau$ by forgetting the tail $f$}. There is a canonical combinatorial morphism $\sigma\rightarrow\tau$. \item Every combinatorial morphism $a:\sigma\rightarrow\tau$ is a composition $a=b\mathbin{{\scriptstyle\circ}} c$, where $b$ is complete and $c$ is a finite composition of morphisms forgetting tails. If $\sigma$ and $\tau$ are stable (Definition~\ref{dosag}), all intermediate graphs in such a factorization are stable. \item Condition~(\ref{commor3}) of Definition~\ref{commor} can be rephrased in a more geometric way---see the remark after Proposition~\ref{cmmgpe}. \end{enumerate} \end{rmks} \begin{defn} A {\em contraction }$\phi:(\tau,\alpha)\rightarrow(\sigma,\beta)$ of $A$-graphs is a contraction of graphs $\phi:\tau\rightarrow\sigma$ such that for every $v\in V_{\sigma}$ we have \begin{enumerate} \item \[g(v)=\sum_{w\in\phi_V^{-1}(v)}g(w)+\dim H^1(|\tau_v|),\] where $\tau_v$ is the graph being contracted onto $v$, \item \[\beta(v)=\sum_{w\in\phi_V^{-1}(v)}\alpha(w).\] \end{enumerate} \end{defn} \begin{defn} \label{dosag} A vertex $v$ of a modular graph with $A$-structure $(\tau,\alpha)$ is called {\em stable}, if $\alpha(v)=0$ implies $2g(v)+|v|\geq3$. Otherwise, $v$ is called {\em unstable}. The $A$-graph $\tau$ is called {\em stable}, if all its vertices are stable. \end{defn} We now come to an important construction which we shall call {\em stable pullback}. Consider the following setup. We suppose given a homomorphism of semigroups $\xi:A\rightarrow B$, a contraction of $A$-graphs $\phi:\sigma\rightarrow\tau$ and a combinatorial morphism $a:(B,\rho)\rightarrow(A,\tau)$ of marked graphs. Moreover, we assume that $\rho$ is a {\em stable }$B$-graph. We shall construct a stable $B$-graph $\pi$, together with a contraction of $B$-graphs $\psi:\pi\rightarrow\rho$ and a combinatorial morphism of marked graphs $b:\pi\rightarrow\sigma$. This $B$-graph $\pi$ will be called the {\em stable pullback }of $\rho$ under $\phi$. \[\begin{array}{ccccc} B & & \pi & \stackrel{\psi}{\longrightarrow} & \rho \\ \ldiagup{\xi} & \phantom{\longrightarrow} & \ldiag{b} & & \rdiag{a} \\ A & & \sigma & \stackrel{\phi}{\longrightarrow} & \tau \end{array}\] According with Remark~\ref{rmc}(\ref{rmcone}), we shall assume that $\phi$ is elementary and contracts the edge $\{f,\overline{f}\}$ of $\sigma$. Let $v_1=\partial_{\sigma}(f)$, $v_2=\partial_{\sigma}(\overline{f})$ and $v_0=\phi(v_1)=\phi(v_2)$. {\em Case I (Contracting a loop). } In this case $v_1=v_2$. Let $w_1,\ldots,w_n$ be the vertices of $\rho$ that map to $v_0$ under $a$. Note that $g(w_i)\geq1$, since $g(v_0)\geq1$. Let $\pi$ be equal to $\rho$ with a loop $\{f_i,\overline{f}_i\}$ attached to $w_i$, for each $i=1,\ldots,n$ and $g_\pi(w_i)=g_\rho(w_i)-1$. Clearly, $\pi$ is stable, the drop in certain genera is made up for by the addition of flags. The morphism $b:\pi\rightarrow\sigma$ is the obvious combinatorial morphism mapping every one of the loops $\{f_i,\overline{f}_i\}$ to $\{f,\overline{f}\}$. The contraction $\psi:\pi\rightarrow\rho$ simply contracts all the added loops. \begin{equation*} \begin{array}{ccc} \beginpictur \setcoordinatesystem units <.3cm,.3cm> point at 3 2.5 \setplotarea x from 0 to 6, y from 0 to 5 \plot 3 4 4 4 / \plot 3 4 4 3 / \plot 3 1 4 1 / \setquadratic \plot 3 4 2 4.75 1 5 / \plot 3 4 2 3.25 1 3 / \plot 3 1 2 1.75 1 2 / \plot 3 1 2 .25 1 0 / \circulararc 180 degrees from 1 2 center at 1 1 \circulararc 180 degrees from 1 5 center at 1 4 \shaderectangleson \setshadegrid span <1mm> \putrectangle corners at 4 0 and 6 5 \put {\circle*{4}} [Bl] at 3 1 \put {\circle*{4}} [Bl] at 3 4 \axis left invisible label {$\scriptstyle\pi$} / \axis right invisible label {\phantom{$\scriptstyle\pi$}} / \axis bottom invisible label {\phantom{.}} / \endpicture & \stackrel{\psi}{\longrightarrow} & \beginpictur \setcoordinatesystem units <.3cm,.3cm> point at 1.5 2.5 \setplotarea x from 0 to 3, y from 0 to 5 \plot 0 1 1 1 / \plot 0 4 1 4 / \plot 0 4 1 3 / \shaderectangleson \setshadegrid span <1mm> \putrectangle corners at 1 0 and 3 5 \put {\circle*{4}} [Bl] at 0 1 \put {\circle*{4}} [Bl] at 0 4 \axis left invisible label {\phantom{$\scriptstyle\rho$}} / \axis right invisible label {$\scriptstyle\rho$} / \axis bottom invisible label {\phantom{.}} / \endpicture \\ \ldiag{b} & & \rdiag{a} \\ \beginpictur \setcoordinatesystem units <.3cm,.3cm> point at 3 2 \setplotarea x from 0 to 6, y from 0 to 4 \plot 3 2 4 3 / \plot 3 2 4 2 / \plot 3 2 4 1 / \setquadratic \plot 3 2 2 2.75 1 3 / \plot 3 2 2 1.25 1 1 / \circulararc 180 degrees from 1 3 center at 1 2 \shaderectangleson \setshadegrid span <1mm> \putrectangle corners at 4 0 and 6 4 \put {\circle*{4}} [Bl] at 3 2 \axis left invisible label {$\scriptstyle\sigma$} / \axis right invisible label {\phantom{$\scriptstyle\sigma$}} / \axis top invisible label {\phantom{.}} / \endpicture & \stackrel{\phi}{\longrightarrow} & \beginpictur \setcoordinatesystem units <.3cm,.3cm> point at 1.5 2 \setplotarea x from 0 to 3, y from 0 to 4 \plot 0 2 1 3 / \plot 0 2 1 2 / \plot 0 2 1 1 / \shaderectangleson \setshadegrid span <1mm> \putrectangle corners at 1 0 and 3 4 \put {\circle*{4}} [Bl] at 0 2 \axis left invisible label {\phantom{$\scriptstyle\tau$}} / \axis right invisible label {$\scriptstyle\tau$} / \axis top invisible label {\phantom{.}} / \endpicture \end{array} \end{equation*} {\em Case II (Contracting a non-looping edge). } In this case $v_1\not=v_2$. Again, let $w_1,\ldots, w_n$ be the vertices of $\rho$ that map to $v_0$ under $a$. First we shall construct an intermediate graph $\pi'$. Let us fix an $i=1,\ldots,n$. Construct $\pi'$ from $\rho$ by replacing $w_i$ with two vertices $w_i'$ and $w_i''$, connected by an edge $\{f_i,\overline{f}_i\}$, such that $\partial(f_i)=w_i'$ and $\partial(\overline{f}_i)=w_i''$. Let $f$ be a flag of $\rho$ such that $\partial_{\rho}(f)=w_i$. If $\partial_{\sigma}\phi^F(a(f))=v_1$ we attach $f$ to $w_i'$ and if $\partial_{\sigma}\phi^F(a(f))=v_2$ we attach $f$ to $w_i''$. Set $g(w_i')=g(v_1)$, $g(w_i'')=g(v_2)$, $\beta(w_i')=\xi(\alpha(v_1))$ and $\beta(w_i'')=\xi(\alpha(v_2))$. This defines the $B$-graph $\pi'$. The problem with $\pi'$ is that it might not be stable. So to construct $\pi$ we proceed as follows. Fix an $i=1,\ldots,n$. If $w_i'$ and $w_i''$ are stable vertices of $\pi'$ we do not change anything. If either of $w_i'$ or $w_i''$ is unstable, we go back to where we started, by contracting $\{f_i,\overline{f}_i\}$ again, obtaining the stable vertex $w_i$. This finally finishes the construction of $\pi$. The contraction $\psi:\pi\rightarrow\rho$ is defined by contracting all the edges that were just inserted into $\rho$ to construct $\pi$. There is an obvious combinatorial morphism $b':\pi'\rightarrow\sigma$ mapping the edge $\{f_i,\overline{f}_i\}$ to $\{f,\overline{f}\}$. Moreover, we define a combinatorial morphism $c:\pi\rightarrow\pi'$ as follows. If $i=1,\ldots,n$ is an index such that either of $w_i'$ or $w_i''$ is an unstable vertex of $\pi'$, we map the vertex $w_i$ of $\pi$ to the stable one of the two, say $w_i'$, to fix notation. If $f$ is a flag of $\rho$ such that $\partial{f}=w_i$, then $f$ is also considered as a flag of $\pi$ and $\pi'$, and under $c$ we map $f$ to itself, if $\partial_{\pi'}(f)=w_i'$, and to $f_i$, otherwise. Finally, $b:\pi\rightarrow\sigma$ is defined as the composition $b=b'\mathbin{{\scriptstyle\circ}} c$. \begin{equation} \label{tadsp} \begin{array}{ccc} \beginpictur \setcoordinatesystem units <.3cm,.3cm> point at 4.5 3 \setplotarea x from 0 to 9, y from 0 to 6 \plot 2 5 7 5 / \plot 2 4 3 5 / \plot 6 5 7 4 / \plot 2 3 3 3 / \plot 2 1 7 1 / \plot 6 1 7 2 / \shaderectangleson \setshadegrid span <1mm> \putrectangle corners at 0 0 and 2 6 \putrectangle corners at 7 0 and 9 6 \put {\circle*{4}} [Bl] at 3 5 \put {\circle*{4}} [Bl] at 6 5 \put {\circle*{4}} [Bl] at 3 3 \put {\circle*{4}} [Bl] at 6 1 \axis left invisible label {$\scriptstyle\pi$} / \axis right invisible label {\phantom{$\scriptstyle\pi$}} / \axis bottom invisible label {\phantom{.}} / \endpicture & \stackrel{\psi}{\longrightarrow} & \beginpictur \setcoordinatesystem units <.3cm,.3cm> point at 3 3 \setplotarea x from 0 to 6, y from 0 to 6 \plot 2 5 4 5 / \plot 2 4 3 5 / \plot 3 5 4 4 / \plot 3 3 2 3 / \plot 2 1 4 1 / \plot 3 1 4 2 / \shaderectangleson \setshadegrid span <1mm> \putrectangle corners at 0 0 and 2 6 \putrectangle corners at 4 0 and 6 6 \put {\circle*{4}} [Bl] at 3 5 \put {\circle*{4}} [Bl] at 3 3 \put {\circle*{4}} [Bl] at 3 1 \axis left invisible label {\phantom{$\scriptstyle\rho$}} / \axis right invisible label {$\scriptstyle\rho$} / \axis bottom invisible label {\phantom{.}} / \endpicture \\ \ldiag{c} & & \\ \beginpictur \setcoordinatesystem units <.3cm,.3cm> point at 4.5 3 \setplotarea x from 0 to 9, y from 0 to 6 \plot 2 5 7 5 / \plot 2 4 3 5 / \plot 6 5 7 4 / \plot 2 3 6 3 / \plot 2 1 7 1 / \plot 6 1 7 2 / \shaderectangleson \setshadegrid span <1mm> \putrectangle corners at 0 0 and 2 6 \putrectangle corners at 7 0 and 9 6 \put {\circle*{4}} [Bl] at 3 5 \put {\circle*{4}} [Bl] at 6 5 \put {\circle*{4}} [Bl] at 3 3 \put {\circle*{4}} [Bl] at 6 3 \put {\circle*{4}} [Bl] at 3 1 \put {\circle*{4}} [Bl] at 6 1 \axis left invisible label {$\scriptstyle\pi'$} / \axis right invisible label {\phantom{$\scriptstyle\pi'$}} / \axis bottom invisible label {\phantom{.}} / \axis top invisible label {\phantom{.}} / \endpicture & & \rdiag{a} \\ \ldiag{b'} & & \\ \beginpictur \setcoordinatesystem units <.3cm,.3cm> point at 4.5 2 \setplotarea x from 0 to 9, y from 0 to 4 \plot 2 2 7 2 / \plot 2 3 3 2 2 1 / \plot 7 3 6 2 7 1 / \shaderectangleson \setshadegrid span <1mm> \putrectangle corners at 0 0 and 2 4 \putrectangle corners at 7 0 and 9 4 \put {\circle*{4}} [Bl] at 3 2 \put {\circle*{4}} [Bl] at 6 2 \axis left invisible label {$\scriptstyle\sigma$} / \axis right invisible label {\phantom{$\scriptstyle\sigma$}} / \axis top invisible label {\phantom{.}} / \endpicture & \stackrel{\phi}{\longrightarrow} & \beginpictur \setcoordinatesystem units <.3cm,.3cm> point at 3 2 \setplotarea x from 0 to 6, y from 0 to 4 \plot 2 2 4 2 / \plot 2 3 3 2 2 1 / \plot 4 3 3 2 4 1 / \shaderectangleson \setshadegrid span <1mm> \putrectangle corners at 0 0 and 2 4 \putrectangle corners at 4 0 and 6 4 \put {\circle*{4}} [Bl] at 3 2 \axis left invisible label {\phantom{$\scriptstyle\tau$}} / \axis right invisible label {$\scriptstyle\tau$} / \axis top invisible label {\phantom{.}} / \endpicture \end{array} \end{equation} Iterating this construction leads to the construction of a stable pullback for arbitrary contractions of $A$-graphs. \begin{rmk} Let \[\begin{array}{ccccc} B & & \pi & \stackrel{\psi}{\longrightarrow} & \rho \\ \ldiagup{\xi} & \phantom{\longrightarrow} & \ldiag{b} & & \rdiag{a} \\ A & & \sigma & \stackrel{\phi}{\longrightarrow} & \tau \end{array}\] be a stable pullback. \begin{enumerate} \item The diagram \[\comdia{V_\pi}{\psi_V}{V_\rho}{b}{}{a}{V_{\sigma}}{\phi_V}{V\t}\] commutes. \item The diagram \[\comdiaback{F_\pi}{\psi^F}{F_\rho}{b}{}{a}{F_{\sigma}}{\phi^F}{F\t}\] does {\em not }commute (except for special cases, e.g.\ if the $B$-graph $\pi'$ constructed above is stable). \end{enumerate} \end{rmk} \begin{prop} \label{cspcn} Stable pullback is independent of the order in which $\phi$ is decomposed into elementary contractions. Moreover, if \[\begin{array}{ccccc} B & & \pi & \stackrel{\psi}{\longrightarrow} & \rho \\ \ldiagup{\xi} & \phantom{\longrightarrow} & \ldiag{b} & & \rdiag{a} \\ A & & \sigma & \stackrel{\phi}{\longrightarrow} & \tau \end{array}\] and \[\begin{array}{ccccc} B & & \pi' & \stackrel{\psi'}{\longrightarrow} & \pi \\ \ldiagup{\xi} & \phantom{\longrightarrow} & \ldiag{b'} & & \rdiag{b} \\ A & & \sigma' & \stackrel{\phi'}{\longrightarrow} & \sigma \end{array}\] are stable pullbacks, then \[\begin{array}{ccccc} B & & \pi' & \stackrel{\psi\mathbin{{\scriptstyle\circ}}\psi'}{\longrightarrow} & \rho \\ \ldiagup{\xi} & \phantom{\longrightarrow} & \ldiag{b'} & & \rdiag{a} \\ A & & \sigma' & \stackrel{\phi\mathbin{{\scriptstyle\circ}}\phi'}{\longrightarrow} & \tau \end{array}\] is a stable pullback, too. \end{prop} \begin{pf} To check that stable pullback is well-defined, it suffices by Remark~\ref{rmc}(\ref{rmcone}) to check that the above construction yields the same result for both orders in which two elementary contractions can be composed. This is a straightforward, though maybe slightly tedious calculation. The compatibility of stable pullback with compositions of contractions follows trivially from the definition. \end{pf} \begin{prop} \label{cspcb} If \[\begin{array}{ccccc} B & & \pi & \stackrel{\psi}{\longrightarrow} & \rho \\ \ldiagup{\xi} & \phantom{\longrightarrow} & \ldiag{b} & & \rdiag{a} \\ A & & \sigma & \stackrel{\phi}{\longrightarrow} & \tau \end{array}\] and \[\begin{array}{ccccc} C & & \pi' & \stackrel{\chi}{\longrightarrow} & \rho' \\ \ldiagup{\eta} & \phantom{\longrightarrow} & \ldiag{b'} & & \rdiag{a'} \\ B & & \pi & \stackrel{\psi}{\longrightarrow} & \rho \end{array}\] are stable pullbacks, then \[\begin{array}{ccccc} C & & \pi' & \stackrel{\chi}{\longrightarrow} & \rho' \\ \ldiagup{\eta\mathbin{{\scriptstyle\circ}}\xi} & \phantom{\longrightarrow} & \ldiag{b\mathbin{{\scriptstyle\circ}} b'} & & \rdiag{a\mathbin{{\scriptstyle\circ}} a'} \\ A & & \sigma & \stackrel{\phi}{\longrightarrow} & \tau \end{array}\] is a stable pullback, too. \end{prop} \begin{pf} Of course, it suffices to consider the case that $\phi$ is an elementary contraction. Then the claim follows immediately from the construction. \end{pf} We are now ready to define the notion of {\em morphism } of marked stable graphs. \begin{defn} \label{dmmsg} Let $(A,\tau)$ and $(B,\sigma)$ be marked stable graphs. A {\em morphism } from $(A,\tau)$ to $(B,\sigma)$ is a quadruple $(\xi,a,\tau',\phi)$, where $\xi:A\rightarrow B$ is a homomorphism of semigroups, $\tau'$ is a stable $B$-graph, $a:\tau'\rightarrow\tau$ makes $(\xi,a)$ a combinatorial morphism of marked graphs, and $\phi:\tau'\rightarrow\sigma$ is a contraction of $B$-graphs. We also say that $(a,\tau',\phi)$ is a morphism of marked stable graphs {\em covering }$\xi$. Let $(\xi,a,\tau',\phi):(A,\tau)\rightarrow(B,\sigma)$ and $(\eta,b,\sigma',\psi):(B,\sigma)\rightarrow(C,\rho)$ be morphisms of stable marked graphs. Then we define the {\em composition } $(\eta,b,\sigma',\psi)\mathbin{{\scriptstyle\circ}}(\xi,a,\tau',\phi):(A,\tau)\rightarrow(C,\rho)$ to be $(\eta\xi,a c,\tau'',\psi\chi)$, where $(c,\tau'',\chi)$ is the stable pullback of $\sigma'$ under $\phi$. \[\begin{array}{ccccccc} C & & \tau'' & \stackrel{\chi}{\longrightarrow} & \sigma' & \stackrel{\psi}{\longrightarrow} & \rho \\ \ldiagup{\eta} & \phantom{\longrightarrow} & \ldiag{c} & & \rdiag{b} & & \\ B & & \tau' & \stackrel{\phi}{\longrightarrow} & \sigma & & \\ \ldiagup{\xi} & & \ldiag{a} & & & & \\ A & & \tau & & & & \end{array}\] \end{defn} \begin{rmks} \begin{enumerate} \item In reality a morphism is an isomorphy class of quadruples as in this definition. But we shall always stick to the abuse of language begun here. \item The composition of morphisms is associative by Propositions~\ref{cspcn} and~\ref{cspcb}. \item Every combinatorial morphism of marked graphs whose source and target are stable defines a morphism of marked stable graphs, but in the {\em opposite } direction. \item Every contraction of $A$-graphs whose source (and hence target) is stable defines a morphism of marked stable graphs (in the same direction). \end{enumerate} \end{rmks} The category of stable marked graphs shall be denoted by ${\frak G}_s$. Let $\AA$ be the category of (additive) semigroups with indecomposable zero element. By projecting onto the first component, we get a functor ${\frak a}:{\frak G}_s\rightarrow\AA$. For $A\in\operatorname{ob}\AA$ let ${\frak G}_s(A)$ be the fiber of ${\frak a}$ over $A$, i.e.\ the category of stable $A$-graphs. \begin{prop} \label{ppes} Let $\tau$ be an $A$-graph. Then there exists a stable $A$-graph $\tau^s$, together with a combinatorial morphism $\tau^s\rightarrow\tau$, such that every combinatorial morphism $\sigma\rightarrow\tau$, where $\sigma$ is a stable $A$-graph, factors uniquely through $\tau^s$. We call $\tau^s$ the {\em stabilization }of $\tau$. \end{prop} \begin{pf} let $\alpha$ denote the $A$-structure of $\tau$. {\em Case I\@. }Assume that $\tau$ has a vertex $v_0$ such that $g(v_0)=0$, $\alpha(v_0)=0$, $v_0$ has a unique flag $f_1$, and $f_2:=j(f_1)\not=f_1$. Let $\tau'\rightarrow\tau$ be the `subgraph' defined by $F_{\tau'}=F\t-\{f_1\}$, $V_{\tau'}=V\t-\{v_0\}$, $\partial_{\tau'}=\partial\t{ \mid } F_{\tau'}$, $j_{\tau'}{ \mid } F_{\tau'}-\{f_2\}=j\t{ \mid } F_{\tau'}-\{f_2\}$ and $j_{\tau'}(f_2)=f_2$. {\em Case II\@. }Assume that $\tau$ has a vertex $v_0$ such that $g(v_0)=0$, $\alpha(v_0)=0$, $v_0$ has exactly two flags, $f_1$ and $f_2$, $f_1$ is a tail of $\tau$ and $f_3:=j(f_2)\not=f_2$. Let $\tau'\rightarrow\tau$ be the `subgraph' defined by $F_{\tau'}=F\t-\{f_1,f_2\}$, $V_{\tau'}=V\t-\{v_0\}$, $\partial_{\tau'}=\partial\t{ \mid } F_{\tau'}$, $j_{\tau'}{ \mid } F_{\tau'}-\{f_3\}=j\t{ \mid } F_{\tau'}-\{f_3\}$ and $j_{\tau'}(f_3)=f_3$. {\em Case III\@. }Assume that $\tau$ has a vertex $v_0$ such that $g(v_0)=0$, $\alpha(v_0)=0$, $v_0$ has exactly two flags, $f_1$ and $f_2$, $\overline{f}_1:=j(f_1)\not=f_1$ and $\overline{f}_2:=j(f_2)\not=f_2$. Let $\tau'\rightarrow\tau$ be the `subgraph' defined by $F_{\tau'}=F\t-\{f_1,f_2\}$, $V_{\tau'}=V\t-\{v_0\}$, $\partial_{\tau'}=\partial\t{ \mid } F_{\tau'}$, $j_{\tau'}{ \mid } F_{\tau'}-\{\overline{f}_1,\overline{f}_2\}=j\t{ \mid } F_{\tau'}-\{\overline{f}_1,\overline{f}_2\}$ and $j_{\tau'}(\overline{f}_1)=\overline{f}_2$. {\em Case IV\@. }Assume that $\tau$ has a vertex $v_0$ such that $2g(v_0)+|v_0|<3$, $\alpha(v_0)=0$ and $F\t(v_0)$ is a union of orbits of $j\t$. Let $\tau'\rightarrow\tau$ be the `subgraph' defined by $F_{\tau'}=F\t-F\t(v_0)$, $V_{\tau'}=V\t-\{v_0\}$, $\partial_{\tau'}=\partial\t{ \mid } F_{\tau'}$ and $j_{\tau'}=j\t{ \mid } F_{\tau'}$. In each of these four cases every combinatorial morphism $\sigma\rightarrow\tau$, with $\sigma$ stable factors uniquely through $\tau'$. By induction on the number of vertices of $\tau$, the graph $\tau'$ has a stabilization, which is thus also a stabilization of $\tau$. If $\tau$ has no vertices $v_0$ of the kind covered by the above four cases, $\tau$ is stable and $\tau$ itself may serve as stabilization of $\tau$. \end{pf} See Section~10 in \cite[Exp.\ ~VI]{sga1}) for the definition of {\em cofibration } of categories. \begin{prop} \label{goscf} The functor ${\frak a}:{\frak G}_s\rightarrow\AA$ is a cofibration. \end{prop} \begin{pf} Let $\xi:A\rightarrow B$ be a homomorphism in $\AA$, and $(\tau,\alpha)$ a stable $A$-graph. We need to construct a stable $B$-graph $\sigma=\xi_{\ast}\tau$, together with a morphism $(a,\tau',\phi):(A,\tau)\rightarrow(B,\sigma)$ covering $\xi$, with the following universal mapping property. Whenever $\eta:B\rightarrow C$ is another homomorphism in $\AA$, $\rho$ is a stable $C$-graph and $(b,\tau'',\psi):(A,\tau)\rightarrow(C,\rho)$ is a morphism covering $\eta\mathbin{{\scriptstyle\circ}}\xi$, there exists a unique morphism $(c,\sigma',\chi):(B,\sigma)\rightarrow(C,\rho)$ covering $\eta$, such that $(c,\sigma',\chi)\mathbin{{\scriptstyle\circ}}(a,\tau',\phi)=(b,\tau'',\psi)$, i.e.\ such that $\tau''$ is the stable pullback of $\sigma'$ under $\phi$. In fact, it is not difficult to see that the stabilization of $(\tau,\xi\mathbin{{\scriptstyle\circ}}\alpha)$ satisfies this universal mapping property. \end{pf} \begin{numrmk} \label{pfabs} Choosing a {\em clivage normalis\'e }(see Definition~7.1 in \cite[Exp.\ {}~VI]{sga1}) of ${\frak G}_s$ over $\AA$ amounts to choosing a pushforward functor $\xi_{\ast}:{\frak G}_s(A)\rightarrow{\frak G}_s(B)$ for any homomorphism $\xi:A\rightarrow B$ in $\AA$. We may call $\xi_{\ast}$ {\em stabilization } with respect to $\xi$. If $B=\{0\}$, we speak of {\em absolute stabilization }(or simply {\em stabilization}, if no confusion seems likely to arise). \end{numrmk} \section{Prestable Curves} We recall the definition of prestable curves. A morphism of prestable curves is defined in such a way that it has degree at most one and contracts at most rational components. \begin{defn} \label{dopsc} A {\em prestable curve }over the scheme $T$ is a flat proper morphism $\pi:C\rightarrow T$ of schemes such that the geometric fibers of $\pi$ are reduced, {\em connected}, one-dimensional and have at most ordinary double points (nodes) as singularities. The {\em genus } of a prestable curve $C\rightarrow T$ is the map $t\mapsto\dim H^1(C_t,\O_{C_t})$, which is a locally constant function $g:T\rightarrow{\Bbb Z}_{\geq0}$. If $L$ is a line bundle on $C$, then the {\em degree } of $L$ is the locally constant function $\deg L:T\rightarrow{\Bbb Z}_{\geq0}$ given by $t\mapsto\chi(L_t)+g-1$. A {\em morphism }$p:C\rightarrow D$ of prestable curves over $T$ is a $T$-morphism of schemes, such that for every geometric point $t$ of $T$ we have \begin{enumerate} \item if $\eta$ is the generic point of an irreducible component of $D_t$, then the fiber of $p_t$ over $\eta$ is a finite $\eta$-scheme of degree at most one, \item if $C'$ is the normalization of an irreducible component of $C_t$, then $p_t(C')$ is a single point only if $C'$ is rational. \end{enumerate} \end{defn} If $V$ is a scheme and $f:C\rightarrow V$ a morphism, then $L\mapsto\deg f^{\ast} L$ defines a locally constant function $T\rightarrow\operatorname{\rm Hom}\nolimits_{\Bbb Z}(\operatorname{\rm Pic}\nolimits V,{\Bbb Z})$ which we shall call the {\em homology class } of $f$, by abuse of language, denoted $f_{\ast}[C]$. If $V$ is a scheme admitting an ample invertible sheaf let \[H_2(V)^+=\{\alpha\in\operatorname{\rm Hom}\nolimits_{\Bbb Z}(\operatorname{\rm Pic}\nolimits V,{\Bbb Z})\mathrel{\mid} \text{$\alpha(L)\geq0$ whenever $L$ is ample}\}.\] Note that $H_2(V)^+$ is a semigroup with indecomposable zero. This is because if $V$ admits an ample invertible sheaf then $\operatorname{\rm Pic}\nolimits V$ is generated by ample invertible sheaves (see Remarque~4.5.9 in \cite{ega2}). So if $f:C\rightarrow V$ is a morphism from a prestable curve into $V$, then the homology class is a locally constant function $T\rightarrow H_2(V)^+$. \begin{lem} \label{tloq} Let $f:X\rightarrow Y$ be a proper surjective morphism of $T$-schemes such that $f_{\ast}\O_X=\O_Y$. Let $g:X\rightarrow U$ be another morphism of $T$-schemes, such that for every geometric point $t$ of $T$ the map $g_t:X_t\rightarrow U_t$ is constant (as a map of underlying Zariski topological spaces) on the fibers of $f_t:X_t\rightarrow Y_t$. Then $g$ factors uniquely through $f$. \end{lem} \begin{pf} This follows easily, for example, from Lemma~8.11.1 in \cite{ega2}. \end{pf} \begin{cor} \label{zc} Let $C$ be a prestable curve over $T$ and $f:C\rightarrow V$ a morphism, where $V$ is a scheme admitting an ample invertible sheaf. Then $f_{\ast}[C]=0$ if and only if $f$ factors through $T$. \qed \end{cor} We shall need the following two results about gluing marked prestable curves at the marks. \begin{prop} \label{glue} Let $T$ be a scheme and $C_1$, $C_2$ two prestable curves over $T$. Let $x_1\in C_1(T)$ and $x_2\in C_2(T)$ be sections such that for every geometric point $t$ of $T$ we have that $x_1(t)$ and $x_2(t)$ are in the smooth locus of $C_{1,t}$ and $C_{2,t}$, respectively. Then there exists a prestable curve $C$ over $T$, together with $T$-morphisms $p_1:C_1\rightarrow C$ and $p_2:C_2\rightarrow C$, such that \begin{enumerate} \item $p_1(x_1)=p_2(x_2)$, \item $C$ is universal among all $T$-schemes with this property. \end{enumerate} The curve $C$ is uniquely determined (up to unique isomorphism) and will be called {\em obtained by gluing $C_1$ and $C_2$ along the sections $x_1$ and $x_2$}, notation \[C=C_1\amalg_{x_1,x_2}C_2.\] If $u:S\rightarrow T$ is a morphism of schemes, then $C_S$ is the curve obtained by gluing $C_{1,S}$ and $C_{2,S}$ along $x_{1,S}$ and $x_{2,S}$. If $g_i$ is the genus of $C_i$, for $i=1,2$, then for the genus $g$ of $C$ we have $g=g_1+g_2$. If, for $i=1,2$, $f_i:C_i\rightarrow V$ is a morphism into a scheme such that $f_1(x_1)=f_2(x_2)$, and $f:C\rightarrow V$ is the induced morphism, we have $f_{\ast}[C]={f_1}_{\ast}[C_1]+{f_2}_{\ast}[C_2]$ in $\operatorname{\rm Hom}\nolimits_{\Bbb Z}(\operatorname{\rm Pic}\nolimits V,{\Bbb Z})$. \qed \end{prop} \begin{prop} \label{glue1} Let $T$ be a scheme and $C$ a prestable curve over $T$. Let $x_1\in C(T)$ and $x_2\in C(T)$ be sections such that for every geometric point $t$ of $T$ we have that $x_1(t)$ and $x_2(t)$ are in the smooth locus of $C_{t}$ and $x_1(t)\not= x_2(t)$. Then there exists a prestable curve $\tilde{C}$ over $T$, together with a $T$-morphism $p:C\rightarrow \tilde{C}$, such that \begin{enumerate} \item $p(x_1)=p(x_2)$, \item $\tilde{C}$ is universal among { all }$T$-schemes with this property. \end{enumerate} The curve $\tilde{C}$ is uniquely determined (up to unique isomorphism) and will be called {\em obtained by gluing $C$ with itself along the sections $x_1$ and $x_2$}, notation \[\tilde{C}=C/{x_1\sim x_2}.\] If $u:S\rightarrow T$ is a morphism of schemes, then $(\tilde{C})_S$ is the curve obtained by gluing $C_{S}$ with itself along $x_{1,S}$ and $x_{2,S}$. If $g$ is the genus of $C$, then for the genus $\tilde{g}$ of $\tilde{C}$ we have $\tilde{g}=g+1$. If $f:C\rightarrow V$ is a morphism into a scheme such that $f(x_1)=f(x_2)$, and $\tilde{f}:\tilde{C}\rightarrow V$ is the induced morphism, we have $\tilde{f}_{\ast}[\tilde{C}]={f}_{\ast}[C]$ in $\operatorname{\rm Hom}\nolimits_{\Bbb Z}(\operatorname{\rm Pic}\nolimits V,{\Bbb Z})$. \qed \end{prop} \begin{defn} \label{mpc} Let $\tau$ be a modular graph. A {\em $\tau$-marked prestable curve over $T$} is a pair $(C,x)$, where $C=(C_v)_{v\in V\t}$ is a family of prestable curves $\pi_v:C_v\rightarrow T$ and $x=(x_i)_{i\in F\t}$ is a family of sections $x_i:T\rightarrow C_{\partial\t(i)}$, such that for every geometric point $t$ of $T$ we have \begin{enumerate} \item $x_i(t)$ is in the smooth locus of $C_{\partial\t(i),t}$, for all $i\in F\t$, \item $x_i(t)\not=x_j(t)$, if $i\not=j$, for $i,j\in F\t$, \item $g(C_{v,t})=g(v)$ for all $v\in V\t$. \end{enumerate} We define a {\em marked prestable curve over $T$} to be a triple $(\tau,C,x)$, where $\tau$ is a modular graph and $(C,x)$ a $\tau$-marked prestable curve over $T$. \end{defn} \section{Stable Maps} We now come to the definition of stable maps, the central concept of this work, which is due to Kontsevich. Fix a field $k$ and let ${\frak V}$ be the category of smooth projective (not necessarily connected) varieties over $k$. Consider the covariant functor \begin{eqnarray*} H_2^+:{\frak V} & \longrightarrow & \AA \\ V & \longmapsto & H_2(V)^+, \end{eqnarray*} where $\AA$ is the category of semigroups with indecomposable zero (see Section~\ref{graphs}). Define the category ${\frak V}{\frak G}_s$ as the fibered product (see Section~3 in \cite[Exp.\ ~VI]{sga1}) \[\comdia{{\frak V}{\frak G}_s}{}{{\frak G}_s}{}{\Box}{{\frak a}}{{\frak V}}{H_2^+}{\AA.}\] To spell this definition out, we have \begin{enumerate} \item objects of ${\frak V}{\frak G}_s$ are pairs $(V,\tau)$, where $V$ is a smooth projective variety over $k$ and $\tau$ is a stable $H_2(V)^+$-graph, \item a morphism $(V,\tau)\rightarrow (W,\sigma)$ is a quadruple $(\xi,a,\tau',\phi)$, where $\xi:V\rightarrow W$ is a morphism of $k$-varieties and $(H_2^+(\xi),a,\tau',\phi)$ is a morphism in ${\frak G}_s$ as defined in Definition~\ref{dmmsg}. \end{enumerate} \begin{numrmk} \label{ugscf} By Corollary~6.9 of \cite[Exp.\ ~VI]{sga1} and Proposition~\ref{goscf} the category ${\frak V}{\frak G}_s$ is a cofibered category over ${\frak V}$. \end{numrmk} \begin{defn} \label{dsm} Let $(V,\tau,\alpha)$ be an object of ${\frak V}{\frak G}_s$ and $T$ a $k$-scheme. A {\em stable $(V,\tau,\alpha)$-map over $T$} is a triple $(C,x,f)$, where $(C,x)$ is a $\tau$-marked prestable curve over $T$ and $f=(f_v)_{v\in V\t}$ is a family of $k$-morphisms $f_v:C_v\rightarrow V$, such that the following conditions are satisfied. \begin{enumerate} \item For every $i\in F\t$ we have $f_{\partial\t(i)}(x_i)=f_{\partial\t(j\t(i))}(x_{j\t(i)})$ as $k$-morphisms from $T$ to $V$. \item For all $v\in V\t$ we have that ${f_v}_{\ast}[C_{v}]=\alpha(v)$ in $H_2(V)^+$. \item For every geometric point $t$ of $T$ and every $v\in V\t$ the {\em stability condition }is satisfied. This means that if $C'$ is the normalization of a component of $C_{v,t}$ that maps to a point under $f_{v,t}:C_{v,t}\rightarrow V_t$, then \begin{enumerate} \item if the genus of $C'$ is zero, then $C'$ has at least three special points, \item if the genus of $C'$ is one, then $C'$ has at least one special point. \end{enumerate} Here, a point of $C'$ is called {\em special}, if it maps in $C_{v,t}$ to a marked point or a node. \end{enumerate} We define a {\em stable map over $T$} to be a sextuple $(V,\tau,\alpha,C,x,f)$, where $(V,\tau,\alpha)$ is an object of ${\frak V}{\frak G}_s$ and $(C,x,f)$ is a stable $(V,\tau,\alpha)$-map over $T$. A {\em morphism }$(V,\tau,\alpha,C,x,f)\rightarrow (W,\sigma,\beta,D,y,h)$ of stable maps over $T$ is a quintuple $(\xi,a,\tau',\phi,p)$, where $(\xi,a,\tau',\phi):(V,\tau,\alpha)\rightarrow (W,\sigma,\beta)$ is a morphism in ${\frak V}{\frak G}_s$ and $p=(p_v)_{v\in V_{\tau'}}$ is a family of morphisms of prestable curves $p_v:C_{a(v)}\rightarrow D_{\phi_V(v)}$, such that the following are true. \begin{enumerate} \item \label{mpc1}For every $i\in F_{\sigma}$ we have $p_{\partial(\phi^F(i))}(x_{a\phi^F(i)})=y_i$, \item \label{pre} If $\{i_1,i_2\}$ is an edge of $\tau'$ which is being contracted by $\phi$, then $p_{v_1}(x_{a(i_1)})=p_{v_2}(x_{a(i_2)})$, where $v_1=\partial i_1$ and $v_2=\partial i_2$. So, in particular, if $v_1\not=v_2$ there exists an induces morphism \[p_{12}:C_{a(v_1)}\amalg_{x_{a(i_1)},x_{a(i_2)}}C_{a(v_2)}\rightarrow D_w,\] where $w=\phi(v_1)=\phi(v_2)$. \item \label{post} With the notation of (\ref{pre}), if $v_1\not=v_2$, the morphism $p_{12}$ is a morphism of prestable curves. \item For every $v\in V_{\tau'}$ the diagram \[\comdia{C_{a(v)}}{f_{a(v)}}{V}{p_v}{}{\xi}{D_{\phi(v)}}{h_{\phi(v)}}{W}\] commutes. \end{enumerate} In this situation we also say that $p:(C,x,f)\rightarrow(D,y,h)$ is a morphism of stable maps {\em covering }the morphism $(\xi,a,\tau',\alpha)$ in ${\frak V}{\frak G}_s$. To define the {\em composition } of morphisms, let $(\xi,a,\tau',\phi,p):(V,\tau,\alpha,C,x,f)\rightarrow(W,\sigma,\beta,D,y,h)$ and $(\eta,b,\sigma',\psi,q):(W,\sigma,\beta,D,y,h)\rightarrow(U,\rho, \gamma,E,z,e)$ be morphisms of stable maps over $T$. We already know how to compose the morphisms $(\xi,a,\tau',\phi)$ and $(\eta,b,\sigma',\psi)$ in ${\frak V}{\frak G}_s$. Use notation as in Definition~\ref{dmmsg}. Then this composition is $(\eta\xi,ac,\tau'',\psi\chi)$. Define the family $r=(r_u)_{u\in V_{\tau''}}$ of morphisms of prestable curves $r_u:C_{ac(u)}\rightarrow E_{\psi\chi(u)}$ as $r_u=q_{\chi(u)}\mathbin{{\scriptstyle\circ}} p_{c(u)}$, which is well-defined, since $\phi_Vc(u)=a\chi_V(u)$. Then we define our composition as \[(\eta,b,\sigma',\psi,q)\mathbin{{\scriptstyle\circ}}(\xi,a,\tau',\phi,p) = (\eta\xi,ac,\tau'',\psi\chi,r).\] \end{defn} \begin{prop} \label{cmsmm} The composition of morphisms of stable maps is a morphism of stable maps. \end{prop} \begin{pf} The proof will be given at the same time as the proof of Theorem~\ref{mbfc} below. \end{pf} \begin{defn} \label{sovgs} Let $V\in\operatorname{ob}{\frak V}$ be a variety, $\beta\in H_2(V)^+$ a homology class and $g,n\geq0$ integers. Then $(V,g,n,\beta)$ shall denote the object $(V,\tau,\beta)$ of ${\frak V}{\frak G}_s$ whose modular graph $\tau$ is given by $F\t=\underline{n}$, $V\t=\{\varnothing\}$, $\partial\t:F\t\rightarrow V\t$ the unique map, $j\t=\operatorname{\rm id}_{\underline{n}}$ and $g(\varnothing)=g$. The $H_2(V)^+$-structure on $\tau$ is given by $\beta(\varnothing)=\beta$. A stable $(V,g,n,\beta)$-map is also called a {\em stable map from an $n$-pointed curve (of genus $g$) to $V$ (of class $\beta$)}. Here we use the notation $\underline{n}=\{1,\ldots,n\}$. \end{defn} \begin{lem} \label{lcsn} Over an algebraically closed field, let $(C,x,f)$ be a stable map from an $n$-pointed curve of genus $g$ to $V$ of class $\beta$ and let $(D,y,h)$ be a stable map from an $m$-pointed curve of genus $g$ to $V$ of class $\beta$, where $m\leq n$. Let $p:C\rightarrow D$ be a morphism such that $p(x_i)=y_i$ for $i\leq m$ and $hp=f$. If $C'\subset C$ is a subcurve (a connected union of irreducible components), such that \begin{enumerate} \item letting $C''$ be the closure of the complement of $C'$ in $C$, the curves $C'$ and $C''$ have exactly one node in common, \item $g(C')=0$, \item $f(C')$ is a point, \item for $i\leq m$ the $x_i$ do not lie on $C'$ except for at most one of them, \end{enumerate} then $p$ maps $C'$ to a point in $D$. \qed \end{lem} Let us denote the category of stable maps over $T$ by $\overline{M}(T)$. It comes together with a functor \[\overline{M}(T)\longrightarrow{\frak V}{\frak G}_s,\] defined by projecting onto the first components. For a morphism $u:S\rightarrow T$, pulling back defines a ${\frak V}{\frak G}_s$-functor \[u^{\ast}:\overline{M}(T)\longrightarrow \overline{M}(S).\] \begin{them} \label{mbfc} For every $k$-scheme $T$ the functor $\overline{M}(T)\rightarrow{\frak V}{\frak G}_s$ is a cofibration, whose fibers are groupoids. In other words, $\overline{M}(T)$ is cofibered in groupoids over ${\frak V}{\frak G}_s$. For every base change $u:S\rightarrow T$ the ${\frak V}{\frak G}_s$-functor $u^{\ast}:\overline{M}(T)\rightarrow \overline{M}(S)$ is cocartesian. \end{them} \begin{pf} To prove that $\overline{M}(T)\rightarrow{\frak V}{\frak G}_s$ is a cofibration, we need to prove the following. Let $(\xi,a,\tau',\phi):(V,\tau)\rightarrow(W,\sigma)$ be a morphism in ${\frak V}{\frak G}_s$ and $(C,x,f)$ a stable $(V,\tau)$-map over $T$. Then there exists a {\em pushforward }$(D,y,h)$ of $(C,x,f)$ under $(\xi,a,\tau',\phi)$. This pushforward comes with a morphism $p:(C,x,f)\rightarrow(D,y,h)$ of stable maps covering $(\xi,a,\tau',\phi)$ and is characterized by the following universal mapping property. Whenever $(\eta,b,\sigma',\psi):(W,\sigma)\rightarrow(U,\rho)$ is another morphism in ${\frak V}{\frak G}_s$, $(E,z,e)$ a stable $(U,\rho)$-map over $T$ and $r:(C,x,f)\rightarrow(E,z,e)$ a morphism of stable maps covering $(\eta\xi,ac,\tau'',\psi\chi):(V,\tau)\rightarrow(U,\rho)$ (in the notation of Definition~\ref{dmmsg}), there exists a unique morphism of stable maps $q:(D,y,h)\rightarrow(E,z,e)$ covering $(\eta,b,\sigma',\psi):(W,\sigma)\rightarrow(U,\rho)$ such that $r=q\mathbin{{\scriptstyle\circ}} p$. \begin{equation} \label{umprq} \begin{array}{ccccc} & & \stackrel{r}{\overtoparrow} & & \\ (C,x,f) & \stackrel{p}{\longrightarrow} & (D,y,h) & \stackrel{q}{\longrightarrow} & (E,z,e) \\ \mid & & \mid & & \mid \\ (V,\tau) & \stackrel{(\xi,a,\tau',\phi)}{\longrightarrow} & (W,\sigma) & \stackrel{(\eta,b,\sigma',\psi)}{\longrightarrow} & (U,\rho) \\ & & {\underbottomarrow\atop(\eta\xi,ac,\tau'',\psi\chi)} & & \end{array} \end{equation} To prove that $u^{\ast}:\overline{M}(T)\rightarrow\overline{M}(S)$ is always cocartesian, we need to prove that this pushforward commutes with base change. Recall that we also wish to prove Proposition~\ref{cmsmm}, i.e.\ that if morphisms of stable maps $p:(C,x,f)\rightarrow(D,y,h)$ and $q:(D,y,h)\rightarrow(E,z,e)$ are given as in (\ref{umprq}), then the composition $r:(C,x,f)\rightarrow(E,z,e)$ is also a morphism of stable maps. Purely formal considerations tell us that to prove these three facts, we may decompose the morphism $(\xi,a,\tau',\phi):(V,\tau)\rightarrow(W,\sigma)$ into a composition of other morphisms in any way we wish and prove the three facts for the factors of this decomposition. We shall thus consider the following five cases. {\em Case I (Changing $V$). } In this case $\sigma=\xi_{\ast}\tau$. This means that $\sigma$ is the pushforward of $\tau$ under $\xi:V\rightarrow W$, using the fact that ${\frak V}{\frak G}_s\rightarrow{\frak V}$ is a cofibration (Remark~\ref{ugscf}). In other words, $\sigma$ is the stabilization of $\tau$ with respect to the induced $H_2(W)^+$-structure (Proposition~\ref{ppes}). Thus $\tau'=\sigma$ and $\phi=\operatorname{\rm id}_{\sigma}$. In all other cases $W=V$ and $\xi$ is the identity. In the next two cases $a=\operatorname{\rm id}\t$ and $\tau'=\tau$. {\em Case II (Contracting and edge). } The contraction $\phi:\tau\rightarrow\sigma$ contracts exactly one edge $\{i_1,i_2\}\subset F\t$ and we have $v_1\not=v_2$, where $v_1=\partial(i_1)$ and $v_2=\partial(i_2)$. To fix notation, let $v_0=\phi(v_1)=\phi(v_2)$. {\em Case III (Contracting a loop). } This is the same as Case~II, except that we have $v_1=v_2$. In the last two cases $\tau'=\sigma$ and $\phi=\operatorname{\rm id}_{\sigma}$. {\em Case IV (Complete combinatorial). } The combinatorial morphism $a:\sigma\rightarrow\tau$ has the property that $a:F_{\sigma}(v)\rightarrow F\t(a(v))$ is a bijection, for all $v\in V_{\sigma}$. {\em Case V (Removing a tail). } In this case, $V_{\sigma}=V\t$, we have given a vertex $v_0\in V\t$ and a {\em tail }$i_0\in F\t(v_0)$ of $\tau$ and we have \begin{enumerate} \item $F_{\sigma}= F\t-\{i_0\}$, \item $\partial_{\sigma}=\partial\t{ \mid } F_{\sigma}$, \item $j_{\sigma}=j\t{ \mid } F_{\sigma}$. \end{enumerate} \renewcommand{\qed}{}\end{pf} Note that the proof of Proposition~\ref{cmsmm} is only interesting (if at all) for Case~II, since only in this case carrying out the composition of $(\xi,a,\tau',\phi)$ and $(\eta,b,\sigma',\psi)$ involves the second case of the construction of stable pullback (Section~\ref{graphs}). {\em Case I\@. } First we note the following trivial lemma. \begin{lem} \label{tcvw} Assume that $\tau$ is stable with respect to the induced $H_2(W)^+$-structure, so that $\sigma=\tau$ and $a=\operatorname{\rm id}\t$. Then if $(C,x,\xi\mathbin{{\scriptstyle\circ}} f)$ satisfies the stability condition it may serve as pushforward of $(C,x,f)$ under $\xi$. \qed \end{lem} We shall now reduce Case~I to Cases~IV and~V\@. By the claimed compatibility with base change, we may construct the pushforward locally, and pass to an \'etale cover of $T$, whenever desirable. Thus we add tails to $\tau$, obtaining $\tilde{\tau}$, and corresponding sections of $C$, obtaining $(C,\tilde{x})$ until $\tilde{\tau}$ with the induced $H_2(W)^+$-structure is stable and $(C,\tilde{x},\xi\mathbin{{\scriptstyle\circ}} f)$ satisfies the stability condition. Then we have the commutative diagram \begin{equation} \label{divgs} \comdia{(V,\tilde{\tau})}{}{(V,\tau)}{}{}{}{(W,\tilde{\tau})}{}{(W,\sigma)} \end{equation} in ${\frak V}{\frak G}_s$. The top row of (\ref{divgs}) is covered by $(C,\tilde{x},f)\rightarrow(C,x,f)$, and clearly $(C,x,f)$ is the pushforward of $(C,\tilde{x},f)$ under $(V,\tilde{\tau})\rightarrow(V,\tau)$ (see also Case~V). The first column of (\ref{divgs}) is covered by $(C,\tilde{x},f)\rightarrow(C,\tilde{x},\xi\mathbin{{\scriptstyle\circ}} f)$, which is a pushforward by Lemma~\ref{tcvw}. Now the pushforward of $(C,\tilde{x},\xi\mathbin{{\scriptstyle\circ}} f)$ under $(W,\tilde{\tau})\rightarrow(W,\sigma)$ will also be the sought after pushforward of $(C,x,f)$ under $(V,\tau)\rightarrow(W,\sigma)$. But $(W,\tilde{\tau})\rightarrow(W,\sigma)$ is covered by Cases~IV and~V, achieving the reduction. \qed {\em Case II\@. } The diagram defining the composition of $\phi$ and $(b,\sigma',\psi)$ is \[\begin{array}{ccccc} \tau' & \stackrel{\chi}{\longrightarrow} & \sigma' & \stackrel{\psi}{\longrightarrow} & \rho \\ \ldiag{c} & & \rdiag{b} & & \\ \tau & \stackrel{\phi}{\longrightarrow} & \sigma. & & \end{array}\] Let us first deal with the proof of Proposition~\ref{cmsmm}. \begin{lem} \label{lc2} For every $i\in F_{\sigma'}$ we have \[q_{\partial(i)}p_{\partial c\chi^F(i)}(x_{c\chi^F(i)})=q_{\partial(i)}p_{\partial\phi^Fb(i)}(x_{\phi^Fb(i)}).\] \end{lem} \begin{pf} Assume that $c\chi^F(i)\not=\phi^Fb(i)$, since otherwise there is nothing to prove. In this case, necessarily, $c\chi^F(i)$ is being contracted by $\phi$. Without loss of generality, let $c\chi^F(i)=i_1$, so the situation is as in the following diagram (cf.\ ~(\ref{tadsp})). \begin{equation} \label{fdgc} \begin{array}{ccc} \beginpictur \setcoordinatesystem units <.5cm,.3cm> point at 4.5 2 \setplotarea x from 0 to 9, y from 0 to 4 \plot 1.5 2 7.5 2 / \plot 6 2 7.5 1 / \shaderectangleson \setshadegrid span <1mm> \putrectangle corners at 0 0 and 1.5 4 \putrectangle corners at 7.5 0 and 9 4 \put {\circle*{4}} [Bl] at 6 2 \put {$\scriptstyle\chi^F(i)$} at 5.5 2.6 \axis left invisible label {$\scriptstyle\tau'$} / \axis right invisible label {\phantom{$\scriptstyle\tau'$}} / \axis bottom invisible label {\phantom{.}} / \endpicture & \stackrel{\chi}{\longrightarrow} & \beginpictur \setcoordinatesystem units <.5cm,.3cm> point at 3 2 \setplotarea x from 0 to 6, y from 0 to 4 \plot 1.5 2 4.5 2 / \plot 3 2 4.5 1 / \shaderectangleson \setshadegrid span <1mm> \putrectangle corners at 0 0 and 1.5 4 \putrectangle corners at 4.5 0 and 6 4 \put {\circle*{4}} [Bl] at 3 2 \put {$\scriptstyle i$} at 2.5 2.6 \axis left invisible label {\phantom{$\scriptstyle\sigma'$}} / \axis right invisible label {$\scriptstyle\sigma'$} / \axis bottom invisible label {\phantom{.}} / \endpicture \\ \ldiag{} & & \\ \beginpictur \setcoordinatesystem units <.5cm,.3cm> point at 4.5 2 \setplotarea x from 0 to 9, y from 0 to 4 \plot 1.5 2 7.5 2 / \plot 6 2 7.5 1 / \shaderectangleson \setshadegrid span <1mm> \putrectangle corners at 0 0 and 1.5 4 \putrectangle corners at 7.5 0 and 9 4 \put {\circle*{4}} [Bl] at 3 2 \put {\circle*{4}} [Bl] at 6 2 \axis left invisible label {$\scriptstyle\tilde{\tau}'$} / \axis right invisible label {\phantom{$\scriptstyle\tilde{\tau}'$}} / \axis top invisible label {\phantom{.}} / \axis bottom invisible label {\phantom{.}} / \endpicture & & \rdiag{b} \\ \ldiag{} & & \\ \beginpictur \setcoordinatesystem units <.5cm,.3cm> point at 4.5 2 \setplotarea x from 0 to 9, y from 0 to 4 \plot 1.5 2 7.5 2 / \plot 3 2 1.5 1 / \plot 7.5 3 6 2 7.5 1 / \shaderectangleson \setshadegrid span <1mm> \putrectangle corners at 0 0 and 1.5 4 \putrectangle corners at 7.5 0 and 9 4 \put {\circle*{4}} [Bl] at 3 2 \put {\circle*{4}} [Bl] at 6 2 \put {$\scriptstyle\phi^Fb(i)$} at 2.5 2.6 \put {$\scriptstyle c\chi^F(i)$} at 5.5 2.6 \put {$\scriptstyle i_1$} at 5.5 1.4 \put {$\scriptstyle i_2$} at 3.5 1.4 \axis left invisible label {$\scriptstyle\tau$} / \axis right invisible label {\phantom{$\scriptstyle\tau$}} / \axis top invisible label {\phantom{.}} / \endpicture & \stackrel{\phi}{\longrightarrow} & \beginpictur \setcoordinatesystem units <.5cm,.3cm> point at 3 2 \setplotarea x from 0 to 6, y from 0 to 4 \plot 1.5 2 4.5 2 / \plot 3 2 1.5 1 / \plot 4.5 3 3 2 4.5 1 / \shaderectangleson \setshadegrid span <1mm> \putrectangle corners at 0 0 and 1.5 4 \putrectangle corners at 4.5 0 and 6 4 \put {\circle*{4}} [Bl] at 3 2 \put {$\scriptstyle b(i)$} at 2.5 2.6 \axis left invisible label {\phantom{$\scriptstyle\sigma$}} / \axis right invisible label {$\scriptstyle\sigma$} / \axis top invisible label {\phantom{.}} / \endpicture \end{array} \end{equation} Here, ${\tau}'$ is the stable pullback and $\tilde{\tau}'$ the intermediate graph used in the construction of $\tau'$. Using the fact that $p$ is a morphism of stable maps we get a morphism $p_{12}:C_{12}\rightarrow D_{v_0}$ of prestable curves, where \[C_{12}=C_{v_1}\amalg_{x_{i_1},x_{i_2}}C_{v_2}.\] Compose this with $q_{\partial(i)}:D_{v_0}\rightarrow E_{\psi\partial(i)}$. Let $f_{12}:C_{12}\rightarrow V$ be the map induced from $f_{v_1}$ and $f_{v_2}$ and $\tilde{x}=x{ \mid } F\t(v_1)\cup F\t(v_2)-\{i_1,i_2\}$. Then $(C_{12},\tilde{x},f_{12})$ is a stable map and \[q_{\partial(i)}\mathbin{{\scriptstyle\circ}} p_{12}:(C_{12},\tilde{x},f_{12})\longrightarrow(E_{\psi\partial(i)},z{ \mid } F_\rho(\psi\partial(i)),e_{\psi\partial(i)})\] is a morphism of stable maps to which Lemma~\ref{lcsn} applies, with $C'=C_{v_2}$ and $x_{\phi^Fb(i)}\in C'$ being the only marked point coming from $F_\rho(\psi\partial(i))$, if there exists such a point at all (this is because $\tau'\not=\tilde{\tau}'$). So by Lemma~\ref{lcsn} $q_{\partial(i)}p_{v_2}(C_{v_2})$ is a point in $E_{\psi\partial(i)}$. To be precise, this holds if $T$ is the spectrum of an algebraically closed field. For the general case, applying Lemma~\ref{tloq} yields that $q_{\partial(i)}\mathbin{{\scriptstyle\circ}} p_{v_2}$ factors thought $T$. In particular, \[q_{\partial(i)}p_{v_2}(x_{\phi^Fb(i)})= q_{\partial(i)}p_{v_2}(x_{i_2})= q_{\partial(i)}p_{v_1}(x_{i_1}),\] which is what we set out to prove. \end{pf} Let us check that $r:(C,x,f)\rightarrow(E,z,e)$ is a morphism of stable maps, i.e.\ satisfies Properties~(1) through~(4) from Definition~\ref{dsm}. {\em Property (1). } Let $i\in F_\rho$. The we have \begin{align*} r_{\partial\chi^F\psi^F(i)}(x_{c\chi^F\psi^F(i)}) &= q_{\partial\psi^F(i)}p_{\partial c\chi^F\psi^F(i)}(x_{c\chi^F\psi^F(i)}) \\ \intertext{by Definition~\ref{dsm},} &= q_{\partial\psi^F(i)}p_{\partial\phi^Fb\psi^F(i)}(x_{\phi^Fb\psi^F(i)}) \\ \intertext{by Lemma~\ref{lc2},} &= q_{\partial\psi^F(i)}(y_{b\psi^F(i)}) \\ &= z_i, \end{align*} since $p$ and $q$ are morphisms of stable maps. {\em Property (2). } Let $\{j_1,j_2\}$ be an edge of $\tau'$ which is being contracted by $\psi\chi$. Let $u_1=\partial j_1$ and $u_2=\partial j_2$. {\em Case 1. } Let $\{j_1,j_2\}$ be contracted by $\chi$. Then $\{c(j_1),c(j_2)\}$ is being contracted by $\phi$. So without loss of generality $c(j_1)=i_1$ and $c(j_2)=i_2$. Then \begin{align*} r_{u_1}(x_{i_1}) & = q_{\chi(u_1)}p_{v_1}(x_{i_1}) \\ & = q_{\chi(u_2)}p_{v_2}(x_{i_2}) \\ & = r_{u_2}(x_{i_2}), \end{align*} since $p$ is a morphism of stable maps and $\chi(u_1)=\chi(u_2)$. {\em Case 2. } If $\{j_1,j_2\}$ is not contracted by $\chi$, then there exists a unique edge $\{j_1',j_2'\}$ of $\sigma'$ being contracted by $\psi$, such that $j_1=\chi^F(j_1')$ and $j_2=\chi^F(j_2')$. Then \begin{align*} r_{u_1}(x_{c(j_1)}) & = q_{\chi(u_1)}p_{c(u_1)}(x_{c(j_1)}) \\ & = q_{\chi(u_1)}p_{\partial\phi^Fb(j_1')}(x_{\phi^Fb(j_1')}) \\ \intertext{by Lemma~\ref{lc2},} & = q_{\chi(u_1)}(y_{b(j_1')}) \\ & = q_{\chi(u_2)}(y_{b(j_2')}) \\ \intertext{since $q$ is a morphism of stable maps,} & = r_{u_2}(x_{c(j_2)}), \end{align*} by symmetry. {\em Property (3). } This follows from the fact that the composition of morphisms of prestable curves is again a morphism of prestable curves. {\em Property (4). } Straightforward. This finishes the proof of Proposition~\ref{cmsmm} in Case~II\@. Let us now construct the pushforward $(D,y,h)$ of $(C,x,f)$ under $\phi$. Let $w\in V_{\sigma}$. If $w\not=v_0$, let $v$ be the unique vertex $v\in V\t$ such that $\phi_V(v)=w$ and set $D_w=C_v$. If $w=v_0$ set \[D_{v_0}=C_{v_1}\amalg_{x_{i_1},x_{i_2}}C_{v_2}.\] This defines a family of prestable curves $D$. For every $v\in V\t$ let $p_v:C_v\rightarrow D_{\phi(v)}$ be the canonical map. Define sections $y_i$, for $i\in F_{\sigma}$, by \[y_i=p_{\partial\phi^F(i)}(x_{\phi^F(i)}).\] Finally, define for every $w\in V_{\sigma}$ a map $g_w:D_w\rightarrow V$ from $f$ (by using Proposition~\ref{glue}, if $w=v_0$). Essentially by definition, $(D,y,h)$ is a stable $(V,\sigma)$-map and $p:(C,x,f)\rightarrow(D,y,h)$ is a morphism of stable maps covering $\phi:(V,\tau)\rightarrow(V,\sigma)$. It remains to check that $(D,y,h)$ satisfies the universal mapping property of a pushforward under $\phi$. So let $r:(C,x,f)\rightarrow(E,z,e)$ as in Diagram~(\ref{umprq}) be given. Let $u\in V_{\sigma'}$. We need to define a unique morphism $q_u:D_{b(u)}\rightarrow E_{\psi(u)}$ such that for every $u'\in V_{\tau'}$, satisfying $\chi(u')=u$, the diagram \[\begin{array}{ccc} C_{c(u')} & & \\ \ldiag{p_{c(u')}} & \sediagr{r_{u'}} & \\ D_{b(u)} & \stackrel{q_u}{\longrightarrow} & E_{\psi(u)} \end{array}\] commutes. If $b(u)\not=v_0$, necessarily, $q_u=r_{u'}$. So let $b(u)=v_0$. If there are two vertices $u_1'$ and $u_2'$ such that $\chi(u_1')=\chi(u_2')=u$, then we have two maps $r_{u_1'}:C_{v_1}\rightarrow E_{\psi(u)}$ and $r_{u_2'}:C_{v_2}\rightarrow E_{\psi(u)}$ giving rise to a unique map $q_u:D_{v_0}\rightarrow E_{\psi(u)}$. If there is only one vertex $u_1'$ of $\tau'$ such that $\chi(u_1')=u$, then we are in a situation as in Diagram~(\ref{fdgc}), and by Lemma~\ref{lcsn} $q_u$ has to map $C_{v_2}\subset D_{v_0}$ to a single point of $E_{\psi(u)}$ and $r_{u_1'}:C_{v_1}\rightarrow E_{\psi(u)}$ suffices to determine $q_u:D_{v_0}\rightarrow E_{\psi(u)}$ uniquely. This defines $q:D\rightarrow E$ satisfying all properties required of a morphism of stable maps, as some routine considerations reveal. This finishes Case~II\@. \qed {\em Case III\@. } This case is similar to Case~II, but much simpler, because the construction of the composition of $\phi$ and $(b,\sigma,\psi)$ is simpler, and thus for every $i\in F_{\sigma'}$ we have $c\chi^F(i)=\phi^Fb(i)$. We use Proposition~\ref{glue1} instead of Proposition~\ref{glue} to construct the pushforward of $(C,x,f)$ under $\phi$, gluing the two sections $x_{i_1}$ and $x_{i_2}$ of $C_{v_1}=C_{v_2}$ to obtain $D_{v_0}$. \qed {\em Case IV\@. } To construct the pushforward, set $D_v=C_{a(v)}$, $p_v:C_{a(v)}\rightarrow D_v$ the identity and $h_v=f_{a(v)}$, for every $v\in V_{\sigma}$. Moreover, for $i\in F_{\sigma}$ set $y_i=x_{a(i)}$. To check that $(D,y,h)$ is a stable map and $p:(C,x,f)\rightarrow(D,y,h)$ a morphism of stable maps, the only fact to check is that for every $i\in F_{\sigma}$ we have $h_{\partial(i)}(y_i)=h_{\partial(j(i))}(y_{j(i)})$, in other words \[f_{\partial a(i)}(x_{a(i)})=f_{\partial(j(i))}(x_{a(j(i))}).\] Here, Condition~(\ref{commor3}) in the definition of combinatorial morphism of $A$-graphs (Definition~\ref{commor}) enters in. It implies this claim together with Corollary~\ref{zc}. The universal mapping property of $(D,y,h)$ is easily verified. \qed {\em Case V\@. } Before we can treat this case, we need a few preparations. \begin{prop} \label{mplsm} Let $(C,x_1,\ldots,x_n,f)$ be a stable map over a field from a curve of genus $g$ to $V$ and $M$ an ample invertible sheaf on $V$. Then \[L=\omega_C(x_1+\ldots+x_n)\otimes f^{\ast} M^{\otimes3}\] is ample on $C$. Here $\omega_C$ is the dualizing sheaf of $C$. \end{prop} \begin{pf} Let us first consider the case that $C$ has no nodes, so that $C$ is irreducible and non-singular. Then is suffices to prove that $\deg L>0$. {\em Case 1. }The image $f(C)$ is a point. Then $\deg f^{\ast} M=0$ and we have \[\deg L=\deg\omega_C+n=2g-2+n\geq1,\] by the stability condition. {\em Case 2. }The image $f(C)$ is not a point. Then $\deg f^{\ast} M\geq1$ and so \[\deg L=2g-2+n+3\deg f^{\ast} M\geq 2g-2+n+3=2g+n+1>0.\] So suppose now that $C$ has a node $P$. Let $q:C'\rightarrow C$ be the curve obtained by blowing up $P$ and let $P_1,P_2\in C'$ be the two points lying over $P$. Let $L'=q^{\ast} L$ and $f'=f\mathbin{{\scriptstyle\circ}} q$. {\em Case 1. }The curve $C'$ is connected. Then $(C',x_1,\ldots,x_n,P_1,P_2,f')$ is a stable map and \[L'=\omega_{C'}(x_1+\ldots+x_n+P_1+P_2)\otimes {f'}^{\ast} M^{\otimes3}.\] {\em Case 2. }The curve $C'$ is disconnected. Let $C_1'$ and $C_2'$ be the two components of $C'$ and $L_1',L_2'$ the restriction of $L'$ to $C_1'$ and $C_2'$, respectively. Let $f_i':C_i'\rightarrow V$ for $i=1,2$ be the map induced by $f'$. Without loss of generality assume that $x_1,\ldots,x_r\in C_1'$ and $x_{r+1},\ldots,x_n\in C_2'$, for some $0\leq r\leq n$ and $P_1\in C_1'$, $P_2\in C_2'$. Then $(C_1',x_1,\ldots,x_r,P_1,f_1')$ and $(C_2',x_{r+1},\ldots,x_n,P_2,f_2')$ are stable maps and \begin{eqnarray*} L_1' & = & \omega_{C_1'}(x_1+\ldots +x_r+P_1)\otimes {f_1'}^{\ast} M^{\otimes3}\\ L_2' & = & \omega_{C_2'}(x_{r+1}+\ldots+x_n+P_2)\otimes {f_2'}^{\ast} M^{\otimes3}. \end{eqnarray*} Thus by induction on the the number of nodes we may assume that $L'$ is ample on $C'$. Let ${\cal F}$ be a coherent sheaf on $C$ and ${\cal F}'=q^{\ast}{\cal F}$. Then there exists an $n_0$ such that for all $n\geq n_0$ we have that ${\cal F}'\otimes{L'}^{\otimes n}(-P_1)$, ${\cal F}'\otimes{L'}^{\otimes n}(-P_2)$ and ${\cal F}'\otimes{L'}^{\otimes n}(-P_1-P_2)$ are generated by global sections. This implies that ${\cal F}\otimes L^{\otimes n}$ is generated by global sections. So $L$ is ample. \end{pf} We will now consider the following setup. Let $(C,x_1,\ldots,x_{n+1},f)$ be a stable map over $T$ from an $(n+1)$-pointed curve of genus $g$ to $V$ of class $\beta\in H_2(V)^+$, where $2g+n\geq3$ if $\beta=0$ (otherwise, $n\geq0$). If $t$ is a geometric point of $T$ and $C'$ a component of $C_t$, then we say that $C'$ {\em is to be contracted}, if, after removing $x_{n+1}$, the normalization of $C'$ violates the stability condition. Equivalently, \begin{enumerate} \item $C'$ is rational without self intersection (so that $C'$ is equal to its normalization), \item $x_{n+1}\in C'$, \item $C'$ has exactly two special points besides $x_{n+1}$, at least one of which is not a marked point, but a node, \item $f_t(C')$ is a single point of $V$. \end{enumerate} Pictorially, the only two possible components to be contracted look as follows. \[\beginpicture \setcoordinatesystem units <1cm,.71cm> point at 4 0 \setplotarea x from -1 to 3, y from -3 to 1 \plot 0 1 0 -3 / \plot -1 0 3 0 / \put {\circle*{8}} [Bl] at 0 -3 \put {$\boldsymbol{\times}$} at 1 0 \put {$\boldsymbol{\times}$} at 2 0 \put {$x_{n+1}$} at 2.2 -.4 \setcoordinatesystem units <1cm,.71cm> point at -3 0 \setplotarea x from -2 to 2, y from -3 to 1 \plot -2 0 2 0 / \setquadratic \plot 0 -3 -.75 -1 -1 1 / \plot 0 -3 .75 -1 1 1 / \put {\circle*{8}} [Bl] at 0 -3 \put {$\boldsymbol{\times}$} at 0 0 \put {$x_{n+1}$} at 0.2 -.4 \endpicture\] Note that every geometric fiber of $\pi:C\rightarrow T$ has at most one component to be contracted. We say a $T$-morphism $q:C\rightarrow U$, for a $T$-scheme $U$, {\em contracts the components to be contracted}, if for every geometric point $t$ of $T$ the map (of underlying Zariski topological spaces) $q_t:C_t\rightarrow U_t$ maps every component to be contracted to a single point in $U_t$. For example, $f:C\rightarrow V_T$ contracts the components to be contracted. \begin{prop} \label{cae} There exists a universal morphism $p:C\rightarrow\tilde{C}$ contracting the components to be contracted. Let $\tilde{f}:\tilde{C}\rightarrow V$ be the unique map given by the universal mapping property of $(\tilde{C},p)$. Then $(\tilde{C},p(x_1),\ldots, p(x_n),\tilde{f})$ is a stable map from an $n$-pointed curve of genus $g$ to $V$ of class $\beta$. \end{prop} \begin{pf} This is a variation on Section~1 of \cite{knudsen}. Let us first prove the proposition for the case that $T$ is the spectrum of an algebraically closed field. Let $C'$ be a component to be contracted. {\em Case 1. }The component $C'$ has one node. We define $\tilde{C}=C-(C'-C)$ and let $p:C\rightarrow \tilde{C}$ be the obvious map. Clearly, $\O_{\tilde{C}}=p_{\ast}\O_C$, so $\tilde{C}$ satisfies the universal mapping property by Lemma~\ref{tloq}. The rest is trivial. {\em Case 2. }The component $C'$ has two nodes. We define $\overline{C}=C-(C'-C)$ and let $P_1$ and $P_2$ be the two points of $\overline{C}$ that intersect $C'$. Then we set $\tilde{C}=\overline{C}/P_1\sim P_2$, i.e.\ we identify the two points $P_1$ and $P_2$. We then proceed similarly as in Case~1. \begin{lem} \label{clocct} Let $T$ be the spectrum of an algebraically closed field and let $\tilde{C}$ be the universal curve contracting the components of $C$ to be contracted. Choose an ample invertible sheaf $M$ on $V$. Let \[L=\omega_C(x_1+\ldots+x_n)\otimes f^{\ast} M^{\otimes3}\] and \[\tilde{L}=\omega_{\tilde{C}}(p(x_1)+\ldots+p(x_n))\otimes \tilde{f}^{\ast} M^{\otimes3}.\] Then for all $k\geq0$ we have \begin{enumerate} \item $\tilde{L}^{\otimes k}=p_{\ast} L^{\otimes k}$, \item $p^{\ast} \tilde{L}^{\otimes k}={L}^{\otimes k}$, \item $R^1f_{\ast} L^{\otimes k}=0$, \item $H^i(\tilde{C},\tilde{L}^{\otimes k})=H^i(C,L^{\otimes k})$, for $i=0,1$. \end{enumerate} \end{lem} \begin{pf} This is analogous to Lemma~1.6 of \cite{knudsen}. \end{pf} \begin{lem} \label{tfll} Let $T$ be the spectrum of an algebraically closed field and let $L$ be defined as in Lemma~\ref{clocct}. Define the open subset $U$ of $C$ by \[U=\{x\in C\mathrel{\mid} \rtext{$x$ is smooth and $x$ is not in any component to be contracted}\}.\] Then for $k$ sufficiently large we have \begin{enumerate} \item $L^{\otimes k}$ is generated by global sections, \label{genglob} \item $H^1(C,L^{\otimes k})=0$, \item $L^{\otimes k}$ is normally generated, \label{normgen} \item $L^{\otimes k}(-P)$ is generated by global sections for all $P\in U$. \label{mpglob} \end{enumerate} (The sheaf $L$ is normally generated if $\Gamma(C,L)^{\otimes k}\rightarrow\Gamma(C,L^{\otimes k})$ is surjective, for all $k\geq1$.) \end{lem} \begin{pf} Let $\tilde{C}$ and $\tilde{L}$ be as in Lemma~\ref{clocct}. Note that one can apply Proposition~\ref{mplsm} to $\tilde{C}$ and $\tilde{L}$. Then the results are implied by Lemma~\ref{clocct}. \end{pf} We can now proceed with the proof of Proposition~\ref{cae} for general base $T$. Choose an ample invertible sheaf $M$ on $V$ and consider on $C$ the invertible sheaf \[L=\omega_{C/T}(x_1+\ldots+x_n)\otimes f^{\ast} M^{\otimes3},\] where $\omega_{C/T}$ is the relative dualizing sheaf of $C$ over $T$. Then form \[{\cal S}=\bigoplus_{k\geq0}\pi_{\ast} L^{\otimes k},\] where $\pi:C\rightarrow T$ is the structure map, and let \[\tilde{C}=\operatorname{\rm Proj}\nolimits{\cal S}.\] {\em Claim 1. }The formation of $\tilde{C}$ commutes with base change. \begin{pf} Clearly, the formation of $L^{\otimes k}$ commutes with base change. That the formation of $\pi_{\ast} L^{\otimes k}$ commutes with base change for $k$ sufficiently large follows from the fact that $H^1(C_t,L^{\otimes k})=0$, for all $t\in T$, by Lemma~\ref{tfll}. For $k=0$ this is trivially true. Thus the formation of \[{\cal S}^{(d)}=\bigoplus_{d\mid k}{\cal S}_k\] commutes with base change, for a suitable $d>0$. This implies the claim for $\tilde{C}$, since \[\tilde{C}=\operatorname{\rm Proj}\nolimits{\cal S}=\operatorname{\rm Proj}\nolimits{\cal S}^{(d)}.\qed\] \renewcommand{\qed}{}\end{pf} {\em Claim 2. }The structure map $\tilde{\pi}:\tilde{C}\rightarrow T$ is flat and projective. \begin{pf} The flatness of $\operatorname{\rm Proj}\nolimits{\cal S}^{(d)}$ follows from the fact that $\pi_{\ast} L^{\otimes k}$ is locally free, for $d\mid k$, which follows from the fact that its formation commutes with base change. By passing to a larger $d$ if necessary, we may assume that for every $k\geq0$ the homomorphism \[\pi_{\ast}(L^{\otimes d})^{\otimes k}\longrightarrow\pi_{\ast}(L^{\otimes dk}) \] is surjective. This may be checked on fibers and thus follows from Lemma~\ref{tfll}(\ref{normgen}). So ${\cal S}^{(d)}$ is generated by ${\cal S}^{(d)}_1$ and hence $\operatorname{\rm Proj}\nolimits{\cal S}^{(d)}$ is projective by Proposition~5.5.1 in \cite{ega2}. \end{pf} {\em Claim 3. }The canonical morphism from an open subset of $C$ to $\tilde{C}$ is everywhere defined, proper and surjective. \begin{pf} This canonical morphism is defined by $\pi^{\ast}{\cal S}\rightarrow\bigoplus_k L^{\otimes k}$, or equivalently by ${\cal S}\rightarrow\bigoplus_k\pi_{\ast} L^{\otimes k}$ (see Section~3.7 in \cite{ega2}). For it to be everywhere defined, it suffices to prove that $\pi^{\ast}\pi_{\ast} L^{\otimes k}\rightarrow L^{\otimes k}$ is an epimorphism, for $k$ sufficiently large. This may be checked on fibers and thus follows from Lemma~\ref{tfll}(\ref{genglob}). That the canonical morphism is dominant follows immediately, since ${\cal S}\rightarrow\bigoplus\pi_{\ast} L^{\otimes k}$ is an isomorphism. That it is proper, is clear. So it has to be surjective. \end{pf} We call this canonical morphism $p:C\rightarrow\tilde{C}$. {\em Claim 4. }Let $x$ be a geometric point of $\tilde{C}$ and $p^{-1}(x)$ the fiber of $p$ over $x$. Then either the cardinality of $p^{-1}(x)$ is one or $p^{-1}(x)$ is a component of $C_{\tilde{\pi}(x)}$ to be contracted. \begin{pf} Without loss of generality we may assume that $T$ is the spectrum of an algebraically closed field. Then with the notation of Lemma~\ref{tfll} and by Property~(\ref{mpglob}) of the same lemma, we have that $p{ \mid } U:U\rightarrow\tilde{C}$ is an open immersion. If $C'$ is to be contracted, then $L{ \mid } C'\cong\O_{C'}$, and so $C'$ is mapped to a point in $\tilde{C}$. These facts clearly imply Claim~4. \end{pf} {\em Claim 5. }We have $p_{\ast}\O_C=\O_{\tilde{C}}$. \begin{pf} With the notation of Claim~4, note that \[H^1(p^{-1}(x),\O_C\otimes_{\O_{\tilde{C}}}\kappa(x))=0,\] since $p^{-1}(x)$ is rational if it is one and not zero dimensional. So by Corollary~1.5 in \cite{knudsen}, the formation of $p_{\ast}\O_C$ commutes with base change in $T$. So to prove the claim, we may assume that $T$ is the spectrum of an algebraically closed field, but then it is clear. \end{pf} Now by Lemma~\ref{tloq} the last three claims imply that $p:C\rightarrow\tilde{C}$ is a universal morphism contracting the components to be contracted. In particular, we get a unique morphism $\tilde{f}:C\rightarrow V$ such that $\tilde{f}\mathbin{{\scriptstyle\circ}} p=f$. The fact that $(\tilde{C},p(x_1),\ldots,p(x_n),\tilde{f})$ is a stable map from an $n$-pointed curve of genus $g$ to $V$ of class $\beta$ may now be checked on fibers, which has already been done. This finishes the proof of Proposition~\ref{cae}. \end{pf} We now proceed with the proof of Theorem~\ref{mbfc} in Case~V\@. Let $n=\# F_{\sigma}(v_0)$. Choose an identification $\underline{n+1}\rightarrow F\t(v_0)$ mapping $n+1$ to $i_0$, the flag being removed. Then $(C_{v_0},x_1,\ldots,x_{n+1},f_{v_0})$ is a stable map to which Proposition~\ref{cae} applies and we let $p_{v_0}:C_{v_0}\rightarrow D_{v_0}$ be the universal morphism contracting the components to be contracted. For $v\not=v_0$ we let $D_v=C_v$ and $p_v:C_v\rightarrow D_v$ be the identity. It is then clear how to define $y$ and $h$ to get a stable map $(D,y,h)$ satisfying the universal mapping property of a pushforward under the graph morphism $\tau\rightarrow \sigma$ given by $a:\sigma\rightarrow \tau$. This completes the proof of Case~V\@. \qed To complete the proof of Theorem~\ref{mbfc} we need to show that if $(V,\tau,\alpha)$ is an object of ${\frak V}{\frak G}_s$ and $p:(C,x,f)\rightarrow(D,y,h)$ is a morphism of stable $(V,\tau,\alpha)$-maps (covering the identity of $(V,\tau,\alpha)$), then $p$ is an isomorphism. This is immediately reduced to the case that $(V,\tau,\alpha)=(V,g,n,\alpha)$ and using Lemma~\ref{tloq} to the case that $T$ is the spectrum of an algebraically closed field. Then it follows from the stability condition that $p$ cannot contract any rational components, so it is injective. To prove that $p$ is surjective use induction on the number of nodes of $D$. So let $P$ be a node of $D$ and let $D'$ be the curve obtained from $D$ by blowing up $P$ and let $p':C'\rightarrow D'$ by the pullback of $p:C\rightarrow D$ under $D'\rightarrow D$. {\em Case 1. } The curve $D'$ is disconnected, $D'=D_1'\amalg D_2'$. Then $C'=C_1'\amalg C_2'$ with induced maps $p_i':C_i'\rightarrow D_i'$, for $i=1,2$. Let $g_i=g(D_i')$ and $\alpha_i=f_{\ast}[D_i']$, for $i=1,2$. Then $g=g_1+g_2$ and $\alpha=\alpha_1+\alpha_2$. Now $g_i(C_i')\leq g_i(D_i')$ and $f_{\ast}[C_i']\leq f_{\ast}[D_i']$ imply that $g_i(C_i')=g_i$ and $f_{\ast}[C_i']=\alpha_i$ and thus we may apply the induction hypothesis to $p_1'$ and $p_2'$, proving the surjectivity of $p$. {\em Case 2. } The curve $D'$ is connected. Then $C'$ is connected, since otherwise we would have two curves contradicting the induction hypothesis. So me may apply the induction hypothesis to $p':C'\rightarrow D'$. This finally completes the proof of Theorem~\ref{mbfc}. \qed \begin{defn} \label{doosf} For a given object $(V,\tau)$ of ${\frak V}{\frak G}_s$, we let $\overline{M}(V,\tau)(T)$ be the fiber of $\overline{M}(T)$ over $(V,\tau)$ under the cofibration of Theorem~\ref{mbfc}. Letting $T$ vary we get a stack $\overline{M}(V,\tau)$ on the category of $k$-schemes with the fppf-topology. For $(V,\tau)=(V,g,n,\beta)$ we denote $\overline{M}(V,\tau)$ by $\overline{M}_{g,n}(V,\beta)$. \end{defn} If $\operatorname{\rm char}\nolimits k\not=0$, let $L$ be a very ample invertible sheaf on $V$. Consider only stable $V$-graphs for which $\beta(v)(L)<\operatorname{\rm char}\nolimits k$, for all $v\in V\t$. If this condition is satisfied, we say that $\tau$ or $(V,\tau)$ is {\em bounded by the characteristic}. If $\operatorname{\rm char}\nolimits k=0$, we call {\em every }$(V,\tau)$ bounded by the characteristic, so that we have uniform terminology. \begin{them} \label{mgnvas} For every $(V,\tau)$, bounded by the characteristic, the stack $\overline{M}(V,\tau)$ is a proper algebraic Deligne-Mumford stack over $k$. \end{them} \begin{pf} The proof will be postponed to a later section (see Corollary~\ref{mgnvass}). \end{pf} Every time we refer to $\overline{M}(V,\tau)$ as a Deligne-Mumford stack, we shall tacitly assume that $(V,\tau)$ is bounded by the characteristic. \begin{rmk} Theorems~\ref{mbfc} and~\ref{mgnvas} give rise to a functor \begin{eqnarray*} \overline{M}:{\frak V}{\frak G}_s & \longrightarrow & (\rtext{\normalshape proper algebraic DM-stacks over $k$})\\ (V,\tau) & \longmapsto & \overline{M}(V,\tau), \end{eqnarray*} by choosing for every $k$-scheme $T$ a {\em clivage normalis\'e }(see Definition~7.1 in \cite[Exp.\ ~VI]{sga1}) of the cofibered category $\overline{M}(T)$ over ${\frak V}{\frak G}_s$. Of course, this functor $\overline{M}$ is essentially independent of the choice of the {\em clivage normalis\'e}. Another way of stating this would be to construct a fibered category $\overline{M}$ over ${\frak V}{\frak G}^{{\text{\normalshape op}}}_s\times(\text{$k$-schemes})$, such that $\overline{M}(V,\tau)(T)$ is the fiber of $\overline{M}$ over $(V,\tau,T)$ and $\overline{M}(T)$ is the fiber of $\overline{M}$ over $T$. \end{rmk} \section{Further Study of $\overline{M}$} \begin{prop} \label{drfu} Let $(V,\tau)$ be an object of ${\frak V}{\frak G}_s$, bounded by the characteristic of $k$. Then the diagonal \[\Delta:\overline{M}(V,\tau)\longrightarrow\overline{M}(V,\tau)\times\overline{M}(V,\tau)\] is representable, finite and unramified. \end{prop} \begin{pf} The assumption that $(V,\beta)$ is bounded by the characteristic implies that all stable maps of class $\beta$ are separable. So by Lemma~\ref{rsmsc} we may reduce the case of stable maps to the case of stable curves, which is well-known. \end{pf} \begin{lem} \label{rsmsc} Let $(C,x,f)$ and $(D,y,h)$ be $n$-pointed stable maps to $V$ over the base $T$, and $t\in T(K)$ a geometric point of $T$. Assume that $f_t:C_t\rightarrow V$ and $h_t:D_t\rightarrow V$ are separable morphisms. Then there exists an \'etale neighborhood $S\rightarrow T$ of $t$, an integer $N$, markings $x'=(x_1',\ldots,x_N')$ of $C_S$ and $y'=(y_1',\ldots,y_N')$ of $D_S$ such that $(C_S,x_S,x')$ and $(D_S,y_S,y')$ are stable marked curves over $S$ and a closed immersion of sheaves on $(\rtext{$S$-schemes})$ \[\operatorname{\underline{\rm Isom}}\nolimits((C,x,f),(D,y,h))_S\longrightarrow\operatorname{\underline{\rm Isom}}\nolimits((C_S,x_S,x'),(D_S,y_S,y')).\] \end{lem} \begin{pf} Without loss of generality assume that $C$ and $D$ have the same genus $g$ and $f$ and $h$ have the same class $\beta$. Choose an embedding $\mu:V\hookrightarrow{\Bbb P}^r$, let $d=\mu_{\ast}\beta$ and reduce to the case $V={\Bbb P}^r$ and $d=\beta$. Let $N=d(r+1)$. Choose linearly independent hyperplanes $H_0,\ldots,H_r$ in ${\Bbb P}^r$ such that for each $i=0,\ldots,r$ \begin{enumerate} \item no special point of $C_t$ or $D_t$ is mapped into $H_{i,K}$ under $f_t$ or $g_t$, \item $f_t$ and $g_t$ are transversal to $H_{i,K}$. \end{enumerate} Then there exists an \'etale neighborhood $S\rightarrow T$ of $t$ such that \begin{enumerate} \item for each $i=0,\ldots,r$ \begin{enumerate} \item $H_{i,S}\cap C_S$ gives rise to $d$ sections $x_{di+1}',\ldots,x_{di+d}'$ of $C_S$ over $S$, \item $H_{i,S}\cap D_S$ gives rise to $d$ sections $y_{di+1}',\ldots,y_{di+d}'$ of $D_S$ over $S$, \end{enumerate} \item $(C_S,x_S,x')$ and $(D_S,y_S,y')$ are marked prestable curves. \end{enumerate} Then $(C_S,x_S,x')$ and $(D_S,y_S,y')$ are in fact stable and there exists an obvious morphism \[\operatorname{\underline{\rm Isom}}\nolimits((C,x,f),(D,y,h))_S\longrightarrow\operatorname{\underline{\rm Isom}}\nolimits((C_S,x_S,x'),(D_S,y_S,y')),\] which is clearly a closed immersion. \end{pf} \begin{lem} \label{lalss} Let $(C,x_1,\ldots,x_{n+1},f)$ be a stable map and $(D,y_1,\ldots,y_n,h)$ the stabilization under forgetting $x_{n+1}$. Let $p:C\rightarrow D$ be the structure morphism. Then any section $y_0$ of $D$ making $(D,y_0,\ldots,y_n)$ a marked prestable curve lifts uniquely to a section $x_0$ of $C$ making $(C,x_0,\ldots,x_n)$ a marked prestable curve. If $y_0$ avoids $p(x_{n+1})$, then $(C,x_0,\ldots,x_{n+1})$ is a marked prestable curve. \end{lem} \begin{pf} Let $V\subset D$ be the open subset consisting of smooth points of $D$ which are not in the image of $y_i$, for any $i=1,\ldots,n$. Let $U=p^{-1}(V)$. Then $p$ induces an isomorphism $p{ \mid } U:U\stackrel{\sim}{\rightarrow} V$. Moreover, $U$ is smooth and $x_{n+1}$ is the only section of $C$ which may meet $U$. \end{pf} \begin{prop} \label{uors} Let $(C,x_1,\ldots,x_{n+1},f)$ and $(\tilde{C},\tilde{x}_1,\ldots,\tilde{x}_{n+1},\tilde{f})$ be stable maps with isomorphic stabilizations forgetting the $(n+1)$-st section. Let $(C,y_1,\ldots,y_n,h)$ be such a stabilization, with structure maps $p:C\rightarrow D$ and $\tilde{p}:\tilde{C}\rightarrow D$. If $p(x_{n+1})=\tilde{p}(\tilde{x}_{n+1})$ then there exists a unique isomorphism $q:C\rightarrow\tilde{C}$ of stable maps such that $\tilde{p}\mathbin{{\scriptstyle\circ}} q=p$. \end{prop} \begin{pf} This is local over the base, so we may freely choose sections as necessary. In fact, choose sections $z_1,\ldots,z_N$ of $D$ in the smooth locus, avoiding $y_1,\ldots,y_n$ and $\Delta=p(x_{n+1})=\tilde{p}(\tilde{x}_{n+1})$ and making $$(D,z_1,\ldots,z_N,y_1,\ldots,y_n)$$ a stable marked curve. By Lemma~\ref{lalss} these lift uniquely to sections $w_1,\ldots,w_N$ of $C$ and $\tilde{w}_1,\ldots,\tilde{w}_N$ of $\tilde{C}$ making $$(C,w_1,\ldots,w_N,x_1,\ldots,x_{n+1})$$ and $$(\tilde{C},\tilde{w}_1,\ldots,\tilde{w}_N,\tilde{x}_1, \ldots,\tilde{x}_{n+1})$$ marked prestable curves. Moreover, these are clearly marked {\em stable } curves with a common stabilization $$(D,z_1,\ldots,z_N,y_1,\ldots,y_n)$$ forgetting the last section, such that $p(x_{n_1})=\tilde{p}(\tilde{x}_{n+1})$. Then they have to be isomorphic by Knutson's theorem (see~\cite{knudsen}) that $\overline{M}_{g,N+n+1}$ is the universal curve over $\overline{M}_{N+n}$. \end{pf} \begin{prop} \label{flpuc} Let $(C,x_1,\ldots,x_n,f)$ be a stable map and $\Delta$ a section of $C$. Then there exists up to isomorphism a unique stable map $(\tilde{C},\tilde{x}_1,\ldots,\tilde{x}_{n+1},\tilde{f})$ such that $(C,x_1,\ldots,x_n,f)$ is the stabilization of $(\tilde{C},\tilde{x}_1,\ldots,\tilde{x}_{n+1},\tilde{f})$ forgetting the $(n+1)$-st section and $p(\tilde{x}_{n+1})=\Delta$, where $p:\tilde{C}\rightarrow C$ is the structure map. \end{prop} \begin{pf} Uniqueness follows from Proposition~\ref{uors}, hence existence is a local question. Thus we may choose sections $z_1,\ldots,z_N$ of $C$ such that $$(C,z_1,\ldots,z_N,x_1,\ldots,x_n)$$ is a stable marked curve. By Knudsen's result again, there exists a stable curve $$(C',z_1',\ldots,z_N',x_1',\ldots,x_{n+1}')$$ whose stabilization forgetting the last section is $$(C,z_1,\ldots,z_N,x_1,\ldots,x_n)$$ and such that $q(x_{n+1}')=\Delta$, where $q:C'\rightarrow C$ is the structure map. Clearly, $$(C',z_1',\ldots,z_N',x_1',\ldots,x_{n+1}',f\mathbin{{\scriptstyle\circ}} q)$$ is a stable map. Then let $(\tilde{C},\tilde{x}_1,\ldots,\tilde{x}_{n+1},\tilde{f})$ be the stabilization of $$(C',z_1',\ldots,z_N',x_1',\ldots,x_{n+1}',f\mathbin{{\scriptstyle\circ}} q)$$ forgetting the sections $z_1',\ldots,z_N'$. By its universal mapping property there exists a morphism $p:\tilde{C}\rightarrow C$ which makes $(C,x_1,\ldots,x_n,f)$ the stabilization of $(\tilde{C},\tilde{x}_1,\ldots,\tilde{x}_{n+1},\tilde{f})$ forgetting $\tilde{x}_{n+1}$. \end{pf} \begin{cor} \label{esuc} Let $C_{g,n}(V,\beta)$ be the universal curve over $\overline{M}_{g,n}(V,\beta)$. Then the canonical morphism $\overline{M}_{g,n+1}(V,\beta)\rightarrow C_{g,n}(V,\beta)$ induced by the $(n+1)$-st section is an isomorphism. \qed \end{cor} \begin{prop} Let $(C,x,f)$ be a stable $(V,g,n,\beta)$-map over $T$. Then the set of $t\in T$ such that $(C,x)$ is a stable marked curve is open in $T$. \end{prop} \begin{pf} The set of such $t$ is the set of all $t\in T$ for which $(C,x)$ is isomorphic to its stabilization. For any morphism of schemes, the set of elements of its source at which it is an isomorphism is always open. Finally, use properness of prestable curves. \end{pf} By this proposition we may define \[U_{g,n}(V,\beta)\subset\overline{M}_{g,n}(V,\beta)\] to be the open substack of those stable maps, whose underlying marked curve is stable. The canonical morphism $U_{g,n}(V,\beta)\rightarrow \overline{M}_{g,n}$ has as fiber over the marked curve $(C,x)$ the scheme of morphisms form $C$ to $V$ of class $\beta$. By results of Grothendieck in \cite{fgaIV} this is a quasi-projective scheme. Hence $U_{g,n}(V,\beta)$ is an algebraic $k$-stack of finite type. Now, for given $n$ there exists an $N>n$ such that $U_{g,N}(V,\beta)\rightarrow\overline{M}_{g,n}(V,\beta)$ is surjective. Since this morphism is flat by Corollary~\ref{esuc}, it is a flat epimorphism, hence a presentation of $\overline{M}_{g,n}(V,\beta)$. Together with Proposition~\ref{drfu} this implies that $\overline{M}_{g,n}(V,\beta)$ is a finite type separated algebraic Deligne-Mumford stack over $k$. This is then true for all objects of ${\frak V}{\frak G}_s$, bounded by the characteristic. \begin{cor} \label{mgnvass} Theorem~\ref{mgnvas} is true. \end{cor} \begin{pf} It only remains to show properness. This is easily reduced to the case $(V,\tau)=({\Bbb P}^r,g,n,d)$ and follows from Proposition~3.3 of \cite{pandh}. \end{pf} \section{An Operadic Picture} \begin{defn} Let $(\tau,\alpha)$ be an $A$-graph. Let $R\t\subset F\t\times F\t$ be defined by $(f,\overline{f})\in R\t$ if and only if one of the conditions \begin{enumerate} \item $\overline{f}=j\t(f)$, \item $\partial f=\partial\overline{f}$ and for $v=\partial f=\partial\overline{f}$ we have $g(v)=\alpha(v)=0$ \end{enumerate} is satisfied. Let $\sim$ be the equivalence relation on $F\t$ generated by $R\t$ and let \[P\t=F\t/\sim.\] (In fact, $P_{(\tau,\alpha)}$ would be better notation, but we will stick with the abuse of notation $P_{\tau}$.) \end{defn} \begin{prop} \label{cmmgpe} Let $a:(B,\sigma)\rightarrow(A,\tau)$ be a combinatorial morphism of marked graphs. Then $a_F:F_{\sigma}\rightarrow F\t$ preserves equivalence. \qed \end{prop} \begin{rmk} In fact, Condition~(\ref{commor3}) of Definition~\ref{commor} may be replaced by requiring $a_F$ to preserve equivalence. \end{rmk} \begin{prop} \label{dafpea} Let $\phi:\tau\rightarrow\sigma$ be a contraction of $A$-graphs. Then $\phi^F:F_{\sigma}\rightarrow F\t$ preserves equivalence. \qed \end{prop} \begin{prop} \label{dpc} If \[\begin{array}{ccccc} B & & \pi & \stackrel{\psi}{\longrightarrow} & \rho \\ \ldiagup{\xi} & \phantom{\longrightarrow} & \ldiag{b} & & \rdiag{a} \\ A & & \sigma & \stackrel{\phi}{\longrightarrow} & \tau \end{array}\] is a stable pullback, then the induced diagram \[\comdiaback{P_\pi}{\psi^F}{P_\rho}{b}{}{a}{P_{\sigma}}{\phi^F}{P\t}\] commutes. \qed \end{prop} By Propositions~\ref{cmmgpe},~\ref{dafpea} and~\ref{dpc}, we have a contravariant functor \[P:{\frak G}_s\longrightarrow(\rtext{finite sets})\] given by $P(A,\tau)=P\t$ on objects. Composing with the functor ${\frak V}{\frak G}_s\rightarrow{\frak G}_s$ we get a contravariant functor \begin{eqnarray*} P:{\frak V}{\frak G}_s & \longrightarrow & (\rtext{finite sets}) \\ (V,\tau) & \longmapsto & P\t. \end{eqnarray*} There is an obvious functor \begin{eqnarray*} {\frak V}\times(\rtext{finite sets}) & \longrightarrow & {\frak V} \\ (V,P) & \longmapsto & V^P, \end{eqnarray*} contravariant in the second argument, and composing with $P$ times the natural functor ${\frak V}{\frak G}_s\rightarrow{\frak V}$ gives rise to a covariant functor \begin{eqnarray*} P:{\frak V}{\frak G}_s & \longrightarrow & {\frak V} \\ (V,\tau) & \longmapsto & V^{P\t}, \end{eqnarray*} still denoted $P$, by abuse of notation. We may consider ${\frak V}$ as a subcategory of the 2-category of proper algebraic Deligne-Mumford stacks over $k$ and consider this as a functor \[P:{\frak V}{\frak G}_s\longrightarrow(\rtext{proper algebraic DM-stacks over $k$}).\] Now fix an object $(V,\tau)$ of ${\frak V}{\frak G}_s$. Let $(C,x,f)$ be a stable $(V,\tau)$-map over $T$. Then $x$ and $f$ define a morphism \begin{eqnarray*} f(x):T & \longrightarrow & V^{F\t} \\ t & \longmapsto & (f(x_i(t)))_{i\in F\t}. \end{eqnarray*} By Corollary~\ref{zc} this morphism $f(x)$ factors through $V^{P\t}\subset V^{F\t}$, so we consider it as a morphism \[f(x):T\longrightarrow V^{P\t}.\] Thus we get a map $\overline{M}(V,\tau)(T)\rightarrow P(V,\tau)(T)$. Since it is compatible with base change $S\rightarrow T$, we have a morphism of $k$-stacks \[\operatorname{\rm ev}(V,\tau):\overline{M}(V,\tau)\longrightarrow P(V,\tau).\] \begin{prop} We have defined a natural transformation of functors from ${\frak V}{\frak G}_s$ to $(\rtext{proper algebraic DM-stacks over $k$})$ \[\operatorname{\rm ev}:\overline{M}\longrightarrow P,\] called {\em evaluation}. \end{prop} In the general framework of $\Gamma$-operads, this allows us to consider (appropriate subfunctors of) $\overline{M}$ and $P$ as a modular operad and a cyclic endomorphism operad, respectively. The evaluation functor then induces a structure of $\overline{M}$-algebra on $V$. \vfill\eject \part{Gromov-Witten Invariants} \section{Isogenies} \begin{defn} Let $\tau$ be a stable $A$-graph. \begin{enumerate} \item The {\em class }of $\tau$ is \[\beta(\tau)=\sum_{v\in V\t}\beta(v).\] \item The {\em Euler characteristic } of $\tau$ is \[\chi(\tau)=\chi(|\tau|)-\sum_{v\in v\t}g(v).\] \item If $|\tau|$ is non-empty and connected the {\em genus }of $\tau$ is \[g(\tau)=1-\chi(\tau).\] \end{enumerate} \end{defn} \begin{defn} Let $\tau$ be a stable $V$-graph, where $V$ is of pure dimension. \begin{enumerate} \item The {\em dimension } of $\tau$ is \[\dim(V,\tau)=\chi(\tau)(\dim V-3)-\beta(\tau)(\omega_V)+\# S\t-\# E\t,\] where $\omega_V$ is the canonical line bundle on $V$. \item The {\em degree } of $\tau$ is \begin{multline*} \deg(V,\tau)=\\ \beta(\tau)(\omega_V)+(\dim V-3)(\chi(\tau^s)-\chi(\tau))+(\# S_{\tau^s}-\# S\t)-(\# E_{\tau^s}-\# E\t), \end{multline*} where $\tau^s$ is the absolute stabilization of $\tau$. \end{enumerate} \end{defn} Note that \[\dim(V,\tau)-\dim(\tau^s)=\chi(\tau^s)\dim V-\deg(V,\tau).\] \begin{defn} The stable $A$-graph with one vertex of genus and class zero and three tails (no edges) shall be called the {\em $A$-tripod}, or simply a {\em tripod}. \end{defn} \begin{defn} Let $a:\tau'\rightarrow\tau$ be a combinatorial morphism of stable $A$-graphs. A {\em tail map } for $a$ is a map $m:S_{\tau'}\rightarrow S\t$. Let $(a,\tau',\phi):\tau\rightarrow\sigma$ be a morphism of stable $A$-graphs. A {\em tail map } for $(a,\tau',\phi)$ is a tail map for $a$. \end{defn} Since for a contraction $\phi:\tau'\rightarrow\sigma$ the map $\phi^F:F_{\sigma}\rightarrow F_{\tau'}$ induces a bijection $\phi^S:S_{\sigma}\rightarrow S_{\tau'}$, there is an obvious way to compose morphisms with tail map of stable $A$-graphs. By abuse we will also call the composition $m\mathbin{{\scriptstyle\circ}}\phi^S$ the tail map of $(a,\tau',\phi,m)$. \begin{defn} \label{dsft} Let $a:\tau'\rightarrow \tau$ be a combinatorial morphism of stable $A$-graphs with tail map $m:S_{\tau'}\rightarrow S\t$. We say that $(a,m)$ is of type {\em stably forgetting a tail}, or that $\tau'$ is obtained form $\tau$ by {\em stably forgetting a tail}, if there exists a tail $f$ of $\tau$ such that \begin{enumerate} \item $\tau'$ is the stabilization of $\tau''$, where $\tau''$ is obtained from $\tau$ by forgetting the tail $f$, \item for all $h\in S_{\tau'}$, such that $a(h)$ is a tail, we have $m(h)=a(h)$, \item $f\not\in m(S_{\tau'})$. \end{enumerate} \end{defn} Note that the tail $f$ is uniquely determined by $(a,m)$ and $f$ determines $(a,m)$ uniquely. We say that $(a,m)$ stably forgets the tail $f$. \begin{numrmk} \label{sft} Every combinatorial morphism (with tail map) of type stably forgetting a tail is of one of the following types (notation of Definition~\ref{dsft}). {\em Type I (Incomplete case). } No stabilization is needed, i.e.\ $\tau'=\tau''$. \[ \beginpictur \setcoordinatesystem units <.3cm,.3cm> point at 3 2 \setplotarea x from 2 to 6, y from 0 to 4 \plot 3 2 4 3 / \plot 2 2 4 2 / \plot 3 2 4 1 / \shaderectangleson \setshadegrid span <1mm> \putrectangle corners at 4 0 and 6 4 \put {\circle*{4}} [Bl] at 3 2 \axis left invisible label {$\scriptstyle\tau$} / \axis right invisible label {\phantom{$\scriptstyle\tau$}} / \endpicture \stackrel{a}{\longleftarrow} \beginpictur \setcoordinatesystem units <.3cm,.3cm> point at 3 2 \setplotarea x from 3 to 6, y from 0 to 4 \plot 3 2 4 3 / \plot 3 2 4 2 / \plot 3 2 4 1 / \shaderectangleson \setshadegrid span <1mm> \putrectangle corners at 4 0 and 6 4 \put {\circle*{4}} [Bl] at 3 2 \axis left invisible label {\phantom{$\scriptstyle\tau'$}} / \axis right invisible label {{$\scriptstyle\tau'$}} / \endpicture \] {\em Type II (Removing a tripod from a tail). }Only in this case does the tail map $m$ contain any information. It serves to `remember' which of the two tails in the following diagram is being forgotten by $a$. \[ \beginpictur \setcoordinatesystem units <.3cm,.3cm> point at 3 2 \setplotarea x from 2 to 6, y from 0 to 4 \plot 2 3 3 2 / \plot 3 2 4 2 / \plot 2 1 3 2 / \shaderectangleson \setshadegrid span <1mm> \putrectangle corners at 4 0 and 6 4 \put {\circle*{4}} [Bl] at 3 2 \axis left invisible label {$\scriptstyle\tau$} / \axis right invisible label {\phantom{$\scriptstyle\tau$}} / \endpicture \stackrel{a}{\longleftarrow} \beginpictur \setcoordinatesystem units <.3cm,.3cm> point at 3 2 \setplotarea x from 3 to 6, y from 0 to 4 \plot 3 2 4 2 / \shaderectangleson \setshadegrid span <1mm> \putrectangle corners at 4 0 and 6 4 \axis left invisible label {\phantom{$\scriptstyle\tau'$}} / \axis right invisible label {{$\scriptstyle\tau'$}} / \endpicture \] {\em Type III (Removing a tripod from an edge). } \[ \beginpictur \setcoordinatesystem units <.3cm,.3cm> point at 3 2 \setplotarea x from 2 to 6, y from 0 to 4 \plot 2 2 3 2 / \circulararc 180 degrees from 4 3 center at 4 2 \shaderectangleson \setshadegrid span <1mm> \putrectangle corners at 4 0 and 6 4 \put {\circle*{4}} [Bl] at 3 2 \axis left invisible label {$\scriptstyle\tau$} / \axis right invisible label {\phantom{$\scriptstyle\tau$}} / \endpicture \stackrel{a}{\longleftarrow} \beginpictur \setcoordinatesystem units <.3cm,.3cm> point at 3 2 \setplotarea x from 3 to 6, y from 0 to 4 \circulararc 180 degrees from 4 3 center at 4 2 \shaderectangleson \setshadegrid span <1mm> \putrectangle corners at 4 0 and 6 4 \axis left invisible label {\phantom{$\scriptstyle\tau'$}} / \axis right invisible label {{$\scriptstyle\tau'$}} / \endpicture \] {\em Type IV (Forgetting a lonely tripod or a lonely elliptic component.) } Only in this case does the number of connected components of the geometric realization change. \[ \beginpictur \setcoordinatesystem units <.3cm,.3cm> point at 3 2 \setplotarea x from 2 to 6, y from 0 to 4 \plot 2 3 3 2 / \plot 2 2 3 2 / \plot 2 1 3 2 / \shaderectangleson \setshadegrid span <1mm> \putrectangle corners at 4 0 and 6 4 \put {\circle*{4}} [Bl] at 3 2 \axis left invisible label {$\scriptstyle\tau$} / \axis right invisible label {\phantom{$\scriptstyle\tau$}} / \endpicture \stackrel{a}{\longleftarrow} \beginpictur \setcoordinatesystem units <.3cm,.3cm> point at 3 2 \setplotarea x from 3 to 6, y from 0 to 4 \shaderectangleson \setshadegrid span <1mm> \putrectangle corners at 4 0 and 6 4 \axis left invisible label {\phantom{$\scriptstyle\tau'$}} / \axis right invisible label {{$\scriptstyle\tau'$}} / \endpicture \] \[ \beginpictur \setcoordinatesystem units <.3cm,.3cm> point at 3 2 \setplotarea x from 2 to 6, y from 0 to 4 \plot 2 2 3 2 / \shaderectangleson \setshadegrid span <1mm> \putrectangle corners at 4 0 and 6 4 \put {\circle*{4}} [Bl] at 3 2 \axis left invisible label {$\scriptstyle\tau$} / \axis right invisible label {\phantom{$\scriptstyle\tau$}} / \endpicture \stackrel{a}{\longleftarrow} \beginpictur \setcoordinatesystem units <.3cm,.3cm> point at 3 2 \setplotarea x from 3 to 6, y from 0 to 4 \shaderectangleson \setshadegrid span <1mm> \putrectangle corners at 4 0 and 6 4 \axis left invisible label {\phantom{$\scriptstyle\tau'$}} / \axis right invisible label {{$\scriptstyle\tau'$}} / \endpicture \] Here, the genus of the vertex displayed in the last diagram is equal to one. \end{numrmk} \begin{defn} Let $(a,\tau',\phi,m):\tau\rightarrow\sigma$ be a morphism of stable $A$-graphs with tail map. We call $(a,\tau',\phi,m)$ an {\em isogeny}, if \begin{enumerate} \item $(a,m)$ is a composition of morphisms of type stably forgetting a tail, \item $\pi_0|\sigma|\rightarrow\pi_0|\tau|$ is bijective. \end{enumerate} We call the isogeny $\Phi:\tau\rightarrow\sigma$ an {\em elementary isogeny}, if it is an elementary contraction, or if $\sigma$ is obtained from $\tau$ by stably forgetting a tail. \end{defn} \begin{note} If $\Phi:\tau\rightarrow\sigma$ is an isogeny of stable $A$-graphs, then $\chi(\sigma)=\chi(\tau)$. An elementary isogeny either contracts a loop, or a non-looping edge or is of type stably forgetting a tail~I, II, or~III. If we write $\Phi:\tau\rightarrow\sigma$ for an isogeny of stable $A$-graphs, we denote the tail map of $\Phi$ by $\Phi^S:S_{\sigma}\rightarrow S\t$. \end{note} \begin{prop} The composition of isogenies is an isogeny. \end{prop} \begin{pf} Let \[\begin{array}{ccccc} A & & \pi & \stackrel{\psi}{\longrightarrow} & \rho \\ \ldiagup{\operatorname{\rm id}} & \phantom{\longrightarrow} & \ldiag{b} & & \rdiag{a} \\ A & & \sigma & \stackrel{\phi}{\longrightarrow} & \tau \end{array}\] be a stable pullback, where $a$ stably forgets the tail $f$ of $\tau$, $\pi_0|\rho|\rightarrow\pi_0|\tau|$ is bijective and $\phi$ is an elementary contraction of stable $A$-graphs. Then $b$ stably forgets the tail $\phi^F(f)$ of $\sigma$. Even if there is a vertex $v_0$ of $\tau$ which does not appear in $\rho$, this vertex $v_0$ cannot be the vertex onto which $\phi$ contracts an edge. \end{pf} Fix a semi-group with indecomposable zero $A$. We shall define a category $\tilde{{\frak G}}_s(A)$ from ${\frak G}_s(A)$, retaining only isogenies and morphisms of type cutting edges, but reversing the direction of the latter, making them morphisms {\em gluing tails}. In fact, define the category $\tilde{{\frak G}}_s(A)$ as follows. Objects of $\tilde{{\frak G}}_s(A)$ are stable $A$-graphs. A morphism $\sigma\rightarrow\tau$ is a triple $(a,\sigma',\Phi)$, where $a:\sigma\rightarrow\sigma'$ is a combinatorial morphism of $A$-graphs of type cutting edges and $\Phi:\sigma'\rightarrow\tau$ is an isogeny of stable $A$-graphs. To compose $(a,\sigma',\Phi):\sigma\rightarrow\tau$ and $(b,\tau',\Psi):\tau\rightarrow\rho$, we need to construct a diagram \begin{equation}\label{dcmei} \begin{array}{ccccc} \sigma'' & \stackrel{\Xi}{\longrightarrow} & \tau' & \stackrel{\Psi}{\longrightarrow} & \rho \\ \ldiagup{c} & & \rdiagup{b} & & \\ \sigma' & \stackrel{\Phi}{\longrightarrow} & \tau & & \\ \ldiagup{a} & & & & \\ \sigma, & & & & \end{array} \end{equation} where $c:\sigma'\rightarrow\sigma''$ is a combinatorial morphism of type cutting edges and $\Xi:\sigma''\rightarrow\tau'$ is an isogeny of stable $A$-graphs. Let $f$ and $\overline{f}$ be two tails of $\tau$ such that $\{b(f),b(\overline{f})\}$ is an edge of $\tau'$. Then construct $\sigma''$ from $\sigma'$ by gluing the two tails $\Phi^S(f)$ and $\Phi^S(\overline{f})$ to an edge. If $b$, cuts more than one edge, iterate this process to construct $\sigma''$. This defines composition of morphisms in $\tilde{{\frak G}}_s(A)$, which is clearly associative. \begin{note} In the situation of (\ref{dcmei}), we get a diagram in ${\frak G}_s(A)$ \[\comdia{\sigma''}{\Xi}{\tau'}{\overline{c}}{}{\overline{b}}{\sigma'}{\Phi}{\tau},\] which is easily seen to commute. Here, $\overline{b}$ and $\overline{c}$ are the morphisms of stable $A$-graphs induced by $b$ and $c$, respectively. \end{note} \begin{defn} \label{ecisg} We call $\tilde{{\frak G}}_s(A)$ the {\em extended category of isogenies of stable $A$-graphs}, or the {\em extended isogeny category }over $A$. The morphisms in $\tilde{{\frak G}}_s(A)$ are called {\em extended isogenies}. An extended isogeny is called {\em elementary}, if it is an elementary isogeny or glues two tails to an edge. \end{defn} Now consider the following situation. Fix a smooth projective variety $V$ of pure dimension. Let $\Phi:\tau\rightarrow\sigma$ be an elementary extended isogeny of stable modular graphs. Let $\sigma'$ be a stable $V$-graph and $b:\sigma\rightarrow\sigma'$ a combinatorial morphism identifying $\sigma$ as the absolute stabilization of $\sigma'$. Note that $b$ is injective on vertices and complete, so that $b:F_{\sigma}(v)\rightarrow F_{\sigma'}(b(v))$ is bijective, for all $v\in V_{\sigma}$. Let $(a_i,\tau_i)_{i\in I}$ be a family of pairs, where $I$ is a finite set and for each $i\in I$ we have a combinatorial morphism $a_i:\tau\rightarrow\tau_i$ identifying $\tau$ as the absolute stabilization of $\tau_i$. Finally, let for every $i\in I$ be given an extended isogeny of stable $V$-graphs $\Phi_i:\tau_i\rightarrow\sigma'$. In particular, for each $i\in I$ we have a diagram of stable marked graphs (but note that the horizontal and vertical morphisms live in different categories) \[\comdia{\tau_i}{\Phi_i}{\sigma'}{\overline{a}_i}{}{\overline{b}}{\tau}{ \Phi}{\phantom{b}\sigma \phantom{b}.}\] We shall now define what we mean by $(a_i,\tau_i,\Phi_i)_{i\in I}$ to be {\em cartesian}, or a {\em pullback } of $\sigma'$ under $\Phi$. We have to distinguish six cases, according to which kind of elementary extended isogeny $\Phi$ is. Let us first consider the case that $\Phi$ is an elementary contraction $\phi:\tau\rightarrow\sigma$, contracting the edge $\{f,\overline{f}\}$ of $\tau$. As usual, let $v_1=\partial f$, $v_2=\partial\overline{f}$ and $v_0=\phi(v_1)=\phi(v_2)$. Let $w_0=b(v_0)$. {\em Case I (Contracting a loop). } In this case $v_1=v_2$. The set $I$ has one element, say $0$, and $(a_0,\tau_0,\Phi_0)$ is cartesian, if $\Phi_0$ is a contraction contracting a single loop $\{a_0(f),a_0(\overline{f})\}$ onto $w_0$. {\em Case II (Contracting a non-looping edge). } In this case $v_1\not=v_2$. We require each $\Phi_i$ to contract exactly one edge, namely $\{a_i(f),a_i(\overline{f})\}$ onto $w_0$. In particular, this means that the only way the various $(a_i,\tau_i,\Phi_i)$ differ is in the classes of $a_i(v_1)$ and $a_i(v_2)$. We require that $(\beta(a_i(v_1)),\beta(a_i(v_2)))_{i\in I}$ be a complete and non-repetitive list of all pairs of elements of $H_2(V)^+$ adding up to $\beta(w_0)$. {\em Case III (Forgetting a tail). } Let us now deal with the case that $\Phi:\tau\rightarrow\sigma$ stably forgets the tail $f\in S\t$. We require $I$ to have one element, say $0$, and call $(a_0,\tau_0,\Phi_0)$ cartesian if $\Phi_0$ stably forgets the tail $a_0(f)$ (and does nothing else). {\em Case IV (Gluing two tails to an edge). } Finally, let us consider the case that $\Phi$ is given by a combinatorial morphism $c:\tau\rightarrow\sigma$, gluing the two tails $f$ and $\overline{f}$ of $\tau$ to an edge $\{c(f),c(\overline{f})\}$ of $\sigma$. Again, $I$ is required to have one element, say $0$, and $(a_0,\tau_0,\Phi_0)$ is called cartesian if $\Phi_0$ glues two tails of $\tau_0$ to an edge of $\sigma'$ (and does nothing else). Moreover, we require that $\Phi_0\mathbin{{\scriptstyle\circ}} a_0=b\mathbin{{\scriptstyle\circ}} c$. An example: \[\begin{array}{ccc} \beginpictur \setcoordinatesystem units <.3cm,.3cm> point at 3 2 \setplotarea x from 2 to 6, y from 0 to 4 \plot 2 3 4 3 / \plot 4 1 3 1 / \shaderectangleson \setshadegrid span <1mm> \putrectangle corners at 4 0 and 6 4 \put {\circle*{4}} [Bl] at 3 3 \axis left invisible label {$\scriptstyle\tau_0$} / \axis right invisible label {\phantom{$\scriptstyle\tau_0$}} / \axis bottom invisible label {\phantom{.}} / \endpicture & \stackrel{\Phi_0}{\longrightarrow} & \beginpictur \setcoordinatesystem units <.3cm,.3cm> point at 3 2 \setplotarea x from 3 to 6, y from 0 to 4 \circulararc 180 degrees from 4 3 center at 4 2 \shaderectangleson \setshadegrid span <1mm> \putrectangle corners at 4 0 and 6 4 \put {\circle*{4}} [Bl] at 3 2 \axis left invisible label {\phantom{$\scriptstyle\sigma'$}} / \axis right invisible label {{$\scriptstyle\sigma'$}} / \axis bottom invisible label {\phantom{.}} / \endpicture \\ \ldiagup{a_0} & & \rdiagup{b} \\ \beginpictur \setcoordinatesystem units <.3cm,.3cm> point at 3 2 \setplotarea x from 2 to 6, y from 0 to 4 \plot 4 1 3 1 / \plot 3 3 4 3 / \shaderectangleson \setshadegrid span <1mm> \putrectangle corners at 4 0 and 6 4 \axis left invisible label {{$\scriptstyle\tau$}} / \axis right invisible label {\phantom{$\scriptstyle\tau$}} / \axis top invisible label {\phantom{.}} / \endpicture & \stackrel{c}{\longrightarrow} & \beginpictur \setcoordinatesystem units <.3cm,.3cm> point at 3 2 \setplotarea x from 3 to 6, y from 0 to 4 \circulararc 180 degrees from 4 3 center at 4 2 \shaderectangleson \setshadegrid span <1mm> \putrectangle corners at 4 0 and 6 4 \axis left invisible label {\phantom{$\scriptstyle\sigma$}} / \axis right invisible label {{$\scriptstyle\sigma$}} / \axis top invisible label {\phantom{.}} / \endpicture \end{array} \] Note that in each case pullbacks exist, even though they are not necessarily unique, even up to isomorphism, in the last two cases. Note also, that for each $i\in I$ we have $\deg(\tau_i)=\deg(\sigma')$. We shall now define still another category, denoted $\tilde{{\frak G}}_s(V)_{{\mbox{\tiny cart}}}$, called the {\em cartesian extended isogeny category }over $V$. \begin{defn} \label{ceicv} Objects of $\tilde{{\frak G}}_s(V)_{\mbox{\tiny cart}}$ are pairs $(\tau,(a_i,\tau_i)_{i\in I})$, where $\tau$ is a stable modular graph, $I$ is a finite set and for each $i\in I$ the pair $(a_i,\tau_i)$ is a stable $V$-graph $\tau_i$, together with a combinatorial morphism $a_i:\tau\rightarrow\tau_i$, identifying $\tau$ as the absolute stabilization of $\tau_i$. A {\em premorphism } from $(\tau,(a_i,\tau_i)_{i\in I})$ to $(\sigma,(b_j,\sigma_j)_{j\in J})$ is a triple $(\Phi,\lambda,(\Phi_i)_{i\in I})$, where $\Phi:\tau\rightarrow\sigma$ is an extended isogeny of stable modular graphs, $\lambda:I\rightarrow J$ is a map and for each $i\in I$ we have an extended isogeny of stable $V$-graphs $\Phi_i:\tau_i\rightarrow\sigma_{\lambda(i)}$. It is clear how to compose such premorphisms and that composition is associative. A {\em morphism }in $\tilde{{\frak G}}_s(V)_{\mbox{\tiny cart}}$ is defined to be a premorphism which can be factored into a composition of elementary morphisms and isomorphisms. It is clear what an isomorphism is. An {\em elementary morphism }is a premorphism $(\Phi,\lambda,(\Phi_i)_{i\in I})$, as above, such that \begin{enumerate} \item $\Phi$ is an elementary extended isogeny, \item for each $j\in J$ we have that $(a_i,\tau_i,\Phi_i)_{i\in\lambda^{-1}(j)}$ is cartesian in the sense defined in Cases~I through~IV, above. \end{enumerate} \end{defn} \begin{numrmk} \label{cfnc} Projecting onto the first component defines a functor \[\tilde{{\frak G}}_s(V)_{\mbox{\tiny cart}}\longrightarrow\tilde{{\frak G}}_s(0).\] Despite the notation, this is not a fibration of categories. \end{numrmk} We shall, in what follows, often shorten the notation $(\tau,(a_i,\tau_i)_{i\in I})$ to $(\tau,(\tau_i)_{i\in I})$ or even $(\tau_i)_{i\in I}$. Call an object $(\tau_i)_{i\in I}$ of $\tilde{{\frak G}}_s(V)_{\mbox{\tiny cart}}$ {\em homogeneous } of degree $n\in{\Bbb Z}$, if for all $i\in I$ we have $\deg(V,\tau_i)=n$. For a stable modular graph $\tau$, we may consider the fiber $\tilde{{\frak G}}_s(V)_{{\mbox{\tiny cart}}/\tau}$ of the functor $\tilde{{\frak G}}_s(V)_{\mbox{\tiny cart}}\rightarrow\tilde{{\frak G}}_s(0)$ over $\tau$. In every such fiber $\tilde{{\frak G}}_s(V)_{{\mbox{\tiny cart}}/\tau}$ we have a functor \[\oplus:\tilde{{\frak G}}_s(V)_{{\mbox{\tiny cart}}/\tau}\times\tilde{{\frak G}}_s(V)_{{\mbox{\tiny cart}}/\tau} \longrightarrow\tilde{{\frak G}}_s(V)_{{\mbox{\tiny cart}}/\tau},\] given by \[(\tau_i)_{i\in I}\oplus(\sigma_j)_{j\in J}=((\tau_i)_{i\in I},(\sigma_j)_{j\in J}),\] where we think of the object on the right hand side as a family parametrized by $I\amalg J$. The functor $\oplus$ satisfies some obvious properties, which we shall not list. Every object $X=(\tau_i)_{i\in I}$ of $\tilde{{\frak G}}_s(V)_{\mbox{\tiny cart}}$ has a unique decomposition $X=\bigoplus_{n\in{\Bbb Z}}X_n$ into homogeneous components. Every morphism in $\tilde{{\frak G}}_s(V)_{\mbox{\tiny cart}}$ respects this decomposition. Finally, $\tilde{{\frak G}}_s(V)_{\mbox{\tiny cart}}$ is a tensor category (in the sense of \cite{delmil}) with tensor product given by \[\otimes:\tilde{{\frak G}}_s(V)_{{\mbox{\tiny cart}}}\times\tilde{{\frak G}}_s(V)_{{\mbox{\tiny cart}}} \longrightarrow \tilde{{\frak G}}_s(V)_{{\mbox{\tiny cart}}},\] which is defined by the formula \[(\tau,(\tau_i)_{i\in I})\otimes(\sigma,(\sigma_j)_{j\in J})=(\tau\times\sigma,(\tau_i\times\sigma_j)_{(i,j)\in I\times J}).\] For two graphs $\sigma$ and $\tau$ we denote by $\sigma\times\tau$ the graph whose geometric realization is the disjoint union of $|\sigma|$ and $|\tau|$. This notion extends in an obvious way to marked graphs. The identity object for $\otimes$ is the one element family with value the empty graph. There are obvious compatibilities between these various structures on $\tilde{{\frak G}}_s(V)_{\mbox{\tiny cart}}$. For example, if $X=\bigoplus_nX_n$ and $Y=\bigoplus_mY_m$ are objects of $\tilde{{\frak G}}_s(V)_{\mbox{\tiny cart}}$, then the decomposition of $X\otimes Y$ into homogeneous components is given by \[X\otimes Y=\bigoplus_r\left(\bigoplus_{n+m=r}X_n\otimes Y_m\right).\] We summarize these properties by saying that $\tilde{{\frak G}}_s(V)_{\mbox{\tiny cart}}$ has $\oplus$, $\otimes$ and $\deg$ structures. A formally similar situation arises, for example, if we consider the category of morphisms of an additive tensor category ${\frak C}$ in which all homomorphism groups are graded. If we denote this morphism category by ${\frak M}{\frak C}$, there is a functor ${\frak M}{\frak C}\rightarrow{\frak C}\times{\frak C}$, given by source and target, whose fibers have a graded $\oplus$-structure as above. Also, ${\frak M}{\frak C}$ becomes a tensor category compatible with $\deg$ and $\oplus$. So ${\frak M}{\frak C}$ has $\oplus$, $\otimes$ and $\deg$ structures. In fact, Gromov-Witten invariants may be thought of as a functor from $\tilde{{\frak G}}_s(V)_{\mbox{\tiny cart}}$ to ${\frak M}{\frak C}$ respecting the $\oplus$, $\otimes$ and $\deg$ structures. In this case ${\frak C}$ will be a category of motives. \begin{defn} \label{doasvg} A full subcategory $\tilde{{\frak T}}_s(A)\subset\tilde{{\frak G}}_s(A)$ is called {\em admissible}, if it satisfies the following axioms. \begin{enumerate} \item If $\Phi:\sigma\rightarrow\tau$ is an extended isogeny in $\tilde{{\frak G}}_s(A)$ and $\tau\in \operatorname{ob}\tilde{{\frak T}}_s(A)$, then $\sigma\in\operatorname{ob}\tilde{{\frak T}}_s(A)$. \item If $\sigma$ and $\tau$ are in $\tilde{{\frak T}}_s(A)$, then so is $\sigma\times\tau$. \end{enumerate} \end{defn} For an admissible subcategory $\tilde{{\frak T}}_s(A)\subset\tilde{{\frak G}}_s(A)$ and a homomorphism $\xi:A\rightarrow B$, the full subcategory $\tilde{{\frak T}}_s(B)\subset\tilde{{\frak G}}_s(B)$ of graphs which are stabilizations of objects of $\tilde{{\frak T}}_s(A)$ is admissible. For a smooth projective variety $V$ of pure dimension, we may construct the full subcategory $\tilde{{\frak T}}_s(V)_{\mbox{\tiny cart}}\subset\tilde{{\frak G}}_s(V)_{\mbox{\tiny cart}}$, called the {\em associated cartesian category}, which may be characterized as the subcategory of $\tilde{{\frak G}}_s(V)_{\mbox{\tiny cart}}$ such that for each object $(\tau,(a_i,\tau_i)_{i\in I})$ we have that $\tau\in\operatorname{ob}\tilde{{\frak T}}_s(0)$ and for all $i\in I$ that $\tau_i\in\operatorname{ob}\tilde{{\frak T}}_s(V)$. Note that $\tilde{{\frak T}}_s(V)_{\mbox{\tiny cart}}$ inherits the $\oplus$, $\otimes$ and $\deg$ structures from $\tilde{{\frak G}}_s(V)_{\mbox{\tiny cart}}$. \begin{examples} {\em I}. Call a marked graph $\tau$ a {\em forest}, if \begin{enumerate} \item $H^1(|\tau|)=0$, \item $g(v)=0$, for all $v\in V\t$. \end{enumerate} Let $\tilde{{\frak T}}_s(A)\subset\tilde{{\frak G}}_s(A)$ be the full subcategory whose objects are forests. Then $\tilde{{\frak T}}_s(A)$ is an admissible subcategory, called the {\em tree level }subcategory of $\tilde{{\frak G}}_s(A)$. {\em II}. Let $\tilde{{\frak T}}_s(A)\subset\tilde{{\frak G}}_s(A)$ be an admissible subcategory. Let $d:A\rightarrow{\Bbb Z}_{\geq0}$ be an additive map and $N>0$ an integer. Then let $\tilde{{\frak T}}_s(A)_{d<N}$ be the full subcategory of $\tilde{{\frak T}}_s(A)$ given by the condition \[\tau\in\operatorname{ob}\tilde{{\frak T}}_s(A)_{d<N}\quad \Longleftrightarrow\quad \mbox{$d(\beta(v))<N$, for all $v\in V\t$}.\] The subcategory $\tilde{{\frak T}}_s(A)_{d<N}\subset\tilde{{\frak G}}_s(A)$ is admissible. If $A=H_2(V)^+$, then a very ample invertible sheaf $L$ on $V$ gives rise to $d:H_2(V)^+\rightarrow{\Bbb Z}_{\geq0}$, by setting $d(\beta)=\beta(L)$. If $\operatorname{\rm char}\nolimits k\not=0$, we shall always pass to $\tilde{{\frak T}}_s(V)_{d<\operatorname{\rm char}\nolimits k}$, in other words assume that \[\tilde{{\frak T}}_s(V)=\tilde{{\frak T}}_s(V)_{d<\operatorname{\rm char}\nolimits k}.\] But for emphasis, we may say that $\tilde{{\frak T}}_s(V)$ is {\em bounded by the characteristic}. \end{examples} \section{Orientations} Fix a smooth projective variety $V$ of pure dimension. Recall the following five basic properties of $\overline{M}$. {\em Property I (Mapping to a point). } Let $\tau$ be a stable $V$-graph of class zero. Then $\tau$ is absolutely stable. The evaluation morphism factors through $V^{\pi_0|\tau|}\subset V^{P\t}$ and the canonical morphism \[\overline{M}(V,\tau)\longrightarrow V^{\pi_0|\tau|}\times\overline{M}(\tau)\] is an isomorphism. This follows immediately from Corollary~\ref{zc}. In particular, $\overline{M}(V,\tau)$ is smooth. Assume that $|\tau|$ is non-empty and connected. Let $(C,x)$ be the universal family of stable marked curves over $\overline{M}(\tau)$. Glue the $(C_v)_{v\in F\t}$ according to the edges of $\tau$ to obtain a stable marked curve $\pi:\tilde{C}\rightarrow\overline{M}(\tau)$ over $\overline{M}(\tau)$. Denote the vector bundle of rank $g(\tau)\dim V$ on $\overline{M}(V,\tau)$ given by $T_V\boxtimes R^1\pi_{\ast}\O_{\tilde{C}}$ by ${\cal T}^{(1)}$. {\em Property II (Products). } Let $\sigma$ and $\tau$ be stable $V$-graphs and $\sigma\times\tau$ the obvious stable $V$-graph whose geometric realization is the disjoint union of $|\sigma|$ and $|\tau|$. There are obvious combinatorial morphisms $\sigma\rightarrow\sigma\times\tau$ and $\tau\rightarrow\sigma\times\tau$ giving rise to morphisms of stable $V$-graphs $\sigma\times\tau\rightarrow\sigma$ and $\sigma\times\tau\rightarrow\tau$ called the {\em projections}. The induced morphism \[\overline{M}(V,\sigma\times\tau)\longrightarrow\overline{M}(V,\sigma) \times\overline{M}(V,\tau)\] is an isomorphism. This follows directly from the definitions. {\em Property III (Cutting edges). } Let $\Phi:\sigma\rightarrow\tau$ be a morphism of stable $V$-graphs of type cutting an edge. So $\Phi$ is induced by a combinatorial morphism $a:\tau\rightarrow\sigma$. Let $f$ and $\overline{f}$ be the tails of $\tau$ that come from the edge of $\sigma$ which is being cut by $\Phi$. So this edge is $\{a(f),a(\overline{f})\}$. The diagram of algebraic $k$-stacks \begin{equation} \label{ceedd} \comdia{\overline{M}(V,\sigma)}{\operatorname{\rm ev}_{\{a(f),a(\overline{f})\}}}{V}{\overline{M}(\Phi)}{ }{\Delta}{\overline{M}(V, \tau)}{\operatorname{\rm ev}_f\times \operatorname{\rm ev}_{\overline{f}}}{V\times V,} \end{equation} where the horizontal maps are evaluations at the indicated flags, is cartesian. In particular, $\overline{M}(\Phi)$ is a closed immersion. Again, this follows directly from the definitions. {\em Property IV (Forgetting tails). } Let $\Phi:\sigma\rightarrow\tau$ be a morphism of stable $V$-graphs stably forgetting a tail. Denote the combinatorial morphism giving rise to $\Phi$ by $a:\tau\rightarrow\sigma$ and the forgotten tail by $f\in F_{\sigma}$. If $\Phi$ is of Type~I (i.e.\ incomplete), let $v=\partial_{\sigma}(f)$. Let $\pi':C'\rightarrow\overline{M}(V,\sigma)$ be the universal curve indexed by $v$ and $x:\overline{M}(V,\sigma)\rightarrow C'$ the universal section given by $f$. Let $\pi:C\rightarrow\overline{M}(V,\tau)$ be the universal curve indexed by the unique vertex $w$ of $\tau$ such that $a(w)=v$. Then by definition there is a commutative diagram \[\comdia{C'}{}{C}{\pi'}{}{\pi}{\overline{M}(V,\sigma)}{\overline{M}(\Phi)}{ \overline{M}(V,\tau),}\] and the section $x$ induces an $\overline{M}(V,\tau)$-morphism \[\overline{M}(V,\sigma)\rightarrow C.\] This is an isomorphism. In particular, $\overline{M}(\Phi)$ is proper and flat of relative dimension one. This follows from Corollary~\ref{esuc}. If $\Phi:\sigma\rightarrow\tau$ removes a tripod, then \[\overline{M}(\Phi):\overline{M}(V,\sigma)\rightarrow\overline{M}(V,\tau)\] is an isomorphism. This is because $\overline{M}(\text{$0$-tripod})=\overline{M}_{0,3}=\operatorname{\rm Spec}\nolimits k$. {\em Property V (Isogenies). } Let \[(\Phi,\lambda,(\Phi_i)_{i\in I}):(\tau,(a_i,\tau_i)_{i\in I})\longrightarrow(\sigma,(b_j,\sigma_j)_{j\in J})\] be a morphism in $\tilde{{\frak G}}_s(V)_{\mbox{\tiny cart}}$, where $\Phi$ (and hence all $\Phi_i$) is an isogeny, i.e.\ free of any tail gluing factors. For each $j\in J$ we have a commutative diagram \[\comdia{\displaystyle\coprod_{i\in I\atop\lambda(i)=j}\overline{M}(V,\tau_i)}{\amalg\overline{M}(\Phi_i)}{ \overline{M}(V,\sigma_j)}{\amalg \overline{M}(\overline{a}_i)}{}{\overline{M}(\overline{b})}{\overline{M}(\tau)}{\overline{M}(\Phi)}{ \overline{M}(\sigma).}\] This diagram should be considered close to being cartesian. See Definition~\ref{domb} for a more precise statement. For the moment let us note that the induced morphism \[\coprod_{i\in I\atop\lambda(i)=j}\overline{M}(V,\tau_i)\longrightarrow\overline{M}(\tau) \times_{\overline{M}(\sigma)} \overline{M}(V,\sigma_j)\] is surjective. If $X$ is a separated algebraic Deligne-Mumford stack, by $A_{\ast}(X)$ we shall mean the rational Chow group of $X$ (see \cite{vistoli}). If $X\rightarrow Y$ is a morphism of separated algebraic Deligne-Mumford stacks, $A^{\ast}(X\rightarrow Y)$ will denote the rational bivariant intersection theory defined in \cite{vistoli}. \begin{defn} \label{domb} Let $\tilde{{\frak T}}_s(V)\subset\tilde{{\frak G}}_s(V)$ be an admissible subcategory (bounded by the characteristic). Let for each $\tau\in\operatorname{ob}\tilde{{\frak T}}_s(V)$ be given a cycle class \[J(V,\tau)\in A_{\dim(V,\tau)}(\overline{M}(V,\tau)).\] This collection of cycle classes is called an {\em orientation } of $\overline{M}$ over $\tilde{{\frak T}}_s(V)$, if the following axioms are satisfied. \begin{enumerate} \item \label{domb1} {\em (Mapping to a point). }We have \[J(V,\tau)=c_{g(\tau)\dim V}({\cal T}^{(1)})\cdot[\overline{M}(V,\tau)],\] for every stable $\tau\in\operatorname{ob}\tilde{{\frak T}}_s(V)$ of class zero such that $|\tau|$ is non-empty and connected. \item \label{domb2} {\em (Products). }In the situation of Property~II we have \[J(V,\sigma\times\tau)=J(V,\sigma)\times J(V,\tau).\] \item \label{domb3} {\em (Cutting edges). }In the situation of Property~III the following is true. Let $[\overline{M}(\Phi)]\in A^{\dim V}(\overline{M}(V,\sigma)\rightarrow\overline{M}(V,\tau))$ be the orientation class of $\overline{M}(\Phi)$ obtained by pullback (using Diagram~(\ref{ceedd})) from the canonical orientation $[\Delta]\in A^{\dim V}(V\rightarrow V\times V)$. Then we have \[J(V,\sigma)=[\overline{M}(\Phi)]\cdot J(V,\tau).\] In other words, \[J(V,\sigma)=\Delta^{!} J(V,\tau),\] where $\Delta^{!}$ is the Gysin homomorphism given by the complete intersection morphism $\Delta$. \item \label{domb4} {\em (Forgetting tails). }In the situation of Property~IV the morphism $\overline{M}(\Phi)$ has a canonical orientation $[\overline{M}(\Phi)]\in A^{\ast}(\overline{M}(V,\sigma)\rightarrow\overline{M}(V,\tau))$. We require that \[J(V,\sigma)=[\overline{M}(\Phi)]\cdot J(V,\tau).\] In other words, \[J(V,\sigma)=\overline{M}(\Phi)^{\ast} J(V,\tau),\] where $\overline{M}(\Phi)^{\ast}$ is given by flat pullback. \item \label{domb5} {\em (Isogenies). }In the situation of Property~V, we have for every $j\in J$ a class \[\overline{M}(\Phi)^{!} J(V,\sigma_j)\in A_{\dim(V,\sigma_j)}(\overline{M}(\tau)\times_{\overline{M}(\sigma)}\overline{M}(V,\sigma_j)),\] since $\overline{M}(\Phi)$ has a canonical orientation, $\overline{M}(\tau)$ and $\overline{M}(\sigma)$ being smooth of pure dimension. We also have a morphism \[h:\coprod_{\lambda(i)=j}\overline{M}(V,\tau_i)\longrightarrow\overline{M}( \tau)\times_{\overline{M}( \sigma)}\overline{M}(V,\sigma_j)),\] which is proper. The requirement is that \[h_{\ast}(\sum_{\lambda(i)=j}J(V,\tau_i))=\overline{M}(\Phi)^{!} J(V,\tau).\] \end{enumerate} \end{defn} \begin{numrmk} \label{afmsl} To check Axiom~(\ref{domb5}), it suffices to do so for $\Phi$ an elementary isogeny, $\#J=1$ and $(a_i,\tau_i,\Phi_i)_{i\in I}$ a pullback. This follows from the projection formula. \end{numrmk} \begin{example} If $\tau$ is a stable $V$-graph such that $|\tau|$ is non-empty and connected, define \[J_0(V,\tau)= \begin{cases} c_{g(\tau)\dim V}({\cal T}^{(1)})\cdot[\overline{M}(V,\tau)] & \text{if $\beta(\tau)=0$,} \\ 0 & \text{otherwise.} \end{cases}\] For an arbitrary stable $V$-graph $\tau$, let $\tau=\tau_1\times\ldots\times\tau_n$, for stable $V$-graphs $\tau_1,\ldots,\tau_n$, such that $|\tau|=|\tau_1|\amalg\ldots\amalg|\tau_n|$ is the decomposition of $|\tau|$ into connected components. Then set \[J_0(V,\tau)=J_0(V,\tau_1)\times\ldots\times J_0(V,\tau_n).\] We claim that $J_0$ is an orientation of $\overline{M}$ over $\tilde{{\frak G}}_s(V)$, called the {\em trivial orientation}. \end{example} \begin{defn} Call a smooth projective variety $V$ {\em convex}, if for every morphism $f:{\Bbb P}^1\rightarrow V$ (defined over an extension $K$ of $k$) we have $H^1({\Bbb P}^1,f^{\ast} T_V)=0$. \end{defn} \noprint{ \begin{prop} Let $\pi:C\rightarrow T$ be a prestable curve of genus zero and $f:C\rightarrow V$ a morphism to a convex variety. Then $R^1\pi_{\ast} f^{\ast} T_V=0$. \end{prop} \begin{pf} \end{pf} } \begin{prop} Let $V$ be convex and $\tau$ a stable $V$-forest. Then $\overline{M}(V,\tau)$ is smooth of dimension $\dim(V,\tau)$. Moreover, the morphism \[\overline{M}(V,\tau)\longrightarrow\overline{M}(\tau^s)\] is flat of relative dimension $\chi(\tau^s)\dim V-\deg(V,\tau)$. \end{prop} \begin{pf} Let us start with some general remarks. Let $\tau$ be an absolutely stable $V$-graph. Then we define \[U(V,\tau)\subset\overline{M}(V,\tau)\] to be the open substack of those stable maps $(C,x,f)$, such that $(C_v,(x_i)_{i\in F\t(v)})$ is a stable marked curve, for all $v\in V\t$. Let $(C,x):T\rightarrow\overline{M}(\tau)$ be a $T$-valued point of $\overline{M}(\tau)$, i.e.\ $(C_v,(x_i)_{i\in F\t(v)})_{v\in V\t}$ is a family of stable marked curves parametrized by $T$. Let $(\tilde{C},\tilde{x})$ be the stable marked curve over $T$ obtained by gluing the $C_v$ according to the edges of $\tau$. The diagram \[\comdia{\operatorname{\rm Mor}\nolimits_T(\tilde{C},V_T)}{}{T}{}{}{}{U(V,\tau)}{}{\overline{M}(\tau)}\] is cartesian. In particular, by Grothendieck \cite{fgaIV}, the morphism $U(V,\tau)\rightarrow\overline{M}(\tau)$ is representable, separated and of finite type. Moreover, let $(C,x,f)$ be a $K$-valued point of $U(V,\tau)$. Let $(\tilde{C},\tilde{x})$ be the marked curve obtained by gluing the $C_v$ and $\tilde{f}:\tilde{C}\rightarrow V$ the morphism induced by the $f_v$. If $H^1(\tilde{C},\tilde{f}^{\ast} T_V)=0$, then $(C,x,f)$ is a smooth point of $U(V,\tau)\rightarrow\overline{M}(\tau)$ and we have \[T_{U(V,\tau)/\overline{M}(\tau)}(C,x,f)=H^0(\tilde{C},\tilde{f}^{\ast} T_V)\] for the relative tangent space. (This is the case, if $\tau$ is a $V$-forest and $V$ is convex.) In this smooth case we may calculate the relative dimension of $U(V,\tau)$ over $\overline{M}(\tau)$ at $(C,x,f)$ as \begin{eqnarray*} \dim_K H^0(\tilde{C},\tilde{f}^{\ast} T_V) & = & \chi(\tilde{f}^{\ast} T_V) \\ & = & \deg\tilde{f}^{\ast} T_V + \operatorname{\rm rk}(\tilde{f}^{\ast} T_V)\chi(\tilde{C}) \\ & = & -\beta(\tau)(\omega_V) + \dim V \chi(\tau) \\ & = & \dim(V,\tau)-\dim(\tau). \end{eqnarray*} Since $\overline{M}(\tau)$ is smooth of dimension $\dim(\tau)$, we get that $U(V,\tau)$ is smooth of dimension $\dim(V,\tau)$ at $(C,x,f)$. Now let $\tau$ be an arbitrary stable $V$-graph. Then there exists an absolutely stable $V$-graph $\tau'$, together with a morphism $\tau'\rightarrow\tau$ of type forgetting tails, such that the morphism \[U(V,\tau')\longrightarrow\overline{M}(V,\tau)\] is surjective, hence a flat epimorphism of relative dimension $\#S_{\tau'}-\#S\t$. So if $U(V,\tau')$ is smooth of dimension $\dim(V,\tau')$, then $\overline{M}(V,\tau)$ is smooth of dimension \[\dim(V,\tau')-\#S_{\tau'}+\#S\t=\dim(V,\tau).\] Finally, by considering the commutative diagram \[\comdia{U(V,\tau')}{}{\overline{M}(V,\tau)}{}{}{ }{\overline{M}(\tau')}{}{\overline{M}(\tau^s),}\] we see that in this case $\overline{M}(V,\tau)\rightarrow\overline{M}(\tau^s)$ is flat of relative dimension $\chi(\tau^s)\dim V-\deg(V,\tau)$. \end{pf} \begin{them} \label{uosc} Let $V$ be a convex variety and $\tilde{{\frak T}}_s(V)\subset\tilde{{\frak G}}_s(V)$ the admissible subcategory of $V$-forests bounded by the characteristic. Then the collection \[J(V,\tau)=[\overline{M}(V,\tau)]\] is an orientation of $\overline{M}$ over $\tilde{{\frak T}}_s(V)$. \end{them} \begin{pf} Let us check the axioms. (1) {\em Mapping to a point. } This follows from the fact that $g(\tau)=0$ and hence \[c_{g(\tau)\dim V}({\cal T}^{(1)})=c_0(0)=1.\] (2) {\em Products. } In complete generality we have for smooth proper Deligne-Mumford stacks $X$ and $Y$ that \[[X\times Y]=[X]\times[Y]\] in $A_{\ast}(X\times Y)$. (3) {\em Cutting edges. } Again we have a general fact to the following effect. Consider the cartesian diagram of separated Deligne-Mumford stacks \[\comdia{X}{f}{V}{j}{}{i}{Y}{}{W,}\] where $i$ and $j$ are regular embeddings such that for the normal bundles we have \[f^{\ast} N_{V/W}=N_{X/Y}.\] Then $i^{!}[Y]=[X]$. If all four participating stacks are smooth and $i$ and $j$ are closed immersions of the same codimension, then these conditions are automatically satisfied (see for example Proposition~17.13.2 in \cite{ega4}). Thus we may apply this fact in our case. More generally, we have that $i^{!}[Y]=[X]$ if all participating stacks are smooth and \[\dim X+\dim W=\dim Y +\dim V.\] (4) {\em Forgetting tails. } Again, there is a general fact that $f^{!}[Y]=[X]$ if $f:X\rightarrow Y$ is a flat morphism of smooth and proper Deligne-Mumford stacks. (5) {\em Isogenies. } In accordance with Remark~\ref{afmsl} we assume that $\Phi$ is an elementary isogeny, $\#J=1$ and that $(a_i,\tau_i,\Phi_i)_{i\in I}$ is a pullback. There are five cases to consider, according to what type of elementary isogeny $\Phi$ is. We use notation as in the definition of pullback. {\em Case I (Contracting a loop). } This case does not occur, since $\sigma$ and $\tau$ are forests. {\em Case II (Contracting an edge). } We will start with some general remarks. Let $\tau$ be a stable $V$-graph, and $v_1,\ldots,v_n$ absolutely stable vertices of $\tau$, i.e.\ vertices $v$ such that $2g(v)+|v|\geq3$. (To avoid ill-defined notation we assume that $n\geq1$.) Let \[U_{v_1,\ldots,v_n}(V,\tau)\subset\overline{M}(V,\tau)\] be the open substack of all those stable maps $(C,x,f)\in\overline{M}(V,\tau)$ such that $$(C_{v_{\nu}},(x_i)_{i\in F\t(v_{\nu})})$$ is a stable marked curve, for all $\nu=1,\ldots,n$. With this notation the diagram \[\comdia{\displaystyle\coprod_{i\in I} U_{a_i(v_1),a_i(v_2)}(V,\tau_i)}{}{U_{b(v_0)}(V,\sigma')}{}{ }{}{\overline{M}(\tau)}{}{\overline{M}(\sigma)}\] is cartesian. Consider for a fixed $i\in I$ the open immersion \[U_{a_i(v_1),a_i(v_2)}(V,\tau_i)\subset\overline{M}(V,\tau_i).\] Let \[Z_{a_i(v_1),a_i(v_2)}(V,\tau_i)\subset\overline{M}(V,\tau_i)\] be the closed complement. We have \[\dim Z_{a_i(v_1),a_i(v_2)}(V,\tau_i)<\dim \overline{M}(V,\tau_i).\] Thus, to prove the equality of two cycles of degree $\dim(V,\tau_i)$ in $\overline{M}(\tau)\times_{\overline{M}(\sigma)}\overline{M}(V,\sigma')$, it suffices to prove the equality of the cycles restricted to $\coprod_i U_{a_i(v_1),a_i(v_2)}(V,\tau_i)$. This reduces us to proving that \[\overline{M}(\Phi)^{!}[U_{b(v_0)}(V,\sigma')]=\sum_i[U_{a_i(v_1), a_i(v_2)}(V,\tau_i)].\] This claim finally follows from the general fact already mentioned in the proof of Axiom~(3). {\em Case III (Forgetting a tail, incompletely). } Let $f\in F\t$ be the forgotten flag, $v=\partial_{\tau_0}(a_0(f))$ and $w\in V_{\sigma'}$ the vertex of $\sigma'$ corresponding to $v$ via $\Phi_0$. We have an open immersion \[U_v(V,\tau_0)\subset \overline{M}(V,\tau_0)\] with closed complement \[Z_v(V,\tau_0)\subset \overline{M}(V,\tau_0)\] of strictly smaller dimension. Thus, as in the previous case, we may reduce to proving that \[\overline{M}(\Phi)^{!}[U_w(V,\sigma')]=[U_v(V,\tau_0)].\] This follows from the fact that the diagram \[\comdia{U_v(V,\tau_0)}{}{U_w(V,\sigma')}{}{}{ }{\overline{M}(\tau)}{\overline{M}(\Phi)}{\overline{M}(\sigma)}\] is cartesian. {\em Cases IV and V (Removing a tripod). } These cases are trivial, since $\overline{M}(\Phi_0)$ and $\overline{M}(\Phi)$ are isomorphisms. \end{pf} \noprint{ \begin{conjecture} \label{gmc} Let $V$ be a Grassmannian variety. Then the collection \[J(V,\tau)=[\overline{M}(V,\tau)]^0\] is an orientation of $\overline{M}$ over $\tilde{{\frak G}}_s(V)$. \end{conjecture} \begin{pf} \end{pf} } \section{Deligne-Mumford-Chow Motives} We shall imitate the usual construction of the category of Chow motives, as described for example in \cite{scholl}. Fix a ground field $k$. Let ${\frak W}$ be the category of smooth and proper algebraic Deligne-Mumford stacks over $k$. For an object $X$ of ${\frak W}$, let $A^{\ast}(X)$ be the rational Chow ring of $X$ defined by Vistoli \cite{vistoli}. Then $A^{\ast}$ is a generalized cohomology theory with coefficient field ${\Bbb Q}$ in the sense of \cite{kleiman}. Moreover, it is a graded global intersection theory with Poincar\'e duality and cycle map in the terminology of \cite{kleiman}. If $X$ and $Y$ are objects of ${\frak W}$ we define $S^d(Y,X)$, the group of {\em correspondences }from $Y$ to $X$ of degree $d$, to be \[S^d(Y,X)=A^{n+d}(Y\times X),\] if $Y$ is purely $n$-dimensional and \[S^d(Y,X)=\bigoplus_i S^d(Y_i,X),\] if $Y=\coprod_i Y_i$ is the decomposition of $Y$ into irreducible components. Note that $S^d(Y,X)\subset A^{\ast}(Y\times X)$. The isomorphism $Y\times X\cong X\times Y$ exchanging components induces an isomorphism \[S^d(Y,X)\cong S^{d+n-m}(X,Y),\] if $\dim Y=n$ and $\dim X=m$. We call this isomorphism {\em transpose }of correspondences. For objects $Z$, $Y$ and $X$ of ${\frak W}$ we define composition of correspondences by the usual formula \[g\mathbin{{\scriptstyle\circ}} f={p_{13}}_{\ast}(p_{12}^{\ast} f\cdot p_{23}^{\ast} g),\] for $f\in S^d(Z,Y)$ and $g\in S^e(Y,X)$. Then $g\mathbin{{\scriptstyle\circ}} f\in S^{d+e}(Z,X)$. The category $\overline{{\frak W}}$ of {\em Deligne-Mumford-Chow motives } (or DMC-motives) is now defined to be the category of triples $(X,p,n)$, where $X\in \operatorname{ob}{\frak W}$, $p\in S^0(X,X)$ such that $p^2=p$ and $n\in{\Bbb Z}$. Homomorphisms are defined by \[\operatorname{\rm Hom}\nolimits_{\overline{{\frak W}}}((Y,q,m),(X,p,n))=p S^{n-m}(Y,X) q.\] Note that $\operatorname{\rm Hom}\nolimits_{\overline{{\frak W}}}((Y,q,m),(X,p,n))\subset S^{n-m}(Y,X)$. Composition of homomorphisms in $\overline{{\frak W}}$ is defined as composition of correspondences. There is a contravariant involution $\overline{{\frak W}}\rightarrow\overline{{\frak W}}$, denoted $M\mapsto M^{\vee}$, defined by $(X,p,n)^{\vee}=(X,^tp,\dim X-n)$, where $^tp$ is the transpose of $p$, on objects and by transpose of correspondences on homomorphisms. \begin{prop} The category $\overline{{\frak W}}$ is a ${\Bbb Q}$-linear pseudo-abelian category. \qed \end{prop} Every morphism $f:X\rightarrow Y$ in ${\frak W}$ defines a correspondence of degree zero $\overline{f}\in S^0(Y,X)$ by \[\overline{f}={\Gamma_f}_{\ast}[X]\in A^{\ast}(Y\times X),\] where $\Gamma_f:X\rightarrow Y\times X$ is the graph of $f$. We define the contravariant functor $h:{\frak W}\rightarrow\overline{{\frak W}}$ by $h(X)=(X,\overline{\operatorname{\rm id}}_X,0)$ and $h(f)=\overline{f}$. We usually write $f^{\ast}$ for $h(f)$ and $f_{\ast}$ for $h(f)^{\vee}$. Let $\ll=(\operatorname{\rm Spec}\nolimits k,\overline{\operatorname{\rm id}},-1)$ be the {\em Lefschetz motive}. We shall use the notation \[M(n)=M\otimes\ll^{-n}.\] We set \[\operatorname{\rm Hom}\nolimits_{\overline{{\frak W}}}^i(M,N)=\operatorname{\rm Hom}\nolimits_{\overline{{\frak W}}}(M\otimes\ll^i,N)\] and \[\operatorname{\rm Hom}\nolimits_{\overline{{\frak W}}}^{\ast}(M,N)=\bigoplus_{i\in{\Bbb Z}}\operatorname{\rm Hom}\nolimits_{\overline{{\frak W}}}^i(M,N).\] The category with the same objects as $\overline{{\frak W}}$, but with homomorphism groups given by $\operatorname{\rm Hom}\nolimits_{\overline{{\frak W}}}^{\ast}(M,N)$ will be called the category of {\em graded }DMC-motives. For a DMC-motive $M$, define \[A^i(M)=\operatorname{\rm Hom}\nolimits(\ll^i,M)\] and \[A^{\ast}(M)=\bigoplus_iA^i(M).\] \begin{prop}[Identity principle] If $f,g:M\rightarrow N$ are two homomorphisms of DMC-motives, such that the induced homomorphisms \[A^{\ast}(M\otimes h(X))\longrightarrow A^{\ast}(N\otimes h(X))\] agree, for all $X\in\operatorname{ob}{\frak W}$, then $f=g$. \qed \end{prop} Let $\overline{{\frak V}}$ be the category of Chow motives (which is defined as $\overline{{\frak W}}$ is above, but starting with ${\frak V}$ instead of ${\frak W}$). There is a natural fully faithful functor $\overline{{\frak V}}\rightarrow\overline{{\frak W}}$. \begin{question} Is the functor $\overline{{\frak V}}\rightarrow\overline{{\frak W}}$ an equivalence of categories? \end{question} Let $H$ be a graded generalized cohomology theory on ${\frak W}$ with a coefficient field $\Lambda$ of characteristic zero, possessing a cycle map such that ${\Bbb P}^1$ satisfies epu (see \cite{kleiman}). Then $H$ induces a covariant functor (called a {\em realization functor\/}) \[\overline{H}:(\rtext{graded DMC-motives})\longrightarrow(\rtext{graded $\Lambda$-algebras}),\] such that for $X\in\operatorname{ob}{\frak W}$ we have $\overline{H}(h(X))=H(X)$ and for a correspondence $\xi\in S^d(Y,X)$ we have an induced homomorphism \begin{eqnarray*} \overline{H}(\xi):H(Y) & \longrightarrow & H(X) \\ \alpha & \longmapsto & {p_X}_{\ast}({p_Y}^{\ast}(\alpha)\cup\operatorname{\rm cl}\nolimits_{Y\times X}(\xi)). \end{eqnarray*} The functor $\overline{H}$ doubles the degree of a homomorphism. The following are examples of such a cohomology theory $H$. \begin{enumerate} \item If $k={\Bbb C}$, consider to $X$ the associated topological stack $X^{\mbox{\tiny top}}$. This is a stack on the category of topological spaces with the \'etale topology. It has an associated \'etale topos $X^{\mbox{\tiny top}}_{\mbox{\tiny \'{e}t}}$. Set \[H_B(X)=H^{\ast}(X^{\mbox{\tiny top}}_{\mbox{\tiny \'{e}t}},{\Bbb Q})\] and call it the {\em Betti cohomology }of $X$. Here $\Lambda={\Bbb Q}$. \item If $\ell\not=\operatorname{\rm char}\nolimits k$ set \[H_{\ell}(X)=H^{\ast}(\overline{X}_{\mbox{\tiny \'{e}t}},{{\Bbb Q}_\ell})={\displaystyle\projlim\limits_n} H^{\ast}(\overline{X}_{\mbox{\tiny \'{e}t}},{\Bbb Z}/\ell^n),\] where $\overline{X}=X\times_{\operatorname{\rm Spec}\nolimits k}\operatorname{\rm Spec}\nolimits \overline{k}$ is the lift of $X$ to an algebraic closure of $k$ and $\overline{X}_{\mbox{\tiny \'{e}t}}$ denotes the \'etale topos of $\overline{X}$. We call $H_{\ell}(X)$ the {\em $\ell$-adic cohomology } of $X$. In this case $\Lambda={{\Bbb Q}_\ell}$. \item If $\operatorname{\rm char}\nolimits k=0$, let $\Omega_X^{\scriptscriptstyle\bullet}$ be the algebraic deRham complex of $X$ and set \[H_{dR}(X)={\Bbb H}^{\ast}(X,\Omega_X^{\scriptscriptstyle\bullet}).\] We call $H_{dR}(X)$ the {\em algebraic deRham cohomology }of $X$. Here $\Lambda=k$. \end{enumerate} \section{Motivic Gromov-Witten Classes} Define the contravariant tensor functor \[h(\overline{M}):\tilde{{\frak G}}_s(0)\longrightarrow(\rtext{DMC-motives})\] by $h(\overline{M})(\tau)=h(\overline{M}(\tau))$ on objects. For a morphism $(a,\sigma',\Phi):\sigma\rightarrow\tau$ we have $\overline{M}(\overline{a}):\overline{M}(\sigma')\rightarrow\overline{M}(\sigma)$ and $\overline{M}(\Phi):\overline{M}(\sigma')\rightarrow\overline{M}(\tau)$. Then let \[h(\overline{M})(a,\sigma',\Phi)=\overline{M}(\overline{a})_{\ast}\mathbin{{\scriptstyle\circ}}\overline{M}(\Phi)^{\ast}.\] This makes sense, because $\overline{M}(\overline{a})_{\ast}$ is of degree zero, $\overline{M}(\overline{a})$ being an isomorphism. This is also why $h(\overline{M})$ is functorial. Now fix a smooth projective variety $V$ of pure dimension and consider the contravariant tensor functor \[h(V)^{\otimes S}{(\chi\dim V)}:\tilde{{\frak G}}_s(0)\longrightarrow(\rtext{DMC-motives})\] defined on objects by \[\tau\longmapsto h(V)^{\otimes S\t}({\chi(\tau)\dim V}).\] For a morphism $(a,\sigma',\Phi):\sigma\rightarrow\tau$ let $E$ be the set of edges of $\sigma'$ which are cut by $a:\sigma\rightarrow\sigma'$. Then we have $V^{S_{\sigma}}=V^{S_{\sigma'}}\times(V\times V)^E$. Let $p:V^{S_{\sigma'}}\times V^E\rightarrow V^{S_{\sigma'}}$ be the projection, $\Delta:V^{S_{\sigma'}}\times V^E\rightarrow V^{S_{\sigma'}}\times (V\times V)^E=V^{S_{\sigma}}$ the identity times the $E$-fold power of the diagonal. Finally, we have an injection $\Phi^S:S\t\rightarrow S_{\sigma'}$ giving rise to $\Phi^S:V^{S_{\sigma'}}\rightarrow V^{S\t}$. We define the homomorphism \[h(V)^{\otimes S\t}{(\chi(\tau)\dim V)}\longrightarrow h(V)^{\otimes S_{\sigma}}{(\chi(\sigma)\dim V)}\] as the composition of the three homomorphisms \[(\Phi^S)^{\ast}:h(V)^{\otimes S\t}{(\chi(\tau)\dim V)}\longrightarrow h(V)^{\otimes S_{\sigma'}}{(\chi(\sigma')\dim V)},\] \[p^{\ast}:h(V)^{\otimes S_{\sigma'}}{(\chi(\sigma')\dim V)}\longrightarrow h(V)^{\otimes S_{\sigma'}\cup E}{(\chi(\sigma')\dim V)}\] and \[\Delta_{\ast}:h(V)^{\otimes S_{\sigma'}\cup E}{(\chi(\sigma')\dim V)}\longrightarrow h(V)^{\otimes S_{\sigma}}{(\chi(\sigma)\dim V)},\] noting that $\chi(\tau)=\chi(\sigma')$ and $\chi(\sigma')=\chi(\sigma)-\# E$. Functoriality is a straightforward check using the identity principle. Pulling back $h(\overline{M})$ and $h(V)^{\otimes S}{(\chi\dim V)}$ to the cartesian extended isogeny category over $V$ via the functor of Remark~\ref{cfnc}, we get two contravariant tensor functors \[\tilde{{\frak G}}_s(V)_{\mbox{\tiny cart}}\longrightarrow(\text{graded DMC-motives}).\] Now let $\tilde{{\frak T}}_s(V)\subset\tilde{{\frak G}}_s(V)$ be an admissible subcategory (bounded by the characteristic) and $J$ an orientation of $\overline{M}$ over $\tilde{{\frak T}}_s(V)$. For every object $\tau$ of $\tilde{{\frak T}}_s(V)$ we have a morphism \[\phi_{(V,\tau)}:\overline{M}(V,\tau)\longrightarrow V^{S_{\tau^s}}\times \overline{M}(\tau^s).\] The first component is given by evaluation, noting that we have a map $F_{\tau^s}\rightarrow F\t$. Then \begin{eqnarray*} {\phi_{(V,\tau)}}_{\ast} J(V,\tau) & \in & S^{\dim(\tau^s)-\dim(V,\tau)}(V^{S_{\tau^s}},\overline{M}(\tau^s)) \\ & = & \operatorname{\rm Hom}\nolimits_{\overline{{\frak W}}}^{\deg(V,\tau)}(h(V^{S_{\tau^s}}){(\chi(\tau^s)\dim V)},h(\overline{M}(\tau^s))). \end{eqnarray*} \begin{defn} Define \[I(V,\tau)={\phi_{(V,\tau)}}_{\ast} J(V,\tau),\] so that we have a homomorphism \[I(V,\tau):h(V)^{\otimes S_{\tau^s}}{(\chi(\tau^s)\dim V)}\longrightarrow h(\overline{M}(\tau^s)){(\deg(V,\tau))}\] of DMC-motives over $k$. We call $I$ the system of {\em Gromov-Witten classes }associated to the orientation $J$. \end{defn} Restricting the two functors $h(\overline{M})$ and $h(V)^{\otimes S}{(\chi\dim V)}$ to $\tilde{{\frak T}}_s(V)_{\mbox{\tiny cart}}$, we get two contravariant tensor functors \[\tilde{{\frak T}}_s(V)_{\mbox{\tiny cart}}\longrightarrow(\text{graded DMC-motives}).\] We shall now define a natural transformation \[I:h(V)^{\otimes S}{(\chi\dim V)}\longrightarrow h(\overline{M}).\] So let $(\tau,(\tau_i)_{i\in I})$ be an object of $\tilde{{\frak T}}_s(V)_{\mbox{\tiny cart}}$, and define \[I(\tau,(\tau_i)_{i\in I})=\sum_{i\in I}I(V,\tau_i):h(V)^{\otimes S_{\tau}}{(\chi(\tau)\dim V)}\longrightarrow h(\overline{M}(\tau)).\] \begin{them} \label{vagwc} The Gromov-Witten transformation $I$ is a natural transformation compatible with the $\oplus$, $\otimes$ and $\deg$ structures. Moreover, \begin{enumerate} \item \label{vag1} {\em (Mapping to a point). } The triangle \[\comtri{h(V)^{\otimes S\t}{(\chi(\tau)\dim V)}}{\rtext{mult}}{h(V){(\chi(\tau)\dim V)}}{I(V,\tau)}{c_{g(\tau)\dim V}({\cal T}^{(1)})}{h(\overline{M}(\tau))}\] commutes, for any stable $V$-graph $\tau$ of class zero in $\tilde{{\frak T}}_s(V)$, such that $|\tau|$ is non-empty and connected. \item \label{vag2} {\em (Divisor). } Let ${\cal L}\in\operatorname{\rm Pic}\nolimits(V)$ be a line bundle, so its Chern class induces a homomorphism $c_1({\cal L}):\ll\rightarrow h(V)$. Let $\Phi:\sigma\rightarrow\tau$ be a morphism in $\tilde{{\frak T}}_s(V)$ of type forgetting a tail, such that the corresponding vertex of $\tau$ is absolutely stable. Then the square \[\begin{array}{ccc} h(V)^{\otimes S_{\sigma^s}}{(\chi(\sigma^s)\dim V)} & \stackrel{I(V,\sigma)}{\longrightarrow} & h(\overline{M}(\sigma^s)){(\deg(V,\sigma))}\\ \ldiagup{c_1({\cal L})} & & \rdiag{\overline{M}(\Phi)_{\ast}} \\ h(V)^{\otimes S_{\tau^s}}{(\chi(\tau^s)\dim V)} \otimes\ll & \stackrel{\beta({\cal L})I(V,\tau)}{\longrightarrow} & h(\overline{M}(\tau^s)){(\deg(V,\tau))}\otimes\ll \end{array}\] commutes. \end{enumerate} \end{them} \begin{rmk} To make this statement more precise, consider to $(\text{graded DMC-motives})$ the associated category of morphisms. Then the natural transformation $I$ may be considered as a functor \[I:\tilde{{\frak T}}_s(V)_{\mbox{\tiny cart}}\longrightarrow(\text{graded morphisms of DMC-motives}).\] Both categories have $\oplus$, $\otimes$ and $\deg$ structures and $I$ preserves them. This essentially means that \begin{enumerate} \item $I((\tau_i)\oplus(\sigma_j))=I((\tau_i))+I((\sigma_j))$, \item $\deg I((\tau_i))=\deg(\tau_i)$, if $(\tau_i)$ is homogeneous, \item $I((\tau,\tau_i)\otimes(\sigma,\sigma_j))=I(\tau,\tau_i)\otimes I(\sigma,\sigma_j)$. \end{enumerate} \end{rmk} \begin{pf} All this follows formally from Definition~\ref{domb} using the identity principle and the bivariant formalism (as explained for example in \cite{fulmacp}). \end{pf} \begin{rmks} \begin{enumerate} \item Applying Theorem~\ref{uosc} we get the tree level system of Gromov-Witten invariants for convex varieties. \item By applying a realization functor, we get Betti, $\ell$-adic or deRham Gromov-Witten classes. \item Theorem~\ref{vagwc} implies all the axioms for Gromov-Witten classes listed in \cite{KM}. Perhaps only Formula~(2.7) is not quite evident. In view of its importance (it implies that the fundamental class remains the identity with respect to quantum multiplication), we will show that it follows from the rest of the axioms. In fact, assume that \begin{equation} \label{leman} \langle I_{0,3,\beta}\rangle (\gamma_1\otimes\gamma_2\otimes e^0) \ne 0. \end{equation} Choose a divisorial class $\delta$ with nonvanishing intersection with $\beta$. In view of the Divisor Axiom, we must then have $$\langle I_{0,4,\beta}\rangle (\gamma_1\otimes\gamma_2\otimes\delta\otimes e^0) \ne 0.$$ In view of (2.6), the last class is the lift of $$\langle I_{0,3,\beta}\rangle (\gamma_1\otimes\gamma_2\otimes\delta ).$$ But this cannot be non-vanishing simultaneously with~(\ref{leman}) because the Grading Axiom does not allow this. More generally, this argument shows that whenever $e^0$ is among the arguments, then $\langle I\rangle = 0$ for $\beta \ne 0$, any genus, any $n.$ Geometrically: `if one of the points on $C$ is unconstrained, the problem cannot have finitely many (and non-zero) solutions.' \noprint{The composition \[h(V)^{\otimes2}{-\dim V}\stackrel{p^{\ast}}{\longrightarrow} h(V)^{\otimes3}{-\dim V}\stackrel{I_{0,3}(V,\beta)}{\longrightarrow}\ll^{\beta(\omega_V)}\] is zero, if $\beta\not=0$. Here $p:V^3\rightarrow V^2$ is a projection.} \end{enumerate} \end{rmks}

9506

alg-geom/9506012

en

https://arxiv.org/abs/alg-geom/9506012

alg-geom/9506012

Dmitri Orlov

A.Bondal and D.Orlov

Semiorthogonal decomposition for algebraic varieties

LaTeX 2.9

A criterion for a functor between derived categories of coherent sheaves to be full and faithful is given. A semiorthogonal decomposition for the derived category of coherent sheaves on the intersection of two even dimensional quadrics is obtained. The behaviour of derived categories with respect to birational transformations is investigated. A theorem about reconstruction of a variety from the derived category of coherent sheaves is proved.

alg-geom

\section*{ SEMIORTHOGONAL DECOMPOSITIONS FOR ALGEBRAIC VARIETIES} \begin{abstract} A criterion for a functor between derived categories of coherent sheaves to be full and faithful is given. A semiorthogonal decomposition for the derived category of coherent sheaves on the intersection of two even dimensional quadrics is obtained. The behaviour of derived categories with respect to birational transformations is investigated. A theorem about reconstruction of a variety from the derived category of coherent sheaves is proved. \end{abstract} \tableofcontents \section{Introduction.} This paper is devoted to study of the derived categories of coherent sheaves on smooth algebraic varieties. Of special interest for us is the case when there exists a functor ${\db M}\longrightarrow {\db X}$ which is full and faithful. It appears that some geometrically important constructions for moduli spaces of (semistable) coherent sheaves on varieties can be interpreted as instances of this situation. Conversely, we are convinced that any example of such a functor is geometrically meaningful. If a functor $\Phi: {\db M}\longrightarrow {\db X}$ is full and faithful, then it induces a semiorthogonal decomposition (see definition in ch.2) of ${\db X}$ with the 2--step chain $\Bigl( {\db M}^{\perp}$, ${\db M} \Bigl)$, where ${\db M}^{\perp}$ is the right orthogonal to ${\db M}$ in ${\db X}$. Decomposing summands of this chain, one can obtain a semiorthogonal decomposition with arbitrary number of steps. Full exceptional sequences existing on some Fano varieties (see \cite{K}) provide with examples of such decompositions. For this case, every step of the chain is equivalent to the derived category of vector spaces or, in other words, sheaves over the point. This leads to the idea that the derived category of coherent sheaves might be reasonable to consider as an incarnation of the motive of a variety, while semiorthogonal decompositions are a tool for simplification of a motive similar to spliting by projectors in the Grothendieck motive theory. Main result of ch.1 is a criterion for fully faithfulness. Roughly speaking, it claims that for a functor ${\db M}\longrightarrow{\db X}$ to be full and faithful it is sufficient to satisfy this property on the full subcategory of the one dimensional skyscraper sheaves and its translations. Let us mention that ${\db M}^{\perp}$ might be zero. In this case we obtain an equivalence of derived categories ${\db M}\stackrel{\sim}{\longrightarrow}{\db X}$. Examples of such equivalences have been considered by Mukai in \cite{Mu}, \cite{Mu2} (see ch.1). In ch.3 we prove such equivalences for some flop birational transformations. Ch.2 is devoted to description of a semiorthogonal decomposition for ${\db X}$, when $X$ is the smooth intersection of two even dimensional quadrics. It appears that if we consider the hyperelliptic curve $C$ which is a double covering of the projective line parametrizing the pencil of quadrics, with ramification in the points corresponding to degenerate quadrics, then ${\db C}$ is embedded in ${\db X}$ as a full subcategory. The orthogonal to ${\db C}$ in ${\db X}$ is decomposed in an exceptional sequence (of linear bundles ). This allows to identify moduli spaces of semistable bundles (of arbitrary rank) on the curve with moduli spaces of complexes of coherent sheaves on the intersection of quadrics. For rank 2 bundles such identification is well known (see \cite{DR}) and was used for computation of cohomologies of moduli spaces \cite{Bar} and for verification of the Verlinde formula. In ch.3 we investigate the behaviour of ${\db X}$ under birational transformations. We prove that for a couple of varieties $X$ and $X^+$ related by some flips the category ${\db {X^+}}$ has a natural full and faithful embedding in ${\db X}$. This suggests the idea that the minimal model program of the birational geometry can be considered as a `minimization' for the derived category of coherent sheaves in a given birational class. We also explore some cases of flops. Considered examples allow us to state a conjecture that the derived categories of coherent sheaves on varieties connected by a flop are equivalent. Examples of varieties having equivalent derived categories appeal to the question: {\it to which extent a variety is determined by its derived category?} In ch.4 we prove a reconstruction theorem, which claims that if $X$ is a smooth algebraic variety with ample either canonical or anticanonical sheaf, then another algebraic variety $X'$ having equivalent the derived category of coherent sheaves ${\db X}\simeq {\db {X'}}$ should be biregulary isomorphic to $X$. As a by-product we obtain a description for the group of auto-equivalences of ${\db X}$ provided $X$ has ample either canonical or anticanonical class. We are grateful to Max--Planck--Institute for hospitality and stimulating atmosphere. Our special thanks go to S.Kuleshov for help during preparation of this paper. The work was partially supported by International Science Foundation Grant M3E000 and Russian Fundamental Research Grant. {\section {Full and faithful functors.}} \hspace*{0.6cm}For a smooth algebraic variety $X$ over an algebraically closed field $k$ of characteristic $0$ by ${\db X}$ (resp., $D^b_{Qcoh}(X)$) we denote the bounded derived category of coherent (resp., quasicoherent) sheaves over $X$. Notations like $f^{*}, f_{*}, \otimes, {\rm Hom}, {\cal H}om$ etc. are reserved for derived functors between derived categories, whereas $R^if_*, {\rm Hom}^i$, etc. (resp., $L^if^*$) denote i--th (resp., (-i)--th) cohomology of a complex obtained by applying $f_*, {\rm Hom}$ etc. (resp., $f^*$); $[n]$ denotes the translation by $n$ functor in a triangulated category. Let $X$ and $M$ be smooth algebraic varieties of dimension $n$ and $m$ respectively, and $E$ an object of ${\db {X \times M}}$. With $E$ one can associate a couple of functors $$ \Phi_{E} : {\db M}\longrightarrow {\db X}, $$ $$ \Psi_{E} : {\db X}\longrightarrow {\db M}. $$ Denote by $p$ and $\pi$ the projections of $M \times X$ to $M$ and $X$ respectively. $$ \begin{array}{ccc} M\times X&\stackrel{\pi }{\longrightarrow}& X\\ \llap{\footnotesize $p$} \downarrow &&\\ M && \end{array} $$ Then $\Phi_{E}$ and $\Psi_{E}$ are defined by the formulas: $$ \Phi_{E}(\cdot):=\pi_* (E \otimes p^*(\cdot)), $$ $$ \Psi_{E}(\cdot):=p_* (E \otimes \pi^*(\cdot)). $$ The main goal of this chapter is the proof of the following \th{Theorem}\label{mai} Let $M$ and $X$ be smooth algebraic varieties and\hfill\\ $E\in{\db {M\times X}}$. Then $\Phi_{E}$ is full and faithful functor, if and only if the following orthogonality conditions are verified: $$ \begin{array}{lll} i) & {\H i, X, \Phi_E({\cal O}_{t_1}), {\Phi_{E}({\cal O}_{t_2})}} = 0 & \qquad \mbox{for every }\: i\;\mbox{ and } t_1\ne t_2.\\ &&\\ ii) & {\H 0, X, \Phi_E({\cal O}_t), {\Phi_E({\cal O}_t)}} = k,&\\ &&\\ & {\H i, X, \Phi_E({\cal O}_t), {\Phi_E({\cal O}_t)}} = 0 , & \qquad \mbox{ for }i\notin [0, dim M]. \end{array} $$ Here $t$, $t_{1}$, $t_{2}$ are points of $M$, ${\cal O}_{t_{i}}$ corresponding skyscraper sheaves. \par\endgroup \bigskip Let us mention that if some full subcategory ${\cal C}\subset {\cal D}$ generates ${\cal D}$ as a triangulated category then for an exact functor ${\cal D}\longrightarrow{\cal D}'$ to be full and faithful it is sufficient to be full on ${\cal C}$. Unfortunately, the class of skyscraper sheaves does not generate $\db{M}$ as a triangulated category if $dimM>0$. At the level of the Grothendieck group $K_0(M)$ they generate only the lowest term of the topological filtration. The proof of the theorem is preceded by a series of assertions concerning functors between and objects from the derived categories of complexes of coherent sheaves on smooth varieties. For any object $E$ from $\db{X}$ we denote by $E^{\vee}$ the dual object: $$ E^{\vee}:={\cal H}om( E,\:{\o X}). $$ \th{Lemma}\label{adj} The left adjoint functor to $\Phi_{E}$ is $$ \Psi_{E^{\vee}\otimes \pi^*\omega_X}[n]:= p_{*}(E^{\vee}\otimes \pi^*\omega_X\otimes \pi^*(\cdot))[n]. $$ \par\endgroup {\bf Proof} is given by a series of natural isomorphisms, which come from the adjoint property of functors and Serre duality: $$ \begin{array}{l} {\h A, {\pi_*(E\otimes p^*B)}}\cong \\ {\h {\pi^*A}, {E\otimes p^* B}}\cong \\ {\h {p^* B}, {\pi^* A\otimes E^{\vee}\otimes \omega_{X\times M}[n+m]}}^* \cong \\ {\h B, {p_*(\pi^*(A\otimes \omega_X[n])\otimes E^{\vee})\otimes \omega_M[m]}}^* \cong\\ {\h {p_*(\pi^*(A\otimes \omega_X[n])\otimes E^{\vee})}, B}. \end{array} $$ \bigskip The next lemma differs from analogous in \cite{H} in what concerns base change (we consider arbitrary $g$ instead of flat one in \cite{H}) and morphism $f$ (we consider only smooth morphism instead of arbitrary one in \cite{H}). \th{Lemma}\label{isof} Let $f: X\to Y $ be a smooth morphism of relative dimension $r$ of smooth projective varieties and $ g:Y'\to Y $ a base change, with $Y^{\prime}$ being a smooth variety. Define $X'$ as the cartesian product $X'=X\times_Y Y'$. $$ \begin{array}{ccc} X^{\prime}=X\times_Y Y^{\prime}& \stackrel{ g'}{\longrightarrow}& X\\ \llap{$f'$}\downarrow && \llap{$f$}\downarrow \\ Y'&\stackrel{g}{\longrightarrow}& Y\\ \end{array} $$ Then there is a natural isomorphism of functors: $$ g^*f_*(\cdot) \simeq {f^{\prime}}_* {g^{\prime}}^*(\cdot). $$ \par\endgroup \par\noindent{\bf\ Proof. } First, note that the right adjoint functors to $g^*f_*$ and $ f'_* g'^*$ are, respectively, $f^{!}g_*$ and $g'_*f'^!$ , where $f^!$ denote the right adjoint functor to $f_*$. We are going to prove that $f^{!}g_*$ and $g'_*f'^!$ are isomorphic. Serre duality gives a natural isomorphism \begin{equation}\label{1} f^!(\cdot)\simeq f^*(\cdot)\otimes \omega_{X/Y}[r]. \end{equation} Hence, \begin{equation}\label{2} f^!g_*(\cdot)\simeq f^*g_*(\cdot)\otimes \omega_{X/Y}[r]. \end{equation} Analogously, $$ g'_*f'^!(\cdot)\simeq g'_*(f'^*(\cdot)\otimes \omega_{X'/Y'}[r])\simeq g'_*(f'^*(\cdot)\otimes g'^*\omega_{X/Y}[r]). $$ The latter isomorphism goes from the fact that for a smooth $f$ differentials are compatible with base change (see \cite{H},III,\S1,p.141). Then, by the projection formula one has \begin{equation}\label{3} g'_* f'^!(\cdot)\simeq g'_* f'^*(\cdot)\otimes \omega_{X/Y}[r]. \end{equation} By the theorem of flat base change (see \cite{H},II,\S5,prop.5.12) one has $$ g'_* f'^*\simeq f^*g_*. $$ Formulas (\ref{2}) and (\ref{3}) imply a functorial isomorphism of $g'_* f'^!(\cdot)$ and $f^! g_*(\cdot)$. Therefore, $g^*f_*(\cdot)$ is isomorphic to $f'_* g'^*(\cdot) $. \bigskip Let $X, Y, Z$ be smooth projective varieties and $I, J, K$ objects of $\db{X\times Y}$, $\db{Y\times Z}$ and $\db{X\times Z},$ respectively. Consider the following diagram of projections \hspace*{5cm}\epsffile{c.eps} and the triple of functors $$ \phi_{I} : {\db X}\longrightarrow {\db Y}, $$ $$ \psi_{J} : {\db Y}\longrightarrow {\db Z}, $$ $$ \chi_{K} : {\db X}\longrightarrow {\db Z}, $$ defined by the formulas $$ \phi_I={\pi_{12}^2}_{*}(I\otimes {\pi_{12}^{1}}^{*}(\cdot)), $$ $$ \psi_J={\pi^3_{23}}_*(J\otimes {\pi^2_{23}}^*(\cdot)), $$ $$ \chi_K={\pi^3_{13}}_*(K\otimes {\pi^1_{13}}^*(\cdot)). $$ The next proposition from \cite{Mu} is an analog for derived categories of the composition law for correspondences (see \cite{M}). \th {Proposition}\label{comp} The composition functor for $\phi_I$ and $\psi_J$ is isomorphic to $\chi_K$ with $$ K={p_{13}}_*({p_{23}}^*J\otimes {p_{12}}^*I). $$ \par\endgroup \par\noindent{\bf\ Proof. } It goes from the following sequence of natural isomorphisms, which uses the projection formula and a base change theorem from \cite{H}: $$ \begin{array}{l} \psi_J \circ \phi_I(\cdot)\cong {\pi^3_{23}}_*(J\otimes {\pi^2_{23}}^*({\pi^2_{12}}_*(I\otimes {\pi^1_{12}}^*(\cdot)))) \cong \\ {\pi^3_{23}}_*(J\otimes {p_{23}}_*({p_{12}}^*(I\otimes {\pi^1_{12}}^*(\cdot))))\cong\\ {\pi^3_{23}}_*{p_{23}}_*({p_{23}}^*J\otimes {p_{12}}^*(I\otimes {\pi^1_{12}}^*(\cdot)))\cong\\ {\pi^3_{13}}_*{p_{13}}_*({p_{23}}^*J\otimes {p_{12}}^*I\otimes {p_{12}}^* {\pi^1_{12}}^*(\cdot))\cong\\ {\pi^3_{13}}_*{p_{13}}_*({p_{23}}^*J\otimes {p_{12}}^*I\otimes {p_{13}}^* {\pi^1_{13}}^*(\cdot))\cong\\ {\pi^3_{13}}_*({p_{13}}_*({p_{23}}^*J\otimes {p_{12}}^*I)\otimes {\pi^1_{13}}^*(\cdot)).\\ \end{array} $$ \th{Proposition}\label{tor} Let $j: Y\hookrightarrow X$ be a smooth irreducible subvariety of codimension $d$ of a smooth algebraic variety $X$, and $K$ a non-zero object of $\db{X}$ satisfying following conditions: a) $i^*_x K=0$, \qquad for any closed point $x\stackrel{i_x}{\hookrightarrow} X\setminus Y$, b) $L^i i^*_x K=0$,\qquad when $ i\notin [0, d]$, for any closed point $x\stackrel{i_x}{\hookrightarrow} Y$. Then i) $K$ is a pure sheaf (i.e. quasiisomorphic to its zero cohomology sheaf), ii) the support of $K$ is $Y$. \par\endgroup \par\noindent{\bf\ Proof. } Let ${\cal H}^q$ be the q--th cohomology sheaf of $K$. Then, for any point $x\stackrel{i_x}{\hookrightarrow} X$ there is spectral sequence with the $E_2$--term consisting of $L^p i^*_x({\cal H}^q)$ and converging to cohomology sheaves of $i_*(K)$: $$ E^{-p,q}_2=L^p i^*_x({\cal H}^q) \Rightarrow L^{p-q} i^*_x(K) $$ Recall that $L^i f^*$ denotes the (--i)--th cohomology of $f^*$ in accordance with notations of the analogous left derived functors between abelian categories. If ${\cal H}^{q_{max}}$ is a non--zero sheaf with maximal $q$, then $L^0 i^*_x{\cal H}^{q_{max}}$ is intact by differentials while going to $E_{\infty}$. By assumptions of the proposition $L^q i^*_x K=0$, for $q>0$ and for any point $x\in X$. This implies $q_{max}\le 0$. Considering the sheaf ${\cal H}^q$ with maximal $q$ among those having the support outside $Y$, one obtains by the same reasoning that all ${\cal H}^q$ actually have their support in $Y$. Let ${\cal H}^{q_{min}}$ be the non--zero sheaf with minimal $q$. The spectral sequence is depicted in the following diagram: \begin{figure}[th] \hspace*{5cm}\epsffile{seq.eps} \end{figure} Consider any component $C\subset Y$ of the support of ${\cal H}^{q_{min}}$. If $c$ is the codimension of $ C$ in $X$, then $L^c i^*_{x_0}({\cal H}^{q_{min}})\ne 0$ for a general closed point $x_0\in C$. It could have been killed in the spectral sequence only by $L^p i^*_x({\cal H}^q)$ with $p\ge c+2$. But for any sheaf $F$ the closed subscheme $S_m(F)$ of points of cohomological dimension $\ge m$ (see \cite{S}) $$ S_m(F)=\Bigl\{ x\in X\;\Bigl|\;L^p i^*_x(F)\ne 0,\quad\mbox{ for some}\:p\ge m\Bigl\} $$ has codimension $\ge m$. Therefore, $S_m({\cal H})$ with $m\ge c+2$ cannot cover $C$, i.e. there exists a point $x_0\in C$, such that $L^c i^*_{x_0}({\cal H}^{q_{min}})$ survives at infinity in the spectral sequence, hence $L^{c-q_{min}} i^*_{x_0}(K)\ne 0$. Then, by assumption b) of the proposition it follows that $c-q_{min}\le d$. Since $C$ belongs to $Y$, $c\ge d$, hence $q_{min}\ge 0$. In other words, $q_{min}=q_{max}$ and $K$ has the only non--trivial cohomology sheaf ${\cal H}^0$. This proves i). Now consider ${\cal L}^i=L^i j^*K$. There is a spectral sequence for composition of $i^*_x$ and $j^*$: $$ E^{-p,-q}_2=L^p i^*_x({\cal L}^q) \Rightarrow L^{p+q} i^*_x(K). $$ Let ${\cal L}^{q_0}$ be a non--zero sheaf with maximal $q$. Since the support of $K$ belongs to $Y$, $q_0\ge d$. Again consider a component of the support for ${\cal L}^{q_0}$. The same reasoning as above shows that if this component is of codimension $b$, then for some point $x_0$ in it, $L^b i^*_{x_0}({\cal L}^{q_0})$ survives in $E_{\infty}$ of the latter spectral sequence. By the assumptions of the proposition we have $q_0+b\le d$. This implies $q_0=d$ and $b=0$. This means that the support of ${\cal L}^d$ is the whole $Y$. It follows that the support of $K$ coincides with $Y$. The proposition is proved. \bigskip {\bf Proof of the Theorem \ref{mai}.} First, let us mention that if $\Phi_E$ is full and faithful functor, then conditions i) and ii) are verified for obvious reasons. Indeed, it is well known fact that extension groups between skyscraper sheaves in $\db{M}$ have the following form: $$ \begin{array}{lll} i) & {\H i, X, {\cal O}_{t_1}, {{\cal O}_{t_2}}} = 0 & \qquad \mbox{for every }\: i\;\mbox{ and } t_1\ne t_2;\\ ii) & {\H i, X, {\cal O}_t , {{\cal O}_t}} = \Lambda^i T_{M,t} , & \qquad \mbox{ for }i\in [0, dim M],\\ & {\H i, X, {\cal O}_t , {{\cal O}_t}} = 0 , & \qquad \mbox{ for }i\notin [0, dim M]. \end{array} $$ Here $t, t_1, t_2$ are points of $M$, $T_{M,t}$ the tangent vector space to $M$ at $t$, and $ \Lambda^i$ the $i$--th exterior power. Fully faithfulness of $\Phi_E$ implies that the same relations are valid for images $\Phi_E({\cal O}_t)$ in $\db{X}$. In what follows we prove the inverse statement. Consider composition of $\Phi_E$ with its left adjoint functor $\Phi_E^*$. We are going to prove that the canonical natural transformation $\alpha : \Phi_E^*\circ\Phi_E\to id$ is an isomorphism of functors. This is equivalent to fully faithfulness of $\Phi_E$. Indeed, for any pair of objects $A, B\in\db{M}$ the natural homomorphism $$ {\h A, B}\longrightarrow {\h \Phi_E A, {\Phi_E B}}\cong{\h \Phi_E^*\Phi_E A, B}, $$ is induced by $\alpha$. By lemma \ref{adj} we have $$ \Phi_E^*\cong \Psi_{E^{\vee}\otimes \pi^*\omega_X}[n]. $$ From proposition \ref{comp} the object $K$ of $\db {M\times M}$, which determines $\Phi_E^*\circ\Phi_E$, is \begin{equation}\label{K} K={q_{13}}_*({q_{23}}^*(E^{\vee}\otimes \pi^*\omega_X)\otimes {q_{12}}^*E)[n], \end{equation} where the morphisms $q_{13}, q_{23}, q_{12}$ and $\pi$ are taken from the following diagram \begin{figure}[th] \hspace*{5cm}\epsffile{d2.eps} \end{figure} We need to prove that $K$ is quasiisomorphic to ${\o\Delta}=\Delta_*{\o M}$, where $\Delta: M\longrightarrow M\times M$ is the diagonal embedding, because ${\o\Delta}$ gives the identity functor on $\db{M}$. Let us consider a commutative diagram $$ \begin{array}{ccc} X&\stackrel{j_{t_1 t_2}}{\longrightarrow}& M\times X\times M\\ \llap{\footnotesize $f$} \downarrow &&\llap{$q_{13}$} \downarrow\\ {\rm Spec} k&\stackrel{i_{t_1 t_2}}{\longrightarrow}& M\times M \end{array} $$ Here $i_{t_1 t_2}$ is the embedding of a geometric point $(t_1 t_2)$ in $M\times M$, and $f : X\to {\rm Spec} k$ the corresponding fibre of $q_{13}$ over this point. This diagram is useful for computing the fibres of $K$ over points of $M\times M$. Indeed, by lemma \ref{isof} $$ i_{t_1 t_2}^*K=i_{t_1 t_2}^*{q_{13}}_*({q_{23}}^*(E^{\vee}\otimes \pi^*\omega_X)\otimes {q_{12}}^*E)[n]= $$ \begin{eqnarray}\label{fib} =f_*j_{t_1 t_2}^*({q_{23}}^*(E^{\vee}\otimes \pi^*\omega_X)\otimes {q_{12}}^*E)[n]=f_*(j_{t_1 t_2}^*{q_{23}}^*(E^{\vee}\otimes \pi^*\omega_X)\otimes j_{t_1 t_2}^*{q_{12}}^*E)[n]. \end{eqnarray} From the commutative diagram \begin{figure}[h] \hspace*{5cm}\epsffile{f2.eps} \end{figure} where $j_{t_1}$ is the embedding $x\mapsto ( t_1, x)$, and from the definition of $\Phi_E$one obtains: \begin{equation}\label{fun1} j_{t_1 t_2}^*{q_{12}}^*E=j_{t_1}^*E=\Phi_E({\cal O}_{t_1}). \end{equation} Analogously, \begin{equation}\label{fun2} j_{t_1 t_2}^*{q_{12}}^*(E^{\vee}\otimes \pi^*\omega_X)=\Phi_E({\cal O}_{t_2})^{\vee}\otimes\omega_X. \end{equation} Formulas (\ref{fib}), (\ref{fun1}), (\ref{fun2}) imply isomorphisms: $$ i_{t_1 t_2}^*K=f_*(\Phi_E({\cal O}_{t_1})\otimes\Phi_E({\cal O}_{t_2})^{\vee}\otimes\omega_X)[n]= $$ \begin{eqnarray}\label{res} =f_*({\cal H}om(\Phi_E({\cal O}_{t_2})\:,\;\Phi_E({\cal O}_{t_1}))\otimes\omega_X)[n]={\h \Phi_E({\cal O}_{t_1}), {\Phi_E({\cal O}_{t_2})}}^*. \end{eqnarray} The last equality comes from Serre duality on $X$. Apply proposition \ref{tor} to the diagonal embedding of $M$ in $M\times M$. By formula (\ref{res}) and assumptions of the theorem, the object $K$ satisfies the hypothesis of the proposition. Therefore, $K$ is a pure sheaf with the support at the diagonal $\Delta M$. The natural transformation $\alpha$ gives rise to a sheaf homomorphism $K\to {\o\Delta}$. It is an epimorphism, because otherwise its image would not generate the stalk of ${\o\Delta}$ at some point $(t,t)$ at the diagonal. But this would imply that $\Phi_E({\cal O}_{\Delta})$ has no endomorphisms ( that is, the trivial object) in contradiction with assumptions of the theorem. Let $F$ be the kernel of this morphism, i.e. there is an exact sequence of coherent sheaves on $M\times M$ : \begin{equation}\label{seq} 0\longrightarrow F \longrightarrow K\longrightarrow {\o\Delta}\longrightarrow 0 \end{equation} We have to prove that $F$ is trivial. Considering the pull back of the short exact sequence to any point from $M\times M$ we obtain a long exact sequence showing that the sheaf $F$ satisfies hypothesis of proposition \ref{tor}. It follows from the proposition that the support of $F$ coincides with the diagonal $\Delta M$. It is sufficient to prove that the restriction of $F$ to the diagonal is zero. Let us consider for this the commutative diagram: $$ \begin{array}{ccc} M\times X&{\longrightarrow}& M\times X\times M\\ \llap{\footnotesize $p$} \downarrow && \downarrow\\ M&\stackrel{\Delta}{\longrightarrow}& M\times M \end{array} $$ where vertical morphisms are natural projections. Applying lemma \ref{isof} to the object $({q_{23}}^*(E^{\vee}\otimes \pi^*\omega_X)\otimes {q_{12}}^*E)[n]$ from $\db{M\times X\times M}$ and formula (\ref{K}) we obtain a formula for the derived functors of the restriction--to--diagonal functor for $K$ : $$ L^i\Delta^*(K)=R^{n-i} p_*(E\otimes E^{\vee}\otimes \pi^*\omega_X). $$ Therefore, by the relative version of Serre duality and hypothesis of the theorem $$ \Delta^*K={\o \Delta}, $$ $$ L^1 \Delta^*(K)=R^1p_*(E\otimes E^{\vee})^{\vee}. $$ Unfortunately, it is not sufficient to know that the restriction of $K$ to the diagonal is $ {\o \Delta}$, because $K$ might not be the push forward along $\Delta$ of a sheaf on $M$ (being, `situated' on some infinitesimal neighborhood of $\Delta(M)$). Furthermore, $L^1 \Delta^*({\o \Delta})=\Omega^1_M$, this means that the long exact sequence, obtained from (\ref{seq}) by tensoring with ${\o \Delta}$ looks as follows: $$ \cdots\longrightarrow R^1p_*(E\otimes E^{\vee})^{\vee}\stackrel{\beta}{\longrightarrow} \Omega^1_M\longrightarrow L^0\Delta^* F\longrightarrow{\o \Delta}\stackrel{\sim}{\longrightarrow}{\o \Delta}\longrightarrow 0. $$ Since the support of $F$ coincides with $\Delta(M)$, so does the support of $L^0\Delta^* F$, i.e. $L^0\Delta^* F$ is not a torsion sheaf. Therefore, if $L^0\Delta^* F$ is not zero, then this exact sequence shows that $$ \beta^* : T_M\longrightarrow R^1p_*(E\otimes E^{\vee}) $$ has a non-trivial kernel. \par \bigskip {\sc Remark.} If one consider $\Phi_E({\cal O}_t)$ as a system of objects from $\db{X}$ parametrized by $M$, then the restriction of $\beta^*$ to any point $t$ from $M$ is , actually, the homomorphism from the deformation theory $$ T_{M,t}\longrightarrow {\H 1, X, {\Phi_E({\cal O}_t)}, {\Phi_E({\cal O}_t)}}. $$ Therefore, a vector field from the kernel of $\beta^*$ gives a direction, which the objects do not change along with. This is in contradiction with the orthogonality assumptions of the theorem. Unfortunately, integrating such an algebraic vector field one might obtain non-algebraic curves. \bigskip For this reason our further strategy is going to find only a formal one--parameter deformation at one point $t_0$ in $M$, along which $E$ has a formal connexion (analog of trivialization), and then to bring this in contradiction with the property of $\Phi_E({\cal O}_{t_0})$ to having the support in point $t_0$, which is a consequence of the orthogonality condition. Consider a point $t_0$ in $M$, $U$ an open neighborhood of $t_0$ and a non-zero at $t_0$ local section $\xi\in H^0(U , T_M\Bigl|_U)$, which belongs to the kernel of $\beta^*$. This vector field $\xi$ defines a formal 1--dimensional subscheme $\mit\Gamma$ of the formal neighborhood $\hat U$ of $t_0$ in $M$. The defining ideal of the subscheme consists of the function on ${\hat U}$ having trivial all iterated derivatives along $\xi$ at point $t_0$ ( the zero derivative being the value of a function at $t_0$ ) : $$ I=\Bigl\{ f\in H^0(\hat U, {\cal O})\; \Bigl| \;\xi^k(f)|_{t_o}=0, \:\mbox{for any }\: k\ge 0 \Bigl\}. $$ It follows that the restriction of $\beta^*$ to the tangent bundle $T_{\mit\Gamma , t_0}$ of $\mit\Gamma$ at $t_0$ is trivial. Denote $E_{\mit\Gamma}$ the restriction of $E$ to $\mit\Gamma\times X$. One has \begin{eqnarray}\label{R} {\H 1, {\mit\Gamma\times X}, {p^*T_{\mit\Gamma} }, {E_{\mit\Gamma}^{\vee}\otimes E_{\mit\Gamma}}}\cong {\H 1, {\mit\Gamma}, {T_{\mit\Gamma}} , {p_*(E_{\mit\Gamma}^{\vee}\otimes E_{\mit\Gamma})}}\cong\nonumber\\ {\H 0 , {\mit\Gamma}, {T_{\mit\Gamma}}, {R^1p_*(E_{\mit\Gamma}^{\vee}\otimes E_{\mit\Gamma})}}, \end{eqnarray} since $T_{\mit\Gamma}$ is free ( of rank 1 ) on ${\mit\Gamma}$. Let us consider the first infinitesimal neighborhood $\Delta^1_{\mit\Gamma}$ of the diagonal $\Delta_{\mit\Gamma} : {\mit\Gamma}\times X\longrightarrow {\mit\Gamma}\times {\mit\Gamma}\times X$. Pulling $E_{\mit\Gamma}$ back to ${\mit\Gamma}\times {\mit\Gamma}\times X$ along the first coordinate, then restricting to $\Delta^1_{\mit\Gamma}$ and then pushing forward along the second coordinate, one obtains the object $J^1(E_{\mit\Gamma})\in \db{{\mit\Gamma}\times X}$ of `first jets' of $E_{\mit\Gamma}$. It is included in an exact triangle : \begin{equation}\label{tr} E_{\mit\Gamma}\otimes p^*\Omega^1_{\mit\Gamma}\longrightarrow J^1(E_{\mit\Gamma})\longrightarrow E_{\mit\Gamma}\stackrel{at_E}{\longrightarrow} E_{\mit\Gamma}\otimes p^*\Omega^1_{\mit\Gamma}[1]. \end{equation} Here $at_E$ is the so called Atiyah class of $E$. It can be considered as an element of ${\H 1, {\mit\Gamma\times X}, {p^*T_{\mit\Gamma} }, {E_{\mit\Gamma}^{\vee}\otimes E_{\mit\Gamma}}}$. Under identification from (\ref{R}) $at_E$ comes into the restriction of $\beta^*$ to $\mit\Gamma$, which is the trivial element of ${\H 0 , {\mit\Gamma}, {T_{\mit\Gamma}}, {R^1p_*(E_{\mit\Gamma}^{\vee}\otimes E_{\mit\Gamma})}}$ by the choice of $\mit\Gamma$. We consider $E_{\mit\Gamma}$ as an element of the derived category of quasicoherent sheaves on X. It is naturally endowed with an additional homomorphism $A\to {\rm End}_X E_{\mit\Gamma}$, where $A$ is an algebra of functions on ${\mit\Gamma}$ ( isomorphic to $k[[t]]$). Such a homomorphism we call by {\it A}--module structure on $E_{\mit\Gamma}$. An {\it A}--module structure on $E_{\mit\Gamma}$ induces {\it A}--module structures on $E_{\mit\Gamma}\otimes p^*\Omega^1_{\mit\Gamma}$ and $J^1(E_{\mit\Gamma})$, so that morphisms from (\ref{tr}) are compatible with them. Like for usual vector bundles there exists a natural homomorphism in $D^b_{Qcoh}(X)$ : $$ E_{\mit\Gamma}\stackrel{\mu}{\longrightarrow}J^1(E_{\mit\Gamma}) , $$ which is a differential operator of the first order with respect to the {\it A}--module structures. Triviality of $at_E$ implies existence of a morphism $$ J^1(E_{\mit\Gamma})\stackrel{\nu}{\longrightarrow}E_{\mit\Gamma}\otimes p^*\Omega^1_{\mit\Gamma}, $$ which is a section of the first morphism from (\ref{tr}). The composition $\nabla=\nu\circ\mu$ defines a morphism of quasicoherent sheaves on $\mit\Gamma\times X$ $$ \nabla : E_{\mit\Gamma}\longrightarrow E_{\mit\Gamma}\otimes p^*\Omega^1_{\mit\Gamma}, $$ which is a connexion on $E_{\mit\Gamma}$ along the fibres of the projection $p_{\mit\Gamma} :{\mit\Gamma}\times X \longrightarrow {\mit\Gamma}$ in the sense that if $t\in A$ is a function on our formal scheme $\mit\Gamma$ , then the following equality for morphisms from $E_{\mit\Gamma}$ to $E_{\mit\Gamma}\otimes p^*\Omega^1_{\mit\Gamma}$ is valid: \begin{equation}\label{con} \nabla{\circ}t - t{\circ}\nabla=dt, \end{equation} here $t$ is identified with the corresponding morphism from $E_{\mit\Gamma}$ to $E_{\mit\Gamma}$ and $dt$ denotes the operator of tensor multiplication by $dt$. Since ${\mit\Gamma}$ is a one-dimensional subscheme, $\Omega^1_{\mit\Gamma}$ is a one-dimensional free {\it A}--module. Hence, for the reason of simplicity we can identify $E_{\mit\Gamma}$ with $E_{\mit\Gamma}\otimes p^*\Omega^1_{\mit\Gamma}$ by means of tensoring with $dt$, where $t\in A$ is a formal parameter on the scheme ${\mit\Gamma}$. Then, formula (\ref{con}) gives the coordinate-impulse relation between $\nabla$ and $t$: \begin{equation}\label{ci} [\nabla, t ]={\bf 1}. \end{equation} \th{Lemma}\label{inv} Under the above identification of $E_{\mit\Gamma}$ with $E_{\mit\Gamma}\otimes p^*\Omega^1_{\mit\Gamma}$ the morphism $\nabla{\circ}t$ is invertible in ${\rm End}_X E_{\mit\Gamma}$ \par\endgroup \par\noindent{\bf\ Proof. } From (\ref{ci}) one has : $$ [\nabla, t ^k]=k t^{k-1}. $$ This gives a formula for the inverse to $\nabla\circ t $ : $$ (\nabla{\circ}t)^{-1}=\sum_{k=0}^{\infty}\frac{(-1)^k t^k{\circ}\nabla^k}{(k+1)!} $$ This formal series correctly defines an endomorphism of $E_{\mit\Gamma}$, because by definition $E_{\mit\Gamma}$ is the limit of a system of objects $E_{\mit\Gamma_n}$ from ${\mit\Gamma}_n\times X$ , where ${\mit\Gamma}_n$ is the n--th infinitesimal neighborhood of $t_0$ in ${\mit\Gamma}$. For every $n$ the formula gives a finite expansion for an endomorphism of $E_{\mit\Gamma_n}$, thus , in the limit, it does an endomorphism of $E_{\mit\Gamma}$ . This proves the lemma. \bigskip Let $E_0$ be the first member of the exact triangle : $$ E_0\stackrel{\rho}{\longrightarrow}E_{\mit\Gamma}\stackrel{\nabla}{\longrightarrow} E_{\mit\Gamma}. $$ It is an object of the derived category of quasicoherent ${\o X}$--modules (`horisontal sections of $E$'). \th{Proposition}\label{flat} i) The composition $\lambda$ of $\rho\times id_A$ with the multiplication morphism $E_{\mit\Gamma}\times A\longrightarrow E_{\mit\Gamma}$: $$ \lambda : E_0\times A \stackrel{\rho\times id_A}{\longrightarrow}E_{\mit\Gamma}\times A\longrightarrow E_{\mit\Gamma} $$ is an isomorphism, in other words it yields trivialization of $E_{\mit\Gamma}$. ii) $E_0$ is quasiisomorphic to a complex of coherent sheaves on $X$. \par\endgroup \par\noindent{\bf\ Proof. } Let us consider the cone $C$ of $\lambda$ : $$ E_0\times A \stackrel{\lambda}{\longrightarrow}E_{\mit\Gamma}\longrightarrow C. $$ Restricting this exact triangle to the fibre $X_0$ of $p_{\mit\Gamma}$ over the closed point of ${\mit\Gamma}$ (which is, of course, naturally identified with $X$), one obtains an exact triangle : $$ E_0\stackrel{\lambda_0}{\longrightarrow}E_{\mit\Gamma}\Bigl|_{X_0}\longrightarrow C\Bigl|_{X_0}. $$ Vanishing of $C\Bigl|_{X_0}$ implies vanishing of $C$, hence, for proving i) we need to show that the left morphism $\lambda_0$ of this triangle is isomorphism. Multiplication by $t$ gives an exact triangle of sheaves on $ \mit\Gamma\times X$ : $$ 0\longrightarrow{\o {\mit\Gamma\times X}}\stackrel{t}{\longrightarrow}{\o {\mit\Gamma\times X}}\longrightarrow{\o {X_0}}\longrightarrow 0 $$ It lifts to an exact triangle: $$ E_{\mit\Gamma}\stackrel{t}{\longrightarrow}E_{\mit\Gamma}\longrightarrow E_{\mit\Gamma}\Bigl|_{X_0}\longrightarrow E_{\mit\Gamma}[1]. $$ Consider an octahedral diagram of exact triangles \cite{BBD}: \begin{figure}[th] \hspace*{5cm}\epsffile{oct1.eps} \end{figure} By lemma \ref{inv} $\nabla{\circ}t$ is an isomorphism. Hence, $G$ is zero object and $\lambda_0$ is an isomorphism. Since $ E\Bigl|_{X_0}$ is the restriction of complex of coherent sheaves to $X_0$, $E_0$ is coherent over $X_0$. Since $C\Bigl|_{X_0}=G[1]$ is zero, so is $C$. Therefore, $\lambda$ is isomorphism. In order to finish the proof of theorem \ref{mai} let us look at the image $L=\Phi^*_E \circ \Phi_E({\o {t_0}})$ of ${\o {t_0}}$ under the functor $$ \Phi^*_E\circ \Phi_E: \db{M}\longrightarrow\db{M}. $$ Recall that by lemma \ref{adj} $$ \Phi^*_E=\Psi_{E^{\vee}\otimes \pi^*\omega_X}[n]. $$ The trivialization of $E$ along ${\mit\Gamma}$ from proposition \ref{flat} gives us a similar trivialization of $E^{\vee}\otimes \pi^*\omega_X$. By the definition of $\Psi$ this implies trivialization along ${\mit\Gamma}$ of any object from the image of $\Phi^*_E$. Since we know that $\Phi^*_E\circ \Phi_E$ is determined by sheaf $K$, having the diagonal $\Delta(M)$ as its support, the image $L$ of a skyscraper sheaf ${\o {t_0}}$ is a non--zero object from $\db{M}$ having $t_0$ as the support. This means that $L$ annihilates by some power $I^k$ of the maximal ideal $I\subset A$. Such an object has a trivialization only if it is zero. This finishes the proof of theorem \ref{mai}. \bigskip The simplest example of a full and faithful functor $\db{M}\longrightarrow\db{X}$ rises in the case when $M$ is a point. In this situation we have the only {\sf object} $E\in \db{X}$, which is an {\sf exceptional} one: $$ \begin{array}{ll} {\H 0, X, E, E}=k, &\\ {\H i, X, E, E}=0, \quad\mbox{for }\: i\ne 0 \end{array} $$ It gives a functor from the derived category of vector spaces over $k$ to $\db{X}$. Mukai in \cite{Mu} and \cite{Mu2} considered two important examples of fully faithful functors between geometric categories. First one is the so called Fourier--Mukai transform. It gives equivalence $$ \db{A}\longrightarrow\db{\hat A} $$ for any abelian variety $A$ and its dual $\hat A$. We briefly recall his construction. Let $A$ be an abelian variety of dimension $g$, $\hat A$ its dual abelian variety and ${\cal P}$ the normalized Poincare bundle on $A\times\hat A$. As $\hat A$ is a moduli space of invertible sheaves on $A$, ${\cal P}$ is a linear vector bundle, and normalization means that both ${\cal P}\Bigl|_{A\times\hat 0}$ and ${\cal P}\Bigl|_{0\times\hat A}$ are trivial. \th{Theorem}{\rm (\cite{Mu}).}\label{ab} The functors $\Phi_{\cal P}:\db{A}\to \db{\hat A}$ and $\Psi_{\cal P}:\db{\hat A}\to \db{A}$ are equivalences of triangulated categories and $$ \Psi_{\cal P}\circ\Phi_{\cal P}\cong (-1_A)^*[g], $$ $$ \Phi_{\cal P}\circ\Psi_{\cal P}\cong (-1_{\hat A})^*[-g], $$ here $(-1_A)^*$ is the auto-equivalence of $\db{A}$ induced by the automorphism of multiplication by $-1$ on $A$. \par\endgroup \par\noindent{\bf\ Proof. } (see \cite{Mu}). In the case of a principally polarized abelian variety $(A, L)$, where $L$ is a polarization, the dual $\hat A$ is identified with $A$. Then $\Phi_{\cal P}$ can be regarded as an auto-equivalence of $\db{A}$. $\Phi_{\cal P}$ in couple with the functor of tensoring by $L$ generates the action of the Artin braid group $B_3$ on three strands. The other example of Mukai is a K3--surface $S$, while $M$ is a moduli space of stable vector bundles. Specifically, for a smooth K3--surface $S$ one consider the Mukai lattice ${\cal M}(S)$, which is the image of the Chern homomorphism $K_0(S)\longrightarrow H^*(S, {\bf C})$ from the Grothendieck group $K_0(S)$ to full cohomology group $H^*(S, {\bf C})$. There is the Euler bilinear form on ${\cal M}(S)$, which for vectors $v$ and $v'$ presented by some sheaves ${\cal F}$ and ${\cal F'}$ is defined by the formula: $$ \chi(v, v')= \sum (-1)^i {\rm dimExt}^i({\cal F, F'}). $$ Since the canonical class is trivial, by Serre duality this form is symmetric. Let $v$ be an isotropic indivisible by integer vector with respect to $\chi$. The coarse moduli space of stable bundles on $S$, corresponding to $v$, is again a smooth K3--surface $S'$. There is a rational correspondence between $S$ and $S'$. If $S'$ is a {\it fine} moduli space, then we have the universal vector bundle $E$ on $S\times S'$. \th{Theorem}{\rm \cite{Mu2}}. Functor $\Phi_{E}:\db{S}\longrightarrow \db{S'}$ is an equivalence of triangulated categories. \par\endgroup In the both examples of equivalences the canonical class of varieties (either of abelian one or of a K3--surface) is trivial. In chapter 3 we construct another example of equivalence between geometric categories using flops. The centre of such transformation is in a sense trivial with respect to the canonical class. An explanation for this phenomenon is given in chapter 4. \section{Intersection of two even dimensional quadrics.} In this chapter we show how theorem 1.1 helps to construct a semiorthogonal decomposition of the derived category of coherent sheaves on the intersection of two even dimensional quadrics, with one summand being the derived category on a hyperelliptic curve and with the others being generated by single exceptional objects. This result can be considered as a categorical explanation for the description, due to Desale and Ramanan, of moduli spaces of rank 2 vector bundles on a hyperelliptic curve as a base of a family of projective subspaces belonging to the intersection of two even dimensional quadrics \cite{DR}. Our construction gives analogous description for any moduli spaces of bundles on the curve by means of families of complexes of coherent sheaves on the intersection locus. We first recall some definitions and facts concerning exceptional sequences, admissible subcategories, Serre functors and semiorthogonal decompositions \cite{Bon}, \cite{BK}. Let ${\cal B}$ be a full subcategory of an additive category. The {\sf right orthogonal} to ${\cal B}$ is the full subcategory ${\cal B}^{\perp}\subset {\cal A}$ consisting of the objects $C$ such that ${\h B, C}=0$ for all $B\in{\cal B}$. The {\sf left orthogonal} ${}^{\perp}{\cal B}$ is defined analogously. If ${\cal B}$ is a triangulated subcategory of a triangulated category ${\cal A}$, then ${}^{\perp}{\cal B}$ and ${\cal B}^{\perp}$ are also triangulated subcategories. \th{Definition}\label{adm} Let ${\cal B}$ be a strictly full triangulated subcategory of a triangulated category ${\cal A}$. We say that ${\cal B}$ is {\sf right admissible} (resp., {\sf left admissible}) if for each $X\in{\cal A}$ there is an exact triangle $B\to X\to C$, where $B\in{\cal B}$ and $C\in{\cal B}^{\perp}$ (resp., $D\to X\to B$, where $D\in{}^{\perp}{\cal B}$ and $B\in{\cal B}$). A subcategory is called {\sf admissible} if it is left and right admissible. \par\endgroup \th{Definition}\label{exc} An {\sf exceptional object} in a derived category ${\cal A}$ is an object $E$ satisfying the conditions ${\rm Hom}^i(E\:,\;E)=0$ when $i\ne0$ and ${\rm Hom}(E\:,\;E)=k$. \par\endgroup \th{Definition}\label{exs} {\sf A full exceptional sequence} in ${\cal A}$ is a sequence of exceptional objects $(E_0,..., E_n)$, satisfying the semiorthogonal condition ${\rm Hom}^.(E_i\:,\;E_j)=0$ when $i>j$, and generating the category ${\cal A}$. \par\endgroup The concept of an exceptional sequence is a special case of the concept of a semiorthogonal sequence of subcategories: \th{Definition}\label{sd} A sequence of admissible subcategories $({\cal B}_0,..., {\cal B}_n)$ in a derived category ${\cal A}$ is said to be {\sf semiorthogonal} if the condition ${\cal B}_j\subset {\cal B}^{\perp}_i$ holds when $j<i$ for any $0\le i\le n$. In addition, a semiorthogonal sequence is said to be {\sf full} if it generates the category ${\cal A}$. In this case we call such a sequence {\sf semiorthogonal decomposition} of the category ${\cal A}$ and denote this as follows: $$ {\cal A}=\Bigl\langle{\cal B}_0,....,{\cal B}_n\Bigl\rangle. $$ \par\endgroup \th{Definition}\label{SF} Let ${\cal A}$ be a triangulated $k$--linear category with finite--dimensional ${\rm Hom's}$. A covariant additive functor $F: {\cal A}\to{\cal A}$ that commutes with translations is called a {\sf Serre functor} if it is a category equivalence and there are given bi--functorial isomorphisms $$ \varphi_{E,G}: {\rm Hom}_{\cal A}(E\:, \;G)\stackrel{\sim}{\longrightarrow}{\rm Hom}_{\cal A}(G\:, \;F(E))^* $$ for $E,G\in{\cal A}$, with the following property: the composite $$ (\varphi_{F(E),F(G)}^{-1})^*{\scriptsize{\circ}}\varphi_{E,G}: {\rm Hom}_{\cal A}(E\:, \;G)\longrightarrow{\rm Hom}_{\cal A}(G\:, \;F(E))^*\longrightarrow{\rm Hom}_{\cal A}(F(E)\:, \;F(G)) $$ coincides with the isomorphism induced by $F$. \par\endgroup \th{Theorem}\label{USF}{\rm \cite{BK}} i) Any Serre functor is exact, ii) Any two Serre functors are connected by a canonical functorial isomorphism. \par\endgroup Let $X$ be a smooth algebraic variety, $n={\rm dim}X$, ${\cal A}=\db{X}$ the derived category of coherent sheaves on $X$, and $\omega_X$ the canonical sheaf. Then the functor $(\cdot)\otimes\omega_X[n]$ is a Serre functor on ${\cal A}$, in view of the Serre--Grothendieck duality: $$ {\rm Ext}^i(F\:,\;G)={\rm Ext}^{n-i}(G\:,\;F\otimes\omega_X)^* $$ Let us fix notations. For vector spaces $U$ and $V$ of dimension 2 and $n=2k$, respectively, we consider a linear embedding: $$U\stackrel{\varphi}{\longrightarrow} S^2V^*.$$ By projectivization $\varphi$ defines a pencil of projective quadrics in ${\bf P}^{n-1}={\bf P}(V)$, parametrized by ${\bf P}^1={\bf P}(U)$. Denote by $X$ the intersection locus of these quadrics. Let $\{q_i\}_{i=1,\ldots,n}\subset {\bf P}^1$ are the points, corresponding to the degenerate quadrics. We assume that all $q_i$ are mutually distinct. This implies that $X$ is a smooth variety and quadrics corresponding to $q_i$ have simple degeneration. Consider a double covering $C\stackrel{p}{\longrightarrow} {\bf P}^1$ with ramification in all points $\{q_i\}$. Then $C$ is a hyperelliptic curve. In order to construct a fully faithful functor $D_{coh}^b(C)\longrightarrow D_{coh}^b(X)$ we find a vector bundle $S$ on $C\times X$ and then use theorem 1.1 for the functor $\Phi_S$ (see ch.1). To outline the idea of constructing the bundle $S$, let us recall that for non-degenerate even dimensional quadric there exist two spinor bundles (c.g.~\cite{KAPR}). Restricting these two bundles to $X$ and varying our quadric in the pencil we obtain that $C$ is the fine moduli space of spinor bundles. Unfortunately, the fine moduli space exists only for a pencil of even dimensional quadrics. For the case of more than two quadrics of arbitrary dimension there appear some global obstructions for gluing together spinor bundles and local problems for extending to points, corresponding to degenerate quadrics. A generalization to the case of more then two quadrics of arbitrary dimension will be given in a forthcoming paper. Let $Y$ (relative grassmanian of maximal isotropic subspaces) be a subvariety in ${\bf P}(U)\times {\rm G}(k,V)$ consisting of the pairs $(q,L)$ such that $L$ is isotropic with respect to the quadric corresponding to $q$ (which we denote by the same letter $q$): $$Y:=\Bigl\{ (q,L)\in{\bf P}^1\times {\rm G}(k,V)|\ q(L)=0 \Bigr\}.$$ The image of $Y$ under the natural map into Albanese variety is isomorphic to $C$. Thus, we have a natural projection $\varphi:Y\longrightarrow C$, which is a smooth projective morphism. Its fibre over a point $c\in C$ is one of two connected components of the maximal isotropic grassmanian, corresponding to the quadric $p(c)$. Of course, the composition $p{\scriptstyle\circ}\varphi$ coincides with the natural projection to ${\bf P}^1$, in other words $p$ and $\varphi$ give the Stein factorization of the projection. Now consider linear subspaces of dimension $k-1$, which belongs to $X$. It is well known that the variety of all such subspaces is isomorphic to Jacobian $J(C)$ of the curve $C$ \cite{R1}. We choose one of them $M$. It gives a section of $\varphi$. Indeed, if one consider a subvariety $C_M\subset Y$ of pairs $(q,L)\in Y$ such that $L$ contains $M$: $$C_M:=\Bigl\{ (q,L)\in Y|\ L\supset M \Bigr\},$$ then $\varphi$ biregulary projects $C_M$ to $C$, because for any non-degenerate (resp., degenerate) quadric from our pencil there exist two (resp., one) containing $M$ maximal isotropic subspaces, which lie in the different components of the grassmanian, corresponding to this quadric. Now consider the subvariety $D\subset Y$ of pairs $(q,L)$ such that $L$ has a non-trivial intersection with $M$: $$D:=\Bigl\{ (q,L)\in Y|\ L\cap M\not=0 \Bigr\}.$$ Then $D$ is a divisor in $Y$. Denote by ${\cal L}={\cal O}(D)$ the corresponding linear bundle on $Y$. Now consider the variety $F$ (of partial isotropic flags) consisting of triples $(q,l,L)\in {\bf P}^1\times {\bf P}(V)\times{\rm G}(k,V)$ such that $l\subset L$ and $q(L)=0$: $$F:=\Bigl\{ (q,l,L)\in {\bf P}^1\times {\bf P}(V)\times{\rm G}(k,V)|\ l\subset L,\ q(L)=0\Bigr\}.$$ Since $l\subset L$, $l$ is a point of the quadric $q$. In other words, the projections of $F$ to the components ${\bf P}^1\times{\bf P}(V)$ and ${\bf P}^1\times{\rm G}(k,V)$ of the product give the couple of maps: $\mu:F\longrightarrow Y,\ \lambda:F\longrightarrow Q,$ where $Q$ is the relative quadric, i.e., the variety of pairs $(q,l)\in{\bf P}^1\times{\bf P}(V)$, such that $q(l)=0$: $$Q:=\Bigl\{ (q,l)\in {\bf P}^1\times {\bf P}(V)|\ q(l)=0\Bigr\}.$$ The projection to the first component gives a map: $Q\longrightarrow{\bf P}^1$. Let $Q_C=Q\times_{{\bf P}^1}C$ be the product of $Q$ over ${\bf P}^1$. Since $\varphi{\scriptstyle\circ}\mu:F\longrightarrow C$ and $\lambda:F\longrightarrow Q$ are maps to the component of the product, by the universal property we have a map $\nu:F\longrightarrow Q_C$. Since $X$ belongs to all quadrics of the pencil we have the natural embedding $X\times{\bf P}^1\hookrightarrow Q$, which lifts up to an embedding $\varepsilon:X\times C\hookrightarrow Q_C$. All these varieties and maps are depicted in the following diagram: \centerline{\epsffile{q.eps}} Now define $S=\varepsilon^*\nu_*\mu^*{\cal L}.$ Actually, $S$ is a vector bundle on $X\times C$. Let us fixed a point $c\in C$. If $q=p(c)$ is a smooth quadric, then the fibre of $S$ over $X\times c\simeq X$ is one of two spinor bundles on $q$, restricted to $X$. If $q=p(c)$ is degenerate, then it is a cone over a quadric of the same dimension as $X$. Then the fibre of $S$ over $X\times c\simeq X$ is the restriction to $X$ of the pull back of the spinor bundle on this even dimensional quadric to the cone. Since $X$ does not meet the singular point of the cone, this restriction is also a vector bundle on $X$. Let us recall the structure of the derived category for a smooth projective quadric due to M.~Kapranov. There exist two for an even dimensional (resp., one for an odd dimensional ) quadric $q$ spinor bundles $S_q$ and $\widetilde S_q$ (resp., $S_q$). The exceptional sequence \begin{equation}\label{qcol} \Bigl({\cal O}(-d+1),{\cal O}(-d+2),\ldots,{\cal O},\widetilde S_q,S_q\Bigr)\qquad{\rm for}\ d\ {\rm even}, \end{equation} \begin{equation}\label{qcol2} \Bigl({\cal O}(-d+1),{\cal O}(-d+2),\ldots,{\cal O},S_q\Bigr)\qquad\qquad{\rm for}\ d\ {\rm odd} \end{equation} is a full strong exceptional sequence on $q$, here $d=\dim q$ (see \cite{KAPR}). For $d$ even, $S_q$ and $\widetilde S_q$ are mutually homologically orthogonal. \th{Theorem}\label{iq} The functor $\Phi_S:D_{coh}^b(C)\longrightarrow D_{coh}^b(X)$ is full and faithful. \par\endgroup \par\noindent{\bf\ Proof. } We shall use facts about spinor bundles on smooth quadrics. We have to verify conditions of theorem 1.1 for fibres of $S$ over $X\times c$. First, let us check the orthogonality conditions. Suppose that $c_1$ and $c_2$ are points of $C$, such that $p(c_1)\not=p(c_2)$, and let $S_1$ and $S_2$ are spinor bundles over corresponding quadrics. There are short exact sequences of sheaves on the projective space ${\bf P}(V)$: $$ 0\longrightarrow V\otimes{\cal O}(-2)\longrightarrow V\otimes{\cal O}(-1)\longrightarrow S^{\vee}_1\longrightarrow 0, $$ $$ 0\longrightarrow W\otimes{\cal O}(-1)\longrightarrow W\otimes {\cal O}\longrightarrow S_2 \longrightarrow 0. $$ Here we identify bundles on quadrics with corresponding coherent sheaves on ${\bf P}(V)$. If any of these quadrics is degenerate, then the same sequences holds beyond singular points of the quadric, being sufficient for what follows. Consider these sequences as resolutions for $S^{\vee}_1$, and $S_2$ and use them for computation $S_1^{\vee}\otimes S_2$. Since the quadrics intersect transversally, there are no torsion groups: $$ {\rm Tor}^i_{{\bf P}(V)}(S_1^{\vee},S_2)=0,\qquad{\rm for}\ i>0. $$ Therefore, we obtain a resolution for $S_1^{\vee}\otimes S_2$ of the following kind: $$ 0\longrightarrow C\otimes {\cal O}(-3)\longrightarrow B\otimes {\cal O}(-2)\longrightarrow A\otimes {\cal O}(-1)\longrightarrow S_1^{\vee}\otimes S_2\longrightarrow 0. $$ Computing cohomologies of $S_1^{\vee}\otimes S_2$ by means of this resolution we obtain the orthogonality conditions for the case $p(c_1)\not=p(c_2):$ $$ {\rm Ext}^i_X(S_1,S_2)={\rm H}^i({\bf P}(V),S_1^{\vee}\otimes S_2)=0. $$ Now suppose that $p(c_1)=p(c_2)$. Then we have to verify orthogonality, while restricted to $X$, between two spinor bundles $S_q$ and $\widetilde S_q$ on a single non-degenerate quadric $q$. Consider the tensor product $S_q^{\vee}\otimes\widetilde S_q$ over $q$. Since $X$, as a divisor in $q$, is equivalent to double hyperplane section, we have an exact sequence of sheaves on $q$: $$ 0\longrightarrow S_q^{\vee}\otimes\widetilde S_q(-2)\longrightarrow S_q^{\vee}\otimes\widetilde S_q \longrightarrow S_q^{\vee}\otimes\widetilde S_q \left|_X\right. \longrightarrow 0. $$ Computing cohomology we easily find that $$ {\rm Ext}^i_X(S_q,\widetilde S_q)={\rm H}^i(q,S_q^{\vee}\otimes\widetilde S_q \left|_X\right.)=0,\qquad{\rm for\ any}\ i. $$ Indeed, $$ {\rm H}^i(S_q^{\vee}\otimes\widetilde S_q )={\rm Ext}^i_X(S_q,\widetilde S_q)=0,\qquad{\rm for\ any}\ i. $$ Then, mutating $\widetilde S_q$ two times to the left in sequence (\ref{qcol}) one can obtain a new exceptional sequence: $$\Bigl({\cal O}(-d+1),\ldots,{\cal O}(-2),\widetilde S_q(-2),{\cal O}(-1),{\cal O},S_q\Bigr).$$ It yields: $$ {\rm H}^i(S^{\vee}_q\otimes \widetilde S_q(-2))={\rm Ext}^i(S_q,\widetilde S_q(-2))=0. $$ Now let us verify condition ii) of theorem 1.1. Suppose $q=p(c)$ is a non-degenerate quadric and $\Phi_S({\cal O}_C)=\widetilde S_q$. We can calculate ${\rm Ext}^i_X(\Phi_S({\cal O}_C),\Phi_S({\cal O}_C)) $ using the following exact sequence on the quadric: $$ 0 \longrightarrow \widetilde S^{\vee}_q\otimes \widetilde S_q(-2) \longrightarrow \widetilde S^{\vee}_q\otimes \widetilde S_q \longrightarrow \widetilde S^{\vee}_q\otimes \widetilde S_q\left |_X\right.\longrightarrow 0. $$ Since $\widetilde S_q$ is an exceptional object in $D^b_{coh}(q)$ and $\widetilde S_q(-2)$ is a double left mutation in the collection (\ref{qcol}), we have $$ {\rm H}^0_q(\widetilde S^{\vee}_q\otimes \widetilde S_q)=k,\quad {\rm H}^i_q(\widetilde S^{\vee}_q\otimes \widetilde S_q)=0\ i\not=0; $$ $$ {\rm H}^2_q(\widetilde S^{\vee}_q\otimes \widetilde S_q(-2))=k,\quad {\rm H}^i_q(\widetilde S^{\vee}_q\otimes \widetilde S_q(-2))=0\ i\not=2. $$ Then the short sequence gives: $$ {\rm Hom}_X( \widetilde S_q, \widetilde S_q)=k,\ \ {\rm Ext}^1_X( \widetilde S_q, \widetilde S_q)=k,\ \ {\rm Ext}^i_X( \widetilde S_q, \widetilde S_q)=0,\ i>1. $$ Similarly for $\Phi_S({\cal O}_C)=S_q$. Now suppose that $q=p(c)$ is a degenerate quadric. Then the projection from the centre of the cone gives a double covering $\pi:X\longrightarrow q'$ from $X$ to a quadric of dimension $\ d-1$. Since $\Phi_S({\cal O}_C)=\pi^*S_{q'}$ is the pull back of the spinor bundle $S_{q'}$ on this quadric along $\pi$, \begin{equation}\label{sq} {\rm Ext}^1(\Phi_S({\cal O}_C),\Phi_S({\cal O}_C))= {\rm H}^i(X,\pi^*(S^{\vee}_{q'}\otimes S_{q'}))= {\rm H}^i(q',\pi_*\pi^*(S^{\vee}_{q'}\otimes S_{q'})). \end{equation} By projection formula we have: $$ \pi_*\pi^*(S^{\vee}_{q'}\otimes S_{q'})=\pi_*{\cal O}_X\otimes S^{\vee}_{q'}\otimes S_{q'}=\Bigl[ {\cal O}_{q'}\oplus {\cal O}_{q'}(-1)\Bigr]\otimes S^{\vee}_{q'}\otimes S_{q'}= $$ \begin{equation}\label{pp} =S^{\vee}_{q'}\otimes S_{q'}\oplus S^{\vee}_{q'}\otimes S_{q'}(-1). \end{equation} Since $S_{q'}$ is exceptional on $q'$ and $S_{q'}(-1)$ is the left mutation of $S_{q'}$ in sequence (\ref{qcol2}), it follows that $$ {\rm H}^0(S^{\vee}_{q'}\otimes S_{q'})=k,\quad {\rm H}^i(S^{\vee}_{q'}\otimes S_{q'})=0 \ \ {\rm for}\ i\not=0; $$ $$ {\rm H}^1(S^{\vee}_{q'}\otimes S_{q'}(-1))=k,\quad {\rm H}^i(S^{\vee}_{q'}\otimes S_{q'})=0 \ \ {\rm for}\ i\not=1. $$ Combining this with (\ref{sq}) and (\ref{pp}), we obtain: $$ {\rm Hom}\Bigl(\Phi_S({\cal O}_C),\Phi_S({\cal O}_C)\Bigr)=k,\qquad {\rm Ext}^1\Bigl(\Phi_S({\cal O}_C),\Phi_S({\cal O}_C)\Bigr)=k, $$ $$ {\rm Ext}^i\Bigl(\Phi_S({\cal O}_C),\Phi_S({\cal O}_C)\Bigr)=0,\quad{\rm for}\ i>1. $$ This concludes the proof of the theorem. Let us recall that we consider the intersection $X$ of two quadrics of dimension $d=n-2.$ \th{Proposition}\label{orthog} The image of $\Phi_s:D^b_{coh}(C)\longrightarrow D^b_{coh}(X)$ is left orthogonal to the exceptional sequence $\sigma= \Bigl({\cal O}_X(-d+3),\ldots,{\cal O}_X\Bigr) $ on $X$. \par\endgroup \par\noindent{\bf\ Proof. } First, the sequence $\Bigl({\cal O}_X(-d+3),\ldots,{\cal O}_X\Bigr)$ is exceptional on $X$. Indeed, from the short exact sequence on a non-degenerate quadric $q$: $$0\longrightarrow {\cal O}_q(i-2) \longrightarrow {\cal O}_q(i)\longrightarrow {\cal O}_X(i) \longrightarrow 0,$$ and from exceptionality of (\ref{qcol}) one can easily find: ${\rm H}^j\Bigl(X,{\cal O}(k)\Bigr)=0$, for any $j$ and $-d+3<k<0$. Similarly to the proof of lemma 1.2 one can show the existence of the right adjoint functor $\Psi:D^b_{coh}(X)\longrightarrow D^b_{coh}(C)$ to $\Phi_S$. Then for any object $A\in D^b_{coh}(C)$ one has $${\rm Hom}_X\Bigl(\Phi_SA,{\cal O}(i)\Bigr)={\rm Hom}\Bigl(A,\Psi({\cal O}(i))\Bigr).$$ We have to show that $\Psi({\cal O}(i))=0$, for ${\cal O}(i)\in \sigma.$ Since there are no non-zero objects in $D^b_{coh}(C)$ which are orthogonal to all skyscraper sheaves ${\cal O}_c,\ c\in C$, it is sufficient to prove that all $\Phi_S({\cal O}_c)$ are orthogonal to $\sigma$. But every $\Phi_S({\cal O}_c)$ is a spinor bundle $S_q$ restricted to $X$ either from a smooth or from a degenerate quadric. In the former case we have an exact sequence $$ 0\longrightarrow S_q^{\vee}(-2)\longrightarrow S_q^{\vee} \longrightarrow S_q^{\vee}\left|_X\right.\longrightarrow 0.$$ Using this sequence and exceptionality of (\ref{qcol}) we easily find that $S_q$ is right orthogonal to $\sigma$. If the quadric is degenerate, then we have a projection $\pi:X\longrightarrow q'$ to a quadric of dimension $d-1$, and $\Phi({\cal O}_c)=\pi^*(S_{q'}).$ Therefore: $${\rm Ext}^j_X\Bigl(\Phi({\cal O}_c),{\cal O}(i) \Bigr)= {\rm Ext}^j_X\Bigl(\pi^*(S_{q'}),{\cal O}(i) \Bigr)=$$ $$={\rm Ext}^j_{q'}\Bigl(S_{q'},\pi_*{\cal O}(i) \Bigr)={\rm Ext}^j_{q'}\Bigl(S_{q'},{\cal O}(i)\oplus {\cal O}(i-1) \Bigr).$$ Because of exceptionality of (\ref{qcol2}) we are done. Now we regard $\db{C}$ as a subcategory in $\db{X}$. As has been shown $\sigma= \Bigl({\cal O}_X(-d+3),\ldots,{\cal O}_X\Bigr) $ lies in the right orthogonal to $\db{X}$. \th{Theorem}\label{qdq} The category $\db{X}$ on the intersection of two quadrics of dimension $d$ is generated as a triangulated category by $\sigma $ and $\db{C}$, in other words there is a semiorthogonal decomposition $$ \db{X}\ =\ \Bigl\langle{\cal O}_X(-d+3),\ldots,{\cal O}_X,\db{C}\Bigl\rangle $$ \par\endgroup \par\noindent{\bf\ Proof. } Consider the subcategory $D\subset \db{X}$, generated by $\sigma $ and $\db{C}$. First, let us mention that the composition of the natural embedding $K_0(D)\otimes k\to K_0(X)\otimes k$ with the Chern character $$ ch:K_0(X)\otimes k\to H^{even}(X,k)=\oplus H^{i,i}(X,k) $$ is a surjective homomorphism from the Grothendieck group of $D$, tensored by $k$, to the sum of the diagonal cohomologies of $X$ with coefficients in $k$. Indeed, the Chern character is a surjective morphism with the kernel, consisting, by Riemann-Roch-Hirzebruch formula, of those $v\in K_0(X)\otimes k$, which are in the (say, right) kernel of the Euler characteristic bilinear form $\chi $ (see ch.1). From the orthogonal decomposition for $D$ one easily finds that the restriction of $\chi $ on $D$ has rank $d$, which coincides with the dimension of $H^*(X,k)$. The surjectivity follows. Since $D$ has a semiorthogonal decomposition by admissible subcategories, it is in turn admissible \cite{BK}. Then, as usually, we suppose that $D^{\perp }$ is not trivial and consider an object $Z\in D^{\perp }$. It follows from above that $ch(Z)=0$. Consider a singular quadric containing $X$. Let $\pi :X\to q$ be a projection from the singular point of this quadric to a non-singular quadric $q$ of dimension $d-1$. There is a semiorthogonal decomposition \begin{equation}\label{dqp} \db{q}\ =\ \Bigl\langle{\cal O}_q(-d+2),\ldots,{\cal O}_q,S_q\Bigl\rangle \end{equation} For any $A\subset \db{q}$ we have an isomorphism: $$ {\rm Hom}_X(\pi ^{*}A,Z)={\rm Hom}_q(A,\pi _{*}Z) $$ Since all but the first element of (\ref{dqp}) after lifting to $X$ belong to the subcategory $D$, it follows that $\pi _{*}Z$ belongs to $\Bigl\langle{\cal O}_q(-d+3),\ldots,{\cal O}_q,S_q\Bigl\rangle ^{\perp }=\Bigl\langle{\cal O}_q(-d+2)\Bigl\rangle$. We aim to prove that $Z$ as an object in $\db{X}$ is quasi-isomorphic to the direct sum of its cohomology sheaves. \th{Lemma} For any couple of coherent sheaves $A,B$ on $X$, such that $\pi_{*}A$ and $\pi_{*}B$ are direct sums of copies of a single linear bundle on $q$, one has $$ {\rm Ext}^{i}_{X}(A,B)=0,\ {\rm for}\ i>1. $$ \par\endgroup \par\noindent{\bf\ Proof. } Let $s$ be the $\pi $-fibrewise involution on $X$. The fibred square $X'=X\times _qX$ of $X$ over $q$ is a union of two copies of $X$, which normally intersects in the (smooth) ramification divisor $H$ of $\pi$ in $X$. These are the diagonal $$ \Delta X=\{(x,x)|x\in X\} $$ and the $s$-diagonal $$ \Delta _sX=\{(x,sx)|x\in X\}. $$ This description implies a short exact sequence of coherent sheaves on $X'$: \begin{equation}\label{diadec} 0\to {\cal O}_{\Delta _sX}(-H)\to {\cal O}_{X'}\to {\cal O}_{\Delta _X}\to 0. \end{equation} Denote by $p_{1}, p_{2}$ the projections of $X'$ to $X$. Take any coherent sheaf $C$ on $X$. Tensoring (\ref{diadec}) with $p_{1}^{*}C$ one obtains: $$ 0\to p_{1}^{*}C\otimes {\cal O}_{\Delta _sX}(-H)\to p_{1}^{*}C\to p_{1}^{*}C\otimes {\cal O}_{\Delta _X}\to 0 $$ Then, applying $p_{2*}$ to this sequence, one has: $$ 0\to s_{*}C(-1)\to p_{2*}p_{1}^{*}C\to C\to 0. $$ Using the flat base change theorem (see \cite{H},II,\S5,prop.5.12) where the morphism and the base change both are $\pi $, we obtain: $$ \pi ^{*}\pi _{*}=p_{2*}p_{1}^{*} $$ Therefore one has an exact sequence for any sheaf on $X$: \begin{equation}\label{ppdec} 0\to s_{*}C(-1)\to \pi ^{*}\pi _{*}C\to C\to 0. \end{equation} Let now $A$ and $B$ be such that $\pi_{*}A$ and $\pi_{*}B$ are sums of copies of a linear bundle on $q$. Without loss of generality we can assume that this linear bundle is trivial. Since $\pi _{*}s_{*}\simeq \pi _{*}$, putting $C=B$ and $C=s_{*}B(-1)$ in (\ref{ppdec}), we obtain exact sequences: $$ 0\to s_{*}B(-1)\to \oplus {\cal O}\to B\to 0 $$ $$ 0\to B(-2)\to \oplus {\cal O}(-1)\to s_{*}B(-1)\to 0. $$ Juxtaposing these two sequences and then repeating the procedure in the same way one obtains a resolution for $B$: $$ \dots \to \oplus {\cal O}(-2)\to \oplus {\cal O}(-1)\to \oplus {\cal O}\to B\to 0. $$ Using this resolution and the fact that $\pi _{*}A$ is a trivial bundle, one obtains a spectral sequence converging to ${\rm Ext}^{\cdot }(B,A(-d+2))$: $$ E^{p,q}_1= {\rm Ext}^q({\cal O}(-p)\:,\; {\cal O}(-d+2)) \Longrightarrow {\rm Ext}^{p+q}( B\:, \; A(-d+2)) $$ which shows that $$ {\rm Ext}^{<d-2}(B,A(-d+2))=0. $$ By the adjunction formula one easily calculates the canonical class of $X$: $$ \omega _X= {\cal O}_X(-d+2). $$ Thus, by Serre duality the equality holds: $$ {\rm Ext}^{i}( A\:, \; B)={\rm Ext}^{d-1-i}( B\:, \; A(-d+2))^{*}. $$ It follows that ${\rm Ext}^{>1}( A\:, \; B)=0$. Since ${\cal O}_q(-d+2)$ is an exceptional sheaf, any object from $\Bigl\langle{\cal O}_q(-d+2)\Bigl\rangle$ is isomorphic to the direct sum of its cohomology sheaves, which in turn are direct sums of copies of ${\cal O}_q(-d+2)$. Let ${\cal H}^i$ be the cohomology sheaves of $Z$. Since $R^{0}\pi_{*}$ is an exact functor, $R^{0}\pi_{*}{\cal H}^i$ are direct sums of copies of ${\cal O}_q(-d+2)$. It follows by the lemma that $$ {\rm Ext}^k({\cal H}^i\: ,\;{\cal H}^j)=0,\ {\rm for} k>1. $$ It is well known that this implies a decomposition of $Z$ into a direct sum of its cohomology sheaves. Since $Z\in D^{\perp }$, it follows that ${\cal H}^i\in D^{\perp }$. Then we have from above that $ch({\cal H}^i)=0$. But a sheaf (not a complex of sheaves) with trivial Chern character is zero. As all cohomologies of $Z$ are zero, then $Z$ is quasi-isomorphic to zero itself. This finishes the proof of the theorem. {\section{Birational transformations.}} The aim of this chapter is to trace behaviour of the derived category of coherent sheaves with respect to birational transformations. It turns out that blowing up and flip transformations have a categorical incarnation as adding or removing of semiorthogonal summands of a quite simple nature. For a flop there are no such summands, thus it produces an equivalence of triangulated categories. The simplest instance of a birational transformation is a blowing up of a variety along a smooth centre. A discription of the derived category of the blow-up in terms of the categories of the variety and of the centre is done in \cite{or} . We give here the treatment with small corrections and with a stress on theorem \ref{mai} in proofs. Let $Y$ be a smooth subvariety of codimension $r$ in a smooth algebraic variety $X$. Denote $\widetilde X$ the smooth algebraic variety obtained by the blowing up of $X$ along the centre $Y$. There exists a fibred square: $$ \begin{array}{ccc} \widetilde Y&\stackrel{j }{\longrightarrow}& \widetilde X\\ \llap{\footnotesize $p$} \downarrow &&\llap{$\pi$} \downarrow\\ Y&\stackrel{i}{\longrightarrow}& X \end{array} $$ where $i$ and $j$ are embeddings of smooth varieties, and $p: \widetilde Y\to Y$ is the projective fibration of the exceptional divisor $\widetilde Y$ in $\widetilde X$ over the centre $Y$. Recall that $\widetilde Y={\bf P}(N_{X/Y})$ is the projectivization of the normal bundle to $Y$ in $X$. \th{Proposition}\label{blow} {\rm (see \cite{or})} The pull back functors $$ \pi^*: \db {X}\longrightarrow \db {\widetilde X} $$ $$ p^*: \db {Y}\longrightarrow \db {\widetilde Y} $$ are full and faithful. \par\endgroup \par\noindent{\bf\ Proof. } The functor $\pi^*$ (resp., $p^*$) is isomorphic to $\Phi_{E_X}$ (resp., $\Phi_{E_Y}$), where $E_X$ (resp., $E_Y$) is the structure sheaf of the incidence subscheme $Z_X$ (resp., $Z_Y$) in $X\times {\widetilde X}$ (resp., $Y\times {\widetilde Y}$). We shall show that proof easily follows from theorem \ref{mai}. Indeed, for a point $x\in X$, $\Phi_{E_X}({\o x})=\pi^*({\o x})$ (resp., for a point $y\in Y$, $\Phi_{E_Y}({\o y})=p^*({\o y})$) is the structure sheaf of the $\pi$--fibre over $x$. Since fibres are disjoint, orthogonality condition i) of theorem \ref{mai} follows. For $\Phi_{E_Y}$ analogously. Since $\widetilde X$ has the same dimension as $X$ and due to the fact that for any couple $(F, E)$ of sheaves on a smooth variety ${\rm Ext}^i(F, E)=0$ for $i$ greater than the dimension of the variety, condition ii) of theorem \ref{mai} for $\Phi_{E_X}$ is verified. Let us remark that for $A\subset B$, a smooth subvariety in a smooth variety, local extension groups for the structure sheaf ${\o A}$ of the subvariety are equal: \begin{equation}\label{ext} {\cal E}xt^i_B({\o A}\:, {\o A})=\Lambda^i N_{B/A}, \end{equation} where $N_{B/A}$ is the normal vector bundle of $A$ in $B$. For fibres $p^*({\o Y})$, which are biregulary isomorphic to projective spaces, the normal bundle is trivial. Thus they have no the higher cohomologies and the spectral sequence from local to global extension groups gives condition ii) of theorem \ref{mai} for $\Phi_{E_Y}$. Another proof uses the projection formula: $$ {\rm Ext}^i(p^*F\:,\:p^*G)={\rm Ext}^i(F\:,\:p_*p^*G)={\rm Ext}^i(F\:,\:p_*{\o {\widetilde Y}}\otimes G). $$ Analogously for $\pi^*$. Combining with the facts that $\pi_*{\o{\widetilde X}}={\o X}$ and $p_*{\o{\widetilde Y}}={\o Y}$ this gives the proof. \th{Proposition}\label{emb} For any invertible sheaf $L$ over $\widetilde Y$, the functor $$ \Phi j_*(L\otimes p^*(\cdot)): \db {Y}\longrightarrow \db {\widetilde X} $$ is full and faithful. \par\endgroup \par\noindent{\bf\ Proof. } Let us verify the hypothesis i) and ii) of the theorem \ref{mai}. For a point $y\in Y$ the image $\Phi ({\cal O}_y)$ is the structure sheaf of the corresponding $p$-fibre (regarded as a sheaf on ${\widetilde X}$). Since $p$-fibres over distinct points are disjoint the orthogonality condition i) of the theorem \ref{mai} follows. Now let us consider the structure sheaf ${\cal O}_F$ of a single $p$-fibre $F\subset {\widetilde X}$. By formula (\ref{ext}) one has the spectral sequence: $$ {\rm E}^{i,j}_2=\oplus \ {\rm H}^i(F, \Lambda ^jN_{{\widetilde X}/F})=\oplus \ {\rm H}^i({\widetilde X},{\cal E}xt^j_{\widetilde X}({\o F}\:, {\o F}))\Longrightarrow \oplus \ {\rm Ext}^{i+j}_{\widetilde X}({\o F}\:, {\o F}). $$ For the normal bundle $N_{{\widetilde X}/F}$ one has the exact sequence: $$ 0\to N_{{\widetilde Y}/F}\to N_{{\widetilde X}/F}\to N_{{\widetilde X}/{\widetilde Y}}|_F\to 0. $$ Obviously, the normal bundle $N_{{\widetilde Y}/F}$ is trivial, while $$ N_{{\widetilde X}/{\widetilde Y}}|_F\ =\ {\cal O}(Y)|_F\ =\ {\cal O}(-1). $$ Since $F$ is a projective space and since there are no extension groups between ${\cal O}(-1)$ and ${\cal O}$ on a projective space, it follows that the above short exact sequence splits. Then the spectral sequence gives conditions ii) of the theorem \ref{mai}. This proves the proposition. \th{Lemma}\label{combar} Let $j:D\to Z$ be the embedding of a smooth divisor in a smooth algebraic variety $Z$ of dimension $n$. Consider for an object $A\in \db{X}$ an exact triangle with the canonical second morphism: $$ \overline A\to j^{*}j_{*}A\to A $$ Then $\overline A\simeq A\otimes {\cal O}_D(-D)[1]$. \par\endgroup \par\noindent{\bf\ Proof. } The functor $j_{*}$ coincides with $\Phi _E$ (in notations of ch.1), where $E={\o G}$ is the structure sheaf of the graph subvariety $G$ for $j$ in $D\times Z$. The adjoint functor $j^{*}$ is isomorphic to $\Psi _{E'}$, where $E'={\o G}$. By proposition \ref{comp} in order to calculate $j^{*}j_{*}$ one has to find ${p_{13}}_*({p_{23}}^*{\o G}\otimes {p_{12}}^*{\o G})$, where $p_{ij}$ are the projections from the product $D\times Z\times D$ along the $k$-th component, where $\{ ijk\} =\{ 123\}$. Note that $$ {p_{12}}^*{\o G}={\o {G\times D}};\ \ {p_{23}}^*{\o G}={\o {D\times G}}, $$ where ${G\times D}$ and ${D\times G}$ are regarded as subvarieties in $D\times Z\times D$ of codimension $n$. These varieties intersect along the subvariety, which is the image of the morphism $({\rm id},j,{\rm id}):D\to D\times Z\times D$. It is of codimension $2n-1$, hence a non-transversal intersection. Fortunately, both ${G\times D}$ and ${D\times G}$ lie in the image of $\Delta_3:D^3\to D\times Z\times D$, where they meet transversally. This helps to compute ${\cal H}^i({\o {G\times D}}\otimes {\o {D\times G}})$. Indeed, one can consider the tensor product ${\o {G\times D}}\otimes {\o {D\times G}}$ as the restriction of ${\o {D\times G}}$ to ${G\times D}$. Restricting first to the divisor $D^3$, we obtain: $$ \begin{array}{l} L^0\Delta_3{\o {D\times G}}={\o {D\times G}},\\ L^1\Delta_3{\o {D\times G}}={\o {D\times G}}(-D^3),\\ L^i\Delta_3{\o {D\times G}}=0, \ {\rm for}\ i>1. \end{array} $$ Then restriction to ${G\times D}\subset D\times Z\times D$ and projection along $Z$ give that the complex ${\cal K}={p_{13}}_*({p_{23}}^*{\o G}\otimes {p_{12}}^*{\o G})\in {\db {D\times Z}}$ has only two cohomology sheaves. Namely, $$ \begin{array}{l} {\cal H}^0({\cal K})={\o {\Delta}},\\ {\cal H}^{-1}({\cal K})={\o {\Delta }}\otimes \pi ^*{\cal O}(-D). \end{array} $$ Therefore one has the exact triangle $$ {\o {\Delta }}\otimes \pi ^*{\cal O}(-D)[1]\to {\cal K}\to {\o {\Delta }}. $$ Applying functors, corresponding to objects from this triangle to arbitrary $A\in {\db D}$ one gets the proof. Denote by $D(X)$ the full subcategory of $\db{\widetilde X}$ which is the image of $\db{X}$ with respect to the functor $\pi^*$ and by $D(Y)_k$ the full subcategories of $\db{\widetilde X}$ which are the images of $\db{Y}$ with respect to the functors $j_*(\o{\widetilde Y}(k)\otimes p^*(\cdot))$. \th{Proposition}\label{ort} The sequence $$ \Bigl( D(Y)_{-r+1}, ... , D(Y)_{-1}, D(X) \Bigr) $$ is a semiorthogonal sequence of admissible subcategories in $\db{\widetilde X}$. \par\endgroup \par\noindent{\bf\ Proof. } 1). Let $j_*A\in D(Y)_k$ and $j_*B\in D(Y)_m$ with $r-2\ge k-m >0$ . That means \begin{equation}\label{fb} A=p^*A'\otimes \o{\widetilde Y}(k),\; B=p^*B'\otimes \o{\widetilde Y}(m), \end{equation} for some $A', B'\in \db{Y}$. We have an exact triangle: \begin{equation}\label{jj} \bar A \longrightarrow j^*j_*A\longrightarrow A, \end{equation} and by lemma \ref{combar} an isomorphism $$ \bar A\cong A\otimes \o{\widetilde Y}(1)[1]. $$ Furthermore, $$ \begin{array}{l} {\h A, B}\cong {\h p^*A'\otimes \o{\widetilde Y}(k), {p^*B'\otimes \o{\widetilde Y}(m)}}\cong \\ {\h p^*A', {p^*B'\otimes \o{\widetilde Y}(m-k)}}\cong {\h A', {p_*(p^*B'\otimes \o{\widetilde Y}(m-k))}}\cong\\ {\h A', {B'\otimes p_*\o{\widetilde Y}(m-k)}}. \end{array} $$ From vanishing of $p_*\o{\widetilde Y}(-n)=0$, with $r-1\ge n >0$ we obtain: $$ {\h A, B}=0. $$ Analogously $$ {\h {\bar A}, B}=0. $$ Then, triangle (\ref{jj}) gives $$ {\h j_*A, {j_*B}}=0. $$ This proves semiorthogonality for the sequence of the subcategories $$ \Bigl( D(Y)_{-r+1}, ... , D(Y)_{-1} \Bigl). $$ 2). If $\pi^*A\in D(X)$ and $j_*B\in D(Y)_m$ for $-r+1\le m\le -1$ with $B$ being of the form (\ref{fb}), then $$ {\h \pi^*A, {j_*B}}\cong {\h A, {\pi_*j_*B}}\cong {\h A, {i_*p_*B}}=0. $$ This is equal to zero, because $p_*B=p_*(p^*B'\otimes \o{\widetilde Y}(m))=B'\otimes p_*\o{\widetilde Y}(m)=0$. \th{Theorem}\label{full} In the above notations, the semiorthogonal sequence of admissible subcategories $$ \Bigl< D(Y)_{-r+1}, ... , D(Y)_{-1}, D(X) \Bigl> $$ generates the category $\db{\widetilde X}$. \par\endgroup \par\noindent{\bf\ Proof. } See \cite{or}. This theorem gives a semiorthogonal decomposition of the derived category $\db{\widetilde X}$ on a blow-up $\widetilde X$. It was used in \cite{or} for constructing a full exceptional sequence in $\db{\widetilde X}$, starting from such sequences on $X$ and $Y$. \bigskip Now we explore the behaviour of the derived categories of coherent sheaves with respect to simplest flip and flop transformations. Let $Y$ be a smooth subvariety of a smooth algebraic variety $X$ such that $Y\cong {\bf P}^k$ and $N_{X/Y}\cong {\o Y}(-1)^{\oplus (l+1)}$ with $l\le k$. If now $\widetilde X$ is a blow-up of $X$ along $Y$, then exceptional divisor $\widetilde Y\cong {\bf P}^k\times {\bf P}^l$ is isomorphic to the product of projective spaces. This allows us to blow down $\widetilde X$ in such a way that$\widetilde Y$ project to the second component ${\bf P}^l$ of the product. As a result we obtain a smooth algebraic variety $X^+$ with subvariety $Y^+\cong {\bf P}^l$. This situation is depicted in the following diagram: \begin{figure}[h] \hspace*{6cm}\epsffile{b.eps} \end{figure} The birational map $X\longrightarrow X^+$ is a simple example of flip, for $l<k$, and flop, for $l=k$, transformations. One can easily calculate that for the restriction $\o{\widetilde X}(\widetilde Y){\Bigl |}_{\widetilde Y}$ there exists an isomorphism $$ {\o {\widetilde X}}(\widetilde Y){\Bigl |}_{\widetilde Y}\cong {\cal O}(-1)\; \epsffile{k.eps}\; {\cal O}(-1), $$ where ${\cal O}(-1)\; \epsffile{k.eps}\; {\cal O}(-1):= p^*{\o Y}(-1)\otimes p^{+*}{\o {Y^+}}(-1)$. For subsequent calculations we need the formula for the canonical sheaf $\omega_{\widetilde X}$ of the blow-up. $$ \omega_{\widetilde X}\cong \pi^*\omega_X\otimes \o{\widetilde X}(l\widetilde Y). $$ Further, by the adjunction formula we know that $$ \omega_X{\Bigl |}_Y\cong \omega_Y\otimes \Lambda^{l+1}N^*_{X/Y}\cong {\o Y}(l-k). $$ Combining these facts we conclude that $$ \omega_{\widetilde X}{\Bigl |}_{\widetilde Y}\cong (\pi^*\omega_X\otimes {\o {\widetilde X}}(l\widetilde Y)){\Bigl |}_{\widetilde Y}\cong p^*(\omega_X\Bigl|_Y)\otimes {\o {\widetilde X}}(l\widetilde Y){\Bigl |}_{\widetilde Y}\cong {\cal O}(-k)\; \epsffile{k.eps}\; {\cal O}(-l). $$ Next is the main theorem of this section. \th{Theorem}\label{fl} In the above notations, the functor $$ \pi_*\pi^{+*}: \db {X^+}\longrightarrow \db {X} $$ is full and faithful. \par\endgroup \par\noindent{\bf\ Proof. } We have to show that for any pair of objects $A, B\subset \db {X^+}$ there is an isomorphism $$ {\h \pi_*\pi^{+*}A, {\pi_*\pi^{+*}B}}\cong {\h A, B}. $$ For the left hand side we have a canonical isomorphism \begin{equation}\label{pi} {\h \pi_*\pi^{+*}A, {\pi_*\pi^{+*}B}}\cong {\h \pi^* \pi_*\pi^{+*}A, {\pi^{+*}B}}. \end{equation} Consider an exact triangle: \begin{equation}\label{nbar} \pi^* \pi_*\pi^{+*}A\longrightarrow \pi^{+*}A\longrightarrow {\bar A}. \end{equation} Applying to it the functor ${\h {\cdot}, {\pi^{+*}B}}$ we find that if \begin{equation}\label{zer} {\h {\bar A}, {\pi^{+*}B}}=0, \end{equation} then the right hand side of (\ref{pi}) is isomorphic to ${\h \pi^{+*}A, {\pi^{+*}B}}$. Since $\pi^+$ is an instance of a blow up morphism, by proposition \ref{blow} we have an isomorphism: $$ {\h \pi^{+*}A, {\pi^{+*}B}}\cong {\h A, B}. $$ Hence we need to check (\ref{zer}). Again by proposition \ref{blow} $\pi^*$ is full and faithful, or, in other words, $\pi_*\pi^*$ is isomorphic to the identity functor. Applying this isomorphism to the object $\pi_*\pi^{+*}A$, we get the first morphism from the exact triangle, obtained by application of functor $\pi_*$ to triangle (\ref{nbar}): $$ \begin{array}{ccccc} \pi_*\pi^*\pi_*\pi^{+*}A&\stackrel{\sim}{ \longrightarrow}&\pi_*\pi^{+*}A&\longrightarrow&\pi_*{\bar A}. \end{array} $$ Therefore, $\pi_*{\bar A}=0$. Consequently, for any object $C\in\db {X^+}$ ${\h \pi^*C , {\bar A}}=0$, i.e. $\bar A\in D(X)^{\perp}$. Recall that by theorem \ref{full} $$ D(X)^{\perp}=\Bigl< D(Y)_{-l}, ... , D(Y)_{-1} \Bigl> $$ is a semiorthogonal decomposition of the category $D(X)^{\perp}$. The notations $D(Y)_{-k}$ are fixed before proposition \ref{ort}. If we choose full exceptional sequences in each $D(Y)_{-i}$, then gathering them together we obtain a full sequence in $D(X)^{\perp}$. The following one will be convinient for us: $$ \begin{array}{lllll} D(X)^{\perp}=\Bigl< &j_*({\cal O}(-k)\; \epsffile{k.eps}\; {\cal O}(-l)),& ... & ....& j_*({\cal O}\; \epsffile{k.eps}\; {\cal O}(-l)),\\ & j_*({\cal O}(-k+1)\; \epsffile{k.eps}\; {\cal O}(-l+1)),& ... & .... & j_*({\cal O}(1)\; \epsffile{k.eps}\; {\cal O}(-l+1)),\\ & ................ & ... & ... & .............. \\ & j_*({\cal O}(-k+l-1)\; \epsffile{k.eps}\; {\cal O}(-1)),&... & ...& j_*({\cal O}(-l-1)\; \epsffile{k.eps}\; {\cal O}(-1))\Bigl> \end{array} $$ Let us divide this sequence in two parts ${\cal A}$ and ${\cal B}$, such that $$ D(X)^{\perp}=\Bigl< {\cal B , A}\Bigl> $$ be a semiorthogonal decomposition for $D(X)^{\perp}$ with ${\cal A}$ and ${\cal B}$ being the subcategories generated by $j_*({\cal O}(i)\; \epsffile{k.eps}\; {\cal O}(s))$ with $i\ge 0$ and $i<0$ respectively. If $1\le i\le k$ and $1\le s\le l$ then the object $ j_*({\cal O}(-i)\; \epsffile{k.eps}\; {\cal O}(-s))$ belongs to simultaneously $D(X)^{\perp}$ and $D(X^+)^{\perp}$. Therefore, applying the functor Hom with the target in this object to exact triangle (\ref{nbar}) we obtain: $$ {\h {\bar A}, { j_*({\cal O}(-i)\; \epsffile{k.eps}\; {\cal O}(-s))}}=0,\qquad \mbox{for } 1\le i\le k \;\mbox{and } 1\le j \le l. $$ Since ${\bar A}\in D(X)^{\perp}$, it immediately follows that $\bar A\in {\cal A}$. Further, we notice that ${\cal A}\otimes\omega_{\widetilde X}\in D(X^+)^{\perp}$, because $\omega_{\widetilde X}{\Bigl |}_{\widetilde Y}\cong {\cal O}(-k)\; \epsffile{k.eps}\; {\cal O}(-l)$ and $l\le k$. Therefore, for $B\in{\db {X^+}}$ $$ {\h \pi^{+*}B , {\bar A \otimes \omega_{\widetilde X}}}=0. $$ Hence by Serre duality (\ref{zer}) follows. This proves the theorem. \bigskip {\sc Remark}. For the case of flop ($l=k$) the functor $\pi_*\pi^{+*}$ is an equivalence of triangulated categories. \bigskip Now we investigate more carefully 3-dimensional flops. Let $f:X\longrightarrow Y$ be a proper birational morphism between compact threefolds, which blow down only an indecomposable curve $C$. Assume that $X$ is smooth and $C\cdot K_X=0$. Then $C\cong{\bf CP }^1$ and $N_{X/C}$ is equal to either ${\cal O}(-1)\oplus{\cal O}(-1)$ or ${\cal O}\oplus{\cal O}(-2)$, or ${\cal O}(1)\oplus{\cal O}(-3)$ (see, e.g. \cite{CKM}). There exist in this situation a smooth compact threefold $X^+$ with a curve $C^+\subset X^+$ and with a morphism $f^+:X^+\longrightarrow Y$, which blows down only the curve $C^+$, and with birational, but not biregular, map $g:X\longrightarrow X^+$, which is embedded in the commutative triangle \begin{figure}[h] \hspace*{6cm}\epsffile{flop.eps} \end{figure} Such $X^+$ is unique (see \cite{K}). Birational map $g$ is called flop; $g$ induces isomorphism $X\setminus C\stackrel{\sim}{\rightarrow}X^+\setminus C^+$. Let us remark that the curve $C^+$ also has trivial intersection with $K_{X^+}$ and its normal bundle is of the same kind as the one on $C$. If the curve $C$ has the normal bundle either of the first or of the second kind, i.e., isomorphic to ${\cal O}(-1)\oplus{\cal O}(-1)$ or ${\cal O}\oplus{\cal O}(-2)$, then, following M.~Reid \cite{R}, we call it $(-2)$-curve. We are going to prove that if $X$ and $X^+$ are related by flop with $(-2)$-curve, then $D^b_{coh}(X)$ is equivalent to $D^b_{coh}(X^+)$. This supplies the following {\bf Conjecture.} If two smooth varieties are related by flop, then the derived categories of coherent sheaves on them are equivalent as triangulated categories. {\bf Comment.} By \cite{K} any birational transformation between two 3-dimensional Calabi--Yau varieties can be decomposed in a sequence of flops. Therefore the conjecture would imply equivalence of the derived categories of any two birationally isomorphic 3-dimensional Calabi--Yau's. \vspace{2ex} Let us remark that there exist examples of flops on threefolds with the normal bundle $N_{X/C}={\cal O}(1)\oplus {\cal O}(-3)$. Consider a smooth compact threefold $X$ with a curve $C\cong {\bf CP}^1$, which is a $(-2)$-curve. Then there exist a flop $X\longrightarrow X^+$. It is known an explicit decomposition for it in the so called `pagoda' of M.~Reid: \begin{figure}[h] \hspace*{6cm}\epsffile{pag.eps} \end{figure} Here $X_1$ is a blow-up of $X$ in a curve $C$, $E_1$ the exceptional divisor of this transformation, which is isomorphic to ${\bf F}_2$. The exceptional section $S_1\hookrightarrow E_1$ also is a $(-2)$-curve on $X_1$, so $X_2$ is the blow-up of $X_1$ in $S_1$ and so on. Finally we obtain a threefold $X_n$ with a divisor $E_n\simeq {\bf F}_2$ and with the section $S_n$, such that $N_{X_n/S_n}={\cal O}(-1)\oplus {\cal O}(-1)$. The blow-up of $X_n$ in $S_n$ is a threefold $\widetilde X$ with the exceptional divisor $E\simeq {\bf P}^1\times {\bf P}^1$. Contracting it in the other direction we obtain $X^+_n$. Further, contracting one-by-one the proper transforms of divisors $E_n,\ldots,E_1$ we break our way through to a threefold $X^+$. See detals in the original paper \cite{R}. All maps $X_i\longrightarrow X_i^+$ are flops too. We are going to proceed by induction on the length of the `pagoda'. Denote by $\pi_i$ the birational morphisms $X_{i+1}\longrightarrow X_i$ and by $\Pi_i$ the morphisms $\widetilde X\longrightarrow X_i$ obtained by composition. Similarly (by $\pi^+_i$ and $\Pi^+_i$ ) for the right side of the `pagoda'. By proposition \ref{blow} the pull back functor for a blowing up is full and faithful, therefore the derived categories $D^b_{coh}(X_i)$ and $D^b_{coh}(X^+_i)$ can be identified with the full admissible subcategories in $D^b_{coh}(\widetilde X)$. Denote them $D(X_i)$ and $D(X^+_i)$ respectively. We have two filtrations on $D^b_{coh}(\widetilde X)$: $$D(X_0)\subset D(X_1)\subset\ldots \subset D(X_n)\subset D^b_{coh}(\widetilde X),$$ $$D(X^+_0)\subset D(X^+_1)\subset\ldots \subset D(X^+_n)\subset D^b_{coh}(\widetilde X).$$ Denote ${\cal A}_i$ (resp., ${\cal A}^+_i$ ) the right orthogonal to $D(X_i)$ (resp., $D(X^+_i)$), i.e. ${\cal A}_i=D(X_i)^{\bot},\ {\cal A}^+_i=D(X^+_i)^{\bot}.$ Denote ${\cal B}_i$ the common part of ${\cal A}_i$ and ${\cal A}^+_i $, i.e. the full subcategory, consisting of the objects, which are right orthogonal to $D(X_i)$ and to $D(X^+_i)$. \th{Proposition}\label{asub} In the above notations one has i) ${\cal B}_i$ are admissible subcategories ii) there exists a semiorthogonal decomposition of the categories ${\cal A}_i$ and ${\cal A}^+_i $ in pairs of admissible subcategories $${\cal A}_i=\Bigl\langle{\cal B}_i,{\cal C}_i\Bigr\rangle\qquad{\cal A}^+_i=\Bigl\langle{\cal D}_i,{\cal B}_i\Bigr\rangle ,$$ such that ${\cal D}_i={\cal C}_i\otimes\omega_{\widetilde X}$ (i.e. subcategory ${\cal D}_i$ consists of those objects, which are twists by the canonical sheaf $\omega_{\widetilde X}$ of the objects from ${\cal C}_i$). \par\endgroup \par\noindent{\bf\ Proof. } We use induction by the length of the `pagoda'. The base of the induction is a flop in a curve with normal bundle ${\cal O}(-1)\oplus{\cal O}(-1)$. In our notations we have the subcategories $D(X_n)$ and $D(X^+_n)$. Moreover, we can choose the following decompositions for ${\cal A}_n$ and ${\cal A}^+_n$ (theorem \ref{full}) $${\cal A}_n=\Bigl\langle {\cal O}_E(-l'-l''),{\cal O}_E(-l'')\Bigr\rangle ,$$ $${\cal A}^+_n=\Bigl\langle {\cal O}_E(-l'-2l''),{\cal O}_E(-l'-l'')\Bigr\rangle ,$$ where $l'$ and $l''$ are fibres of the projections of $E$ to $S_n$ and $S^+_n$ respectively. It follows that ${\cal B}_n$ is a subcategory generated by one exceptional object ${\cal O}_E(-l'-l'')$, hence admissible, and ${\cal C}_n$ and ${\cal D}_n$ are also generated by one object ${\cal O}_E(-l'')$ and ${\cal O}_E(-l'2-l'')$ respectively. We have the formula for the restriction of $\omega_{\widetilde X}$ to $E$: $$\omega_{\widetilde X}\left|_E\right.={\cal O}(-l'-l'').$$ It follows that ${\cal D}_n={\cal C}_n\otimes\omega_{\widetilde X}.$ Now suppose that for $i>0$ we have already proved that ${\cal B}_i$ are admissible, and that ${\cal A}_i$ and ${\cal A}_i^+$ have semiorthogonal decompositions $$ {\cal A}_i=\Bigl\langle {\cal B}_i,\; {\cal C}_i\Bigl\rangle\quad\mbox{and }\; {\cal A}_i^+=\Bigl\langle {\cal D}_i,\; {\cal B}_i\Bigl\rangle $$ $$ \mbox{with}\quad{\cal D}_i={\cal C}_i\otimes\omega_{\tilde X}. $$ Let us prove it for $i=0$. Again by theorem \ref{full} we have the decomposition $$ D(X_1)=\Bigl\langle \Pi^*_1{\o {E_1}}(-s_1-2l_1),\; \Pi^*_1{\o {E_1}}(-s_1-l_1), \;D(X_0)\Bigl\rangle, $$ where $s_1$ is the class of the exceptional section of $E_1\simeq{\bf F}_2$ and $l_1$ is a fibre. The decomposition for ${\cal A}_0$ follows: $$ {\cal A}_0=\Bigl\langle {\cal A}_1,\; \Pi^*_1{\o {E_1}}(-s_1-2l_1),\; \Pi^*_1{\o {E_1}}(-s_1-l_1)\Bigl\rangle. $$ Now we shall show that $$ \Pi^*_1{\o {E_1}}(-s_1-2l_1)=\Pi^{+*}_1{\o {E_1^+}}(-s_1^+-2l_1^+). $$ There exists an exact sequence on $X_1$: $$ 0\longrightarrow {\o {E_1}}(-s_1-2l_1)\longrightarrow{\o {E_1}}\longrightarrow{\o \Gamma}\longrightarrow 0, $$ where $\Gamma$ is a curve from the linear system $|s_1+2l_1|$ on $E_1$. The main point here is that $\Gamma\cap S_1=\emptyset$, i.e. $\Gamma$ does not intersect with the locus for the blowing up of $X_1$. Therefore $$ \Pi^*_1{\o {\Gamma}}\cong\Pi^{+*}_1{\o {\Gamma}} $$ (we identify the curve $\Gamma$ with its proper transform on $X^+_1$). Moreover, $\Pi^*_1{\o {E_1}}={\o Z}$, where $Z=\bigcup^n_{i=1}E_i\bigcup E$, i.e. ${\o Z}=\Pi^{+*}_1{\o {E_1^+}}$. Again using the fact that the pull back functor for a blowing up is full and faithful, we immediately obtain \begin{equation}\label{=1} \Pi^*_1{\o {E_1}}(-s_1-2l_1)=\Pi^{+*}_1{\o {E_1^+}}(-s_1^+-2l_1^+). \end{equation} Consider again the decomposition for ${\cal A}_0$: $$ {\cal A}_0=\Bigl\langle {\cal A}_1,\; \Pi^*_1{\o {E_1}}(-s_1-2l_1),\; \Pi^*_1{\o {E_1}}(-s_1-l_1)\Bigl\rangle= $$ $$ =\Bigl\langle {\cal B}_1,\; {\cal C}_1, \Pi^*_1{\o {E_1}}(-s_1-2l_1),\; \Pi^*_1{\o {E_1}}(-s_1-l_1)\Bigl\rangle. $$ For any object $C\subset{\cal C}_1$ one has: $$ {\h C, {\Pi^*_1{\o {E_1}}(-s_1-2l_1)}}\cong{\h \Pi^*_1{\o {E_1}}(-s_1-2l_1), {C\otimes\omega_{\tilde X}[3]}}^*\cong $$ $$ \cong{\h \Pi^{+*}_1{\o {E_1^+}}(-s_1^+-2l_1^+), {C\otimes\omega_{\tilde X}[3]}}^*=0. $$ The last equation is due to the fact that $C\otimes\omega_{\tilde X}\in{\cal D}_1$, i.e. it belongs to $D(X^+_1)^{\perp}$. Therefore, the subcategory ${\cal C}_1$ and the object $ \Pi^*_1{\o {E_1}}(-s_1-2l_1)$ are both sides mutually orthogonal. This means that we can exchange their positions in the semiorthogonal decomposition $$ {\cal A}_0=\Bigl\langle {\cal B}_1,\; \Pi^*_1{\o {E_1}}(-s_1-2l_1),\; {\cal C}_1, \;\Pi^*_1{\o {E_1}}(-s_1-l_1)\Bigl\rangle . $$ It follows from (\ref{=1}) that the object $\Pi^*_1{\o {E_1}}(-s_1-2l_1)$ is orthogonal from the right to both $D(X_0)$ and $D(X_0^+)$. Therefore, we have the semiorthogonal decomposition for ${\cal B}_0$: $$ {\cal B}_0=\Bigl\langle {\cal B}_1,\; \Pi^*_1{\o {E_1}}(-s_1-2l_1) \Bigl\rangle, $$ hence ${\cal B}_0$ is admissible. For ${\cal C}_0$ we have the decomposition: $$ {\cal C}_0=\Bigl\langle {\cal C}_1,\; \Pi^*_1{\o {E_1}}(-s_1-l_1)\Bigl\rangle. $$ Now let us consider a subcategory ${\cal A}_0^+$. We can choose the following decomposition for it $$ {\cal A}_0^+=\Bigl\langle {\cal A}_1^+,\; \Pi^{+*}_1{\o {E_1^+}}(-s_1^+-3l_1^+),\; \Pi^{+*}_1{\o {E_1^+}}(-s_1^+-2l_1^+)\Bigl\rangle= $$ $$ =\Bigl\langle {\cal D}_1,\; {\cal B}_1, \Pi^{+*}_1{\o {E_1^+}}(-s_1^+-3l_1^+),\; \Pi^{+*}_1{\o {E_1^+}}(-s_1^+-2l_1^+)\Bigl\rangle. $$ Now we shall show that \begin{equation}\label{=2} \Pi^{+*}_1{\o {E_1^+}}(-s_1^+-3l_1^+)\cong \Pi^*_1{\o {E_1}}(-s_1-l_1)\otimes\omega_{\tilde X}. \end{equation} It follows from here that: first, the objects ${\cal B}_1$ and $\Pi^{+*}_1{\o {E_1^+}}(-s_1^+-3l_1^+)$ are both sides mutually orthogonal. Therefore, one can exchange their positions in the decomposition for ${\cal A}_0^+$, because for any object $B\in{\cal B}_1$ $$ {\h B, {\Pi^{+*}_1{\o {E_1^+}}(-s_1^+-3l_1^+)}}\cong{\h {\Pi^*_1{\o {E_1}}(-s_1-l_1)[-3]}, B}^*=0. $$ second, for ${\cal D}_0$ one has the decomposition $$ {\cal D}_0=\Bigl\langle {\cal D}_1,\; \Pi^{+*}_1{\o {E_1^+}}(-s_1^+-3l_1^+)\Bigl\rangle. $$ This implies that ${\cal D}_0={\cal C}_0\otimes\omega_{\tilde X}$, because for ${\cal D}_1$ we have such decomposition by induction, hence (\ref{=2}) allows us to claim this for ${\cal D}_0$. Thus, the proof of the proposition follows from the \th{Lemma}\label{congr} In the above notations one has: $$ \Pi^{+*}_1{\o {E_1^+}}(-s_1^+-3l_1^+)\cong \Pi^*_1{\o {E_1}}(-s_1-l_1)\otimes\omega_{\tilde X}. $$ \par\endgroup \par\noindent{\bf\ Proof. } Standart calculations for blow-ups give that the restriction of $\omega_{ X_1}$ to $E_1$ is an invertible sheaf ${\o {E_1}}(-s_1-2l_1)$. Further, $\Pi_1^*\omega_{X_1}\cong\Pi_1^{+*}\omega_{X_1^+}$, because $$ \omega_{\tilde X}\cong\Pi_1^*\omega_{X_1}\otimes{\o {\tilde X}}(2E_2+\cdots 2E_n +E) $$ and $$ \omega_{\tilde X}\cong\Pi_1^{+*}\omega_{X_1^+}\otimes{\o {\tilde X}}(2E_2+\cdots 2E_n +E) $$ (we denote by common letter $E_i$ the exceptional divisor on $X_i$ as well as its proper transforms on $X_{i+1},...,{\widetilde X}$). One has $$ {\o {E_1}}(-s_1-l_1)={\o {E_1}}(l_1)\otimes\omega_{X_1}, $$ $$ {\o {E_1^+}}(-s_1^+-3l_1^+)={\o {E_1^+}}(-l_1^+)\otimes\omega_{X_1^+}. $$ Therefore, it is sufficient to show that $$ \Pi^{+*}_1{\o {E_1^+}}(-l_1^+)\cong \Pi^*_1{\o {E_1}}(l_1)\otimes\omega_{\tilde X}. $$ This fact we shall prove also by induction on the length of the `pagoda'. The base of the induction: Consider $X_n$ and ${\o {E_n}}(l_n)$. Then for $\Pi^*_n{\o {E_n}}(l_n)$ we have the exact sequence: $$ 0\longrightarrow{\o {E_n}}(-s_n+l_n)\longrightarrow\Pi^*_n{\o {E_n}}(l_n)\longrightarrow {\o E}(l')\longrightarrow 0. $$ Let us twist it by $\omega_{\tilde X}$. We know that $$ \omega_{\tilde X}\cong\Pi_n^*\omega_{X_n}\otimes{\o {\tilde X}}(E), $$ and $$ \omega_{X_n}\Bigl|_{E_n}={\o {E_n}}(-s_n-2l_n), {\o {\tilde X}}(E)\Bigl|_{E_n}={\o {E_n}}(s_n), $$ $$ \omega_{\tilde X}={\o E}(-l'-l''). $$ Therefore $$ 0\longrightarrow {\cal O}_{E_n}(-s_n-l_n)\longrightarrow \Pi^*_n {\cal O}_{E_n}(l_n)\otimes\omega_{\widetilde X}\longrightarrow{\cal O}_{E}(-l'')\longrightarrow 0. $$ From the other hand, for $\Pi^{+*}_n {\cal O}_{E^+_n}(-l^+_n)$ we have the short exact sequence $$ 0\longrightarrow {\cal O}_{E^+_n}(-s^+_n-l^+_n)\longrightarrow \Pi^{+*}_n {\cal O}_{E^+_n}(-l^+_n)\longrightarrow{\cal O}_{E}(-l'')\longrightarrow 0. $$ Keeping in mind that $E_n^+=E_n$ on $\widetilde X$ and that ${\rm Ext}^1({\cal O}_E(-l')\;,\; {\cal O}_E(-s_n-l_n))$ is one dimensional we conclude that $$ \Pi_n^*{\cal O}_{E_n}(l_n)\otimes \omega_{\widetilde X}\cong\Pi_n^{+*}{\cal O}_{E^+_n}(-l^+_n). $$ To make one step of the induction is in this case practically the same as to check the base of the induction. Namely, for $\pi_1^*{\cal O}_{E_1}(l_1)$ one has the short exact sequence on $X_2$: $$ 0\longrightarrow {\cal O}_{E_1}(-s_1+l_1)\longrightarrow \pi^{*}_1 {\cal O}_{E_1}(l_1)\longrightarrow{\cal O}_{E_2}(l_2)\longrightarrow 0. $$ Lifting it up to ${\widetilde X}$ and twisting by $\omega_{\widetilde X}$ one obtains: $$ 0\longrightarrow {\cal O}_{E_1}(-s_1-l_1)\longrightarrow \Pi^*_1 {\cal O}_{E_1}(l_1)\otimes \omega_{\widetilde X}\longrightarrow\Pi^*_2{\cal O}_{E_2}(l_2)\otimes\omega_{\widetilde X}\longrightarrow 0. $$ (we use here the fact that $E_1\cap E_2=\emptyset$, for $i>2$). By hypothesis of the induction $$ \Pi_2^*{\cal O}_{E_2}(l_2)\otimes\omega_{\widetilde X}\cong \Pi_2^{+*}{\cal O}_{E^+_2}(-l_2), $$ and the sheaf ${\cal O}_{E_1}(-s_1-l_1)$ coincides with ${\cal O}_{E_1^+}(-s_1^+-l_1^+)$. Using as above that ${\rm Ext}^1(\Pi_2^*{\cal O}_{E_2^+}(-l_2)\;,\; {\cal O}_{E_1^+}(-s_1^+-l^+_1))$ is of dimension 1, we obtain: $$ \Pi_1^*{\cal O}_{E_1}(l_1)\otimes\omega_{\widetilde X}\cong \Pi_1^{+*}{\cal O}_{E^+_1}(-l_1). $$ This proves the lemma and, consequently, proposition \ref{asub}. \th{Theorem}\label{efl} The functor $$ \Pi_*\Pi^{+*}:\db{X^+}\longrightarrow\db{X} $$ is an equivalence of triangulated categories. \par\endgroup \par\noindent{\bf\ Proof. } Let us consider two objects $A, B\in\db{X^+}$, then we have: $$ {\h \Pi_*\Pi^{+*}A, {\Pi_*\Pi^{+*}B}}\cong{\h \Pi^*\Pi_*\Pi^{+*}A, {\Pi^{+*}B}}. $$ There exists an exact triangle: $$ \Pi^*\Pi_*\Pi^{+*}A\longrightarrow\Pi^{+*}A\longrightarrow{\bar A}. $$ Since by proposition \ref{blow} $\Pi_*\Pi^*$ is isomorphic to the identity functor on $\db{X}$, one can easily see that ${\bar A}\in D(X)^{\perp}$. Moreover the group of the homomorphisms from ${\bar A}$ to any object of the subcategory ${\cal B}_0$ is trivial, for so are the groups of the homomorphisms from the other members of the exact triangle. Therefore ${\bar A}\in{\cal C}$. It follows that $$ {\h {\bar A}, {\Pi^{+*}B}}\cong{\h {\Pi^{+*}B}, {{\bar A}\otimes\omega_{\tilde X}[3]}}^*=0, $$ because ${\bar A}\otimes\omega_{\tilde X}\in{\cal D}\subset{D(X^+)}^{\perp}$. Therefore, $$ {\h \Pi_*\Pi^{+*}A, {\Pi_*\Pi^{+*}B}}\cong{\h \Pi^{+*}A, {\Pi^{+*}B}}\cong{\h A, B}. $$ The latter isomorphism is due to fully faithfulness of $\Pi^{+*}$ (by proposition \ref{blow}). This proves that $\Pi_*\Pi^{+*}$ is full and faithful. Now suppose that $\Pi_*\Pi^{+*}$ is not an equivalence. Then $\Pi_*\Pi^{+*}\db{X^+}$ is a full subcategory in $\db{X}$. It is admissible, that is, there exists a non--zero left orthogonal to it. Let $Z\in{}^{\perp}\Pi_*\Pi^{+*}\db{X^+}$ be a non--zero object. Then $$ {\h Z, {\Pi_*\Pi^{+*}A}}=0\quad\mbox{for any }\:A\in\db{X^+}. $$ It follows that $$ {\h {\Pi^{+*}A} , {\Pi^*Z\otimes\omega_{\tilde X}[3]}}^* \cong {\h \Pi^*Z, {\Pi^{+*}A}}\cong{\h Z, {\Pi_*\Pi^{+*}A}}=0, $$ i.e. $\Pi^*Z\otimes\omega_{\tilde X}\in{\cal A}_0^+=\langle{\cal D}_0,\;{\cal B}_0\rangle$. Let $K\in {\cal D}_0$, then $K=K'\otimes\omega_{\tilde X}$ with $K'\in{\cal C}_0$. Further, $$ {\h \Pi^*Z\otimes\omega_{\tilde X}, K}\cong {\h \Pi^*Z, {K'}}=0, $$ as $K'\in{\cal C}_0\subset D({X})^{\perp}$. It follows that $\Pi^*Z\otimes\omega_{\tilde X}\in{\cal B}_0\subset D(X)^{\perp}$, in other words, for every $M\in\db{X}$ $$ {\h \Pi^*M, {\Pi^*Z\otimes\omega_{\tilde X}}}=0. $$ From the other hand, by Serre duality one has: $$ {\h \Pi^*Z[-3], {\Pi^*Z\otimes\omega_{\tilde X}}}\cong{\h \Pi^*Z, {\Pi^*Z}}^*\ne 0. $$ This proves that ${}^{\perp}\Pi_*\Pi^{+*}\db{X^+}$ is zero. Therefore $\Pi_*\Pi^{+*}$is an equivalence of categories. \bigskip {\sc Remark.} The theorem on existence of flip is valid only in the category of Moishezon varieties. Though we have considered here only algebraic varieties, all the same works with minor changes in the Moishezon case. {\section{Reconstruction of a variety from the derived category of coherent sheaves.}} We have seen above that there exist examples of different varieties having equivalent the derived categories of coherent sheaves. Does it mean that $\db{X}$ is a weak invariant of a variety? In this chapter we are going to show that this is not the case. Specifically, we prove that a variety is uniquely determined by its category, if its anticanonical (Fano case) or canonical (general type case) class is ample. In fact, the proof indicates that obstructions to the reconstruction mostly due to (partial) triviality of the canonical class. The idea is that for good, in the above sense, varieties we can recognize the one--dimensional skyskraper sheaves in $\db{X}$, using nothing but the triangulated structure of the category. The main tool for this is the Serre functor (see ch.2). Let $D$ be a k--linear triangulated category. Denote by $F_D$ the Serre functor in $D$ (in case it exists). Recall, that if $D=\db{X}$, where $X$ is an algebraic variety of dimension $n$, then by Serre--Grothendieck duality: \begin{equation}\label{Ser} F_{\db{X}}(\cdot)= (\cdot)\otimes \omega_X[n], \end{equation} where $\omega_X$ is the canonical sheaf on $X$. \th{Definition}\label{po} An object $P\in D$ is called {\sf point object} of codimention $s$, if $$ \begin{array}{ll} i)& F_D(P)\simeq P[s],\\ ii)& {\rm Hom}^{<0}(P\:,\; P)=0,\\ iii)& {\rm Hom}^{0}(P\:,\; P)=k. \end{array} $$ \par\endgroup \th{Proposition}\label{rp} Let $X$ be a smooth algebraic variety of dimension $n$ with the ample canonical or anticanonical sheaf. Then an object $P\in \db{X}$ is a point object, iff $P\cong{\o x}[r]$ is isomorphic (up to translation) to a one-dimensional skyscraper sheaf of a closed point $x\in X$. \par\endgroup {\sc Remark.} Since $X$ has an ample invertible sheaf it is projective. \par\noindent{\bf\ Proof. } Any one--dimensional skyscraper sheaf obviously satisfies properties of a point object of the same codimension as the dimension of the variety. Suppose now that for some object $P\in\db{X}$ properties i)--iii) of definition \ref{po} are verified. Let ${\cal H}^i$ are cohomology sheaves of $P$. It immediately follows from i) that $s=n$ and ${\cal H}^i\otimes \omega_X={\cal H}^i$. Since $\omega_X$ is an ample or antiample sheaf, we conclude that ${\cal H}^i$ are finite length sheaves, i.e. their support are isolated points. Sheaves with the support in different points are homologically orthogonal, therefore any such object decomposes into direct some of those having the support of all cohomology sheaves in a single point. By iii) the object $P$ is indecomposable. Now consider the spectral sequence, which calculates ${\rm Hom}^m( P\:, \; P)$ by ${\rm Ext}^i( {\cal H}^j\:,\; {\cal H}^k)$: $$ E^{p,q}_2= \bigoplus_{k-j=q}{\rm Ext}^p({\cal H}^j\:,\; {\cal H}^k) \Longrightarrow {\rm Hom}^m( P\:, \; P). $$ Let us mention that for any two finite length sheaves having the same single point as their support, there exists a non--trivial homomorphism from one to the other, which sends generators of the first one to the socle of the second. Considering ${\rm Hom}^m({\cal H}^j \:, \; {\cal H}^k)$ with minimal $k-j$, we observe that this non--trivial space survives at $E_{\infty}$, hence by ii) $k-j=0$. That means that all but one cohomology sheaves are zero. Moreover, iii) implies that this sheaf is a one--dimensional skyscraper. This concludes the proof. Now having the skyscrapers we are able to reconstruct the invertible sheaves. \th{Definition}\label{inv} An {\sf object} $l\in D$ is called {\sf invertible} if for any point object $P\in D$ there exists $s\in {\bf Z}$ such that $$ \begin{array}{lll} i)& {\rm Hom}^s(L\:,\; P)=k,&\\ ii)& {\rm Hom}^i(L\:,\; P)=0, & \quad\mbox{for}\;i\ne s\\ \end{array} $$ \par\endgroup \th{Proposition}\label{rin} Let $X$ be a smooth irreducible algebraic variety. Assume that all point objects have the form ${\o x}[s]$ for some $x\in X, s\in {\bf Z}$. Then an object $L\in D$ is invertible, iff $L\cong {\cal L}[p]$ for some linear vector bundle ${\cal L}$ on $X$. \par\endgroup \par\noindent{\bf\ Proof. } For a linear bundle ${\cal L}$ we have: $$ {\rm Hom}({\cal L}\:, \;{\o x})=k, \quad{\rm Ext}^i({\cal L}\:, \; {\o x})=0 \;\mbox{if }\:i\ne 0. $$ Therefore, if $L={\cal L}[s]$, then it is an invertible object. Now let ${\cal H}^i$ are the cohomology sheaves for an invertible object ${\cal L}$. Consider the spectral sequence, which calculates ${\rm Hom}^.({\cal L}\:, \;{\o x})$ for a point $x\in X$ by means of ${\rm Hom}^i({\cal H}^j\:, \;{\o x})$: $$ {\rm Hom}^p({\cal H}^q\:, \;{\o x})\Longrightarrow{\rm Ext}^{p-q}({\cal L}\:, \;{\o x}). $$ Let ${\cal H}^{q_0}$ be the non--zero cohomology sheaf with maximal index. Then for any point $x\in X$ from the support of ${\cal H}^{q_0}$ ${\rm Hom}({\cal H}^{q_0}\:, \;{\o x})\ne 0$. But both ${\rm Hom}({\cal H}^{q_0}\:, \;{\o x})$ and ${\rm Ext}^1({\cal H}^{q_0}\:, \;{\o x})$ are intact by differential of the spectral sequence. Therefore, by definition of an invertible object we obtain that for any point $x$ from the support of ${\cal H}^{q_0}$ $$ \begin{array}{ll} a)& {\rm Hom}({\cal H}^{q_0}\:,\;{\o x})=k,\\ b)& {\rm Ext}^1({\cal H}^{q_0}\:,\;{\o x})=0. \end{array} $$ Since $X$ is smooth and connected it follows the ${\cal H}^{q_0}$ is a locally free one dimensional sheaf on $X$. This implies that ${\rm Ext}^i({\cal H}^{q_0}\:,\;{\o x})=0$ for $i>0$ and ${\rm Hom}({\cal H}^{q_0-1}\:,\;{\o x})$ are intact by differentials of the spectral sequence. This means that ${\rm Hom}({\cal H}^{q_0-1}\:,\;{\o x})=0$, for any $x\in X$, i.e. ${\cal H}^{q_0-1}=0$. Repeting this argument for ${\cal H}^q$ with smaller $q$, we easily see that all ${\cal H}^q$, except $q=q_0$, are zero. This proves the proposition. Now we are ready to prove the reconstruction theorem. Linear bundles help us to `glue' points together. \th{Theorem}\label{rec} Let $X$ be a smooth irreducible projective variety with ample canonical or anticanonical sheaf. If $D=\db{X}$ is equivalent as a triangulated category to $\db{X'}$ for some other smooth algebraic variety $X'$, then $X$ is isomorphic to $X'$. \par\endgroup This theorem is stronger than just a reconstruction for a variety with ample canonical or anticanonical sheaf from its derived category. One have to be careful: since $X'$ might not have ample canonical or anticanonical sheaf, the situation is not symmetric with respect to $X$ and $X'$. We divide the proof in several steps, so that the reconstruction procedure was transparent. \par\noindent{\bf\ Proof. } Step 1. Denote ${\cal P}_D$ the set of isomorphism classes of the point objects in $D$, ${\cal P}_X$ the set of isomorphism classes of objects in $\db{X}$ $$ {\cal P}_X:=\Bigl\{ {\o x}[k]\;\Bigl |\: x\in X, k\in {\bf Z}\Bigl\}. $$ By proposition \ref{rp} ${\cal P}_D\cong{\cal P}_X$. Obviously, ${\cal P}_X'\subset {\cal P}_D$. Suppose that there is an object $P\subset {\cal P}_D$, which is not contained in ${\cal P}_X'$. Since ${\cal P}_D\cong{\cal P}_X$, any two objects in ${\cal P}_D$ either are homologically mutually orthogonal or belong to a common orbit with respect to the translation functor. It follows that $P\in\db{X'}$ is orthogonal to any skyscraper sheaf ${\cal O}_{x'}, x'\in X'$. Hence $P$ is zero. Therefore, ${\cal P}_X'= {\cal P}_D= {\cal P}_X$. Step 2. Denote by ${\cal L}_D$ the set of isomorphism classes of invertible objects in $D$, ${\cal L}_X$ the set of isomorphism classes of objects in $\db{X}$ $$ {\cal L}_X:=\Bigl\{ L[k]\;\Bigl |\: L\:\mbox{is linear bundle on } X , k\in {\bf Z}\Bigl\}. $$ By step 1 both varieties $X$ and $X'$ satisfy the assumptions of proposition \ref{rin}. It follows that ${\cal L}_X= {\cal L}_D= {\cal L}_X'$. Step 3. Let us fix some invertible object $L_0$ in $D$, which is a linear bundle on $X$. By step 2 $L_0$ can be considered, up to translation, as a linear bundle on $X'$. Moreover, changing if necessary, the equivalence $\db{X}\simeq\db{X'}$, by the translation functor, we can assume that $L_0$, regarded as an object on $X'$, is a genuine linear bundle. Obviously, by step 1 the set $p_D\subset P_D$ $$ p_D:=\Bigl\{ P\in P_D \;\Bigl |\: {\h L, P}=k\Bigl\} $$ coincides with both sets $p_X=\{{\o x}, x\in X\}$ and $p_X'=\{{\cal O}_{x'}, x'\in X'\}$. This gives us a pointwise identification of $X$ with $X'$. Step 4. Let now $l_X$ (resp., $l_X'$) be the subset in ${\cal L}_D$ of linear bundles on $X$ (resp., on $X'$). They can be recognized from the triangulated structure of $D$ as follows: $$ l_X'=l_X=l_D:=\Bigl\{ L\in {\cal L}\;\Bigl |\:{\h L, P}=k \:\mbox{for any } P\in p_D \Bigl\}. $$ For $\alpha\in{\h L_1, {L_2}}$, where $L_1, L_2\in l_D$, and $P\in p_D$, denote by $\alpha^*_P$ the induced morphism: $$ \alpha^*_P: {\h L_2, P}\longrightarrow {\h L_1, P}, $$ and by $U_{\alpha}$ the subset of those objects $P$ from $p_D$ for which $\alpha^*_P\ne 0$. By \cite{Il} any algebraic variety has an ample system of linear bundles. This means that $U_{\alpha}$, for all ${\alpha}, L_1, L_2$, gives a base for the Zariski topologies on both $X$ and $X'$. This means that the topologies on $X$ and $X'$ coincide. Step 5. Since codimension of all point objects are equal to dimension of $X$ and of $X'$, we have $dimX=dimX'$. Then, formula (\ref{Ser}) for the Serre functor shows that the operations of twisting by the canonical sheaf of $X$ and $X'$ induce equal transformations on the set $l_D$. Let $L_i=F^iL_0[-ni]$. Then $\{L_i\}$ is the orbit of $L_0$ with respect to twisting by the canonical sheaf. Since $\omega_X$ is ample or antiample, the set of all $U_{\alpha}$, where ${\alpha}$ runs all elements from ${\h L_i, {L_j}}, i,j\in {\bf Z}$, is the base of the Zariski topology on $X$, hence, by step 4, on $X'$. That means that canonical sheaf of $X'$ is also ample or, respectively, antiample. This means that if we consider the graded algebra $A$ with graded components $$ A_i={\h L_0, {L_i}} $$ and with obvious ring structure, then ${\bf Proj}A=X=X'$. This finishes the proof. The problem of reconstructing of a variety from its derived category is closely related to the problem of computing the group of auto-equivalences for $\db{X}$. For ample canonical or anticanonical class we have the following \th{Theorem}\label{aut} Let $X$ be a smooth irreducible projective variety with ample canonical or anticanonical class. Then the group of isomorphism classes of exact auto-equivalences $\db{X} \to \db{X} $ is generated by the automorphisms of the variety, the twists by linear bundles and the translations. \par\endgroup \par\noindent{\bf\ Proof. } Assume for definiteness that the canonical class is ample. Let us look more carefully at the proof of theorem \ref{rec} for the case $X=X'$. At step 3 we can choose $L_{0}=\cal{O}$, and using twists by linear bundles and translations, we can assume that our functor takes $\cal{O}$ to $\cal{O}$. Then, step 5 gives us an automorphism of the canonical ring. Since the canonical class is ample, automorphisms of the ring are in one-to-one correspondence with those of the variety. Therefore, composing our functor with an automorphism of the variety, we can assume that it induces the trivial automorphism of the canonical ring. Thus we have a functor, which takes the trivial linear bundle and any power of the canonical bundle to themselves and preserves homomorphisms between all these bundles. Such a functor is isomorphic to the identity functor. Indeed, it preserves the abelian subcategory of pure sheaves, because the sheaves can be characterized as the objects having trivial ${\rm Hom}^{i}$, for $i\ne 0$, from a sufficiently negative power of the canonical sheaf. Any sheaf has a resolution by direct sums of powers of the canonical class. Our functor takes such a resolution to isomorphic one, i.e. any sheaf goes to isomorphic one. Since the sheaves generate the derived category, we are done. The problem of computing the group of auto-equivalences for the case of non-ample canonical or anticanonical class seems to be of considerable interest.

9506

alg-geom/9506019

en

https://arxiv.org/abs/alg-geom/9506019

alg-geom/9506019

Lothar Goettsche

Geir Ellingsrud and Lothar G\"ottsche

Wall-crossing formulas, Bott residue formula and the Donaldson invariants of rational surfaces

We correct some missing attributions and citations. In particular this applies to the cited paper of Kotschick and Lisca "Instanton invariants via topology", which contains some ideas which have been important for this work. AMSLaTeX

We study the Donaldson invariants of rational surfaces and their dependence on the chambers in the ample cone. We build on a previous joint paper in which we have expressed the change of the Donaldson invariants on an algebraic surface $S$ under crossing a so-called good wall in terms of certain intersection numbers on Hilbert schemes of points on $S$. In the current paper we assume that $S$ is rational. Thish enables us to apply the Bott residue theorem to evaluate these intersection numbers with the help of a computer. Combining this with the blowup formulas we obtain an algorithm for computing the Donaldson invariants of all rational surfaces in almost all chambers. In particular we compute all the $SU(2)$- and $SO(3)$-Donaldson invariants of the projective plane of degree at most 50.

alg-geom

P(n){P(n)} \def{\text{\rom{\bf b}}}{{\text{\rom{\bf b}}}} \defS^{(\bb)}{S^{({\text{\rom{\bf b}}})}} \def{\text{\rom{\bf E}}}{{\text{\rom{\bf E}}}} \def\Inc#1{Z_{#1}} \def\inc#1{\zeta_{#1}} \def\boh#1{pt_{#1} \def\Boh#1{Pt_{#1}} \def\mah#1{al_{#1} \def\Mah#1{Al_{#1}} \def\pr#1{W_{#1}} \def\alpha{\alpha} \def\Gamma{\Gamma} \def\bar \Gamma{\bar \Gamma} \def\gamma{\gamma} \def\bar \eta{\bar \eta} \def{\cal Z}{{\cal Z}} \def{pt}{{pt}} \def\alpha{\alpha} \def{S^{(n)}}{{S^{(n)}}} \def{S^{(m)}}{{S^{(m)}}} \def\<{\langle} \def\>{\rangle} \def{\text{\rom{Hilb}}}{{\text{\rom{Hilb}}}} \def{\cal W}{{\cal W}} \def{\hbox{\rom{Tor}}}{{\hbox{\rom{Tor}}}} \def{\hbox{\rom{Ext}}}{{\hbox{\rom{Ext}}}} \def{\text{\rm {Sym}}}{{\text{\rm {Sym}}}} \def\stil#1{\tilde S^#1} \def\pi{\pi} \defg{g} \def\phi{\varphi} \def\tilde{\tilde} \def{\bar\al}{{\bar\alpha}} \begin{document} \title[Donaldson invariants of rational surfaces] {Wall-crossing formulas, Bott residue formula and the Donaldson invariants of rational surfaces} \author{Geir Ellingsrud} \address{Mathematical Institute\\University of Oslo\{\Bbb P}.~O.~Box~1053\\ N--0316 Oslo, Norway} \email{ellingsr@@math.uio.no} \keywords{Moduli spaces, Donaldson invariants, Hilbert scheme of points} \author{Lothar G\"ottsche} \address{Max--Planck--Institut f\"ur Mathematik\\Gottfried--Claren--Stra\ss e 26\\ D-53225 Bonn, Germany} \email{lothar@@mpim-bonn.mpg.de} \maketitle\ \section{Introduction} The Donaldson invariants of a smooth $4$ manifold $M$ depend by definition on the choice of a Riemannian metric. In case $b^+(M)>1$ they however turn out to be independent of the metric as long as it is generic, and thus they give $C^\infty$-invariants of $M$. In case $b_+(M)=1$ the invariants have been introduced and studied by Kotschick in [Ko]. It turns out that the positive cone of $M$ has a chamber structure, and Kotschick and Morgan show in \cite{K-M} that the invariants only depend on the chamber of the period point of the metric. Now let $S$ be a smooth algebraic surface with geometric genus $p_g(S)=0$, irregularity $q(S)=0$, and let $H$ be an ample divisor on $S$. Let $M^S_H(c_1,c_2)$ be the moduli space of $H$-Gieseker semistable rank $2$ sheaves on $S$ with Chern classes $c_1$ and $c_2$. In the recent paper \cite{E-G} we studied the variation of $M^S_H(c_1,c_2)$ and that of the corresponding Donaldson invariants under change of the ample divisor $H$. For the Donaldson invariants this corresponds to restricting our attention from the positive cone of $S$ to the subcone of ample classes. We imposed a suitable additional condition on the walls between two chambers and called walls satisfying this condition good walls. We showed that if the polarisation $H$ passes through a good wall $W$ defined by a cohomology class $\xi\in H^2(S,{\Bbb Z})$, then $M^S_H(c_1,c_2)$ changes by a number of flips. Following \cite{K-M} we wrote the change of the degree $N$ Donaldson invariant as a sum of contributions $\delta_{\xi,N}$ with $\xi$ running through the set of cohomology classes defining $W$. We then used our flip description to compute $\delta_{\xi,N}$ in terms of Segre classes of certain standard bundles ${\cal V}_{\xi,N}$ over a Hilbert scheme of points ${\text{\rom{Hilb}}}^{d_{\xi,N}}(S\sqcup S)$ on two disjoint copies of $S$ (here $d_{\xi,N}=(N+3+\xi^2)/4$). We proceeded to compute the leading terms of $\delta_{\xi,N}$ explicitely and formulated a conjecture about the precise shape of $\delta_{\xi,N}$, related to a conjecture from \cite{K-M}. We will in future refer to any formula for $\delta_{\xi,N}$ as a wall-crossing formula. Most of the results of \cite{E-G} were also obtained independently in \cite{F-Q}, and a flip description of the change of the moduli spaces was obtained independently for varieties of arbitrary dimension and sheaves of arbitrary rank in \cite{M-W}. In \cite{H-P} a Feynman path integral aproach to this problem is developed, and some of the leading terms of the wall-crossing formulas are determined. The current paper is a continuation of \cite{E-G}. We specialize to the case that the surface $S$ is rational. The first advantage is that now almost always all walls are good and so the formulas from \cite{E-G} almost always apply. The main reason for restricting our attention to rational surfaces is that they allow us to use an additional powerful tool: the Bott residue formula. A rational surface can always be deformed to a surface admitting an action of a two-dimensional algebraic torus $\Gamma$ with only a finite number of fixpoints. As the Donaldson invariants are in particular deformation invariants, we can assume that $S$ admits such an action of $\Gamma$. It is easy to see that this action will lift to the Hilbert schemes ${\text{\rom{Hilb}}}^{d_{\xi,N}}(S\sqcup S)$, and that the standard bundles ${\cal V}_{\xi,N}$ are equivariant for the induced action. Furthermore also the induced action will only have a finite number of fixpoints, and the same is true for a general $1$-parameter subgroup $T$ of $\Gamma$. The weights of the action of $T$ on the tangent spaces of ${\text{\rom{Hilb}}}^{d_{\xi,N}}(S\sqcup S)$ and on the fibres of ${\cal V}_{\xi,N}$ at the fixpoints can be determined explicitely from the corresponding weights on $S$. So we can apply the Bott residue formula to this situation and, given $N$ and $\xi$ and the weights on $S$, we always have an algorithm to compute the change $\delta_{\xi,N}$ explicitely. This algorithm involves very many computations, so we use a suitable Maple program. Now let $S$ be a rational ruled surface with projection $t:S\longrightarrow {\Bbb P}_1$. Let $F$ be the class of a fibre of $t$ and assume that the intersection number $c_1.F$ is $1$. Then \cite{Q2} shows that, given $c_2\in H^2(S,{\Bbb Z})$, there always exists a special chamber $\cal C_0$ such that for $H$ in $\cal C_0$ the moduli space $M^S_H(c_1,c_2)$ is empty. In particular the corresponding $SO(3)$-invariant is zero on $\cal C_0$. This already gives us an algorithm for computing all the $SO(3)$-invariants corresponding to first Chern classes $c_1$ with $c_1.F=1$ on $S$. Given a chamber $\cal C$ we obtain the value of the invariant by just summing up all the changes for all the walls between $\cal C$ and $\cal C_0$. At this point we can combine our methods with an additional ingredient: The blowup formulas, which relate the Donaldson invariants of an algebraic surface $S$ with those of the blowup $\widehat S$ of $S$ in a point. In the case of the projective plane ${\Bbb P}_2$ we obtain an algorithm for computing all the $SO(3)$ and $SU(2)$-invariants. Let $\rho:\widehat {\Bbb P}_2\longrightarrow {\Bbb P}_2$ be the blow up of ${\Bbb P}_2$ in a point, let $H,F$ and $E$ be the hyperplane class, the fibre of the projection $\widehat{\Bbb P}_2\longrightarrow {\Bbb P}_1$ and the exceptional divisor respectively. We obtain the $SO(3)$-invariants of ${\Bbb P}_2$ by first computing the invariants on $\widehat {\Bbb P}_2$ corresponding to $c_1=\rho^*(H)$ and applying the $SU(2)$-blowup formulas. Similarly we obtain the the $SU(2)$-invariants of ${\Bbb P}_2$ by first computing those on $\widehat {\Bbb P}_2$ corresponding to $c_1=E$ and applying the $SO(3)$-blowup formulas. Notice that in both cases $c_1.F=1$ on $\widehat {\Bbb P}_2$, so that the algorithm of the previous paragraph applies. Using a suitable Maple program we have computed all the $SO(3)$- and $SU(2)$-invariants of ${\Bbb P}_2$ of degree smaller then $50$. $SO(3)$- and $SU(2)$-invariants of ${\Bbb P}_2$ and rational ruled surfaces had already been computed by several authors (see e.g. \cite{L-Q} \cite{E-LP-S} and \cite{K-L}) using a variety of methods. In \cite{K-L} Kotschick and Lisca have already made use of the blowup formulas in combination with the wall-crossing formulas. Their computations also involve for the first time the $4$-dimensional class. Their results agree with ours up to diffenent conventions. Our paper is partially motivated by and built on \cite{K-L}. In particular we found there the correct formulation and the references for the blowup formulas in the case $b_+=1$. We then go back to the wall-crossing formulas. Assuming the conjecture from \cite{E-G} about the shape of $\delta_{\xi,N}$ we are able to determine (again with a suitable Maple program) the first $5$ leading terms of $\delta_{\xi,N}$ and using an additional conjecture even the first $7$ leading terms. Furthermore, again using the conjecture, we determine $\delta_{\xi,N}$ on a rational ruled surface for $d_{\xi,N}\le 8$. By explicitely determining the corresponding $\delta_{\xi,N}$ we show that on a rational ruled surface the conjecture and all the formulas are correct for all walls $\xi$ and all $N$ with $N\le 40$ and $d_{\xi,N}\le 8$. Now we compute the Donaldson invariants for rational ruled surfaces $S$ by again combining the wall-crossing formulas with the blowup formulas. We apply this algorithm to compute all the invariants on $S$ of degree at most $35$. The result shows that the special chamber $\cal C_0$, where the invariants corresponding to first Chern class $c_1$ with $c_1.F=1$ vanish, is also special for all other $c_1$. We obtain that in the chamber $\cal C_0$ the Donaldson invariants can be expressed as a polynomial in the linear form $L_F$ defined by $F$ and the quadratic form $q_S$. This polynomial is independent of $S$, and there is a simple relationship between the polynomials for different $c_1$. Finally we observe that by combining the results obtained so far with the blowup formulas, we obtain an algorithm for computing all the $SO(3)$ and $SU(2)$-invariants for all rational surfaces $S$ for all polarisations in a reasonably big part of the ample cone of $S$. This can be seen as a generalization of the result of \cite{K-L} that the Donaldson invariants of ${\Bbb P}_2$ and ${\Bbb P}_1\times {\Bbb P}_1$ are determined by the wall-crossing formulas on some blowups. The explicit computations of the wall-crossing formulas and the Donaldson invariants of rational surfaces gives us a lot of empirical data about the shape of these invariants. We have therefore tried to find some patterns in the results and so the paper also contains a number of conjectures and questions. Several of these can already be motivated by the results of \cite{K-L}. We would like to thank Dieter Kotschick for sending us the preprint \cite{K-L}, which was quite important for our work, and also for some useful comments. Furthermore the second author would like to thank S.A. Str\o mme for a sample Maple program for computations on Hilbert schemes of points. \section{Background material} In this paper let $S$ be a rational surface over ${\Bbb C}$. For such a surface the natural map from the group of divisors modulo rational equivalence to $H^2(S,{\Bbb Z})$ is an isomorphism. So, for $\xi\in H^2(S,{\Bbb Z})$, we will often write ${\cal O}_S(\xi)$ for the line bundle associated to a divisor with class $\xi$. For a polarization $H$ of $S$ we denote by $M^S_H(c_1,c_2)$ the moduli space of torsion-free sheaves $E$ on $S$ which are H-semistable (in the sense of Gieseker and Maruyama) of rank $2$ with $c_1(E)=c_1$ and $c_2(E)=c_2$. \begin{nota} For a sheaf ${\cal F}$ on a scheme $X$ and a divisor $D$ let ${\cal F}(D):={\cal F}\otimes {\cal O}_X(D)$. If $X$ is a smooth variety of dimension $n$, we denote the cup product of two elements $\alpha$ and $\beta$ in $H^*(X,{\Bbb Z})$ by $\alpha\cdot\beta$ and the degree of a class $\alpha\in H^{2n}(X,{\Bbb Z})$ by $\int_X\alpha$. For $\alpha,\beta\in H^2(S,{\Bbb Z})$ let $\<\alpha\cdot\beta\>:=\int_S\alpha\cdot\beta$. We write $\alpha^2$ for $\<\alpha\cdot\alpha\>$ and, for $\gamma\in H^2(S,{\Bbb Z})$, we put $\<\alpha,\gamma\>:=\<\alpha\cdot \check\gamma\>$, where $\check\gamma$ is the Poincar\'e dual of $\gamma$. We denote by $q_S$ the quadratic form on $H_2(S,{\Bbb Z})$ and, for a class $\eta\in H^2(S,{\Bbb Q})$ by $L_\eta$ the corresponding linear form on $H_2(S,{\Bbb Q})$. \end{nota} \begin{conve} \label{convent} When we are considering surfaces $S$ and $X$ with a morphism $f:X\longrightarrow S$, that is either canonical or clear from the context, then for a cohomology class $\alpha\in H^*(Y,{\Bbb Z})$ (or a line bundle $L$ on $Y$) we will very often also denote the pull-back via $f$ by $\alpha$ (resp. $L$). (Very often $f$ will be a sequence of blowups. In particular if $X$ is a surface which is obtained by ${\Bbb P}_2$ by a number of blowups, then we denote by $H$ the pullback of the hyperplane class. Similarly on ${\Bbb P}_1\times {\Bbb P}_1$ or a variety obtained from ${\Bbb P}_1\times {\Bbb P}_1$ by a number of blowups, we denote by $F$ and $G$ the classes of the fibres of the projections to the two factors.) \end{conve} \bigskip \subsection{Walls and chambers} (see \cite{Q1}, \cite{Q2}, \cite{Go}, \cite{K-M} and \cite{E-G}.) \begin{defn}\label{defwall} Let $C_S$ be the ample cone in $H^2(S, {\Bbb R})$. For $\xi\in H^2(S,{\Bbb Z})$ let $$W^\xi:=C_S\cap\big \{ x\in H^2(S, {\Bbb R}) \bigm| \<x\cdot\xi\>=0\big\}.$$ We shall call $W^\xi$ a wall of type $(c_1,c_2)$, and say that it is defined by $\xi$ if the following conditions are satisfied: \begin{enumerate} \item $\xi+c_1$ is divisible by $2$ in $NS(S)$, \item $c_1^2-4c_2\le \xi^2<0$, \item there is a polarisation $H$ with $\< H\cdot \xi\>=0$. \end{enumerate} In particular $d_{\xi,N}:= (4c_2-c_1^2+\xi^2)/4$ is a nonnegative integer. An ample divisor $H$ is said to lie in the wall $W$ if $[H]\in W$. If $D$ is a divisor with $[D]=\xi$, we will also say that $D$ defines the wall $W$. A {\it chamber} of type $(c_1,c_2)$ or simply a chamber, is a connected component of the complement of the union of all the walls of type $(c_1,c_2)$. We will call a wall $W$ {\it good}, if $D+K_S$ is not effective for any divisor $D$ defining the wall $W$. If $(c_1,c_2)$ are given, we call a polarization $L$ of $S$ generic if it does not lie on a wall of type $(c_1,c_2)$. On a rational surface $S$ we will call a divisor $L$ {\it good} if $\<L\cdot K_S\><0$, and we denote by $C_{S,g}$ the real cone of all good ample divisors. We see that any wall $W$ intersecting $C_{S,g}$ is a good wall. Let $L_-$ and $L_+$ be two divisors on $S$. We denote by $W_{(c_1,c_2)}(L_-,L_+)$ the set of all $\xi\in H^2(S,{\Bbb Z})$ defining a wall of type $(c_1,c_2)$ and satisfying $\<\xi\cdot L_-\><0<\<\xi\cdot L_+\>$. We notice that for $L_-$ and $L_+$ good all the walls $W^\xi$ defined by $\xi\in W_{(c_1,c_2)}(L_-,L_+)$ are good. \end{defn} \subsection {The change of the Donaldson invariants in terms of Hilbert schemes} In [Ko] the Donaldson invariants have been introduced for $4$-manifolds $M$ with $b_+(M)=1$. In [K-M] it has been shown that in case $b_+(M)=1$, $b_1(M)=0$ they depend only on the chamber of the period point of the metric in the positive cone of $H^2(M,{\Bbb R})$. We want to use conventions from algebraic geometry, which differ by a sign from the usual conventions for Donaldson invariants and furthermore by a factor of a power of $2$ from the conventions of \cite{Ko}. \begin{nota} Let $S$ be a simply connected algebraic surface with $p_g(S)=0$. Let $N:=4c_2-c_1^2-3$ be a nonnegative integer. We denote by $A_N(S)$ the set of polynomials of weight $N$ on $H_{2}(S,{\Bbb Q})\oplus H_{0}(S,{\Bbb Q})$, where we give weight $2-i$ to a class in $H_{2i}(S,{\Bbb Q})$. Let $\gamma^S_{c_1,N,g}$ be the Donaldson polynomial of degree $N$ with respect to a generic Riemannian metric $g$ associated to the principal $SO(3)$-bundle $P$ on $S$ whose second Stiefel-Whitney class $w_2(P)$ is the reduction of $c_1$ mod $2$ (in the conventions of e.g. \cite{F-S}). Then $\gamma^S_{c_1,N,g}$ is a linear map $A_N(S)\longrightarrow {\Bbb Q}$. If $N$ is not congruent to $-c_1^2-3$ modulo $4$, then by definition $\gamma^S_{c_1,N,g}=0$. If $g$ is the Fubini-Studi metric associated to generic ample divisor $L$ on $S$ we denote $\Phi^{S,L}_{c_1,N}:=(-1)^{(c_1^2+\<c_1\cdot K_S \>)/2}\gamma^S_{c_1,N,g}$. We denote $\Phi^{S,L}_{c_1}:=\sum_{N\ge 0} \Phi^{S,L}_{c_1,N}$. We denote by ${pt}\in H_0(S,{\Bbb Z})$ the class of a point. Sometimes we will consider the Donaldson invariants as polynomials on $H_2(S,{\Bbb Q})$ by putting $\Phi^{S,L}_{c_1,N,r}(\alpha):=\Phi^{S,L}_{c_1,N}({pt}^r\alpha^{N-2r})$ for $\alpha\in H_2(S,{\Bbb Q})$. \end{nota} If the moduli space $M_L(c_1,c_2)$ fulfills certain properties (in particular there is a universal sheaf ${\cal U}$ over $S\times M_H(c_1,c_2)$), then for $\alpha_1,\ldots \alpha_r\in H_{2i}(S,{\Bbb Q})$ we have $$\Phi^{S,L}_{c_1,N}(\alpha_1\ldots\alpha_r)=\int_{M_L(c_1,c_2)} \nu(\alpha_1)\cdot\ldots\cdot\nu(\alpha_r)$$ where $\nu(\alpha)=(c_2({\cal U})-c_1^2({\cal U})/4)/\alpha$ (\cite{Mo}, \cite{Li}).) We will use a result from \cite{E-G} (also proved independently in \cite{F-Q}), We state it only for rational surfaces. Note that there are some changes in notation. \begin{defn} \label{wallchange} Let $\xi\in H^2(S,{\Bbb Z})$ be a class defining a good wall of type $(c_1,c_2)$. For $N:=4c_2-c_1^2-3$ we denote $d_{\xi,N}:=(N+3+\xi^2)/4$, $e_{\xi,N}:=-\<\xi\cdot(\xi-K_S)\>/2+d_{\xi,N}+1$. Assume now that $(c_1,c_2)$ are fixed. Let $$T_\xi:={\text{\rom{Hilb}}}^{d_{\xi,N}}(S\sqcup S)= \coprod_{n+m=d_{\xi,N}}{\text{\rom{Hilb}}}^n(S)\times {\text{\rom{Hilb}}}^m(S).$$ be the Hilbert scheme of $d$ points on $2$ disjoint copies of $S$. Let $q:S\times T_\xi\longrightarrow T_\xi$ and $p:S\times T_\xi\longrightarrow T_\xi$ be the projections. Let $V_\xi$ be the sheaf $p^*({\cal O}_S(-\xi)\oplus{\cal O}_S(-\xi+K_S))$ on $S\times T_\xi$. Let ${\cal Z}_{1}$ (resp.${\cal Z}_{2}$) be the subscheme of $S\times T_\xi$ which restricted to each component $S\times{\text{\rom{Hilb}}}^n(S)\times {\text{\rom{Hilb}}}^m(S)$ is the pullback of the universal subscheme $Z_n(S)$ (resp. $Z_m(S)$) from the first and second (resp. first and third) factor. Let ${\cal I}_{{\cal Z}_{1}}$, ${\cal I}_{{\cal Z}_{2}}$ be the corresponding ideal sheaves and $[{\cal Z}_1]$ and $[{\cal Z}_2]$ their cohomology classes. For $\alpha\in H_i(S,{\Bbb Q})$ let $\widetilde\alpha:=([{\cal Z}_1]+[{\cal Z}_2])/\alpha\in H^{4-i}(T_\xi,{\Bbb Q})$ Then for $\alpha=\alpha_1\cdot\ldots\cdot\alpha_{N-2r}{pt}^r\in A_N(S)$ (with $\alpha_i\in H_2(S,{\Bbb Q})$) we put \begin{eqnarray*} \delta_{\xi,N}(\alpha)&:=&\int\limits_{T_\xi}\left(\left(\prod_{i=1}^{N-2r} (\<\alpha_i,\xi/2\>+\widetilde\alpha_i)\right)(-1/4+\widetilde{pt})^r s({\hbox{\rom{Ext}}}^1_{q}({\cal I}_{{\cal Z}_{1}}, {\cal I}_{{\cal Z}_{2}}\otimes V_\xi)\right),\end{eqnarray*} where $s(\cdot)$ denotes the total Segre class. We denote $\delta_{\xi}:=\sum_{N>0}\delta_{\xi,N}$. We will also denote for $\alpha\in H_2(S,{\Bbb Q})$ by $\delta_{\xi,N,r}(\alpha):=\delta_{\xi,N}({pt}^r\alpha^{N-2r})$. \end{defn} \begin{thm}\label{wallchange1}\cite{E-G},\cite{F-Q}\label{donch1} Let $S$ be a rational surface. Let $c_1\in H^2(S,{\Bbb Z})$ and $c_2\in {\Bbb Z}$. Let $H_-$ and $H_+$ be ample divisors on $S$, such that all the walls defined by elements of $W_{(c_1,c_2)}(H_-,H_+)$ are good. Then for all $\alpha\in A_N(S)$ we have $$\Phi^S_{H_+,N}(\alpha)-\Phi^S_{H_-,N}(\alpha)= \sum_{\xi\in W_{(c_1,c_2)}(H_-,H_+)} (-1)^{e_{\xi,N}} \delta_{\xi}(\alpha).$$ \end{thm} \subsection{Blowup formulas} We briefly recall the blowup formulas in the context of algebraic surfaces. In the case $b_+(S)>1$, when the invariants do not depend on the chamber structure, they have been shown e.g. in \cite{O}, \cite{L} and in the most general form in \cite{F-S}. In the case $b_+(S)=1$ we cite these results after \cite{K-L}. By \cite{T} the formulas of \cite{F-S} also hold for $S$ with $b_+(S)=1$, if the chamber structure is properly taken into account. Let $S$ be an algebraic surface with $b_+=1$ and let $\epsilon:\widehat S\longrightarrow S$ be the blowup in a point. Let $E\in H^2(S,{\Bbb Z})$ be the class of the exceptional divisor. Let $c_1\in H^2(S,{\Bbb Z})$ and $c_2\in H^4(S,{\Bbb Z})$ and put $N=4c_2-c_1^2-3$. Let $\cal C\subset \cal C_S$ be a chamber of type $(c_1,c_2)$, let $\cal C_E\subset C_{\widehat S}$ be a chamber of type $(c_1+E,c_2)$, and let $ \cal C_0\subset C_{\widehat S}$ be a chamber of type $(c_1,c_2)$. Following \cite{Ko} we say that the chambers $\cal C$ and $ \cal C_E$ (resp. $\cal C$ and $ \cal C_0$) are related chambers if $\epsilon^*(\cal C)$ is contained in the closure $\overline \cal C_E$ (resp in $\overline \cal C_0$). \begin{thm} There are universal polynomials $S_k(x)$ and $B_k(x)$ such that for all related chambers $\cal C$ and $ C_E$ (resp, $\cal C$ and $ \cal C_0$) as above, all $k\le N$ and all $\alpha\in A_{N-k}(S)$ we have \begin{eqnarray} \label{sobl}\Phi^{\widehat S, \cal C_E}_{c_1-E}(\check E^k\alpha)=- \Phi^{\widehat S, \cal C_E}_{c_1+E}(\check E^k\alpha) &=&\Phi^{S,\cal C}_{c_1}(S_k({pt})\alpha),\\ \label{subl}\Phi^{\widehat S,\cal C_0}_{c_1}(\check E^k\alpha) &=&\Phi^{ S,\cal C}_{c_1}(B_k({pt})\alpha). \end{eqnarray} (Note the different sign convention). The $S_k(x)$ and $B_k(x)$ can be given in terms of the coefficients of of the $q$-development of certain $\sigma$-functions. \end{thm} We refer to (\ref{sobl}) as $SO(3)$-blowup formulas and to \ref{subl} as $SU(2)$-blowup formulas. We will use that the $S_k(x)$ and the $B_k(x)$ are determined by recursive relations: (a1) $S_{2k}(x)=0$ for all $k$, (b1) $S_1(x)=1$, $S_3(x)=-x$, $S_5(x)=x^2+2$, $S_7(x)=-x^3-6x$, (a2) $B_{2k+1}(x) =0$ for all $k$, (b2) $B_{0}(x)=1$, $B_{2}(x) =0$, $B_{4}(x)=-2$ and, in both cases, the recursive relation \begin{eqnarray*}\label{rec}&&\sum_{i=0}^h{h\choose i} \big( U_{h+4-i}U_{i}-4U_{h+3-i}U_{i+1} +6U_{h+2-i}U_{i+2}-4U_{h+1-i}U_{i+3}+U_{h-i}U_{i+4}\big)\\ &&= -4\sum_{i=0}^h{h\choose i}\big( xU_{h+2-i}U_{i}+ xU_{h-i}U_{i+2}-2xU_{h+1-i}U_{i+1}+U_{h-i}U_{i}\big), \end{eqnarray*} with either $U_i=S_i(x)$ or $U_i=B_i(x)$ (see e.g. \cite{F-S},\cite{K-L}). \subsection{The walls for rational surfaces} Now let $S$ be a rational surface. We want to collect some information about the set of walls in the ample cone $C_S$. The following is easy to see: \begin{rem} \begin{enumerate} \item If $S$ is a rational ruled surface then $C_S=C_{S,g}$, i.e. all walls are good. \item If $S$ is obtained from ${\Bbb P}_2$ by a sequence of blow ups with exceptional divisors $E_1,\ldots,E_r$ then $C_{S,g}=C_S\cap\big \{ a(H-a_1E_1-\ldots -a_rE_r) \bigm | a>0,a_i>0,\ \sum_i a_i<3\big\}$. \end{enumerate} \end{rem} \begin{lem}\label{finwall} For any pair $(H_-,H_+)$ of ample divisors on a rational surface $S$ and all $c_1\in Pic(S)$ and $c_2\in H^2(S,{\Bbb Z})$ the set $W_{(c_1,c_2)}(H_-,H_+)$ is finite. \end{lem} \begin{pf} The set $\{ tH_-+(1-t)H_+ \ |\ t\in [0,1]\}$ is a compact subset of $C_S$. Therefore by \cite{F-M} corollary 1.6 it intersects only finitely many walls of type $(c_1,c_2)$. \end{pf} We now give a list of all walls for $S=\widehat{\Bbb P}_2$ and $S={\Bbb P}_1\times{\Bbb P}_1$ which will be used repeatedly in our computations. We denote by $F=H-E$ the class of a fibre of $\widehat {\Bbb P}_2\longrightarrow {\Bbb P}_1$. We also denote by $F$ the fibre of the projection to the first factor of ${\Bbb P}_1\times {\Bbb P}_1$ and by $G$ the class of the fibre of the projection to the second factor. The verifications are elementary. \begin{rem} \label{wallp1p1} \begin{eqnarray*} W^{\widehat {\Bbb P}_2}_{0,c_2}(F,H-\delta E)&=&\big\{ 2aH-2bE\bigm| b>a>\delta b,\, b^2-a^2\le c_2\big\},\\ W^{\widehat {\Bbb P}_2}_{E,c_2}(F,H-\delta E)&=&\big\{ 2aH-(2b-1)E\bigm| b>a>\delta (b-1/2),\,b(b-1)-a^2\le c_2\big\},\\ W^{\widehat {\Bbb P}_2}_{H,c_2}(F,H-\delta E)&=&\big\{ (2a-1)H-2bE\bigm| b\ge a>\delta b+1/2,\,b^2-a(a-1)\le c_2\big\},\\ W^{\widehat {\Bbb P}_2}_{F,c_2}(F,H-\delta E)&=&\big\{ (2a-1)H-(2b-1)E\bigm| b> a>\delta (b-1/2)+1/2,\,b(b-1)-a(a-1)\le c_2\big\},\\ W^{ {\Bbb P}_1\times{\Bbb P}_1}_{0,c_2}(F,F+\delta G)&=&\big\{ 2aF-2bG\bigm| 0<b<a\delta,\,2ab\le c_2 \big\},\\ W^{ {\Bbb P}_1\times{\Bbb P}_1}_{F,c_2}(F,F+\delta G)&=&\big\{ (2a-1)F-2bG\bigm| 0<b<(a-1/2)\delta,\,(2a-1)b\le c_2 \big\},\\ W^{ {\Bbb P}_1\times{\Bbb P}_1}_{G,c_2}(F,F+\delta G)&=&\big\{ 2aF-(2b-1)G\bigm| 0<b<a\delta+1/2,\,(2b-1)a\le c_2 \big\},\\ W^{ {\Bbb P}_1\times{\Bbb P}_1}_{F+G,c_2}(F,F+\delta G)&=&\big\{ (2a-1)F-(2b-1)G\bigm| 0<b<(a-1/2)\delta+1/2,\,2ab-a-b\le c_2 \big\}. \end{eqnarray*} \end{rem} \subsection{Botts formula} Now we recall the Bott residue formula (see e.g. \cite{B},\cite{A-B},\cite{E-S2},\cite{C-L1},\cite{C-L2}). Let $X$ be a smooth projective variety of dimension $n$ with an algebraic action of the multiplicative group ${\Bbb C}^*$ such that the fixpoint set $F$ is finite. Differentiation of the action induces a global vector field $\xi\in H^0(X,T_X)$, and $F$ is precisely the zero locus of $\xi$. Hence the Koszul complex on the map $\xi^\vee:\Omega_X\longrightarrow {\cal O}_X$ is a locally free resolution of ${\cal O}_F$. For $i\ge 0$ denote by $B_i$ the cokernel of the Koszul map $\Omega_X^{i+1}\longrightarrow \Omega_X^i$. It is well known that $H^j(X,\Omega^i_X)=0$ for $i\ne j$. So there are natural exact sequences for all $i$: $$0\longrightarrow H^i(X,\Omega_X^i)\mapr{p_i} H^i(X,B_i)\mapr{r_i} H^{i+1}(X,B_{i+1})\longrightarrow 0.$$ In particular there are natural maps $q_i=r_{i-1}\circ\ldots\circ r_0:H^0(F,{\cal O}_F)\longrightarrow H^i(X,B_i)$. \begin{defn} Let $f:F\longrightarrow {\Bbb C}$ be a function and $c\in H^i(X,\Omega^i_X)$. We say that $f$ represents $c$ if $q_{i+1}(f)=0$ and $q_i(f)=p_i(c).$ \end{defn} If $f_1$ represents $a_1\in H^i(X,\Omega^i_X)$ and $f_2$ represents $a_2\in H^j(X,\Omega^j_X)$, then $f_1f_2$ represents $a_1\cdot a_2 \in H^{i+j}(X,\Omega^{i+j}_X)$. The following result enables us to compute the degree of polynomials of weight in the Chern classes of equivariant vector bundles on $X$. Let ${\cal E}$ be an equivariant vector bundle of rank $r$ on $X$. At each fixpoint $x\in F$ the fibre ${\cal E}(x)$ splits as a direct sum of one-dimensional representations of ${\Bbb C}^*$. Let $\tau_1(E,x),\ldots\tau_r({\cal E},x)$ denote the corresponding weights, and for all $k\ge 0$ let $\sigma_k({\cal E},x)\in {\Bbb Z}$ be the $k$-th elementary symmetric function in the $\tau_i({\cal E},x)$. \begin{thm}\label{Bottres} \begin{enumerate} \item The $k$-th Chern class $c_k({\cal E})\in H^k(X,\Omega_X^k)$ of ${\cal E}$ can be represented by the function $x\mapsto \sigma_k({\cal E},x)$. \item The composition $H^0({\cal O}_F)\longrightarrow H^n(X,\Omega^n_X)\mapr{res}{\Bbb C}$ maps $f:F\longrightarrow {\Bbb C}$ to $\sum_{x\in F} f(x)/\sigma_n(T_X,x)$. \end{enumerate} \end{thm} \section{Application of the Bott residue formula} In this section we want to see how the Bott residue formula can be used to compute $\delta_{\xi,N}$ for a class $\xi$ defining a wall on a rational surface $X$. Let $\Gamma={\Bbb C}^*\times {\Bbb C}^*$ be an algebraic 2-torus and let $\lambda$ and $\mu$ be two independent primitive characters of $\Gamma$. We identify the representation ring of $\Gamma$ with the ring of Laurent polynomials in $\lambda$ and $\mu$. For a variety $Y$ with an action of $\Gamma$ we will denote by $F_Y$ the set of fixpoints. \subsection{\label{torS} Actions of a torus on rational surfaces } We are going to define actions with finitely many fixpoints of $\Gamma$ on $X={\Bbb P}_2$, $X={\Bbb P}_1\times {\Bbb P}_1$ and inductively on surfaces $X=X_r$, where $X_0={\Bbb P}_2$ or $X_0={\Bbb P}_1\times {\Bbb P}_1$ and $X_i$ is the blowup of a fixpoint of the $\Gamma$-action on $X_{i-1}$. We also define a lift of the action of $\Gamma$ to all line bundles on $X$. These actions will have the following properties: \begin{enumerate} \item Each fixpoint $p\in F_X$ has an invariant neighbourhood $A_p$ isomorphic to ${\Bbb A}^2=spec([k[x,y])$ on which $\Gamma$ acts by $t\cdot x=\alpha_px$, $ t\cdot y=\beta_py$ for two independent characters $\alpha_p$ and $\beta_p$ of $\Gamma$, and the $A_p$ cover $X$. \item For each line bundle $L\in Pic(X)$ the restriction $L|_{A_p}$ has a nowhere vanishing section $s_{L,p}$, with $t\cdot s_{L,p}=\gamma_{L,p}s_{L,p}$ for $\gamma_{L,p}$ a character of $\Gamma$. \end{enumerate} {\it (a) The case of ${\Bbb P}_2$.} Let $T_0,T_1,T_2$ be homogeneous coordinates on ${\Bbb P}_2$. Let $\Gamma$ act on ${\Bbb P}_2$ by $t\cdot T_0= T_0$, $t\cdot T_1=\lambda T_1$, $t\cdot T_2=\mu T_2$. The action of $\Gamma$ has $3$ fixpoints $p_0:=(1\!\!:\!\! 0\!\! : \!\!0)$, $p_1:=(0\!\! : \!\!1\!\! : \!\!0)$ and $p_2:=(0\!\! : \!\!0\!\! : \!\!1)$. The sets $A_{p_i}:=D(T_i)$ (i.e. the locus where $T_i\ne 0$) are affine invariant neighbourhoods. In appropriate coordinates $x,y$ on $A_{p_0}$ (resp. $A_{p_1}$,$A_{p_2}$), the induced action of $\Gamma$ is $t\cdot (x,y)=(\lambda x,\mu y)$ (resp. $t\cdot (x,y)=(\lambda^{-1} x,\mu\lambda^{-1} y)$, $t\cdot (x,y)=(\mu^{-1}x,\lambda\mu^{-1} y)$). Furthermore on $A_{p_i}$ the monomial $T_i^n$ defines a trivializing section of ${\cal O}_{{\Bbb P}_2}(n)$ with $t\cdot T_0^n=T_0^n$, $t\cdot T_1^n=\lambda^nT_1^n$, $t\cdot T_2^n=\mu^nT_2^n$. {\it (b) The case of ${\Bbb P}_1\times {\Bbb P}_1$.} Let $X_0,X_1$ and $Y_0,Y_1$ be homogeneous coordinates on the two factors. Let $\Gamma$ act on ${\Bbb P}_1\times{\Bbb P}_1$ by $t\cdot X_0=X_0$, $t\cdot X_1=\lambda X_1$, $t\cdot Y_0=Y_0$ and $t\cdot Y_1=\mu Y_1$. This action has $4$ fixpoints $p_{ij}:=V(X_{1-i})\cap V(Y_{1-j})$ (i.e. the locus where $X_{1-i}=Y_{1-j}= 0$), which have affine neightbourhoods $A_{p_{ij}}=D(X_i)\cap D(Y_j)$. In the appropriate coordinates $x,y$ on $A_{p_{ij}}$ the action is given by $t\cdot (x,y)=(\lambda^{1-2i}x,\mu^{1-2j}y)$ ($i$ and $j\in \{0,1\}$). Finally a trivializing section of ${\cal O}(n,m)$ on $A_{p_{ij}}$ is $X_i^n Y_i^m$ with $t\cdot (X_i^nY_j^m)=\lambda^{in}\mu^{jm}X_i^nY_j^m$. {\it (c) The blowup.} Now assume that $Y$ is a surface obtained from ${\Bbb P}_1\times {\Bbb P}_1$ or ${\Bbb P}_2$ by successively blowing up fixpoints of the action of $\Gamma$, and assume that the action is extended to $Y$, so that it still has finitely many fixpoints, and that the assumptions (1) and (2) above are satisfied. Let $p\in F_Y$ be a fixpoint. Let $A_{p}$ be an affine neighbourhood of $p$ with coordinates $x,y$ on which $\Gamma$ acts by $t\cdot (x,y)=(\alpha x,\beta y)$ for two independent characters $\alpha,\beta$ of $\Gamma$. Let $X$ be the blowup of $Y$ in $p$, and denote by $E$ the exceptional divisor and by $\widehat A$ the blow up of $A_{p}$ at $p$. We can identify $E={\Bbb P}(\<x,y\>^\vee)$, and the induced action of $\Gamma$ on $E$ has $2$ fixpoints $q_0:=(1\!\! : \!\!0)$ and $q_1:=(0\!\! : \!\!1)$, which are the fixpoints of $\Gamma$ on $X$ over $p$. There are affine neighbourhoods $A_{q_0}=\widehat A\cap D(x) $ and $A_{q_1}=\widehat A\cap D(y)$ of $q_0$ and $q_1$ in $X$, with coordinates $(x,y/x)$ and $(y,x/y)$. The action $t\cdot (x,y/x)=(\alpha x,\beta\alpha^{-1}y/x)$, $t\cdot (y,x/y)=(\beta y,\alpha\beta^{-1}x/y)$ extends the action of $\Gamma$ on $Y\setminus \{p\}$ to $X$. Let $L$ be a linebundle on $Y$ with a trivializing section $s_{L,q}$ near each $q\in F_Y$ with $t\cdot s_{L,q}=\gamma_{L,p_i}s_{L,q}$. Then $L\otimes {\cal O}(kE)$ has for $i\ne 0$ still $s_{L,q}$ as a trivializing section near $q$ (with $q\ne p$), and near $q_0$ (resp. $q_1$) such a section is $s_0=s_{L,p}\otimes y^{-k}$ (resp. $s_1=s_{L,p}\otimes x^{-k}$) with $t\cdot s_0=\gamma_{L,p}\beta^{-k}s_0$ (resp. $t\cdot s_1=\gamma_{L,p}\alpha^{-k}s_1$). \subsection{ The induced action on the Hilbert scheme } We assume that $S$ is a surface obtained by blowing up ${\Bbb P}_2$ or ${\Bbb P}_1\times {\Bbb P}_1$ repeatedly, with an action of $\Gamma$ as above. We fix a positive integer $d$ and want to study the induced action of $\Gamma $ on the Hilbert scheme ${\text{\rom{Hilb}}}^d(S\sqcup S)$ and on certain "standard bundles" on ${\text{\rom{Hilb}}}^d(S\sqcup S)$, which appear in the wall-crossing formula \ref{wallchange}. The induced action on ${\text{\rom{Hilb}}}^d(S\sqcup S)$ is given by $t\cdot (Y,Z)=(t\cdot Y,t\cdot Z)$, where for a subscheme $Z\subset S$ we denote by $t\cdot Z$ the subscheme with ideal $t\cdot {\cal I}_{Z/S}:=\{t\cdot f\ |\ f\in {\cal I}_{Z/S}\}$. Now let $F_S:=\{ p_1,\ldots, p_m\}$ be the set of fixpoints on $S$, and, for all $i$, let $A_i$ be the invariant affine neighbourhood of $p_i$ with coordinates $x_i,y_i$, such that $t\cdot x_i=\alpha_ix_i,$ $t\cdot y_i=\beta_i y_i$ for two independent characters $\alpha_i$ and $\beta_i$. As the characters $\alpha_i$ and $\beta_i$ are independent, it is easy to see that a subscheme $Z\in {\text{\rom{Hilb}}}^n(S)$ is fixed by the induced action of $\Gamma$ if and only if $supp(Z)\subset F_S$ and if, for all $i$, denoting by $Z_i$ the part of $Z$ with support $p_i$, all the ideals ${\cal I}_{Z_i/A_i}$ are generated by monomials in $x_i$ and $y_i$. We denote by $F_{{\text{\rom{Hilb}}}^d(S\sqcup S)}$ the fixpoints on ${\text{\rom{Hilb}}}^d(S\sqcup S)$. \begin{defn} A partition of a nonnegative integer $n$ is a sequence $\alpha=(a_0,\ldots a_r)$ with $\alpha_0\ge \ldots \ge a_{r-1}\ge a_r=0$ and $\sum a_i=n$. We identify $(a_0,\ldots a_r)$ and $(a_0,\ldots a_r,0)$. Let $P_{2m}(d)$ be the set of sequences $(P_1,\ldots,P_m,Q_1,\ldots Q_m)$ where the $P_i$ and $Q_i$ are all partitions of numbers $n_i$ and $m_i$ with $\sum (n_i +m_i)=d$. We see that $P_{2m}(d)$ and $F_{{\text{\rom{Hilb}}}^d(S\sqcup S)}$ are in one-one correspondence, with $(P_1,\ldots,P_m,Q_1,\ldots Q_m)$ corresponding to $(Y_1\sqcup\ldots\sqcup Y_m,Z_1\sqcup\ldots\sqcup Z_m)$, where for $P_i=(a_0,\ldots,a_r)$, $Q_i=(b_0,\ldots,b_r)$ the subschemes $Y_i$ and $Z_i$ are supported at $p_i$ and defined by ${\cal I}_{Y_i/A_i}=(y_i^{a_0},x_iy_i^{a_1}, \ldots x_i^sy_i^{a_s},x^{s+1})$ and ${\cal I}_{Z_i/A_i}=(y_i^{b_0},x_iy_i^{b_1}, \ldots x_i^ry_i^{b_r},x^{r+1})$. \end{defn} \subsection{The action on some standard bundles.} We now want to determine the action of $\Gamma$ on some standard bundles on ${\text{\rom{Hilb}}}^d(S\sqcup S)$ which appear in the wall-crossing formula \ref{wallchange}. Let $\xi$ define a good wall. We denote by $V$ the vector bundle ${\cal O}_S(-\xi)\oplus{\cal O}_S(-\xi+K_S)$. Then by the results of \cite{E-G} and \cite{F-Q} ${\hbox{\rom{Ext}}}^1_q({\cal I}_{{\cal Z}_1},{\cal I}_{{\cal Z}_2}\otimes p^*V)$ is a locally free sheaf on ${\text{\rom{Hilb}}}^d(S\sqcup S)$, which is compatibel with base change, i.e. its fibre over $(Y,Z)\in {\text{\rom{Hilb}}}^d(S\sqcup S)$ is ${\hbox{\rom{Ext}}}^1({\cal I}_{Y},{\cal I}_{Z}\otimes V)$. Furthermore the $\Gamma$-linearisation of ${\cal O}_S(\xi)$ from \ref{torS} determines in a canonical way a $\Gamma$-linearisation of ${\hbox{\rom{Ext}}}^1_q({\cal I}_{{\cal Z}_1},{\cal I}_{{\cal Z}_2}\otimes p^*V)$. It also induces an action of $\Gamma$ on $H^1(S,V)$. Now let $(Y,Z)\in F_{{\text{\rom{Hilb}}}^d(S\sqcup S)}$ be a point corresponding to $(P_1,\ldots,P_m,Q_1,\ldots Q_m)$. We will determine the action on the fibre ${\hbox{\rom{Ext}}}^1({\cal I}_{Y},{\cal I}_{Z}\otimes V)$. We denote by $V(p_i)$ the fibre of $V$ over the fixpoint $p_i$ considered as a representation of $\Gamma$. \begin{lem}\label{stanbott} For partitions $P:=(a_0,\ldots,a_r)$, $Q:=(b_0,\ldots,b_r)$ we denote $$E_{P,Q}(x,y):=\sum_{1\le i\le j\le r}\left( \sum_{s=a_j}^{a_{j-1}-1} x^{i-j-1}y^{b_{i-1}-s-1}+ \sum_{s=b_j}^{b_{j-1}-1} x^{j-i}y^{s-a_{i-1}}\right).$$ Then in the representation ring of $\Gamma$ we have the identities \begin{eqnarray}\label{for1} T_{Hilb^d(S\sqcup S)}(Y,Z)&=& \sum_{i=0}^m (E_{P_i,P_i}(\alpha_i,\beta_i)+E_{Q_i,Q_i}(\alpha_i,\beta_i)),\\ \label{for2} {\hbox{\rom{Ext}}}^1({\cal I}_{Y},{\cal I}_{Z}\otimes V)&=& H^1(S,V)+\sum_{i=0}^m V(p_i)\cdot E_{P_i,Q_i}(\alpha_i,\beta_i)). \end{eqnarray} \end{lem} \begin{pf} (\ref{for1}) follows directly from \cite{E-S1}. \noindent{\it Claim:} In the representation ring of $\Gamma$ we have the identity $${\hbox{\rom{Ext}}}^1({\cal I}_{Y},{\cal I}_{Z}\otimes V)=H^1(S,V)+ H^0(S,{\cal Ext}^1({\cal I}_Y,{\cal I}_Z)\otimes V)+H^0(S,{\cal O}_{Z}\otimes V) -H^0(S,{\cal Hom}({\cal O}_Y,{\cal O}_Z)\otimes V).$$ {\it Proof of the Claim:} As $\xi$ defines a good wall, we have $H^2(S,{\cal Hom}({\cal I}_{Y },{\cal I}_{Z})\otimes V)=H^0(S,{\cal Hom}({\cal I}_{Z },{\cal I}_{Y}) \otimes V^\vee(K_S))=0$ and $H^0(S,{\cal Hom}({\cal I}_{Z },{\cal I}_{Y})\otimes V)=0.$ Therefore the low-term exact sequence of the local to global spectral sequence $H^p({\cal Ext}^q({\cal I}_Y,{\cal I}_Z\otimes V))\Rightarrow {\hbox{\rom{Ext}}}^{p+q}({\cal I}_{Y},{\cal I}_{Z}\otimes V)$ gives in the representation ring of $\Gamma$ $${\hbox{\rom{Ext}}}^1({\cal I}_Y,{\cal I}_Z(V)) =H^0(S,{\cal Ext}^1({\cal I}_Y,{\cal I}_Z)\otimes V)+H^1(S,{\cal Hom}({\cal I}_Y,{\cal I}_Z)\otimes V).$$ We have an exact sequence $$0\longrightarrow {\cal I}_{Z}\longrightarrow {\cal Hom}({\cal I}_{Y},{\cal I}_{Z})\longrightarrow {\cal Hom}({\cal O}_{Y},{\cal O}_{Z})\longrightarrow 0.$$ So, tensoring by $V$, taking the long exact sequence of cohomology and using the vanishing of $H^0({\cal Hom}({\cal I}_Y,{\cal I}_Z)\otimes V)$ and $H^1({\cal Hom}({\cal O}_{Y },{\cal O}_{Z})\otimes V)$, we get in the representation ring of $\Gamma$ the identity $$H^1({\cal Hom}({\cal I}_{Y},{\cal I}_{Z})\otimes V)=H^1(S,{\cal I}_Z\otimes V) -H^0({\cal Hom}({\cal O}_{Y},{\cal O}_{Z})\otimes V).$$ Finally we use the sequence $0\longrightarrow {\cal I}_{Z}\otimes V\longrightarrow V \longrightarrow {\cal O}_{Z}\otimes V\longrightarrow 0$ and the vanishing of $H^0(S,V)$ and $H^1(S,{\cal O}_Z\otimes V)$ to replace $H^1(S,{\cal I}_Z\otimes V)$ by $H^0(S,{\cal O}_Z\otimes V)+H^1(S,V)$. This shows the claim. We denote by ${\cal F}$ the virtual $\Gamma$-sheaf ${\cal Ext}^1({\cal I}_Y,{\cal I}_Z)+{\cal O}_{Z} -{\cal Hom}({\cal O}_Y,{\cal O}_Z)$. We have to show that $H^0(S,{\cal F}\otimes V) =\sum_{i=0}^m V(p_i)\cdot E_{P_i,Q_i}(\alpha_i,\beta_i).$ If we denote by ${\cal F}_i$ the part of ${\cal F}$ with support $p_i$, then $H^0(S,{\cal F}\otimes V)=\sum_{i=1}^m H^0(S,{\cal F}_i\otimes V).$ We can therefore assume that $supp(Y)=supp(Z)$ is one fixpoint $p$. Let $x$ and $y$ be coordinates near $p$ as before and $R:={\Bbb C}[x,y]$. Let $J:=(y^{a_0},xy^{a_1},\ldots,x^{r+1})$ (resp. $I:=(y^{b_0},xy^{b_1},\ldots,x^{r+1})$) be the ideal of $Y$ (resp. $Z$). We denote by $F$ the virtual $R$-$\Gamma$-module corresponding to ${\cal F}$. In the representation ring of $\Gamma$ we have $$H^0(S,{\cal F}\otimes V)= F\cdot V(p).$$ So we finally have to show that in the representation ring of $\Gamma$ we have $F=E_{(a_0,\ldots,a_r),(b_0,\ldots b_r)}(\lambda,\mu)$. The exact sequences \begin{eqnarray*} &&0\longrightarrow I\longrightarrow {\hbox{\rom{Hom}}}_R(J,I)\longrightarrow {\cal Hom}(R/J,R/I)\longrightarrow 0\\ &&0\longrightarrow I\longrightarrow R\longrightarrow R/I\longrightarrow 0 \end{eqnarray*} give $F={\hbox{\rom{Ext}}}_R^1(J,I)-{\hbox{\rom{Hom}}}_R(J,I)+R$ in the representation ring of $\Gamma$. Following \cite{E-S1} we denote by $R[\alpha,\beta]$ the ring $R$ with $\Gamma$-operation defined by $t(x^iy^j):=x^{i-\alpha}y^{j-\beta}$. We put $A_0:=\bigoplus_{i=0}^r R[i,a_i],$ $B_0:=\bigoplus_{j=0}^r R[j,b_j],$ $A_1:=\bigoplus_{i=1}^r R[i,a_{i-1}],$ $B_1:=\bigoplus_{j=1}^r R[j,a_{j-1}]$. Then we have $\Gamma$-equivariant free resolutions $0\longrightarrow A_1\longrightarrow A_0\longrightarrow J\longrightarrow 0$ and $0\longrightarrow B_1\longrightarrow B_0\longrightarrow I\longrightarrow 0$. So the total complex $$A_0^\vee\otimes B_1\mapr{\alpha}A_1^\vee\otimes B_1\oplus A_0^\vee\otimes B_0\longrightarrow A_1^\vee\otimes B_0$$ associated to the double complex ${\hbox{\rom{Hom}}}_R({\cal A}_{\bullet},B_{\bullet})$ computes the ${\hbox{\rom{Ext}}}^i_R(J,I)$, hence $F=R+A_1^\vee\otimes B_1+A_0^\vee\otimes B_0-A_0^\vee\otimes B_1-A_1^\vee\otimes B_0.$ Again following \cite{E-S1} we write $n_i:=(i,a_{i-1})$, $d_i:=(i,a_i)$, $m_j:=(j,b_{j-1})$ and $e_j:=(j,b_j)$. Then a calculation analogous to \cite{E-S1} shows $$F=R+\sum_{{1\le i\le r}\atop {0\le j\le r}} R[e_j-n_i]-\sum_{{1\le i\le r}\atop {1\le j\le r}} R[m_j-n_i] -\sum_{{0\le i\le r}\atop {0\le j\le r}}R[e_j-d_i]+ \sum_{{0\le i\le r}\atop {1\le j\le r}}R[m_j-d_i].$$ Putting \begin{eqnarray*} K_{i,j}&:=&R[m_j-d_{i-1}]-R[m_j-n_i]-R[e_j-d_{i-1}]+R[e_j-n_i],\\ L_{i,j}&:=&R[m_i-d_{j}]-R[m_i-n_j]-R[e_{i-1}-d_{j}]+R[e_{i-1}-n_j], \end{eqnarray*} a calculation analogous to \cite{E-S1} gives $$F=\sum_{1\le i\le j\le r} (K_{i,j}+L_{i,j}),\ K_{i,j}=\sum_{s={a_i}}^{a_{i-1}} \lambda^{i-j-1}\mu^{b_{i-1}-s-1}\hbox{ and } L_{i,j}=\sum_{s={b_j}}^{b_{j-1}} \lambda^{j-i}\mu^{s-a_{i-1}}, $$ and the result follows. \end{pf} We want to use the easy fact that representation of cohomology classes is compatible with equivariant pullback: Let $X$ and $Y$ be smooth projective varieties with an action of ${\Bbb C}^*$ with finitely many fixpoints and let $\mu:X\longrightarrow Y$ be an equivariant surjective morphism. Then $\mu$ induces a morphism $\mu|_{F_X}:F_X\longrightarrow F_Y$. \begin{lem} \label{compat} $f\in {\cal O}_{F_Y}$ represents a cohomology class $c\in H^j(Y,\Omega_Y^j)$ if and only if $(\mu|_{F_X})^*f$ represents $\mu^*c$. \end{lem} \begin{lem}\label{altild} Let $\alpha\in H^k(S,{\Bbb Z})$ be a class represented by $f:F_S\longrightarrow {\Bbb C}$. Then $\widetilde \alpha$ (see \ref{wallchange}) on ${\text{\rom{Hilb}}}^d(S\sqcup S)$ is represented by $$\widetilde f:P_{2m}(d)\longrightarrow {\Bbb C}, ((P_i),(Q_i))\mapsto \prod_{i=1}^m (n_i+m_i) f(p_i),$$ where $P_i\in P(n_i)$ and $Q_i\in P(m_i)$. \end{lem} \begin{pf} Let $${\text{\rom{Hilb}}}^{d-1,d}(S\sqcup S):=\big\{(Z_{d-1},Z_d)\in {\text{\rom{Hilb}}}^{d-1}(S\sqcup S) \times {\text{\rom{Hilb}}}^{d}(S\sqcup S)\bigm | Z_{d-1}\subset Z_d\big\}$$ with the reduced induced structure. Then ${\text{\rom{Hilb}}}^{d-1,d}(S\sqcup S)$ is smooth and we have a diagram $${\text{\rom{Hilb}}}^d(S\sqcup S)\mapl{\varphi}{\text{\rom{Hilb}}}^{d-1,d}(S\sqcup S)\mapr{\psi} (S\sqcup S)\times {\text{\rom{Hilb}}}^{d-1}(S\sqcup S)\mapr{\eta} S\times {\text{\rom{Hilb}}}^{d-1}(S\sqcup S)$$ Here $\psi$ is the blowup along the universal family $Z_d(S\sqcup S)$ \cite{E} and $\eta$ is induced by the identity map on $S$ and ${\text{\rom{Hilb}}}^d(S\sqcup S)$. It is easy to see from the definitions that $\varphi^*\widetilde\alpha=\psi^*\eta^*(p_1^*\alpha+p_2^*\bar \alpha)$, where $p_1$ and $p_2$ are the projections of $S\times {\text{\rom{Hilb}}}^{d-1}(S\sqcup S)$ onto its two factors and $\bar \alpha$ is the class corresponding to $\widetilde \alpha$ if we replace $d$ by $d-1$. It is easy to see that $\varphi$, $\psi$ and $\eta$ are equivariant for the natural lifts of the action of $\Gamma$ on $S$, furthermore the fixpoint sets $F_{{\text{\rom{Hilb}}}^d(S\sqcup S)}$, $F_{{\text{\rom{Hilb}}}^{d-1,d}(S\sqcup S)}$ and $F_{S\times {\text{\rom{Hilb}}}^{d-1}(S\sqcup S)}$ are all finite. In fact we can identify $$F_{{\text{\rom{Hilb}}}^{d-1,d}(S\sqcup S)} =\big \{((S_i,T_i),(P_i,Q_i))\in P_{2m}(d-1)\times P_{2m}(d)\bigm| P_i\ge S_i, \ Q_i\ge T_i \hbox{ for all } i\big\},$$ where for partitions $P=(a_1,\ldots ,a_r),$ $Q=(b_1,\ldots b_r)$ we denote by $P\ge Q$ that $a_i\ge b_i$ for all $i$. Obviously $F_{S\times {\text{\rom{Hilb}}}^{d-1}(S\sqcup S)}= F_S\times P_{2m}(d-1)$ and with this identification $\varphi$ and $\eta\circ\psi$ are the obvious maps. Now, applying lemma \ref{compat} to $\varphi$ and $\eta\circ\psi$, the result follows by easy induction. \end{pf} We can now put our results together: \begin{nota} Fix a one-parameter subgroup $T$ of $\Gamma$. Let $\xi$ define a good wall on $S$. For any line bundle $L$ on $S$ denote by $w_i(L)$ the weight of the induced action of $T$ on the fibre $L(p_i)$. Let $L_1$ and $L_2$ be two line bundles with $\<L_1\cdot L_2\>={pt}$ (e.g. if $S$ is a blow up of ${\Bbb P}_2$ then we take $L_1=L_2=H$). Furthermore denote by $w(x_i)$, $w(y_i)$ the weight of the action of $T$ on $x_i$, $y_i$. We denote for partitions $P=(a_0,\ldots a_r)$ and $Q=(b_0,\ldots b_r)$ of numbers $n$ and $m$ \begin{eqnarray*} \overline F_{P,Q}(u,v)&:=&\prod_{1\le i\le j \le r} \prod_{s=a_j}^{a_{j-1}-1} ((i-j-1)u+(b_{i-1}-s-1)v) \prod_{s=b_j}^{b_{j-1}-1} ((j-i)u+(s-a_{i-1})u)\\ F^z_{P,Q}(u,v,t)&:=&\prod_{1\le i\le j \le r} \prod_{s=a_j}^{a_{j-1}-1} (1+z((i-j-1)u+(b_{i-1}-s-1)v+t))\\&&\qquad \prod_{s=b_j}^{b_{j-1}-1} (1+z((j-i)u+(s-a_{i-1})v+t)) \end{eqnarray*} By lemma \ref{stanbott}, when putting the correct weights $F_{P,Q}(u,v)$ will represent the top Chern class of ${\text{\rom{Hilb}}}^d(S\sqcup S)$ and $F^z_{P,Q}(u,v,t)$ the total Chern class of ${\hbox{\rom{Ext}}}^1_q({\cal I}_{{\cal Z}_1},{\cal I}_{{\cal Z}_2}\otimes p^*V)$. \end{nota} \begin{thm}\label{Botthilb} Let $\alpha_1,\ldots,\alpha_{N-2r}\in H^2(S,{\Bbb Z})$. If $T$ is sufficiently general, then \begin{eqnarray*}&&\label{bottformel} \delta_{\xi}(\alpha_1\alpha_2\ldots\alpha_{N-2r}{pt}^r)= {\hbox{\rm Coeff}}_{z^{2d}}\Bigg(\sum_{((P_i),(Q_i))\in P_{2m}(d)}\\&& \Bigg( \prod_{k=1}^{N-2r}\Big(\<\xi,\alpha_k\>/2+\sum_{i=1}^m w_i(\alpha_k)(n_i+m_i)z\Big) \Big(-1/4+\sum_{i=1}^m w_i(L_1)w_i(L_2)(n_i+m_i)z^2\Big)^r\cdot\\ &&\Bigg(\prod_{i=1}^m\Big(\overline F_{(P_i,P_i)}(w(x_i),w(y_i)) \overline F_{Q_i,Q_i}(w(x_i),w(y_i))\cdot\\ &&F^z_{P_i,Q_i}(w(x_i),w(y_i),-w_i(\xi)) F^z_{P_i,Q_i}(w(x_i),w(y_i),-w_i(\xi))+w_i(K_S))\Bigg)^{-1}\Bigg). \end{eqnarray*} \end{thm} \begin{pf} The Chern classes of ${\cal V}_\xi={\hbox{\rom{Ext}}}^1_q({\cal I}_{{\cal Z}_1},{\cal I}_{{\cal Z}_2}\otimes p^*V)$ are the same as those of the virtual bundle ${\cal V}_\xi-H^1(S,V)\otimes {\cal O}_{T_{\xi}}$. Therefore the result just follows by putting together lemma \ref{stanbott}, lemma \ref{altild} and applying the Bott residue formula \ref{Bottres}. Notice that $T$ is sufficiently general if none of the denominators vanish. \end{pf} This formula can be implemented as a Maple program. \section{The Donaldson invariants of the projective plane} In this section we want to compute the $SU(2)$- and the $ SO(3)$-invariants of the projective plane ${\Bbb P}_2$ by first computing on the blowup $\widehat {\Bbb P}_2$ and then using the blowup formulas. In order to get started we need the following easy result of \cite{Q2}: \begin{lem}\label{vancham} Let $S$ be a rational ruled surface, $F$ the class of a fibre and $E$ the class of a section. Fix $(c_1,c_2)\in H^2(S,{\Bbb Z})\times H^4(S,{\Bbb Z})$ with $\<c_1\cdot F\>=1$. Then, for all $\epsilon>0$ which are sufficiently small, we have $M_{F+\epsilon E}(c_1,c_2)=\emptyset$. In particular we get for $N:=4c_2-c_1^2-3$ that $\Phi^{S,F+\epsilon E}_{c_1,N}=0$. \end{lem} We will denote by $E$ the exceptional divisor on $\widehat {\Bbb P}_2$ and by $H$ the (pullback of) the hyperplane class on ${\Bbb P}_2$. \subsection{The $SU(2)$-case} We first consider the $SO(3)$-invariants on $\widehat {\Bbb P}_2$ with respect to Chern classes $(E,c_2)$ and put $N:=4c_2-3$. For $0<\epsilon <<1$ the polarisation $L_\epsilon:=H-\epsilon E$ of $\widehat {\Bbb P}_2$ lies in a chamber of type $(E,c_2)$ which is related to the polarisation $H$ of ${\Bbb P}_2$. Thus (\ref{sobl}) gives $$\Phi^{{\Bbb P}_2,H}_{0,N}(H^{N-2r}{pt}^r)= \Phi^{\widehat{\Bbb P}_2,L_{\epsilon}}_{E,N+1}(\check E\check H^{N-2r}{pt}^r).$$ On the other hand we know by lemma \ref{vancham} that $\Phi^{\widehat {\Bbb P}_2,L_{1-\epsilon}}_{E,N+1}=0$, for $L_{1-\epsilon}:=H-(1-\epsilon) E$. Thus we get $$\Phi^{{\Bbb P}_2,H}_{0,N}(H^{N-2r}{pt}^r)= \sum_{\xi\in W^{\widehat {\Bbb P}_2}_{E,c_2}(H-E,H)}(-1)^{e_{\xi,N+1}} \delta_{\xi,N+1}(\check E \check H^{N-2r}{pt}^r),$$ where $W^{\widehat {\Bbb P}_2}_{E,c_2}(H-E,H)$ is known by remark \ref{wallp1p1}. Now we compute the $\delta_{\xi,N+1}(\check E \check H^{N-2r}pt^r)$ with a maple program using the Bott residue theorem (i.e. theorem \ref{Botthilb}). For $N:=4i+1$ we denote $$A_N:=\sum_{j=0}^{2N} \Phi^{{\Bbb P}_2,H}_{0,N}(\check H^{N-2j}{pt}^j)h^{N-2j}p^j.$$ Then our result is: \begin{thm}\label{p2su} The $SU(2)$-invariants of ${\Bbb P}_2$ are \par {\tolerance 10000\small\parindent 0pt\raggedright $A_1=-{\frac {3\,h}{2}}$, $A_5={h}^{5}-p{h}^{3}-{\frac {13\,{p}^{2}h}{8}}$, $A_9=3\,{h}^{9}+{\frac {15\,p{h}^{7}}{4}} -{\frac {11\,{p}^{2}{h}^{5}}{16}}- {\frac {141\,{p}^{3}{h}^{3}}{64}}-{\frac {879\,{p}^{4}h}{256}}$, \par $A_{13}=54\,{h}^{13}+24\,p{h}^{11}+{\frac {159\,{p}^{2}{h}^{9}}{8}}+{\frac {51 \,{p}^{3}{h}^{7}}{16}}-{\frac {459\,{p}^{4}{h}^{5}}{128}}-{\frac {1515 \,{p}^{5}{h}^{3}}{256}}-{\frac {36675\,{p}^{6}h}{4096}}$, \par $A_{17}=2540\,{h}^{17}+694\,p{h}^{15}+{\frac {487\,{p}^{2}{h}^{13}}{2}}+{ \frac {2251\,{p}^{3}{h}^{11}}{16}}+{\frac {2711\,{p}^{4}{h}^{9}}{64}}- {\frac {5\,{p}^{5}{h}^{7}}{16}}-{\frac {3355\,{p}^{6}{h}^{5}}{256}}-{ \frac {143725\,{p}^{7}{h}^{3}}{8192}}-{\frac {850265\,{p}^{8}h}{32768} }$, \par $ A_{21}=233208\,{h}^{21}+45912\,p{h}^{19}+10625\,{p}^{2}{h}^{17}+3036\,{p}^{3} {h}^{15}+{\frac {41103\,{p}^{4}{h}^{13}}{32}}+{\frac {1741\,{p}^{5}{h} ^{11}}{4}}+{\frac {5619\,{p}^{6}{h}^{9}}{64}} -{\frac {20379\,{p}^{7}{h }^{7}}{1024}}-{\frac {754141\,{p}^{8}{h}^{5}}{16384}}-{\frac {904239\, {p}^{9}{h}^{3}}{16384}}-{\frac {10504593\,{p}^{10}h}{131072}}$, \par $A_{25}=35825553\,{h}^{25}+{\frac {21975543\,p{h}^{23}}{4}}+{\frac {15224337\, {p}^{2}{h}^{21}}{16}}+{\frac {12159687\,{p}^{3}{h}^{19}}{64}}+{\frac { 11618625\,{p}^{4}{h}^{17}}{256}}+{\frac {15077511\,{p}^{5}{h}^{15}}{ 1024}} +{\frac {19602561\,{p}^{6}{h}^{13}}{4096}}+{\frac {20676279\,{p} ^{7}{h}^{11}}{16384}}+{\frac {11107665\,{p}^{8}{h}^{9}}{65536}}-{ \frac {28437201\,{p}^{9}{h}^{7}}{262144}}-{\frac {169509159\,{p}^{10}{ h}^{5}}{1048576}}-{\frac {757633329\,{p}^{11}{h}^{3}}{4194304}} -{ \frac {4334081031\,{p}^{12}h}{16777216}}$, \par $A_{29}=8365418914\,{h}^{29}+1047342410\,p{h}^{27}+{\frac {1157569571\,{p}^{2} {h}^{25}}{8}}+{\frac {357034013\,{p}^{3}{h}^{23}}{16}}+{\frac { 499796309\,{p}^{4}{h}^{21}}{128}}+{\frac {25506259\,{p}^{5}{h}^{19}}{ 32}}+{\frac {423516455\,{p}^{6}{h}^{17}}{2048}}+{\frac {245576651\,{p} ^{7}{h}^{15}}{4096}+{\frac {537423737\,{p}^{8}{h}^{13}}{32768}}+{ \frac {118590907\,{p}^{9}{h}^{11}}{32768}}}+{\frac {131266019\,{p}^{10} {h}^{9}}{524288}}-{\frac {498648655\,{p}^{11}{h}^{7}}{1048576}}-{ \frac {4800905323\,{p}^{12}{h}^{5}}{8388608}}-{\frac {2551074181\,{p}^ {13}{h}^{3}}{4194304}}-{\frac {115237180987\,{p}^{14}h}{134217728}}$, \par $A_{33}=2780195996868\,{h}^{33}+293334321858\,p{h}^{31}+{\frac {67261095005\,{ p}^{2}{h}^{29}}{2}}+{\frac {67539891519\,{p}^{3}{h}^{27}}{16}}+{\frac {37480404303\,{p}^{4}{h}^{25}}{64}}+{\frac {1455758501\,{p}^{5}{h}^{23 }}{16}}+{\frac {258640401\,{p}^{6}{h}^{21}}{16}}+{\frac {14239101477\, {p}^{7}{h}^{19}}{4096}+{\frac {14274421501\,{p}^{8}{h}^{17}}{16384}}+ {\frac {7420640919\,{p}^{9}{h}^{15}}{32768}}}+{\frac {7179481275\,{p}^{ 10}{h}^{13}}{131072}}+{\frac {10830752675\,{p}^{11}{h}^{11}}{1048576}} -{\frac {218792349\,{p}^{12}{h}^{9}}{4194304}}-{\frac {8125524657\,{p} ^{13}{h}^{7}}{4194304}}-{\frac {34310453897\,{p}^{14}{h}^{5}}{16777216 }}-{\frac {561608698989\,{p}^{15}{h}^{3}}{268435456}}-{\frac { 3135392459541\,{p}^{16}h}{1073741824}}$, \par $A_{37}=1253558847090600\,{h}^{37}+114049802084088\,p{h}^{35}+11151310348527\, {p}^{2}{h}^{33}+{\frac {2356053433779\,{p}^{3}{h}^{31}}{2}}+{\frac { 4330247481231\,{p}^{4}{h}^{29}}{32}}+{\frac {136302640305\,{p}^{5}{h}^ {27}}{8}}+{\frac {19010805303\,{p}^{6}{h}^{25}}{8}}+{\frac {5969251539 \,{p}^{7}{h}^{23}}{16}}+{\frac {560820990153\,{p}^{8}{h}^{21}}{8192}}+ {\frac {120608801397\,{p}^{9}{h}^{19}}{8192}}+{\frac {227818311585\,{p }^{10}{h}^{17}}{65536}}+{\frac {108448319625\,{p}^{11}{h}^{15}}{131072 }}+{\frac {380576939595\,{p}^{12}{h}^{13}}{2097152}}+{\frac { 30600598425\,{p}^{13}{h}^{11}}{1048576}}-{\frac {24269489295\,{p}^{14} {h}^{9}}{8388608}}-{\frac {128245327215\,{p}^{15}{h}^{7}}{16777216}}-{ \frac {3958008534873\,{p}^{16}{h}^{5}}{536870912}}-{\frac { 3928400321367\,{p}^{17}{h}^{3}}{536870912}}-{\frac {43427017514031\,{p }^{18}h}{4294967296}}$, \par $A_{41}=739328941273642584\,{h}^{41}+59025071651407086\,p{h}^{39}+{\frac { 10043994491097141\,{p}^{2}{h}^{37}}{2}}+{\frac {3657609707934747\,{p}^ {3}{h}^{35}}{8}}+{\frac {1431875545375857\,{p}^{4}{h}^{33}}{32}}+{ \frac {605936823728685\,{p}^{5}{h}^{31}}{128}}+{\frac {279056393051655 \,{p}^{6}{h}^{29}}{512}}+{\frac {141081414974709\,{p}^{7}{h}^{27}}{ 2048}}+{\frac {79278076181247\,{p}^{8}{h}^{25}}{8192}}+{\frac { 50786721057147\,{p}^{9}{h}^{23}}{32768}}+{\frac {37720541142561\,{p}^{ 10}{h}^{21}}{131072}}+{\frac {31629775641351\,{p}^{11}{h}^{19}}{524288 }}+{\frac {28182330795381\,{p}^{12}{h}^{17}}{2097152}}+{\frac { 24970918823121\,{p}^{13}{h}^{15}}{8388608}}+{\frac {20134588411731\,{p }^{14}{h}^{13}}{33554432}}+{\frac {10829868897921\,{p}^{15}{h}^{11}}{ 134217728}}-{\frac {9966987184029\,{p}^{16}{h}^{9}}{536870912}}-{ \frac {63772382356485\,{p}^{17}{h}^{7}}{2147483648}}-{\frac { 230109186457887\,{p}^{18}{h}^{5}}{8589934592}}-{\frac {891115248823257 \,{p}^{19}{h}^{3}}{34359738368}}-{\frac {4881669867807723\,{p}^{20}h}{ 137438953472}}$, \par $A_{45}=554194295294679879984\,{h}^{45}+39362965900726633056\,p{h}^{43}+ 2959973227900487391\,{p}^{2}{h}^{41}+{\frac {472790595509415927\,{p}^{ 3}{h}^{39}}{2}}+{\frac {321971749339669677\,{p}^{4}{h}^{37}}{16}}+{ \frac {29333848336377675\,{p}^{5}{h}^{35}}{16}}+{\frac { 45985006658693745\,{p}^{6}{h}^{33}}{256}}+{\frac {9746506854402795\,{p }^{7}{h}^{31}}{512}}+{\frac {9002198684193567\,{p}^{8}{h}^{29}}{4096}} +{\frac {285870961148823\,{p}^{9}{h}^{27}}{1024}}+{\frac { 2602166222135403\,{p}^{10}{h}^{25}}{65536}}+{\frac {844639104181119\,{ p}^{11}{h}^{23}}{131072}}+{\frac {1247528861178057\,{p}^{12}{h}^{21}}{ 1048576}}+{\frac {252908840631825\,{p}^{13}{h}^{19}}{1048576}}+{\frac {855800427325917\,{p}^{14}{h}^{17}}{16777216}}+{\frac {356809084455699 \,{p}^{15}{h}^{15}}{33554432}}+{\frac {532554096813723\,{p}^{16}{h}^{ 13}}{268435456}}+{\frac {28716835828749\,{p}^{17}{h}^{11}}{134217728}} -{\frac {398090966021613\,{p}^{18}{h}^{9}}{4294967296}}-{\frac { 983201250012705\,{p}^{19}{h}^{7}}{8589934592}}-{\frac { 6735895639287969\,{p}^{20}{h}^{5}}{68719476736}}-{\frac { 51089630811025563\,{p}^{21}{h}^{3}}{549755813888}}-{\frac { 1110523927325938473\,{p}^{22}h}{8796093022208}} $, \par $A_{49}= 515844680321852815028832\,{h}^{49}+32961783591325975299120\,p{h}^{47}+ 2217961739175425919036\,{p}^{2}{h}^{45}+$\par${\frac {315133878583666626003 \,{p}^{3}{h}^{43}}{2}}+{\frac {94814551550640933927\,{p}^{4}{h}^{41}}{ 8}}+{\frac {3787623211889821185\,{p}^{5}{h}^{39}}{4}}+{\frac { 645212007742318605\,{p}^{6}{h}^{37}}{8}}+$\par${\frac {3765282218997607935\, {p}^{7}{h}^{35}}{512}}+{\frac {1477531861770819915\,{p}^{8}{h}^{33}}{ 2048}}+{\frac {313794615563382465\,{p}^{9}{h}^{31}}{4096}}+{\frac { 145420138664361261\,{p}^{10}{h}^{29}}{16384}}+$\par${\frac { 148925406295325835\,{p}^{11}{h}^{27}}{131072}}+{\frac { 85668471069217551\,{p}^{12}{h}^{25}}{524288}}+{\frac { 13954513249092609\,{p}^{13}{h}^{23}}{524288}}+{\frac { 10148990043310371\,{p}^{14}{h}^{21}}{2097152}}+$\par${\frac { 31811088804006807\,{p}^{15}{h}^{19}}{33554432}}+{\frac { 25711943256475155\,{p}^{16}{h}^{17}}{134217728}}+{\frac { 10166014410124383\,{p}^{17}{h}^{15}}{268435456}}+{\frac { 7042540335027723\,{p}^{18}{h}^{13}}{1073741824}}+$\par${\frac { 4488743637561879\,{p}^{19}{h}^{11}}{8589934592}}-{\frac { 14409338464147941\,{p}^{20}{h}^{9}}{34359738368}}-{\frac { 30196613342204865\,{p}^{21}{h}^{7}}{68719476736}}-{\frac { 198392485810791933\,{p}^{22}{h}^{5}}{549755813888}}-$\par${\frac { 5911762450029857199\,{p}^{23}{h}^{3}}{17592186044416}}-{\frac { 31891120592767324251\,{p}^{24}h}{70368744177664}} $}. \end{thm} \begin{rem} Note that the coefficients of the monomial $h^{N-2j}p^j$ of $ A_{N}$ are not well-defined for $j>(N-5)/4$ because they do not lie in the stable range. We would like to thank Dieter Kotschick for pointing this out. One might however view the above formulas as a definition of these addititonal terms. One also sees that many of the invariants out of the stable range are negative whereas all those inside the stable range are positive (this was also pointed out to us by Dieter Kotschick). \end{rem} \subsection{The $SO(3)$-case} We consider first the $SO(3)$-invariants on $\widehat {\Bbb P}_2$ with respect to Chern classes $(H,c_2)$, and we put $N:=4c_2-4$. For $0<\epsilon <<1$ the polarisation $L_\epsilon:=H-\epsilon E$ of $\widehat {\Bbb P}_2$ lies in a chamber of type $(H,c_2)$ related to the polarisation $H$ of ${\Bbb P}_2$. Thus (\ref{subl}) gives $$\Phi^{{\Bbb P}_2,H}_{H,N}(\check H^{N-2r}{pt}^r) =\Phi^{\widehat{\Bbb P}_2,L_{\epsilon}}_{H,N}(\check H^{N-2r}{pt}^r).$$ Putting $L_{1-\epsilon}:=H-(1-\epsilon) E$ we obtain $\Phi^{\widehat {\Bbb P}_2,L_{1-\epsilon}}_{H}(\check H^{N-2r}{pt}^r)=0$. Thus we get $$\Phi^{{\Bbb P}_2,H}_{H}(\check H^{N-2r} {pt}^r)= \sum_{\xi\in W^{\widehat{\Bbb P}_2}_{H,c_2}(H-E,H)}(-1)^{e_{\xi,N}} \delta_{\xi,N}(\check H^{N-2r}{pt}^r),$$ and, using lemma \ref{wallp1p1}, we can again carry out the computation with Botts formula. For $N:=4i$ we denote $$B_N:=2^{2i}\sum_{j=0}^{2i} \Phi^{{\Bbb P}_2,H}_{H}(\check H^{N-2j}{pt}^j)h^{N-2j}p^j.$$ Then we obtain: \begin{thm}\label{p2so} The $SO(3)$-Donaldson invariants of ${\Bbb P}_2$ are \par {\tolerance 10000\small\raggedright $B_{0}=1$ , $B_{4}=3\,{h}^{4}+5\,{h}^{2}p+19\,{p}^{2}$, $B_{8}/8=29\,{h}^{8}+19\,{h}^{6}p+17\,{h}^{4}{p}^{2}+23\,{h}^{2}{p}^{3}+85\,{p} ^{4}$, \par $B_{12}=69525\,{h}^{12}+26907\,{h}^{10}p+12853\,{h}^{8}{p}^{2}+7803\,{h}^{6}{p }^{3}+6357\,{h}^{4}{p}^{4}+8155\,{h}^{2}{p}^{5}+29557\,{p}^{6}$, \par $B_{16}/8=6231285\,{h}^{16}+1659915\,{h}^{14}p+519777\, {h}^{12}{p}^{2}+194439\,{ h}^{10}{p}^{3}+88701\,{h}^{8}{p}^{4}+51027\,{h}^{6} {p}^{5}+39753\,{h}^ {4}{p}^{6}+49519\,{h}^{2}{p}^{7}+176837\,{p}^{8}$, \par $B_{20}=68081556995\,{h}^{20}+13571675125\,{h}^{18}p+3084569555\,{h}^{16}{p}^{ 2}+808382629\,{h}^{14}{p}^{3}+247407779\,{h}^{12}{p}^{4}+89811541\,{h} ^{10}{p}^{5}+39553139\,{h}^{8}{p}^{6}+21987589\,{h}^{6}{p}^{7}+ 16652099\,{h}^{4}{p}^{8}+20329653\,{h}^{2}{p}^{9}+71741715\,{p}^{10}$, \par $B_{24}/8=19355926872345\,{h}^{24}+3046788353175\,{h}^{22}p+535206161485\,{h}^{ 20}{p}^{2}+105824308635\,{h}^{18}{p}^{3}+23774344785\,{h}^{16}{p}^{4}+ 6132120911\,{h}^{14}{p}^{5}+1838332965\,{h}^{12}{p}^{6}+651103923\,{h} ^{10}{p}^{7}+279395017\,{h}^{8}{p}^{8}+151590087\,{h}^{6}{p}^{9}+ 112496445\,{h}^{4}{p}^{10}+135266955\,{h}^{2}{p}^{11}+472659585\,{p}^{ 12}$, \par $B_{28}=536625215902182969\,{h}^{28}+69259301021976999\,{h}^{26}p+ 9817859613586809\,{h}^{24}{p}^{2}+1538955926660199\,{h}^{22}{p}^{3}+ 268722697637049\,{h}^{20}{p}^{4}+52689438785319\,{h}^{18}{p}^{5}+ 11702994789369\,{h}^{16}{p}^{6}+2974340336103\,{h}^{14}{p}^{7}+ 875889126201\,{h}^{12}{p}^{8}+304140743847\,{h}^{10}{p}^{9}+ 127923966585\,{h}^{8}{p}^{10}+68135251815\,{h}^{6}{p}^{11}+49776298425 \,{h}^{4}{p}^{12}+59127015975\,{h}^{2}{p}^{13}+204876497145\,{p}^{14}$, \par $B_{32}/8=332465777488176686045\,{h}^{32}+36176961518799287203\,{h}^{30}p+ 4270943660527526777\,{h}^{28}{p}^{2}+550013108311246927\,{h}^{26}{p}^{ 3}+77722220365607813\,{h}^{24}{p}^{4}+12129004922528395\,{h}^{22}{p}^{ 5}+2104879834580993\,{h}^{20}{p}^{6}+409294250544727\,{h}^{18}{p}^{7}+ 89934657950957\,{h}^{16}{p}^{8}+22556396083123\,{h}^{14}{p}^{9}+ 6542216760905\,{h}^{12}{p}^{10}+2235172850335\,{h}^{10}{p}^{11}+ 925169690645\,{h}^{8}{p}^{12}+485534741275\,{h}^{6}{p}^{13}+ 350230091345\,{h}^{4}{p}^{14}+411833933095\,{h}^{2}{p}^{15}+ 1416634092797\,{p}^{16}$, \par $B_{36}=17982292064097834276691197\,{h}^{36}+1685376850354867108198203\,{h}^{ 34}p+169728914674713290425549\,{h}^{32}{p}^{2}+18446964561578451602667 \,{h}^{30}{p}^{3}+2174127485943121961373\,{h}^{28}{p}^{4}+ 279319741333450241435\,{h}^{26}{p}^{5}+39339602087475090285\,{h}^{24}{ p}^{6}+6111138005878747467\,{h}^{22}{p}^{7}+1054025359144892989\,{h}^{ 20}{p}^{8}+203321142108471291\,{h}^{18}{p}^{9}+44233113780975117\,{h}^ {16}{p}^{10}+10964566444466603\,{h}^{14}{p}^{11}+3139014782527197\,{h} ^{12}{p}^{12}+1058019835991643\,{h}^{10}{p}^{13}+432158763674797\,{h}^ {8}{p}^{14}+224042778598923\,{h}^{6}{p}^{15}+159901382125437\,{h}^{4}{ p}^{16}+186411458197691\,{h}^{2}{p}^{17}+637107121682253\,{p}^{18}$, \par $B_{40}/8=19983831593150830258093037499 \,{h}^{40}+1640698532032417214980201941\, {h}^{38}p+143617787626796457582947271\,{h}^{36}{p}^{2}+ 13451663520190902994423761\,{h}^{34}{p}^{3}+1353428584063925323593987 \,{h}^{32}{p}^{4}+146907352128976242766365\,{h}^{30}{p}^{5}+ 17282999997688436388975\,{h}^{28}{p}^{6}+2214864601846913417145\,{h}^{ 26}{p}^{7}+310874334747308389131\,{h}^{24}{p}^{8}+48070219333713236901 \,{h}^{22}{p}^{9}+8241254396581767639\,{h}^{20}{p}^{10}+ 1577751227160324321\,{h}^{18}{p}^{11}+340134212696649171\,{h}^{16}{p}^ {12}+83440287229631085\,{h}^{14}{p}^{13}+23620292992955391\,{h}^{12}{p }^{14}+7869891016663881\,{h}^{10}{p}^{15}+3178622918644059\,{h}^{8}{p} ^{16}+1630875748081269\,{h}^{6}{p}^{17}+1153440155417319\,{h}^{4}{p}^{ 18}+1334613223327473\,{h}^{2}{p}^{19}+4535236702668195\,{p}^{20}$, \par $B_{44}/8=226901192268190530686926956861797\,{h}^{44}+ 16542462134525153318253326085835\,{h}^{42}p+$ \par $1277706977403778580365852666661\,{h}^{40}{p}^{2}+ 104862798979925329727378003659\,{h}^{38}{p}^{3}+$\par$ 9174416297780080293761973989\,{h}^{36}{p}^{4}+ 858689743856030000767365835\,{h}^{34}{p}^{5}+ 86310585758469215596920485\,{h}^{32}{p}^{6}+9355633875773319246298315 \,{h}^{30}{p}^{7}+1098557533992391977544805\,{h}^{28}{p}^{8}+$\par$ 140418552503311458801355\,{h}^{26}{p}^{9}+19640467303990766625317\,{h} ^{24}{p}^{10}+3023185118099492260555\,{h}^{22}{p}^{11}+$\par$ 515310612119604105701\,{h}^{20}{p}^{12}+97958161753205459659\,{h}^{18} {p}^{13}+20943663715791766949\,{h}^{16}{p}^{14}+$\par$5090445779293122763\,{ h}^{14}{p}^{15}+1426864216020459365\,{h}^{12}{p}^{16}+ 470672723850968779\,{h}^{10}{p}^{17}+188268529044707621\,{h}^{8}{p}^{ 18}+95733138877112011\,{h}^{6}{p}^{19}+67173305015551205\,{h}^{4}{p}^{ 20}+77210866621686475\,{h}^{2}{p}^{21}+261019726029655205\,{p}^{22}$,\par $B_{48}/64=401623524463671616144253869033873677\,{h}^{48}+ 26294009028509419866433950400817907\,p{h}^{46}+$ \par $1814139310232228402229320933713849\,{p}^{2}{h}^{44}+ 132233743700306798807714195145903\,{p}^{3}{h}^{42}+$ \par $ 10210502184961866655190088128661\,{p}^{4}{h}^{40}+ 837650587235973991054920612155\,{p}^{5}{h}^{38}+$ \par $ 73245138148540706205679224225\,{p}^{6}{h}^{36}+ 6850202117661264825075213975\,{p}^{7}{h}^{34}+$ \par $ 687815006512629065815415005\,{p}^{8}{h}^{32} +74447573563889724907246275\,{p}^{9}{h}^{30} +8724562938113746968261705\,{p}^{10}{h}^{28}+$ \par $ 1112238016349638764497855\,{p}^{11}{h}^{26} +155030522200663787180517\,{p}^{12}{h}^{24} +23757432754397656762251\,{p}^{13}{h}^{22} +$ \par $ 4027259666817871766449\,{p}^{14}{h}^{20}+ 760537452815695217703\,{p}^{15}{h}^{18} +161380994483655259053\,{p}^{16}{h}^{16} +$ \par $ 38900268404198860691\,{p}^{17}{h}^{14}+10809116358008226777 \,{p}^{18}{h}^{12}+3534337177551658959\,{p}^{19}{h}^{10}+$ \par $ 1401766305125084725\,{p}^{20}{h}^{8}+707191549935960795\,{p}^{21}{h}^{ 6}+492754565374149825\,{p}^{22}{h}^{4}+563040363143655095\,{p}^{23}{h} ^{2}+1894476461608956285\,{p}^{24}$. } \end{thm} \begin{conj} For all $n$ there is a nonnegative integer $l(n)$ such that $B_{4n}/2^{l(n)}$ is a polynomial in $h$ and $p$ all of whose coefficients are odd positive integers. \end{conj} \begin{rem} $B_0$ to $B_{16}$ were already computed in \cite{K-L} also using blowup and wall-crossing formulas, showing that ${\Bbb P}_2$ is not of simple type. Apart from slightly different conventions their results agree with ours. Their results and the computations of the $SU(2)$ invariants by various other authors have been quite useful to check the correctness of our programs -- and thus of the computations in section 3 -- in earlier stages of our work. The conjecture above could already have been made on the basis of their result. \end{rem} \section{Wall-crossing formulas} In our paper \cite{E-G} we formulated a conjecture about the shape of the wall-crossing formula, compatible with the conjecture of Kotschick and Morgan \cite{K-M}. Here we state a slightly stronger form of the conjecture which is also supported by the computations in \cite{E-G}. \begin{conj}\label{wallconj} In the polynomial ring on $H^2(S,{\Bbb Q})$ we have $$\delta_{\xi,N,r}=(-4)^{-r}\sum_{k=0}^d {(N-2r)!\over (N-2r-2d+2k)!(d-k)!}Q_{k}(N,d,r,K_S^2) L_{\xi/2}^{N-2r-2d+2k}q_S^{d-k},$$ where $Q_{k}(N,d,r,K_S^2)$ is a polynomial of degree $k$ in $N,d,r,K_S^2$, which is independent of $S$ and $\xi$. \end{conj} We now will show, that, assuming the conjecture, we can compute several of the $Q_{k}(N,d,r,K^2_S)$. This computation will also give a check of the conjecture in many specific cases. For all $i\ge 0$ we put $$P_i(N,d,r,K^2):={-2d+2N+2K^2-24r+{\frac{13+3i}{2}}\choose i} +(3N-288r){-2d+2N+2K^2-24r+{\frac{7+3i}{2}} \choose i-2}.$$ \begin{prop}\label{wallprop1} If conjecture \ref{wallconj} is true, then for $i=0,1,2,3,4$ we can write $Q_{i}(N,d,r,K^2)=P_i(N,d,r,K^2)+R_{i}(N,d,r,K^2),$ where $R_{i}(N,d,r,K^2)=0$ for $i<2$ and \newline {\raggedright $ R_2(N,d,r,K^2)={\frac {69}{8}}$, \newline $ R_3(N,d,r,K^2)= -13\,d+29\,N+17\,K^2-5452\,r+91$, \newline $ R_4(N,d,r,K^2)= {\frac {35\,{d}^{2}}{4}} -{\frac {99\,dN}{2}}-{\frac {51\,d(K^2)}{2} }+10802\,dr+{\frac {181\,{N}^{2}}{4}}+{\frac {115\,N(K^2)}{2}}- 12050\,Nr+{\frac {67\,{(K^2)}^{2}}{4}}-10898\,(K^2)\,r+169836\,{ r}^{2}-{\frac {537\,d}{4}}+{\frac {2821\,N}{8}}+{\frac {761\,(K^2)} {4}}-146495\,r+{\frac {72005}{128}}.$} \end{prop} \begin{pf} We assume conjecture \ref{wallconj}. Let $X={\Bbb P}_1\times {\Bbb P}_1$ or a blow up of ${\Bbb P}_1\times {\Bbb P}_1$ in finitely many points. We denote by $F$ and $G$ the pullbacks of the fibres of the two projections from ${\Bbb P}_1\times {\Bbb P}_1$ to ${\Bbb P}_1$. For a class $\xi=F-sG$ in $H^2(X,{\Bbb Z})$ defining a wall and an integer $d\ge 0$, let $N:=4d+2s-3$. Then on $X$ we can determine the coefficients $a_{k}$ of $L_{\xi/2}^{N-2r-2d+2k}q_X^{d-k}$ in $\delta_{\xi,N,r}$ as follows: We can assume that $X$ has an action of ${\Bbb C}^*$ with finitely many fixpoints as in \ref{torS}. For $x$ an indeterminant we put $\alpha:=-xF+G$ and compute the polynomial $\delta_{\xi,N,r}(\alpha^{N-2r})$ in $x$ using the Bott residue formula on ${\text{\rom{Hilb}}}^d(X\sqcup X)$, from which we can compute the $a_k$. Now we can compute the polynomials $Q_k(N,d,r,K_X^2)$ as follows: We consider all nonnegative integers $d,w,b,r$ with $d\ge k$ and $d+w+b+r\le 2k$. Let $X$ be the blow up of ${\Bbb P}_1\times {\Bbb P}_1$ in $b$ points. With the method of the last paragraph we compute the coefficient $c_{d,w,b,r}$ of $L_{\xi/2}^{N-2r-2d+2k}q_X^{d-k}$ in $\delta_{\xi,N,r}$ on $X$, where $N:=4d+2w+1$ and $\xi=F-(w+2)G$. Using all the $c_{d,w,b,r}$ we obtain a system of ${k+4 \choose 4}$ linear equations for the coefficients of the $N^id^jr^t(K_S^2)^s$ (with $0\le i+j+t+s\le k$) in $Q_k(N,d,r,K_S^2)$. Solving this system of equations we obtain our result. All the computations are again carried out using a suitable Maple program. \end{pf} The formulas suggest the following conjecture: \begin{conj} \label{wallconj1} \begin{enumerate} \item For all $i$ the polynomial $Q_i(N,d,r,K^2)$ is of the form $Q_{i}(N,d,r,K^2)=P_{i}(N,d,r,K^2)+R_{i}(N,d,r,K^2),$ where $R_{i}(N,d,r,K^2)$ is a polynomial in $N,d,r,K^2$ of degree $i-2$. \item If we view $R_i(N,d,r,K^2)$ as a polynomial in $N,-d,-r,K^2$, then all its coefficients are positive and the same is true for $Q_i(N,d,r,K^2)$. \end{enumerate} \end{conj} \begin{prop}\label{wallprop2} If conjecture \ref{wallconj} and part (1) of conjecture \ref{wallconj1} are true then \newline {\raggedright $R_5(N,d,r,K^2)= 11\,{{(K^2)}}^{3}+57\,{{(K^2)}}^{2}N-25\,{{(K^2)}}^{2}d-10892\,{{ (K^2)}}^{2}r+90\,{(K^2)}\,{N}^{2}-98\,{(K^2)}\,Nd-24088\,{(K^2)}\, Nr+17\,{(K^2)}\,{d}^{2}+21592\,{(K^2)}\,dr+339600\,{(K^2)}\,{r}^{2} +44\,{N}^{3}-82\,{N}^{2}d-13304\,{N}^{2}r+41\,N{d}^{2}+23896\,Ndr+ 363792\,N{r}^{2}-3\,{d}^{3}-10700\,{d}^{2}r-338448\,d{r}^{2}-2525760\, {r}^{3}+198\,{{(K^2)}}^{2}+744\,{(K^2)}\,N-276\,{(K^2)}\,d-303600\, {(K^2)}\,r+618\,{N}^{2}-612\,Nd-333888\,Nr+78\,{d}^{2}+301008\,dr+ 5457888\,{r}^{2}+1213\,{(K^2)}+2506\,N-729\,d-3101884\,r+2490 $, \newline \smallskip $R_6(N,d,r,K^2)= {\frac {65\,{{(K^2)}}^{4}}{12}}+{\frac {113\,{{(K^2)}}^{3}N}{3}}-{ \frac {49\,{{(K^2)}}^{3}d}{3}}-{\frac {21772\,{{(K^2)}}^{3}r}{3}}+{ \frac {179\,{{(K^2)}}^{2}{N}^{2}}{2}}-97\,{{(K^2)}}^{2}Nd-24076\,{{ (K^2)}}^{2}Nr+{\frac {33\,{{(K^2)}}^{2}{d}^{2}}{2}}+21580\,{{(K^2)} }^{2}dr+339528\,{{(K^2)}}^{2}{r}^{2}+{\frac {263\,{(K^2)}\,{N}^{3}}{ 3}}-163\,{(K^2)}\,{N}^{2}d-$ \par $26596\,{(K^2)}\,{N}^{2}r+81\,{(K^2)}\,N{ d}^{2}+47768\,{(K^2)}\,Ndr+727440\,{(K^2)}\,N{r}^{2}-{\frac {17\,{ (K^2)}\,{d}^{3}}{3}}-21388\,{(K^2)}\,{d}^{2}r-676752\,{(K^2)}\,d{r} ^{2}-5050944\,{(K^2)}\,{r}^{3}+{\frac {365\,{N}^{4}}{12}}-{\frac {247 \,{N}^{3}d}{3}}-{\frac {29332\,{N}^{3}r}{3}}+{\frac {147\,{N}^{2}{d}^{ 2}}{2}}+26404\,{N}^{2}dr+389208\,{N}^{2}{r}^{2}-{\frac {65\,N{d}^{3}}{ 3}}-23692\,N{d}^{2}r-725136\,Nd{r}^{2}-5327424\,N{r}^{3}+{\frac {{d}^{ 4}}{12}}+{\frac {21196\,{d}^{3}r}{3}}+337224\,{d}^{2}{r}^{2}+5041728\, d{r}^{3}+24147648\,{r}^{4}+{\frac {821\,{{(K^2)}}^{3}}{6}}+{\frac { 3127\,{{(K^2)}}^{2}N}{4}}-{\frac {565\,{{(K^2)}}^{2}d}{2}}-314198\,{ {(K^2)}}^{2}r+1306\,{(K^2)}\,{N}^{2}-{\frac {2567\,{(K^2)}\,Nd}{2}} -691258\,{(K^2)}\,Nr+{\frac {309\,{(K^2)}\,{d}^{2}}{2}}+623020\,{ (K^2)}\,dr+11250984\,{(K^2)}\,{r}^{2}+{\frac {7987\,{N}^{3}}{12}}- 1154\,{N}^{2}d-380192\,{N}^{2}r+{\frac {2007\,N{d}^{2}}{4}}+685594\,Nd r+12076284\,N{r}^{2}-{\frac {53\,{d}^{3}}{6}}-308822\,{d}^{2}r- 11204904\,d{r}^{2}-93985056\,{r}^{3}+{\frac {249499\,{{(K^2)}}^{2}}{ 192}}+{\frac {529411\,{(K^2)}\,N}{96}}-{\frac {48265\,{(K^2)}\,d}{32 }}-{\frac {156136081\,{(K^2)}\,r}{24}}+{\frac {944107\,{N}^{2}}{192}} -{\frac {134657\,Nd}{32}}-{\frac {170651209\,Nr}{24}}+{\frac {41627\,{ d}^{2}}{192}}+{\frac {154931089\,dr}{24}}+{\frac {538005841\,{r}^{2}}{ 4}}+{\frac {1027343\,{(K^2)}}{192}}+{\frac {5033035\,N}{384}}-{\frac {406799\,d}{192}}-{\frac {1032193219\,r}{16}}+{\frac {7872921}{1024}}.$} \end{prop} \begin{pf} The method of the proof is very similar to that of proposition \ref{wallprop1}. Using conjecture \ref{wallconj1} we can reduce the number of equations of the linear system we have to solve. For the computation of $R_k$ we have in the notations of the proof of proposition \ref{wallprop1} only to consider $d,w,b,r$ with $d\ge k$ and $d+w+b+r\le 2k-2$. \end{pf} \begin{prop} Assume conjecture \ref{wallconj}. Then for $d\le 8$ and $K^2=8$ also conjecture \ref{wallconj1} holds. Furthermore we have \newline {\raggedright $R_7(N,7,r,8)={\frac {242\,{N}^{5}}{15}}-{\frac {16136\,{N}^{4}r}{3}}+277280\,{N}^{3 }{r}^{2}-5613696\,{N}^{2}{r}^{3}+50448384\,N{r}^{4}-{\frac {843153408 \,{r}^{5}}{5}}+584\,{N}^{4}-{\frac {926104\,{N}^{3}r}{3}}+14133648\,{N }^{2}{r}^{2}-215219520\,N{r}^{3}+1068715008\,{r}^{4}+{\frac {49259\,{N }^{3}}{6}}-{\frac {27036430\,{N}^{2}r}{3}}+327411912\,N{r}^{2}- 2834114784\,{r}^{3}+55637\,{N}^{2}-{\frac {491711972\,Nr}{3}}+ 3453170368\,{r}^{2}+{\frac {5432731\,N}{30}}-{\frac {7607891522\,r}{5} }+226878$, \newline $R_7(N,8,r,8)={\frac {242\,{N}^{5}}{15}}-{\frac {16136\,{N}^{4}r}{3}}+277280\,{N}^{3 }{r}^{2}-5613696\,{N}^{2}{r}^{3}+50448384\,N{r}^{4}-{\frac {843153408 \,{r}^{5}}{5}}+526\,{N}^{4}-{\frac {867848\,{N}^{3}r}{3}}+13357680\,{N }^{2}{r}^{2}-204584256\,N{r}^{3}+1020478464\,{r}^{4}+{\frac {40547\,{N }^{3}}{6}}-{\frac {24608014\,{N}^{2}r}{3}}+301879416\,N{r}^{2}- 2636445408\,{r}^{3}+{\frac {85365\,{N}^{2}}{2}}-{\frac {444885076\,Nr} {3}}+3161672456\,{r}^{2}+{\frac {1989263\,N}{15}}-{\frac {6895690692\, r}{5}}+162324$, \newline $R_8(N,8,r,8)={\frac {69\,{N}^{6}}{10}}-{\frac {35428\,{N}^{5}r}{15}}+147976\,{N}^{4 }{r}^{2}-3940224\,{N}^{3}{r}^{3}+52663680\,{N}^{2}{r}^{4}-{\frac { 1749924864\,N{r}^{5}}{5}}+{\frac {4618156032\,{r}^{6}}{5}}+{\frac { 17993\,{N}^{5}}{60}}-164166\,{N}^{4}r+9819288\,{N}^{3}{r}^{2}- 222171264\,{N}^{2}{r}^{3}+2193801408\,N{r}^{4}-{\frac {39855794688\,{r }^{5}}{5}}+{\frac {342159\,{N}^{4}}{64}}-{\frac {75235799\,{N}^{3}r}{ 12}}+{\frac {675173309\,{N}^{2}{r}^{2}}{2}}-5810679060\,N{r}^{3}+ 32006797596\,{r}^{4}+{\frac {9540439\,{N}^{3}}{192}}-{\frac {678142435 \,{N}^{2}r}{4}}+{\frac {28388710281\,N{r}^{2}}{4}}-68700098862\,{r}^{3 }+{\frac {646487951\,{N}^{2}}{2560}}-{\frac {602240211743\,Nr}{192}}+{ \frac {11796298005871\,{r}^{2}}{160}}+{\frac {10191068747\,N}{15360}}- {\frac {19195182347591\,r}{640}}+{\frac {23061793325}{32768}}$.} \end{prop} \begin{pf} The method is again similar to that of the proof of proposition \ref{wallprop1}. Now we carry out our computations on $X={\Bbb P}_1\times{\Bbb P}_1$. In the notation of the proof of proposition \ref{wallprop1} we have therefore $b=0$. For the computation of $R_k$ we consider nonnegative integers $d,w,r$ with $d+w+r\le 2k$ and $k\le d \le 8$. \end{pf} \begin{prop}\label{wallrul} Let $S$ be a rational ruled surface, then for $N\le 40$ and $d\le 8$ the conjectures \ref{wallconj} and \ref{wallconj1} are correct (and therefore also all the formulas above). \end{prop} \begin{pf} Any rational ruled surface $X$ is a degeneration of either or ${\Bbb P}_1\times {\Bbb P}_1$ or $\widehat {\Bbb P}_2$, and under the degeneration the ample cone of $X$ corresponds to a subcone of the ample cone of ${\Bbb P}_1\times {\Bbb P}_1$ (resp. $\widehat {\Bbb P}_2$). Therefore it is enough to prove the result for ${\Bbb P}_1\times {\Bbb P}_1$ and $\widehat {\Bbb P}_2$. We let $c_1$ run through $0,F,G,F+G$ on ${\Bbb P}_1\times {\Bbb P}_1$ and through $0,H,E,F$ on $\widehat {\Bbb P}_2$ ($F=H-E$). For $S={\Bbb P}_1\times{\Bbb P}_1$ and $S=\widehat {\Bbb P}_2$ we consider for all integers $d$ with $0\le d\le 8$ the set $W_{S,d}$ of all classes $\xi$, which define a wall of type $(c_1,c_2)$, such that $N:=4d-\xi^2-3\le 40$ and $\<\xi\cdot F\><0$. It is easy to see that \begin{eqnarray*} W_{{\Bbb P}_1\times{\Bbb P}_1,d}&=&\Big\{\xi=aF-bG\Bigm| a>0,b>0,2ab\le 40-4d+3\Big\},\\ W_{{\widehat{\Bbb P}_2},d}&=&\Big\{\xi=bH-aE\Bigm| a>b>0, a^2-b^2\le 40 -4d+3\}. \end{eqnarray*} For all $d\le 8 $ and all $\xi\in W_{S,d}$ we again compute all the coefficients of $L_{\xi/2}^{N-2r-2d+2k}q_{S}^{d-k}$ with the method of the first paragraph of the proof of proposition \ref{wallprop1}. \end{pf} \section{The Donaldson Invariants of birationally ruled surfaces.} In this section we will show that our algorithm for computing the wall-crossing formula $\delta_{\xi,N}$ and the blowup formulas enable us to compute all the Donaldson invariants of all rational surfaces $X$ for all generic polarisations lying in a suitable subcone of the ample cone of $X$. \subsection{The case of rational ruled surfaces} In this case we can indeed determine the Donaldson invariants for all generic polarisations. For simplicity we will only compute the restriction of the Donaldson invariants to ${\text{\rm {Sym}}}^N(H_2(S,{\Bbb Q}))$. In \cite{K-L} some invariants of ${\Bbb P}_1\times {\Bbb P}_1$ have been computed also using blowup and wall-crossing formulas. Their results show e.g. that there is no chamber, for which ${\Bbb P}_1\times {\Bbb P}_1$ is of simple type. Our results again agree with theirs and earlier results e.g. in \cite{L-Q}. \begin{thm}\label{donrul} Let $S$ be a rational ruled surface, $F$ the class of a fibre and $q_S$ the quadratic form on $H_2(S,{\Bbb Z})$. We denote by $F_\epsilon$ the polarisation $F+\epsilon E$, where $E$ is the class of a section with nonpositive selfintersection. \noindent $(1)$ For $\epsilon>0$ sufficiently small we have $\Phi^{S,F_\epsilon}_{E,N}=0$ and $\Phi^{S,F_\epsilon}_{E+F,N}=0.$ \noindent $(2)$ For $\epsilon>0$ sufficiently small we have for $E_N:=\Phi^{S,F_\epsilon}_{0,N,0}$: \newline {\raggedright $E_5=-L_F^5+5/2L_F^3q_S-5/2L_Fq_S^2$, \newline $E_9=40L_F^9-108L_F^7q_S+108L_F^5q_S^2-42L_F^3q_S^3$, \newline $E_{13}=-9345L_F^{13}+26949L_F^{11}q_S-31590L_F^9q_S^2 +18018L_F^7q_S^3-4290L_F^5q_S^4$, \newline {\small $E_{17}=7369656L_F^{17}-22136040L_F^{15}q_S +28474320L_F^{13}q_S^2-19734960L_F^{11}q_S^3 +7425600L_F^9q_S^ 4-1225224L_F^7q_S^5$, \newline $E_{21}=-14772820744L_F^{21}+45586042992L_F^{19}q_S-62181472500L_F^{17}q_S^2 +48231175860L_F^{15}q_S^3-$ \newline $ 22562971200L_F^{13}q_S^4+6074420688L_F^{11}q_S^5-740703600L_F^9q_S^6$, \newline $E_{25}=63124363433664L_F^{25}-198545836440000L_F^{23}q_S +281925714232800L_F^{21}q_S^2- 235199340734400L_F^{19}q_S^3+$ \newline $ 125056219068000L_F^{17}q_S^4-42588214875360L_F^{15}q_S^5+ 8649138960000L_F^{13}q_S^6-813136737600L_F^{11}q_S^7$, \newline $E_{29}=-509894102555251905L_F^{29}+1626742370158553130L_F^{27}q_S -2378321090933081112L_F^{25}q_S^2+$ \newline $ 2087846466793743600L_F^{23}q_S^3-1207966082767844400L_F^{21}q_S^4 +473530658232013200 L_F^{19}q_S^5-$ \newline $ 123363365393268000L_F^{17}q_S^6+19623703009790880L_F^{15}q_S^7- 1467326424564000L_F^{13}q_S^8,$\newline $E_{33}=7135482220088837442520\,L_F^{33}-23016295766978863295760\,L_F^{31}q_S+ 34404291587748659734080\,L_F^{29}q_S^{2}-$ \newline $31360607908598315276160\,L_F^{27}q_S^{3}+19266231547036209415680\,L_F^{25} q_S^{4}- 8299005150626510918400\,L_F^{23}q_S^{5}+$ \newline $ 2515398487826672448000\,L_F^{ 21}q_S^{6}-519339581441771650560\,L_F^{19}q_S^{7}+66567414222758592000 \,L_F^{17}q_S^{8}-$ \newline $ 4055565288690115200\,L_F^{15}q_S^{9}.$}} \noindent $(3)$ For $\epsilon>0$ sufficiently small and all $N\le 33$ we have, writing $\Phi^{S,F_\epsilon}_{0,N,0}$ as a polynomial $E_N(L_F,q_S)$ in $L_F$ and $q_S$, $$\Phi^{S,F_\epsilon}_{F,N,0}=E_N(L_F,q_S) -E_N(L_F/2,q_S).$$ \end{thm} \begin{pf} (1) is just lemma \ref{vancham}. \noindent (2) and (3): We fix $N:=4c_2-3$ with $c_2>1$. We will just compute the corresponding Donaldson invariants explicitely. As any pair $(S,L)$ consisting of a Hirzebruch surface $S=\Sigma_n$ and $L=aF+bE\in Pic(S)$ (where $E$ is a section with selfintersection $-n\le 0$) can be deformed to either $({\Bbb P}_1\times {\Bbb P}_1, aF+b(E-nF/2))$ or $(\widehat {\Bbb P}_2,aF+b(E-(n-1)F/2))$ we see that we can assume that $S={\Bbb P}_1\times {\Bbb P}_1$ or $S=\widehat {\Bbb P}_2$ and $c_1=F$ or $c_1=0$. (a) $S={\Bbb P}_1\times{\Bbb P}_1$, $c_1=F$ (we will always denote by $F$ and $G$ the fibres of the projections to the two factors). By (1) we have for $\epsilon>0$ sufficiently small $\Phi^{{\Bbb P}_1\times{\Bbb P}_1,G+\epsilon F}_{F,N}=0$. Therefore we get $$\Phi^{{\Bbb P}_1\times{\Bbb P}_1,F_{\epsilon}}_{F,N}= -\sum_{\xi\in W^{{\Bbb P}_1\times {\Bbb P}_1}_{F,c_2}(F,G)} (-1)^{e_{\xi,N}}\delta_{\xi,N}.$$ So the invariants can be computed using proposition \ref{wallrul} and remark \ref{wallp1p1}. (b) $S=\widehat {\Bbb P}_2$, $c_1=F$. Let $\epsilon>0$ be sufficiently small. By the $SO(3)$-blow up formula we have for all $\alpha\in A_{N-i}({\Bbb P}_2)$: $$\Phi^{\widehat {\Bbb P}_2,H-\epsilon E}_{F,N}(\check E^i\alpha) =\Phi^{{\Bbb P}_2,H}_H(S_i({pt})\alpha),$$ and the $SO(3)$-invariants of ${\Bbb P}_2$ have been determined in theorem \ref{p2so}. Therefore $$\Phi^{\widehat {\Bbb P}_2,F_\epsilon }_{F,N}(\check E^i\alpha)=\Phi^{{\Bbb P}_2,H}_H(S_i({pt})\alpha) -\sum_{\xi\in W^{\widehat {\Bbb P}_2}_{F,c_2}(F,H)} (-1)^{e_{\xi,N}}\delta_{\xi,N}(\check E^i\alpha),$$ So the sum can be computed using proposition \ref{wallrul} and remark \ref{wallp1p1}. (c) $S=\widehat{\Bbb P}_2$, $c_1=0$. Let $\widetilde {\Bbb P}_2$ be the blowup of ${\Bbb P}_2$ in two points with exceptional divisors $E_1$ and $E_2$. Then $\widetilde {\Bbb P}_2$ is also the blow up of ${\Bbb P}_1\times {\Bbb P}_1$ in a point. We denote the exceptional divisor by $E$. We denote by $F$ the pullback of $F=H-E_1$ from $\widehat {\Bbb P}_2$ (which coincides with the pullback of $F$ from ${\Bbb P}_1\times {\Bbb P}_1$). We have $F=E_2+E$. We also denote by $G$ the pullback of $G$ from ${\Bbb P}_1\times {\Bbb P}_1$ and have $G=E_1+E$. For $1>>\epsilon>>\mu>0$, let $H_1:=F+\epsilon G-\mu E$ and $H_2:=F+\epsilon G- (\epsilon-\mu)E$. Then $H_2$ is a polarisation of $\widetilde {\Bbb P}_2$ which lies in a $(E_2,c_2)$-chamber related to the $(0,c_2)$-chamber of $F+\epsilon E_1$ on $\widehat {\Bbb P}_2$. Thus by the $SO(3)$-blowup formula we have $$\Phi^{\widehat {\Bbb P}_2,F+\epsilon E_1}_{0,N}(\check F^i\check E_1^{N-i})= -\Phi^{\widetilde{\Bbb P}_2,H_2}_{E_2,N+1}(\check E_2 \check F^i (\check G-\check E)^{N-i})= -\Phi^{\widetilde{\Bbb P}_2,H_2}_{F-E,N+1} ((\check F-\check E) \check F^i (\check G-\check E)^{N-i}).$$ We have $$ \Phi^{\widetilde{\Bbb P}_2,H_1}_{F-E,N+1}- \Phi^{\widetilde{\Bbb P}_2,H_2}_{F-E,N+1} =\sum_{\xi\in W^{\widetilde{\Bbb P}_2}_{F-E,c_2}(H_2,H_1)} (-1)^{e_{\xi,N+1}}\delta_{\xi,N+1},$$ and for $\epsilon$ sufficiently small is is easy to see that $$W^{\widetilde{\Bbb P}_2}_{F-E,c_2}(H_2,H_1)= \big\{(2a-1)F-(2b-1)E\bigm| b>a>0, b(b-1)\le c_2\big \}.$$ So $\Phi^{\widetilde{\Bbb P}_2,H_1}_{F-E,N+1}- \Phi^{\widetilde{\Bbb P}_2,H_2}_{F-E,N+1}$ can be computed by the Bott residue formula. Finally $H_1$ lies in a $(F-E,c_2)$-chamber on $\widetilde {\Bbb P}_2$ related to the $(F,c_2)$-chamber of $F+\epsilon G$ on ${\Bbb P}_1\times {\Bbb P}_1$. So, by the $SO(3)$-blowup formula (with exceptional divisor $E$), we get for $\alpha\in A_{N+1-i}( {\Bbb P}_1\times {\Bbb P}_1)$ $$\Phi^{\widetilde{\Bbb P}_2,H_1}_{F-E,N+1}(\alpha \check E^i)= \Phi^{ {\Bbb P}_1\times{\Bbb P}_1,F+\epsilon E}_{F}(\alpha S_i({pt})),$$ and the last is computed by the method of (a). Now we put everything together to get our result. (d) $S={\Bbb P}_1\times {\Bbb P}_1$, $c_1=0$. This case is very similar to (c), only with the role of ${\Bbb P}_1\times{\Bbb P}_1$ and $\widehat {\Bbb P}_2$ exchanged. We use the same notations as in (c). Now $H_1$ is a polarisation of $\widetilde {\Bbb P}_2$ which lies in a $(E,c_2)$-chamber related to the $(0,c_2)$-chamber of $F+\epsilon G$ on ${\Bbb P}_1\times {\Bbb P}_1$. Thus by the $SO(3)$-blowup formula we have $$\Phi^{{\Bbb P}_1\times{\Bbb P}_1,F+\epsilon G}_{0,N}(\check F^i\check G^{N-i})= -\Phi^{\widetilde{\Bbb P}_2,H_1}_{E,N+1}(\check E \check F^i \check G^{N-i})= -\Phi^{\widetilde{\Bbb P}_2,H_1}_{F-E_2,N+1} ((\check F-\check E_2) \check F^i (\check F+\check E_1-\check E_2)^{N-i}).$$ We have $$ \Phi^{\widetilde{\Bbb P}_2,H_1}_{E,N+1}- \Phi^{\widetilde{\Bbb P}_2,H_2}_{E,N+1} =\sum_{\xi\in W^{\widetilde{\Bbb P}_2}_{E,c_2}(H_2,H_1)} (-1)^{e_{\xi,N+1}}\delta_{\xi,N+1},$$ and for $\epsilon$ sufficiently small is is easy to see that $$W^{\widetilde{\Bbb P}_2}_{E,c_2}(H_2,H_1)= \big\{(2aF-(2b-1)E\bigm| b-1/2>a>0, b(b-1)\le c_2\big \}.$$ So $\Phi^{\widetilde{\Bbb P}_2,H_1}_{E,N+1}- \Phi^{\widetilde{\Bbb P}_2,H_2}_{E,N+1}$ can be computed by the Bott residue formula. Finally $H_2$ lies in a $(E,c_2)$-chamber on $\widetilde {\Bbb P}_2$ related to the $(F,c_2)$-chamber of $F+\epsilon E_1$ on $\widehat {\Bbb P}_2$. So, by the $SO(3)$-blowup formula (with exceptional divisor $E_2$), we get for $\alpha\in A_{N+1-i}(\widehat {\Bbb P}_2)$ $$\Phi^{\widetilde{\Bbb P}_2,H_2}_{E,N+1}(\alpha \check E_2^i)= \Phi^{\widehat {\Bbb P}_2,F+\epsilon E}_{F}(\alpha S_i({pt})),$$ and the last is computed by the method of (b). \end{pf} \begin{conj} For $S$ a rational ruled surface we have in the notation of theorem \ref{donrul} for all $N=4c_2 -3 $ with $c_2\ge 2$: \begin{enumerate} \item $\Phi^{S,F_\epsilon}_{0,N,0}$ and $\Phi^{S,F_\epsilon}_{F,N,0}$ are polynomials $E_{0,N}(L_F,q_S)$ and $E_{F,N}(L_F,q_S)$ in $L_F$ and $q_S$, which are independent of $S$. \item $E_{0,N}(L_F,q_S)$ and $E_{F,N}(L_F,q_S)$ are divisible by $L_F^{N-2c_2}$. \item $E_{F,N}(L_F,q_S)=E_{0,N}(L_F,q_S) -E_{0,N}(L_F/2,q_S).$ \end{enumerate} \end{conj} \begin{rem} We keep the notation of theorem \ref{donrul}. Notice that theorem \ref{donrul} and proposition \ref{wallrul} determines all the $SU(2)$- and $SO(3)$- Donaldson invariants of a rational ruled surface $S$ of degree at most $35$ for all generic polarisations: Fix $(c_1,c_2)$ and put $N:=4c_2-c_1^2-3$. If $L$ is a generic polarisation then $$\Phi^{L,S}_{c_1,N}=\Phi^{F_\epsilon,S}_{c_1,N}+\sum_{\xi\in W^S_{c_1,c_2}(F,L)} (-1)^{e_{\xi,N}}\delta_{\xi,N}.$$ This sum is given for $N\le 35$ by theorem \ref{donrul}, remark \ref{wallp1p1} and proposition \ref{wallrul}. \end{rem} \medskip \subsection{The Donaldson invariants of blowups of ${\Bbb P}_2$} We want to finish by showing that our methods give an algorithm for computing all the Donaldson invariants for all rational surfaces $X$ at least for polarisations lying in a reasonably big subcone $C^{g}$ of the ample cone $C_X$ of $X$. In \cite{K-L} it is shown that the Donaldson invariants of ${\Bbb P}_2$ and ${\Bbb P}_1\times {\Bbb P}_1$ can be determined from the wall-crossing formulas on some blowups, and our results can be seen as a generalization of this. A rational surface $X$, which is neither ${\Bbb P}_2$ nor ruled can be deformed to a a blowup ${\Bbb P}_2(x_1,\ldots x_r)$ of ${\Bbb P}_2$ in finitely many general points. Under this deformation $C_X$ corresponds to a (in general strict) subcone of the ample cone $C_{{\Bbb P}_2(x_1,\ldots x_r)}$. We can therefore restrict our attention to $X={\Bbb P}_2(x_1,\ldots x_r)$. \begin{thm} There exists an algorithm computing all the $SU(2)$- and $SO(3)$-Donaldson invariants of ${\Bbb P}_2(x_1,\ldots x_r)$ with respect to all generic polarisations in a nonempty open subcone $C^{g}$ of the ample cone of ${\Bbb P}_2(x_1,\ldots x_r)$. \end{thm} \begin{pf} Let $S=Y_r$, where $Y_0={\Bbb P}_2$ and $Y_i$ is obtained from $Y_{i-1}$ by blowing up a point such that each $Y_i$ carries an action of an algebraic $2$-torus with finitely many fixpoints satisfying conditions (1) and (2) of section \ref{torS}. This just means that each $Y_i$ is obtained from $Y_{i-1}$ by blowing up a fixpoint. We can deform $S$ to ${\Bbb P}_2(x_1,\ldots x_r)$, but under this deformation the good ample cone $C_S^g$ of $S$ will in general correspond to a proper subcone $C^g$ of $C_{{\Bbb P}_2(x_1,\ldots x_r)}$. Note that $C^g$ contains a neighbourhood of the hyperplane class $H$. It is enough to prove that there is such an algorithm computing the Donaldson invariants of $S$ for all generic polarisations in $C_S^g$. Fix $c_1\in Pic(S)$ and $c_2\in H^2(S,{\Bbb Z})$. Let $N:=4c_2-c_1^2-3$. Let $H_1$ and $H_2$ be two good generic polarisations of $S$. Then by lemma \ref{finwall} the set $W_{c_1,c_2}(H_1,H_2)$ is finite and consists only of good walls. Therefore $$\Phi_{c_1,N}^{S,H_2}=\Phi_{c_1,N}^{S,H_1}+\sum_{\xi\in W^S_{c_1,c_2}(H_1,H_2)} (-1)^{e_{\xi,N}}\delta_{\xi,N},$$ and all the $\delta_{\xi,N}$ can be determined explicitely by applying the Bott residue formula. So it is enough to determine $\Phi_{c_1,N}^{S,H_0}$ for one good polarisation $H_0$. We will denote by $E_1,\ldots,E_r$ the exceptional divisors of $S$ over ${\Bbb P}_2$. {\it First case $c_1\ne 0$.} Denote $c_1=a H+b_1 E_1+\ldots + b_r E_r$, with each of $a,b_1,\ldots,b_r$ lying in $\{ 0,1\}$. We denote $D_i:=a H+b_1 E_1+\ldots b_i + E_i$ By reordering the $E_i$ we can assume $a\ne 0$ or $b_1\ne 0$. Let $F:=H-E_1$. Then for $1>>\epsilon>>\delta_2>>\ldots >>\delta_r$ the divisors $H_{i}:=F+\epsilon E_1-(\delta_2 E_2+\ldots+\delta_i E_i) $ are polarisations on $Y_i$ lying in a chamber of type $(D_i,c_2)$ related to the chamber of type $(D_{i-1},c_2)$ of $H_{i-1}:=F+\epsilon E_1-(\delta_2 E_2+\ldots\delta_{i-1} E_{i-1})$ on $Y_{i-1}$. So the blowup formulas give $\Phi^{Y_i,H_{i}}_{D_i}(\alpha \check E_i^j) =\Phi^{Y_{i-1},H_{i-1}}_{D_{i-1}}(\alpha S_j({pt}))$ if $b_i=1$ (resp. $\Phi^{Y_i,H_{i}}_{D_i}(\alpha \check E_i^j)=\Phi^{Y_{i-1},H_{i-1}}_{D_{i-1}}(\alpha B_j({pt}))$ if $b_i=0$) for all $\alpha\in A_{N-j}(Y_{i-1})$. The proof of theorem \ref{donrul} gives an algorithm for computing $\Phi^{Y_1,H_{1}}_{D_1}(\alpha)$ for all $\alpha\in A_*(Y_{1})$. Thus by induction we get the desired algorithm. {\it Second case $c_1=0$.} Let $\widehat S$ be the blow up of $S$ in a point and let $E$ be the exceptional divisor. By the first case we have an algorithm for computing the Donaldson invariant $\Phi_{E,N}^{\widehat S,H_\epsilon}$ for a polarisation $H_\epsilon=H_0-\epsilon E$ on $\widehat S$ lying in a related chamber to that of a generic good polarisation $H_0$ of $S$. Then the $SO(3)$-blowup formula gives $\Phi_{0}^{S,H_0}(\alpha)=\Phi_{E}^{\widehat S,H_\epsilon}(\check E\alpha)$, and the result follows. \end{pf}

9506

alg-geom/9506004

en

https://arxiv.org/abs/alg-geom/9506004

alg-geom/9506004

Steven Duplij

Steven Duplij

On Alternative Supermatrix Reduction

22 pages, Standard LaTeX with AmS fonts

Lett.Math.Phys. 37 (1996) 385

10.1007/BF00312670

KL-TH-95/15

We consider a nonstandard odd reduction of supermatrices (as compared with the standard even one) which arises in connection with possible extension of manifold structure group reductions. The study was initiated by consideration of the generalized noninvertible superconformal-like transformations. The features of even- and odd-reduced supermatrices are investigated on a par. They can be unified into some kind of "sandwich" semigroups. Also we define a special module over even- and odd-reduced supermatrix sets, and the generalized Cayley-Hamilton theorem is proved for them. It is shown that the odd-reduced supermatrices represent semigroup bands and Rees matrix semigroups over a unit group.

alg-geom

\section{Introduction} According to the general theory of $G$-structures \cite{che1,gui,kobayashi} various geometries are obtained by a reduction of a structure group of a manifold to some subgroup $G$ of the tangent space endomorphisms. In the local approach using coordinate description this means that one should reduce a corresponding matrix in a given representation to some reduced form as a matter of fact. In the most cases this form is triangle, because of the simple observation from the ordinary matrix theory that the triangle matrices preserve the shape and form a subgroup. In supersymmetric theories, despite of appearance of odd subspaces and anticommuting variables, the choice of the reduction shape remained the same\cite{howe,lot,ros/sch1,schw4}, and a ground reason of this was the fully identity of the supermatrix multiplication with the ordinary one, and consequently the shape of the matrices from a subgroup was the same. However in fine search of nontrivial supersymmetric manifestations one can observe that the closure of multiplication can be also achieved for other shapes, but due to existence of zero divisors in the Grassmann algebra or in the ring over which a theory is defined. So the meaning of the reduction itself could be extended principally. Evidently, that some ''good'' properties of the transformations could be lost in this direction, but opening of new possibilities, beauty and interesting and unusual features which are distinctive for supersymmetric case only are the sufficient price for the surprises arisen and reason for them to investigate. This paper was initiated by the study of superconformal symmetry semigroup extensions \cite{dup6,dup7}. Indeed superconformal transformations \cite {bar/fro/sch1,cra/rab,coh} appear as a result of the reduction of the structure group matrix to the triangle form \cite{gid/nel3,gid/nel1}. Also, the transition functions on semirigid surfaces \cite{dis/nel,gov/nel/won} (see \cite{dol/ros/sch}) occurred in the description of topological supergravity \cite{gov/nel/rey} have the same shape. In \cite{dup6} we considered an alternative version of the reduction. The superconformal-like transformations obtained in this way have many unusual features, e. g. they are noninvertible and twist parity of the tangent space in the supersymmetric basis\footnote{ This situation is different from the case of $Q$-manifolds \cite {ale/kon/sch/zab}, where changing parity of the tangent space is done by hand from the first definitions. }. Here we study the alternative reduction of supermatrices from a more abstract viewpoint without connecting a special physical model. \section{Preliminaries} Let $\Lambda $ be\footnote{ The standard notations can be found in \cite{berezin,lei1}, and here we list some of them needed only. } a commutative Banach ${\Bbb Z}_2$-graded superalgebra over a field ${\Bbb K}$ (where ${\Bbb K}={\Bbb R,}$ ${\Bbb C}$ or ${\Bbb Q}_p$) with a decomposition into the direct sum: $\Lambda =\Lambda _0\oplus \Lambda _1$. The elements $a$ from $\Lambda _0$ and $\Lambda _1$ are homogeneous and have the fixed even and odd parity defined as $\left| a\right| \stackrel{def}{=}\left\{ i\in \left\{ 0,1\right\} ={\Bbb Z}_2|\,a\in \Lambda _i\right\} $. The even homomorphism ${\frak m}_b:\Lambda \rightarrow {\Bbb B}$ , where ${\Bbb B}$ is a purely even algebra over ${\Bbb K}$, is called a body map, if for any other purely even algebra ${\Bbb A}$ and any homomorphism ${\frak m}_a:\Lambda \rightarrow {\Bbb A}$ there is an even homomorphism ${\frak m}_{ab}:{\Bbb B}% \rightarrow {\Bbb A}$ such that ${\frak m}_a={\frak m}_{ab}\circ {\frak m}_b$. The kernel of ${\frak m}_b$ is ${\Bbb S\ }\equiv \ker {\frak m}_b\stackrel{def}{=}% \left\{ a\in \Lambda |\,{\frak m}\left( a\right) =0\right\} $ and is called the soul sector of $\Lambda $. If there are exists an embedding ${\frak n}:% {\Bbb B\hookrightarrow \Lambda }$ such that ${\frak m}\circ {\frak n}=id$, then $% \Lambda $ admits the body and soul decomposition $\Lambda ={\Bbb B}\oplus {\Bbb S}$, and a soul map can be defined as ${\frak m}_s:\Lambda \rightarrow {\Bbb S}$. Usually the isomorphism ${\Bbb B\ }\cong {\Bbb K}$ is implied (which is not necessary in general and can lead to very nontrivial behavior of the body). This is the case when $\Lambda $ is modeled with the Grassmann algebras $\wedge \left( N\right) $ having $N$ generators \cite {rog1,rab/cra1,vla/vol} or $\wedge \left( \infty \right) $ \cite {rog4,boy/git,khr2}, or with the free graded-commutative Banach algebras $% \wedge _BE$ over Banach spaces \cite{jad/pil,pes7,bry1}. The soul ${\Bbb S}$ is obviously a proper two-sided ideal of $\Lambda $ which is generated by $% \Lambda _1$. In case $\Lambda $ is a Banach algebra (with a norm $\left| \left| \cdot \right| \right| $) soul elements are quasinilpotent \cite{iva2}% , which means $\forall a\in {\Bbb S},\lim \limits_{n\rightarrow \infty }\left| \left| a\right| \right| ^{1/n}=0$. But in the infinite-dimensional case quasinilpotency of the soul elements does not necessarily lead to their nilpotency ($\forall a\in {\Bbb S}\,\exists n,\,a^n=0$) \cite{pes5}. These facts allow us to consider noninvertible morphisms on a par with invertible ones (in some sense), which gives, in proper conditions, many interesting and nontrivial results (see \cite{dup6,dup10,dup11}). The $\left( p|q\right) $-dimensional linear model superspace $\Lambda ^{p|q}$ over $\Lambda $ (in the sense of \cite{rog1,vla/vol,khr2,dewitt}) is the even sector of the direct product $\Lambda ^{p|q}=\Lambda _0^p\times \Lambda _1^q$. The even morphisms $\mbox{{\rm Hom}}_0\left( \Lambda ^{p|q},\Lambda ^{m|n}\right) $ between superlinear spaces $\Lambda ^{p|q}\rightarrow \Lambda ^{m|n}$ are described by means of $\left( m+n\right) \times \left( p+q\right) $-supermatrices (for details see \cite {berezin,lei1}). In various physical applications supermatrices are reduced to some suitable form which is necessary for concrete consideration. For instance, in the theory of super Riemann surfaces \cite{fri,ros/sch/vor1} the $\left( 1+1\right) \times \left( 1+1\right) $-supermatrices describing holomorphic morphisms of the tangent bundle have a triangle shape \cite {gid/nel1,gid/nel3}. Here we consider a special alternative reduction of supermatrices and study its features. We note that the supermatrix theory per se has many own problems \cite{bac/fel1,ebn,hus/nie} and unexpected conclusions (e.g. the lowering of the degree of characteristic polynomials comparing to the standard Cayley-Hamilton theorem\cite{urr/mor,urr/mor1}). For transparency and clarity we confine ourselves to $\left( 1+1\right) \times \left( 1+1\right) $-supermatrices\footnote{which will allow us not to melt ideas by large formulas, and only this size will be used in the following consideration} , and generalization to the $\left( m+n\right) \times \left( p+q\right) $ case is straightforward and can be mostly done by means of simple changing of notations. \section{Structure of $\mbox{{\rm Mat}} _\Lambda \left( 1|1\right) $} In the standard basis in $\Lambda ^{1|1}$ the elements from $\mbox{{\rm Hom}}% _0\left( \Lambda ^{1|1},\Lambda ^{1|1}\right) $ are described by the $\left( 1+1\right) \times \left( 1+1\right) $-supermatrices \cite{berezin} \begin{equation} \label{1}M\equiv \left( \begin{array}{cc} a & \alpha \\ \beta & b \end{array} \right) \in \mbox{{\rm Mat}} _\Lambda \left( 1|1\right) \end{equation} \noindent where $a,b\in \Lambda _0,\,\alpha ,\beta \in \Lambda _1$ (in the following we use Latin letters for elements from $\Lambda _0$ and Greek letters for ones from $\Lambda _1$). For sets of matrices we also use corresponding bold symbols, e. g. $\ {\bf M}\stackrel{def}{=}\left\{ M\in \mbox{{\rm Mat}} _\Lambda \left( 1|1\right) \right\} $. In this simple $% \left( 1|1\right) $ case the supertrace defined as $\mbox{str}:\mbox{% Mat}_\Lambda \left( 1|1\right) \rightarrow \Lambda _0$ and Berezinian defined as $\mbox{{\rm Ber}} :\mbox{{\rm Mat}} _\Lambda \left( 1|1\right) \setminus \left\{ M|\,{\frak m}_b\left( b\right) =0\right\} \rightarrow \Lambda _0$ are \begin{equation} \label{2}\mbox{str}M=a-b, \end{equation} \begin{equation} \label{3}\mbox{{\rm Ber}} M=\frac ab+\frac{\beta \alpha }{b^2}. \end{equation} Now we define two types of possible reductions of $M$ on a par and study some of their properties simultaneously. \begin{definition} {\sl Even-reduced supermatrices} are elements from $\mbox{{\rm Mat}}_\Lambda \left( 1|1\right) $ having the form \begin{equation} \label{4}S\equiv \left( \begin{array}{cc} a & \alpha \\ 0 & b \end{array} \right) \in \mbox{{\rm RMat}} _\Lambda ^{\,even}\left( 1|1\right) . \end{equation} \noindent {\sl Odd-reduced supermatrices} are elements from $\mbox{{\rm Mat}} % _\Lambda \left( 1|1\right) $ having the form \begin{equation} \label{5}T\equiv \left( \begin{array}{cc} 0 & \alpha \\ \beta & b \end{array} \right) \in \mbox{{\rm RMat}} _\Lambda ^{\,odd}\left( 1|1\right) . \end{equation} \end{definition} The name of the odd-reduced supermatrices follows naturally from $\mbox{{\rm Ber}} T=\beta \alpha /b^2\Rightarrow \left( \mbox{{\rm Ber}} T\right) ^2=0$ and \begin{equation} \label{5a}\mbox{{\rm Ber}} T^2=\mbox{{\rm Ber}} \left( \begin{array}{cc} \alpha \beta & \alpha b \\ \beta b & b^2-\alpha \beta \end{array} \right) =0. \end{equation} The explanation of the ground of the notations $S$ and $T$ comes from the fact that the even-reduced supermatrices give superconformal transformations which describe morphisms of the tangent bundle over the super Riemann surfaces \cite{gid/nel1}, while the odd-reduced supermatrices give the superconformal-like transformations twisting the parity of the $\left( 1|1\right) $ tangent superspace in the standard basis (see \cite{dup6,dup10}). \begin{assertion} ${\bf M}$ is a direct sum of diagonal ${\bf D}$ and anti-diagonal (secondary diagonal) ${\bf A}$ supermatrices (the even and odd ones in the notations of \cite{berezin}) \begin{equation} \label{8}{\bf M=D\oplus A}, \end{equation} \noindent where \begin{equation} \label{9}D{\bf \equiv }\left( \begin{array}{cc} a & 0 \\ 0 & b \end{array} \right) \in {\bf D}\equiv \mbox{{\rm Mat}} _\Lambda ^{\,Diag}\left( 1|1\right) , \end{equation} \begin{equation} \label{10}A{\bf \equiv }\left( \begin{array}{cc} 0 & \alpha \\ \beta & 0 \end{array} \right) \in {\bf A}\equiv \mbox{{\rm Mat}} _\Lambda ^{\,Adiag}\left( 1|1\right) , \end{equation} \noindent and ${\bf D\subset S}$ and ${\bf A\subset T}$. \end{assertion} For the reduced supermatrices one finds \begin{equation} \label{st}{\bf S\cap T}=\left( \begin{array}{cc} 0 & \alpha \\ 0 & b \end{array} \right) {\bf \neq \emptyset .} \end{equation} Nevertheless, the following observation explains the fundamental role of $% {\bf S}$ and ${\bf T}$. \begin{proposition} The Berezians of even- and odd-reduced supermatrices are additive components of the full Berezinian \begin{equation} \label{st1}\mbox{{\rm Ber}}M=\mbox{{\rm Ber}}S+\mbox{{\rm Ber}}T. \end{equation} \end{proposition} The first term in (\ref{st1}) covers all subgroups of even-reduced supermatrices from $\mbox{Mat}_\Lambda \left( 1|1\right) $, and only it was considered in the applications. But the second term is dual to the first in some sense and corresponds to all subsemigroups of odd-reduced supermatrices from $\mbox{Mat}_\Lambda \left( 1|1\right) $% \footnote{ The relation (\ref{st1}) is a supersymmetric version of the obvious equality $\det M=\det D+\det A$, when $D$ and $A$ from (\ref{8}) and (\ref{9}) are ordinary matrices. The problem is that for $A$ being a supermatrix $\mbox{{\rm Ber}}\,A$ is not defined at all. }. \section{Invertibility and ideals of $\mbox{{\rm Mat}} _\Lambda \left( 1|1\right) $} Denote the set of invertible elements of ${\bf M}$ by ${\bf M^{*}}$, and $% {\bf I=M\setminus M^{*}}$. In \cite{berezin} it was proved that ${\bf M^{*}}% =\left\{ M\in {\bf M}|\,{\frak m}_b\left( a\right) \neq 0\wedge {\frak m}% _b\left( b\right) \neq 0\right\} $. Then similarly ${\bf S^{*}}=\left\{ S\in {\bf S}|\,{\frak m}_b\left( a\right) \neq 0\wedge {\frak m}_b\left( b\right) \neq 0\right\} $ and ${\bf T^{*}}=\emptyset $, i. e. the odd-reduced matrices are noninvertible and ${\bf T}\subset {\bf I}$. Consider the invertibility structure of $\mbox{{\rm Mat}} _\Lambda \left( 1|1\right) $ in more detail. Let us denote \begin{equation} \label{10a} \begin{array}{ccc} {\bf M^{\prime }} & = & \left\{ M\in {\bf M|\,}{\frak m}_b\left( a\right) \neq 0\right\} , \\ {\bf M^{\prime \prime }} & = & \left\{ M\in {\bf M|\,}{\frak m}_b\left( b\right) \neq 0\right\} , \\ {\bf I^{\prime }} & = & \left\{ M\in {\bf M|\,}{\frak m}_b\left( a\right) =0\right\} , \\ {\bf I^{\prime \prime }} & = & \left\{ M\in {\bf M|\,}{\frak m}_b\left( b\right) =0\right\} . \end{array} \end{equation} \noindent Then ${\bf M}={\bf M^{\prime }\cup I^{\prime }=M^{\prime \prime }\cup I^{\prime \prime }}$ and ${\bf M^{\prime }\cap I^{\prime }=\emptyset }$ , ${\bf M^{\prime \prime }\cap I^{\prime \prime }=\emptyset }$, therefore $% {\bf M}^{*}={\bf M^{\prime }}\cap {\bf M^{\prime \prime }}$ and ${\bf T}% \subset {\bf M}^{\prime \prime }$. The Berezinian $\mbox{{\rm Ber}} M$ is well-defined for the matrices from ${\bf M}^{\prime \prime }$ only and is invertible when $M\in {\bf M}^{*}$, but for the matrices from ${\bf M}% ^{\prime }$ the inverse $\left( \mbox{{\rm Ber}} M\right) ^{-1}$ is well-defined and is invertible when $M\in {\bf M}^{*}$ too \cite{berezin}. Under the ordinary matrix multiplication the set ${\bf M}$ is a semigroup of all $\left( 1|1\right) $ supermatrices \cite{mca1,put2,put1}, and the set ${\bf M}^{*}$ is a subgroup of ${\bf M}$. In the standard basis ${\bf M}^{*}$ represents the general linear group $GL_\Lambda \left( 1|1\right) $ \cite{berezin}. According to the general definitions \cite{cli/pre1} a subset ${\bf I}% \subset {\bf M}$ is called a right (left) ideal of the semigroup ${\bf M}$, if ${\bf I}\cdot {\bf M}\subset {\bf I}$ (${\bf M}\cdot {\bf I}\subset {\bf I% }$), where the point denotes the standard matrix set multiplication: ${\bf % A\cdot B}\stackrel{def}{=}\left\{ \bigcup AB|\,A\in {\bf A},\,B\in {\bf B}% \right\} $. An isolated ideal satisfies the relation \cite{cli/pre1} \begin{equation} \label{11}M_1M_2\in {\bf I}\Rightarrow M_1\in {\bf I}\vee M_2\in {\bf I,} \end{equation} \noindent and a filter ${\bf F}$ of the semigroup ${\bf M}$ is defined by \begin{equation} \label{12}M_1M_2\in {\bf F}\Rightarrow M_1\in {\bf F\wedge }M_2\in {\bf F}. \end{equation} \begin{proposition} 1) The sets ${\bf I}$, ${\bf I^{\prime }}$ and ${\bf I^{\prime }}$ are isolated ideals of ${\bf M}$. 2) The sets ${\bf M}^{*}$, ${\bf M}^{^{\prime }}$ and ${\bf M}^{^{\prime \prime }}$ are filters of the semigroup ${\bf M}$. 3) The sets ${\bf M^{\prime }}$ and ${\bf M^{\prime \prime }}$ are not subgroups\footnote{as it was incorrectly translated in \cite {berezin}, pp. 95, 103 (in the original Russian edition, Moscow, 1983, pp. 89, 93, the sets ${\bf M^{\prime }}$ and ${\bf M^{\prime \prime }}$, denoted as $G^{\prime }Mat\left( 1,1|\Lambda \right) $ and $G^{\prime \prime }Mat\left( 1,1|\Lambda \right) $ respectively, are called semigroups). }, but subsemigroups of ${\bf M}$, which are ${\bf M^{\prime }=M}^{*}\cup {\bf J% }^{\prime }$ and ${\bf M^{\prime \prime }=M}^{*}\cup {\bf J^{\prime \prime }} $ with the isolated ideals ${\bf J}^{\prime }={\bf M^{\prime }}\setminus {\bf M}^{*}={\bf M^{\prime }}\cap {\bf I^{\prime \prime }}$ and ${\bf J}% ^{\prime \prime }={\bf M^{\prime \prime }}\setminus {\bf M}^{*}={\bf % M^{\prime \prime }}\cap {\bf I^{\prime }}$ respectively. 4) The ideal of the semigroup ${\bf M}$ is \begin{equation} \label{13}{\bf I}={\bf I}^{\prime }\cup {\bf J}^{\prime }={\bf I}^{\prime \prime }\cup {\bf J}^{\prime \prime }. \end{equation} \end{proposition} \begin{proof} Let $M_3=M_1M_2$, then $a_3=a_1a_2+\alpha _1\beta _2$ and $% b_3=b_1b_2+\beta _1\alpha _2$. Taking the body part we derive \begin{equation} \label{14}{\frak m}_b\left( a_3\right) ={\frak m}_b\left( a_1\right) {\frak m}% _b\left( a_2\right) , \end{equation} \begin{equation} \label{15}{\frak m}_b\left( b_3\right) ={\frak m}_b\left( b_1\right) {\frak m}% _b\left( b_2\right) . \end{equation} 1) The left-hand side of (\ref{14}) and (\ref{15}) vanishes iff the first or second multiplier of the right-hand side equals zero. Then use (\ref{11}). 2) The left-hand side of (\ref{14}) and (\ref{15}) does not vanish iff the first and second multiplier of the right-hand side does not equal zero. Then use (\ref{12}). 3) ${\bf J}^{\prime }$ consists of the matrices with ${\frak m}_b\left( a\right) \neq 0$, but ${\frak m}_b\left( b\right) =0$, and ${\bf J^{\prime \prime }}$ consists of the matrices with ${\frak m}_b\left( a\right) =0$, but $% {\frak m}_b\left( b\right) \neq 0$. 4) The ideal of ${\bf M}$ consists of the matrices with ${\frak m}_b\left( a\right) =0$ or ${\frak m}_b\left( b\right) =0$. Then use 3) and the definitions. \end{proof} \begin{remark} Since the ideals ${\bf J^{\prime }}$ and ${\bf % J^{\prime \prime }}$ are isolated, i. e. ${\bf J^{\prime }\cap M}^{*}={\bf % J^{\prime }\cap M}^{*}=\emptyset $, they cannot be represented as sequences of elements from the group ${\bf M}^{*}$ (viz. no one noninvertible element can be derived from a sequence of invertible ones, see, e. g. \cite{iva2}), and so the statement ''that any element of $G^{\prime \prime }Mat\left( p,q|\Lambda \right) $ (the semigroup ${\bf M}^{\prime \prime }$ here, and so the notation $G^{\prime \prime }...$ misleads) is the limit of a sequence from $GMat\left( p,q|\Lambda \right) $ (the group ${\bf M}^{*}$ here)'' holds only for invertible elements from ${\bf M}^{\prime \prime }$, i.e. belonging ${\bf M^{*}}$, and it means that elements from ${\bf M}^{*}$ can be obtained from a sequence of elements from ${\bf M}^{*}$, which is simply a group action. \end{remark} \begin{assertion} For the odd-reduced matrices from (\ref{10a}) it follows ${\bf T}\subset {\bf I}^{\prime }$ and ${\bf A}\cap {\bf M^{\prime }=A}\cap {\bf M^{\prime \prime }=\emptyset }$. \end{assertion} \section{Multiplication properties of odd-reduced supermatrices} In general the odd-reduced matrices do not form a semigroup, since \begin{equation} \label{15a}T_1T_2=\left( \begin{array}{cc} \alpha _1\beta _2 & \alpha _1b_2 \\ b_1\beta _2 & b_1b_2+\beta _1\alpha _2 \end{array} \right) \neq T. \end{equation} But from (\ref{15a}) it follows that \begin{equation} \label{15b} \begin{array}{ccc} {\bf T}\cdot {\bf T}\cap {\bf T}\neq \emptyset & \Rightarrow & \alpha \beta =0, \\ {\bf T}\cdot {\bf T}\cap {\bf S}\neq \emptyset & \Rightarrow & \beta b=0, \end{array} \end{equation} \noindent which can take place, because of the existence of zero divisors in $\Lambda $. \begin{proposition} 1) The subset ${\bf T}^{SG}\subset {\bf T}$ of the odd-reduced matrices satisfying $\alpha \beta =0$ form an odd-reduced subsemigroup of ${\bf M}$. 2) In the odd-reduced semigroup ${\bf T}^{SG}$ the subset of matrices with $% \beta =0$ is a left ideal, and one with $\alpha =0$ is a right ideal, the matrices with $b=0$ form a two-sided ideal. \end{proposition} \subsection{Semigroup band representations} Let \begin{equation} \label{b0}Z_\alpha \left( t\right) =\left( \begin{array}{cc} 0 & \alpha t \\ \alpha & 1 \end{array} \right) \in {\bf Z}_\alpha {\bf \subset T}^{SG}, \end{equation} i.e. ${\bf Z}_\alpha $ is a set of the odd-reduced matrices parameterized by the even parameter $t \in \Lambda_0$. Then ${\bf Z}_\alpha $ is a semigroup under the matrix multiplication ($\alpha $ numbers the semigroups) which is isomorphic to a one parameter semigroup with the multiplication \begin{equation} \label{b1}\left\{ t_1\right\} *_\alpha \left\{ t_2\right\} =\left\{ t_1\right\} . \end{equation} This semigroup is called a right zero semigroup ${\cal Z}_R=\left\{ \bigcup \left\{ t\right\} ;*_\alpha \right\} $ and plays an important role (together with the left zero semigroup ${\cal Z}_L$ defined in a dual manner) in the general semigroup theory (e.g., see \cite{cli/pre1}, Theorem 1.27, and \cite {howie}). Let \begin{equation} \label{b11}B_\alpha \left( t,u\right) =\left( \begin{array}{cc} 0 & \alpha t \\ \alpha u & 1 \end{array} \right) \in {\bf B}_\alpha {\bf \subset T}^{SG}, \end{equation} then ${\bf B}_\alpha $ is a matrix semigroup (numbered by $\alpha $) which is isomorphic to a two $\Lambda_0$-parametric semigroup ${\cal B}=\left\{ \bigcup \left\{ t,u\right\} ;*_\alpha \right\} $, where the multiplication is \begin{equation} \label{b2}\left\{ t_1,u_1\right\} *_\alpha \left\{ t_2,u_2\right\} =\left\{ t_1,u_2\right\} . \end{equation} Here every element is an idempotent (as in the previous case too), and so this is a rectangular band multiplication \cite{howie,petrich2}. Let $C_\alpha \left( t,u,v\right) =\left( \begin{array}{cc} 0 & \alpha t \\ \alpha u & v \end{array} \right) \in {\bf C}_\alpha {\bf \subset T}^{SG}$, then ${\bf C}_\alpha $ is a matrix semigroup isomorphic to a semigroup ${\cal B}_G=\left\{ \bigcup \left\{ t,u,v\right\} ;*_\alpha \right\} $ where the multiplication is \begin{equation} \label{b3}\left\{ t_1,u_1,v_1\right\} *_\alpha \left\{ t_2,u_2,v_2\right\} =\left\{ t_1v_2,u_2v_1,v_1v_2\right\} . \end{equation} The parameter $v$ describes the difference of an element from an idempotent, since $\left\{ t,u,v\right\} ^2-\left\{ t,u,v\right\} =\left\{ t\left( v-1\right) ,u\left( v-1\right) ,v\left( v-1\right) \right\} $. \begin{assertion} The one and two parametric subsemigroups of the semigroup of odd-reduced supermatrices ${\bf T}^{SG}$ having vanishing Berezinian represent semigroup bands, viz. the left and right zero semigroups and rectangular bands. \end{assertion} \begin{theorem} The continuous supermatrix representation of the Rees matrix semigroup over a unit group $G=e$ (see \cite{cli/pre1,howie}) is given by formulas (\ref{b0}% ) and (\ref{b11}). \end{theorem} \subsection{''Square root'' of even-reduced supermatrices} Consider the second equation in (\ref{15b}). \begin{proposition} The elements $T^{\sqrt{S}}$from the subset ${\bf T}^{\sqrt{S}}\subset {\bf T} $ of the odd-reduced matrices satisfying $\beta b=0$ can be interpreted as ''square roots'' of the even-reduced matrices $S$. \end{proposition} \begin{example} 1) Let $T^{\sqrt{S}}=\left( \begin{array}{cc} 0 & \alpha \\ \beta & \beta \gamma \end{array} \right) \in {\bf T}^{\sqrt{S}}$, then $\left( T^{\sqrt{S}}\right) ^2=\alpha \beta \left( \begin{array}{cc} 1 & \gamma \\ 0 & -1 \end{array} \right) \in {\bf S}\setminus {\bf S}^{*}$. 2) If $\gamma =0$ in 1), then we obtain $A^2=\alpha \beta \left( \begin{array}{cc} 1 & 0 \\ 0 & -1 \end{array} \right) \in {\bf D}\setminus {\bf D}^{*}$ (see (\ref{9}), (\ref{10}) and compare with $D^2=\left( \begin{array}{cc} a^2 & 0 \\ 0 & b^2 \end{array} \right) $). This could be accepted as a definition of a square root of $% \alpha \beta $ in some sense. Thus we have \begin{equation} \label{15c} \begin{array}{ccc} {\bf D\cdot D} & = & {\bf D}, \\ {\bf A\cdot A} & = & {\bf D}, \end{array} \end{equation} \noindent and the second relation could be formally considered as an ''odd branch'' of the root $\sqrt{D}$. \end{example} \section{Unification of reduced supermatrices} Now we try to unify the even- and odd-reduced matrices (\ref{4}) and (\ref{5}% ) into a common abstract object. To begin with consider the multiplication table of all introduced sets including the even-reduced matrices products \begin{equation} \label{16} \begin{array}{ccc} {\bf S\cdot S} & = & {\bf S,} \\ {\bf D\cdot D} & = & {\bf D,} \\ {\bf % D\cdot S} & = & {\bf S,} \\ {\bf S\cdot D} & = & {\bf S,} \end{array} \end{equation} \noindent and ones for the odd-reduced matrices \begin{equation} \label{17} \begin{array}{ccc} {\bf A\cdot T} & = & {\bf S}, \\ {\bf A\cdot S} & = & {\bf T,} \\ {\bf T}% \cdot {\bf A} & = & {\bf S}^{st}, \\ {\bf S\cdot A} & = & {\bf T}^\Pi , \\ {\bf S\cdot T} & = & {\bf S}\cup {\bf T} \\ {\bf T}\cdot {\bf S} & = & {\bf % T.} \end{array} \end{equation} \noindent Here $st:\mbox{{\rm Mat}} _\Lambda \left( 1|1\right) \rightarrow \mbox{{\rm Mat}} _\Lambda \left( 1|1\right) $ is a supertranspose \cite{berezin}, i. e. $\left( \begin{array}{cc} a & \alpha \\ \beta & b \end{array} \right) ^{st}=\left( \begin{array}{cc} a & \beta \\ -\alpha & b \end{array} \right) $. Also we use the $\Pi $-transpose \cite{manin1} defined by $\Pi $$% : \mbox{{\rm Mat}} _\Lambda \left( 1|1\right) \rightarrow \mbox{{\rm Mat}} % _\Lambda \left( 1|1\right) $ and \begin{equation} \label{pc}\left( \begin{array}{cc} a & \alpha \\ \beta & b \end{array} \right) ^\Pi =\left( \begin{array}{cc} b & \beta \\ \alpha & a \end{array} \right) . \end{equation} Note that the sets of matrices ${\bf S}$ and ${\bf T}$ are not closed under $% st$ and $\Pi $ operations, but ${\bf S}^{st}\cap {\bf S=D}$ and ${\bf T}^\Pi \cap {\bf T}={\bf A}$. First we observe from the first two relations of (\ref{17}) that ${\bf A}$ plays a role of the left type-changing operator ${\bf A}:{\bf S}\rightarrow {\bf T}$ and ${\bf A}:{\bf T\rightarrow S}$, while ${\bf D}$ does not change the type. Next from the first two relations of (\ref{16}) it is obviously seen that the sets ${\bf S}$ and ${\bf D}$ are subsemigroups. Unfortunately, due to the next to last relation of (\ref{17}) the set ${\bf T}$ has no clear abstract meaning. However, the last relation ${\bf T}\cdot {\bf S}=% {\bf T}$ is important from another viewpoint: any odd-reduced morphism $% \Lambda ^{1|1}\rightarrow \Lambda ^{1|1}$ corresponding to ${\bf T}$ can be represented as a product of odd- and even-reduced morphisms, such that \begin{center} \begin{equation} \label{d1}\setlength{\unitlength}{.15in}% \begin{picture}(4,4) \put(0,3.1){\makebox(1,1){\large}} \put(3.1,3.05){\makebox(1,1){\large}} \put(3.1,0){\makebox(1,1){\large}} \put(0.3,2){\small${\bf T}$} \put(1.1,3){\vector(1,-1){1.8}} \put(2,3.8){\small${\bf S}$} \put(1,3.5){\vector(1,0){2.3}} \put(3.5,3.1){\vector(0,-1){1.8}} \put(3.7,2){\small${\bf T}$} \end{picture} \end{equation} \end{center} \noindent is a commutative diagram. This decomposition is crucial in the application to the superconformal-like transformations construction (see \cite{dup6}). \subsection{Reduced supermatrix set semigroup} To unify the introduced sets (\ref{16}) and (\ref{17}) we consider the triple products \begin{equation} \label{18} \begin{array}{ccc} {\bf S}\cdot {\bf A}\cdot {\bf T} & = & {\bf S,} \\ {\bf T}\cdot {\bf A}% \cdot {\bf T} & = & {\bf T,} \\ {\bf S}\cdot {\bf D}\cdot {\bf S} & = & {\bf % S,} \\ {\bf T}\cdot {\bf D}\cdot {\bf S} & = & {\bf T.} \end{array} \end{equation} Here we observe that the matrices ${\bf A}$ and ${\bf D}$ play the role of ''sandwich'' elements in a special ${\bf S}$ and ${\bf T}$ multiplication. Moreover, the sandwich elements are in one-to-one correspondence with the right sets on which they act, and so they are ''sensible from the right''. Therefore, it is quite natural to introduce the following \begin{definition} {\sl A sandwich right sensible product} of the reduced supermatrix sets $% {\bf R}={\bf S,T}$ is \begin{equation} \label{19}{\bf R}_1\odot {\bf R}_2\stackrel{def}{=}\left\{ \begin{array}{cc} {\bf R}_1\cdot {\bf D\cdot R}_2, & {\bf R}_2={\bf S,} \\ {\bf R}_1\cdot {\bf % A}\cdot {\bf R}_2, & {\bf R}_2={\bf T.} \end{array} \right. \end{equation} \end{definition} In terms of the sandwich product instead of (\ref{18}) we obtain \begin{equation} \label{20} \begin{array}{ccc} {\bf S\odot T} & = & {\bf S,} \\ {\bf T\odot T} & = & {\bf T,} \\ {\bf % S\odot S} & = & {\bf S,} \\ {\bf T\odot S} & = & {\bf T.} \end{array} \end{equation} \begin{proposition} The $\odot $-multiplication is associative. \end{proposition} \begin{proof} Consider the relations: \begin{equation} \label{21} \begin{array}{ccccc} \left( {\bf T\odot S}\right) {\bf \odot T} & = & \left( {\bf T\cdot D\cdot S}% \right) {\bf \cdot A\cdot T} & = & {\bf T\cdot D\cdot S\cdot A\cdot T,} \\ {\bf T\odot \left( S\odot T\right) } & = & {\bf T\cdot D\cdot \left( S\cdot A\cdot T\right) } & = & {\bf T\cdot D\cdot S\cdot A\cdot T,} \end{array} \end{equation} \noindent where the last equalities follow from the associativity of the ordinary matrix multiplication. Therefore, $\left( {\bf T\odot S}\right) {\bf \odot T}={\bf T\odot \left( S\odot T\right) }$. Other associativity relations can be proved in a similar way\footnote{ We stress here that the associativity does not follow from the associativity of the supermatrix multiplication only, but is a consequence of the special and nontrivial set multiplication table (\ref{20}). }. \end{proof} \begin{definition} The elements ${\bf S}$ and ${\bf T}$ form a semigroup under $\odot $% -multiplication (\ref{19}), which we call a {\sl reduced matrix set semigroup% } and denote ${\cal RMS}_{set}$. \end{definition} Comparing (\ref{b1}) and (\ref{20}) we observe that the reduced matrix set semigroup can be viewed as a right zero semigroup having two elements. \begin{assertion} The reduced matrix set semigroup is isomorphic to a special right zero semigroup, i. e. ${\cal RMS}_{set}\cong {\cal Z}_R=\left\{ {\bf R=S},{\bf % T;\odot }\right\} $. \end{assertion} \subsection{Scalars, anti-scalars and generalized modules} Now we introduce the analog of $\odot $-multiplication for the reduced matrices per se (not for sets). First we define the structure of generalized $\Lambda $-module in $\mbox{{\rm Hom}}_0\left( \Lambda ^{1|1},\Lambda ^{1|1}\right) $ in some alternative way, the even part of which is described in \cite{lei1} (in the ordinary matrix theory this is a trivial fact that the product of a matrix and a number is equal to a product of a matrix and a diagonal matrix having this number on the diagonal). \begin{definition} In $\mbox{{\rm Mat}} _\Lambda \left( 1|1\right) $ a {\sl scalar} (matrix) $E\left( x\right) $ and {\sl anti-scalar} (matrix) ${\cal E}\left( \chi \right) $ are defined by \begin{equation} \label{22} \begin{array}{c} E\left( x\right) \stackrel{def}{=} \left( \begin{array}{cc} x & 0 \\ 0 & x \end{array} \right) \in {\bf D}=\mbox{{\rm Mat}} _\Lambda ^{Diag}\left( 1|1\right) ,\;x\in \Lambda _{0,} \\ {\cal E}\left( \chi \right) \stackrel{def}{=} \left( \begin{array}{cc} 0 & \chi \\ \chi & 0 \end{array} \right) \in {\bf A}=\mbox{{\rm Mat}} _\Lambda ^{Adiag}\left( 1|1\right) ,\;\chi \in \Lambda _{1.} \end{array} \end{equation} \end{definition} \begin{assertion} The Berezin's queer subalgebra $Q_\Lambda \left( 1\right) \equiv \left( \begin{array}{cc} x & \chi \\ \chi & x \end{array} \right) \subset \mbox{{\rm Mat}} _\Lambda \left( 1|1\right) $ \cite {berezin} is a direct sum of the scalar and anti-scalar \begin{equation} \label{23}Q_\Lambda \left( 1\right) =E\left( x\right) \oplus {\cal E}\left( \chi \right) . \end{equation} \end{assertion} \begin{assertion} The anti-scalars anticommute ${\cal E}\left( \chi _1\right) {\cal E}\left( \chi _2\right) +{\cal E}\left( \chi _2\right) {\cal E}\left( \chi _1\right) =0$, and so they are nilpotent. \end{assertion} \begin{proposition} The structure of the generalized $\Lambda _0\oplus \Lambda _1$-module in \\ $\mbox{{\rm Hom}}_0\left( \Lambda ^{1|1},\Lambda ^{1|1}\right) $ is defined by action of the scalars and anti-scalars (\ref{22}). \end{proposition} This means that everywhere we exchange the multiplication of supermatrices by even and odd elements from $\Lambda $ with the multiplication by the scalar matrices and anti-scalar ones (\ref{22}). The relations containing the scalars are well-known \cite{lei1}, but for the anti-scalars we obtain new dual ones. Consider their action on elements ${\bf M\in }\mbox{{\rm Mat}} % _\Lambda \left( 1|1\right) $ in more detail. First we need \begin{definition} {\sl Left }${\cal P}$ {\sl and right }${\cal Q}$ {\sl anti-transpose }are $ \mbox{{\rm Hom}}_0\left( \Lambda ^{1|1},\Lambda ^{1|1}\right) \rightarrow \mbox{{\rm Hom}}_1\left( \Lambda ^{1|1},\Lambda ^{1|1}\right) $ mappings acting on $M\in {\bf M}$ as \begin{equation} \label{t1}\left( \begin{array}{cc} a & \alpha \\ \beta & b \end{array} \right) ^{{\cal P}}=\left( \begin{array}{cc} \beta & b \\ a & \alpha \end{array} \right) , \end{equation} \begin{equation} \label{t2}\left( \begin{array}{cc} a & \alpha \\ \beta & b \end{array} \right) ^{{\cal Q}}=\left( \begin{array}{cc} \alpha & a \\ b & \beta \end{array} \right) . \end{equation} \end{definition} \begin{corollary} The anti-transpose is a square root of the parity changing operator (\ref{pc}% ) in the following sense \begin{equation} \label{atr}{\cal PQ}={\cal QP}=\Pi . \end{equation} \end{corollary} \begin{assertion} The anti-transpose satisfy \begin{equation} \label{atr1} \begin{array}{ccc} \left( {\cal E}\left( \chi \right) M\right) ^{{\cal P}} & = & \chi M \\ \left( {\cal E}\left( \chi \right) M\right) ^{{\cal Q}} & = & \chi M^\Pi \\ \left( M{\cal E}\left( \chi \right) \right) ^{{\cal P}} & = & M^\Pi \chi \\ \left( M{\cal E}\left( \chi \right) \right) ^{{\cal Q}} & = & M\chi \end{array} \end{equation} \end{assertion} Thus the concrete realization of the right, left and two-sided generalized $% \Lambda _0\oplus \Lambda _1$-modules in $\mbox{{\rm Hom}}_0\left( \Lambda ^{1|1},\Lambda ^{1|1}\right) $ is determined by the actions \begin{equation} \label{mod1} \begin{array}{ccc} {\cal E}\left( \chi \right) M & = & \chi M^{ {\cal P}}, \\ M{\cal E}\left( \chi \right) & = & M^{ {\cal Q}}\chi , \\ {\cal E}\left( \chi _1\right) M{\cal E}\left( \chi _2\right) & = & \chi _1M^\Pi \chi _2, \end{array} \end{equation} \noindent together with the standard $\Lambda $-module structure \cite{lei1} \begin{equation} \label{mod2} \begin{array}{ccc} E\left( x\right) M & = & xM, \\ ME\left( x\right) & = & Mx, \\ E\left( x_1\right) ME\left( x_2\right) & = & x_1Mx_2. \end{array} \end{equation} \begin{corollary} The generalized $\Lambda _0\oplus \Lambda _1$-module relations are \begin{equation} \label{mod3} \begin{array}{ccc} \left( E\left( x\right) M\right) N & = & E\left( x\right) \left( MN\right) \\ \left( ME\left( x\right) \right) N & = & M\left( E\left( x\right) N\right) \\ M\left( NE\left( x\right) \right) & = & \left( MN\right) E\left( x\right) \\ \left( {\cal E}\left( \chi \right) M\right) N & = & {\cal E}\left( \chi \right) \left( MN\right) \\ \left( M{\cal E}\left( \chi \right) \right) N & = & M\left( {\cal E}\left( \chi \right) N\right) \\ M\left( N{\cal E}\left( \chi \right) \right) & = & \left( MN\right) {\cal E}\left( \chi \right) \end{array} \end{equation} \noindent where $M,N\in \mbox{{\rm Mat}} _\Lambda \left( 1|1\right) $. \end{corollary} \begin{proposition} The structure of the generalized $\Lambda _0\oplus \Lambda _1$-module in \\ $\mbox{{\rm Hom}}_1\left( \Lambda ^{1|1},\Lambda ^{1|1}\right) $ is determined by the analogous actions of {\sl odd scalar} \begin{equation} \label{24o}E\left( \chi \right) \stackrel{def}{=}\left( \begin{array}{cc} \chi & 0 \\ 0 & -\chi \end{array} \right) \in \mbox{{\rm Hom}}_1\left( \Lambda ^{1|1},\Lambda ^{1|1}\right) \end{equation} and {\sl odd anti-scalar} \begin{equation} \label{25o}{\cal E}\left( x\right) \stackrel{def}{=}\left( \begin{array}{cc} 0 & x \\ x & 0 \end{array} \right) \in \mbox{{\rm Hom}}_1\left( \Lambda ^{1|1},\Lambda ^{1|1}\right) \end{equation} respectively\footnote{ {}From (\ref{24o}) and (\ref{25o}) it is clear why we use the name ''anti-'' and not ''odd'' (as in \cite{berezin}) for the secondary diagonal matrices $% {\cal E}\left( \chi \right) $. }. \end{proposition} \subsection{Reduced supermatrix sandwich semigroup} One way to unify the even- (\ref{4}) and odd-reduced (\ref{5}) supermatrices into an object analogous to a semigroup is consideration of the sandwich multiplication similar to (\ref{19}), but on the level of matrices (not sets), by means of the scalars and anti-scalars as sandwich matrices. Indeed, the ordinary matrix product can be written as $M_1M_2=M_1E\left( 1\right) M_2$. But we cannot find an analog of this relation using anti-scalar, because among $\chi \in \Lambda _1$ there is no unity. Therefore, the only possibility to include ${\cal E}\left( \chi \right) $ into equal play is consideration of sandwich elements (\ref{22}) having arbitrary (or fixed by other special conditions) both arguments $x$ and $% \chi $. Thus we naturally come to \begin{definition} {\sl A sandwich right sensible} $\Lambda _0\oplus \Lambda _1$% {\sl -product} of the reduced supermatrices $R=S{\bf ,}T$ is \begin{equation} \label{26}R_1\star _XR_2\stackrel{def}{=}\left\{ \begin{array}{cc} R_1E\left( x\right) R_2, & R_2=S {\bf ,} \\ R_1{\cal E}\left( \chi \right) R_2, & R_2=T, \end{array} \right. \end{equation} \noindent where $X=\left\{ x,\chi \right\} \in ${\sl $\Lambda _0\oplus \Lambda _1$}. \end{definition} The $\star _X$-multiplication table coincides with (\ref{20}). The associativity can be proved similar to (\ref{21}). Therefore, we have \begin{proposition} Under {\sl $\Lambda _0\oplus \Lambda _1$}-multiplication the reduced matrices form a semigroup which we call {\sl a reduced matrix sandwich semigroup }${\cal RMSS}$. \end{proposition} \begin{assertion} The reduced matrix sandwich semigroup is isomorphic to a special right zero semigroup, i. e. ${\cal RMSS}\cong {\cal Z}_R=\left\{ R=\bigcup S\bigcup T% {\bf ;}\star _X\right\} $. \end{assertion} \subsection{Direct sum of reduced supermatrices} Another way to unify the reduced supermatrices is consideration of the connection between them and the generalized $\Lambda _0\oplus \Lambda _1$% -modules. \begin{definition} The {\sl reduced supermatrix direct space }${\cal RMDS}$ is a direct sum of the even-reduced supermatrix space and the odd-reduced one. \end{definition} In terms of sets we have ${\bf R_{\oplus }=S}\oplus {\bf T}$. \begin{assertion} In ${\cal RMDS}$ the scalar is the Berezin's queer subalgebra $Q_\Lambda \left( 1\right) $ (see (\ref{23})). \end{assertion} \begin{theorem} In ${\cal RMDS}$ the scalars play the same role for the even-reduced supermatrices, as the anti-scalars for the odd-reduced ones. \end{theorem} \begin{corollary} The eigenvalues of even- (\ref{4}) and odd-reduced (\ref{5}) supermatrices should be found from different equations, viz. \begin{equation} \label{eig1} \begin{array}{ccc} SV & = & E\left( x\right) V, \\ TV & = & {\cal E}\left( \chi \right) V, \end{array} \end{equation} \noindent where $V$ is a column vector, and they are \begin{equation} \label{eig2} \begin{array}{cc} x_1=a, & x_2=b, \\ \chi _1=\alpha , & \chi _2=\beta . \end{array} \end{equation} \noindent (see (\ref{4}) and (\ref{5})). \end{corollary} \begin{definition} The characteristic functions for even- and odd-reduced supermatrices are defined in ${\cal RMDS}$ by \begin{equation} \label{27} \begin{array}{ccc} H_S^{even}\left( x\right) & = & \mbox{{\rm Ber}} \left( E\left( x\right) -S\right) , \\ H_T^{odd}\left( \chi \right) & = & \mbox{{\rm Ber}} \left( {\cal E% }\left( \chi \right) -T\right) . \end{array} \end{equation} \end{definition} \begin{remark} In the standard $\Lambda $-module over $\mbox{{\rm Mat}} _\Lambda \left( 1|1\right) $ \cite{berezin} one derives characteristic functions and eigenvalues for any matrix (and for odd-reduced too) from the first equations in (\ref{eig1}) and (\ref{27}) and obtains different results (see, e. g. \cite{kob/nag1,urr/mor1}). \end{remark} Using (\ref{4}), (\ref{5}) we easily found \begin{equation} \label{28} \begin{array}{ccc} H_S^{even}\left( x\right) & = &{\displaystyle \frac{\left( x-a\right) \left( x-b\right) }{% \left( x-b\right) ^2}}, \\ H_T^{odd}\left( \chi \right) & = &{\displaystyle \frac{\left( \chi -\alpha \right) \left( \chi -\beta \right) }{b^2}}. \end{array} \end{equation} Here we observe the full symmetry between even- and odd-reduced supermatrices (for this purpose the cancellation in the first equation was avoided) and consistency with their $\Lambda _0\oplus \Lambda _1$% -eigenvalues (\ref{eig2}). The characteristic polynomial\footnote{ For a nonsupersymmetric matrix $M$ it evidently coincides with the characteristic function $P_M\left( x\right) =H_M\left( x\right) \equiv \det \left( Ix-M\right) $, where $I$ is a unity matrix. } of a supermatrix $M$ is defined by $P_M\left( M\right) =0$ and in complicated cases is constructed from the parts of the characteristic function $H_M\left( x\right) $ according to a special algorithm \cite {kob/nag1,urr/mor1}. Due to existence of zero divisors in $\Lambda $ the degree of $P_M\left( x\right) $ can be less than $n=p+q$ , $M\in \mbox{Mat% }_\Lambda \left( p|q\right) $. But this algorithm is not applicable for the odd-reduced and secondary diagonal supermatrices. As before, we introduce two dual characteristic polynomials and, using (\ref{28}), obtain the Cayley-Hamilton theorem in ${\cal RMDS}$. \begin{theorem}[The generalized Cayley-Hamilton theorem] The characteristic polynomials in the reduced supermatrix direct space are \begin{equation} \label{29} \begin{array}{ccc} P_S^{even}\left( x\right) & = & \left( x-a\right) \left( x-b\right) , \\ P_T^{odd}\left( \chi \right) & = & \left( \chi -\alpha \right) \left( \chi -\beta \right) . \end{array} \end{equation} \noindent and $P_S^{even}\left( S\right) =0$ for any $S$, but $% P_T^{odd}\left( T\right) =0$ for nilpotent $b$ only. \end{theorem} \begin{proof} The even case is well-known, but for clarity we repeat it too, demonstrating the avoiding of multiplication of a matrix by a constant and using instead the scalars and anti-scalars (\ref{22}), i. e. the introduced $\Lambda _0\oplus \Lambda _1$-module structure. Thus, considering simultaneously the even and odd cases we obtain \begin{equation} \label{30}P_S^{even}\left( S\right) =\left( S-E\left( a\right) \right) \left( S-E\left( b\right) \right) =\left( \begin{array}{cc} 0 & \alpha \\ 0 & b \end{array} \right) \left( \begin{array}{cc} a & \alpha \\ 0 & 0 \end{array} \right) =0, \end{equation} \begin{equation} \label{31} \begin{array}{c} P_T^{odd}\left( T\right) =\left( T- {\cal E}\left( \alpha \right) \right) \left( T-{\cal E}\left( \beta \right) \right) = \\ \left( \begin{array}{cc} 0 & 0 \\ \beta -\alpha & b \end{array} \right) \left( \begin{array}{cc} 0 & \alpha -\beta \\ 0 & b \end{array} \right) =\left( \begin{array}{cc} 0 & 0 \\ 0 & b^2 \end{array} \right) =0. \end{array} \end{equation} \end{proof} \section{Conclusions } We conclude that almost all above constructions are universal and ideas mostly do not depend on size of the supermatrices under consideration. In particular case of superconformal-like transformations it would be interesting to use the alternative reduction introduced here in building the objects analogous to super Riemann or semirigid surfaces, which can also lead to new topological-like models.

9506

alg-geom/9506021

en

https://arxiv.org/abs/alg-geom/9506021

alg-geom/9506021

Herbert Lange

H. Lange

A Vector Bundle of Rank 2 on P1 x P3

20 pages, LaTeX

The paper studies a rank 2 vector bundle on P1 x P3. Similarly to the Horrocks - Mumford bundle on P4 this vector bundle encodes a lot of geometric information. It is defined via the Serre construction by an abelian surface in P1 x P3. The bundle is stable with respect to O(1,1), its jumping lines are determined. Morover it is shown that it is closely related to a double structue on P1 x P1 embedded in P1 x P3 as well as to a certain pencil of quadrics of degree 4 in P3.

alg-geom

\section{Abelian surfaces in ${\bf P_1 \times P_3}$} It is shown in [L] that there is a two--parameter family of abelian surfaces admitting an embedding into $\Bbb{P}_1 \times \Bbb{P}_3$. Moreover, it is proven that every abelian surface in $\Bbb{P}_1 \times \Bbb{P}_3$ is a member of this family. Suppose $\varphi =$ $(\varphi_1, \varphi_3) : X \hookrightarrow P := \Bbb{P}_1 \times \Bbb{P}_3$ is such an embedding of an abelian surface $X$ over the field of complex numbers. In this section we recall some of its properties from [L] and derive some additional facts, which are needed subsequently.\\ \par Let $h_i$ denote the pullback of the hyperplane section class of $\Bbb{P}_i$ to $P$ for $i = 1$ and $3$.\\ \par {\bf (1)} the class $[X]$ of $X$ in $H^4(P, \Bbb{Z})$ is $$ [X] = 8h_1 h_3 + 6h^2_3 . $$ \par {\bf (2)} There is a commutative diagram $$ \begin{array}{rcl} 0 \rightarrow E \rightarrow X&\stackrel{q}{\longrightarrow} & F \rightarrow 0\\[1ex] & \varphi_1 \! \searrow & \downarrow \overline{\varphi_1}\\[1ex] &&\Bbb{P}_1 \end{array} $$ where the row is an exact sequence of abelian varieties with elliptic curves $E$ and $F$ and $\overline{\varphi}_1$ is a double covering.\\ \par {\bf (3)} Let $L_i$ denote the line bundle on $X$ defining the morphism $\varphi_i : X \rightarrow \Bbb{P}_i$. Then $\varphi_i$ is given by the complete linear system $|L_i|$.\\ \par {\bf (4)} The line bundle $L_3$ is of type $(1,4)$ on $X$ and $\varphi_3 : X \rightarrow \Bbb{P}_3$ is a birational morphism onto a singular octic in $\Bbb{P}_3$.\\ \par {\bf (5)} For every $t \in \Bbb{P}_1$ consider the scheme theoretical intersection $X_t := X \cap \{ t \} \times \Bbb{P}_3$ as a curve in $\Bbb{P}_3$, i.e. $$ X_t = \varphi_3 (\varphi^{-1}_1 (t)) . $$ Let $t_1, \ldots , t_4$ denote the 4 ramification points of the double covering $\overline{\varphi_1} : F \rightarrow \Bbb{P}_1$. \bigskip \noindent {\bf Lemma 1.1.} (i) {\it For every $t \not= t_1, \ldots , t_4$ the curve $X_t$ is a disjoint union of \, $2$ smooth plane cubics $E_t$ and $E'_t$ in $\Bbb{P}_3$ both isomorphic to $E$.\\} (ii) {\it For $\nu = 1, \ldots, 4$ the curve $X_{t_\nu}$ is a double curve in $\Bbb{P}_3$, not lying in a plane, with support a plane cubic $E_{t_\nu}$ isomorphic to } $E$. \bigskip \noindent {\bf Proof. } (i) This follows from (2) and (3), since $\varphi :X \hookrightarrow P$ embeds the curve $X_t$ into $\{ t \} \times \Bbb{P}_3$.\\ (ii) The assertion about the support follows again from (2) and (3). That $X_{t_{\nu}}$ is a double curve follows from (2) and the fact that $t_{\nu}$ is a ramification point of the double covering $\overline{\varphi_1}$. It is only to show that the linear system $|L_3| \big| X_{t_{\nu}}$is of dimension 3.\\ Since dim$|L_3|=3$, we have obviously dim$(|L_3|\big| X_{t_v}) \leq 3$. For the converse inequality consider $X$ as a family of curves $(X_t)_{t \in \Bbb{P}_1}$ over $\Bbb{P}_1$. Since dim$(|L_3| \big|X_t) =3$ for a general $t \in \Bbb{P}_1$, the assertion follows by semicontinuity. \hfill $\Box$\\ \par {\bf (6)} For every point $x \in F$ let $P_{2,x}$ denote the plane in $\Bbb{P}_3$ spanned by the plane cubic $\varphi_3(q^{-1}(x)) \,\, ( = E_t $ or $ E'_t$ if $t = \overline{\varphi}_3 (x)).$ \bigskip \noindent {\bf Lemma 1.2.} $P_{2,x} \not= P_{2,y}$ {\it for all} $x,y \in F, x \not= y$. \bigskip \noindent {\bf Proof.} Suppose $P_{2,x} = P_{2,y}$. Let $D \in |L_3|$ denote the divisor corresponding to the plane $P_{2,x} = P_{2,y} $ in $\Bbb{P}_3$. then $$ D =q^{-1} (x) + q^{-1} (y) + \widetilde{D} $$ with elliptic curves $q^{-1}(x)$ and $q^{-1} (y)$ and a curve $\widetilde{D}$. It follows $$ 8 = D^2 = D \cdot q^{-1} (x) + D \cdot q^{-1} (y) + D \cdot \widetilde{D} = 6 + D \cdot \widetilde{D} $$ and hence $$ 2 = D \cdot \widetilde{D} = q^{-1} (x) \widetilde{D} + q^{-1} (y) \widetilde{D} + \widetilde{D}^2 . $$ Since all summands on the right hand side are nonnegative, one of the summands must be 0. But then $\widetilde{D}$ is a disjoint union of translates of $E$ and all summands are $0$, a contradiction. \hfill $\Box$\\ \par {\bf (7)} Let $x' \in F$ denote the conjugate of $x \in F$ with respect to the double covering $\overline{\varphi_1}$. Then, denoting $t = \overline{\varphi_1} (x)$, $$ Q_t := P_{2,x} + P_{2,x'} $$ is a quadric in $\Bbb{P}_3$, of rank 2 for all $t \not= t_1, \ldots, t_4$ and of rank 1 for $t = t_\nu$. Consider the pencil $\{ Q_t | t \in \Bbb{P}_1 \}$ of these quadrics. \bigskip \noindent {\bf Lemma 1.3.} $\{Q_t | t \in \Bbb{P}_1 \}$ {\it is a rational pencil of quadrics of rank $\leq 2$ and degree $ d\geq 2$ in $\Bbb{P}_3$, i.e. of the form $\{ \lambda^d Q_0 + \lambda^{d-1} Q_1 +\ldots + \mu^d Q_d | (\lambda, \mu) \in \Bbb{P}_1 \}$ with quadrics $Q_0, \ldots, Q_d $ in }$\Bbb{P}_3$. \bigskip \noindent {\bf Proof.} Suppose that $\{ Q_t | t \in \Bbb{P}_1 \}$ is a linear pencil and let $Q$ and $Q'$ denote 2 different quadrics of rank 2 in it. We may choose the coordinates of $\Bbb{P}_3$ such that $Q$ is given by the matrix diag$(1,1,0,0)$. Then $Q'$ is given by a matrix $ \left( \begin{array}{cccc} a_0 & a_1 &0&0\\ a_1&a_2 &0&0\\ 0&0&0&0\\ 0&0&0&0 \end{array} \right)$, since all quadrics $Q_t$ are of rank $\leq 2$. Then the linear system is given by the matrices\\ $$ \left\{ \left( \begin{array}{cccc} \lambda a_0 + \mu & \lambda a_1 &0 & 0\\ \lambda a_1 & \lambda a_2 + \mu &0 & 0\\ 0&0&0&0\\ 0&0&0&0 \end{array} \right) i (\lambda, \mu) \in \Bbb{P}_1 \right\} .$$ Since det${{\lambda a_0 + \mu \quad \lambda a_1}\choose{\lambda a_1 \quad \lambda a_2 + \mu}} =0$ has only 2 solutions, the pencil admits only 2 quadrics of rank 1, a contradiction. \hfill $\Box$ \bigskip \noindent For a quadric $Q_t$ of rank 2 in $\Bbb{P}_3$ let $\ell_t$ denote the singular line of $Q_t$. \bigskip \noindent {\bf Lemma 1.4.} {\it The lines $\ell_t, t \in \Bbb{P}_1$ form a one--dimensional family of lines in } $\Bbb{P}_3$. \bigskip \noindent {\bf Proof.} If this is not the case, then all quadrics of the pencil contain the same singular line $\ell$ and the pencil $\{Q_t | t \in \Bbb{P}_1 \}$ is given by a curve $q$ of degree $d \geq 2$ in the plane $\pi$ parametrizing the quadrics in $\Bbb{P}_3$ with singular line $\ell$. On the other hand the quadrics in $\pi$ containing a fixed plane in $\Bbb{P}_3$ form a line in $\pi$. This line intersects $q$ in $d \geq 2$ points, which contradicts Lemma 1.2. \hfill $\Box$ \medskip \noindent We will see later (see Proposition 5.3) that the pencil $\{ Q_t | t \in \Bbb{P}_1 \}$ is of degree 4. \section{Symmetries of $X$ in ${\bf P_1 \times P_3}$. } Let $\varphi = (\varphi_1, \varphi_3) :X \rightarrow \Bbb{P}_1 \times \Bbb{P}_3$ be an embedding of an abelian surface $X$ defined by line bundles $L_1$ and $L_3$ on $X$ as in Section 1. In this section we want to determine the group of all translations of $X$ which extend to elements of $P G L_1(\Bbb{C}) \times P G L_3 (\Bbb{C})$.\\ \par Recall that for any line bundle $L$ on $X$ the subgroup of all $x \in X$ with $t^{\ast}_x L \simeq L$ is denoted by $K(L)$. Here $t_x : X \rightarrow X$ denotes the translation of the abelian variety $X$ by $x$. If $L$ is base point free with corresponding morphism $\varphi_L : X \rightarrow \Bbb{P}_m$, then $K(L)$ is the largest subgroup of translations of $X$ which are induced by automorphisms of $\Bbb{P}_m$. If $L'$ is a second base point free line bundle on $X$ with corresponding morphism $\varphi_{L'} : X \rightarrow \Bbb{P}_n$, let $K(L, L')$ denote the subgroup of all translations of $X$ which are induced by elements of $P GL_m(\Bbb{C}) \times P G L_n (\Bbb{C})$. Clearly such an element induces a translation $t_x$ on $X$ if and only if $x \in K(L) \cap K(L')$. \medskip \noindent Let $X_2$ denotes the subgroup of 2--division points of $X$. Then we have \bigskip \noindent {\bf Proposition 2.1.} $\quad K(L_1, L_3) = K(L_3) \cap X_2 . $ \bigskip \noindent {\bf Proof.} By construction of $\varphi_1 : X \rightarrow \Bbb{P}_1$ there is a line bundle $\ell_1$ on the elliptic curve $F$ such that $L_1 = q^{\ast} \ell_1$ with $q : X \rightarrow F$ as in (2). We have $K(\ell_1) = F_2$, since $\ell_1$ is of degree 2 on $F$. Hence $K(L_1) = q^{\ast} K(\ell_1)$ contains $X_2$, and thus $$ K(L_3) \cap X_2 \subseteq K(L_3) \cap K(L_1) = K(L_1, L_3). $$ $L_3$ being of type $(1,4)$ implies $K(L_3) \simeq \Bbb{Z} / 4 \Bbb{Z} \times \Bbb{Z} / 4 \Bbb{Z}$. Hence it suffices to show that no element of order 4 in $K(L_3)$ is contained in $K(L_1)$. Suppose $x \in K(L_1) \cap K(L_3)$ is of order 4. Then $t^{\ast}_x (L_1 \otimes L_3) \simeq t^{\ast}_x L_1 \otimes t^{\ast}_x L_3 \simeq L_1 \otimes L_3$, i.e. $x \in K(L_1 \otimes L_3)$. But $$ (L_1 \otimes L_3)^2 = 2(L_1 \cdot L_3) + (L^2_3)= 20 $$ since $(L_1 \cdot L_3) =6$ (see [L], Lemma 1.1). Hence $L_1 \otimes L_3$ is of type $(1,10)$ and $K(L_1 \otimes L_3) \simeq \Bbb{Z}/ 10 \Bbb{Z} \otimes \Bbb{Z} / 10 \Bbb{Z}$. Hence $K(L_1 \otimes L_3)$ does not contain an element of order 4. \hfill$\Box$ \medskip \noindent Since $L_3$ is of type $(1,4)$, Proposition 2.1 implies $$ K(L_1, L_3) \simeq \Bbb{Z} / 2 \Bbb{Z} \times \Bbb{Z} / 2 \Bbb{Z} $$ and one may choose coodinates of $\Bbb{P}_1$ and $\Bbb{P}_3$ in such a way (see [CAV], p. 169) that generators $\sigma$ and $\tau$ of $K(L_1, L_3)$ are given by $$ \begin{array}{ccl} \sigma & = & \left( \left( \begin{array}{cc} 0&1\\ 1&0 \end{array} \right) , \left( \begin{array}{cccc} 0& 0& 1& 0\\ 0& 0& 0&1\\ 1 & 0& 0& 0\\ 0 & 1 & 0& 0 \end{array} \right) \right) \in G L_2 (\Bbb{C}) \times G L_4 (\Bbb{C})\\[2ex] \tau & = & \left( \left( \begin{array}{rr} 1&0\\ 0 & -1 \end{array} \right), \left( \begin{array}{ccrr} 1& 0 & 0&0\\ 0 & -1 & 0 & 0\\ 0 & 0 & 1& 0\\ 0 &0& 0& -1 \end{array} \right) \right) \in G L_2 (\Bbb{C}) \times G L_4 ( \Bbb{C}). \end{array} $$ The {\it Heisenberg group of the pair} $(L_1, L_3)$ is by definition the subgroup $H(L_1, L_3)$ of $G L_2(\Bbb{C}) \times GL_4(\Bbb{C})$ generated by $\sigma$ and $\tau$. The Heisenberg group $H(L_1, L_3)$ acts in the usual way on the pair of line bundles $(L_1, L_3)$. \bigskip \noindent {\bf Proposition 2.2.} {\it The Heisenberg group $H(L_1, L_3)$ is the dihedral group $D_8$ with $8$ elements. It fits into an exact sequence} $$ 0 \rightarrow \mu_2 \rightarrow H(L_1, L_3) \rightarrow K(L_1, L_3) \rightarrow 0 $$ {\it where $\mu_2$ denotes the group of order} 2 {\it generated by the commutator } $\sigma \tau \sigma^{-1} \tau^{-1}$. \bigskip \noindent {\bf Proof:} $H_1 (L_1, L_2)$ is generated bt $\tau$ and the element $\kappa := \sigma \tau$ of order 4 with relation $\tau \kappa \tau = \kappa^{-1}$. \hfill $\Box$ \section{The vector bundle ${\cal{E}}$ on ${\bf P_1 \times P_3}$} Let $\varphi = (\varphi_1, \varphi_3) : X \rightarrow P = \Bbb{P}_1 \times \Bbb{P}_3$ be an embedding as in Section 1. Denote $ {\cal{L}} = {\cal{O}}_P(2,4)$. Then ${\cal{L}} = \omega^{-1}_P$ and there is an isomorphism $$ \xi : {\cal{L}} \otimes \omega_P \otimes {\cal{O}}_X \rightarrow \omega_X = {\cal{O}}_X . $$ According to the Serre--construction (which applies in this case, see [H], Remark 1.1.1) the pair $(X, \xi)$ determines uniquely a triplet $({\cal{E}}, s, \psi)$ with a rank--2 vector bundle ${\cal{E}}$ on $P$, a section $ s \in H^0 (P, {\cal{E}})$ such that $(s) = X$ and an isomorphism $\psi : \bigwedge^2 {\cal{E}} \rightarrow {\cal{L}}$.\\ \par The Koszul complex of the section $s$ is $$ 0 \rightarrow \bigwedge^2 {\cal{E}}^{\ast} \rightarrow {\cal{E}}^{\ast} \stackrel{s}{\rightarrow} I_X \rightarrow 0 $$ where $I_X$ denotes the ideal sheaf of $X$ in $P$. Tensoring with $\bigwedge^2 {\cal{E}} = {\cal{L}} = {\cal{O}}_P (2,4)$ and using ${\cal{E}} ={\cal{E}}^{\ast}(\bigwedge^2 {\cal{E}})$ we obtain the exact sequence $$ 0 \rightarrow {\cal{O}}_P \stackrel{s}{\rightarrow} {\cal{E}} \rightarrow I_X (2,4) \rightarrow 0 . \eqno(1) $$ Then we have, using Section 1 (1) $$ c_1 ({\cal{E}}) = 2 h_1 + 4 h_3 \eqno(2) $$ and $$ c_2 ({\cal{E}}) = 8 h_1 h_3 + 6h^2_3 .\eqno(3) $$ \par Consider the ample line bundle ${\cal{H}} = {\cal{O}}_P(1,1)$ on $P$. A vector bundle ${\cal{F}}$ of rank 2 on $P$ is called ({\it semi--) stable with respect to} ${\cal{H}}$ if for every invertible subsheaf ${\cal{L}} $ of ${\cal{F}}$ $$ {\cal{L}} \cdot {\cal{H}}^3 \stackrel{<}{(=)} \frac{1}{2} \det {\cal{F}} \cdot {\cal{H}}^3 . $$ \bigskip \noindent {\bf Proposition 3.1.} ${\cal{E}}$ {\it is stable with respect to} ${\cal{H}}$.\\ \par For the proof we need the following Lemma \bigskip \noindent {\bf Lemma 3.2.} (i) {\it Suppose $h^0 (I_X (a,b)) \not= 0$ for some integers $a$ and $b$. Then $a \geq 0$ and} $b \geq 2$. \begin{itemize} \item[(ii)] $h^0 (I_X (0,b)) \not= 0$ {\it if and only if} $b \geq 8$. \item[(iii)] $h^0(I_X(a,2)) = 0$ {\it for }$a \leq 2.$ \end{itemize} \bigskip \noindent {\bf Proof.} (i) Since $h^0( {\cal{O}}_P (a,b)) =0$ if $a < 0$, it suffices to show that $b \geq 2$. For a general $t \in \Bbb{P}_1$ the intersection $X_t = X \cap \{ t \} \times \Bbb{P}_3$ is a disjoint untion of 2 plane cubics in $\Bbb{P}_2$ (see Section 1 (5)). In particular $h^0(\Bbb{P}_3, I_{X_t} (1)) =0$. So the exact sequence $$ 0 \rightarrow I_X (a,1) \rightarrow I_{X_t} (1) \rightarrow I_{X_t} (1) / I_X (a,1) \rightarrow 0 $$ implies $h^0 (I_X (a,1)) =0$ for all $a$.\\ (ii ): $h^0 (I_X (0,b)) \not= 0$ means that $X \subset \Bbb{P}_1 \times V_b$ where $V_b$ is a surface of degree $b$ in $\Bbb{P}_3$. Now that the composed map $\varphi_3 : X \hookrightarrow \Bbb{P}_1 \times \Bbb{P}_3 \rightarrow \Bbb{P}_3$ is birational onto a (singular) octic in $\Bbb{P}_3$ (see Section 1(4)). This implies the assertion.\\ (iii): It suffices to show $h^0(I_X (2,2))= 0$. But $h^0(I_X (2,2)) \not= 0$ would mean (using the results of Section 4) that the pencil $\{ Q_t | t \in \Bbb{P}_1 \}$ of Section 1 (7) is quadratic, contradicting Proposition 5.3 below. \hfill $\Box$ \bigskip \noindent {\bf Proof of Proposition 3.1.} Suppose ${\cal{E}}$ is not stable with respect to ${\cal{H}}$ and let ${\cal{O}}_P(a,b)$ be a subinvertible sheaf of ${\cal{E}}$ violating the stability of ${\cal{E}}$, i.e. $$ a + 3b = {\cal{O}}_P (a,b) \cdot {\cal{H}}^3 \geq \frac{1}{2} \det {\cal{E}} \cdot {\cal{H}}^3 = 7 . $$ Then the composed map $$ {\cal{O}}_P(a,b) \rightarrow {\cal{E}} \rightarrow I_X (2,4) $$ is nonzero, implying $$ h^0(I_X(2-a, 4-b)) \not= 0. $$ Lemma 3.2 implies that either $a \leq 1$ and $b \leq 2$ and $a \leq -1$ if $b=2$ or $a=2$ and $b \leq -4$. But in the first case $7 \leq a+3b \leq 5$ and in the second case $7 \leq a + 3b \leq - 10$, a contradiction. \hfill$\Box$ \bigskip \noindent {\bf Remark 3.3.} The line bundle ${\cal{O}}_P (m,n)$ is ample if and only if $m > 0$ and $n > 0$. Defining the (semi--) stability of a rank--2 vector bundle on $P$ with respect to ${\cal{O}}_P(m,n)$ in the same way as for ${\cal{H}}$. Then an analogous proof yields that ${\cal{E}}$ is stable (respectively semistable) with respect to ${\cal{O}}_P (m,n)$ if and only if $n < 18 m $ (respectively $n \leq 18 m)$. \medskip \noindent In order to lift the action of the Heisenberg group $H(L_1, L_3)$ of the pair $(L_1, L_3)$ to the vector bundle ${\cal{E}}$, we need the cohomology of ${\cal{E}} (-2, -4)$. \bigskip \noindent {\bf Lemma 3.4} $$ h^i({\cal{E}}(-2, -4)) = \left\{ \begin{array}{lcr} 0 && i \not= 2\\ &\mbox{\it for} &\\ 2 && i=2 \end{array} \right. .$$ \medskip \noindent {\bf Proof:} Serre duality says $h^{4-i} ({\cal{E}}(-2, -4)) = h^i ({\cal{E}}(-2, -4))$. Hence it suffices to compute $h^i({\cal{E}} ( -2, -4))$ for $i \leq 2$. The exact sequence (1) yields $$ h^i( {\cal{E}} (-2, -4)) = h^i(I_X) $$ for $i = 0,1,2$. Now the exact sequence $0 \rightarrow I_X \rightarrow {\cal{O}}_P \rightarrow {\cal{O}}_X \rightarrow 0$ implies the assertion. \hfill $\Box$ \bigskip \noindent {\bf Proposition 3.5:} {\it The vector bundle ${\cal{E}}$ admits an action of the Heisenberg group $H(L_1, L_3)$ uniquely determined upto a constant by the embedding $X \hookrightarrow P$.} \bigskip \noindent {\bf Proof:} An element $\mu \in H(L_1, L_3)$ may be considered as an automorphism of $P$. Moreover $\mu$ defines an isomorphism $\varphi_{\mu} : I_X (2,4) \widetilde{\rightarrow} \mu^{\ast} I_X(2,4)$. Consider the diagram $$ \begin{array}{llcccccr} 0 \rightarrow & {\cal{O}}_P & \stackrel{s}{\rightarrow} &{\cal{E}}& \rightarrow & I_X(2,4)& \rightarrow 0\\ & ||& &&& \downarrow \varphi_{\mu} &\\ 0 \rightarrow & {\cal{O}}_P & \rightarrow & \mu^{\ast} {\cal{E}}& \rightarrow & \mu^{\ast} I_X(2,4) & \rightarrow 0 \end{array} $$ It induces an exact sequence $$ 0 \rightarrow \, \mbox{Hom} ({\cal{E}}, \mu^{\ast} {\cal{E}}) \rightarrow \, \mbox{Hom} ({\cal{E}}, \mu^{\ast} I_X(2,4)) \rightarrow \, \mbox{Ext}^1({\cal{E}}, {\cal{O}}_P) $$ But Ext$^1({\cal{E}}, {\cal{O}}_P) = H^1({\cal{E}}^{\ast}) = H^1({\cal{E}}(-2, -4))=0$ according to Lemma 3.4. This implies the assertion, the uniqueness coming from the fact that the embedding $X \hookrightarrow P$ determines the section $s$ upto a constant. \hfill $\Box$ \section{Restriction to ${\bf \{ t \} \times P_3}$.} For any point $t \in \Bbb{P}_1$ consider the restriction $$ {\cal{E}}_t := {\cal{E}} | \{ t \} \times \Bbb{P}_3 $$ as a rank--2 vector bundle on $\Bbb{P}_3$. \,\, $X_t$ is a local complete intersection curve in $\Bbb{P}_3$ and the triplet $({\cal{E}}_t, s | \{ t \} \times \Bbb{P}_3, \psi | \{ t \} \times \Bbb{P}_3 )$ corresponds to the pair $(X_t, \xi | \{ t \} \times \Bbb{P}_3)$ via the Serre--construction. Hence we get the following exact sequence directly by restricting (1) of Section 2. $$ 0 \rightarrow {\cal{O}}_{\Bbb{P}_3} \stackrel{s_t}{\rightarrow} {\cal{E}}_t \rightarrow I_{X_t} (4) \rightarrow 0. \eqno(1) $$ This implies $$ \begin{array}{l} c_1 ({\cal{E}}_t) =4\\ c_2 ({\cal{E}}_t) = 6 \end{array} $$ \medskip \noindent {\bf Proposition 4.1} {\it For any $t \in \Bbb{P}_1 $ the vector bundle ${\cal{E}}_t$ is semistable but not stable.} \vspace{1cm} \newline {\bf Proof.} The quadric $Q_t$ of Lemma 1.3 contains the curve $X_t$ and is in fact the only quadric in $\Bbb{P}_3$ containing $X_t$. Hence $h^0 (I_{X_t} (2)) =1$ and the exact sequence $0 \rightarrow {\cal{O}}_{\Bbb{P}_3} (-2) \rightarrow E_t (-2) \rightarrow I_{X_t} (2) \rightarrow 0$ implies $h^0 (E_t(-2)) =1.$ This means that $E_t$ is not stable. On the other hand $h^0(E_t(-3)) = 0$ means that $E_t$ is semistable.\hfill $\Box$\\ \par Let $\sigma_t \in H^0 (E_t(-2))$ denote the non--zero section, uniquely determined upto a constant, and $Y_t = (\sigma_t)$ the corresponding zero variety in $\Bbb{P}_3$. Then we have the exact sequence $$ 0 \rightarrow {\cal{O}}_{\Bbb{P}_3} \stackrel{\sigma_t}{\rightarrow} {\cal{E}}_t (-2) \rightarrow I_{Y_t} \rightarrow 0 . \eqno(2) $$ \smallskip \noindent {\bf Lemma 4.2.} $Y_t$ {\it is a curve of degree $2$ in $\Bbb{P}_3$ with $p_a (Y) = -3$ and} $\omega_{Y_t} = {\cal{O}}_{Y_t} (-4)$. \vspace{1cm} \newline {\bf Proof.} According to [H], Proposition 2.1 we have $\deg Y = c_2 (E(-2))=2$ and $2p_a(Y)-2= c_2(E(-2)) \cdot (c_1(E(-2))-4) = - 8$, i.e. $p_a(Y) = -3$. The assertion for $\omega_{Y_t}$ follows from the adjunction formula.\hfill $\Box$ \vspace{0.51cm} \par Let $Z_t$ denote the reduced curve underlying $Y_t$.\\ \bigskip \noindent {\bf Lemma 4.3} (i): \,\, $Z_t$ {\it is a line in $\Bbb{P}_3$ and $Y_t$ is a multiplicity } 2 {\it structure on} $Z_t$. \\ (ii): {\it Choose the coordinates $x_i , \quad i = 0, \ldots , 3 $ of $\Bbb{P}_3$ in such a way that $Z_t$ is the line $x_0 = x_1 =0.$ then $Y_t$ is given by the homogeneous ideal} $$ (x_0^2, x_0 x_1, x^2_1, f \cdot x_0 + g \cdot x_1) $$ {\it where $f$ and $g$ are forms of degree $3$ in $x_2$ and $x_3$ without common zero}.\\ \bigskip \noindent {\bf Proof.} (i): Since $\deg Y_t =2$, it suffices to show that $Y_t$ is irreducible. But otherwise $Y_t$ would be reduced and hence $p_a(Y) \geq -1$.\\ (ii): According to Ferrand's theorem (see [H], Theorem 1.5 with $m=4$) there is an exact sequence $$ 0 \rightarrow I_{Y_t} \rightarrow I_{Z_t} \rightarrow {\cal{O}}_{Z_t} (2) \rightarrow 0. $$ Consider the conormal bundle sequence $0 \rightarrow N^{\ast}_{Y_t|\Bbb{P}_3} |_{ Z_t} \rightarrow N^{\ast}_{Z_t|\Bbb{P}_3} \rightarrow N^{\ast}_{Z_t|Y_t} \rightarrow 0$. According to (i) \quad $N^{\ast}_{Z_t|\Bbb{P}_3} = {\cal{O}}_{\Bbb{P}_1} (1)^{\oplus 2}$. Hence we get an exact sequence $$ 0 \rightarrow I_{Y_t} | \Bbb{P}_1 \rightarrow {\cal{O}}_{\Bbb{P}_1} (1)^{\oplus 2} \stackrel{u}{\rightarrow} {\cal{O}}_{\Bbb{P}_1} (2) \rightarrow 0 . $$ The map $u$ is given by 2 forms $f$ and $g$ of degree 3 on $\Bbb{P}_1$ without common zeros. This implies that $I_{Y_t} | \Bbb{P}_1$ is generated by $f x_0 + g x_1$.\hfill $\Box$\\ \par Suppose $t \not= t_{\nu}$ for $\nu = 1, \ldots , 4$. Then $X_t$ consists of 2 disjoint plane cubics $E_t$ and $E'_t$. Denote by $P_t$ and $P'_t$ the planes spanned by $E_t$ and $E'_t$. Then we have\\ \bigskip \noindent {\bf Proposition 4.4.} {\it The line $Z_t$ is the line of intersection $P_t \cap P'_t$ and the cubic forms $f$ and $g$ are given by $E_t \cap Z_t$ and } $E'_t \cap Z_t$.\\ \bigskip \noindent {\bf Proof.} Restricting the exact sequence (1) to the plane $P_t$ we obtain an exact sequence $$ 0 \rightarrow {\cal{O}}_{P_t} (3) \rightarrow {\cal{E}}_t | P_t \rightarrow I_{E'_t \cap P_t} (1) \rightarrow 0. \eqno(3) $$ since $P_t$ contains the plane cubic $E_t$. Computing the generic splitting type of ${\cal{E}}_t | P_t$ using (2), we see that $P_t$ contains the line $Z_t$ (but not the double curve $Y_t)$. Similarly $P'_t$ contains $Z_t$, i.e. $P_t \cap P'_t = Z_t$. Restricting (2) to $P_t$ and comparing it with (3) gives the assertion for the forms $f$ and $g$. \hfill $\Box$ \bigskip \noindent \section{The Double Structure and the Pencil of Quadrics} The quadrics of the rational pencil $\{Q_t | t \in \Bbb{P}_1 \}$ of Section 1 fill up a 3--fold $Q \in \Bbb{P}_1 \times \Bbb{P}_3$ containing $X$. According to Lemma 1.3 \, \, $\{ Q_t| t \in \Bbb{P}_1 \}$ is a rational pencil of degree $d \geq 2$ in $\Bbb{P}_3$. Hence $Q$ is a hypersurface of bidegree $(d,2)$ in $\Bbb{P}_1 \times \Bbb{P}_3$. Obviously $Q$ is the only hypersurface of bidegree $(d,2)$ in $\Bbb{P}_1 \times \Bbb{P}_3$ containing $X$. This means $$ h^0 (I_X (d, 2)) =1. $$ From the exact sequence $ 0 \rightarrow {\cal{O}}_{\Bbb{P}_1 \times \Bbb{P}_3} (d-2, -2) \rightarrow {\cal{E}} (d-2, -2) \rightarrow I_X(d,2) \rightarrow 0 $ we deduce $$ h^0({\cal{E}} (d-2, -2)) =1. $$ Hence there is an exact sequence, unique upto a multiplicative constant $$ 0 \rightarrow {\cal{O}}_{\Bbb{P}_1 \times \Bbb{P}_3} \stackrel{\sigma}{\rightarrow} {\cal{E}}(d-2, -2) \rightarrow I_Y(2d-2, 0) \rightarrow 0 \eqno(1) $$ with a surface $Y \subseteq \Bbb{P}_1 \times \Bbb{P}_3$ of class $$ [Y] = 4h_1 h_3 + 2h^2_3. $$ For every point $t \in \Bbb{P}_1$ the exact sequence (1) restricts to the exact sequence (2) of Section 4. Hence for every $t \in \Bbb{P}_1$ the surface $Y \subseteq \Bbb{P}_1 \times \Bbb{P}_3$ restricts to the curve $Y_t \subseteq \{ t \} \times \Bbb{P}_3$. According to Lemma 4.3 \, \, $Y_t$ is a double structure on a line $Z_t$ in $\{ t \} \times \Bbb{P}_3$ for all $t$. This implies\\ \bigskip \noindent {\bf Proposition 5.1.} {\it The surface $Y \subseteq \Bbb{P}_1 \times \Bbb{P}_3$ is a double structure on a $\Bbb{P}_1$--bundle $Z \subseteq \Bbb{P}_1 \times \Bbb{P}_3$ over $\Bbb{P}_1$. The bundle map $Z \rightarrow \Bbb{P}_1$ coincides with the projection onto the first factor of $\Bbb{P}_1 \times \Bbb{P}_3$. The class of $Z$ in $H^4(\Bbb{P}_1 \times \Bbb{P}_3,\Bbb{Z})$ is $[Z] = 2h_1 h_3 + h^2_3$.}\\ \bigskip \noindent In other words $Z$ is a certain Hirzebruch surface $ \Sigma$ embedded into $\Bbb{P}_1 \times \Bbb{P}_3$ with class $2h_1 h_3 + h^2_3$ such that the bundle map $\Sigma \rightarrow \Bbb{P}_1$ coincides with the projection onto the first factor. The following lemma classifies these embeddings.\\ \par Let $\Sigma_e$ denote the Hirzebruch surface with invariant $e \geq 0$. There is a section $C_0 $ of $p : \Sigma_e \rightarrow \Bbb{P}_1$ such that $$ \mbox{Pic} (\Sigma_e) \simeq \Bbb{Z} [C_0] + \Bbb{Z}[f] $$ where $f$ denotes a fibre of $p$, with intersection numbers $$ C^2_0 = -e \quad , \quad C_0 \cdot f =1 \quad , \quad f^2 =0. $$ Suppose $i = (\varphi_1, \varphi_3): \Sigma_e \hookrightarrow \Bbb{P}_1 \times \Bbb{P}_3$ is an embedding such that $$ [\Sigma_e ] = 2h_1 h_3 + h^2_3 \,\, \mbox{in} \,\, H^4(\Bbb{P}_1 \times \Bbb{P}_3, \Bbb{Z}) \leqno(i) $$ $$ \varphi_1 : \Sigma_e \rightarrow \Bbb{P}_1 \,\, \mbox{is given by the linear system} \quad |f|. \leqno(ii) $$ Then we have \bigskip \noindent {\bf Lemma 5.2.} {\it Either $e =0$ and $\varphi_3$ is given by the linear system $|C_0 +f|$ or $e = 2$ and $\varphi_3$ is given by the linear system $|C_0 + 2f|$. Conversely, in these cases $i = (\varphi_1 , \varphi_3)$ is an embedding.} \bigskip \noindent {\bf Proof.} As usual we identify classes in $H^i( \Sigma_e, \Bbb{Z})$ with their images in $H^i (\Bbb{P}_1 \times \Bbb{P}_3, \Bbb{Z})$. Condition (ii) means $[ \Sigma_e] \cdot h_1 = f$ which implies $$ f \cdot h_1 =0 \quad \mbox{and} \quad f \cdot h_3 =1. \eqno(2) $$ Let $\varphi_3 : \Sigma_e \rightarrow \Bbb{P}_3$ be given by a sublinear system of $| \alpha C_0 + \beta f|$. Necessarily $\alpha \geq 0$ and $\beta \geq 0$. $$ 2h_1 h^2_3 + h^3_3 = [ \ \Sigma_3] \cdot h_3 = \alpha C_0 + \beta f . \eqno(3) $$ Using (2) this implies $$ \alpha C_0 \cdot h_1 =1 $$ and $$ \alpha C_0 \cdot h_3 = 2 - \beta. $$ This implies $$ C_0 \cdot h_1 = 1 \quad \mbox{and} \quad \alpha =1 . $$ Hence $$ \beta \leq 2 . $$ On the other hand $$ 2 = [ \Sigma_e] \cdot h^2_3 = (h_3 | \Sigma_e)^2 = (C_0 + \beta f)^2 = - e + 2 \beta $$ i.e. $$ \beta = \frac{e}{2} +1 . $$ We remain with the 2 cases $e = 0, \, \beta =1$ and $e = 2, \, \beta = 2$. \\ Conversely assume that the surface and the linear systems are of one of the 2 cases. In the first case $\varphi_3$ embeds $\Sigma_0 = \Bbb{P}_1 \times \Bbb{P}_1$ as a smooth quadric in $\Bbb{P}_3$. In the second case $\varphi_3$ embeds $\Sigma_2 - C_0$ and contracts $C_0$ to a quadric cone in $\Bbb{P}_3$. But then $\varphi_1$ is injective on $C_0$ and the differential of $(\varphi_1, \varphi_3)$ has maximal rank. So in both cases $i = (\varphi_1, \varphi_3)$ is an embedding. \hfill $\Box$ \\ \par The surface $Y$ is a double structure on $Z$ with $Z$ as in Lemma 6.2. Let ${\cal{L}}$ denote the ideal of $Z$ in ${\cal{O}}_Y$. So we have the exact sequence $$ 0 \rightarrow {\cal{L}} \rightarrow {\cal{O}}_Y \rightarrow {\cal{O}}_Z \rightarrow 0 \eqno(4) $$ Note that ${\cal{L}}$ is a line bundle on $X$, since ${\cal{L}}/{\cal{L}}^2 =0$. Using the Riemann--Roch Theorem for the vector bundle ${\cal{E}} (a, b), $ saying $$ \chi( {\cal{E}} (a,b)) = - 6 + 12 a + \frac{34}{3} b + 6 b^2 + \frac{41}{3} a b + \frac{2}{3} b^3 + 4 ab^2 + \frac{1}{3} ab^3 \eqno(5) $$ for all $a,b \in \Bbb{Z}$ one can determine ${\cal{L}}$ and $d$: \bigskip \noindent {\bf Proposition 5.3.} {\it Suppose $Y \subseteq \Bbb{P}_1 \times \Bbb{P}_3$ is a double structure on $Z$ ($\simeq \Sigma_e, e = 0$ or $2$) defined by the line bundle ${\cal{L}}$ on $Z$. Then} \begin{itemize} \item[(a)] $ {\cal{L}} = \left\{ \begin{array}{lcc} {\cal{O}}_Z(2 C_0 - 2f) && e = 0\\ & \mbox{\it if} &\\ {\cal{O}}_Z(2 C_0) && e=2 \end{array} \right. $ \item[(b)] {\it The pencil of quadrics $\{Q_t | t \in \Bbb{P}_1 \}$ in $\Bbb{P}_3$ is of degree} $4$. \end{itemize} \par \bigskip \noindent {\bf Proof.} Let ${\cal{L}} = {\cal{O}}_Z (x C_0 + yf)$ for some $x, y \in \Bbb{Z}$. For $\alpha, \beta \in \Bbb{Z}$ we have $$ {\cal{O}}_Z \otimes {\cal{O}}_{\Bbb{P}_1 \times \Bbb{P}_3} (\alpha, \beta) = \left \{ \begin{array}{lcc} {\cal{O}}_Z (\beta C_0 +(\alpha + \beta)f) && e=0\\ & \mbox{if}&\\ {\cal{O}}_Z (\beta C_0 +(\alpha + 2 \beta )f) && e=2 \end{array} \right. $$ and from (1) and (4) we get the exact sequences $$ 0 \rightarrow {\cal{O}}_{\Bbb{P}_1 \times \Bbb{P}_3} (a - d+2, b+2) \rightarrow {\cal{E}}(a,b) \rightarrow I_Y (a+d, b+2) \rightarrow 0 \eqno(6) $$ $$ 0 \rightarrow I_Y (a+d, b+2) \rightarrow {\cal{O}}_{\Bbb{P}_1 \times \Bbb{P}_3} (a+d, b+2) \rightarrow {\cal{O}}_Y (a+d, b+2) \rightarrow 0\eqno(7) $$ and $$ \scriptstyle{ 0 \rightarrow {\cal{O}}_Z((x+b+2) C_0 +(y+a+b+d+2)f) \rightarrow {\cal{O}}_Y (a+d, b+2) \rightarrow {\cal{O}}_Z((b+2)C_0 +(a+b+d+2)f) \rightarrow 0} \eqno(8) $$ if $e = 0$ and $$ \scriptstyle{ 0 \rightarrow {\cal{O}}_Z ((x+b+2)C_0 +(y+a+2b+d+4)f)\rightarrow {\cal{O}}_Y (a+d, b+2) \rightarrow {\cal{O}}_Z((b+2) C_0+ (a+2b+d+4)f) \rightarrow 0 }\eqno(9) $$ if $e=2$. Using Riemann--Roch for line bundles on $\Bbb{P}_1 \times \Bbb{P}_3$ and $Z$, equations (6) -- (9) yield $$ \scriptstyle{ \chi({\cal{E}}(a,b))=\chi({\cal{O}}_{\Bbb{P}_1 \times \Bbb{P}_3} (a-d+2, b+2)) + \chi({\cal{O}}_{\Bbb{P}_1 \times \Bbb{P}_3} (a+d, b+2)) - \chi({\cal{O}}_Y(a+d, b+2))} $$ $$ \scriptstyle{ =(a-d+3){{b+5}\choose{3}} +(a+d+1) {{b+5}\choose{3}} -} $$ $$ \scriptstyle{ \left\{ \begin{array}{l} \scriptstyle{ - \chi({\cal{O}}_Z((x+b+2)C_0 +(y+a+b+d+2)f))-\chi({\cal{O}}_Z((b+2) C_0+(a+b+d+2)f)) \quad \mbox{if} \quad \, e=0}\\ \scriptstyle{ - \chi({\cal{O}}_Z((x+b+2)C_0+(y+a+2b+d+4)f)) - \chi({\cal{O}}_Z((b+2) C_0 + (a+2b +d+4)f)) \quad \mbox{if} \,\quad {e=2} } \end{array} \right. } $$ $$ \scriptstyle{ = (2a+4){{b+5}\choose{3}} + \left\{ \begin{array}{l} \scriptstyle{ -(x+b+3)(y+a+b+d+3)-(b+3)(a+b+d+3) \,\quad \mbox{if}\,\quad e=0}\\ \scriptstyle{ -(x+b+3)(y-x+a+b+d+3)-(b+3)(a+b+d+3) \,\quad \mbox{if} \,\quad e=2.} \end{array} \right. } $$ Comparing this with (5) we obtain $$ (x-2)a + (x+y+2d-8)b +(y+d+3)(x+3)+3d -37 =0 \quad \,\, \mbox{if} \,\, e=0 $$ and $$ (x-2)a +(y+2d-8)b+(y-x+d +3)(x+3)+3d-37 =0 \quad \,\, \mbox{if} \,\, e=2 $$ as polynomials in $a$ and $b$. \bigskip \noindent Hence if $e=0 : x =2, y = 6-2d$ and $2d -8 =0$ and if $e=2 : x=2, y=8 -2d$ and $2d -8 =0$. \hfill $\Box$\\ \par K. Hulek gave a different proof of Proposition 5.3 using the normal bundle sequence of $Z \subseteq Y \subseteq P$. The following proposition saying that the case $Z = \Sigma_2$ does not occur in our situation is also due to him. \vspace{1cm} \newline {\bf Proposition 5.4.} (Hulek): {\it Suppose $Z ( = \Sigma_e)$ embedded in $P = \Bbb{P}_1 \times \Bbb{P}_3$ as above admits a double structure $Y$ in $P$, then} $ e = 0$. \bigskip \noindent {\bf Proof:} Suppose $Z = \Sigma_2 \subset P$ admits a double structure $Y \subseteq P$. According to Lemma 5.2 the morphism $\varphi_3 | Z : Z \rightarrow \Bbb{P}_3$ is the contraction of the curve $C_0$ with image a quadric in $\Bbb{P}_3$. Hence $C_0$ considered as a curve in $P$ is the preimage of a point in $\Bbb{P}_3$ under the natural projection $P = \Bbb{P}_1 \times \Bbb{P}_3 \rightarrow \Bbb{P}_3$. This yields $$ N_{C_0|P} = {\cal{O}}_{C_0}^{\oplus3}. $$ Since $N_{C_0|Z} = {\cal{O}}_{C_0}(-2)$, the exact sequence $0 \rightarrow N_{C_0|Z} \rightarrow N_{C_0|P} \rightarrow N_{Z|P} \Big|C_0 \rightarrow 0$ implies that there are 2 possibilities for $N_{Z|P} \Big| C_0$, namely $$ N_{Z|P} \Big| C_0 = \left\{ \begin{array}{cc} {\cal{O}}_{C_0} \oplus {\cal{O}}_{C_0}(2)&\\ & \mbox{or}\\ {\cal{O}}_{C_0}(1)^{\oplus 2} & \end{array} \right. \eqno(10) $$ \smallskip \noindent On the other hand the line bundle ${\cal{L}}^{-1}$ on $Z$ of Proposition 5.3 may be considered as the normal bundle of $Z$ in $Y$. The exact sequence of normal bundles for $Z \subseteq Y \subseteq P$ shows that ${\cal{L}}^{-1}$ is a subbundle of $N_{Z|P}$.\\ But $$ {\cal{L}}^{-1}|C_0 = {\cal{O}}_Z (-2 C_0)| C_0 = {\cal{O}}_{C_0} (4) $$ which cannot be a subbundle of $N_{Z|P} \Big| C_0$ according to (10). \hfill $\Box$ \vspace{0.61cm} \newline So for the rest of the paper we may assume that $Y$ is a double structure on $Z = \Bbb{P}_1 \times \Bbb{P}_1$, embedded in $P$ as in Lemma 5.2. For the computation of the cohomology of ${\cal{E}}$ it will be important to know for which pairs $(a,b)$ the scheme $Y$ is $(a,b)$--normal in $P$. \vspace{0.61cm} \newline Recall that a closed subscheme $V \subseteq \Bbb{P}_1 \times \Bbb{P}_3$ is called $(a,b)$--{\it normal} for some nonnegative integers $a$ and $b$ if the canonical map $H^0({\cal{O}}_{\Bbb{P}_1 \times \Bbb{P}_3} (a,b)) \rightarrow H^0({\cal{O}}_V (a,b))$ is surjective. Note first\\ \bigskip \noindent {\bf Lemma 5.5} {\it Let $ i = (\varphi_1, \varphi_3) : Z \rightarrow \Bbb{P}_1 \times \Bbb{P}_3$ be the embedding of Lemma } 5.2. {\it Then } $Z$ {\it is $(a,b)$--normal in $\Bbb{P}_1 \times \Bbb{P}_3$ for all} $(a,b) \geq 0$.\\ \bigskip \noindent {\bf Proof.} Consider the commutative diagram $$ \begin{array}{ccc} H^0( {\cal{O}}_{\Bbb{P}_1 \times \Bbb{P}_3}(a,b)) & \stackrel{(\varphi_1, \varphi_3)^{\ast}}{\makebox[1.2cm]{\rightarrowfill}} & H^0({\cal{O}}_Z (a,b))\\[2ex] (\varphi_1 \times \varphi_3)^{\ast} \searrow && \nearrow \Delta^{\ast}\\[2ex] & H^0(Z \times Z, \varphi^{\ast}_1 {\cal{O}}_{\Bbb{P}_1} (a) \otimes \varphi_3^{\ast} {\cal{O}}_{\Bbb{P}_3} (b)) & \end{array} $$ where $\Delta : Z \rightarrow Z \times Z$ denotes the diagonal map. It suffices to show that for all $a, b \geq 0:$ \begin{itemize} \item[(i)] $ ( \varphi_1 \times \varphi_3)^{\ast}$ is surjective \item[(ii)] $\Delta^{\ast}$ is surjective. \end{itemize} \underline{(i)}: By K\"unneth $(\varphi_1 \times \varphi_3)^{\ast}$ identifies with $$ \varphi^{\ast}_1 \otimes \varphi^{\ast}_3 : H^0 ({\cal{O}}_{\Bbb{P}_1} (a)) \otimes H^0 ({\cal{O}}_{\Bbb{P}_3} (b)) \rightarrow H^0 ({\cal{O}}_Z (af) \otimes H^0({\cal{O}}_Z (b C_0 +bf). $$ But $\varphi^{\ast}_1$ is an isomorphism and $\varphi^{\ast}_3$ is surjective since $\varphi_3$ is the Veronese embedding $\varphi_3 : Z = \Bbb{P}_1 \times \Bbb{P}_1 \rightarrow \Bbb{P}_3$, which is projectively normal. \\ \underline{(ii)}: Let $\Bbb{P}^i_1$ denote the $i$--th component of $Z = \Bbb{P}_1 \times \Bbb{P}_1$. According to K\"unneth the following diagram commutes $$ \begin{array}{ccc} H^0({\cal{O}}_Z (af))\otimes H^0({\cal{O}}_Z(b C_0 +bf) & \stackrel{\Delta^{\ast}}{ \longrightarrow} & H^0 ({\cal{O}}_Z (b C_0 + (a+b)f))\\[1ex] || && || \\[1ex] H^0({\cal{O}}_{\Bbb{P}^1_1} (a)) \otimes H^0({\cal{O}}_{\Bbb{P}^1_1} (b) \otimes H^0({\cal{O}}_{\Bbb{P}^2_1} (b)) & \longrightarrow & H^0({\cal{O}}_{\Bbb{P}^1_1} (a+b)) \otimes H^0 ({\cal{O}}_{\Bbb{P}^2_1} (b)) \end{array} $$ But this is surjective, since the multiplication map $H^0({\cal{O}}_{\Bbb{P}_1} (a)) \otimes H^0({\cal{O}}_{\Bbb{P}_1} (b)) \longrightarrow H^0 ({\cal{O}}_{\Bbb{P}_1} (a+b))$ is surjective for all $a, b \geq 0$. \hfill $\Box$\\ \bigskip \noindent {\bf Proposition 5.6.} {\it In the case $e=0$ the embedding $Y \rightarrow \Bbb{P}_1 \times \Bbb{P}_3$ is $(a,b)$--normal for all} $a \geq 0, b \geq 3$ or $(a,b) = (0,1).$\\ \bigskip \noindent {\bf Proof.} Using $(4)$ we have the following commutative diagram with exact rows $$ \begin{array}{lccccccccr} 0 \rightarrow & I_{Z| \Bbb{P}_1 \times \Bbb{P}_3}(a,b) & \rightarrow & {\cal{O}}_{\Bbb{P}_1 \times \Bbb{P}_3} (a,b) & \rightarrow & {\cal{O}}_Z(a,b) & \rightarrow 0\\[1ex] & \downarrow && \downarrow && || &\\[1ex] 0 \rightarrow & {\cal{L}}(a,b) & \rightarrow & {\cal{O}}_Y (a,b) & \rightarrow & {\cal{O}}_Z(a,b) & \rightarrow 0. \end{array} $$ Since $Z$ is $(a,b)$--normal in $\Bbb{P}_1 \times \Bbb{P}_3$, this yields $$ \begin{array}{lcccccr} 0 \rightarrow & H^0(I_{Z|\Bbb{P}_1 \times \Bbb{P}_3} (a,b)) & \rightarrow & H^0({\cal{O}}_{\Bbb{P}_1 \times \Bbb{P}_3} (a,b)) & \rightarrow & H^0 ({\cal{O}}_Z (a,b)) & \rightarrow 0 \\[1ex] &\downarrow \varphi & & \downarrow && ||\\[1ex] 0 \rightarrow & H^0({\cal{L}}(a,b)) & \rightarrow & H^0({\cal{O}}_Y (a,b)) & \rightarrow & H^0({\cal{O}}_Z(a,b)) & \rightarrow 0. \end{array} $$ Hence it suffices to show that $\varphi$ is surjective. But $$ {\cal{L}}(a,b) = {\cal{O}}_Z ((b+2) C_0 + (a+b -2) f). $$ So $\varphi$ is surjective if and only if the canonical map $$ H^0(N^{\ast}_{Z| \Bbb{P}_1 \times \Bbb{P}_3} (b C_0 +( a+b)f)) \rightarrow H^0({\cal{O}}_Z ((b+2) C_0 +(a+b -2)f) $$ is surjective, where $N^{\ast}_{Z| \Bbb{P}_1 \times \Bbb{P}_3}$ denotes the conormal bundle of $Z$ in $\Bbb{P}_1 \times \Bbb{P}_3$. Now det$N^{\ast}_{Z| \Bbb{P}_1 \times \Bbb{P}_3} = {\cal{O}}_Z(-2 C_0 - 4f)$ and the normal bundle sequence for $Z \subset Y \subset P$ gives $$ \scriptstyle{0 \rightarrow {\cal{O}}_Z((b-4) C_0 +(a+b-2)f) \rightarrow N^{\ast}_{Z| \Bbb{P}_1 \times \Bbb{P}_3} (b C_0 +(a+b)f)) \rightarrow {\cal{O}}_Z((b+2)C_0 + (a+b-2)f) \rightarrow 0 .} $$ But $$ \begin{array}{rcl} h^1({\cal{O}}_Z (b-4) C_0 +(a+b - 2)f)& = &h^1({\cal{O}}_{\Bbb{P}_1}(b-4)) \cdot h^0({\cal{O}}_{\Bbb{P}_1}(a+b-2))\\ &+&h^0 ({\cal{O}}_{\Bbb{P}_1} (b-4)) \cdot h^1({\cal{O}}_{\Bbb{P}_1} (a+b-2))\\ &=& 0 \quad \mbox{if} \quad b \geq 3 \quad \mbox{or} \quad (a,b) = (0,1). \hspace{2cm} \Box \end{array} $$ \bigskip \noindent {\bf Remark 5.7 } The proof of Proposition 5.6 yields \\ for $b=2, a \geq 0: \quad \mbox{codim} (H^0({\cal{O}}_{\Bbb{P}_1 \times \Bbb{P}_3} (a,2)) \rightarrow H^0 ({\cal{O}}_Y (a,2)) \leq a+1$\\ for $ b=1, a \geq 0: \quad \mbox{codim} (H^0({\cal{O}}_{\Bbb{P}_1 \times \Bbb{P}_3} (a,1)) \rightarrow H^0({\cal{O}}_Y (a,1)) \leq 2a \qquad $ and\\ for $b =0, a \geq 1: \quad \mbox{codim} (H^0({\cal{O}}_{\Bbb{P}_1 \times \Bbb{P}_3} (a,0)) \rightarrow H^0({\cal{O}}_Y (a,0) \leq 3 (a-1)$. \section{Splitting type of ${\cal{E}}$} There are 2 types of lines in $\Bbb{P}_1 \times \Bbb{P}_3$. In this section lines $\{ t \} \times \ell $ (respectively $\Bbb{P}_1 \times \{ x \}$) are called {\it horizontal} (respectively {\it vertical}) {\it lines} in $\Bbb{P}_1 \times \Bbb{P}_3$. Restriction of the vector bundle ${\cal{E}}$ to these lines leads to the notion of {\it horizontal} (respectively {\it vertical}) {\it generic splitting types} as well as {\it horizontal} (respectively {\it vertical}) {\it jumping lines.} \bigskip \noindent {\bf Proposition 6.1 } (a) {\it The horizontal generic splitting type of ${\cal{E}}$ is } $(2,2)$.\\ (b) {\it The vertical generic splitting type of ${\cal{E}}$ is} $(1,1)$. \bigskip \noindent {\bf Proof.} (a) follows from the fact that ${\cal{E}}_t$ is semistable but not stable with $c_1 ({\cal{E}}_t) =4$ on $\Bbb{P}_3$ (see Section 4).\\ (b): According to Section 1(4) the projection $X \subseteq \Bbb{P}_1 \times \Bbb{P}_3 \rightarrow \Bbb{P}_3$ is birational onto its image. Hence there is a a vertical line $\ell = \Bbb{P}_1 \times \{ x \}$ intersecting $X$ transversally in exactly one point. Restricting the exact sequence Section 2(1) to $\ell$ gives $$ 0 \rightarrow {\cal{O}}_{\ell} \rightarrow {\cal{E}}| \ell \rightarrow {\cal{O}}_{\ell}/I_{\ell \cap A} \otimes {\cal{O}}_{\ell} (2) \rightarrow 0. $$ Hence ${\cal{E}} | \ell = {\cal{O}}_{\ell} (1) \otimes {\cal{O}}_{\ell} (1)$. \hfill $\Box$ \bigskip \noindent {\bf Proposition 6.2.} (a) {\it The horizontal jumping lines of ${\cal{E}}$ are exactly the horizontal lines $\ell$ in $\Bbb{P}_1 \times \Bbb{P}_3$ intersecting the surface $Z \subset \Bbb{P}_1 \times \Bbb{P}_3$ of Proposition} 5.1.\\ (b) {\it If $\ell$ intersects $Z$ transversally, then} ${\cal{E}}|\ell = {\cal{O}}_{\ell} (4) \oplus {\cal{O}}_{\ell} (0)$.\\ \par In particular all horizontal jumping lines are "higher" jumping lines. \bigskip \noindent {\bf Proof.} Let $\ell = \{ t \} \times \Bbb{P}_1$ intersect the line $Z_t$ in $\{ t \} \times \Bbb{P}_3$ transversally. Then $\ell$ intersects the double structure $Y_t$ on $Z_t$ with multiplicity 2. Restricting the exact sequence (2) of Section 4 to $\ell$ we get $$ 0 \rightarrow {\cal{O}}_{\ell} \rightarrow E_t (-2)| \ell \rightarrow {\cal{O}}_{\ell}/I_{\ell \cap Y_t} \oplus {\cal{O}}_{\ell} \rightarrow 0 $$ which fills up to an exact sequence $$ 0 \rightarrow {\cal{O}}_{\ell} (2) \rightarrow E_t (-2) | \ell \rightarrow {\cal{O}}_{\ell} (-2) \rightarrow 0 $$ which necessarily splits. This proves (b). \vspace{1ex} \newline {\it As for (a):} Since ${\cal{E}}_t$ is semistable on $\Bbb{P}_3$, according to a theorem of Barth (see [O.S.S] p. 228) the jumping lines of ${\cal{E}}_t$ on $\Bbb{P}_3$ form a divisor in the Grassmannian Gr$(1,3)$ and there is an effective divisor $D_{{\cal{E}}_t}$ of degree $c_2({\cal{E}}_t(-2))=2$ with support the set of jumping lines $S_{{\cal{E}}_t}$ in $Gr (1,3)$.\\ On the other hand: The hypersurface $H_{Z_t} = \{ \ell \in Gr(1,3) | \ell \cap Z_t = \emptyset \}$ consists of jumping lines. Since every such jumping line is "higher", we get $D_{E_t} = 2H_{Z_t}$. This implies the assertion. \hfill $\Box$\\ \par Finally let us determine the vertical jumping lines of ${\cal{E}}$. In order to state the result recall from [L] that the coordinates of $\Bbb{P}_3$ can be chosen in such a way that the map $\varphi_3 : X \rightarrow \varphi_3 (X) \subseteq \Bbb{P}_3$ is an embedding outside the coordinate planes, whereas $\varphi_3(X)$ is singular along the coordinate planes. Choosing the coordinates of $\Bbb{P}_3$ in this way, we have \bigskip \noindent {\bf Proposition 6.3.} {\it The vertical jumping lines are exactly the lines $\Bbb{P}_1 \times \{ x \}$ with $x$ contained in a coordinate plane.\\ (b) For a general vertical jumping line }$\ell : {\cal{E}} | \ell = {\cal{O}}_{\ell} (2) \oplus {\cal{O}}_{\ell} (0)$. \bigskip \noindent For the proof let $S_{{\cal{E}}} \subseteq \Bbb{P}_3$ denote the set of jumping lines and consider the projection $q : \Bbb{P}_1 \times \Bbb{P}_3 \rightarrow \Bbb{P}_3$. \bigskip \noindent {\bf Lemma 6.4.} $S_{{\cal{E}}} = \mbox{supp} (R^1_{q_{\ast}} {\cal{E}} (-2, -2)).$ \bigskip \noindent {\bf Proof.} Let $U \subset \Bbb{P}_3$ be open affine. For $x \in U$ denote $\ell = \Bbb{P}_1 \times \{ x \}$ and $I_{\ell}$ the ideal sheaf of $\ell$. Then $H^2 (\Bbb{P}_1 \times U, {\cal{E}} (-2, -2) \otimes I_{\ell}) =0$, since $\Bbb{P}_1 \times U$ can be covered by 2 open affine sets. Hence the canonical map $H^1(\Bbb{P}_1 \times U, {\cal{E}}(-2, -2) ) \rightarrow H^1(\ell, {\cal{E}} (-2, -2)|\ell)$ is surjective. This implies that the base change homomorphism $$ R^1 q_{\ast} {\cal{E}} (-2, -2) (x) \rightarrow H^1 (\ell, {\cal{E}}(-2, -2)|\ell) $$ is an isomorphism. On the other hand $\ell = \Bbb{P}_1 \times \{ x \}$ is a vertical jumping line if and only if $h^1(\ell, {\cal{E}} (-2, -2)| \ell) > 0$. \hfill $\Box$\\ \par In order to define a scheme structure on $S_{{\cal{E}}}$ choose a resolution $$ 0 \rightarrow {\cal{K}} \rightarrow \bigoplus^{r+2}_{i=1} {\cal{O}}_{\Bbb{P}_1 \times \Bbb{P}_3} (-a_i, -b_i) \rightarrow {\cal{E}} (-2, -2) \rightarrow 0 \eqno(1) $$ with $a_i, b_i > 0$ for all $i$ and ${\cal{K}}$ is a locally free sheaf of rank $r$ on $\Bbb{P}_1 \times \Bbb{P}_3$. The sequence $$ 0 \rightarrow R^1_{q_{\ast}} {\cal{K}} \stackrel{\varphi}{\rightarrow} \bigoplus^{r+2}_{i=1} R^1_{q_{\ast}} {\cal{O}}(-a_i, -b_i) \stackrel{\psi}{\rightarrow} R^1_{q_{\ast}} {\cal{E}}(-2, -2) \rightarrow 0 \eqno(2) $$ is exact: $\varphi$ is injective since $q_{\ast} {\cal{E}}(-2, -2)$ is torsion free and 0 outside $S_{{\cal{E}}}$ and $\psi$ is surjective since $R^2_{q_{\ast}} {\cal{K}} =0$. \par The sheaves $R^1_{q_{\ast}} {\cal{K}}$ and $R^1_{q_{\ast}} {\cal{O}}(-a_i, -b_i)$ are locally free by the base change theorem since $h^1( {\cal{O}}_{\ell} (-a_i, -b_i))$ and $h^1({\cal{K}}| \ell) = -\chi({\cal{K}}| \ell)$ are independent of $\ell$. \par Hence $\varphi$ is a homomorphism of locally free sheaves of the same rank on $\Bbb{P}_3$ and $$ J_{\varphi} := \mbox{Im} (\det \varphi) \otimes \det \left( \bigoplus^{r+2}_{i=1} R^1_{q_{\ast}} {\cal{O}} (-a_i, -b_i)\right)^{\ast} \subseteq {\cal{O}}_{\Bbb{P}_3} $$ is an invertible ideal sheaf in ${\cal{O}}_{\Bbb{P}_3}$ with supp$({\cal{O}}_{\Bbb{P}_3} / J_{\varphi}) = S_{{\cal{E}}}$. Hence $D_{{\cal{E}}} := (S_{{\cal{E}}}, {\cal{O}}_{\Bbb{P}_3}/J_{\varphi})$ is a divisor on $\Bbb{P}_3$. \bigskip \noindent {\bf Lemma 6.5.} deg$D_{{\cal{E}}} =4$.\\ \medskip \noindent {\bf Proof.} Let $\Bbb{P}_1 \subseteq \Bbb{P}_3$ be a general line. Since the divisor $D_{{\cal{E}}}$ intersects the line $\Bbb{P}_1$ in a divisor of the same degree, we may restrict the whole situation to $\Bbb{P}_1 \times \Bbb{P}_1$ and compute the degree of the corresponding divisor on $\Bbb{P}_1$. \par By abuse of notation we denote the restricted objects to $\Bbb{P}_1 \times \Bbb{P}_1$ by the same letter. In particular we have the restricted sequences (1) on $\Bbb{P}_1 \times \Bbb{P}_1$ and (2) on $\Bbb{P}_1$.\\ If $h_1 = [p^{\ast} {\cal{O}}_{\Bbb{P}_1} (1)]$ and $h_2 = [q^{\ast} {\cal{O}}_{\Bbb{P}_1} (1)]$, then from (1) we get $$ c_1 (K)= - \sum_i (a_i h_1 + b_i h_2) + 2 h_1 \eqno(3) $$ $$ c_2 (K) = \sum_{i \not=j} a_i b_j - 4 -2 \sum_i b_i \eqno(4) $$ Applying flat base change and the projection formula, we get $$ \begin{array}{rl} R^1 q_{\ast} {\cal{O}} (-a_i, -b_i) & = R^1_{q_{\ast} } p^{\ast} {\cal{O}}_{\Bbb{P}_1} (- a_i) \otimes {\cal{O}}_{\Bbb{P}_1} (- b_i)\\[1ex] &= H^1 ({\cal{O}}_{\Bbb{P}_1} (- a_i)) \otimes {\cal{O}}_{\Bbb{P}_1} (-b_i) \end{array} $$ and hence $$ c_i (R^1 q_{\ast} {\cal{O}}(-a_i, -b_i)) =-b_i (a_i - 1). $$ The Theorem of Grothendieck--Riemann--Roch for the morphism $q: \Bbb{P}_1 \times \Bbb{P}_1 \rightarrow \Bbb{P}_1$ and equations (3) and (4) give $$ \begin{array}{ll} c_1(R^1 q_{\ast} {\cal{K}}) &= - c_1 ({\cal{K}})h_1 - \frac{1}{2} (c^2_1 ({\cal{K}}) - 2c_2({\cal{K}}))\\[1ex] & = - \displaystyle{ \sum_i a_i b_i + \sum_i b_i - 4} \end{array} $$ It follows that $$ \begin{array}{ll} \deg D_{{\cal{E}}} & = - c_1 \left( \bigoplus^{r+2}_{i=1} R^1 q_{\ast} {\cal{O}}(-a_i, -b_i)\right) - c_1(R^1_{q_{\ast}} {\cal{K}})\\[1ex] & = \displaystyle{- \sum_i b_i (a_i -1) + \sum_i a_i b_i - \sum_i b_i +4 = 4} \hspace{5.0cm} \Box \end{array} $$ {\bf Proof of Proposition 6.3.} Let $H_1, \ldots , H_4$ denote the 4 coordinate planes in $\Bbb{P}_3$. For a general point $x$ of $\varphi_3 (X) \cap H_i$ the vertical line $\ell = \Bbb{P}_1 \times \{ x \}$ intersects $X \subset \Bbb{P}_1 \times \Bbb{P}_3$ transversally in 2 points. Restriction of the exact sequence (1) of Section 2 to $\ell$ gives ${\cal{E}} | \ell = {\cal{O}}_{\ell}(2) \oplus {\cal{O}}_{\ell} (2)$. Hence the divisor $\varphi_3(X) \cap H_i \subset D_{{\cal{E}}}$ for $i = 1, \ldots , 4$. Since $H_1 + \ldots + H_4$ is the only divisor of degree $\leq 4$ in $\Bbb{P}_3$ containing the 4 curves $\varphi_3 (X) \cap H_i$, the assertion follows. \hfill $\Box$ \section{Problems} Of course concerning the vector bundle ${\cal{E}}$ on $\Bbb{P}_1 \times \Bbb{P}_3$ one can ask all the questions which have been studied for the Horrocks--Mumford bundle on $\Bbb{P}_4$. The most important problems are \bigskip \noindent (1) {\bf Cohomology of ${\cal{E}}$:} Compute the dimensions of the groups $H^i({\cal{E}}(a,b))$ for all $a,b$ and $i$. I computed most of these groups, unfortunately the list is not complete. In particular the most important dimension $h^0({\cal{E}})$ is not known. \bigskip \noindent (2) {\bf Automorphisms of ${\cal{E}}$:} According to Proposition 3.5 the group of automorphisms of the vector bundle ${\cal{E}}$ contains the dihedral group $D_8 = H_1(L_1, L_3)$. Does ${\cal{E}}$ admit further automorphisms? Recall that the automorphism group of the Horrocks--Mumford bundle is the normalizer of the Heisenberg group in $S\ell_5(\Bbb{C})$. Is the analogue statement valid in the case of ${\cal{E}}$? \bigskip \noindent (3) {\bf Moduli:} Describe the moduli spaces of vector bundles with the invariants of ${\cal{E}}$. This question is closely related to (1) and (2). For example, if $h^0({\cal{E}}) \leq 2$, then ${\cal{E}}$ admits moduli, since there is a two--dimensional family of abelian surfaces in $\Bbb{P}_1 \times \Bbb{P}_3$. \bigskip \noindent (4) {\bf Other constructions:} It seems not too difficult to construct the vector bundle ${\cal{E}}$ out of the double structure $Y$ on $\Bbb{P}_1 \times \Bbb{P}_1$ in $\Bbb{P}_3 \times \Bbb{P}_3$ of Section 5. Are there other constructions? \bigskip \noindent (5) {\bf Degenerations:} It is not difficult to see that every smooth zero set of a global section of ${\cal{E}}$ is an abelian surface $A$ in $\Bbb{P}_1 \times \Bbb{P}_3$. Determine the degenerations of $A$ in $\Bbb{P}_1 \times \Bbb{P}_3$. Is every such degeneration the zero set of a section of (one of the vector bundles) ${\cal{E}}$? \bigskip \noindent (6) {\bf Classification:} Are there other rank--2 vector bundles on $\Bbb{P}_1 \times \Bbb{P}_3$ apart from the bundles derived from ${\cal{E}}$ or the bundles of the introduction? \vspace{2cm} \newline {\bf Literature:} \begin{quote}\begin{itemize} \item[\mbox{\makebox[1.2cm][l]{[CAV]}}] {\bf H. Lange, Ch. Birkenhake:} {\it Complex Abelian Varieties}; Grundlehren 302, Springer 1992 \item[\mbox{\makebox[1.2cm][l]{[H]}}] {\bf R. Hartshorne:} {\it Stable Vector Bundles of Rank 2 on} $\Bbb{P}_3$; Math. Ann. 238 (1978), 229 -280 \item[\mbox{\makebox[1.2cm][l]{[L]}}] {\bf H. Lange:} {\it Abelian surfaces in }$\Bbb{P}_1 \times \Bbb{P}_3$. Arch. Math. 63 (1994), 80 -- 84. \item[\mbox{\makebox[1.2cm][l]{[O.S.S]}}] {\bf Ch. Okonek, M. Schneider, H. Spindler:} {\it Vector Bundles on Complex Projective Spaces}; Progr. in Math. 3, Birkh\"auser 1980 \end{itemize} \end{quote} \end{document} te:x

9506

alg-geom/9506010

fr

https://arxiv.org/abs/alg-geom/9506010

alg-geom/9506010

Francois Lauze

Francois Lauze

Rang maximal pour $T_P^n$

LateX 2e, in french, no more accents, major revision

In this paper, I compute the last non-trivial term of the minimal free resolution of the homogeneous ideal of $s$ points of $P^n$ in sufficiently general position, for any $s$, showing that this term is the one conjectured by the Minimal Resolution Conjecture of Anna Lorenzini. I use a geometrical method, the "vectorial m\'ethode d'Horace" developped by Andr\'e Hirschowitz and Carlos Simpson to get a proof of the MRC for $s$ large enough.

alg-geom

\subsection*{ \begin{center} \thepara.~#1 \end{center} } } \title{Rang maximal pour $\tp{n}$} \author{Fran{\c{c}}ois Lauze \\ U.N.S.A Laboratoire de Math{\'e}matiques J. Dieudonn{\'e}, U.R.A 168\\ Parc Valrose, 06108 Nice Cedex 2\\ e-mail : [email protected] } \date{} \newcommand{\item[$\bullet$]}{\item[$\bullet$]} \newcommand{\demo} {{\bf D{\'e}monstration : }} \begin{document} \maketitle \begin{abstract} Let ${\bf k}~$ an algebraically closed field, ${\bf P}^n$ the n-dimensional projective space over ${\bf k}~$ and $\tp{n}$ the tangent vector bundle of ${\bf P}^n$. In this paper I prove the following result : for every integer $\ell$, for every non-negative integer $s$, if $Z_s$ is the union of $s$ points in sufficiently general position in ${\bf P}^n$, then the restriction map $H^0({\bf P}^n,\tp{n}(\ell))\rightarrow H^0(Z_s,\tp{n}(\ell)_{|Z_s})$ has maximal rank. This result implies that the last non-trivial term of the minimal free resolution of the homogeneous ideal of $Z_s$ is the conjectured one by the Minimal Resolution Conjecture of Anna Lorenzini (cf~\cite{mrc}). \end{abstract} \paragraphe{Introduction} Soit {\bf k} un corps alg{\'e}briquement clos et ${\bf P}^n = {\bf P}^n_{\bf k}$ l'espace projectif de dimension $n$ sur {\bf k} et $S = {\bf k}[x_0, \ldots,x_n]$ l'anneau des coordonn{\'e}es homog{\`e}nes sur ${\bf P}^n$. Soit $a$ un entier non n{\'e}gatif et $R := \{P_1,\ldots,P_a\} \subset {\bf P}^n$ des points en position suffisamment g{\'e}n{\'e}rale, ${\cal I}_R$ le faisceau d'id{\'e}aux de $R$ et $I = \Gamma_*({\cal I}_R)$ l'id{\'e}al homog{\`e}ne de $Z$ dans $S$. Si on pose $d := min\{\ell,h^0({\bf P}^n,{\cal I}_R(\ell)) > 0\}$, alors $I$ est engendr{\'e} par $I_d\oplus I_{d+1}$ o{\`u} $I_s$ d{\'e}signe la partie homog{\`e}ne de degr{\'e} $s$ de $I$ (Th{\'e}orie de Castelnuovo-Mumford, \cite{mumford}, p 99). La r{\'e}solution minimale libre de $I$ s'{\'e}crit alors : $$ 0\rightarrow L_n \rightarrow \ldots \rightarrow L_p \rightarrow \ldots \rightarrow L_0\rightarrow I\rightarrow 0 $$ avec $$ L_p = S(-d-p-1)^{a_p}\oplus S(-d-p)^{b_p} $$ et $$ a_p := h^1({\bf P}^n,\Omega_{{\bf P}^n}^{p+1}(d+p+1)\otimes {\cal I}_R)\;,\; b_p := h^0({\bf P}^n,\Omega_{{\bf P}^n}^{p}(d+p)\otimes {\cal I}_R). $$ Puisque l'application de restriction \begin{equation} \label{eq1} H^0({\bf P}^n,\op{}{n}(\ell)) \rightarrow H^0(R,{\cal O}_R(\ell)) \end{equation} est de rang maximal, c'est {\`a} dire surjective ou injective, les nombres $b_0$, $a_{n-1}$ et $b_n$ sont connus. Les nombres $a_p$ et $b_p$ ont {\'e}t{\'e}s calcul{\'e}s dans~\cite{hirsim} pour tout nombre $a$ de points suffisamment grand et tout entier p. L'entier $a_n$ est {\'e}videmment nul et puisque l'application~(\ref{eq1}) est de rang maximal, on a \begin{itemize} \item[$\bullet$] $b_n = max\{0,h^0({\bf P}^n,\op{}{n}(d-1)) - a\} = 0$, par d{\'e}finition de $d$, \item[$\bullet$] $a_{n-1} = max\{0,a-h^0({\bf P}^n,\op{}{n}(d-1))\} = a - \left(\!\!\!\begin{array}{c} n\!\!+\!\!d\!\!-\!\!1\\ n \end{array}\!\!\!\right)$, pour les m{\^e}mes raisons. \end{itemize} Le but de cet article est de calculer, sans restriction sur le nombre de points, les entiers $a_{n-2}$ et $b_{n-1}$. On va {\'e}tablir le th{\'e}or{\`e}me suivant :\vspace{3mm} \begin{theoreme} \label{th1} Avec les notations pr{\'e}c{\'e}dentes : \begin{itemize} \item[$\bullet$] $a_{n-2} = max\{0,na - h^0({\bf P}^n,\Omega_{{\bf P}^n}^{n-1}(d+n-1))\}$ \item[$\bullet$] $b_{n-1} = max\{0,h^0({\bf P}^n,\Omega_{{\bf P}^n}^{n-1}(d+n-1))-na\}$, \end{itemize} c'est {\`a} dire que l'application de restriction $$ H^0({\bf P}^n,\Omega_{{\bf P}^n}^{n-1}(d+n-1)) \rightarrow H^0(R,\Omega_{{\bf P}^n}^{n-1}(d+n-1)|_R) $$ est de rang maximal, i.e. surjective ou injective (puisque $rg(\Omega^{n-1}_{{\bf P}^n})=n)$. \end{theoreme} Des cas particuliers de ce th{\'e}or{\`e}me ont {\'e}t{\'e} d{\'e}montr{\'e}s dans \cite{GGR}, \cite{GO1}, \cite{GO2}, \cite{GM} ainsi que dans \cite{Roberts}. Une d{\'e}monstration g{\'e}n{\'e}rale se trouve dans \cite{truval}, mais selon Charles Walter, elle contient un point litigieux au milieu de la page 116 (`` ...it is clear that the monomial $\gamma$ is greater than any other $M-$product ...''). La d{\'e}monstration donn{\'e}e ici est plus de nature g{\'e}om{\'e}trique qu'alg{\'e}brique et est un raffinement de la m{\'e}thode d'Horace introduite par A. Hirschowitz et C. Simpson dans \cite{hirsim} pour prouver la ``Minimal Resolution Conjecture'' (MRC) de \cite{mrc} pour tout nombre de points suffisamment grand et en position suffisamment g{\'e}n{\'e}rale dans ${\bf P}^n$. \paragraphe{R{\'e}duction du probl{\`e}me} Pour {\'e}tablir le th{\'e}or{\`e}me~\ref{th1} il suffit {\'e}videment de prouver le suivant. \begin{letheoreme} Pour tout entier non-n{\'e}gatif $a$ il existe des points $P_1,\dots,P_a$ de ${\bf P}^n$, tels que tout entier $\ell$, le morphisme d'{\'e}valuation $$ \sigma_{\ell,a} : H^0({\bf P}^n,\tp{n}(\ell)) \rightarrow \tp{n}(\ell)|_{P_1}\oplus\ldots\oplus \tp{n}(\ell)|_{P_a} $$ est de rang maximal. \end{letheoreme} (on utilise l'isomorphisme $\Omega_{{\bf P}^n}^{n-1}(\ell) \simeq T_{{\bf P}^n}(\ell-n-1)$.)\vspace{5mm} On va d'abord r{\'e}duire le probl{\`e}me de rang maximal {\`a} d{\'e}calage $\ell$ fix{\'e} {\`a} celui d'une bijection, ceci gr{\^a}ce {\`a} des arguments standards (cf \cite{harthirsch}, \cite{Ballico}, \cite{thIda}). Pour plus de simplicit{\'e}, on posera \begin{itemize} \item[$\bullet$] $t_n(\ell) := h^0({\bf P}^n,\tp{n}(\ell))$, \item[$\bullet$] $o_n(\ell) := h^0({\bf P}^n,\op{}{n}(\ell))$. \end{itemize} \begin{lelemme} Soient $q = q(\ell)$ et $r = r(\ell)$ respectivement le quotient et le reste de la {division eu-} clidienne de $t_n(\ell)$ par $n$. Alors pour que le th{\'e}or{\`e}me 1 soit vrai, il suffit qu'il existe un point $P_{q+1}$ tel que pour tout quotient $$ \tp{n}(\ell)_{P_{q+1}} \rightarrow B \rightarrow 0 $$ de dimension $r$, il existe des points $P_1,\dots,P_{q+1}$ tels que le morphisme d'{\'e}valuation $$ \tau_\ell : H^0({\bf P}^n,\tp{n}(\ell)) \rightarrow \tp{n}(\ell)|_{P_1} \oplus\ldots\oplus\tp{n}(\ell)|_{P_q}\oplus B $$ soit bijectif. \end{lelemme} \demo Supposons donc qu'il existe des points $P_1,\dots,P_{q+1}$ tels que $\tau_\ell$ soit bijectif. Alors si $z \leq q$, on a une application $\sigma_{\ell,z}$, compos{\'e}e de $\tau_{\ell}$ et de la surjection $$ \begin{array}{c} \tp{n}(\ell)|_{P_1}\oplus\ldots\oplus \tp{n}(\ell)|_{P_q}\oplus B \\ \downarrow \\ \tp{n}(\ell)|_{P_1}\oplus\ldots\oplus \tp{n}(\ell)|_{P_z} \end{array} $$ donc surjective. Soit maintenant $z'$ un entier plus grand que $q+1$. Soient alors $P_{q+2},\dots P_{z'}$ des points distincts, et diff{\'e}rents de $P_1,\dots,P_{q+1}$. A l'aide du quotient $\tp{n}(\ell)_{P_{q+1}}\rightarrow B \rightarrow 0$, on d{\'e}duit que le morphisme $\tau_\ell$ se factorise par $\sigma_{\ell,z'}$ et la projection $$ \begin{array}{c} \tp{n}(\ell)|_{P_1}\oplus\ldots\oplus \tp{n}(\ell)|_{P_{z'}}\\ \downarrow \\ \tp{n}(\ell)|_{P_1}\oplus\ldots\oplus \tp{n}(\ell)|_{P_q} \oplus B. \end{array} $$ On en d{\'e}duit donc que $\sigma_{\ell,z'}$ est injective. \hspace{\fill Pour obtenir le r{\'e}sultat escompt{\'e} pour $a_{n-2}$ et $b_{n-1}$ il suffit donc de montrer le \begin{letheoreme} \label{thtpn} Pour tout entier $\ell$, il existe des points $P_1,\dots P_{q+1}$ tels que le morphisme $\tau_\ell$ soit bijectif. \end{letheoreme} Remarquons que le th{\'e}or{\`e}me est trivialement vrai si $\ell \leq -2$. on fera donc dans la suite l'hypoth{\`e}se que $\ell \geq -1$.\newpage \paragraphe{M{\'e}thodes d'Horace vectorielles} Dans ce paragraphe on introduit la m{\'e}thode d'Horace pour les probl{\`e}mes d'{\'e}valuation de sections de fibr{\'e}s vectoriels en des points et des ``fractions de points''. Cette m{\'e}thode, essentiellement bas{\'e}e sur les transformations {\'e}l{\'e}mentaires de fibr{\'e}s vectoriels, fut introduite en 1984 dans une lettre d'A. Hirschowitz {\`a} R. Hartshorne pour montrer que si $P_1,\dots,P_{28}$ sont des points en position g{\'e}n{\'e}rale dans ${\bf P}^3$, l'application naturelle $$ H^0({\bf P}^3,\Omega_{{\bf P}^3}(5)) \rightarrow \Omega_{{\bf P}^3}(5)|_{P_1} \oplus \dots \oplus \Omega_{{\bf P}^3}(5)|_{P_{28}} $$ est bijective. Elle a aussi {\'e}t{\'e} utilis{\'e}e par M. Id{\`a} dans \cite{thIda} cette fois-ci avec des droites et des points pour calculer la r{\'e}solution minimale des id{\'e}aux d'arrangement de droites en position g{\'e}n{\'e}rale dans ${\bf P}^3$, et par O.F. Rahavandrainy dans~\cite{felix} pour calculer des r{\'e}solutions de fibr{\'e}s instantons. La pr{\'e}sentation suivante est celle de A. Hirschowitz et C. Simpson (cf.~\cite{hirsim}). Fixons quelques notations qu'on utilisera r{\'e}guli{\`e}rement par la suite. Soit $X$ une vari{\'e}t{\'e} projective lisse, et $X'$ un diviseur non-singulier de $X$. Soit ${\cal F}$ un faisceau localement libre sur $X$ et $$ 0 \rightarrow {\cal F} '' \rightarrow {\cal F}|_{X'} \rightarrow {\cal F} ' \rightarrow 0 $$ une suite exacte stricte de faisceaux localement libres sur $X'$. On notera ${\cal E}$ le noyau du morphisme ${\cal F} \rightarrow {\cal F} '$; on notera que ${\cal E}$ est localement libre sur $X$ et qu'on a la suite exacte $$ 0 \rightarrow {\cal F} '(-X') \rightarrow {\cal E}|_{X'} \rightarrow {\cal F} '' \rightarrow 0. $$ {~}\newline\newline \begin{flushleft} \underline{La m{\'e}thode d'Horace "simple" :} \end{flushleft} Le lemme suivant est une cons{\'e}quence triviale du lemme du serpent. \begin{lemme} \label{lemA} Supposons donn{\'e} une application lin{\'e}aire bijective $\lambda : H^0(X',{\cal F} ' )\rightarrow L.$ Supposons que $H^1(X,{\cal E} )=0$. Soit $\mu : H^0 (X, {\cal F} )\rightarrow M$ une application lin{\'e}aire. Alors, pour que le morphisme $$ H^0(X,{\cal F})\rightarrow M\oplus L $$ soit de rang maximal, il faut et il suffit que le morphisme $$ H^0(X,{\cal E})\rightarrow M $$ le soit. \end{lemme} \newpage \begin{flushleft} \underline{La m{\'e}thode d'Horace "diff{\'e}rentielle":} \end{flushleft} Le lemme suivant est le lemme 1 de~\cite{hirsim}. \begin{lemme} \label{lemB} On se donne maintenant une application lin{\'e}aire surjective $\lambda :H^0(X',{\cal F} ' )\rightarrow L$ et supposons qu'il existe un point $Z'\in X'$ tel que l'application $H^0 (X', {\cal F} ' )\rightarrow L \oplus {\cal F} ' _{Z'}$ soit injective. Supposons encore que $H^1(X,{\cal E} )=0$. Alors il existe un quotient ${\cal E} _{Z'}\rightarrow D$ avec noyau contenu dans ${\cal F} ' (-X')_{Z'}$, de dimension $dim (D) = r({\cal F} ) -dim (ker (\lambda )),$ ayant la propri{\'e}t{\'e} suivante. Soit $\mu : H^0 (X, {\cal F} )\rightarrow M$ une application lin{\'e}aire. Pour qu'il existe $Z\in X$ tel que l'application $$ H^0 (X, {\cal F} )\rightarrow M \oplus L \oplus {\cal F} _Z $$ soit de rang maximal, il suffit que $$ H ^0 ( X, {\cal E} )\rightarrow M \oplus D $$ le soit. \end{lemme}~\newline Consid{\'e}rons maintenant ${\cal G}$ un quotient localement libre de ${\cal F}$ et ${\cal K}$ le noyau du morphisme ${\cal F} \rightarrow {\cal G}$. Posons ${\cal H} = ({\cal E}+{\cal K})/{\cal K}$ le sous-faisceau de ${\cal G}$ engendr{\'e} par ${\cal E}$ et ${\cal G}'$ le quotient ${\cal G}/{\cal H}$. G{\'e}om{\'e}triquement on a $$ {\bf P}({\cal G}') = {\bf P}({\cal G})\cap{\bf P}({\cal F}') \subset {\bf P}({\cal F}). $$ On supposera ${\cal G}'$ localement libre sur $X'$ de sorte que ${\cal H}$ est localement libre sur $X$.\newline Le lemme suivant est alors une g{\'e}n{\'e}ralisation du pr{\'e}c{\'e}dent. \begin{lemme} \label{lemC} Soit $\lambda : H^0(X',{\cal F}') \rightarrow L$ une application lin{\'e}aire surjective. On suppose qu'il existe $Y' \in X'$ tel que l'application $H^0(X',{\cal F}')\rightarrow L\oplus {\cal G}'_{Y'}$ soit injective. On suppose encore que $H^1(X,{\cal E})=0$. Alors il existe un quotient ${\cal H}_{Y'}\rightarrow D$ avec noyau contenu dans ${\cal G}(-X')$ et de dimension $rg({\cal G})-dim(ker(\lambda))$ ayant la propri{\'e}t{\'e} suivante. Soit $\mu : H^0 (X, {\cal F} )\rightarrow M$ une application lin{\'e}aire. Pour qu'il existe $Y\in X$ tel que l'application $$ H^0 (X, {\cal F} )\rightarrow M \oplus L \oplus {\cal G}_Y $$ soit de rang maximal, il suffit que $$ H ^0 ( X, {\cal E} )\rightarrow M \oplus D $$ le soit. \end{lemme} \demo Puisque l'application $H^0(X',{\cal F}')\rightarrow L\oplus {\cal G}'_{Y'}$ est injective, l'application $$ H^0(X',{\cal F}')\rightarrow L\oplus {\cal F}'_{Y'} $$ l'est aussi. Ces conditions ainsi que la premi{\`e}re hypoth{\`e}se du lemme entra{\^\i}nent que le morphisme $$ ker(\lambda)\otimes_k{\cal O}_{X'} \rightarrow {\cal F}'\;\;\;(resp.\; ker(\lambda)\otimes_k{\cal O}_{X'} \rightarrow {\cal G}') $$ est injectif et que son image ${\cal A}' \subset {\cal F}'$ (resp. ${\cal A}'' \subset {\cal G}'$) est un sous-fibr{\'e} de ${\cal F}'$ (resp. ${\cal G}'$) au voisinage de $Y'$. Notons que $rg({\cal A}') = rg({\cal A}'') = dim(ker(\lambda))$ et que ${\cal A}''$ est l'image de ${\cal A}'$ par le morphisme surjectif ${\cal F}'\rightarrow {\cal G}'$. Notons ${\cal B}$ le noyau de ${\cal F}\rightarrow {\cal F}'/{\cal A}'$. Alors le noyau de $H^0(X,{\cal F})\rightarrow L$ est {\'e}gal {\`a} $H^0(X,{\cal B})$. Soit ${\cal C}$ l'image de ${\cal B}$ par le morphisme ${\cal F} \rightarrow {\cal G}$. Soit $D$ l'image du morphisme ${\cal H}_{Y'} \rightarrow {\cal C}_{Y'}$. On a une suite exacte $$ 0\rightarrow D \rightarrow {\cal C}_{Y'} \rightarrow {\cal A}''_{Y'} \rightarrow 0. $$ Puisque ${\cal A}'' \subset {\cal G}'$ est un sous-fibr{\'e} de rang $dim(ker(\lambda))$ au voisinage de $Y'$, et ${\cal C}$ est localement libre au voisinage de $Y'$ avec $rg({\cal C}) = rg({\cal G})$, on a $$ dim(D) = rg({\cal G}) - dim(ker(\lambda)). $$ Si $Y$ est un point de $X-X'$ les fibres ${\cal C}_Y$ et ${\cal G}_Y$ s'identifient. Le fait que $H^1(X,{\cal E})$ est nul entra{\^\i}ne que l'application $H^0(X,{\cal F}) \rightarrow H^0(X',{\cal F}')$ est surjective, en particulier la suite $$ 0\rightarrow H^0(X,{\cal B}) \rightarrow H^0(X,{\cal F}) \rightarrow L \rightarrow 0 $$ est exacte, et puisque la suite $$ 0 \rightarrow M\oplus{\cal C}_Y \rightarrow M\oplus L \oplus {\cal G}_Y \rightarrow L \rightarrow 0 $$ l'est aussi, on obtient alors que le morphisme $$ H^0(X,{\cal F}) \rightarrow M\oplus L \oplus {\cal G}_Y $$ est de rang maximal si et seulement si $$ H^0(X,{\cal B}) \rightarrow M \oplus {\cal C}_Y $$ l'est aussi. On va maintenant sp{\'e}cialiser et prendre $Y = Y'$. Puisque $H^1(X,{\cal E})=0$, on a une suite exacte $$ 0\rightarrow H^0(X,{\cal E}) \rightarrow H^0(X,{\cal B}) \rightarrow H^0(X,{\cal A}') \rightarrow 0 $$ Du diagramme commutatif {\`a} lignes exactes $$\diagram 0\rto & {\cal E} \rto\dto & {\cal B} \rto\dto& {\cal A}'\rto\dto & 0\\ 0\rto & {\cal H}\rto & {\cal C} \rto & {\cal A}'' \rto & 0 \enddiagram$$ on d{\'e}duit que le morphisme naturel compos{\'e} $$ H^0(X,{\cal E}) \subset H^0(X,{\cal B}) \rightarrow {\cal C}_{Y'} \rightarrow {\cal A}''_{Y'} $$ est nul. On en d{\'e}duit que le morphisme naturel compos{\'e} $$ H^0(X,{\cal B}) \rightarrow M\oplus{\cal C}_{Y'} \rightarrow {\cal A}''_{Y'} $$ se factorise par $H^0(X,{\cal A}') \rightarrow {\cal A}''_{Y'}$. Le morphisme $H^0(X,{\cal E})\rightarrow M\oplus {\cal C}_{Y'}$ se factorise alors par le noyau de $M\oplus {\cal C}_{Y'} \rightarrow {\cal A}_{Y'}$, c'est {\`a} dire $M \oplus D$. Le noyau du morphisme $H^0(X,{\cal B}) \rightarrow M\oplus {\cal C}_{Y'}$ est contenu dans le noyau de $$ H^0(X,{\cal F})\rightarrow L \oplus {\cal A}''_{Y'} \subset L\oplus {\cal G}'_{Y'}. $$ Par hypoth{\`e}se, il est donc contenu dans le noyau de $H^0(X,{\cal F})\rightarrow H^0(X,{\cal F}')$, c'est {\`a} dire $H^0(X,{\cal E})$ et donc les noyaux des deux morphismes suivants son {\'e}gaux : $$ H^0(X,{\cal E}) \rightarrow M\oplus D,\;\; H^0(X,{\cal B}) \rightarrow M\oplus {\cal C}_{Y'} $$ Puisque le morphisme $H^0(X',{\cal A}') \rightarrow {\cal A}''_{Y'}$ est surjectif, on conclut alors que pour que $$ H^0(X,{\cal B}) \rightarrow M\oplus {\cal C}_{Y'} $$ soit de rang maximal, il suffit que $$ H^0(X,{\cal E}) \rightarrow M \oplus D $$ le soit. Mais la condition de maximalit{\'e} du rang {\'e}tant ouverte, on en d{\'e}duit alors qu'il existe $Y \in X-X'$ tel que $$ H^0(X,{\cal B}) \rightarrow M\oplus {\cal C}_Y $$ soit de rang maximal, ce qui donne la conclusion cherch{\'e}e.\hspace{\fill \paragraphe{Les {\'e}nonc{\'e}s R, RB et un lemme d'Horace} \subsubsection*{Enonc{\'e} ${\bf R}({\cal F};a)$} Soit $a$ un entier non-n{\'e}gatif. L'{\'e}nonc{\'e} ${\bf R}({\cal F} ; a)$ veut dire que pour tout entier non-n{\'e}gatif $z$, il existe des points $T_1,\dots,T_a \in X$, tels que pour tous quotients $$ {\cal F} _{T_i}\rightarrow A_i \rightarrow 0, $$ il existe des points $P_1,\dots,P_z \in X$, tels que le morphisme d'{\'e}valuation \begin{eqnarray*} H^0 (X,{\cal F}) &\rightarrow& A_1 \oplus \ldots \oplus A_a \oplus \\ & & {\cal F}_{P_1}\oplus \dots \oplus{\cal F}_{P_z} \end{eqnarray*} est de rang maximal. \subsubsection*{Enonc{\'e} ${\bf RB}({\cal F},{\cal F}',z,y;\alpha,\beta)$} Soit $X'$ un diviseur lisse connexe de $X$ et soient $z,y,\alpha$ et $\beta$ des entiers non-n{\'e}gatifs v{\'e}rifiant \begin{enumerate} \item[$\bullet$] $rg({\cal F})z + rg({\cal F}')y + \alpha + \beta = h^0(X,{\cal F})$ \item[$\bullet$] $rg({\cal F}')y + \alpha + b \leq h^0(X,{\cal F}')$ \item[$\bullet$] $0\leq \alpha\leq rg({\cal F}')$ \item[$\bullet$] si $\beta \not=0$, $rg({\cal F}')\leq \beta < rg({\cal F})$, \end{enumerate} o{\`u} $b = b(\beta)$ vaut $r'$ si $\beta$ est non nul et 0 sinon. L'{\'e}nonc{\'e} ${\bf RB}({\cal F},{\cal F}',z,y;\alpha,\beta)$ signifie alors la chose suivante. Il existe des points $T$ et $V$ de $X'$, tels que pour tout quotient $$ {\cal F}'_T \rightarrow A \rightarrow 0 $$ de dimension $\alpha$ et si $\beta \not=0$,pour tout quotient d{\'e}pendant rationnellement de $y$ points g{\'e}n{\'e}raux de $X'$, $$ {\cal F}_V\rightarrow B \rightarrow 0 $$ de dimension $\beta$ avec noyau contenu dans ${\cal F}''_V$, (si $\beta = 0$ on pose $B= 0$), il existe $y$ points $Y_1,\dots,Y_y \in X'$ et $z$ points $P_1,\dots,P_z \in X$ tel que le morphisme d'{\'e}valuation \begin{eqnarray*} H^0(X,{\cal F}) &\rightarrow & A \oplus B \oplus \\ & & {\cal F}'_{Y_1}\oplus\dots\oplus {\cal F}'_{Y_y}\oplus \\ & & {\cal F}_{P_1}\oplus\dots\oplus {\cal F}_{P_z} \end{eqnarray*} soit bijectif.\newline\newline Soient $z,y,\alpha$ et $\beta$ des entiers v{\'e}rifiant les conditions de l'{\'e}nonc{\'e} ${\bf RB}({\cal F},{\cal F}',z,y;\alpha,\beta)$. On rappelle qu'on avait d{\'e}fini l'entier $b=b(\beta)$ comme valant $r'$ si $\beta$ est non nul et 0 sinon. Posons alors \begin{itemize} \item[$\bullet$] $t := h^0(X',{\cal F}')- r'y-\alpha - b$, \item[$\bullet$] $y'$ le quotient de la division euclidienne de $t$ par $r'$, $\delta$ le reste, \item[$\bullet$] $\zeta := 1$ si $\delta \not=0$ et $0$ sinon, \item[$\bullet$] $\beta' := \zeta(r-\delta)$ \item[$\bullet$] $\alpha' := \beta - r'$ si $\beta \not= 0$ et 0 sinon. \item[$\bullet$] $z' := z-y'-\zeta$ qu'on supposera non-n{\'e}gatif. \end{itemize} o{\`u} $r := rg({\cal F})=rg({\cal E})$ et $r' := rg({\cal F}')$. On peut alors {\'e}noncer le lemme suivant : \begin{lemme} \label{lemRB} Soient $z,y,\alpha$ et $\beta$ des entiers non-n{\'e}gatifs v{\'e}rifiant les conditions de l'{\'e}nonc{\'e} ${\bf RB}({\cal F},{\cal F}',z,y;\alpha,\beta)$. On d{\'e}finit $a(\alpha) = 0$ si $\alpha$ est nul, 1 sinon. Supposons que $H^1(X,{\cal E})=0$ et que (avec les notations pr{\'e}c{\'e}dant le lemme) les {\'e}nonc{\'e}s $$ {\bf R}({\cal F}';a)\;et\;{\bf RB}({\cal E},{\cal F}'',z',y';\alpha',\beta') $$ soient vrais. Alors l'{\'e}nonc{\'e} ${\bf RB}({\cal F},{\cal F}',z,y;\alpha,\beta)$ l'est aussi. \end{lemme} \demo On ne va traiter que le cas o{\`u} $\beta \not= 0$, le cas $\beta=0$ {\'e}tant laiss{\'e} au lecteur. l'hypoth{\`e}se ${\bf R}({\cal F}';a)$ entra{\^\i}ne qu'il existe $T$ et $V \in X'$, tels que pour tout quotient $$ {\cal F}'_U\rightarrow A \rightarrow 0 $$ de dimension $\alpha$ et pour tout quotient $$ {\cal F}_V\rightarrow B \rightarrow 0, $$ avec noyau contenu dans ${\cal F}''_V$ il existe des points $Y_1,\dots Y_y \in X'$ et $Z_1,\dots Z_{y'}$ tel que le morphisme d'{\'e}valuation \begin{eqnarray*} \lambda : H^0(X'{\cal F}') &\rightarrow& A\oplus \\ & & {\cal F}'_{Y_1}\oplus\dots\oplus{\cal F}'_{Y_y}\oplus\\ & & {\cal F}'_{Z_1}\oplus\dots\oplus{\cal F}'_{Z_{y'}}\\ & & {\cal F}'_V \end{eqnarray*} soit de rang maximal, donc surjectif ici. $\bullet$ Si $\lambda$ est bijectif, (c'est en particulier le cas si $rg({\cal F}') = 1$), on entre dans le cadre du lemme~\ref{lemA} et on conclut que pour tous $Z_{y'+1},\dots,Z_z \in X$, le morphisme \begin{eqnarray*} H^0(X,{\cal F}) &\rightarrow& A\oplus B \oplus \\ & & {\cal F}'_{Y_1}\oplus\dots\oplus{\cal F}'_{Y_y}\oplus \\ & & {\cal F}_{Z_1}\oplus\dots\oplus{\cal F}_{Z_z} \end{eqnarray*} est bijectif pourvu que, en d{\'e}signant par $B''$ l'image de ${\cal F}''_V$ dans $B$, le morphisme \begin{eqnarray*} \epsilon : H^0(X,{\cal E}) &\rightarrow& B'' \oplus\\ & & {\cal F}''_{Z_1} \oplus\dots\oplus{\cal F}''_{Z_{y'}}\oplus \\ & & {\cal E}_{Z_{y'+1}}\oplus\dots\oplus{\cal E}_{Z_z} \end{eqnarray*} le soit. L'hypoth{\`e}se ${\bf RB}({\cal E},{\cal F}'',z',y',\alpha',\beta')$ garantit qu'il existe un point $V$ et des points $Z_1,\dots,Z_z$ tel que $\epsilon$ soit de rang maximal. L'existence de tels choix pour l'une ou l'autre des hypoth{\`e}ses entra{\^\i}ne l'existence d'ouverts de choix qui sont irr{\'e}ductibles, puisqu'ils sont des ouverts de produit de $X$ ou $X'$. L'ouvert des choix de la premi{\`e}re hypoth{\`e}se intersecte donc celui de la seconde. On notera le fait important que $B''$ d{\'e}pend a priori de $Y_1,\dots,Y_y$, lesquels {\em n'interviennent pas} dans $\epsilon$.\newline $\bullet$ Si $\lambda$ n'est pas bijectif, il existe un point $\bar{Z}$ de $X'$ tel que le morphisme \begin{eqnarray*} H^0(X'{\cal F}') &\rightarrow& A\oplus \\ & & {\cal F}'_{Y_1}\oplus\dots\oplus{\cal F}'_{Y_y}\oplus\\ & & {\cal F}'_{Z_1}\oplus\dots\oplus{\cal F}'_{Z_{y'}}\oplus\\ & & {\cal F}'_V\oplus \\ & &{\cal F}'_{\bar{Z}} \end{eqnarray*} est injectif. On entre alors dans le cadre du lemme~\ref{lemB} et on conclut qu'il existe un quotient ${\cal E}_{\bar{Z}}\rightarrow D$, de dimension $\beta'$, avec noyau contenu dans ${\cal F}'_{\bar{Z}}$, avec la propri{\'e}t{\'e} suivante. Pour tout ensemble de points $Z_{y'+2},\dots,Z_z$, et tout quotient $B$, il existe $Z_{y'+1}$ dans $X$ tel que le morphisme \begin{eqnarray*} H^0(X,{\cal F}) &\rightarrow& A\oplus B \oplus \\ & & {\cal F}'_{Y_1}\oplus\dots\oplus {\cal F}'_{Y_y}\oplus\\ & & {\cal F}_{Z_1}\oplus\dots\oplus {\cal F}_{Z_{y'}}\oplus\\ & & {\cal F}_{Z_{y'+1}}\oplus \\ & & {\cal F}_{Z_{y'+2}}\oplus\dots\oplus {\cal F}_{Z_z} \end{eqnarray*} est bijectif, pourvu que, en d{\'e}signant par $B''$ l'image de ${\cal F}''_V$ dans $B$, le morphisme \begin{eqnarray*} \epsilon : H^0(X,{\cal E}) &\rightarrow& B'' \oplus D\oplus\\ & & {\cal F}''_{Z_1} \oplus\dots\oplus{\cal F}''_{Z_{y'}}\oplus \\ & & {\cal E}_{Z_{y'+2}}\oplus\dots\oplus{\cal E}_{Z_z} \end{eqnarray*} le soit. L'hypoth{\`e}se ${\bf RB}({\cal E},{\cal F}'',z',y',\alpha',\beta')$ et l'argument donn{\'e} dans le cas pr{\'e}c{\'e}dent permettent alors de conclure.\hspace{\fill \paragraphe{L'{\'e}nonc{\'e} MB et un autre lemme d'Horace} \subsubsection*{Enonc{\'e} ${\bf MB}({\cal F},{\cal G},z,y;a)$} Soit maintenant $$ {\cal F}\rightarrow{\cal G}\rightarrow 0 $$ une suite exacte stricte de faisceaux localement libres et $z,y$ et $a$ des entiers v{\'e}rifiant \begin{enumerate} \item[$\bullet$] $rg({\cal F})z + rg({\cal G})y +a = h^0(X,{\cal F})$, \item[$\bullet$] $rg({\cal G})(z+y) + a \leq h^0(X',{\cal F}')$ \item[$\bullet$] $a < rg({\cal G})$. \end{enumerate}\vspace{3mm} L'{\'e}nonc{\'e} ${\bf MB}({\cal F},{\cal G},z,y;a)$ signifie alors la chose suivante : Il existe un point $V$ tel que pour tout quotient $$ {\cal G}_T \rightarrow A \rightarrow 0 $$ de dimension $a$, il existe des points $Z_1,\dots,Z_z$ et des points $Y_1,\dots,Y_y$ dans $X$ tels que le morphisme d'{\'e}valuation \begin{eqnarray*} H^0(X,{\cal F}) &\rightarrow& A \oplus\\ & & {\cal G}_{Y_1}\oplus\dots\oplus{\cal G}_{Y_y}\\ & & {\cal F}_{Z_1}\oplus\dots\oplus{\cal F}_{Z_z} \end{eqnarray*} soit bijectif. On va maintenant donner un lemme d'Horace permettant d'obtenir un {\'e}nonc{\'e} {\bf RB} {\`a} partir d'un {\'e}nonc{\'e} de type {\bf R} et d'un {\'e}nonc{\'e} de type {\bf MB}. \begin{lemme} \label{lemMB} Soient $z,y$ et $a$ des entiers v{\'e}rifiant les conditions de l'{\'e}nonc{\'e} ${\bf RB}({\cal F},{\cal F}',z,y;a,0)$. Posons $$ z' = \frac{h^0(X',{\cal F}|_{X'})-r'y-a}{r} $$ qu'on suppose {\em entier} et supposons que $H^1(X,{\cal F}(-X'))=0$ et que les {\'e}nonc{\'e}s $$ {\bf R}({\cal F}(-X');0)\;\;et\;\;{\bf MB}({\cal F}|_{X'},{\cal F}',z',y,a) $$ soient vrais. Alors l'{\'e}nonc{\'e} ${\bf RB}({\cal F},{\cal F}',z,y;a,0)$ l'est. \end{lemme} \demo C'est une cons{\'e}quence triviale du lemme~\ref{lemA} et de la suite exacte $$ 0\rightarrow{\cal F}(-X')\rightarrow{\cal F}\rightarrow{\cal F}_{|X'}\rightarrow 0. $$ \hspace{\fill \paragraphe{Le th{\'e}or{\`e}me principal} Le th{\'e}or{\`e}me suivant, contient dans sa partie (i) le th{\'e}or{\`e}me \ref{thtpn} avec $z = q(\ell),y=a=0$ et $b = r(\ell)$. \begin{theoreme} \label{thmprinc} Soit $n$ un entier $\geq 1$. Alors \begin{description} \item[(i)] on a ${\bf RB}(\tp{n}(\ell),\op{}{n-1}(\ell+1),z,y;0,b)$ avec $y \leq o_{n-1}(\ell+1)$ et $b \in \{0,\ldots,n-1\}$, \item[(ii)] on a ${\bf RB}(\op{n}{n}(\ell+1),\tp{n-1}(\ell),z,y;a,0)$ avec $a \in \{0,\ldots, n-2\}$, \item[(iii)] on a ${\bf R}(\tp{n\!-\!1}(\ell);1)$, \item[(iv)] on a ${\bf MB}(\op{(n\!+\!1)}{n}(\ell+1),\tp{n}(\ell);z,y;a)$ avec $z\geq o_n(\ell)$ et $a \leq n-1$. \end{description} \end{theoreme} \begin{rem} Dans l'{\'e}nonc{\'e} (ii) on a toujours $z \geq o_n(\ell)$. En effet,des deux premi{\`e}res conditions des {\'e}nonc{\'e}s {\bf RB} on tire que $$ nz \geq no_n(\ell+1)-t_{n-1}(\ell) = t_{n}(\ell-1) $$ et ce dernier est plus grand que $no_n(\ell)$. \end{rem} Le reste de cet article sera consacr{\'e} {\`a} la preuve de ce th{\'e}or{\`e}me. Celle-ci se fera par r{\'e}currence sur l dimension. Pour amorcer cette r{\'e}currence, notons d'abord que pour $n=1$, les assertions (i) {\`a} (iii) sont trivialement vraies. On va maintenant donner une preuve de l'assertion (iv) dans ce cas. Cette assertion s'{\'e}crit ainsi. \newline On a \begin{itemize} \item[$\bullet$] ${\bf MB}(\op{2}{1}(\ell+1),\op{}{1}(\ell+2),o_1(\ell+1),0;0)$, \item[$\bullet$] ${\bf MB}(\op{2}{1}(\ell+1),\op{}{1}(\ell+2),o_1(\ell),1;1)$. \end{itemize} Le premier de ces {\'e}nonc{\'e}s est trivialement vrai. Montrons alors le second. Consid{\'e}rons donc des points {\it distincts} $Z_1,\dots,Z_{o_1(\ell)},Y$ et $U$ de ${\bf P}^1$. Il faut alors montrer que le morphisme \begin{eqnarray*} \mu : H^0({\bf P}^1,\op{2}{1}(\ell+1))&\rightarrow& \op{2}{1}(\ell+1)_{Z_1}\oplus\ldots\oplus \op{2}{1}(\ell+1)_{Z_{o_1(\ell)}}\oplus \\ & & \op{}{1}(\ell+2)_Y\oplus \op{}{1}(\ell+2)_U \end{eqnarray*} est bijectif. On va utiliser pour ce faire la suite exacte suivante $$ 0\rightarrow\op{}{1}(\ell)\rightarrow \op{2}{1}(\ell+1)\rightarrow \op{}{1}(\ell+2)\rightarrow 0, $$ ainsi que celle qui s'en d{\'e}duit en cohomologie. Cette suite permet aussi, si $Z\in {\bf P}^1$ de d{\'e}composer $\op{2}{1}(\ell+1)_Z$ en $\op{}{1}(\ell)_Z\oplus \op{}{1}(\ell+2)_Z$. Posons alors \begin{eqnarray*} M &=& \op{}{1}(\ell)_{Z_1}\oplus\ldots\oplus \op{}{1}(\ell)_{Z_{o_1(\ell)}}\,,\\ L &=& \op{}{1}(\ell+2)_{Z_1}\oplus\ldots\oplus \op{}{1}(\ell+2)_{Z_{o_1(\ell)}}\oplus \\ & & \op{}{1}(\ell+2)_Y\oplus\op{}{1}(\ell+2)_U\,. \end{eqnarray*} Alors les applications $H^0({\bf P}^1,\op{}{1}(\ell))\rightarrow M$ et $H^0({\bf P}^1,\op{}{1}(\ell+1))\rightarrow L$ sont bijectives. On en d{\'e}duit alors que $\mu $ l'est. Le th{\'e}or{\`e}me est donc montr{\'e} pour $n=1$. On supposera donc dans la suite $n\geq 2$. \vspace{6mm} On proc{\`e}de maintenant {\`a} la preuve des parties (i) et (ii) du th{\'e}or{\`e}me. Elle se fait par r{\'e}ductions successives, pour aboutir {\`a} un {\'e}nonc{\'e} du type ${\bf MB}(\op{n}{n-1}(m+1),\tp{n-1}(m),z,y;a:\alpha)$ avec $z\geq o_{n-1}(m)$ et $\alpha \leq n-2$, qui est vrai par hypoth{\`e}se de r{\'e}currence. On utilisera pour ce faire les trois lemmes de r{\'e}duction suivants. \newline \begin{lemme} \label{lemred1} Soient $z,y$ et $b$ des entiers non n{\'e}gatifs v{\'e}rifiant les hypoth{\`e}ses de l'{\'e}nonc{\'e} $$ {\bf RB}(\tp{n}(\ell),\op{}{n}(\ell),z,y;0,b). $$ Posons $\beta = 0$ si $b=0$, 1 sinon. Alors, pour que l'{\'e}nonc{\'e} ci-dessus soit vrai, il suffit que $$ {\bf RB}(\op{n}{n}(\ell+1),\tp{n-1}(\ell), z-o_n(\ell+1)+y+\beta,o_n(\ell+1)-y-\beta;b-\beta,0) $$ le soit. \end{lemme} \demo C'est une instanciation du lemme~\ref{lemRB} (l'{\'e}nonc{\'e} ${\bf R}(\op{}{n\!-\!1}(\ell+1);1)$ est trivialement vrai). {\it Il faut noter que le quotient intervenant dans l'{\'e}nou{\'e} r{\'e}duit est ind{\'e}pendant des points intervenant dans cet {\'e}nonc{\'e}.} \newline {~}\newline Soient maintenant $z,y$ et $a$ des entiers non n{\'e}gatifs v{\'e}rifiant les hypoth{\`e}ses de l'{\'e}nonc{\'e} ${\bf RB}(\op{n}{n}(\ell+1),\tp{n-1}(\ell),z,y;a,0)$. Posons \begin{itemize} \item[$\bullet$] $t = t_n(\ell)-3y-a$, \item[$\bullet$] $u$ le reste de la division euclidienne de $t$ par n-1, \item[$\bullet$] $b'$ = 0 si $u=0$, $b' = n-u$ sinon, \item[$\bullet$] $y'= (t-u)/(n-1)$ si $b' \not= 0$, $y= t/(n-1)$ sinon, \item[$\bullet$] $z'=z- y'-1$ si $b' \not= 0$, $z=z-y'$ sinon. \end{itemize} \begin{lemme} \label{lemred2} Supposons $t \leq n\,o_n(\ell)$.Alors pour que l'{\'e}nonc{\'e} ci-dessus soit vrai, il suffit que $$ {\bf RB}(\tp{n}(\ell-1),\op{}{n}(\ell),z',y';0,b') $$ le soit. \end{lemme} \demo C'est une instanciation du lemme~\ref{lemRB} (on utilise l'hypoth{\`e}se de r{\'e}currence sur ${\bf R}(\tp{n\!-\!1}(\ell);1))$.\newline Supposons maintenant que $t > n\,o_n(\ell)$. On r{\'e}{\'e}crit cette derni{\`e}re in{\'e}galit{\'e} en tenant compte de la relation $nz+(n-1)y+a = no_n(\ell+1)$. On trouve que $z > o_n(\ell) + o_{n-1}(\ell)$. On a alors le lemme suivant. \begin{lemme} \label{alakon} Pour que l'{\'e}nonc{\'e} ${\bf RB}(\op{n}{n}(\ell+1),\tp{n}(\ell),z,y;a,0)$ soit vrai, il suffit que l'{\'e}nonc{\'e} $$ {\bf MB}(\op{n}{n\!-\!1}(\ell+1),\tp{n\!-\!1}(\ell),z-o_n(\ell),y;a) $$ le soit. \end{lemme} \demo C'est une instanciation du lemme~\ref{lemMB}.\newline Pour d{\'e}montrer (ii) on utilise soit le lemme~\ref{alakon} si l'{\'e}nonc{\'e} entre dans ses hypoth{\`e}ses, soit successivement les lemmes~\ref{lemred2} et \ref{lemred1} autant qu'il est n{\'e}cessaire pour obtenir soit un {\'e}nonc{\'e} trivialement vrai, soit rentrer dans les hypoth{\`e}ses du lemme~\ref{alakon}. On utilise alors l'hypoth{\`e}se de r{\'e}currence sur les {\'e}nonc{\'e}s {\bf MB} pour conclure.\hspace{\fill \vspace{6mm} Pour d{\'e}montrer (i) on utilise le lemme~\ref{lemred1} qui permet alors de ce r{\'e}duire {\`a}~(ii) et on conclut comme pr{\'e}c{\'e}demment.\hspace{\fill {~}\vspace{5mm} On va maintenant proc{\'e}der {\`a} la d{\'e}monstration de la partie (iii) du th{\'e}or{\`e}me. C'est une cons{\'e}quence de l' {\'e}nonc{\'e} $$ {\bf RB}(\tp{n}(\ell),\op{}{n-1}(\ell+1),q(\ell),0;0,r(\ell)). $$ L'{\'e}nonc{\'e} ci-dessus dit que pour tout $P_{q+1}\in {\bf P}^{n-1}$,et pour tout quotient $$ \tp{n}(\ell)_{Z_{q(\ell)+1}} \rightarrow B \rightarrow 0 $$ de dimension $r(\ell)$, il existe $P_1,\dots,P_q \in {\bf P}^n$ tels que le morphisme d'{\'e}valuation $$ H^0({\bf P}^n,\tp{n}(\ell)) \rightarrow \tp{n}(\ell)_{Z_1} \oplus\ldots\oplus \tp{n}(\ell)_{Z_{q(\ell)}}\oplus B $$ soit bijectif. Il suffira de montrer l'assertion pour $q(\ell)$ et $q(\ell)+1$ points lorsque $a \leq r(\ell)$ et pour $q(\ell)-1$ et $q(\ell)$ points lorsque $a > r(\ell)$. Soit $\tp{n}(\ell)_{Z_{q+1}}\rightarrow A\rightarrow 0$ un quotient. de dimension $a$. Supposons $a \leq r(\ell)$. Il existe alors un quotient $\tp{n}(\ell)_{Z_{q+1}}\rightarrow B\rightarrow 0$ tel que $A$ soit un quotient de $B$. On conclut alors d'apr{\`e}s ce qu'on a vu plus haut qu'il existe des points $Z_1,\dots Z_{q(\ell)}$ tels que, en notant $L$ l'espace vectoriel $$ \tp{n}(\ell)_{Z_1} \oplus\ldots\oplus \tp{n}(\ell)_{Z_q} $$ l'application $$ H^0({\bf P}^n,\tp{n}(\ell)) \rightarrow L\oplus A $$ soit surjective. Par semi-continuit{\'e} on d{\'e}duit qu'il existe un point $T \not= Z_{q+1}$ et un quotient $\tp{n}(\ell)_T\rightarrow Q \rightarrow 0$ de dimension $r(\ell)$ tel que l'application $$ H^0({\bf P}^n,\tp{n}(\ell)) \rightarrow L\oplus Q $$ soit bijective. De la surjection $L\oplus A \oplus \tp{n}(\ell)_Q\rightarrow L\oplus Q$ on d{\'e}duit l'injectivit{\'e} de $$ H^0({\bf P}^n,\tp{n}(\ell)) \rightarrow L\oplus\tp{n}(\ell)_Q\oplus A. $$ Le cas $a >r(\ell)$ se traite avec des arguments similaires et est laiss{\'e} au lecteur. \hspace{\fill {~}\vspace{1cm} Il reste maintenant, pour conclure la d{\'e}monstration du th{\'e}or{\`e}me {\`a} prouver la partie (iv). Pour cela, on distinguera, dans l'{\'e}nonc{\'e} $$ {\bf MB}(\op{(n\!+\!1)}{n}(\ell+1),\tp{n}(\ell);z,y;a) $$ quatre cas : $z = o_n(\ell+1)$, qui est trivialement vrai, $o_n(\ell-1)+o_{n-1}(\ell+1) \leq z < o_n(\ell+1)$, qu'on r{\'e}duira trivialement {\`a} un autre {\'e}nonc{\'e} du m{\^e}me type mais de degr{\'e} plus petit, le cas $o_n(\ell)<z < o_n(\ell-1)+o_{n-1}(\ell+1)$ et enfin le cas o{\`u} $z=o_n(\ell)$. On va donc prouver les trois cas restants. \newline \newline \underline{Second cas :} $o_n(\ell-1)+o_{n-1}(\ell+1) \leq z < o_n(\ell+1)$. Choisissons un hyperplan ${\bf P}^{n-1} \subset {\bf P}^n$ et soient $Z_1,\dots, Z_{o_{n-1}(\ell+1)}\in {\bf P}^{n-1}$ en position g{\'e}n{\'e}rale. L'application d'{\'e}valuation des sections $$ \begin{array}{c} H^0({\bf P}^{n-1},\op{(n+1)}{n}(\ell+1)_{|{\bf P}^{n-1}})\\ \downarrow \\ \op{(n\!+\!1)}{n}(\ell+1)_{|{\bf P}^{n-1}\,Z_1}\oplus\dots\oplus \op{(n\!+\!1)}{n}(\ell+1)_{|{\bf P}^{n-1}\,Z_{o_{n-1}(\ell+1)}} \end{array} $$ est alors bijective. Gr{\^a}ce au lemme~\ref{lemA}, avec la suite exacte $$ 0\rightarrow \op{(n\!+\!1)}{n}(\ell)\rightarrow\op{(n\!+\!1)}{n}(\ell+1) \rightarrow \op{(n\!+\!1)}{n}(\ell+1)_{|{\bf P}^{n-1}}\rightarrow 0 $$ on se r{\'e}duit alors {\`a} l'{\'e}nonc{\'e} $$ {\bf MB}(\op{(n\!+\!1)}{n}(\ell),\tp{n}(\ell-1);z-o_{n-1}(\ell+1),y;a). $$ \newline \newline \underline{Troisi{\`e}me cas :} $z=o_n(\ell)$. On utilisera pour cela la suite d'Euler pour le fibr{\'e} tangent et celle qui s'en d{\'e}duit en cohomologie : $$ \begin{array}{c} 0\rightarrow \op{}{n}(\ell) \rightarrow \op{(n+1)}{n}(\ell+1) \rightarrow \tp{n}(\ell)\rightarrow 0 ,\\ 0\rightarrow H^0({\bf P}^n,\op{}{n}(\ell))\rightarrow H^0({\bf P}^n,\op{(n+1)}{n}(\ell+1)) \rightarrow H^0({\bf P}^n,\tp{n}(\ell)) \rightarrow 0. \end{array} $$ Remarquons que Si $P \in {\bf P}^n$, alors, gr{\^a}ce {\`a} la suite d'Euler, $\op{(\!n\!+\!1\!)}{n}(\!\ell\!+\!1)_P$ se d{\'e}compose en $\op{}{n}(\ell)_P\oplus\tp{n}(\ell)_P$. Les entiers $o_n(\ell)+y$ et $a$ v{\'e}rifient les hypoth{\`e}ses de l'{\'e}nonc{\'e} ${\bf RB}(\tp{n}(\ell),\op{}{n-1}(\ell+1),,0;0,a)$. Soit donc $V$ un point de ${\bf P}^{n-1}$ tel que pour tout quotient $$ \tp{n}(\ell)_V \rightarrow A \rightarrow 0 $$ il existe des points $Z_1,\dots,Z_{o_n(\ell)},Y_1,\dots,Y_y \in {\bf P}^n$ tels que en notant $L$ l'espace vectoriel $$ A\oplus\tp{n}(\ell)_{Z_1}\oplus\dots\oplus\tp{n}(\ell)_{Z_{o_n(\ell)}}\oplus \tp{n}(\ell)_{Y_1}\oplus\dots\oplus\tp{n}(\ell)_{Y_y} $$ l'application $H^0({\bf P}^n,\tp{n}(\ell))\rightarrow L$ soit bijective. Si $P_1,\dots,P_{o_n(\ell)} \in {\bf P}^n$ sont en position suffisamment g{\'e}n{\'e}rale, en notant $M$ l'espace vectoriel $$ \op{}{n}(\ell)_{P_1}\oplus\dots\oplus\op{}{n}(\ell)_{P_{o_n(\ell)}} $$ l'application $H^0({\bf P}^n,\op{}{n}(\ell))\rightarrow M$ est bijective. Or les ouverts de choix des $Z_i$ et des $P_j$ sont des ouverts de $({\bf P}^n)^{o_n(\ell)}$ et donc s'intersectent. On peut donc supposer $P_i = Z_i$ et on conclut alors par le lemme du serpent que l'application $$ H^0({\bf P}^n,\op{(n+1)}{n}(\ell+1))\rightarrow M\oplus L $$ est bijective. \newline \newline \underline{Quatri{\`e}me cas :} $o_n(\ell)<z<o_n(\ell-1)+o_{n-1}(\ell+1)$. C'est le plus technique d'entre eux et c'est pour les besoins cette d{\'e}monstration qu'on a introduit le lemme~\ref{lemC}. On va d'abord introduire les faisceaux et les suites exactes utilis{\'e}es dans la d{\'e}monstration. On a la suite exacte $$ 0\rightarrow \op{n}{n}(\ell+1)\oplus\op{}{n}(\ell)\rightarrow \op{(n+1)}{n}(\ell+1)\rightarrow\op{}{n-1}(\ell+1)\rightarrow 0 $$ et par transformation {\'e}l{\'e}mentaire $$ 0\rightarrow \op{(n+1)}{n}(\ell)\rightarrow \op{n}{n}(\ell+1) \oplus\op{}{n}(\ell)\rightarrow \op{n}{n-1}(\ell+1)\rightarrow 0. $$ On a aussi le diagramme commutatif {\`a} lignes et colonnes exactes $$\diagram 0\rto& \op{n}{n}(\ell\!+\!1)\oplus\op{}{n}(\ell)\dto\rto& \op{(n\!+\!1)}{n}(\ell\!+\!1)\dto\rto&\op{}{n-1}(\ell\!+\!1)\ddouble\rto& 0\\ 0\rto& \op{n}{n}(\ell\!+\!1)\rto\dto& \tp{n}(\ell)\dto \rto & \op{}{n-1}(\ell\!+\!1)\rto& 0\\ & 0 & 0 & \enddiagram$$ et en appliquant la transformation {\'e}l{\'e}mentaire pr{\'e}c{\'e}dente, le diagramme suivant commutatif {\`a} lignes et colonnes exactes. $$\diagram 0\rto& \op{(n+1)}{n}(\ell)\dto\rto&\op{n}{n}(\ell+1)\oplus \op{}{n}(\ell)\rto\dto&\op{n}{n-1}(\ell+1)\rto\dto&0\\ 0\rto& \tp{n}(\ell-1)\rto\dto & \op{n}{n}(\ell+1) \rto\dto & \tp{n-1}(\ell)\rto\dto&0\\ & 0 & 0 & 0 & \enddiagram$$ On posera $\alpha = 0$ si $a = 0$ et $\alpha = 1$ sinon. On notera encore ${\bf P}^{n-1}$ un hyperplan de ${\bf P}^n$. Soit $d$ l'entier $z - o_n(\ell)$ et $y'$ l'entier $o_{n-1}(\ell+1)-d-\alpha$. Les hypoth{\`e}ses entra{\^\i}nent que $d$ et $y'$ sont tous deux non-n{\'e}gatifs. Supposons $\alpha\not=0$. Alors il existe $V\in {\bf P}^{n-1}$ tel que pour tout quotient $$ \tp{n}(\ell)_V\rightarrow A \rightarrow 0 $$ de dimension $a$ avec noyau contenu dans $\tp{n-1}(\ell)$, il existe des points $Z_1,\dots,Z_d, Y_1,\dots,Y_{y'} \in {\bf P}^{n-1}$ tel que l'application \begin{eqnarray*} H^0({\bf P}^{n-1},\op{}{n-1}(\ell+1)) &\rightarrow& \op{}{n-1}(\ell+1)_V \oplus \\ & & \op{}{n-1}(\ell+1)_{Z_1}\oplus\dots\oplus\op{}{n-1}(\ell+1)_{Z_d}\oplus \\ & & \op{}{n-1}(\ell+1)_{Y_1}\oplus\dots\oplus\op{}{n-1}(\ell+1)_{Y_{y'}} \end{eqnarray*} soit bijective et si $\alpha = 0$, l'application \begin{eqnarray*} H^0({\bf P}^{n-1},\op{}{n-1}(\ell+1)) &\rightarrow& \op{}{n-1}(\ell+1)_{Z_1}\oplus\dots\oplus\op{}{n-1}(\ell+1)_{Z_d}\oplus \\ & & \op{}{n-1}(\ell+1)_{Y_1}\oplus\dots\oplus\op{}{n-1}(\ell+1)_{Y_{y'}} \end{eqnarray*} l'est dans ce cas. On rentre alors dans le cadre du lemme~\ref{lemA}et on conclut que, pour que l'{\'e}nonc{\'e} ${\bf MB}(\op{(n\!+\!1)}{n}(\ell\!+\!1),\tp{n}(\ell),z,y;a)$ soit vrai, il suffit qu'il existe des points $Z_{d+1},\dots,Z_z$, $ Y_{y'+1},\dots,Y_y \in {\bf P}^n$, tel que, en d{\'e}signant par $A'$ l'image de $\tp{n-1}(\ell)_V$ dans $A$ si $A\not=0$, $A'=0$ sinon, que l'application \begin{eqnarray*} \label{monmu} \mu :H^0({\bf P}^n,\op{(\!n\!+\!1\!)}{n}(\!\ell\!+\!1\!)) &\rightarrow & A' \oplus \\ & & \tp{n-1}(\ell)_{Y_1}\oplus\dots\oplus\tp{n-1}(\ell)_{Y_{y'}}\oplus \\ & & \op{n}{n-1}(\ell+1)_{Z_1}\oplus\dots\oplus\op{n}{n-1}(\ell+1)_{Z_d}\oplus\\ & & \op{n}{n}(\ell+1)_{Y_{y'+1}}\oplus\dots\oplus\op{n}{n}(\ell+1)_{Y_y}\oplus \\ &&\op{n}{n}(\ell+1)\oplus\op{}{n}(\ell)_{Z_{d+1}}\oplus\dots\oplus \op{n}{n}(\ell+1)\oplus\op{}{n}(\ell)_{Z_{o_n(\ell)}} \end{eqnarray*} soit bijective. Notons que $a' := dim(A') = 0$ si $a \leq 1$, $a-1$ sinon. Soit alors $e$ l'entier $no_{n-1}(\ell+1) - dn - (n-1)y'-a'$, $f$ et $g$ respectivement le quotient et le reste de la division euclidienne de $e$ par $n-1$. On va alors diviser la preuve de la bijectivit{\'e} de $\mu $ en deux cas, le plus simple {\'e}tant celui o{\`u} $g=0$, le second, $g\not= 0$ utilisera le lemme 3. \newline Dans le premier cas, le lecteur v{\'e}rifiera que les entiers $d,y'+f$ et $a'$ v{\'e}rifient les hypoth{\`e}ses de l'{\'e}nonc{\'e} ${\bf MB}(\op{n}{n-1}(\ell+1),\tp{n-1}(\ell),d,y'+f,a')$, qui est suppos{\'e} vrai par hypoth{\`e}se de r{\'e}currence. Il existe donc un point $V \in {\bf P}^{n-1}$ tel que pour tout quotient $$ \tp{n-1}(\ell)_V \rightarrow A' \rightarrow 0 $$ il existe des points $Z_1,\dots,Z_d, Y_1,\dots ,Y_{y'+f}$ tel que l'application \begin{eqnarray*} H^0({\bf P}^{n-1},\op{n}{n-1}(\ell+1)) & \rightarrow & A'\oplus \\ & & \tp{n-1}(\ell)_{Y_1}\oplus \dots\oplus \tp{n-1}(\ell)_{Y_{y'+f}}\oplus\\ & & \op{n}{n-1}(\ell+1)_{Z_1}\oplus\dots\oplus\op{n}{n-1}(\ell+1)_{Z_d} \end{eqnarray*} soit bijective. On rentre alors dans le cadre du lemme~\ref{lemA} et on en conclut qu'il existe des points $Y_{y'+f+1},\dots,Y_y,Z_{d+1},\dots,Z_z \in {\bf P}^n$ tel que $\mu $ soit bijective pourvu que l'application \begin{eqnarray*} \epsilon: H^0({\bf P}^n,\op{(n+1)}{n}(\ell))&\rightarrow& \op{}{n-1}(\ell)_{Y_{y'+1}}\oplus\dots\oplus\op{}{n-1}(\ell)_{Y_{y'+f}}\oplus \\ &&\tp{n}(\ell-1)_{Y_{y'+f+1}}\oplus\dots\oplus\tp{n}(\ell-1)_{Y_y}\oplus\\ &&\op{(n+1)}{n}(\ell)_{Z_{d+1}}\oplus\dots\oplus\op{(n+1)}{n}(\ell)_{Z_z} \end{eqnarray*} le soit. Remarquons alors que $z-d=o_n(\ell-1)$. En utilisant alors un argument similaire {\`a} celui de la preuve du troisi{\`e}me cas de (iv), pour que $\epsilon$ soit bijective, il suffit que l'application \begin{eqnarray*} H^0({\bf P}^n,\tp{n}(\ell-1)))&\rightarrow& \op{}{n-1}(\ell)_{Y_{y'+1}}\oplus\dots\oplus\op{}{n-1}(\ell)_{Y_{y'+f}}\oplus \\ &&\tp{n}(\ell-1)_{Y_{y'+f+1}}\oplus\dots\oplus\tp{n}(\ell-1)_{Y_y}\oplus\\ &&\tp{n}(\ell-1)_{Z_{d+1}}\oplus\dots\oplus\tp{n}(\ell-1)_{Z_z} \end{eqnarray*} le soit. Or l'existence de choix de points pour que ce dernier {\'e}nonc{\'e} soit v{\'e}rifi{\'e} est garanti par l'{\'e}nonc{\'e} ${\bf RB}(\tp{n}(\ell-1), \op{}{n-1}(\ell),o_n(\ell)-1+y-y'-f,f;0,0)$. Pour conclure, on utilise alors l'irreductibilit{\'e} des espaces de param{\`e}tres pour garantir la s{\'e}rie de choix que l'on vient de faire.\newline Reste maintenant {\`a} traiter le cas o{\`u} $g$ est non nul. On va montrer que si $V\in {\bf P}^n$, alors pour tout quotient $$ \tp{n}(\ell)\rightarrow A'\rightarrow 0 $$ il existe des points $Z_1,\dots,Z_d$, $Y_1,\dots,Y_{y'+f}$ et un point $\bar{Y}$ in ${\bf P}^{n-1}$ tel que en notant $L$ l'espace vectoriel $$ \tp{n-1}(\ell)_{Y_1}\oplus \dots\oplus \tp{n-1}(\ell)_{Y_{y'+f}}\oplus \op{n}{n-1}(\ell+1)_{Z_1}\oplus\dots\oplus\op{n}{n-1}(\ell+1)_{Z_d} $$ l'application $ \lambda : H^0({\bf P}^{n-1},\op{n}{n-1}(\ell+1))\rightarrow L\oplus A' $ est surjective et que l'application d{\'e}duite $ \lambda' : H^0({\bf P}^{n-1},\op{n}{n-1}(\ell+1))\rightarrow L\oplus A'\oplus \tp{n-1}(\ell)_{\bar{Y}} $ est injective. Pour ce faire on va alors distinguer suivant la valeur de $g+a'$. \begin{itemize} \item[$\bullet$] $g+a' \leq n-1$. Alors les entiers $d,y'+f$ et $g+a'$ v{\'e}rifient les conditions de l'{\'e}nonc{\'e} $$ {\bf MB}(\op{n}{n-1}(\ell+1),\tp{n-1}(\ell),d,y'+f,g+a'). $$ \item[$\bullet$] $g+a'>n-1$. Alors les entiers $d,y'+f+1$ et $g+a'-(n-1)$ v{\'e}rifient les conditions de l'{\'e}nonc{\'e} $$ {\bf MB}(\op{n}{n-1}(\ell+1),\tp{n-1}(\ell),d,y'+f+1,g+a'-(n-1)). $$ \end{itemize} Ces deux {\'e}nonc{\'e}s sont suppos{\'e}s vrais par hypoth{\`e}se de r{\'e}currence. Consid{\'e}rons le cas $g+a'\leq n-1$. D'apr{\`e}s l'hypoth{\`e}se correspondante, il existe $T\in {\bf P}^{n-1}$ tel que pour tout quotient $$ \tp{n-1}(\ell)_T\rightarrow G \rightarrow 0 $$ de dimension $g+a'$, il existe des points $Z_1,\dots,Z_d$, $Y_1,\dots,Y_{y'+f} \in {\bf P}^{n-1}$ tel que, en notant $N$ l'espace vectoriel $$ \tp{n-1}(\ell)_{Y_1}\oplus\dots\oplus\tp{n-1}(\ell)_{Y_{y'+f}}\oplus \op{n}{n-1}(\ell+1)_{Z_1}\oplus\dots\oplus\op{n}{n-1}(\ell+1)_{Z_d} $$ l'application $$ \xi : H^0({\bf P}^{n-1},\op{n}{n-1}(\ell)) \rightarrow G\oplus N $$ soit bijective. On pose alors $V=T$. Tout quotient $\tp{n-1}(\ell)\rightarrow A'$ se factorise par un quotient $G$ de dimension $g+a'$. On en d{\'e}duit alors la surjectivite de $\lambda$, avec $L=N$. Montrons l'injectivit{\'e} de $\lambda'$. Par semi-continuit{\'e}, il existe un ouvert $U \subset {\bf P}^{n-1}$, un point $\bar{Y} \not= T$ et un quotient $\tp{n-1}(\ell)_{\bar{Y}}\rightarrow G'$ de dimension $g'+ a$ tel que l'application $ H^0({\bf P}^{n-1},\op{n}{n-1}(\ell)) \rightarrow G'\oplus N$ soit aussi bijective. De la surjection $N\oplus A'\oplus \tp{n-1}(\ell)_{\bar{Y}} \rightarrow N\oplus G'$ on tire l'injectivit{\'e} de $\lambda'$.\newline Le cas $g+a'>n-1$ se traite de fa{\c{c}}on similaire, il suffit seulement de remarquer qu'on a {\'e}videment $g+a'-(n-1) \leq a'$.\newline On rentre alors dans le cadre du lemme~\ref{lemC} et on conclut qu'il existe un quotient $$ \tp{n-1}(\ell)_{\bar{Y}}\rightarrow D \rightarrow 0 $$ de dimension $\delta=n-dim\,Ker\lambda$ avec noyau contenu dans $\tp{n-1}(\ell-1)_{\bar{Y}}$ jouissant de la propri{\'e}t{\'e} suivante. Pour tout choix de points $Z_{d+1},\dots,Z_z$, $Y_{y'+f+2},\dots,Y_y \in {\bf P}^n$, il existe un point $Y_{y'+f +1} \in {\bf P}^n$ tel que l'application $\mu $ correspondante (page \pageref{monmu}) soit bijective, pourvu que l'application \begin{eqnarray*} \epsilon: H^0({\bf P}^n,\op{(n+1)}{n}(\ell))&\rightarrow& \op{}{n-1}(\ell)_{Y_{y'+1}}\oplus\dots\oplus\op{}{n-1}(\ell)_{Y_{y'+f}}\oplus \\ && D\oplus \\ &&\tp{n}(\ell-1)_{Y_{y'+f+2}}\oplus\dots\oplus\tp{n}(\ell-1)_{Y_y}\oplus\\ &&\op{(n+1)}{n}(\ell)_{Z_{d+1}}\oplus\dots\oplus\op{(n+1)}{n}(\ell)_{Z_z} \end{eqnarray*} le soit. En utilisant alors un argument similaire {\`a} celui du troisi{\`e}me cas de la preuve de (vi) on r{\'e}duit cette derni{\`e}re assertion {\`a} la preuve de l'{\'e}nonc{\'e} $$ {\bf RB}(\tp{n-1}(\ell-1),\op{n-1}(\ell),o_n(\ell-1)+y-y'-f-1,f;0,\delta) $$ Dans les deux cas $g=0$ et $g\not=0$, le lecteur pourra v{\'e}rifier, pour que ce qui pr{\'e}c{\`e}de ait un sens, que $y\geq y'+f+1$, ({\`a} noter que les hypoth{\`e}ses $n\geq 2$ et $\ell\geq 0$ sont n{\'e}cessaires pour ce faire). Ceci conclut donc la preuve du th{\'e}or{\`e}me~\ref{thmprinc}.\hspace{\fill

9506

alg-geom/9506018

en

https://arxiv.org/abs/alg-geom/9506018

alg-geom/9506018

Lothar Goettsche

Lothar G\"ottsche

Modular forms and Donaldson invariants for 4-manifolds with $b_+=1$

I correct a number of missing attributions and citations. In particular this applies to the cited paper of Kotschick and Lisca "Instanton invariants via topology", which contains some ideas which have been important for this work. AMSLaTeX

We study the Donaldson invariants of simply connected $4$-manifolds with $b_+=1$, and in particular the change of the invariants under wall-crossing. We assume the conjecture of Kotschick and Morgan about the shape of the wall-crossing terms (which Oszva\'th and Morgan are now able to prove), and are determine a generating function for the wall-crossing terms in terms of modular forms. As an application we determine all the Donaldson invariants of the projective plane in terms of modular forms. The main tool are the blowup formulas, which are used to obtain recursive relations.

alg-geom

P(n){P(n)} \def{\text{\rom{\bf b}}}{{\text{\rom{\bf b}}}} \defS^{(\bb)}{S^{({\text{\rom{\bf b}}})}} \def{\text{\rom{\bf E}}}{{\text{\rom{\bf E}}}} \def\Inc#1{Z_{#1}} \def\inc#1{\zeta_{#1}} \def\boh#1{pt_{#1} \def\Boh#1{Pt_{#1}} \def\mah#1{al_{#1} \def\Mah#1{Al_{#1}} \def\pr#1{W_{#1}} \def\alpha{\alpha} \def\Gamma{\Gamma} \def\bar \Gamma{\bar \Gamma} \def\gamma{\gamma} \def\bar \eta{\bar \eta} \def{\cal Z}{{\cal Z}} \def{pt}{{pt}} \def\alpha{\alpha} \def{S^{(n)}}{{S^{(n)}}} \def{S^{(m)}}{{S^{(m)}}} \def\<{\langle} \def\>{\rangle} \def{\text{\rom{Hilb}}}{{\text{\rom{Hilb}}}} \def{\cal W}{{\cal W}} \def{\hbox{\rom{Tor}}}{{\hbox{\rom{Tor}}}} \def{\hbox{\rom{Ext}}}{{\hbox{\rom{Ext}}}} \def\stil#1{\tilde S^#1} \def\pi{\pi} \defg{g} \def\phi{\phi} \def\tilde{\tilde} \def{\bar\al}{{\bar\alpha}} \begin{document} \title[Modular forms and Donaldson invariants] {Modular forms and Donaldson invariants for 4-manifolds with $b_+=1$} \keywords{Blowup-formulas, Donaldson invariants, modular forms} \author{Lothar G\"ottsche} \address{Max--Planck--Institut f\"ur Mathematik\\Gottfried--Claren--Stra\ss e 26\\ D-53225 Bonn, Germany} \email{lothar@@mpim-bonn.mpg.de} \maketitle\ \section{Introduction} The Donaldson invariants of a smooth simply connected $4$-manifold $X$ depend by definition on the choice of a Riemannian metric $g$. In case $b^+(X)>1$ they turn out to be independent of the metric as long as it is generic, and thus give $C^\infty$-invariants of $X$. We study the case $b_+(X)=1$, where the invariants have been introduced in [Ko]. We denote by $\Phi^{X,g}_{c_1,N}$ the Donaldson invariant of $X$ with respect to a lift $c_1\in H^2(X,{\Bbb Z})$ of $w_2(P)$ for an $SO(3)$ bundle $P$ on $X$ with $-p_1(P)-3=N$. Kotschick and Morgan showed in \cite{K-M} that the invariants only depend on the chamber of the period point of $g$ in the positive cone $H^2(X,{\Bbb R})^+$ in $H^2(X,{\Bbb R})$. For two metrics $g_1,g_2$, which do not lie on a wall they express $\Phi^{X,g_1}_{c_1,N}-\Phi^{X,g_2}_{c_1,N}$ as the sum over certain wall-crossing terms $\delta^X_{\xi,N}$, where $\xi$ runs over all classes in $H^2(X,{\Bbb Z})$ which define a wall between $g_1$ and $g_2$. They also make the following conjecture. \begin{conj}\label{KMconj} \cite{K-M} $\delta^X_{\xi,N}$ is a polynomial in the multiplication by $\xi$ and the quadratic form $Q_X$ on $H_2(X,{\Bbb Z})$ whose coefficients depend only on $\xi^2$, $N$ and the homotopy type of $X$. \end{conj} John Morgan and Peter Ozsv\'ath have told me that they are now able to prove the conjecture \cite{M-O}. In previous joint papers \cite{E-G1},\cite{E-G2} with Geir Ellingsrud we have studied the wall-crossing terms $\delta^S_{\xi,N}$ in the case of algebraic surfaces $S$ with $p_g=0$. In \cite{E-G1} we expressed (for so called good walls) the $\delta^S_{\xi,N}$ in terms of Chern classes of some "standard" bundles on Hilbert schemes of points on $S$, and proceeded to compute the leading $6$ terms of $\delta^S_{\xi,N}$ (similar results were also obtained in \cite{F-Q}). In \cite{H-P} a Feynman path integral aproach to this problem is developed, and some of the leading terms of the wall-crossing formulas are determined. In \cite{E-G2}, which builds on \cite{E-G1}, we restrict to the case of rational surfaces and use the Bott residue formula to compute the $\delta^S_{\xi,N}$ explicitly (with help of the computer). As an application, using also the blowup formulas, we computed e.g. the Donaldson invariants of ${\Bbb P}_2$ of degree smaller then $50$. In \cite{K-L} the wall-crossing formulas had already been used in combination with the blowup formulas to compute Donaldson invariants of ${\Bbb P}_2$ and ${\Bbb P}_1\times{\Bbb P}_1$ and to show in particular that neither ${\Bbb P}_2$ nor ${\Bbb P}_1\times {\Bbb P}_1$ is of simple type. Their calculations also showed that the blowup formulas impose restrictions on the wall-crossing formulas, although this is not pursued systematically there. The authors did however expect that this can be used to determine many (and possibly all) the wall-crossing formulas for rational surfaces. In the current paper we want to show that in fact, assuming conjecture \ref{KMconj}, one can determine the $\delta^X_{\xi,N}$ completely for all $X$ and all walls in $H^2(X,{\Bbb R})^+$ by use of the blowup formulas. We will determine a universal generating function $\Lambda(L,Q,x,t,\tau)$ which expresses all $\delta^X_{\xi,N}$ for all $X$, $N$ and $\xi$. Here $\tau$ is a parameter from the complex upper half plane, $L$,$Q$ and $x$ stand for the multiplication with $\xi$, the quadratic form and the class of a point, and the exponent of $t$ is the signature of $X$. It turns out that $\Lambda(L,Q,x,t,\tau)$ is an exponentional expression in certain modular forms (with respect to $\tau$). As an application of our results we also get a modular forms expressions for all the Donaldson invariants of the projective plane ${\Bbb P}_2$. Already in \cite{K-L} it had been shown that the Donaldson invariants of ${\Bbb P}_2$ and ${\Bbb P}_1\times {\Bbb P}_1$ are determined by the wall-crossing formulas on the blowup of ${\Bbb P}_2$ in two points. We use instead a simple fact due to Qin: on a rational ruled surface the Donaldson invariants with respect to a first Chern class $c_1$ with odd restriction to a fibre vanish for a special chamber $\cal C_F$. The results of this paper should be seen in comparison with the new developments of Seiberg-Witten theory \cite{S-W},\cite{W1} which suggest a connection between the Donaldson invariants and modular forms: the Donaldson invariants (and also the Seiberg-Witten invariants) are seen as degenerations of supersymmetric theories, parametrized by the "$u$-plane" (i.e. the modular curve $\Bbb H/\Gamma(2)$). In fact Witten informed me that he is currently trying to determine wall-crossing formulas and the Donaldson invariants of the projective plane by integrating over the $u$-plane (see also \cite{W2}). The results should also be related to the current work \cite{P-T} towards proving the conjectural relationship between Seiberg-Witten and Donaldson invariants. The main tool for getting our result are the blowup formulas, which for $4$-manifolds with $b_+=1$ I learned from \cite{K-L}. Let $\widehat X:=X\#\overline{\Bbb P}_2$ (e.g. if $X$ is an algebraic surface, we can take $\widehat X$ to be the blowup of $X$ at a point). The idea is very simple: If $\cal C$ is a chamber in $H^2(X,{\Bbb R})^+$ and $\widehat \cal C$ is a related chamber (see below), then there is a formula relating the Donaldson invariants of $X$ with respect to $\cal C$ and those of $\widehat X$ with respect to $\widehat \cal C$. So let now $\cal C_-$ and $\cal C_+$ be two chambers separated by the wall $W^\xi$, then in general there are several walls between the related chambers $\widehat \cal C_-$ and $\widehat \cal C_+$ on $\widehat X$, but it is very easy to determine them. We can therefore express the wall-crossing term $\delta^X_{\xi,N}$ as follows. We apply the blowup formulas to the related chambers $\cal C_-,\widehat \cal C_-$ and $\cal C_+,\widehat \cal C_+$ and add up the wall-crossing terms for all walls between $\widehat\cal C_-$ and $\widehat \cal C_+$. This gives recursive relations. After encoding our information into a generating function $\Lambda_X(L,Q,x,t,\tau)$, these recursive relations translate into differential equations, which enable us to determine $\Lambda_X$ up to multiplication by a universal function $\lambda(\tau)$. Unlike the case of the blowup formulas in \cite{F-S}, the modular forms enter the formulas already as the coefficients of the differential equations; they arize as theta functions for lattices describing the walls between related chambers. In order to finally determine $\lambda(\tau)$ we consider the particular case $X={\Bbb P}_1\times{\Bbb P}_1$. The above mentioned result of Qin now says that, for first Chern class $c_1=F+G$, the sum of the classes of the fibres in the two different directions, there are always two different chambers $\cal C_F$, $\cal C_G$ of type $(c_1,N)$ where the corresponding Donaldson invariants vanish. Therefore the sum of the $\delta^X_{\xi,N}$ for all classes $\xi$ defining walls between $\cal C_F$ and $\cal C_G$ must be zero. This fact gives us an additional recursion relation, and with this we can finally determine $\lambda(\tau)$. If we assume only a weaker form of the conjecture, namely if we allow $\delta^X_{\xi,N}$ to depend on $X$, rather than just on the homotopy type, then we still get our result for $X$ a rational surface. If we assume the conjecture and the blowup formulas also in the case that $X$ is not simply connected but $b_1(X)=0$, then we can partially extend our result also to this case. I am very thankful to Don Zagier, who proved lemma \ref{residue} for me. I would also like to thank John Morgan and Stefan Bauer for very useful conversations. I would like to thank Dieter Kotschick for sending me the preprint \cite{K-L}, which was very important both for \cite{E-G2} and for this work. This paper grew out of the joint work \cite{E-G1}, \cite{E-G2} with Geir Ellingsrud. Motivated by this work, and based also on \cite{K-L}, I slowly realized the importace of the blowup formulas in this context. Also the explicit formulas for the wall-crossing in \cite{E-G2} were very important for me to keep confidence in my computations. \section{Background material} In this paper we will denote by $X$ a simply connected smooth $4$-manifold with $b_+(X)=1$ and $b_2(X)\ge 2$. We will assume conjecture \ref {KMconj}. \begin{nota} For elements $A\in H^2(X,{\Bbb Q})$ and $\alpha\in H_2(X,{\Bbb Q})$ we denote by $A\cdot \alpha\in {\Bbb Q}$ the canonical pairing, by $\check A\in H_2(X,{\Bbb Z})$ the Poincar\'e dual and by $A^2$ the number $A\cdot\check A$. We denote by $Q_X$ the quadratic form on $H_2(X,{\Bbb Z})$ and, for a class $\eta\in H^2(X,{\Bbb Q})$, by $L_\eta$ the linear form $\alpha\mapsto \eta\cdot\alpha$ on $H_2(X,{\Bbb Q})$. If there is no risk of confusion we denote by $a$ the reduction of $A\in H^2(X,{\Bbb Z})$ modulo $2$. For a smooth four-manifold $X$ we denote by $\widehat X$ the connected sum $X\#\overline{\Bbb P}_2$ of $X$ with ${\Bbb P}_2$ with the reversed orientation, (e.g. if $X$ is a smooth complex surface, then $\widehat X$ is the blowup of $X$ in a point). Let $E$ be the image of the generator of $H^2(\overline{\Bbb P}_2,{\Bbb Z})$ in $H^2(\widehat X,{\Bbb Z})$. We will will always identify $H_2(X,{\Bbb Z})$ with the kernel of $L_E$ on $H_2(\widehat X,{\Bbb Z})$. We write $e$ for the reduction of $E$ modulo $2$. Let $g$ be a Riemannian metric on $X$, and $P$ and $SO(3)$ principal bundle with first Pontrjagin class $p_1(P)=-(N+3)$. We denote by $\Phi^{X,g}_{c_1,N}$ the Donaldson invariant corresponding to $P$, the metric $g$ and the lift $c_1\in H^2(X,{\Bbb Z})$ of $w_2(P)$. We use the conventions of e.g. \cite{F-S} which coincide up to a power of $2$ with the conventions of \cite{Ko}. If $X$ is an algebraic surface and $H$ an ample divisor we will write $\Phi^{X,H}_{c_1,N}$ for the invariant with respect to the Fubini-Studi metric induced by $H$. Let $p\in H_0(X,{\Bbb Z})$ be the class of a point. Let $A_N(X)$ be the set of polynomials of weight $N$ in $H_2(X,{\Bbb Q})\oplus H_0(X,{\Bbb Q})$, where $\alpha\in H_2(X,{\Bbb Q})$ has weight $1$ and $p$ has weight $2$. Then $\Phi^{X,g}_{c_1,N}$ is a linear map $A_N(X)\longrightarrow {\Bbb Q}$. We put $\Phi^{X,g}_{c_1,N}:=0$ if $N$ is not congruent to $-c_1^2+3$ modulo $4$ and $\Phi^{X,g}_{c_1}:=\sum_{N\ge 0}\Phi^{X,g}_{c_1,N}$. \end{nota} \subsection{Walls and chambers} \begin{defn}\label{defwall} (see e.g. \cite{Ko}, \cite{K-M}) Let $w\in H^2(X,{\Bbb Z}/2{\Bbb Z})$ and $N$ a nonnegative integer. Let $H^2(X,{\Bbb R})^+$ be the positive cone in $H^2(X, {\Bbb R})$. For $\xi\in H^2(X,{\Bbb Z})$ let $$W^\xi:= \{ x\in H^2(X, {\Bbb R})^+ \bigm| \xi\cdot\check x =0\big\}.$$ We shall call $W^\xi$ a wall of type $(w,N)$, and say that it is defined by $\xi$, if $w$ is the reduction of $\xi$ modulo $2$, $N+3$ is congruent to $\xi^2$ modulo $4$ and $-(N+3)\le \xi^2<0$. Note that any class $\xi\in H^2(X,{\Bbb Z})$ with $\xi^2<0$ will define a wall of type $(w,N)$ for suitable $N$ and $w$ the reduction of $\xi$ modulo $2$; we will in this case say that $\xi$ defines a wall of type $(N)$. A chamber of type $(w,N)$ is a connected component of the complement of the walls of type $(w,N)$ in $H^2(X, {\Bbb R})^+$. For a Riemannian metric $g$ on $X$ we denote by $\omega(g)\in H^2(X,{\Bbb R})^+$ the corresponding period point. If $(w,N)$ are given, a metric is called generic if its period point $\omega(g)$ does not lie on a wall of type $(w,N)$. For $A_-,A_+\in H^2(X,{\Bbb R})$ we denote by $W_{w,N}^X(A_-,A_+)$ the set of all $\xi\in H^2(X,{\Bbb Z})$ defining a wall of type $(w,N)$ with $\xi\cdot \check A_-<0<\xi\cdot \check A_+$. We put $$W_{w}^X(A_-,A_+):=\bigcup_{N\ge 0} W_{w,N}^X(A_-,A_+).$$ \end{defn} \begin{thm} \label{wallchange}\cite{K-M} Let $c_1\in H^2(X,{\Bbb Z})$ and $w$ the reduction of $c_1$ modulo $2$. For all $\xi\in H^2(X,{\Bbb Z})$ defining a wall of type $(w,N)$ we put $\varepsilon(c_1,\xi,N):=(5N+3+\xi^2+(\xi-c_1)^2)/4$. There exists $\delta^X_{\xi,N}:{\hbox{\rm {Sym}}}^N(H_2(X,{\Bbb Q}))\longrightarrow {\Bbb Q}$ such that for all generic metrics $g_+$ and $g_-$ with $\omega(g_+)$ and $\omega(g_-)$ in the same connected component of $H^2(X,{\Bbb R})^+$ $$\Phi^{X,g_+}_{c_1,N}-\Phi^{X,g_-}_{c_1,N}= \sum_{\xi\in W_{w,N}^X(\omega(g_-),\omega(g_+))} (-1)^{\varepsilon(c_1,\xi,N)}\delta^X_{\xi,N}.$$ Furthermore, if $\omega(g_1)=-\omega(g)$, then $\Phi^{X,g_1}_{c_1,N}=-\Phi^{X,g}_{c_1,N}$. \end{thm} \begin{rem}\label{sign} \begin{enumerate} \item Our sign conventions are different from those of \cite{K-M} and \cite{K-L}. In fact the sign is chosen in order to give the leading term $L_\xi^{N-2d}Q_X^d$ (with $d=(N+3+\xi^2)/4$) of $\delta^X_{\xi,N}$ a positive coefficient. \item In the future we will always implicitely assume that all the metrics that we consider have their period point in the same connected component of $H^2(X,{\Bbb R})^+$. \item By theorem \ref{wallchange} we can write $\Phi^{X,\cal C}_{c_1,N}:= \Phi^{X,g}_{c_1,N}$ for any metric $g$ with $\omega(g)$ in the chamber $\cal C$. \end{enumerate} \end{rem} \subsection{Blowup formulas} The blowup formulas relate the Donaldson invariants of a $4$-manifold $Y$ and $\widehat Y=Y\#\overline{\Bbb P}_2$. In the case $b_+(Y)>1$, when the invariants do not depend on the chamber structure, they have been shown e.g. in \cite{O}, \cite{L} and in the most general form in \cite{F-S}. In the case when $X$ is a simply connected $4$-manifold with $b_+=1$ I learned the blowup formulas from \cite{K-L}. They then depend on the chamber structure. \begin{defn}\label{related}(see \cite{Ko}). Let $\cal C\subset H^2(X,{\Bbb R})^+$ be a chamber of type $(w,N)$. A chamber $\cal C_0\subset H^2(\widehat X,{\Bbb R})^+$ of type $(w,N)$ (resp. $\cal C_e\subset H^2(\widehat X,{\Bbb R})^+$ of type $(w+e,N+1)$ is said to be related to $\cal C$ if and only if $\cal C$ is contained in the closure $\overline \cal C_0$ (resp in $\overline\cal C_e$). \end{defn} By \cite{T} the formulas of \cite{F-S} also hold for $X$ with $b_+(X)=1$, we will however only need a quite easy special case (see e.g. \cite{Ko} and \cite{S}, ex. 11). \begin{thm} \label{blowup} Let $\cal C\subset H^2(X,{\Bbb R})^+$ be a chamber of type $(w,N)$, and let $\cal C_0\subset H^2(\widehat X,{\Bbb R})^+$ (resp. $\cal C_e\subset H^2(\widehat X,{\Bbb R})^+$) be related chambers of types $(w,N)$ (resp. $(w+e,N+1)$). Then for all $\alpha\in A_N(X)$ and $\beta\in A_{N-2}(X)$ for which both sides are defined we have \begin{align}\tag*{$(0)_b$} \Phi^{X,\cal C}_{c_1,N}(\alpha)&=\Phi^{\widehat X,\cal C_0}_{c_1,N}(\alpha),\\ \tag*{$(1)_b$}\Phi^{X,\cal C}_{c_1,N}(\alpha) &=\Phi^{\widehat X,\cal C_e}_{c_1+E,N+1}(\check E\alpha),\\ \tag*{$(2)_b$}\Phi^{\widehat X,\cal C_0}_{c_1,N}(\check E^2\beta)&=0,\\ \tag*{$(3)_b$}\Phi^{X,\cal C}_{c_1,N}(x\beta)&=-\Phi^{\widehat X, \cal C_e}_{c_1+E,N+1}(\check E^3\beta). \end{align} \end{thm} \subsection{Extension of wall-crossing formulas} We want to extend theorem \ref{wallchange} from ${\hbox{\rm {Sym}}}^N(H_2(X,{\Bbb Q}))$ to $A_N(X)$. For this we have to extend the definition of $\delta^X_{\xi,N}$. In the case that $\xi$ is divisible by $2$ (i.e. $w=0$) we also have to extend the definition of $\Phi^{X,g}_{c_1,N}$ to classes not in the stable range. For technical reasons we also redefine the $\delta^X_{\xi,N}$ in case the intersection form on $H_2(X,{\Bbb Z})$ is even or the rank of $H_2(X,{\Bbb Z})$ is at most $2$. It should not be difficult to prove that this definition agrees with that of \cite{K-M}, but we only need that theorem \ref{wallchange} still holds. \begin{defn}\label{extend} \begin{enumerate} \item Let $N=4c_2-3$ for $c_2\in {\Bbb Z}$. Let $\cal C$ be a chamber of type $(0,N)$ in $H^2(\widehat X,{\Bbb R})^+$ and $\cal C_e$ a related chamber of type $(e,N+1)$ on $\widehat X$. Then we put for all $\alpha\in A_N(X)$ $$\Phi^{X,\cal C}_{0,N}(\alpha):=\Phi^{\widehat X,\cal C_e}_{E,N+1}(\check E\alpha).$$ Note that $(1)_b$ above guaranties that our definition restricts to the standard definition if $\alpha$ is in the stable range. \item Let $\xi\in H^2(X,{\Bbb Z})$ with $\xi^2<0$. We extend the definition of $\delta^X_{\xi,N}$ by putting $\delta^X_{\xi,N}:=0$ if $\xi$ does not define a wall of type $(N)$ (i.e. if $\xi^2$ is not congruent ot $N+3$ modulo $4$ or $N+3+\xi^2<0$). \item Assume now that the intersection form on $H_2(X,{\Bbb Z})$ is even or the rank of $H_2(X,{\Bbb Z})$ is at most $2$ or that $\xi$ is divisible by $2$ in $H^2(X,{\Bbb Z})$. Then we put for $\alpha\in {\hbox{\rm {Sym}}}^N(H_2(X,{\Bbb Q}))$ $$\delta^X_{\xi,N}(\alpha):=\sum_{n\in{\Bbb Z}}(-1)^{n-1} \delta^{\widehat X}_{\xi+(2n+1)E,N+1}(\check E\alpha).$$ Note that by (1) the sum runs in fact only through integers $n$ with $(2n+1)^2\le N+4+\xi^2$. \item Assume that $\delta^Y_{\eta,N}(p^r\beta)$ is already defined for all $m$ for $Y=X\#m\overline {\Bbb P}_2$ for all $N$, all $\xi\in H^2(Y,{\Bbb Z})$ with $\xi^2<0$ and all $\beta\in {\hbox{\rm {Sym}}}^{N-2r}(H_2(Y,{\Bbb Q}))$. Then we put $$\delta_{\xi,N}^Y(p^{r+1}\alpha):=\sum_{{n\in{\Bbb Z}}}(-1)^{n} \delta^{\widehat Y}_{\xi+(2n+1)E,N+1}(\check E^3p^r\alpha)$$ for all $\alpha\in {\hbox{\rm {Sym}}}^{N-2r-2}(H_2(Y,{\Bbb Q}))$. Again by (1) the sum runs only through $n$ with $(2n+1)^2\le N+4+\xi^2$. \end{enumerate} We note that by definition $\delta_{\xi,N}^X=0$ if $\xi$ does not define a wall of type $(N)$. Finally we put $$\delta^X_\xi:=\sum_{N\ge 0} \delta^X_{\xi,N}.$$ If $t$ is an indeterminate, we write $\delta^X_\xi(\sum_N \alpha_N t^N)$ for $\sum_N \delta^X_{\xi,N}(\alpha_N) t^N$, and similarly for $\Phi^{X,g}_{c_1}$. \end{defn} \begin{rem}\label{c10} There is a small subtlety about the definition of $\Phi^{X,\cal C}_{0,N}$ in (1). If $w\ne 0$, then, given a chamber $\cal C$ in $H^2(X,{\Bbb R})^+$ of type $(w,N)$, there is a unique related chamber $\cal C_e$ in $H^2(X,{\Bbb R})^+ $ of type $(w+e,N+1)$ consisting of all $ \mu+a E $ with $\mu\in \cal C$ and $a\in {\Bbb R}$ sufficiently small. If $w=0$, however, $E$ defines a wall of type $(e,N+1)$ separating two chambers $\cal C_e^+$ (corresponding to $a>0$) and $\cal C_e^-$ (corresponding to $a<0$) of type $(e,N+1)$, which are both related to $\cal C$. $\Phi_{0,N}^{X,\cal C}$ is still well-defined, as $\delta_{(2n+1)E,N+1}^{\widehat X}(\check E^{2k+1}\alpha)=0$ for $N$ congruent to $1$ modulo $4$: By conjecture \ref{KMconj} (and the extension \ref{point} to $A_N(X)$) $\delta_{E,N+1}^{\widehat X}(p^r \bullet)$ is a polynomial in $L_E$ and $Q_{\widehat X}$, $N+1$ is even and $E\cdot \alpha=0$. Similarly, if $w\ne 0$, then there is a unique related chamber $\cal C_0$ in $H^2(X,{\Bbb R})^+ $ of type $(w,N)$. If $w=0$, then there are two related chambers separated by a wall defined by $2E$, but $\delta_{2nE,N}^{\widehat X}(\check E^{2k}\alpha)=0$. \end{rem} \subsection{Vanishing on rational ruled surfaces} An important rule both in the proof of the main theorem and in the application to Donaldson invariants of the projective plane is played by the following elementary vanishing result. Let $S$ be a rational ruled surface, and let $F,E\in H^2(S,{\Bbb Z})$ be the classes of a fibre of the projection to ${\Bbb P}_1$ and a section respectively. For an ample divisor $H$ let $M^S_{H}(c_1,c_2)$ be the moduli space of $H$-stable torsion-free sheaves with Chern classes $(c_1,c_2)$. \begin{lem} \label{qin}\cite{Q2} Assume $c_1\cdot F=1$, then $M^S_{F+\epsilon E}(c_1,c_2)$ is empty for all sufficiently small $\epsilon >0$. In particular, given $N\ge 0$, we get $\Phi^{S,F+\epsilon E}_{c_1,N}=0$ for all sufficiently small $\epsilon >0$. \end{lem} \section{Main Theorem} We want to express the wall-crossing formulas in terms of the $q$-development of certain modular forms. We start by reminding the reader of some notations and elementary facts (see e.g. \cite{H-B-J}, \cite{R}). \begin{nota} Let $\Bbb H=\{\tau\in {\Bbb C} \,|\, Im(\tau)>0\}$ be the complex upper half plane. We denote $q=e^{2\pi i\tau}$ and $q^{1/n}=e^{2\pi i\tau/n}$. For a positive integer $n$ let $$\sigma_k(n):=\sum_{d|n}d^k\quad \hbox{and} \quad \sigma_1^{odd}(n):=\sum_{d|n,\ d\ odd}d^k.$$ Let $\eta(\tau):=q^{1/24}\prod_{n>0} (1-q^n)$ be the Dirichlet eta-function, and let $\Delta(\tau)=\eta(\tau)^{24}$ be the discriminant. We denote $$\theta(\tau):=\sum_{n\in {\Bbb Z}}q^{n^2}$$ the theta function for the latice ${\Bbb Z}$. We also have the Eisenstein series $$G_{2}(\tau):=-1/24+\sum_{n\ge 1}\sigma_{1}(n)q^n$$ and the $2$-division value $$e_3(\tau):=1/12+2\sum_{n\ge 1} \sigma_1^{odd}(n)q^{n/2}$$ We put $f(\tau):=\eta(2\tau)^3/\theta(\tau).$ Then $\eta(2\tau)$, $\theta(\tau)$, $G_{2}(2\tau)$, $e_3(2\tau)$ and $f(\tau)$ are modular forms of weights $1/2$, $1/2$, $2k$, $2$ and $1$ respectively for certain subgroups of $SL(2,{\Bbb Z})$. We will denote $ d\log_q(g):=g^{-1}{d g/ dq}$. Note that $$\leqno{(*)} \quad d\log_q(g_1g_2)= d\log_q(g_1)+ d\log_q(g_2) \hbox{\ \ \ and \ \ \ } d\log_q(g_1/g_2)= d\log_q(g_1)- d\log_q(g_2).$$ \end{nota} \begin{rem}\label{modular} We will use the following identities \begin{eqnarray*} &{(1)\ }&\eta(2\tau)^3=\sum_{n\in{\Bbb Z}} (-1)^n (n+1/2)q^{(n+1/2)^2},\\ &{(2)\ }&\theta(\tau)={\eta(2\tau)^5\over \eta(\tau)^2\eta(4\tau)^2},\quad\quad\quad\quad\quad\quad {(3)\ } f(\tau)={\eta(\tau)^2\eta(4\tau)^2\over \eta(2\tau)^2 },\\ &{(4)\ }& q\, d\log_q(\eta(2\tau))=-2G_2(2\tau),\quad\quad\quad {(5)\ } q\, d\log_q(\theta(\tau))=-2G_2(2\tau)-e_3(2\tau). \end{eqnarray*} \end{rem} \begin{pf} (1) and (2) are standard facts, following e.g. from the Jacobi identity. (3) follows from (2). (4) follows by an easy calculation using $(*)$, and, using also (2), the proof of (5) is similar. \end{pf} The main result of this paper is the following. \begin{thm} \label{mainthm} Let $X$ be a simply connected $4$-manifold with $b_+=1$ and signature $\sigma(X)$. Let $\xi\in H^2(X,{\Bbb Z})$ with $\xi^2<0$. For $\alpha\in H_2(X,{\Bbb Z})$ denote \begin{eqnarray*} g^X_{\xi}(\alpha z,x,\tau)&:=&\exp\Big(({\xi/2}\cdot\alpha)z/ f(\tau)-(Q_X(\alpha)/2)z^2 (2G_2(2\tau)+e_3(2\tau))/f(\tau)^2\\ &&\quad -3xe_3(2\tau)/f(\tau)^2\Big) \theta(\tau)^{\sigma(X)}f(\tau){\Delta(2\tau)^2 \over\Delta(\tau)\Delta(4\tau)}.\end{eqnarray*} Then $$ \delta^X_\xi\big(\exp(\alpha z+px)\big) =\hbox{res}_{q=0}(q^{-\xi^2/4}g^X_{\xi}(\alpha z,x,\tau){dq/ q}). $$ \end{thm} \begin{rem} \label{power} \begin{enumerate} \item One can see that this expression for $\delta_{\xi}^X$ is not compatible with the simple type condition. In particular given $c\in H^2(X,{\Bbb Z})$ a $4$-manifold $X$ with $b_+=1$ will be of c-simple type at most for some special points in the closure $\overline C_X$ of the positive cone of $X$. It had already been shown in \cite{K-L} that ${\Bbb P}_2$ is not of simple type and that there is no chamber for which ${\Bbb P}_1\times {\Bbb P}_1$ is of simple type. It is easy to see from this that rational algebraic surfaces $X$ can be of simple type at most for special points in $\overline C_X$. \item The expression $\hbox{res}_{q=0}(q^{-\xi^2/4}g^X_{\xi}(\alpha z,x,\tau)dq/q)$ is just the coefficient of $q^{\xi^2/4}$ of $g^X_{\xi}(\alpha z,x,\tau)$. The current formulation is however more intrinsic. Note also that $dq/q=2\pi i d\tau$. \item We see that the coefficient $g_{N-2r,r}$ of $z^{N-2r}x^r$ in $g^X_{\xi}(\alpha z,x,\tau)$ is $q^{-(N+3)/4}$ multiplied with a power series in $q$ In particular, if $\xi$ defines a wall of type $(N)$, then $q^{-\xi^2/4}g_{N-2r,r}$ is a Laurent series in $q$. If $\xi$ with $\xi^2<0$ does not define a wall of type $(N)$, then the constant term of $q^{-\xi^2/4}g_{N-2r,r}$ is zero. \item It would be interesting to know whether for classes $\xi\in H^2(X,{\Bbb Z})$ with $\xi^2\ge 0$ the expression $\hbox{res}_{q=0}(q^{-\xi^2/4}g^X_{\xi}(\alpha z,x,\tau)dq/q)$ has a geometrical or gauge-theoretical meaning. \end{enumerate} \end{rem} As a reasonably straightforward application of theorem \ref{mainthm} we can determine all the Donaldson invariants of the projective plane ${\Bbb P}_2$. \begin{thm}\label{donp2} We denote by $\sqrt{i}$ a primitive $8$-th root of unity and by $H$ in $H^2({\Bbb P}_2,{\Bbb Z})$ the hyperplane class. Put $$e_n(z,x,\tau):=\exp\big((n/2) \sqrt{i} z /f(\tau)-i(z^2/ 2) (2G_2(2\tau)+e_3(2\tau))/f(\tau)^2-3ix e_3(2\tau)/f(\tau)^2\big) {\Delta(2\tau)^2\over\Delta(\tau)\Delta(4\tau)}.$$ Then \begin{align*} \tag*{$(1)$} &\Phi_{H}^{{\Bbb P}_2}(\exp(\check Hz+px))= \hbox{res}_{q=0}\Big(\sum_{{n>0 \ \text{odd}}\atop {a>n}\ \text{even}} (-1)^{(n+1)/2} q^{(a^2-n^2)/4}e_n(z,x,\tau)f(\tau)\Big){dq/q}.\\ \tag*{$(2)$} &\Phi_{0}^{{\Bbb P}_2}(\exp(\check Hz+px))=\hbox{res}_{q=0} \Big(\sum_{{{n>0}\ \text{even} }\atop{{a>n}\ \text{odd}}} (-1)^{(a-1)/2} q^{(a^2-n^2)/4}{a\over 2\sqrt{i}}e_n(z,x,\tau)\Big){dq/ q}. \end{align*} In $(2)$ we have used definition \ref{extend} to define $\Phi_{0}^{{\Bbb P}_2}(\check H^{N-2r}p^r)$ for $r\ge (N-5)/4$. One can check that (up to different sign conventions) $(1)$ and $(2)$ agree with the explicit computations in \cite{K-L} and \cite{E-G2}. \end{thm} \begin{pf} (of theorem \ref{donp2} from theorem \ref{mainthm}). Let $Y$ be the blowup of ${\Bbb P}_2$ in a point, and let $E\in H^2(Y,{\Bbb Z})$ be the class of the exceptional divisor. Let $F=H-E$ be the class of a fibre of the ruling $Y\longrightarrow {\Bbb P}_1$. Fix a nonnegative integer $N$. By lemma \ref{qin} we get for $\epsilon>0$ sufficiently small $\Phi^{Y,F+\epsilon E}_{H,N}=0=\Phi^{Y,F+\epsilon E}_{E,N+1}$. On the other hand the chamber of $H-\epsilon E$ is related to the polarisation $H$ of ${\Bbb P}_2$. Thus we obtain by the blowup formulas $(0)_b$ and $(1)_b$ that $\Phi^{{\Bbb P}_2}_{H,N}=\Phi^{Y,H-\epsilon E}_{H,N}$ and $\Phi^{{\Bbb P}_2}_{0,N}=\Phi^{Y,H-\epsilon E}_{E,N+1}$. So we get by theorem \ref{wallchange} (and lemma \ref{wallchange1} below) the formulas \begin{eqnarray*} \Phi_{H}^{{\Bbb P}_2}(\exp(\check Hz+px))&=&\sum_{\xi\in W^Y_h(F,H)} \sqrt{i}^{(\xi^2+3)+(\xi-H)^2}\delta^Y_{\xi}(\exp(-\sqrt{i}\check Hz+i px)),\\ \Phi_{0}^{{\Bbb P}_2}(\exp(\check Hz+px))&=&\sum_{\xi\in W^Y_e(F,H)} \sqrt{i}^{(\xi^2+3)+(\xi-E)^2} \delta^Y_{\xi}(-\sqrt{i}\check E \exp(-\sqrt{i}\check Hz+ipx)). \end{eqnarray*} It is easy to see that \begin{eqnarray*} W^Y_h(F,H)&=&\big\{(2n-1)H-2aE\bigm| a\ge n\in{\Bbb Z}_{>0}\big\},\\ W^Y_e(F,H)&=&\big\{ 2nH-(2a-1)E\bigm| a>n\in{\Bbb Z}_{>0}\big\}. \end{eqnarray*} For $\xi=nH-aE$ we get $-\xi^2/4=(a^2-n^2)/4$. Furthermore $\sqrt{i}^{(\xi^2+3)+(\xi-H)^2}=(-1)^{(n+1)/2}$ if $n$ is odd and $a$ is even and $\sqrt{i}^{(\xi^2+3)+(\xi-E)^2} =i^{a+2}$ if $n$ is even and $a$ is odd. Thus, replacing $-\sqrt{i}$ by $\sqrt{i}$, (1) follows directly by applying theorem \ref{mainthm}. (2) follows the same way using that \begin{eqnarray*} \delta^Y_{\xi}(-\sqrt{i}\check E \exp(-\sqrt{i}\check Hz+i px))&=&{d\over dw} \Big(\delta^Y_\xi(\exp(-\sqrt{i}(\check Ew+\check Hz)+i px))\Big)\Big|_{w=0}\\ &=&\hbox{res}_{q=0}\Big(q^{-\xi^2/4} {d\over dw}(g^{\widehat{\Bbb P}_2}_{\xi}(-\sqrt{i}(\check Ew+\check Hz),ix,\tau))\big|_{w=0}\Big). \end{eqnarray*} \end{pf} \begin{rem} The arguments in section 6 of \cite{E-G2} show that, using theorem \ref{mainthm} and the blowup formulas, we can get explicit generating functions for all the Donaldson invariants of all rational surfaces $S$ in all chambers of $H^2(S,{\Bbb R})^+$. In \cite{K-L} it had been shown (also using the blowup formulas) that the wall-crossing terms on ${\Bbb P}_2\#2\overline{\Bbb P}_2$ determine the Donaldson invariants on ${\Bbb P}_2$ and ${\Bbb P}_1\times {\Bbb P}_1$. \end{rem} \section{Proof of the main theorem} We give a brief outline of the argument. Let $\xi\in H^2(X,{\Bbb Z})$ define a wall of type $(w,N)$, and let $\cal C_-$ and $\cal C_+$ be the two chambers separated by $W^\xi$. The related chambers $\cal C_{-0}$ and $\cal C_{+0}$ of type $(w,N)$ (resp. $\cal C_{-e}$ and $\cal C_{+e}$ of type $(w+e,N+1)$) on $\widehat X$ are now separated by several walls. We can express $\delta_{\xi,N}^X$ by first applying the blowup formula to the pair $\cal C_-$, $\cal C_{-0}$ of related chambers, then summing up the wall-crossing formulas for all walls between $\cal C_{-0}$ and $\cal C_{+0}$ and finally applying again the blowup formula for $\cal C_+$, $\cal C_{+0}$ (and similarly for $\cal C_{-e}$, $\cal C_{+e}$). The blowup formulas $(0)_b$--$(3)_b$ from \ref{blowup} will give relations $(0)_r$--$(3)_r$ between the $\delta_{\xi,N}^X$ and the $\delta_{\xi,N}^{\widehat X}$. Using conjecture \ref{KMconj} we encode this information (for all blowups of $X$) in a suitable generating function $\Lambda_X$ in several variables. Then we can translate $(0)_r$--$(3)_r$ into differential equations $(0)_d$--$(3)_d$ for $\Lambda_X$, which determine $\Lambda_X$ up to multiplication by a function $\lambda_X(\tau)$. We finally determine $\lambda_X(\tau)$ by specializing to the case $X={\Bbb P}_1\times {\Bbb P}_1$ and applying lemma \ref{qin}. \begin{lem} \label{wallchange1} Let $w$ be the reduction modulo $2$ of $c_1\in H^2(X,{\Bbb Z})$, and let $N$ be a nonnegative integer. Let $g_-$ and $g_+$ be two metrics on $X$, whose period points $\omega(g_-)$ and $\omega(g_+)$ do not lie on a wall of type $(w,N)$. We denote $W:=W^X_{w,N}(\omega(g_-),\omega(g_+))$. Then we have for all $\alpha\in A_N(X)$ and $\beta\in A_{N-2}(X)$: \begin{align} \Phi^{X,g_+}_{c_1,N}(\alpha)-\Phi^{X,g_-}_{c_1,N}(\alpha)& =\sum_{\xi\in W} (-1)^{\varepsilon(c_1,\xi,N)}\delta^X_{\xi,N}(\alpha)\tag*{$(a)$},\\ \tag*{$(0)_r$} \Phi^{X,g_+}_{c_1,N}(\alpha)-\Phi^{X,g_-}_{c_1,N}(\alpha)& =\sum_{\xi\in W} (-1)^{\varepsilon(c_1,\xi,N)}\sum_{n\in {\Bbb Z}}\delta^{\widehat X}_{\xi+2nE}(\alpha),\\ \tag*{$(1)_r$}\Phi^{X,g_+}_{c_1,N}(\alpha)-\Phi^{X,g_-}_{c_1,N}(\alpha)& =\sum_{\xi\in W} (-1)^{\varepsilon(c_1,\xi,N)}\sum_{n\in {\Bbb Z}}(-1)^{n-1}\delta^{\widehat X}_{\xi+(2n+1)E,N+1}(\check E\alpha),\\ \tag*{$(2)_r$} 0&=\sum_{\xi\in W} (-1)^{\varepsilon(c_1,\xi,N)}\sum_{n\in {\Bbb Z}}\delta^{\widehat X}_{\xi+2nE,N}(\check E^2\beta),\\ \tag*{$(3)_r$} \Phi^{X,g_+}_{c_1,N}(p\beta)-\Phi^{X,g_-}_{c_1,N}(p\beta)& =\sum_{\xi\in W} (-1)^{\varepsilon(c_1,\xi,N)} \sum_{n\in {\Bbb Z}}(-1)^{n}\delta^{\widehat X}_{\xi+(2n+1)E,N+1}(\check E^3\beta). \end{align} (a) says that theorem \ref{wallchange} extends to our definition of $\delta^X_{\xi,N}$. \end{lem} \begin{pf} We assume that $N$ is congruent to $3-c_1^2$ modulo $4$ (otherwise both sides of $(a)$, $(0)_r$--$(3)_r$ are trivially zero). Let $\cal C_-$ and $\cal C_+$ be the chambers of type $(w,N)$ of $\omega(g_-)$ and $\omega(g_+)$ respectively. Let $\cal C_{-0}$ and $\cal C_{+0}$ (resp. $\cal C_{-e}$ and $\cal C_{+e}$) be related chambers in $H^2(\widehat X,{\Bbb R})^+$ of type $(w,N)$ (resp. $ (w+e,N+1)$). \noindent{\it Claim:} \begin{eqnarray*}W_{w,N}^{\widehat X}(\cal C_{-0},\cal C_{+0})&=& \big\{\xi+2nE\bigm |\xi\in W,\ n\in {\Bbb Z},\ n^2\le (N+3+\xi^2)/4\big\},\\ W_{w+e,N+1}^{\widehat X}(\cal C_{-e},\cal C_{+e})&=& \big\{\xi+(2n+1)E\bigm| \xi\in W,\ n\in {\Bbb Z},\ (2n+1)^2\le N+4+\xi^2\big\}. \end{eqnarray*} In the case that $w=0$, we assume that $\cal C_{-0}$ and $\cal C_{+0}$ (resp. $\cal C_{-e}$ and $\cal C_{+e}$) lie on the same side of $W^{2E}$ (resp. $W^E$). The claim is essentially obvious: Any $\eta\in W_{w,N}^{\widehat X}(\cal C_{-0},\cal C_{+0})$ must be of the form $\xi+\alpha E$ for $\xi\in W$. By the definition of a wall we see that $\alpha$ must be an even integer $2n$ with $n^2\le (N+3+\xi^2)/4$. On the other hand it is obvious that all $\xi+2n E$ with $\xi\in W$ and $n^2\le (N+3+\xi^2)/4$ lie in $W_{w,N}^{\widehat X}(\cal C_{-0},\cal C_{+0})$. For $W_{w+e,N+1}^{\widehat X}(\cal C_{-e},\cal C_{+e})$ we argue analogously. Using this description of $W_{w,N}^{\widehat X}(\cal C_{-0},\cal C_{+0})$ and $W_{w+e,N+1}^{\widehat X}(\cal C_{-e},\cal C_{+e})$, we see that for $X$ with $b_2(X)>2$ and odd intersection form and $\alpha\in {\hbox{\rm {Sym}}}^N(H^2(X,{\Bbb Q}))$ and $\beta\in {\hbox{\rm {Sym}}}^{N-2}(H^2(X,{\Bbb Q}))$, the formulas $(0)_r$--$(3)_r$ are just straightforward translations of $(0)_b$--$(3)_b$ (note that $\varepsilon(c_1,\xi,N)-\varepsilon(c_1+E,\xi+(2n+1)E,N+1)$ is congruent to $n-1$ modulo $2$). Now assume that the intersection form of $X$ is even or $b_2(X)\le 2$ or $\xi$ is divisible by $2$ in $H^2(X,{\Bbb Z})$. Then definition \ref{extend}, the description of $W_{w,N}^{\widehat X}(\cal C_{-e},\cal C_{+e})$ and $(1)_b$ imply immediately that $(a)$ holds for all $\alpha\in {\hbox{\rm {Sym}}}^N(H^2(X,{\Bbb Q}))$. We show $(0)_r$--$(3)_r$ for $\alpha\in {\hbox{\rm {Sym}}}^N(H^2(X,{\Bbb Q}))$ (we only carry out the case of $(0)_r$, the other cases are analogous.) Let $\widetilde X:=\widehat X\# \overline {\Bbb P}_2$, we denote by $F$ the generator of $H^2(\overline {\Bbb P}_2,{\Bbb Z})$. Then by definition \ref{extend} and $(0)_r$ for $\widehat X$ we get \begin{eqnarray*} \Phi^{X,\cal C_+}_{c_1,N}(\alpha)-\Phi^{X,\cal C_-}_{c_1,N}(\alpha) &=&\sum_{\xi\in W} \sum_{m\in {\Bbb Z}} (-1)^{m+1} \delta^{\widehat X}_{\xi+(2m+1)E,N+1}(\check E\alpha)\\ &=&\sum_{\xi\in W}\sum_{n\in {\Bbb Z}} \sum_{m\in {\Bbb Z}} (-1)^{m+1} \delta^{\widetilde X}_{\xi+2nF+(2m+1)E,N+1}(\check E\alpha)\\ &=&\sum_{\xi\in W} \sum_{n\in {\Bbb Z}} \delta^{\widehat X}_{\xi+2nE,N}(\alpha). \end{eqnarray*} Now let $X$ be general. We assume $(a)$, $(0)_r$--$(3)_r$ for all blowups $Y$ of $X$ and all classes $\alpha=p^l\beta$ with $\beta\in {\hbox{\rm {Sym}}}^{k}(H_2(Y,{\Bbb Q}))$ for some $k$. Then $(3)_r$ implies immediately $(a)$ for $p\alpha$. The proof of $(0)_r$--$(3)_r$ for $p\alpha$ is analogous to the last section. We only carry out the case of $(1)_r$. Let $\widetilde Y:=\widehat Y\# \overline {\Bbb P}_2$, we denote by $F$ the generator of $H^2(\overline {\Bbb P}_2,{\Bbb Z})$. We get by definition \ref{extend} \begin{eqnarray*} \Phi^{Y,\cal C_+}_{c_1,N}(p\alpha)-\Phi^{Y,\cal C_-}_{c_1,N}(p\alpha) &=&\sum_{\xi\in W} \sum_{n\in {\Bbb Z}}(-1)^{n} \delta^{\widehat Y}_{\xi+(2n+1)E,N+1} (\check E^3\alpha)\\ &=&\sum_{\xi\in W}\sum_{n\in {\Bbb Z}} \sum_{m\in {\Bbb Z}} (-1)^{n}(-1)^{m-1} \delta^{\widetilde Y}_{\xi+(2n+1)E+(2m+1)F,N+2}(\check F\check E^3\alpha)\\ &=&\sum_{\xi\in W} \sum_{m\in {\Bbb Z}} (-1)^{m-1} \delta^{\widehat Y}_{\xi+(2m+1)E,N+1}(\check Ep\alpha). \end{eqnarray*} \end{pf} \begin{lem}\label{combin} For $\xi\in H^2(X,{\Bbb Z})$ we get $$\exp\big(L_{(\xi+nE)/2}+Q_{\widehat X}\big)(\check E^k\bullet)= \sum_{s+2t=k} (n/2)^s(-1)^{s+t}{k!\over s!t!} \exp(L_{\xi/2}+Q_{ X}),$$ as a map $\sum_{N\ge 0} {\hbox{\rm {Sym}}}^N(H_2(X,{\Bbb Q}))\longrightarrow {\Bbb Q}$. \end{lem} \begin{pf} $$ \exp\big(L_{(\xi+nE)/2}+Q_{\widehat X}\big)(\check E^k\bullet)= {d^k\over dw^k} \exp\big((L_{\xi}-nw)/2+Q_X-w^2\big)\big|_{w=0},$$ and the result follows by induction. \end{pf} \begin{rem}\label{point} Using definition \ref{extend}, lemma \ref{combin} and easy induction we see that conjecture \ref{KMconj} implies that $\delta^X_{\xi,N}(p^r\bullet)$ is a polynomial in $L_{\xi/2}$ and $Q_X$ with coefficients only depending on $N$, $\xi^2$, $r$ and the homotopy type of $X$. \end{rem} \begin{defn}\label{pcoeff} For all $b\ge 0$ let $X(b):=X\# b\overline{\Bbb P}_2$. Let $l,k,r,b\in{\Bbb Z}$, put $N:=l+2k+2r$, and assume that there exists a class $\xi\in H^2(X(b),{\Bbb Z})$ with $w=\xi^2/4<0$. Then we put $$P(l,k,r,b,w):={l!k!\over (l+2k)!}\hbox{Coeff}_{L_{\xi/2}^lQ_{X(b)}^k}\delta^{X(b)}_{\xi,N}(p^r\bullet). $$ (By definition $P(l,k,r,b,w)$ will be zero if $\xi$ does not define a wall of type $(N)$ or if one of $l,k,r,b$ is negative). Note that $P(l,k,r,b,w)$ is well defined: By conjecture \ref{KMconj} and remark \ref{point} $\delta^{X(b)}_{\xi,N}(p^r\bullet)$ is a polynomial in $L_{\xi/2}$ and $Q_{X(b)}$. As $b_2(X)>1$, the monomials $L_{\xi/2}^lQ_{X(b)}^k$ are linearly independent as linear maps ${\hbox{\rm {Sym}}}^{l+2k}(H_2(X,{\Bbb Q}))\longrightarrow {\Bbb Q}$, therefore the coefficients of $L_{\xi/2}^lQ_{X(b)}^k$ in $\delta^{X(b)}_{\xi,N}(p^r\bullet)$ are well-defined. Finally, again by conjecture \ref{KMconj} they depend only on the numbers $l,k,r,b,w$. \end{defn} \begin{lem} For all $(l,k,r,b,w)$ with $b\ge 0$, if the left hand side of the equations below is well-defined, then the right hand side is also, and \begin{align*} \tag*{$(0)_s$} P(l,k,r,b,w)&=\sum_{n\in {\Bbb Z}}P(l,k,r,b+1,w-n^2),\\ \tag*{$(1)_s$} P(l,k,r,b,w)& =\sum_{n\in {\Bbb Z}}(-1)^n(n+1/2)P(l+1,k,r,b+1,w-(n+1/2)^2),\\ \tag*{$(2)_s$} \sum_{n\in {\Bbb Z}} n^2 P(l,k,r,&b+1,w-n^2)=2\sum_{n\in {\Bbb Z}} P(l-2,k+1,r,b+1,w-n^2),\\ \tag*{$(3)_s$} P(l,k,r+1,b,w)& =\sum_{n\in {\Bbb Z}}(-1)^{n+1}\Big((n+1/2)^3P(l+3,k,r,b+1,w-(n+1/2)^2)\\ & \quad -6(n+1/2)P(l+1,k+1,r,b+1,w-(n+1/2)^2)\Big). \end{align*} \end{lem} \begin{pf} Take $(l,k,r,b,w)$ such that there exists a class $\xi\in H^2(X(b),{\Bbb Z})$ with $w:=\xi^2/4<0$. Let $N:=l+2k+2r$. We can assume that $\xi$ defines a wall of type $(N)$ (otherwise both sides of $(0)_s$--$(3)_s$ are trivially zero). Assume first that $b_2(X)>2$ and that in addition the intersection form on $H_2(X,{\Bbb Z})$ is odd, or $b>0$. Then we can find an $\eta$ which is not divisible in $H^2(X(b),{\Bbb Z})$ with $\eta^2=\xi^2$. (The intersection form is $(1)\oplus(-1)^{\oplus b_2(X(b))-1}$, therefore we can find orthogonal classes $h$, $e_1$, $e_2$ with $Q_{X(b)}(h)=1=-Q_{X(b)}(e_1)=-Q_{X(b)}(e_2)$, and we put $\eta:=nh+(n+1)e_1$ (resp. $\eta:=nh+(n+1)e_1+e_2$) if $\xi^2=-(2n+1)$ (resp. $\xi^2=-(2n+2)$) for $n\in{\Bbb Z}_{\ge 0}$.) We can therefore assume that $\xi$ is not divisible in $H^2(X(b),{\Bbb Z})$. Let $\cal C_-$ and $\cal C_+$ be the two chambers separated by $W^\xi$, with $\xi\cdot \check a_-<0<\xi\cdot \check a_+$ for $a_-\in \cal C_-$ and $a_+\in \cal C_+$. Assume that $N+3+4\xi^2<0$. Then $W^{X(b)}_{w,N}(a_-,a_+)=\{\xi\}$. Therefore we can replace $\Phi^{X(b),\cal C_+}_{c_1,N}-\Phi^{X(b),\cal C_-}_{c_1,N}$ in $(0)_r$--$(3)_r$ by $(-1)^{\varepsilon(c_1,\xi,N)} \delta^{X(b)}_{\xi,N}$. Now we apply lemma \ref{combin} and the definition of the $P(l,k,r,b,w)$ to obtain the result. Finally, if $m:=N+3+4\xi^2\ge 0$ we make induction over $m$. So we assume that the result is true for all $m'< m$. Then $W^{X(b)}_{w,N}(a_-,a_+)=\{\xi\}\cup W_{m}$, where the classes $\eta\in W_{m}$ satisfy $N+3+\eta^2< m$. So by induction the result holds for all $\eta\in W_{m}$ and thus by lemma \ref{wallchange1} also for $\xi$. \end{pf} We want to use the $P(l,k,r,b,w)$ as the coefficients of a power series, which should solve a system of differential equations. This does not work directly, because in the moment we have only coefficients with $w<0$. So we have to "complete" the coefficients, i.e. to define the $P(l,k,r,b,w)$ for all $l,k,r,b,w$ by making use of relation $(1)_s$. \begin{defn}\label{series} For all $l,k,r,b\in {\Bbb Z}$ and all $w\in {1\over 4}{\Bbb Z}$ define inductively $P(l,k,r,b,w)$ by \begin{enumerate} \item If $w=\xi^2/4<0$ for $\xi\in H^2(X(b),{\Bbb Z})$, then apply definition \ref{pcoeff}. \item We put $$P(l,k,r,b,w):=\sum_{n\in{\Bbb Z}}(-1)^n(n+1/2)P(l+1,k,r,b+1,w-(n+1/2)^2),$$ whenever the right hand side is already defined inductivly by (1) and (2). Note that the sum is again finite. \end{enumerate} We check that the $P(l,k,r,b,w)$ are well-defined. For this we have to see (a), that (1) and (2) give the same $P(l,k,r,b,w)$ whenever both apply, but this is the contents of relation $(1)_s$; and (b), that the above definition determines $P(l,k,r,b,w)$ for each $5$-tuple $(l,k,r,b,w)\in {\Bbb Z}^4\times {1\over 4}{\Bbb Z}$. If $w\le 0$, then there exist on $X(1)$ for all $n\in {\Bbb Z}$ classes $\eta_n$ with $\eta_n^2=4w-(2n+1)^2<0$ (as the intersection form on $X(1)$ is odd and of rank $\ge 3$), and thus $P(l,k,r,b,w)$ is defined by (2). Now assume that $P(l,k,r,b,w')$ is defined for all $l,k,r,b$ and all $w'<w$. Then we use again (2) to define $P(l,k,r,b,w)$. We put $$\Lambda_X(L,Q,x,t,\tau):= \sum_{(l,k,r,b)\in {\Bbb Z}^4}\sum_{w\in {1\over 4}{\Bbb Z}} P(l,k,r,b,w){ L^lQ^kx^rt^bq^w\over l!k!r!b!},$$ where again $\tau\in \Bbb H$ and $q=e^{2\pi i\tau}$. \end{defn} $\Lambda_X$ now encodes all the wall-crossing formulas for all blowups of $X$. \begin{rem} Let $\xi\in H^2(X(b),{\Bbb Z})$ be a class with $\xi^2<0$. Then for all $\alpha\in H_2(X(b),{\Bbb Q})$ $$\delta_{\xi}^{X(b)}(\exp(\alpha z+p x))=\hbox{res}_{q=0} \left({\partial^k \over \partial t^k}(q^{-\xi^2/4}\Lambda_X((\xi/2\cdot\alpha) z,Q_X(\alpha)z^2,x,t,\tau)dq/q\right)\Big|_{t=0}.$$ \end{rem} \begin{pf} This follows directly from the definition. \end{pf} \begin{lem} $\Lambda_X$ satisfies the differential equations \begin{align} \theta(\tau){\partial\over \partial t}\Lambda_X&=\Lambda_X,\tag*{$(0)_d$}\\ \eta(2\tau)^3 {\partial\over \partial L}{\partial\over \partial t} \Lambda_X&=\Lambda_X,\tag*{$(1)_d$}\\ 2\theta(\tau){\partial\over \partial Q}\Lambda_X&=(q{d\over dq}\theta(\tau)){\partial^2\over \partial L^2}\Lambda_X,\tag*{$(2)_d$}\\ {\partial\over \partial x}\Lambda_X&=(q{d\over dq}\eta(2\tau)^3) {\partial^3\over \partial L^3}{\partial\over \partial t}\Lambda_X- 6\eta(2\tau)^3 {\partial\over \partial L} {\partial\over \partial Q}{\partial\over \partial t}\Lambda_X.\tag*{$(3)_d$} \end{align} \end{lem} \begin{pf} We first want to see that the relations $(0)_s$--$(3)_s$ hold for all $(l,k,r,b,w)\in {\Bbb Z}^4\times {1\over 4}{\Bbb Z}$, i.e. that the recursive definition is compatible with $(0)_s$--$(3)_s$. The proof is similar in all cases, so we just do $(0)_s$. We assume that $(0)_s$ holds for all $(l,k,r,b,w')$ with $w'<w$. Then we get \begin{eqnarray*} P(l,k,r,b,w)&=&\sum_{n\in {\Bbb Z}}(-1)^n(n+1/2)P(l+1,k,r,b+1,w-(n+1/2)^2)\\ &=&\sum_{n,m\in {\Bbb Z}}(-1)^n(n+1/2)P(l+1,k,r,b+2,w-(n+1/2)^2-m^2)\\ &=&\sum_{m\in {\Bbb Z}}P(l+1,k,r,b+2,w-m^2). \end{eqnarray*} We now translate $(0)_s$--$(3)_s$ into differential equations $(0)_d$--$(3)_d$: \begin{eqnarray*} \Lambda_X&=&\sum_{(l,k,r,b,w)} P(l,k,r,b,w){ L^lQ^kx^rt^bq^w\over l!k!r!b!}\\ &=&\sum_{(l,k,r,b,w)} \sum_{n\in{\Bbb Z}} P(l,k,r,b,w){L^lQ^kx^rt^{b-1}q^{w+n^2}\over l!k!r!(b-1)!}\\ &=&\theta(\tau){\partial\over \partial t}\Lambda_X. \end{eqnarray*} Similarly we get \begin{eqnarray*} \Lambda_X&=& \sum_{(l,k,r,b,w)} \sum_{n\in{\Bbb Z}} (-1)^n(n+1/2)P(l,k,r,b,w){L^{l-1}Q^kx^rt^{b-1}q^{w+(n+1/2)^2}\over (l-1)!k!r!(b-1)!}\\ &=&\sum_{n\in{\Bbb Z}}(-1)^n(n+1/2)q^{(n+1/2)^2} {\partial\over \partial L}{\partial\over \partial t}\Lambda_X, \end{eqnarray*} and $(1)_d$ follows from remark \ref{modular}. Furthermore \begin{eqnarray*} 0&=&\sum_{(l,k,r,b,w)} P(l,k,r,b,w)\sum_{n\in{\Bbb Z}} \left({n^2L^{l-2}Q^k\over (l-2)!k!}-{2L^{l}Q^{k-1}\over l!(k-1)!}\right){x^rt^{b-1}q^{w+n^2}\over r!(b-1)!}\\ &=&(q {d\over dq}\theta(\tau)) {\partial^2\over \partial L^2}{\partial\over \partial t}\Lambda_X -2{\partial\over \partial Q}{\partial\over \partial t}\Lambda_X. \end{eqnarray*} Finally we get \begin{eqnarray*} {\partial\over \partial x}\Lambda_X &=&\sum_{(l,k,r,b,w)}P(l,k,r+1,b,w){L^{l}Q^kx^rt^{b}q^{w}\over l!k!r!b!}\\ &=&\sum_{(l,k,r,b,w)}P(l,k,r,b,w) \sum_{n\in{\Bbb Z}} (-1)^{n+1}\left({(n+1/2)^3L^{l-3}Q^k\over (l-3)!k!}\right.\\ &&\qquad -\left.{6(n+1/2)L^{l-1}Q^{k-1}\over (l-1)!(k-1)!}\right){x^rt^{b-1}q^{w+(n+1/2)^2}\over r!(b-1)!}\\ &=&(q{d\over dq}\eta(2\tau)^3) {\partial^3\over \partial L^3}{\partial\over \partial t}\Lambda_X- 6\eta(2\tau)^3{\partial\over \partial L}{\partial\over \partial Q}{\partial\over \partial t}\Lambda_X. \end{eqnarray*} \end{pf} \begin{lem}\label{expex} Putting $\lambda_X(\tau):=\Lambda_X(0,0,0,0,\tau)$ we obtain $$\Lambda_X=\exp\Big(L/f(\tau)-(Q/2)(2G_2(2\tau) +e_3(2\tau))/f(\tau)^2-3xe_3(2\tau)/f(\tau)^2- \theta(\tau)t\big)\lambda_X(\tau).$$ \end{lem} \begin{pf} Using remark \ref{modular} we can reformulate $(0)_d$--$(3)_d$ as \begin{eqnarray*} {\partial\over \partial t}\Lambda_X&=&\Lambda_X/\theta(\tau),\qquad {\partial\over \partial L}\Lambda_X=\Lambda_X/f(\tau),\\ {\partial\over \partial Q}\Lambda_X&=&{1\over 2}q\,(d\log_q(\theta(\tau)))\Lambda_X/f(\tau)^2,\\ {\partial\over \partial x}\Lambda_X &=&q\,(d\log_q(\eta(2\tau)^3))\Lambda_X/f(\tau)^2 -3q(d\log_q(\theta(\tau)))\Lambda_X/f(\tau)^2. \end{eqnarray*} So the result follows by remark \ref{modular}. \end{pf} To finish the proof of theorem \ref{mainthm} we now only have to identify $\lambda_X$. \begin{lem}\label{norm} $\lambda_X=\Delta(2\tau)\exp(-\theta(\tau)\sigma(X))/f(\tau)^{11}= f(\tau)\Delta(2\tau)^2/(\Delta(\tau)\Delta(4\tau)) \,\exp(-\theta(\tau)\sigma(X)).$ \end{lem} \begin{pf} We first show that is is enough to prove this result in case $X={\Bbb P}_1\times {\Bbb P}_1$. We note that by lemma \ref{expex} the statement for a variety $Y$ and $\widehat Y=Y\#\overline {\Bbb P}_2$ are equivalent. It is therefore enough to show it for $\widehat X$. $\widehat X$ has odd intersection form and $a:=b_2(\widehat X)-1\ge 2$. So it is homotopy-equivalent to ${\Bbb P}_2\# a\overline {\Bbb P}_2=({\Bbb P}_1\times{\Bbb P}_1)\#(a-1)\overline{\Bbb P}_2$. As $\delta_{\xi}^X$ only depends on the homotopy type of $X$, it is enough to show the result for ${\Bbb P}_1\times{\Bbb P}_1$. This is in fact the only time in our argument where we use that $\delta_{\xi}^X$ depends on the homotopy type of $X$, rather then on $X$ itself. Let $F,G\in H^2({\Bbb P}_1\times{\Bbb P}_1,{\Bbb Z})$ be the classes of the fibres of the two projections to ${\Bbb P}_1$. Let $k\in {\Bbb Z}_{>0}$ and $N:=4k-1$. Then lemma \ref{qin} gives that $\Phi^{{\Bbb P}_1\times{\Bbb P}_1,F+\epsilon G}_{F+G,N}=\Phi^{{\Bbb P}_1\times{\Bbb P}_1,G+\epsilon F}_{F+G,N}=0$ for all sufficiently small $\epsilon>0$. In particular we have for all $k>0$ $$(-1)^{k+1}\sum_{\xi\in W^{{\Bbb P}_1\times{\Bbb P}_1}_{f+g}(F,G)}(-1)^{\varepsilon(F+G,\xi,4k-1)} \delta_\xi^{{\Bbb P}_1\times{\Bbb P}_1}(2\check G^{4k-1})=0.$$ Here $W^{{\Bbb P}_1\times{\Bbb P}_1}_{f+g}(F,G)=\big\{(2n-1)F-(2m-1)G\bigm| n,m\in {\Bbb Z}_{>0}\big\}$, and $(-1)^{k+1+\varepsilon(F+G,(2n-1)F-(2m-1)G,4k-1)}=(-1)^{n+m}$. Applying lemma \ref{expex} we get \begin{equation}\hbox{res}_{q=0}\Big(\sum_{n,m\in{\Bbb Z}_{>0}}(-1)^{n+m}q^{{1\over 2}(2n-1)(2m-1)} (2n-1)^{4k-1}f(\tau)^{-4k+1}\lambda_{{\Bbb P}_1\times{\Bbb P}_1}(\tau){dq\over q}\Big) =0.\tag{*} \end{equation} Note that, by remark \ref{power}, $\lambda_{{\Bbb P}_1\times{\Bbb P}_1}= q^{-3/4}\bar\lambda$ where $\bar\lambda=\sum l_i q^i$ is a power series in $q$. Also $f(\tau)=q^{1/4}\bar f$ with $\bar f$ a power series in $q$ with constant term $1$. It is well-known (\cite{Ko},\cite{K-M}) that $\delta^{{\Bbb P}_1\times{\Bbb P}_1}_{F-3G}((2\check G)^3)=1$. Thus we get $l_0=1$. $(*)$ gives for each $k\ge 1$ the recursive relation $$\sum_{n,m>0} (-1)^{n+m}(2n-1)^{4k-1}\hbox{Coeff}_{q^{k-2nm+n+m}}( \bar \lambda/\bar f^{4k-1})=0,$$ i.e., putting $\lambda_k:=\sum_{j<k} l_jq^j$ we obtain $$l_k=-\sum_{n,m>0} (-1)^{n+m}(2n-1)^{4k-1}\hbox{Coeff}_{q^{k-2nm+n+m}}(\bar \lambda_k/\bar f^{4k-1}).$$ So we see that $\lambda_{{\Bbb P}_1\times{\Bbb P}_1}$ is uniquely determined by $(*)$. We put $$H_k(\tau):=\sum_{n,m\in{\Bbb Z}_{>0}}(-1)^{n+m}q^{{1\over 4}(2n-1)(2m-1)} (2n-1)^{4k-1}\Delta(\tau)/f(\tau/2)^{4k+10}.$$ Then the lemma follows from the following lemma (the proof of which is due to Don Zagier). \end{pf} \begin{lem}\label{residue} $\hbox{res}_{q=0}H_k(\tau){dq\over q}=0$. \end{lem} \begin{pf} We start by rewriting $H_k(\tau)$. \begin{eqnarray*} \sum_{n,m>0}(-1)^{n+m}q^{(n-1/2)(m-1/2)}(2n-1)^{4k-1} =\sum_{d\ \text{\rm odd}}^\infty(-1)^{(d-1)/2}\sigma_{4k-1}(d)q^{d/4}\\ ={1\over 2i}(G_{4k}((\tau+1)/ 4)-G_{4k}((\tau-1)/ 4))=:\widetilde G_{4k}(\tau), \end{eqnarray*} where $G_{4k}(\tau)$ is the Eisenstein series. We write $\phi:=f(\tau/2)^{2}=\left(\eta(\tau/2)\eta(2\tau)/\eta(\tau)\right)^4.$ Then we have $H_k(\tau)={\widetilde G_{4k}(\tau)\Delta(\tau)/\phi^{2k+5}}$. We want to show that $H_k(\tau)$ is a modular form of weight $2$ for the $\theta$-group $$\Gamma_{\theta}:=\big\{ A\in SL(2,{\Bbb Z})\bigm| A\equiv \left(\begin{matrix} 1& 0\\0 & 1\end{matrix}\right) or A\equiv \left(\hbox{$\begin{matrix} 0 & 1\\1& 0\end{matrix}$}\right)\hbox{ modulo } 2 \big\}.$$ The operation of $\Gamma_{\theta}$ is generated by $\tau\mapsto \tau+2$ and $\tau\mapsto {-1/\tau}$. We see that $\widetilde G_{4k}(\tau+2)= -\widetilde G_{4k}(\tau)$. Now we write $${(-1/\tau+1)/4}={(\tau-1)/4 \over 4(\tau-1)/4+1}, \ \ {(-1/\tau-1)/4}={(\tau+1)/4 \over -4(\tau+1)/4+1},$$ and use that $G_{4k}(\tau)$ is a modular form of weight $4k$ for $SL(2,{\Bbb Z})$, to obtain that $$\widetilde G_{4k}(-1/\tau)={1\over 2i}\big(\tau^4G_{4k}((\tau-1)/ 4) -\tau^4 G_{4k}((\tau+1)/ 4)\big)=-\tau^4\widetilde G_{4k}(\tau). $$ Furthermore we see by $\phi(\tau)^6=\Delta(\tau/2)\Delta(2\tau)/ \Delta(\tau)$, that $\phi(-{1/\tau})^6= \tau^{12 }\phi(\tau)^6$, i.e. $\phi(-{1/\tau})=\omega \tau^{2 }\phi(\tau)$ for a $6$-th root of unity $\omega$. Putting $\tau:=i$ we get $\phi(i)=-\omega\phi(-{1/ i})$, i.e. $\omega=-1$. We also obviously have $\phi(\tau+2)=-\phi(\tau)$. Putting this together and using the fact that $\Delta(\tau)$ is a modular form of weight $12$ for $SL(2,{\Bbb Z})$ we finally see that $H_k(\tau)$ is a modular form of weight $2$ for $\Gamma_\theta$. In other words $H_k(\tau){dq/ q}=2\pi i H_k(\tau){d\tau }$ is a differential form on the rational curve $\overline {\Bbb H/\Gamma_\theta}$, holomorphic out of the cusps $\tau=1$ and $\tau=\infty$ (i.e. $q=0$). We show that $H_k(\tau)$ is holomorphic at $\tau=1$. $\Delta(\tau)$ and $\widetilde G_{4k}(\tau)$ are obviously holomorphic at $\tau=1$. We now put $\tau:=1-1/z$ and use again that $\Delta(\tau)$ is a modular form of weight $12$ for $SL(2,{\Bbb Z})$ and write $1/ 2-1/ (2 z)={(z-1)/2\over 2(z-1)/2+1}$ to obtain $$\phi(\tau)^6={\Delta(1/ 2- 1/ (2 z))\Delta(-2/z)\over \Delta(-1/z)}= {z^{12}\Delta((z-1)/ 2)(z/ 2)^{12}\Delta(z/ 2)\over z^{12}\Delta(z)}=-(z/ 2)^{12}{\Delta(z)^2\over \Delta(2z)}. $$ So for $z=\infty$, (i.e. $\tau=1$) the modular form $\phi(\tau)$ is holomorphic and does not vanish. Thus also $H_k(\tau)$ is holomorphic at $\tau=1$. Thus the residue theorem implies that $\hbox{res}_{q=0}( 2\pi iH_k(\tau){d\tau })=0.$ \end{pf} \begin{rem} \label{homotop} As noted above, we have used that by conjecture \ref{KMconj} $\delta_{\xi,N}^X$ depends only on the homotopy type $X$ rather then just on $X$ only in the reduction above to ${\Bbb P}_1\times{\Bbb P}_1$. In particular, without assuming this, our proof still shows theorem \ref{mainthm} for $X$ a rational surface, and therefore also theorem \ref{donp2}. \end{rem} \section{Possible generalizations} It should be possible to prove the blowup formulas and also conjecture \ref{KMconj} for $4$-manifolds $X$ with $b_+(X)=1$ and $b_1(X)=0$ (i.e dropping the assumption that $X$ is simply-connected). If we assume these generalizations, then all our arguments in the proof of theorem \ref{mainthm} work in this more general case except for the reduction to ${\Bbb P}_1\times {\Bbb P}_1$ at the beginning of the proof of lemma \ref{norm}. So we get \begin{cor} Assume that the blowup formulas \ref{blowup} and conjecture \ref{KMconj} hold for all for $4$-manifolds $Y$ with $b_+(Y)=1$ and $b_1(Y)=0$. Then for all $X$ with $b_+(X)=1$ and $b_1(X)=0$, all $\xi$ in $H^2(X,{\Bbb Z})$ with $\xi^2<0$ and all $\alpha\in H_2(X,{\Bbb Q})$ we have $$\delta_{\xi}^{X}(\exp(\alpha z+p x))=\hbox{res}_{q=0}(g^X_{\xi}(\alpha z, x,\tau)\lambda_{[X]}(\tau)\Delta(\tau)\Delta(4\tau)/(f(\tau)\Delta(2\tau)^2)) dq/q,$$ where $g^X_{\xi}$ is the generating function from theorem \ref{mainthm} and $\lambda_{[X]}(\tau)$ is $q^{-3/4}$ multiplied with an unknown power series $\bar\lambda_{[X]}(q)$ in $q$, which depends only on the equivalence class $[X]$, where $X$ and $Y$ are equivalent if $X\#k\overline{\Bbb P}_2$ and $Y\#k\overline{\Bbb P}_2$ are homotopy equivalent for some $k$. \end{cor} The results of \cite{E-G1} suggest that the dependence of $\lambda_{[X]}(\tau)$ on $X$ should be very simple. \begin{conj} $\lambda_{[X]}(q)=n_2f(\tau)\Delta(2\tau)^2/(\Delta(\tau)\Delta(4\tau)),$ for $n_2$ the number of $2$-torsion points in $H^2(X,{\Bbb Z})$. \end{conj}

9506

alg-geom/9506020

en

https://arxiv.org/abs/alg-geom/9506020

alg-geom/9506020

Ian Grojnowski

I. Grojnowski (Yale)

Instantons and affine algebras I: The Hilbert scheme and vertex operators

14 pages, AmsTex

This is the first in a series of papers which describe the action of an affine Lie algebra with central charge $n$ on the moduli space of $U(n)$-instantons on a four manifold $X$. This generalises work of Nakajima, who considered the case when $X$ is an ALE space. In particular, this describes the combinatorial complexity of the moduli space as being precisely that of representation theory, and thus will lead to a description of the Betti numbers of moduli space as dimensions of weight spaces. This Lie algebra acts on the space of conformal blocks (i\.e\., the cohomology of a determinant line bundle on the moduli space) generalising the ``insertion'' and ``deletion'' operations of conformal field theory, and indeed on any cohomology theory. In the particular case of $U(1)$-instantons, which is essentially the subject of this present paper, the construction produces the basic representation after Frenkel-Kac. Then the well known quadratic nature of $ch_2$, $$ch_2 = \frac{1}{2} c_1\cdot c_1 - c_2 $$ becomes precisely the formula for the eigenvalue of the degree operator, i\.e\. the well known quadratic behaviour of affine Lie algebras.

alg-geom

\part{\Cal P} \define\ce{\Cal E} \define\ts{\widetilde{\Sigma}} \define\hilb#1{\widetilde{S^{#1}X}} \define\hilbs#1#2{\widetilde{S^{#1}_{#2}X}} \define\tv#1{\Cal T_{\cv,{#1}}} \define\quot{\frak Q \frak u \frak o \frak t} \define\qbinom#1#2{\thickfracwithdelims[]\thickness0#1#2} \head Introduction \endhead This is the first in a series of papers devoted to describing the action of an affine Lie algebra on the moduli space of instantons on an algebraic surface $X$. This paper, which is only an announcement, is concerned with the ``boundary'' of moduli space; the subsequent papers will describe the action on the interior. We describe the idea briefly. Let $X$ be an algebraic surface, $\mm$ the moduli space of $U(c)$-instantons on $X$ (see below for precise definitions). $\mm$ is not connected; it decomposes into $\mm = \coprod \mm_{c_1,ch_2}$, where $\mm_{c_1,ch_2}$ denotes those instantons with fixed first Chern class equal to $c_1\in H^2(X,\bz)$ and second Chern character equal to $ch_2 \in \bq = H^4(X,\bq)$. Let $\Sigma\subseteq X$ be an algebraic curve. Associated to $\Sigma$ we have various correspondences $ \mm @<<< \Cal P_\Sigma @>>> \mm$. Such a correspondence induces maps $H(\mm) \to H(\mm)$ for any cohomology theory $H$. To describe these individual maps is very complicated. However, these maps satisfy very simple commutation relations, namely those defining a Lie algebra. Hence, shifting our point of view slightly, we see that $H(\mm)$ is a representation of this Lie algebra. This explains the complexity of the individual maps---they are the same as the (known) complexity of describing the action of a Lie algebra on the individual weight spaces of a representation. What then needs to be described is the correspondences, the Lie algebra they generate, and which representations occur. The Lie algebras are {\it affine} Lie algebras, defined by the lattice $H^2(X,\bz)$, or various sublattices such as the lattices of algebraic cycles (Neron-Severi group). These lattices become the ``finite part'' of the weight lattice, with the degree operator taking value in $H^4(X,\bz)$. (The presence of {\it affine} Lie algebras, whose characters are known to be modular forms, is reassuringly consistent with the remarkable work \cite{VW}, which predicts this behaviour because of $S$-duality). The description of the representation will have to wait for a future paper. I hope that it is irreducible, and determined by the K\"ahler cone and Kronheimer-Mrowka basic classes (choice of ``positive roots'' and ``highest weight''). However, the most important invariant of a representation is its central charge, and in this paper we show that this charge is precisely the rank of the instanton (i\.e\. $U(n)$-instantons give rise to level $n$ representations). The most basic example of a correspondence is the ``elementary modifications''; i.e\. given a divisor $ i : \Sigma \hookrightarrow X$ we modify a vector bundle along $\Sigma$, that is consider the correspondence $$ \Cal P_\Sigma^n = \{ 0\to \aaa_1\to\aaa_2\to i_*\Cal E \to 0\mid \aaa_r \in \mm, \Cal E \in Pic^n\Sigma \} $$ where $Pic^n\Sigma$ is the moduli space of holomorphic line bundles on $\Sigma$ with degree $n$. In the case where $X$ is a curve, divisors are points, and the analogous correspondences are precisely the geometric Hecke operators of Drinfeld. Points however cannot interact, whereas curves on surfaces most definitely do---their interaction being precisely described by the lattice $H^2(X,\bz)$. It was the basic observation of Nakajima, in the case of an ALE space $X$, that the interactions of the correspondences are described by the Serre relations. Variants of these correspondences $\Cal P_\Sigma^n$ are the subject of the sequel to this paper. This paper is concerned with a simpler correspondence, which removes a point from an instanton to produce a new one. As a vector bundle modified along a point becomes only a torsion free sheaf, and not a vector bundle, one should think of this correspondence as acting along the ``boundary'' of the moduli space. We find that the algebra generated by this correspondence is essentially an affine Heisenberg Lie algebra (\S3, 5). Obviously, these correspondences are well known in the literature. For example, the elementary modifications along a divisor appear in \cite{MO}, where they stratify $\mm$ by what strongly looks like paths to the highest weight vector of a representation (i\.e\. by the ``crystal'' basis), and notably in \cite{KM} where they are used to impose enough relations on the Donaldson polynomials to determine them in terms of certain basic classes. Even our simple correspondence of inserting a point is a common technical tool; see for example \cite{GL}. Thus our main contribution is to insist that one should study the algebra of these correspondences, and that this is easy. Finally, these correspondences act on {\it any} cohomology theory. In the most intersting case, the cohomology of $\mm$ with coefficents in a determinantal line bundle, one produces the action of an affine algebra on the space of conformal blocks\footnote{This paragraph is the consequence of conversations with Greg Moore; see \cite{LMNS}.}. This space, for which there is now a dimension formula \cite{LMNS} generalising that of Verlinde for curves, has staggering implications for representation theory. Acknowledgements: It should be clear that this paper is inspired by Nakajima's fantastic work \cite{Na}. It is a pleasure to acknowledge helpful conversations with A\. Beilinson, R\. Dijkgraaf, L\. Fastenberg, I\. Frenkel, D\. Gieseker, P\. Kronheimer, Jun Li, A\. Losev, G\. Moore, N\. Nekrasov and S\. Shatashvili; and support both intellectual and moral from I\. Frenkel and G\. Moore, without which this work would not have occurred. Portions of these results were announced at talks at UCLA in November 1994 and UNC-Chapel Hill in April 1995. \head 1. Algebraic Preliminaries \endhead Let $V$ be a complex vector space, $t^{-1}V[t^{-1}] = V \otimes_{\bc}t^{-1}\bc[t^{-1}]$ the associated space of loops at $V$ which vanish at $\infty$. We make this a graded vector space by setting $\text{deg}(v\otimes t^n)= -n $. We write $v_n $ for $v\otimes t^n$. Let $S = S(t^{-1}V[t^{-1}])= \oplus S_n$ be the graded polynomial algebra in infinitely many variables. We make this a Hopf algebra by defining $\Delta v_n = v_n \otimes 1 +1 \otimes v_n$, for $v\in V$. Then $S$ is a free commutative and cocommutative Hopf algebra. Conversely, given such a Hopf algebra $S$, we can reconstruct $V\otimes t^{-n}$ uniquely as the space of primitive elements of degree $n$. Now suppose we are given a graded symmetric bilinear form $(,)$ on $S$ such that multiplication and comultiplication are adjoint, i\.e\. such that $(S_n, S_m) = 0$ if $n\neq m$, and $(xy,z)= (x\otimes y,\Delta z)$ (where $S\otimes S$ inherits a bilinear form by $(x\otimes y,a\otimes b)=(x,a)(y,b)$). Such a form is completely specified by its values on the primitive elements, i\.e\. by the values $(\alpha^i_n,\alpha^j_n)$, where ${\alpha^i}$ runs through a basis of $V$. Given a non-degenerate such form, we can define the action of $tV[t] = V\otimes t\bc[t]$ on $S$, by defining $v_n$ to be the adjoint of $v_{-n}$ for $n>0$, i\.e\. $$(v_nx,y)=(x,v_{-n}y), \qquad n>0.$$ As $\Delta$ is an algebra homomorphism, it follows that $v_n$ acts as a derivation on $S$, $$\multline (v_n(xy),z) = (xy,v_{-n}z)= (x\otimes y,\Delta(v_{-n}z)) = (x\otimes y,(v_{-n}\otimes 1 + 1 \otimes v_{-n})\Delta z) \\ = (v_nx\otimes y + x\otimes v_ny, \Delta z) = (v_n(x)y + xv_n(y),z) \endmultline$$ and hence that the Heisenberg Lie algebra $(\oplus_{n\neq 0}V\otimes t^{n}) \oplus \bc$ acts on $S$, where $[v_n,v_m] = \delta_{n,-m}. (v_n,v_n)$ if $n > 0$. Given $v\in V$, we define new elements $h^v_n \in S_n$ by $$ H^v(t)= \sum_{n\geq 0} h^v_nt^n = \exp\left(\sum_{n\geq 1} v_n t^n/n\right).$$ Then if the elements $\alpha^i$ form a basis of $V$, it is well known that the elements $h^{\alpha^i}_n$ are algebraically independent and generate $S$. As $t^{-1}V[t^{-1}]$ consists of primitive elements, the $H^v(t)$ are `group-like', i\.e\. $$ \Delta h_n^v = \sum_{0\leq a \leq n} h^v_a \otimes h^v_{n-a} $$ and the inner product is given by $$ \sum_{n,m \geq 0} (h^v_n,h^w_m) t^ns^m = \exp\left(\sum_{n\geq 1} (v_n,w_n)/n \cdot (ts)^n/n\right). $$ This infinite family of Heisenberg Lie algebras just constructed is still rather flabby; however inside this space of algebras (parametrised by maps from $\bz_+$ to non-degenerate quadratic forms on $V$) there are certain remarkable families with much larger symmetries; namely the vertex algebras \cite{B,FLM} and $q$-vertex algebras \cite{FJ}. We suppose given a lattice $L$ with non-degenerate symmetric even bilinear form, i\.e\. $(\alpha,\alpha)\in 2\bz$ for $\alpha \in L$, and put $V= L\otimes_\bz\bc$. Define $(,)$ on $t^{-1}V[t^{-1}]$ by $$ (v_n,w_m) = n (v,w) \delta_{n,m}. $$ With this inner product, $$ \sum_{n,m \geq 0} (h^v_n,h^w_m) t^ns^m = (1-ts)^{-(v,w)}. $$ We call $S$ the {\it Fock space modeled on the lattice L}. We also suppose given a two-cocycle $\epsilon : L \times L \to \bz/2\bz$, and define the group algebra of $L$ twisted by $\epsilon$, $\bc\{L\}$, as in \cite{FK,FLM}. Define $\Cal F = S \otimes_\bc \bc\{L\}$, a $\bz_+ \times L$ graded vector space which carries an action of both the Heisenberg Lie algebra $(\oplus_{n\neq 0}V\otimes t^{n}) \oplus \bc$ and $\bc\{L\}$. Let $V$ act on $\bc\{L\}$ by $v\cdot e^\lambda = (v,\lambda) e^\lambda$, where $\lambda \in L$, $e^\lambda$ denotes the corresponding element of the group algebra, and $v\in V$. Then in these circumstances we have the well known result that a far larger algebra acts on $\Cal F$, namely \proclaim{Theorem \cite{FLM,B}} $\Cal F$ is a vertex algebra, and if $L$ is positive definite, a vertex operator algebra \footnote{ If $L$ is $\bz/2$-graded we may also make all these definitions, as long as we work in the $\bz/2$-graded category. So $S$ is the free super-commutative algebra on $t^{-1}V[t^{-1}]$, i\.e\. a tensor product of an exterior algebra and a polynomial algebra, $\Cal F$ is a super-vertex algebra, etc.} \endproclaim For example, in the particular case where $L$ is positive definite and spanned by the roots $ \Delta = \{\alpha \in L \mid (\alpha,\alpha) =2 \} $, then the space of vectors of conformal weight $1$ is isomorphic to the simple Lie algebra $\lieg$ with roots $\Delta$, and $\Cal F$ is the basic representation of $\ghat$ \cite{FK}. If $L$ is of arbitrary signature things become much more complicated. $$ $$ At present, there is apparantly no general definition for a quantum vertex algebra. But if $L$ is positive definite and spanned by the roots, then we can follow \cite{FJ} and define, for $c\in \bn$ $$ (\alpha^i_n,\alpha^j_m) = n\delta_{nm} [nc(\alpha^i,\alpha^j)]/[n] $$ where $[n] = \frac{q^n-q^{-n}}{q-q^{-1}}$, and $\alpha^1,\dots,\alpha^l$ are a basis of simple roots. (There is an additional choice here, that of a positive cone in $V$). When $c=1$ this gives $\Cal F$ the structure of the basic representation for $\uqghat$ \cite{FJ}. This $q$ arises in our situation when there is some ``weight'' structure on the cohomology theory, for example a $\cstar$-action on $X$. This does occur for ALE spaces (see \cite{N, Gr1}) but we will stick to $q=1$ for the present paper. \head 2. Motivic algebras \endhead We write this section with the minimal generality needed for this paper. Suppose $X$ and $Y$ are two smooth proper varieties, and $Z\subseteq X\times Y$ is a subvariety, i\.e\. a correspondence between $X$ and $Y$. We write this $X @<<< Z @>>> Y$, and $\pi_X$, $\pi_Y$ for the two projections from $X\times Y$ to $X$ or $Y$. Then if $H$ is any ``reasonable'' cohomology theory we obtain honest maps $$ R : H(X) \to H(Y), \qquad \bar{R} : H(Y) \to H(X) $$ which are adjoint with respect to the natural inner product on $H(X)$ and $H(Y)$; $(Ra,b) = (a,\bar{R}b)$. Here we define $R(a) = (\pi_Y)_*(\pi_X^*a \cdot [Z])$, $\bar{R}(b) = (\pi_X)_*([Z]\cdot \pi_Y^*b)$, and $(a,a') = \int_* a\cdot a'$, for $a,a' \in H(X)$, $b\in H(Y)$ and where for any space $X$, $\int : X \to pt$ denotes the projection to a point, and $[Z]$ denotes the class of $Z$ in $H(X\times Y)$. We mention some reasonable cohomology theories: i) Usual homology or cohomology $H^*$; topological $K$-theory,..., cobordism, all with (say) complex coefficients. ii) Write $\Cal F(X)$ for the ring of constructible functions from $X$ to $\bc$. If $\pi:X\to Y$ is a map, $f\in \Cal F(Y), g\in \Cal F(X)$, define $(\pi^*f)(x)= f(\pi(x))$ and $(\pi_*g)(y) = \sum_{a\in\bc}a\chi(\pi^{-1}(y)\cap g^{-1}(a))$, where $\chi$ denotes the Euler characteristic of cohomology with compact supports, and define $[Z]$, for $Z\subseteq X$ a subvariety, as the characteristic function of $Z$: $[Z](x) = 1$ if $x\in Z$, and $[Z](x) = 0$ otherwise. iii) If $\dim Z = (\dim X + \dim Y)/2$, then $$ H^{\half \dim X}(X) \overset R \to{\underset {\bar{R}} \to \rightleftarrows } H^{\half \dim Y}(Y)$$ where $H^*$ is the usual cohomology. If $X$ is compact K\"ahler (respectively complex algebraic or symplectic) we can consider the subspace of $H^{\half\dim X}(X)$ spanned by the $(p,p)$-classes (respectively, algebraic or Lagrangian cycles). Denote any of these subspaces $\hlamb(X)$. As long as $Z$ is algebraic or Lagrangian as appropriate, the topological $R$, $\bar{R}$ preserve these subspaces and the theories $\hlamb(X)$ are ``reasonable'' cohomology theories. $$ $$ As part of our definition of reasonable we require that the Kunneth map $H(X)\otimes H(Y) \to H(X\times Y)$, $a\otimes b \mapsto \pi_X^*a \cdot \pi_Y^*b$ is an isomorphism. Here, if $H$ is $\bz/2$-graded then we take $\otimes$ in the $\bz/2$-graded sense also (as in example (i)). One can continue this list of theories as one pleases \cite{JKS}. The above theories all take values in vector spaces, as our goal is to produce representations of algebras, but if the correspondences act non-trivially on the entire motives of $X$ and $Y$ one should also consider functors which do not factor through cycles homologically equivalent to zero. An example not on this list, but which I hope to return to, is homology of $X$ with coefficients in a given sheaf. In our case below, the sheaf should be taken to be a determinant line bundle on the moduli space of torsion free sheaves, so that its cohomology is the space of conformal blocks. Then the geometric Hecke operators (quantum group symmetries) we produce act on the representation theory of the double loop groups in some as yet unknown way. \head 3. Hilbert Schemes \endhead We recall some well known facts about the Hilbert scheme. Let $X$ be a smooth algebraic surface, $S^nX = X^n/S_n$ the $n$'th symmetric power of $X$. For $n>1$, $S^nX$ is singular. Write $\hilb{n}$ for the Hilbert scheme of $X$, i\.e\. for the variety parameterising closed zero dimensional subschemes of $X$ of length $n$, and let $\pi:\hilb{n}\to S^nX$ be the canonical morphism sending a subscheme to its support. Then $\hilb{n}$ actually exists as a separated variety; it is smooth of dimension $2n$ \cite{Gro,Fo}, $\pi$ is proper and produces a desingularisation of $S^nX$. If $X$ is symplectic then so is $\hilb{n}$, if $X$ is hyper-K\"ahler than $\hilb{n}$ is the hyper-K\"ahler resolution of the stack (orbifold) $S^nX$. We phrase everything below in terms of the variety $\hilb{n}$, but it is often much better to work directly with the smooth stack $S^nX$. If $x\mapsto nx$ denotes the diagonal map $X\to X^n \to S^nX$, then $\pi^{-1}(nx)$ is irreducible, and of dimension $n-1$ \cite{Br}. This fact was used in \cite{GS} to compute the Hodge numbers of $\hilb{n}$ in terms of those of $X$; a description of the Euler numbers which is in the spirit of this paper can be found in \cite{VW}. Let $\part_n$ denote the set of partitions of $n$. If $\alpha = (1^{\alpha_1}2^{\alpha_2}\cdots) \in\part_n$, so $\sum_i i\alpha_i = n$, write $\ell(\alpha) = \sum_i\alpha_i$. We have an obvious stratification of $S^nX$ by $\part_n$; the strata $S^n_\alpha X$ has complex dimension $2\ell(\alpha)$. Write $\hilbs{n}{\alpha}$ for $\pi^{-1}(S^n_\alpha X)$. Then $\aaa \in \hilbs{n}{\alpha}$ if $\aaa $ is isomorphic to a direct sum $\oplus\aaa_{i,r}$, where $1\leq r \leq \alpha_i$, each $\aaa_{i,r} \in \hilb{i}$ has support a single point with multiplicity $i$, $\pi(\aaa_{i,r}) = i\gamma_{i,r}$, and the points $\gamma_{i,r}$ are distinct. We have $\dim \hilbs{n}{\alpha} = n + \ell(\alpha)$. The open strata $S^n_{(1^n)}$, $\hilbs{n}{(1^n)}$ we also denote $(S^nX)^0$, $(\hilb{n})^0$; $\pi$ restricted to $(\hilb{n})^0$ is an isomorphism. $$ $$ Define $$ \multline \Lambda^0 = \Lambda^0_{ab} = \{ (\aaa_1,\aaa_2,\aaa_3)\in \hilb{a}\times\hilb{a+b}\times\hilb{b}\mid \\ \aaa_2 \in (\hilb{a+b})^0, \text{ and there is an exact sequence } 0\to\aaa_1\to\aaa_2\to\aaa_3\to 0\} \endmultline$$ and define $\Lambda$ to be the closure of $\Lambda^0$ in $\hilb{a}\times\hilb{a+b}\times\hilb{b}$. Observe that i) If $(\aaa_1,\aaa_2,\aaa_3) \in \Lambda^0$, then $\aaa_1, \aaa_2$ and $\aaa_3$ are all in the open stratum of their Hilbert schemes. ii) We have $(\aaa_1,\aaa_2,\aaa_3) \in \Lambda_{ab}$ if and only if $(\aaa_3,\aaa_2,\aaa_1) \in \Lambda_{ba}$. iii) Writing $+ : S^aX\times S^bX \to S^{a+b}X$ for the obvious morphism, we have $\pi(\aaa_2) = \pi(\aaa_1)+\pi(\aaa_3)$, if $(\aaa_1,\aaa_2,\aaa_3) \in \Lambda$. In fact $\Lambda$ is just the correspondence of varieties produced by the ``obvious'' correspondence of stacks $$ \Lambda^{\text{stack}} =\{ (\aaa_1,\aaa_2,\aaa_3) \mid \aaa_2 = \aaa_1 + \aaa_3 \}. $$ This is the basic motivic object, from which everything else follows. iv) The dimension of $\Lambda$ is $2(a+b)$, i.e. half the dimension of the ambient space. In fact, \proclaim{Lemma 1} If $X$ is symplectic, then $\Lambda$ is Lagrangian (where we change the sign of the symplectic form on $\hilb{a+b}$ in $\hilb{a}\times\hilb{a+b}\times\hilb{b}$ as is usual). \endproclaim Now, let $H$ be a reasonable cohomology theory as in \S2. Write $$ S = \oplus_{n\geq 0} H(\hilb{n}). $$ Define, for $x\in \hilb{a}, y\in\hilb{b}$, the product of $x$ and $y$, $$xy= (\pi_{a+b})_*((\pi_a,\pi_b)^*(x\otimes y) \cdot [\Lambda]),$$ and for $z \in \hilb{a+b}$, define $$\Delta_{ab}z = (\pi_a,\pi_b)_*([\Lambda]\cdot \pi_{a+b}^*z) \in H(\hilb{a})\otimes H(\hilb{b})$$ and $\Delta = \sum_{a+b =n }\Delta_{ab}$. Also define a non-degenerate inner product $(,): H(\hilb{n}) \times H(\hilb{m})\to H(pt)$ by $(x,y) = \delta_{nm}\int_*x\cdot y$. \proclaim{Theorem 2} Equipped with this multiplication and comultiplication, $S$ is a commutative and cocommutative Hopf algebra. In other words, multplication and comultipication are associative, adjoint with respect to the inner product, and (graded) commutative (here, if $H$ is $\bz/2$-graded, so is each $S_n$). $\Delta$ is an algebra homomorphism. \endproclaim The only statement that requires proof is that $\Delta$ is an algebra homomorphism; the rest comes free with the formalism. This is easiest proved by using the stacks $\hilb{n}$ and the obvious correspondendence between them; then if the cohomology theory is the ``orbifold cohomology'' (which here is essentially $K$-theory of $X^n$, equivariant with respect to the symmetric group) this induces our correspondence in the homology of $\hilb{n}$. In fact, in this form the theorem makes sense for {\it any} variety $X$ (of any dimension), and the orbifold cohomology\footnote{i.e\. let $\Cal F = \oplus_n K^{S_n}(X^n,\bc)$, for $X$ any variety. Then the theorem is that $\Cal F$ is a Fock space modeled on $H^*(X,\bc)$, with multiplication and comultiplication defined by the obvious correspondence. Like everything else stated here, the proof of this fact will appear in the longer version of this paper.}; the remarkable thing about surfaces is the additional intepretation in terms of honest seperated smooth varieties. $$ $$ We would now like to describe generators for $S$, and identify $S$ with a Fock space as in \S1. $$ $$ Let $\Sigma\subseteq X$ be a curve, $S^n\Sigma$ its $n$'th symmetric power. If $\Sigma$ is algebraic, we can canonically identify $S^n\Sigma$ with the Hilbert scheme of zero dimensional subschemes of $\Sigma$ of length $n$, so we can also regard $S^n\Sigma$ as contained in $\hilb{n}$. It is a smooth subvariety, and Lagrangian if $X$ is symplectic. Remarkably, the following generalisation appears to be new: Write $\ts$ for the subspace of points $x\in \hilb{n}$ such that $\pi(x)\in S^n\Sigma$, and $\ts^0_\lambda$ for $\ts\cap \hilbs{n}{\lambda}$, where $\lambda\in \part_n$, and $\hilbs{n}{\lambda}$ is the piece of stratification defined above. Explicitly, $\aaa \in \hilb{n}$ is in $\ts^0_\lambda$ if $\lambda=(1^{\alpha_1}2^{\alpha_2}\cdots)$, and $\aaa $ is isomorphic to a direct sum $\oplus\aaa_{i,r}$, where $1\leq r \leq \alpha_i$, each $\aaa_{i,r} \in \hilb{i}$ has support a single point with multiplicity $i$, $\pi(\aaa_{i,r}) = i\gamma_{i,r}$, and the points $\gamma_{i,r}$ are distinct. Write $\ts_\lambda$ for the closure of $\ts_\lambda^0$ in $\hilb{n}$. \proclaim{Proposition 3} i) $\ts$ has pure dimension $n$, $\ts^0_\lambda$ has pure dimension $n$, and the $\ts_\lambda$, $\lambda\in \part_n$, are precisely the irreducible components of $\ts$. ii) If $X$ is symplectic, then $\ts$ is a Lagrangian submanifold. \endproclaim \comment \demo{Proof} As the $\hilbs{n}{\lambda}$ stratify $\hilb{n}$, the $\ts^0_\lambda$ partition $\ts$. Each $\ts^0_\lambda$ is fibred over $S^n\Sigma \cap \hilbs{n}{\lambda}$ which has dimension $\ell(\lambda)$. The fibres have dimension $\sum\alpha_i(i-1)= n - \ell(\lambda)$, if $\lambda=(1^{\alpha_1}2^{\alpha_2}\cdots)$. Whence (i). As for (ii), observe $\Lambda$ is Lagrangian, as are curves $\Sigma \subseteq X$; that multiplication as defined above (via the correspondences) sends Lagrangian subvarieties to Lagrangian subvarieties, and that $\ts$ is just $\Sigma\cdots\Sigma$ ($n$ times). (ii) follows. \enddemo \endcomment See \S6 below for more remarks on $\ts$. Write $h_n^\Sigma$ for the class of $S^n\Sigma \hookrightarrow \hilb{n}$ inside $S$. (Note that as a consequence of the proposition, we can define this class even if $\Sigma$ is not algebraic, though we do not need to.) We adopt the convention that $h_0^\Sigma = 1$, for all $\Sigma$. \proclaim{Proposition 4} i) The elements $h^\Sigma_n$ are group-like, i.e. $$ \Delta h^\Sigma_n = \sum_{a+b=n} h^\Sigma_a\otimes h^\Sigma_b.$$ ii) Let $\Sigma$, $\Sigma'$ be two curves in $X$. Then $$ \sum_{n,m \geq 0} (h^\Sigma_n,h^{\Sigma'}_m) t^ns^m = (1-ts)^{-(\Sigma,\Sigma')} $$ where $(\Sigma,\Sigma')$ denotes the inner product in $H(X)$, as usual. \endproclaim Now, for $n \geq 0$ define $\Gamma^X = \Gamma^X_n = \hilbs{n}{(n)}$, a closed irreducible subvariety of $\hilb{n}$. So $\Gamma^X=\{\aaa\in\hilb{n} \mid \pi(\aaa) = nx,\text{ for some }x\in X\} = X \times_{S^nX} \hilb{n}$. Write $\pi:\Gamma^X\to X$, and define for any submanifold $Z\subseteq X$, $\Gamma^Z = \Gamma^Z_n \subseteq \Gamma^X_n$ by $\Gamma^Z = \pi^{-1}(Z)$, i\.e\. so that the diagram $$\CD \Gamma^Z @>>> \Gamma^X @>>> \hilb{n} \\ @V{\pi}VV @V{\pi}VV @V{\pi}VV \\ Z @>>> X @>{n}>> S^nX \endCD $$ is Cartesian. In particular, if $\Sigma \subseteq X$ is an algebraic curve, $\Gamma^\Sigma = \ts_{(n)}$ in the notation above. Also write $r^Z_n$ for the class of $\Gamma^Z_n$ in $H(\hilb{n})$. We then have \proclaim{Lemma 5} There exists a function $f : \bz\to\bz$, and polynomials in infinitely many variables $p^n_a(x_1,x_2,\dots), q^n_b(x_1,x_2,\dots)$ such that for any submanifolds $Z,Z'$ of $X$, $$\align & \text{i) } (r^Z_n,r^{Z'}_n) = (Z,Z')f(n), \text{ and } \\ & \text{ii) } \Delta r^Z_n = \sum_{a+b=n} p^n_a(r^Z_1,r^Z_2,\dots) \otimes q^n_b(r^Z_1,r^Z_2,\dots) \endalign $$ \endproclaim Now observe that for a fixed algebraic curve $\Sigma$, the algbera generated by the $r^\Sigma_n$, $n\in\bz_+$, is the same as the algebra generated by the $h^\Sigma_n$, $n\in\bz_+$. As a consequence of lemma 5, we can write $h^\Sigma_n$ as a polynomial in the $r^\Sigma_i$ with ``universal'' coefficients; i.e. coefficients independent of $\Sigma$. This allows us to define $h^Z_n$, for any $Z\subseteq X$ by the same formulae. It then follows from lemma 5 and proposition 4 that the $h^Z_n$ are also group like, and satisfy the same inner product formulae as in proposition 4. Take $H(X) = H^*(X)$ (usual cohomology), so $S = \oplus H^*(\hilb{n})$, and let $Z_i$, $i=1,\dots,l$ run through submanifolds of $X$ such that the classes $[Z_i]$ form a basis in $H^*(X,\bc)$. \proclaim{Proposition 6} The elements $r^{Z_i}_n$ freely generate $S$ as an algebra. \endproclaim Now let $X$ be projective, so that the lattice $H^*(X,\bz)/\text{torsion}$ is a non-degenerate lattice with respect to the form $(,)$. We have thus proved \proclaim{Theorem 7} $S$ forms a Fock space modeled on the lattice $H^*(X,\bz)/\text{torsion}$. \endproclaim If $X$ is affine, one easily shows that if we take $H(X) = H_{\half\dim X}(X,\bc)$ to be middle dimensional Borel-Moore homology, that the lattice $L = H_{\half\dim X}(X,\bz)$ is non-degenerate and $S$ (for this cohomology theory) is a Fock space modeled on $L$. This is precisely the case that occurs for $X$ an ALE space. For smooth projective $X$, let us agree to write $S^\Lambda = \oplus\hlamb(\hilb{n})$, where $\hlamb$ as in \S2 denotes either toplogical, $(n,n)$, holomorphic or Lagrangian middle dimensional cycles, and let us still write $S = \oplus H^*(\hilb{n})$. Clearly $S^\Lambda$ is a Hopf subalgebra of $S$, and we can describe it explicitly using the above theorem. For example, consider the Hodge decomposition of $H^*(\hilb{n})$; write $S^{ab}_n=H^{a,b}(\hilb{n})$. Then multiplication in $S$ preserves all this grading: $S^{ab}_n\cdot S^{cd}_m \subseteq S^{a+c,b+d}_{n+m}$. The generators $r_n^Z\in S_n$ have degree $(n-1,n-1) + \deg Z$, i\.e\. if $[Z]\in H^{p,q}(X,\bc)$ then $r_n^Z\in S_n^{n+p-1,n+q-1}$. This gives another proof of \cite{Got,3.1}, independent of \cite{GS}. Observe that $S^\Lambda$ is usually not a Fock space modeled on a lattice. For example, if the odd cohomology of $X$ vanishes, the generating function $\dim S^{n,n}_n z^n$ is the coefficent of $u^0$ in $$ \prod_{n\geq 1} \big((1-z^nu)(1-z^nu^{-1})\big)^{-h^{2,0}}(1-z^n)^{-h^{1,1}}$$ which is not the generating function of a Fock space. \head 4. Vertex algebras and $U(1)$-instantons \endhead Let $X$ be a smooth projective surface. A torsion free sheaf $\ce$ is a coherent sheaf of $\CO_X$-modules which is torsion free as a $\CO_X$-module. If $\ce^*$ denotes the dual of $\ce$, we have a canonical exact sequence $ 0 \to \ce\to\ce^{**}\to \Cal Q\to 0$, where $\Cal Q$ is coherent of finite length, and $\ce^{**}$ is locally free. We say $\ce$ has rank $c$ if $\ce^{**}$ does. Let $H$ be a fixed very ample divisor on $X$, and $\mm$ (resp\. $\mmb$) the space of $H$-stable rank $c$ torsion free sheaves (resp\. $H$-semistable torsion free sheaves, modulo the usual equivalence relation \cite{Gi}). Then $\mmb$ is a projective variety, and $\mm \subseteq \mmb$ an open subvariety \cite{Gi}. Clearly $\mmb = \coprod_{c_1,k} \mmb_{c_1,k}$, where $\mmb_{c_1,k}$ consists of torsion free sheaves $\ce$ with Chern classes $c_1(\ce) = c_1, ch_2(\ce)=k$. Also, $c_1(\ce) $ lands in the lattice of algebraic cycles $c_1(\ce) \in \hlamb(X,\bz)$. In the sequel we will be concerned with elementary modifications, and hence the interior of moduli space. For now, as we are only concerned with the boundary of moduli space, let us suppose $c=1$. Further, for simplicity, suppose $\pi_1(X)=0$. Then $H^1(X,\bz)=H^3(X,\bz)=0$, and $H^2(X,\bz)$ is torsion free. Suppose also that $X$ is spin, so $H^2(X,\bz)$ is even. Now, in this case line bundles have no moduli, and so the space of torsion free sheaves is isomorphic to $$ \oplus_{\lambda\in \hlamb (X,\bz)} (\oplus_n \hilb{n}) \otimes e^\lambda.$$ Observe that the moduli of torsion free sheaves $\ce$ such that $\ce^{**}$ is isomorphic to $\Cal L$ is canonically isomorphic to those with $\ce^{**}$ isomorphic to $\Cal L'$; the map is just $\ce \mapsto \ce\otimes (\Cal L'\otimes \Cal L^{-1})$. This defines the action of the lattice $\hlamb(X,\bz)$ on the moduli space of rank 1 torsion free sheaves. (If $X$ is not simply connected, we must replace this action by elementary modifications; this will be explained in the sequel). Let us write $L = H^*(X,\bz)$, and $ \mm' = (\oplus_n \hilb{n})\times L$, $$\Cal F = \oplus H^*(\hilb{n},\bc)\otimes \bc\{L\} = S \otimes \bc\{L\}.$$ Then $\mm'$ is ``almost'' the moduli space of topological $U(1)$-instantons; i.e\. the hyper-K\"ahler resolution of the ideal ASD-connections on toplogical $U(1)$-bundles. (It would be interesting to give a purely algebraic construction of this space). It differs from this by the term $H^0(X,\bz) + H^4(X,\bz)$, which I do not know how to intepret geometrically. Yet. In any case, the results of \S3 tell us \proclaim{Theorem} $\Cal F$ is a vertex algebra. \endproclaim \noindent and that if $X$ is an ALE space, so $L=H^2(X,\bz)$ is negative definite, that the analogously defined $\Cal F$ (which is now precisely the homology of rank 1 torsion free sheaves) is just the basic representation of $\ghat$, where $\ghat$ is the affine Lie algebra associated to $L$. \indent Also, we remark that the theorem is true for an arbitrary compact K\"ahler 4-manifold, where we use the orbifold cohomology. \head 5. Torsion sheaves of rank $c$ produce central charge $c$ \endhead Let $\cv$ be a vector bundle of rank $c$ on $X$, and let $\tv{n}$ consist of the stack of torsion free sheaves $\ce$ such that $\ce^{**}$ is isomorphic to $\cv$. Then if $\cv$ is stable (resp. semistable), so is any $\ce\in\tv{n}$ (though not conversely), and in that case $\tv{n}$ is a smooth separated variety. Define $\cq_\cv = \oplus_n H(\tv{n})$, where $H$ is any reasonable cohomology theory. One may handle $\cq_\cv$ as one handles $S$ in \S3, by defining an action of $S$ on $\cq_\cv$, $S\otimes \cq_\cv @>m_\cv>> \cq_\cv$ induced by the correspondence $$ \multline \Lambda^0 =\{ (\aaa_1,\aaa_2,\aaa_3) \in \tv{a}\times\tv{a+b} \times \hilb{b} \mid \\ \aaa_2 \in \tv{a+b}^0, \text{ and there is an exact sequence } 0\to\aaa_1\to\aaa_2\to\aaa_3\to 0\} \endmultline$$ and $\Lambda $ is the closure of $\Lambda^0$. Here, $\tv{n}^0$ consists of those torsion free sheaves $\ce$ which fit into an exact sequence $\ce\to\cv\to \Cal Q$, where $\Cal Q \in (\hilb{n})^0$. Then one may proceed exactly as in \S3, and show \proclaim{Theorem} $\cq_\cv$ is a module for the Heisenberg Lie algebra $ (\oplus_{n\neq 0} H^*(X,\bc) \otimes t^n) \oplus \bc$ with central charge $c$. \endproclaim It is pleasant to calculate $$[h^\Sigma_1,h^{\Sigma`}_{-1}] = c(\Sigma,\Sigma')$$ directly. This follows directly from the easy fact that if $\ce$ is any torsion free sheaf of generic rank $c$, and $\bc_0$ denotes the skyscraper sheaf at a point $0\in X$, then \proclaim{Lemma} $\dim Ext^1(\bc_0,\ce) + c = \dim Hom(\ce,\bc_0)$. \endproclaim In the particular case that $c=1$, this may be intepreted as (and in fact follows from) the fact that there is one way more to add a square to a partition than to remove a square. (As one may complete $X$ at 0, to get $\bc[[x,y]]$. This admits a $\cstar$-action, such that $x^iy^j$ have distinct weights $i+j \leq n$ . Then the fixpoints of this $\cstar$ action on the Hilbert scheme of length $n$ subschemes are just in 1-1 correspondence with partitions of $n$, and this punctual Hilbert scheme partitions into vector bundles over these isolated fixpoints. This makes the lemma obvious in this case, and gives yet another reason why the Hodge theory of $H^*(\hilb{n})$ is so simple). \head 6. Remarks on Curves \endhead Suppose $\Sigma \subseteq X$ is an algebraic curve. Replace $X$ with the normal bundle to $\Sigma$ in $X$, so that $X$ admits a contracting $\cstar$ action with fixpoints $\Sigma$, that is if $x\in X$, $\lim_{t\to 0} t\cdot x$ exists and is in $\Sigma$. This $\cstar$ action induces one on $\hilb{n}$, and we define $$ \Cal U = \{ \aaa\in\hilb{n} \mid \lim_{t\to 0} t\cdot x \in S^n\Sigma \} $$ where $S^n\Sigma\hookrightarrow \hilb{n}$ as in \S3. \proclaim{Proposition} i) $\Cal U$ is open in $\hilb{n}$, and $\Cal U$ is a rank $n$ vector bundle on $S^n\Sigma$. ii) Suppose $X= T^*\Sigma$. Then $\Cal U$ canonically identifies with $T^*(S^n\Sigma)$. Under this identification, the Lagrangian subvariety $\ts \cap \Cal U$ identifies with Laumon's global nilpotent cone, a Lagrangian subvariety in $T^*(S^n\Sigma)$ \cite{La}. \endproclaim Because of this proposition, a perverse sheaf on $S^n\Sigma$ with nilpotent characteristic variety (for example, conjecturally any automorphic sheaf) gives rise to a cycle in $\hlamb(\hilb{n})$ via the characteristic cycle map. Its also worth remarking that if $\Sigma$ is the affine line, $S^n\Sigma$ canonically identifies with the variety of {\it regular} conjugacy classes in $\frak g\frak l_n$ (via the characteristic polynomial). Thus $S^n\Sigma$, for $\Sigma$ a curve of genus $g$, which classically \footnote{{`classically' here means after the work of Drinfeld.}} one regards as a genus $g$ generalisation of regular conjugacy classes, is here generalised `microlocally' to produce two dimensional analogues of conjugacy classes. (As the curve $\Sigma$ varies, this really does feel two dimensional). Note that we do {\it not} want to consider the stack of coherent sheaves of length $n$ here (the analogue of the stack of all conjugacy classes in $\frak g\frak l_n$), as in our case the central charge would vary: $\quot_{V,r}$ has central charge the rank of $V$, where $V$ is a vector bundle on $X$. \head 7. Remarks on Nakajima's quiver varieties and \cite{Gr1} \endhead Let $\frak g$ be a Kac-Moody Lie algebra, with symmetric Cartan matrix. In \cite{L1}, Lusztig defined a variety, the moduli space of representations of the quiver associated to $\frak g$, and a Lagrangian subvariety $\Lambda$ such that the middle dimensional cycles on $\Lambda$ ($\hlamb(\Lambda)$ in the notation of \S2) realises the universal Verma module for $\frak g$. In \cite{Na} Nakajima constructed a modified quiver variety $\na(w)$, depending on a highest weight $w$ and a $\xi\in \frak h$, where $\frak h$ is the ``real'' Cartan subalgebra of $\lieg$, with the following properties: i) If $\xi$ is generic, then $\hlamb(\na(w))$ realises the irreducible integrable highest weight module with highest weight $w$, and the Chevalley generators of $\lieg $ act on $\na(w)$ by corrspondences. ii) If $\lieg$ is of affine type, and $tr\,\xi=0$, then $\na(w)$ is the moduli of $U(n)$-instantons on the ALE space $X_\xi$, with monodromy at $\infty$ determined by $w$. (We refer to \cite{KN,N} for all these terms). In other words, in case (i) Nakajima ``cuts down'' a Verma module to get an irreducible highest weight module. Unfortunately, (i) and (ii) cannot occur simultaneously\footnote{{This was explained to me by Greg Moore.}}. For example, $\na(0)$ is a point if $\xi$ is generic, but if $\xi$ is generic trace free it is the Hilbert scheme on $X_\xi$. So, if we care about the moduli of instantons on an ALE space, we must do some extra work from \cite{Na}. Obviously, this is the content of this paper, which complements \cite{Na} even in the case of an ALE space. (Using \cite{Lu2,Na} one can obtain the results above in the quiver language directly. This will appear in \cite{Gr1}). The point of this series of papers is to use elementary representation theory to obtain information about the moduli space of instantons. In the quiver variety case (for $\xi$ generic), one may reverse this, and use the geometry of quiver varieties to obtain new information about quantum affine algebras. Specifically, let $\lieg$ be a Kac-Moody algebra, and $\uqghat$ the associated quantum affine algebra at central charge 0 \cite{Dr,Gr2}. If $\frak g$ is finite dimensional, we can consider the category of finite dimensional representations of $\uqghat$; for the definition for general $\lieg $ see \cite{Gr1} (these representations have the property that they restrict to a direct sum of integrable highest weight representations of $U_q\lieg\hookrightarrow \uqghat$). Then, as was discovered by Drinfeld, these representations are {\it not} deformations of the analogous representations of $\ghat$ (the ``evaluation representations'' and their tensor products); smaller terms must be added. The reason for this is that the varieties $\na(w)$ are not zero dimensional; i.e. $H^*(\na(w))$ is $\hlamb(\na(w))$ plus smaller terms. Geometrically, we take as our reasonable cohomology theory $K^{GL_W\times\cstar}(\na(w))$, which takes full account of the geometric symmetries of $\na(w)$. Then in \cite{Gr1} it is proved that $K^{GL_W\times\cstar}(\na(w))$, admits an action of $\uqghat$ (see also \cite{Gr2}). This explains the occurance of the middle homology in \cite{Na}. Also, to continue the advertisement of \cite{Gr1}, we construct all of $\uqghat$. Namely, we take equivariant cohomology of the Lagrangian subvariety of $\na(w)\times\na(w)$ consisiting of pairs with the same moment map image. This constructs a piece of $\uqghat$ (and the same variety was independantly discovered by Nakajima in \cite{Na2}, where he used it to construct a piece of the enveloping algebra of $\lieg$). These pieces fit togethor via the coproduct---write $w=w'+w''$. This defines a $\cstar$ action on $\na(w)$, with fixpoints of the form $\na(w')\times\na(w'')$. Then one may define a coproduct via localisation to the fixpoints, and this coproduct fits these pieces togethor to produce $\uqghat$. This coproduct, at the $q=1$ non-affine level, realises the map $L_{w'}\otimes L_{w''} \to L_{w'+w''}$, where $L_w$ is the irreducible highest weight module for $\lieg$ with highest weight $w$. Finally, we describe the irreducible modules (even at roots of unity) in terms of certain perverse sheaves with nilpotent characteristc variety on Nakajima's moduli space $\Cal M_0(w)$. In the case $\frak g =\frak g\frak l_n$, this construction of the algebra is due to Ginzburg and Vasserot \cite{GV}; the moduli space is due to Beilinson-Lusztig-MacPherson \cite{BLM}; and the coproduct appeared in \cite{Gr3}. That such a geometric picture of the representation theory of $\uqghat$ should exist was conjectured by Drinfeld, on the basis of Kazhdan and Lusztig's description of the representation theory of affine Hecke algebras, which is similar to this. This was explained to me by G\. Lusztig, in 1991. \Refs \widestnumber\key{LMNS} \ref\key BLM \by A. A. Beilinson, G. Lusztig and R. MacPherson \paper A geometric setting for the quantum deformation of $GL_n$ \jour Duke Math. J.\vol 62\yr 1990 \pages 655-677\endref \ref\key B\by R. Borcherds\paper Vertex algebras, Kac-Moody algebras and the monster \jour Proc. Natl. Acad. Sci. \yr 1986 \vol 83 \pages 3068--3071 \endref \ref\key Br \by J. Briancon\paper Description de $Hilb^n\bc\{x,y\}$ \jour Invent. Math\vol 41 \pages 45--89 \yr 1977 \endref \ref\key Dr \by V. Drinfeld\paper A new realisation of Yangians and quantized affine algebras \jour Soviet Math. Dokl. \vol 36\yr 1988 \pages 212--216\endref \ref\key LMNS\by A. Losev, G. Moore, N. Nekrasov and S. Shatashvili \paper Four-dimensional avatars of two dimensional CFT \paperinfo talk at USC, March 1995, and paper in preparation \endref \ref\key FJ \by I. Frenkel and N. Jing \paper Vertex representations of quantum affine algebras \jour Proc. Natl. Acad. Sci. \yr 1988 \vol 85 \pages 9373--9377 \endref \ref\key FK \by I. Frenkel and V. Kac \paper Basic representation of affine Lie algebras and dula resonance models \jour Invent. Math \yr 1980 \vol 62 \pages 23--66 \endref \ref\key FLM \by I. Frenkel, J. Lepowsky and A. Meurman \book Vertex operator algebras and the monster \publ Pure and applied math, Academic Press \vol 134 \yr 1988 \endref \ref\key Fo \by J. Fogarty\paper Algebraic families on an algebraic surface \jour Am. J. Math \vol 90 \yr 1968 \pages 511--521 \endref \ref\key Gi\by D. Gieseker \paper On the moduli of vector bundles on an algebraic surface \jour Annals of Math \vol 106 \yr 1977 \pages 45--60 \endref \ref\key GL \by D. Gieseker and J. Li \paper Irreducibility of moduli of rank 2 vector bundles on algebraic surfaces \jour Jour. Diff. Geom. \vol 40 \yr 1994 \pages 23--104 \endref \ref\key GV\by V. Ginzburg and E. Vasserot \paper Langlands reciprocity for affine quantum groups of type $A_n$ \jour International Math. Research Notes \vol 3\yr 1993\pages 67--85 \endref \ref\key Got\by L. Gottsche\paper The Betti numbers of the Hilbert scheme of points on a smooth projective surface \jour Math Ann. \yr 1990 \vol 286 \pages 193--207 \endref \ref\key GS \by L. Gottsche and W. Soergel\paper Perverse sheaves and the cohomology of the Hilbert scheme of a smooth algebraic surface\jour Math Ann. \yr 1993 \vol 296 \pages 235--245 \endref \ref\key Gr1\by I. Grojnowski\paper Representations of quantum affine algebras \paperinfo Yale University course notes (1994), book in preparation \endref \ref\key Gr2\by I. Grojnowski\paper Affinizing quantum algebras: from $D$-modules to $K$-theory \paperinfo preprint 1994, posted to q-alg \endref \ref\key Gr3\by I. Grojnowski\paper The coproduct for quantum $GL_n$ \paperinfo preprint 1992\endref \ref\key Gro \by A. Grothendieck \paper Techniques de construction et theorems d'existance en geometrie algebrique IV: Les schemas de Hilbert \jour Seminaire Bourbaki, Expose 221 \yr 1960 \endref \ref\key JKS \by U. Jannsen, S. Kleiman, J. P. Serre (eds) \paper Motives \jour Proc. Symp Pure Math \vol 55 \yr 1994 \endref \ref\key KM\by P. Kronheimer and T. Mrowka \paper Embedded surfaces and the structure of Donaldson polynomial invariants \paperinfo preprint 1994 \endref \ref\key KN\by P. Kronheimer and H. Nakajima \paper Yang-Mills instantons on ALE gravitational Instantons \jour Math Ann. \yr 1990 \vol 288 \pages 263-307\endref \ref\key La\by G. Laumon \paper Un analogue global du cone nilpotent \jour Duke Math Jour \vol 57 \yr 1988 \pages 647--671 \endref \ref\key L1 \by G. Lusztig \paper Quivers, perverse sheaves and quantized enveloping algebras\jour J. Amer. Math. Soc. \vol 4 \yr 1991 \pages 365-421\endref \ref\key L2 \by G. Lusztig \paper Affine quivers and canonical bases \jour Publ. Math IHES \yr 1992 \pages 111-163 \vol 76 \endref \ref\key MO\by J. Morgan and K. O'Grady \paper Differential topology of complex surfaces \jour Springer LNM 1545 \yr 1993 \endref \ref\key Na\by H. Nakajima\paper Instantons on ALE spaces, quivers, and Kac-Moody algebras \jour Duke Math Jour. \yr 1995 \vol 76 \pages 365--416 \endref \ref\key VW\by C. Vafa and E. Witten \paper A strong coupling test of $S$-duality \paperinfo hep-th/9408074 \yr 1994 \endref \endRefs \enddocument

9506

alg-geom/9506011

en

https://arxiv.org/abs/alg-geom/9506011

alg-geom/9506011

Dr. P. E. Newstead

V. Balaji, L. Brambila Paz, and P. E. Newstead

Stability of the Poincar\'e bundle

16pp. Hard copy available from Dr. P. E. Newstead, Dept. of Pure Maths., University of Liverpool, Liverpool, L69 3BX, England. LaTeX 2.09

Let $C$ be a nonsingular projective curve of genus $g\ge2$ defined over the complex numbers, and let $M_{\xi}$ denote the moduli space of stable bundles of rank $n$ and determinant $\xi$ on $C$, where $\xi$ is a line bundle of degree $d$ on $C$ and $n$ and $d$ are coprime. It is shown that the universal bundle $\cu_{\xi}$ on $C\times M_{\xi}$ is stable with respect to any polarisation on $C\times M_{\xi}$. It is shown further that the connected component of the moduli space of $\cu_{\xi}$ containing $\cu_{\xi}$ is isomorphic to the Jacobian of $C$.

alg-geom

\section*{Introduction} In the study of moduli spaces of stable bundles on an algebraic curve $C$, various bundles on the moduli space or on the product of the moduli space with $C$ arise in a natural way. An interesting question to ask about any such bundle is whether it is itself stable in some sense. More precisely, let $C$ be a nonsingular projective curve of genus $g\ge2$ defined over the complex numbers, and let $M = M_{n,d}$ denote the moduli space of stable bundles of rank $n$ and degree $d$ on $C$, where $n$ and $d$ are coprime. For any line bundle $\xi$ of $\deg d$ on $C$, let $M_{\xi}$ denote the subvariety of $M$ corresponding to bundles with determinant $\xi$. There exists on $C \times M$ a universal (or Poincar\'e) bundle ${\cal U}$ such that ${\cal U} {\mbox{\Large $|$}}_{C \times \{ m\}}$ is the bundle on $C$ corresponding to $m$. Moreover the bundle ${\cal U}$ is determined up to tensoring with a line bundle lifted from $M$. The direct image of ${\cal U}$ on $M$ is called the Picard sheaf of ${\cal U}$; for $d> n (2g-2)$, this sheaf is a bundle. It was shown recently by Y.~Li [Li] that, if $d>2gn$, this bundle is stable with respect to the ample line bundle corresponding to the generalised theta divisor (cf. [DN]). (Recall here that, if $H$ is an ample divisor on a projective variety $X$, the {\it degree} $\deg E$ of a torsion-free sheaf $E$ on $X$ is defined to be the intersection number $[c_1(E) \cdot H^{\dim X-1}]$. $E$ is said to be {\it stable} with respect to $H$ (or $H$-{\it stable}) if, for every proper subsheaf $F$ of $E$, $$ \frac{\deg F}{{\rm rank}\, F} < \frac{\deg E}{{\rm rank}\, E}.$$ The definition depends only on the polarisation defined by $H$.) This extends previously known results for the case $n=1$ ([U, Ke1, EL]). We remark that the question of stability of the Picard sheaf of ${\cal U}_{\xi}$ is still open (cf. [BV]). In this paper, we investigate the stability of the Poincar\'e bundle ${\cal U}$ and its restriction ${\cal U}_{\xi}$ to $M_{\xi}$ using methods similar to those of [Li]. Our main results are \vspace{8pt} \noindent {\bf Theorem 1.5.} ${\cal U}_{\xi}$ is stable with respect to any polarisation on $C \times M_{\xi}$. \vspace{8pt} \noindent {\bf Theorem 1.6.} ${\cal U}$ is stable with respect to any polarisation of the form $$a \alpha + b \Theta ,~~ a,b > 0$$ where $\alpha$ is ample on $C$ and $\Theta$ is the generalised theta divisor on $M$. (Note that $C$ has a unique polarisation whereas $M$ does not.) These results are proved in \S 1. In \S 2 we discuss some properties of the bundles ${\rm End}\, {\cal U}_{\xi}$ and ${\rm ad}\, {\cal U}_{\xi}$. It is reasonable to conjecture that ${\rm ad}\, {\cal U}_{\xi}$ is also stable but we are able to prove this only in the case $n=2$. Finally, in \S 3 we consider the deformation theory of ${\cal U}_{\xi}$ using the results in \S 2. The main result we prove is that the only deformations of ${\cal U}_{\xi}$ are those of the form ${\cal U}_{\xi} \otimes p_C^* L$, where $L$ is a line bundle of degree 0 on $C$. More precisely, let $H$ be any ample divisor on $C \times M_{\xi}$ and let $M({\cal U}_{\xi})$ denote the moduli space of $H$-stable bundles with the same numerical invariants as ${\cal U}_{\xi}$ on $C \times M_{\xi}$; then \vspace{8pt} \noindent {\bf Theorem 3.1.} The connected component $M({\cal U}_{\xi})_0$ of $M({\cal U}_{\xi})$ containing \{${\cal U}_{\xi}$\} is isomorphic to the Jacobian $J(C)$, the isomorphism $J(C) \longrightarrow M({\cal U}_{\xi})_0$ being given by $$L \longmapsto {\cal U}_{\xi} \otimes p_C^* L,$$ where $p_C: C \times M_{\xi} \longrightarrow C$ is the projection. \vspace{8pt} \noindent {\bf Acknowledgement.} Most of the work for this paper was carried out during a visit by the first two authors to Liverpool. They wish to acknowledge the generous hospitality of the University of Liverpool. \renewcommand{\thesection}{\S \arabic{section}} \section{Stability of $\cal U$} \renewcommand{\thesection}{\arabic{section}} We begin with some lemmas which are probably well known, but which we could not find in the literature. \begin{lema}$\!\!\!${\bf .}~ \label{l11} Let $X$ and $Y$ be smooth projective varieties of the same dimension $m$. Let $f: X - - \rightarrow Y$ be a dominant rational map defined outside a subset $Z \subset X$ with ${\rm codim}\,_XZ\geq 2$. Suppose that $D_X$ and $D_Y$ are ample divisors on $X$ and $Y$ such that $f^*D_Y|_{X-Z} \simeq D_X |_{X-Z}$. Let $E$ be a vector bundle on $Y$ such that $f^* E$ extends to a vector bundle $F$ on $X$. If $F$ is $D_X$-semi-stable (resp. stable) on $X$, then $E$ is $D_Y$-semi-stable (resp. stable) on $Y$. \end{lema} \noindent {\bf Proof.} The proof is fairly straight forward (cf. [Li, pp.548, 549]). Let rank $E = n$. Suppose that $V$ is a torsion-free quotient of $E$, $$ E \longrightarrow V \longrightarrow 0 \eqno (1) $$ with rank $V = r < n$. When $F$ is $D_X$-semi-stable, we need to check the inequality: $$ \frac{\deg V}{r} \geq \frac{\deg E}{n}. \eqno (2) $$ {}From (1), we have $$ f^* E \longrightarrow f^* V \longrightarrow 0. $$ If $G = f^*V/(torsion)$, then $$ \deg f^* V \geq \deg G. \eqno (3) $$ (Note that $\deg f^* V = [c_1(f^*V)\cdot D_X^{m-1}]$, where $c_1(f^*V)$ makes sense since $f^*V$ is defined outside a subset of codimension 2.) Moreover we have $$ 0 \longrightarrow G^* \longrightarrow f^* (E)^* $$ on X-Z. Let $\widetilde{G}$ be a subsheaf of $F^*$ extending $G^*$. Since ${\rm codim}\,_X Z \geq 2$, we have $\deg \widetilde{G} = \deg G^* = - \deg G$. By semi-stability of $F^*$, $$ \frac{\deg\widetilde{G}}{r} \leq \frac{\deg F^*}{n}. \eqno (4) $$ Note that $$ \begin{array}{rcl} \deg E & = & [c_1(E) . D_Y^{m-1}] \\ [2mm] & = & [c_1 (f^*E) . D_X^{m-1}] (\deg f)^{-1} \\ [2mm] & = & (\deg f^* E) (\deg f)^{-1}.\\ \end{array} $$ The same applies to $V$, so we have $$ \begin{array}{rcl} n \deg V - r \deg E & = & (n \deg f^* V - r \deg f^* E) (\deg f)^{-1} \\ [2mm] & \geq & (n \deg G - r \deg F) (\deg f)^{-1} \mbox{~~~~by (3)} \\ [2mm] & = & (- n \deg \widetilde{G} + r \deg F^* ) ( \deg f)^{-1} \\ & \geq & 0 \mbox{~~~~by (4)}. \end{array} $$ This proves (2). The proof in the stable case is similar. \begin{lema}$\!\!\!${\bf .}~ \label{l12} Let $X^m$ and $Y^n$ be smooth projective varieties, and $D_X$ and $D_Y$ ample divisors on $X$ and $Y$. Let $\eta = a D_X + b D_Y$, $a,b >0$. Suppose that $E$ is a vector bundle on $X \times Y$, such that for generic $x \in X$, $y \in Y$, $E_x \simeq E|_{\{x\} \times Y}$ and $E_y \simeq E |_{X \times \{y\}}$ are respectively $D_Y$-semi-stable and $D_X$-semi-stable. Then $E$ is $\eta$-semi-stable. Further, if either $E_x$ or $E_y$ is stable, then $E$ is stable. \end{lema} {\bf Proof.} Let $F \subset E$ be a subsheaf. Since ${\rm Sing}\, F$ has codimension $\geq 2$, we can choose $x \in X$ and $y \in Y$ such that ${\rm Sing}\, F_x$ and ${\rm Sing}\, F_y$ also have codimension $\geq 2$. Thus any torsion in $F_x$ or $F_y$ is supported in codimension $\geq 2$ and does not contribute to $c_1(F_x)$ or $c_1(F_y)$. Let ${\rm rank}\, (E) = n$, ${\rm rank}\, (F) = r$. Then we need to show that $$ \frac{\deg F}{r} \leq \frac{\deg E}{n} \eqno (5) $$ assuming $E_x$ and $E_y$ are semi-stable. Now, $$ \begin{array}[b]{rcl} \deg E & = & c_1(E) \cdot [aD_X + bD_Y]^{m+n-1} \\ [2mm] & = & c_1(E) [\lambda D_X^m \cdot D_Y^{n-1} + \mu D_X^{m-1}\cdot D_Y^n] \mbox{~~~~for some}~~ \lambda ,\mu > 0 \\ [2mm] & = & [c_1(E_x) + c_{1,1} (E) + c_1(E_y)] \cdot [\lambda D_X^m \cdot D_Y^{n-1} + \mu D_X^{m-1}\cdot D_Y^n] \\ [2mm] & = & [c_1 (E_x) \cdot D_Y^{n-1}] \cdot \lambda D_X^m + [c_1(E_y)\cdot D_X^{m-1}] \cdot \mu D_Y^n. \\ \end{array} \eqno (6) $$ We have a similar expression for $\deg F$, $$ \deg F = [c_1 (F_x) \cdot D_Y^{n-1}] \cdot \lambda D_X^m + [c_1(F_y) \cdot D_X^{n-1}] \cdot \mu D_Y^n. \eqno (7) $$ (5) follows trivially by comparing the terms in (6) and (7) and using semi-stability of $E_x$ and $E_y$. The rest of the lemma follows in a similar fashion. Before stating the next lemma, we recall very briefly some facts on spectral curves. For details see [BNR] and [Li]. Let $K = K_C$ be the canonical bundle and let $W = \oplus_{i=1}^n H^0(C,K^i)$. Let $s=(s_1,\cdots ,s_n) \in W$, and let $C_s$ be the associated spectral curve. Then we have a morphism $$ \pi : C_s \longrightarrow C $$ of degree $n$, such that for $x \in C$, the fibre $\pi^{-1}(x)$ is given by points $y \in K_x$ which are zeros of the polynomial $$ f(y) = y^n + s_1(x) \cdot y^{n-1} + \cdots + s_n(x) . $$ The condition that $x$ be unramified is that the resultant $R(f,f')$ of $f$ and its derivative $f'$ be non-zero at the point $(s_1(x),\cdots ,s_n(x))$. Note that $R(f,f')$ is a polynomial in the $s_i(x)$, $i=1,\cdots ,n$. \begin{lema}$\!\!\!${\bf .}~ \label{l13} Given $x \in C$, there exists a smooth spectral curve $C_s$ such that the covering map $\pi : C_s \longrightarrow C$ is unramified at $x$. \end{lema} {\bf Proof.} Note first that, if $x \in C$, there exists $s = (s_1,\cdots ,s_n) \in W$ such that $$ R(f,f') (s_1(x),\cdots ,s_n(x)) \neq 0. $$ Indeed, since $|K^i|$ has no base points, given any $(\alpha_1,\cdots ,\alpha_n) \in \bigoplus_{i=1}^n K_x^i$, there exist $s_i \in H^0(C,K^i)$ such that $s_i(x) = \alpha_i$, $ i=1,\cdots ,n$. Observe that this is clearly an open condition on $W$. Further, the subset $\{ s \in W ~|~ C_s$ is smooth$\}$ is a non-empty open subset of $W$ [BNR, Remark 3.5] and the lemma follows. Let $M_{\xi}$ and $\cal U_{\xi}$ be as in the introduction, and let $\Theta_{\xi}$ denote the restriction of the generalized theta divisor to $M_{\xi}$. \begin{propo}$\!\!\!${\bf .}~ \label{l14} Let $\cal U_{\xi}$ be the Poincar\'e bundle on $C \times M_{\xi}$ and $x \in C$. Then the bundle $$ {\cal U}_{\xi ,x}\cong {\cal U}_{\xi} |_{\{x\} \times M_{\xi}} $$ is $\Theta_{\xi}$-semi-stable on $M_{\xi}$. \end{propo} {\bf Proof.} For the point $x \in C$ above, choose a spectral curve $C_s$ by Lemma 1.3, so that $$ \pi : C_s \longrightarrow C $$ is unramified at $x$. Let $\pi^{-1} (x) = \{ y_1,\cdots ,y_n \}$, $y_i$ being distinct points in $C_s$. Let $J^{\delta}(C_s)$ denote the variety of line bundles of degree $$\delta=d-\deg\pi_*({\cal O}_{C_s})$$ on $C_s$, and let $P_s$ denote the subvariety of $J^{\delta}(C_s)$ consisting of those line bundles $L$ for which the vector bundle $\pi_*L$ has determinant $\xi$. ($P_s$ is a translate of the Prym variety of $\pi$.) Let ${\cal L}$ denote the restriction of the Poincar\'e bundle on $C_s \times J^{\delta}(C_s)$ to $C_s\times P_s$. Then [BNR, proof of Proposition 5.7], we have a dominant rational map defined on an open subset $T_s$ of $P_s$ such that ${\rm codim}\, (P_s - T_s) \geq 2$, and $$ \phi : T_s \longrightarrow M_{\xi} $$ is generically finite. The morphism $\phi$ on $T_s$ is defined by the family $(\pi\times 1)_*{{\cal L}}$ on $C_s\times T_s$; so, by the universal property of ${\cal U}_{\xi}$, we have $$ (\pi \times 1)_* {\cal L} \simeq (1 \times \phi )^* {\cal U}_{\xi}\otimes p_T^*L_0\eqno (8) $$ for some line bundle $L_0$ on $T_s$. (Here $p_T : C_s\times T_s\longrightarrow T_s$ is the projection.) By $(8)$ we have $$ \phi^* {\cal U}_{\xi ,x} \simeq [(\pi \times 1)_* {\cal L} ]_x\otimes L_0^{-1} ~{\rm on} ~T_s. $$ But $[(\pi \times 1)_* {\cal L}]_x \simeq \oplus_{i=1}^n {\cal L}_{y_i}$ on $P_s$. Hence $$ \phi^* {\cal U}_{\xi ,x} \simeq \oplus_{i=1}^n ({\cal L}_ {y_i}\otimes L_0^{-1}) ~{\rm on}~ T_s. \eqno (9) $$ We observe that the ${\cal L}_{y_i}\otimes L_0^{-1}$ are the restrictions to $T_s$ of algebraically equivalent line bundles on $P_s$. Further one knows [Li, Theorem 4.3] that $\phi^* \Theta_{\xi}$ is a multiple of the restriction of the usual theta divisor on $J(C_s)$ to $T_s$. Now we are in the setting of Lemma 1.1 and we conclude that ${\cal U}_{\xi ,x}$ is semi-stable with respect to $\Theta_{\xi}$ on $M_{\xi}$. \begin{guess}$\!\!\!${\bf .}~ \label{th15} The Poincar\'e bundle ${\cal U}_{\xi}$ on $C \times M_{\xi}$ is stable with respect to any polarisation. \end{guess} {\bf Proof.} Since $\hbox{Pic}\, M_{\xi} = {\bf Z}$, $\hbox{Pic}\, (C \times M_{\xi}) = \hbox{Pic}\, C \oplus \hbox{Pic}\, M_{\xi}$. Thus, any polarisation $\eta$ on $C \times M_{\xi}$ can be expressed in the form $$\eta = a \alpha + b \Theta_{\xi}, ~~~ a,b > 0.$$ for some ample divisor $\alpha$ on $C$. By Proposition 1.4, ${\cal U}_{\xi ,x}$ is semi-stable with respect to $\Theta_{\xi}$ for all $x \in C$ and by definition ${\cal U} _{\xi}|_{C\times\{m\}}$ is stable with respect to any polarisation on $C$. Hence by Lemma 1.2, ${\cal U}_{\xi}$ is stable with respect to $\eta$ on $C \times M_{\xi}$. Note that Proposition 1.4 remains true if we replace $M_{\xi}$, ${\cal U}_{\xi}$ and $\Theta_{\xi}$ by $M$, ${\cal U}$ and $\Theta$. (The key point is that [Li, Theorem 4.3] is valid in this context). We deduce at once \begin{guess}$\!\!\!${\bf .}~ ${\cal U}$ is stable with respect to any polarisation of the form $$a \alpha + b \Theta ,~~ a,b > 0,$$ where $\alpha$ is ample on $C$ and $\Theta$ is the generalized theta divisor on $M$. \end{guess} \begin{rema}$\!\!\!${\bf .}~\rm \label{r16} Since $C \times M$ is a K\"ahler manifold, then by a theorem of Donaldson-Uhlenbeck-Yau, ${\cal U}$ admits an Hermitian-Einstein metric. One can expect that the restriction of this metric to each factor is precisely the metric on the factor. It would be interesting to know an explicit description of the metric on ${\cal U}$. Note that [Ke2] contains such a description for the Picard sheaf in the case $g=1$, $n=1$. \end{rema} \renewcommand{\thesection}{\S \arabic{section}} \section{Some properties of End ${\cal U}_{\xi}$} \renewcommand{\thesection}{\arabic{section}} Our first object in this section is to calculate the dimension of some of the cohomology spaces of ${\rm End}\, {\cal U}_{\xi}$. We denote by $p: C \times M_{\xi} \longrightarrow M_{\xi}$ and $p_C: C \times M_{\xi} \longrightarrow C$ the projections. \begin{propo}$\!\!\!${\bf .}~ Let $h^i ({\rm End}\, {\cal U}_{\xi}) = \dim H^i ({\rm End}\, {\cal U}_{\xi})$. Then, $$h^0({\rm End}\, {\cal U}_{\xi} ) = 1, ~~ h^1({\rm End}\, {\cal U}_{\xi}) =g, ~~h^2({\rm End}\, {\cal U}_{\xi}) = 3g-3.$$ \end{propo} {\bf Proof.} We can write ${\rm End}\, {\cal U}_{\xi} \cong {\cal O} \oplus {\rm ad}\, {\cal U}_{\xi}$; hence $$H^i(C \times M_{\xi}, {\rm End}\, {\cal U}_{\xi}) = H^i(C\times M_{\xi}, {\cal O} ) \oplus H^i (C \times M_{\xi}, {\rm ad}\, {\cal U}_{\xi} ).$$ Since ${\cal U}_{\xi} {\mbox{\Large $|$}}_{C \times \{ m\}}$ is always stable, we have $R_p^0({\rm ad}\, {\cal U}_{\xi}) = 0$. So by the Leray spectral sequence and the fact that $R_p^1({\rm ad}\, {\cal U}_{\xi})$ is the tangent bundle $T M_{\xi}$ of $M_{\xi}$, we have $$\begin{array}{rl} H^i (C \times M_{\xi},{\rm ad}\, {\cal U}_{\xi} ) & \cong H^{i-1} (M_{\xi}, R_p^1({\rm ad}\, {\cal U}_{\xi})) \\ [2mm] & \cong H^{i-1} (M_{\xi},TM_{\xi}) \\ \end{array}$$ By [NR, Theorem 1], this space is 0 if $i \neq 2$, and has dimension $3g-3$ if $i=2$. On the other hand, since $M_{\xi}$ is unirational, it follows from the K\"unneth formula that $$H^i (C \times M_{\xi} , {\cal O} ) = H^i (C,{\cal O}_C ).$$ This space has dimension 1 if $i=0, ~g$ if $i=1$ and 0 otherwise. The proposition follows. \noindent \begin{rema}$\!\!\!${\bf .}~\rm \label{r22} In fact the proof shows that $$h^i ({\rm End}\, {\cal U}_{\xi} ) = 0~~{\rm if}~~ i > 2.$$ \end{rema} \begin{lema}$\!\!\!${\bf .}~ \label{l23} Let $L \in J (C)$ and suppose that $E \cong E \otimes L$ for all $E \in M_{\xi}$. Then ${\cal U}_{\xi} \cong {\cal U}_{\xi} \otimes p_C^* L$. \end{lema} {\bf Proof.} Since $E \cong E \otimes L$ and ${\cal U}_{\xi} \otimes p_C^* L$ is a family of stable bundles, there is a line bundle $L_1$ over $M_{\xi}$ such that $${\cal U}_{\xi} \cong {\cal U}_{\xi} \otimes p_C^*L \otimes p^* L_1.$$ Fix $x \in C$, then ${\cal U}_{\xi ,x} \cong {\cal U}_{\xi ,x} \otimes L_1$ over $M_{\xi}$. Hence $c_1 ({\cal U}_{\xi ,x}) = c_1 ({\cal U}_{\xi ,x}) + nc_1 (L_1 )$ so $nc_1(L_1) = 0$. But $\hbox{Pic}\, {\cal M}_{\xi} \cong {\bf Z}$ (see [DN]); so $c_1 (L_1)=0$, which implies that $L_1$ is the trivial bundle. The next lemma will also be required in \S 3. \begin{lema}$\!\!\!${\bf .}~ \label{l24} If ${\cal U}_{\xi} \cong {\cal U}_{\xi} \otimes p_C^*L$ then $L \cong {\cal O}_C$. \end{lema} {\bf Proof.} If ${\cal U}_{\xi} \cong {\cal U}_{\xi} \otimes p_C^*L$ then $$\begin{array}{rcl} {\cal O} \oplus {\rm ad}\, {\cal U}_{\xi} & \cong & {\rm End}\, {\cal U}_{\xi} \\ [2mm] & \cong & {\rm End}\, {\cal U}_{\xi} \otimes p_C^* L \\ [2mm] & \cong & p_C^* L \oplus {\rm ad}\, {\cal U}_{\xi} \otimes p_C^* L. \\ \end{array}$$ Hence $H^0(C \times M_{\xi}, p_C^*L)$ and $H^0(C \times M_{\xi}, {\rm ad}\, {\cal U}_{\xi} \otimes p_C^*L)$ cannot both be zero. Suppose there is a non-zero section $\phi: {\cal O} \longrightarrow {\rm ad}\, {\cal U}_{\xi} \otimes p_C^* L$. For some $x \in C$, the restriction of $\phi$ to $\{ x\} \times M_{\xi}$ will define a non-zero section of ${\rm ad}\, {\cal U}_{\xi,x}$, which is a contradiction since $H^0(M_{\xi}, {\rm ad}\, {\cal U}_{\xi ,x})=0$ (see [NR, Theorem 2]). Hence $H^0(C \times M_{\xi}, {\rm ad}\, {\cal U}_{\xi} \otimes p_C^* L) = 0$. Therefore $H^0(C \times M_{\xi}, p_C^*L) \ne0$. Since $\deg L=0$, this implies $L \cong {\cal O}_C$. \begin{rema}$\!\!\!${\bf .}~\rm The proof of Lemma 2.4 fails when $g=1$ since [NR, Theorem 2] is not then valid. In fact, Lemma 2.4 and the remaining results of this section are false for $g=1$. \end{rema} We show next that a general stable bundle $E$ is not isomorphic to $E\otimes L$ unless $L\cong{\cal O}_C$. \begin{propo}$\!\!\!${\bf .}~ \label{p25} There exists a proper closed subvariety $S$ of $M_{\xi}$ such that, if $E\not\in S$, then $$E\cong E\otimes L\Longrightarrow L\cong{\cal O}_C.$$ \end{propo} {\bf Proof.} For any $L$, the subset $S_L = \{ {E \in M_{\xi} |E \cong E \otimes L}\}$ is a closed subvariety of $M_{\xi}$. If $L\not\cong{\cal O}_C$, then, by Lemmas 2.3 and 2.4, $S_L$ is a proper subvariety. On the other hand, $S_L$ can only be non-empty if $L^n\cong{\cal O}_C$; so only finitely many of the $S_L$ are non-empty. Since $M_{\xi}$ is irreducible, the union $S= \bigcup\{S_L | L\not\cong{\cal O}_C\}$is a proper subvariety of $M_{\xi}$ as required. \begin{rema}$\!\!\!${\bf .}~\rm It follows at once from Proposition 2.6 that the action of $J(C)$ on $M$ defined by $E\longmapsto E\otimes L$ is faithful. Another proof of this fact has been given in [Li, Theorem 1.2 and Proposition 1.6]. As the following proposition shows, our set $S$ is analogous to the set $S$ of [Li, Theorem 1.2]. \end{rema} \begin{propo}$\!\!\!${\bf .}~ \label{p27} Let $S$ be as above and $E\in M_{\xi}$. Then $E \in S$ if and only if ${\rm ad}\, E$ has a line sub-bundle of degree zero. \end{propo} {\bf Proof.} The trivial bundle cannot be a subbundle of ${\rm ad}\, E$. If $L \in J(C)$ is a subbundle of ${\rm ad}\, E$, so is it of ${\rm End}\, E$, therefore $$H^0(C, {\rm End}\, E \otimes L^*) \neq 0.$$ Hence, there is a non-zero map $\phi : E \otimes L \longrightarrow E$, which is an isomorphism since $E \otimes L$ and $E$ are stable bundles of the same slope. Hence $E\in S$. Conversely, suppose $E \cong E \otimes L$ with $L \not \cong {\cal O}_C$. The isomorphism ${\cal O}_C \oplus {\rm ad}\, E \cong L \oplus {\rm ad}\, E \otimes L$ implies that ${\rm ad}\, E \otimes L$ has a section, i.e.\ there is a non-zero map $\phi: L^* \longrightarrow {\rm ad}\, E$. Since $L^*$ and ${\rm ad}\, E$ have the same slope and ${\rm ad}\, E$ is semi-stable (being a subbundle of a semi-stable bundle ${\rm End}\, E$ with the same slope), $\phi$ is an inclusion. \begin{coro}$\!\!\!${\bf .}~ If $E$ is a general stable bundle of rank $2$ and determinant $\xi$, then ${\rm ad}\, E$ is stable. \end{coro} {\bf Proof.} Note that ${\rm ad}\, E$ has rank 3 and degree $0$ and is semi-stable. By the Proposition, ${\rm ad}\, E$ has no line subbundle of degree 0. On the other hand ${\rm ad}\, E$ is self-dual, so it cannot have a quotient line bundle of degree 0. \begin{rema}$\!\!\!${\bf .}~\rm \label{r38} It would be interesting to know if ${\rm ad}\, E$ is stable for a general stable bundle $E$ of rank greater than $2$. It is certainly true that ${\rm ad}\, E$ is semistable and also that it is stable as an orthogonal bundle [R]. \end{rema} \begin{guess}$\!\!\!${\bf .}~ If $n=2$, ${\rm ad}\, {\cal U}_{\xi}$ is stable with respect to any polarisation on $C \times M_{\xi}$. \end{guess} {\bf Proof.} In view of Corollary 2.9 and Lemma 1.2, we need only prove that ${\rm ad}\, {\cal U}_{\xi ,x}$ is semi-stable for some $x \in C$. The argument is the same as for Proposition 1.4; indeed (9) shows at once that $\phi^* {\rm ad}\, {\cal U}_{\xi ,x}$ can be expressed as a direct sum of restrictions to $T_s$ of algebraically equivalent line bundles on $P_s$. For $n > 2$, we can show similarly that ${\rm ad}\, {\cal U}_{\xi}$ is semi-stable. \renewcommand{\thesection}{\S \arabic{section}} \section{Deformations} \renewcommand{\thesection}{\arabic{section}} As in the introduction, let $H$ be any ample divisor on $C \times M_{\xi}$, let $M({\cal U}_{\xi})$ denote the moduli space of $H$-stable bundles with the same numerical invariants as ${\cal U}_{\xi}$ on $C \times M_{\xi}$, and let $M({\cal U}_{\xi})_0$ denote the connected component of $M({\cal U}_{\xi})$ which contains ${\cal U}_{\xi}$. One can define a morphism $$\beta : J (C) \longrightarrow M({\cal U}_{\xi})_0$$ by $$\beta (L) = {\cal U}_{\xi} \otimes p_C^*L.$$ Our object in this section is to prove \begin{guess}$\!\!\!${\bf .}~ $\beta$ is an isomorphism. \end{guess} \begin{rema}$\!\!\!${\bf .}~\rm Note that this implies in particular that $M({\cal U}_{\xi})_0$ is independent of the choice of $H$, and is a smooth projective variety of dimension $g$. Since $h^2({\rm End}\, {\cal U}_{\xi}) \neq 0$, there is no a priori reason why this should be so. \end{rema} {\bf Proof of Theorem 3.1.} By Lemma 2.4, $\beta$ is injective. Moreover the Zariski tangent space to $M({\cal U}_{\xi})_0$ at ${\cal U}_{\xi} \otimes p_C^*L$ can be identified with $H^1(C \times M_{\xi}, {\rm End}\, {\cal U}_{\xi})$, which has dimension $g$ by Proposition 2.1. It follows that, at any point of ${\rm Im }~\beta$, $M({\cal U}_{\xi})_0$ has dimension precisely $g$ and is smooth. Hence by Zariski's Main Theorem, $\beta$ is an open immersion. Since $J(C)$ is complete, it follows that $\beta$ is an isomorphism.

9506

alg-geom/9506002

en

https://arxiv.org/abs/alg-geom/9506002

alg-geom/9506002

Alan Durfee

Alan H. Durfee

The index of $grad f(x,y)$

A thoroughly revised and hopefully more readable version; the main results are the same. 35 pages with 7 figures

Let $f(x,y)$ be a real polynomial of degree $d$ with isolated critical points, and let $i$ be the index of $grad f$ around a large circle containing the critical points. An elementary argument shows that $|i| \leq d-1$. In this paper we show that $ i \leq max \{1, d-3 \}$. We also show that if all the level sets of $f$ are compact, then $i = 1$, and otherwise $ |i| \leq \dr -1 $ where $\dr$ is the sum of the multiplicities of the real linear factors in the homogeneous term of highest degree in $f$. The technique of proof involves computing $i$ from information at infinity. The index $i$ is broken up into a sum of components $i_{p,c}$ corresponding to points $p$ in the real line at infinity and limiting values $c \in \realinf$ of the polynomial. The numbers $i_{p,c}$ are computed in three ways: geometrically, from a resolution of $f(x,y)$, and from a Morsification of $f(x,y)$. The $i_{p,c}$ also provide a lower bound for the number of vanishing cycles of $f(x,y)$ at the point $p$ and value $c$.

alg-geom

\section{#1}} \newcounter{mycounter}[section] \renewcommand{\themycounter}{\arabic{section}.\arabic{mycounter}} \newenvironment{theorem}% {\medskip \refstepcounter{mycounter} {\bf \noindent Theorem \themycounter.}\begin{em}}% {\end{em} \medskip } \newenvironment{proposition}% {\medskip \refstepcounter{mycounter} {\bf \noindent Proposition \themycounter. \ } \begin{em} }% {\end{em} \medskip } \newenvironment{lemma}% {\medskip \refstepcounter{mycounter} {\bf \noindent Lemma \themycounter. \ } \begin{em} }% {\end{em} \medskip } \newenvironment{corollary}% {\medskip \refstepcounter{mycounter} {\bf \noindent Corollary \themycounter. \ } \begin{em} }% {\end{em} \medskip } \newenvironment{formula}% {\medskip \refstepcounter{mycounter} {\bf \noindent Formula \themycounter. \ } \begin{em} }% {\end{em} \medskip } \newenvironment{remark}% {\medskip \refstepcounter{mycounter} {\bf \noindent Remark \themycounter. \ }}% {\medskip } \newenvironment{conjecture}% {\medskip \refstepcounter{mycounter} {\bf \noindent Conjecture \themycounter. \ }}% {\medskip } \newenvironment{definition}% {\medskip \refstepcounter{mycounter} {\bf \noindent Definition \themycounter. \ }}% {\medskip } \newenvironment{example}% {\medskip \refstepcounter{mycounter} {\bf \noindent Example \themycounter. \ }}% {\medskip } \newenvironment{problem}% {\medskip \refstepcounter{mycounter} {\bf \noindent Problem \themycounter. \ }}% {\medskip } \newenvironment{xproof}% {\medskip \noindent {\bf Proof. \ }}% {$\Box$ \medskip } \newcommand{}{} \begin{document} \maketitle \begin{abstract} Let $f(x,y)$ be a real polynomial of degree $d$ with isolated critical points, and let $i$ be the index of $grad \, f$ around a large circle containing the critical points. An elementary argument shows that $|i| \leq d-1$. In this paper we show that $ i \leq max \{ 1, d-3 \}$. We also show that if all the level sets of $f$ are compact, then $i = 1$, and otherwise $ |i| \leq d_{{\bf R}} -1 $ where $d_{{\bf R}}$ is the sum of the multiplicities of the real linear factors in the homogeneous term of highest degree in $f$. The technique of proof involves computing $i$ from information at infinity. The index $i$ is broken up into a sum of components $i_{p,c}$ corresponding to points $p$ in the real line at infinity and limiting values $c \in \real \cup \{\infinity \} $ of the polynomial. The numbers $i_{p,c}$ are computed in three ways: geometrically, from a resolution of $f(x,y)$, and from a Morsification of $f(x,y)$. The $i_{p,c}$ also provide a lower bound for the number of vanishing cycles of $f(x,y)$ at the point $p$ and value $c$. \end{abstract} \section{Introduction} Let $f(x,y)$ be a real polynomial with isolated critical points. Let $i$ be the index of the gradient vector field of $f(x,y)$ around a large circle $C$ centered at the origin and containing the critical points, oriented in the counterclockwise direction. If the critical points of $f$ are nondegenerate, then the index $i$ is the number of local extrema minus the number of saddles. What bounds can be placed on the index $i$ in terms of the degree $d$ of the polynomial? It follows easily from Bezout's theorem that \cite[Proposition 2.5]{REU} $$\vert i \vert \leq d-1$$ It is easy to find polynomials satisfying the lower bound of this inequality; for example if $f = l_1 \dots l_d$ where the $l_i$ are equations of lines in general position, then $i = 1-d$, as can be seen by looking at how the gradient vector field turns on the circle $C$, or by counting critical points \cite[Section 4]{REU}. The upper bound is more mysterious. In the first place, polynomials with $i >1$ are hard to find. (The dubious reader should try to do so!) A simple example with two local extrema and no other critical points ($i=2$) is $f(x,y) = y^5+x^2y^3-y$. A polynomial of degree five can have as many as sixteen critical points in the complex plane; a generic polynomial of degree five will have exactly this number. The above polynomial, however, has only four critical points in the plane (two real and two complex), so it is not generic. In fact this behavior is typical for polynomials with $i>1$ \cite[Theorem 6.2]{REU}. There are polynomials of degree $d$ with $i$ arbitrarily large (see Example \ref{many-max-min-ex}), but they have $i \approx (1/3)d$. So evidently there is a large gap between the theoretical upper bound and examples. One of the goals of this paper is to give a modest improvement of this upper bound. We will show \medskip \noindent {\bf Theorem \ref{max-theorem}.} If $f(x,y)$ is a real polynomial of degree $d$ with isolated critical points, and $i$ is the index of $grad \, f$ around a large circle containing the critical points, then $$ i \leq max \{ 1, d-3 \}$$ \medskip In particular this result implies that the minimum degree for a polynomial with $i > 1$ is five, as in the example above. In fact, the bound is often better. (See, for example, Proposition \ref{i-upper-bound}.) Let the {\em real degree} $d_{{\bf R}}$ of $f$ be the sum of the multiplicities of the real linear factors in the homogeneous term of highest degree in $f$. (Thus $d_{{\bf R}} \leq d$.) We will also show \medskip \noindent {\bf Theorem \ref{dr-theorem}.} If all the level sets of the polynomial $f(x,y)$ are compact, then $i = 1$. Otherwise $$|i| \leq d_{{\bf R}} -1 $$ \medskip It is easy to find polynomials realizing the lower bound (Corollary \ref{corollary-dr}), but the upper bound still appears high. The basic idea of the proofs is to compute the index $i$ from ``information at infinity''. We write $i$ as $$i = 1 + \sum_{\scriptstyle p \in \reallineinfinity \atop \scriptstyle c \in \real \cup \{\infinity \} } i_{p,c}$$ The terms $i_{p,c}$ are defined as follows: The number $\pm 1/2$ is assigned to a point $q$ where the circle $C$ is tangent to a level set of the polynomial according as whether the level set is locally inside or outside $C$ at $q$. The circle is then made larger and larger. The point $q$ where the level set is tangent to the circle approaches a limiting point $p$ on the line at infinity in real projective space, and the value of the polynomial $f(q)$ approaches a limiting value $c$. The term $i_{p,c}$ is the sum of all the numbers $\pm 1/2$ associated to $p$ and $c$ in this manner. This material is in Section 3. We also show (Proposition \ref{generic-h}) that the family of circles can be replaced by the level sets of any reasonable function, and the $i_{p,c}$ will remain the same. The polynomial $f$ extends to a function on projective space which is not well-defined at certain points on the line at infinity. Blowing up these points gives a well-defined function $\tilde{f}$. We use this technique in Section 4 to derive some simple properties of the level curves of $f$. In Section 5, we use Morse theory to show that the $i_{p,c}$ can be computed from the critical points of $\tilde{f}$ and information about the exceptional sets. The process of blowing up and computing the index is easy to carry out in specific examples. The polynomial $f$ can also be deformed into what we call a ``Morsification'', a polynomial whose real critical points are nondegenerate and whose homogeneous term of highest degree has no repeated real linear factors (Section 6). There is a simple formula relating the index of the original polynomial, the index of the new polynomial, and the index of the newly created critical points. The deformation process is not too well understood, and this section contains some examples and a conjecture. The computations of these sections are used in Section 7 to establish bounds on the $i_{p,c}$. These local bounds are sharp. The global bounds on $i$ follow from the local bounds and some delicate arguments. However, the global bounds are not sharp and there still is a big gap between the global bounds and the examples. In Section 8 we relate $i_{p,c}$ to the ``jump'' at $c$ in the Milnor number of the family $f(x,y) = t$ at the point $p$ on the line at infinity. Throughout this paper the techniques are those of basic topology (Morse Theory) and basic algebraic geometry (Bezout's theorem, explicit computation of intersection multiplicities, etc.) Computer algebra programs were used to find critical points, countour plots and the $i_{p,c}$. Although many of the results and techniques are valid in higher dimensions, the exposition is in dimension two for reasons of clarity. The author's interest in these questions started in 1989 when he worked with a group of undergraduates in the Mount Holyoke Summer REU \cite{REU}. Another group of students continued this work in 1992; one of their results was the construction of polynomials with an arbitrarily large number of local maxima and no other critical points \cite{Robertson-P}. (These polynomials have $ i \approx d/4$.) Shustin \cite{Shustin-96} has studied polynomials all of whose critical points lie in the complex plane. He finds polynomials of this type with almost all arbitrarily prescribed numbers of local maxima, minima and saddles. These polynomials have $i = 1-d_{{\bf R}}$ and, in particular, $i \leq 1$. They are stable in the sense that nearby polynomials have the same number and type of critical points. The primary focus of this paper is polynomials $f$ with $i > 1$; these polynomials are not stable. In fact, \cite[Theorem 6.2]{REU} says that $$i \leq \frac{1}{2} m + 1$$ where $m$ is the sum over $p$ in the line at infinity in real projective space of the intersection multiplicities at $p$ of the completions of $f_x = 0$ and $f_y = 0$. This paper is real counterpart of the study by many people of ``critical points at infinity'' for complex polynomials; see \cite{Durfee-P96} for further references. Research on this paper was partially supported by NSF grant DMS-8901903, and a grant from the International Research and Exchanges Board (IREX), with funds provided by the National Endowment for the Humanities and the United States Information Agency. The research was carried out over the past eight years at Martin-Luther University, Halle, the University of Nijmegen, Warwick University and the Massachusetts Institute of Technology; the author would like to thank them for their hospitality. Some notation which will be used throughout the paper: We let $${\bf L} = \{ [x,y,z] \in {\bf P}^2: z=0 \}$$ be the line at infinity in real projective space ${\bf P}^2$, and $\complexlineinfinity$ be the line at infinity in complex projective space ${\bf C} {\bf P} ^2$. We use $d$ for the degree of the polynomial $f(x,y)$, and $f_d$ for the homogeneous term of degree $d$ in $f$. \section{A polynomial zoo} A number of polynomials with strange properties are used as examples throughout this paper. These are described in this section. \begin{example} \label{std-crpt-ex} The polynomial $y(xy-1)$, which has no critical points in the plane, is the standard example of a polynomial with a ``critical point at infinity'' (at $[1,0,0]$). The ``critical value'' (jump in the Milnor number) is at 0. This polynomial perhaps first appeared in \cite{Broughton-83}. \end{example} \begin{example} \label{std-nocrpt-ex} The polynomial $x(y^2-1)$ has saddles at $(0,1)$ and $(0,-1)$. The family of level curves at $[1,0,0]$ is equisingular; there is no ``critical point'' at $[1,0,0]$. \end{example} \begin{example} \label{parabola-ex} The parabola $y^2-x$ is the simplest example of a polynomial with a ``critical point'' at $[1,0,0]$ with ``critical value'' $\infty$ \cite{Durfee-P96}. \end{example} \begin{example} \label{max-min-ex} The polynomial $y^5 + x^2y^3 - y$ has a local minimum at $(0,-1/\sqrt[4]{5})$, a local maximum at $(0,1/\sqrt[4]{5})$ and no other critical points. This polynomial was found by an REU group of undergraduates at Mount Holyoke College in the summer of 1989 \cite{REU}. \end{example} \begin{example} \label{many-max-min-ex} The polynomial $(y(x^2+1)-1)(y(x^2+2)-1) \dots (y(x^2+k)-1)$ has $k-1$ local extrema and no other real critical points; for $k = 2$ there is a local minimum, for $k=3$ there is a local minimum and a local maximum, for $k=4$ there are two local minima and a local maximum, and so forth. This polynomial was also found by the REU group \cite{REU}. \end{example} \begin{example} \label{two-min-ex} The polynomial $(xy^2 - y -1)^2 + (y^2-1)^2$ from \cite{Mathmag-85} has local minima at $(2,1)$ and $(0, -1)$, and no other critical points. Note the asymmetry of this polynomial compared with the previous ones. \end{example} \begin{example} \label{only-crpt-ex} The polynomial $x^2(1+y)^3+y^2$ has its sole critical point at the origin. This critical point is a local maximum, but not an absolute maximum \cite{Calvert-Vam}. \end{example} \begin{example} \label{kras-ex} The polynomial $y -(xy-1)^2$ has a saddle at $(-1/2,0)$ and no other critical points. At $[1,0,0]$ the level set $f = 0$ has one branch, but the general level set has two branches \cite{Krasinski-91}. \end{example} \begin{example} \label{two-parabola-ex} The polynomial $f(x,y) = (x-y^2)((x-y^2)(y^2+1) - 1)$ has its zero locus along the parabola $x = y^2$ and the curve $x = y^2 + 1/(y^2+1)$ which is asymptotic to this parabola. Its only critical point is a minimum at $(1/2,0)$. The level curves intersect $\reallineinfinity$ only at $[1,0,0]$, and they are tangent to $\reallineinfinity$ at this point. (The ``curve of tangencies'' (see the next section) is also tangent to $\reallineinfinity$ at $[1,0,0]$.) \end{example} \section{A formula for $i$ from the geometry of $grad \, f$} Let $f(x,y)$ be a real polynomial with isolated critical points. (Note that $f$ is thus not constant.) Let $i$ be the index of the gradient vector field of $f(x,y)$ around a large circle $C$ centered at the origin and containing the critical points, oriented in the counterclockwise direction. (Recall that the index is the topological degree of the map $C \to S^1$ defined by $ t \mapsto grad \, f(\alpha(t)) / |grad \, f(\alpha(t))|$, where $t \mapsto \alpha(t) $ is a parameterization of $C$.) This section contains the fundamental geometric decomposition of the index $i$ (Proposition \ref{geometric-index-formula}). To each point $q \in C$ where a smooth level curve of $f$ is tangent to $C$ at $q$ we assign the number $\pm 1/2$ or 0 as follows: If the level curve of $f$ is outside the circle $C$ near $q$, this number is $-1/2$. If it is inside $C$ near $q$, the number is $+1/2$. (These conditions are topological; the tangency may be algebraically degenerate.) If one side is outside and the other inside, or if the level set is contained in $C$ near $q$ (in which case $C$ is a connected component of the level set), the number is $0$. (See Figure \ref{u-sign}; the circle $C$ is dotted, and the level curves of $f$ are solid lines.) \begin{figure} \postscript{u-sign.eps}{0.7} \caption{Assigning $\pm 1/2$ or 0 to a point of tangency} \label{u-sign} \end{figure} The points $q$ where the level sets of $f$ are tangent to $C$ are the zeros of the (real) {\em curve of tangencies} $$f_x y - f_y x = 0$$ This curve may have reducible components. Choose the circle $C$ large enough so that it contains the compact components and the isolated singular points of the curve of tangencies and their points of common tangency. In the exterior of $C$ the curve of tangencies is a union of connected components. Each component $\gamma$ is a smooth arc which goes to infinity; we call this an {\em end} of the curve of tangencies. Choose the circle $C$ large enough so that the numbers $\pm 1/2, 0$ assigned above are constant along each end $\gamma$. (This is possible since the intersection multiplicity of $C$ and the level sets of $f$ is constant along each end $\gamma$ for $C$ large.) Let $i(\gamma)$ be the number $\pm 1/2$ or $0$ assigned to $\gamma$ in this fashion. Let $p(\gamma) \in \reallineinfinity$ be the endpoint of the closure of $\gamma$, and let $c(\gamma) \in \real \cup \{\infinity \} $ be the limiting value of $f(q)$ as $q$ goes to infinity along $\gamma$. \begin{lemma} \label{lemma-cgamma} For each end $\gamma$ of the curve of tangencies, the number $c(\gamma)$ exists. In fact, the function $f$ restricted to $\gamma$ is strictly increasing or decreasing. \end{lemma} \begin{xproof} Let $\Gamma(f) \subset {\bf P}^2 \times {\bf P}$ be the closure of the graph of $f$. The end $\gamma$ lifts uniquely to $\Gamma(f)$, intersecting the fiber over $p$ at a point $(p,c)$. The number $c$ is $c(\gamma)$. The function $f$ is strictly increasing or decreasing since $\gamma$ is perpendicular to the level sets of $f$. \end{xproof} For $p \in {\bf L}$ and $c \in \real \cup \{\infinity \} $, we let $$i_{p,c} = \sum i(\gamma)$$ where the sum is over all ends $\gamma$ with $p(\gamma) = p$ and $c(\gamma) = c$. We also let $$i_p = \sum_{c \in \real \cup \{\infinity \} } i_{p,c}$$ and $$i_{{\bf L}, \infty} = \sum_{p \in \reallineinfinity} i_{p,\infty}$$ \begin{lemma} The numbers $i_{p,c}$ are integers (not just half-integers). \end{lemma} \begin{xproof} The curve of tangencies can be lifted to $\Gamma(f)$, the graph of $f$. A real branch of this curve at $(p,c) \in \Gamma(f)$ is a pair of ends $\gamma \neq \gamma'$. If $i(\gamma) = 0$, then the intersection of the level sets of $f$ with the family of circles is degenerate along $\gamma$, and hence $i(\gamma') = 0$ as well. \end{xproof} \begin{proposition} \label{geometric-index-formula} If $f(x,y)$ is a real polynomial with isolated critical points, then $$i = 1 + \sum_{\scriptstyle p \in \reallineinfinity \atop \scriptstyle c \in {\bf R}} i_{p,c} + i_{{\bf L}, \infty}$$ \end{proposition} \begin{xproof} If no connected component of a level set of $f$ is contained in the large circle $C$, then we have that \begin{equation} \label{i-eqn} i = 1 + \sum i(\gamma) \end{equation} where the sum is over all ends $\gamma$ of the curve of tangencies: This is clearly true if all the points of tangency are regular values for the map $C \to S^1$ defined by $ t \mapsto grad \, f(\alpha(t)) / |grad \, f(\alpha(t))|$, where $t \mapsto \alpha(t) $ is a parameterization of $C$. If a point of tangency is not a regular value for this map (eg for $f(x,y) = y^3 + x$, or $y^4 + x$), then a small (topological) deformation shows that it still holds. The expression of the proposition is just a decomposition of (\ref{i-eqn}). Now suppose that a connected component of a level set $f=c$ is contained in $C$. We may assume without loss of generality that $c \gg 0$. Since one component of $f=c$ is compact, all components are compact by Proposition \ref{geom-prop-noncpt}. Thus by Proposition \ref{index-prop}, $i = 1$ (which is obvious here), $i_{{\bf L}, \infty} = 0$ and $i_{p,c} = 0$ for all $p \in {\bf L}$ and $c \in {\bf R}$. \end{xproof} A corollary of the Proposition is that \begin{equation} \label{i-ip-formula} i = 1 + \sum_{p \in \reallineinfinity} i_p \end{equation} The process of decomposing the index for the polynomial $f(x,y) = y(xy-1)$ of Example \ref{std-crpt-ex} is pictured in Figure \ref{std-crpt-index}. (The circle is dotted, the solid lines are the level sets of $f$, the dashed lines are the ends $\gamma$ of the curve of tangencies, and the numbers are $i(\gamma)$.) \begin{figure} \postscript{std-crpt-index.eps}{0.7} \caption{The index computation for the polynomial $y(xy-1)$} \label{std-crpt-index} \end{figure} A geometrically obvious example of the decomposition of the index is for a polynomial $f$ for which the real linear factors of $f_d$ are irreducible. In this case $i = 1 - d_{{\bf R}}$, $i_{p,c}= 0$ for $p \in \reallineinfinity$ and $c \in {\bf R}$, and $i_{\reallineinfinity, \infty} = -d_{{\bf R}}$. (This will be proved formally in Corollary \ref{corollary-dr}.) The invariants of Proposition \ref{geometric-index-formula} for selected polynomials are given in Table 1; all the nonzero $i_{p,c}$ for $c \in {\bf R}$ are listed. \begin{table} \label{index-invariants} \begin{tabular}{|l|r|l|l|l|r|} \hline $f(x,y)$ & $i_{{\bf L}, \infty}$ & $p \in f \cap \reallineinfinity$ & $c$ & $i_{p,c}$ & $i$ \\ \hline \hline Example \ref{std-crpt-ex}: $y(xy-1)$ &$-2$ & $[1,0,0]$ & $0$ & 1 & 0 \\ \hline Example \ref{std-nocrpt-ex}: $x(y^2-1)$ &$ -3$ & & & &$-2$ \\ \hline Example \ref{parabola-ex}: $y^2-x$ &$-1$ & & & & 0 \\ \hline Example \ref{max-min-ex}: $y^5 + x^2y^3-y$ & $-1$ & $[1,0,0]$ & $0$ & 2 &2 \\ \hline Example \ref{two-parabola-ex} & $-1$ & $[1,0,0]$ & $0$ & 1 & 1 \\ \hline Example \ref{kras-ex}: $y - (xy-1)^2$ & $-2$ & $[1,0,0]$ & 0 & 0 & $-1$ \\ \hline Example \ref{two-min-ex} & $-1$ & $[1,0,0]$ & 1 & 1 & 2 \\ \cline{4-5} & & & 2 & 1 & \\ \hline $y(x^2y-1)$ & $-2$ & $[1,0,0]$ & 0 & 1 & 0 \\ \hline \end{tabular} \caption{Index invariants of selected polynomials} \end{table} Note that the sum of the $i(\gamma)$'s making up $i_{p,c}$ is over ends $\gamma$ where $grad \, f$ points both out of and into the circle $C$; the process of decomposing the index described below does not work if the sum is just over those points where the gradient points out, as can be seen in the example $f(x,y) = y(x^2y-1)$. It is useful to have both the expression of Proposition \ref{geometric-index-formula} where the limiting value $c=\infty$ is separated out and put into $i_{\reallineinfinity, \infty}$ (see, for example, Proposition \ref{res-index-formula}), as well as the expression of Equation (\ref{i-ip-formula}), where these values are grouped by $p$ into $i_p$ (see, for example, Lemma \ref{i-d-local-estimate}). The decomposition of $i$ into the $i_{p,c}$ reflects the geometry of $f$ near infinity. It is apparently not related to the finite critical points of the polynomial and their critical values. \section{Resolutions and the geometry of level sets} In this section we describe the resolution of the points of indeterminacy of a polynomial on the line at infinity, and use this concept to establish some simple properties of its affine level curves. A polynomial $$f: {\bf R}^2 \to {\bf R} $$ extends to a map of real projective spaces $$\hat{f} : {\bf P}^2 \to {\bf P} $$ which is undefined at a finite number of points on the line at infinity ${\bf L}$. By blowing up these points one gets a manifold $M$ and a map $$ \pi : M \to {\bf P}^2 $$ such that the map $$ \tilde{f} : M \to {\bf P} $$ lifting $\hat{f}$ is everywhere defined. We call the map $ \tilde{f}$ a {\em resolution of $f$}. (We avoid the use of minimal resolutions, though this concept could be used to provide alternate proofs of some of the results below.) Any resolution $ \tilde{f} : M \to {\bf P} $ factors through $$\bar{f} : \Gamma(f) \to {\bf P} $$ where $\Gamma(f)$ is the graph of $f$ as defined above; note that the function $\bar{f}$ is everywhere defined. For example, a resolution of $y(xy-1)$ is given in Figure \ref{std-crpt-res-g}. \begin{figure} \postscript{std-crpt-res-g.eps}{0.7} \caption{A resolution of $y(xy-1)$} \label{std-crpt-res-g} \end{figure} The proper transforms of level curves of $f$ are ordinary lines; the exceptional sets are thick lines. The number $c^m$ next to a divisor means that at each smooth point of the divisor there are local coordinates $(u,v)$ in a neighborhood of the point such that the divisor is $u=0$ and $\tilde{f}(u,v) =(u-c)^m$. Let $f$ be a polynomial, and let $\tilde{f}$ be a resolution of $f$. By $A \gg 0$, we mean as usual that $A$ is large, but more precisely in this context we mean that $A$ is greater than the absolute value of all the critical values of $f$, and that if $|t| \geq A$, then the level sets $\tilde{f} = t$ are smooth and transversally intersect the exceptional sets of $\tilde{f}$. In particular, this means that the topological type of the level sets $f(x,y) = A$ and $f(x,y) = -A$ are independent of $A$. Fix a polynomial $f$. Let $$\reallineinfinity' = \{ [a,b,0] \in \reallineinfinity : f_d(a,b) = 0 \} $$ and let $$\reallineinfinity'' = \{ p \in \reallineinfinity : \mbox{ There is a $t \in {\bf R}$ with $p$ in the closure of $f(x,y) = t$} \} $$ Both $\reallineinfinity'$ and $\reallineinfinity''$ are finite sets of points. \begin{lemma} \label{geom-lemma-1} If $p \in \reallineinfinity''$, then \begin{enumerate} \item The point $p$ is in the closure of level sets $|f| = A$ for $A \gg 0$. \item If $\tilde{f}$ is a resolution of $f$, there is an exceptional set over $p$ on which $\tilde{f}$ is not constant. \item $p \in \reallineinfinity'$. \end{enumerate} \end{lemma} \begin{xproof} Choose a resolution $\tilde{f}$ of $f$. There is an exceptional set $E$ with $\pi(E) = p$ such that $\tilde{f} = t$ intersects $E$. The function $\tilde{f}$ restricted to $E$ is a real rational function. If $\tilde{f}$ restricted to $E$ is not constant, then there is a $q \in E$ such that $\tilde{f}$ restricted to $E - \{q\}$ is a polynomial. This implies (1) and (2) in this case. Now suppose that $\tilde{f}$ restricted to $E$ is constant. The (real) exceptional sets over $p$ form a connected tree. Since the value of $\tilde{f}$ is $t$ at one point on the tree, and is infinity at the points where the tree intersects the proper transform of ${\bf L}$, there is a component in the tree where $\tilde{f}$ takes a continum of large values. This implies (1) and (2) in this case. Part (3) follows since the level curves in ${\bf C}^2$ of the complexified polynomial $f$ intersect $\complexlineinfinity$ at exactly the zeros of the complexified $f_d$. \end{xproof} Note that the converse of (3) above is not true: For the polynomial $f(x,y) = y^4 + x^2$, for example, $f_4(1,0) = 0$, but $[1,0,0]$ is not contained in the closure of any real level curve. Given a real polynomial $f(x,y)$, we let $l''$ be the number of points in $\reallineinfinity''$. If $\tilde{f}$ is a resolution of $f$, we let $\xi_{{\bf L},nc}(\tilde{f})$ be the number of (real) exceptional sets $E$ of $\tilde{f}$ such that $\tilde{f} | E$ is nonconstant. (In the pictures, these exceptional sets are cross hatched by level curves of the polynomial.) \begin{corollary} \label{l-xi} If $\tilde{f}$ is a resolution of $f$, then $l'' \leq \xi_{\reallineinfinity, nc}(\tilde{f})$. \end{corollary} The inequality may be strict: The polynomial $x(y+1)(y+2) \dots (y+k)$ has $l''=2$ and $\xi_{\reallineinfinity, nc}(\tilde{f}) = k+1$. \begin{proposition} \label{geom-prop-cpt} Let $f(x,y)$ be a real polynomial, let $\tilde{f}$ be a resolution of $f$, and let $A \gg 0$. The following are equivalent: \begin{enumerate} \item The set $|f| = A$ is compact. \item The set $f = t$ is compact, for all $t$. \item $\xi_{\reallineinfinity, nc}(\tilde{f}) = 0$. \end{enumerate} Furthermore, if any of the above is true, then \begin{enumerate} \item One of $f = A$ and $f = -A$ is homeomorphic to a circle, and the other is empty. \item $f_d$ has no real linear factors ($d_{{\bf R}} = 0$). \end{enumerate} \end{proposition} \begin{xproof} (1) implies (2) by Lemma \ref{geom-lemma-1}. (2) is equivalent to (3) since $f=t$ is compact for all $t$ if and only if $\tilde{f} | E = \infty$ for all exceptional sets $E$. The additional conclusions are obvious. \end{xproof} \begin{proposition} \label{geom-prop-noncpt} Let $f(x,y)$ be a real polynomial, let $\tilde{f}$ be a resolution of $f$, and let $A \gg 0$. If $|f| = A$ is not compact, then \begin{enumerate} \item All the connected components of $|f| = A$ are noncompact. \item The number of connected components of $|f| = A$ is $2\xi_{\reallineinfinity, nc}(\tilde{f})$. \end{enumerate} \end{proposition} \begin{xproof} The geometry of the resolution implies (1). If $E$ is an exceptional set on which $\tilde{f}$ is not constant, then $\tilde{f}$ restricted to $E$ either takes the value $+A$ exactly twice, the value $-A$ exactly twice or the values $+A$ and $-A$ each once. Since the level sets $f = \pm A$ are transverse to $E$, this proves (2). \end{xproof} \section{A formula for $i$ in terms of a resolution} This section gives a formula (Proposition \ref{res-index-formula}) for computing the index $i$ and the terms $i_{p,c}$ in the decomposition of $i$ in terms of a resolution of the polynomial. In Section 7 this proposition will play a role in finding bounds on $i$. \begin{lemma} \label{lemma-i-p-c} Let $c \in {\bf R}$. If $p \in {\bf L}$ is not in the closure of the level set $f = c$, then $i_{p,c} = 0$. (Furthermore, $i^{abs}_{p,c} = 0$, in the notation of Section 7.) \end{lemma} \begin{xproof} Suppose $i_{p,c} \neq 0$. There is an end $\gamma$ of the curve of tangencies passing through $p$, and $f$ has limiting value $c$ along $\gamma$. Let $\tilde{f}: M \to {\bf P}^2$ be a resolution of $f$. The curve $\gamma$ lifts to $M$ and passes through some $q \in M$ with $\pi(q) = p$. Also $\tilde{f}(q) = c$. Hence the closure of $f=c$ intersects some exceptional set over $p$, so $p$ is in the closure of $f=c$. This is a contradiction. \end{xproof} The precise meaning of the notation $A \gg 0$ can be found in the previous section. \begin{proposition} \label{index-prop} If $f(x,y)$ is a real polynomial with isolated critical points, and if $|f| = A$ is compact for $A \gg 0$, then \begin{enumerate} \item $i = 1$. \item $i_{{\bf L}, \infty} = 0$. \item $i_{p,c} = 0$ for all $p \in {\bf L}$ and $c \in {\bf R}$. \end{enumerate} \end{proposition} \begin{xproof} By Proposition \ref{geom-prop-cpt}, either $f(x,y) = A$ or $f(x,y) = -A$ is homeomorphic to a circle; let us assume the former. Clearly $i = 1$ and $i_{{\bf L}, \infty} = 0$. Also $f(x,y) = c$ is compact for all $c$, so \ref{lemma-i-p-c} implies that $i_{p,c} = 0$. \end{xproof} Let $\tilde{f}$ be a resolution of $f$. For $p \in {\bf L}$ and $c \in {\bf R} \cup \{\infty\}$, let \begin{itemize} \item $i_{p,c}(\tilde{f})$ be the sum of the indices of $\tilde{f}$ at critical points $q \in M$ of $\tilde{f}$ such that $\tilde{f}(q) = c$ and $\pi(q) = p$. \item $\xi_{p,c}(\tilde{f})$ be the number of (real) exceptional sets $E$ of $\tilde{f}$ with $\pi(E) =p$ and $\tilde{f}|E = c$. \end{itemize} Recall that $\xi_{{\bf L},nc}(\tilde{f})$ is the number of (real) exceptional sets on which $\tilde{f}$ is nonconstant. \begin{proposition} \label{res-index-formula} If $f(x,y)$ is a real polynomial with isolated critical points and $\tilde{f}$ is a resolution of $f$, then \begin{enumerate} \item $i = 1 - \sum_{\scriptstyle p \in \reallineinfinity \atop \scriptstyle c \in {\bf R}} \bigl( i_{p,c}(\tilde{f}) + \xi_{p,c}(\tilde{f}) \bigr) - \xi_{{\bf L},nc}(\tilde{f})$ \item $ i_{p,c} = - i_{p,c}(\tilde{f}) - \xi_{p,c}(\tilde{f})$, for $p \in {\bf L}$ and $c \in {\bf R}$. \item $ i_{\reallineinfinity,\infty} = - \xi_{\reallineinfinity,nc}(\tilde{f})$ \end{enumerate} \end{proposition} By Proposition \ref{geometric-index-formula}, any two parts of this proposition imply the third, but proving each part separately is more instructive. In fact, Part (1) follows from a straight-forward application of Morse theory, Part (2) follows from Morse theory on a manifold with boundary, and Part (3) follows from the geometry of the large level curves. \begin{xproof} Proof of (1): Let \begin{itemize} \item $i_{{\bf L}}(\tilde{f}) = \sum_{\scriptstyle p \in \reallineinfinity \atop \scriptstyle c \in {\bf R}} i_{p,c}(\tilde{f})$ \item $\xi_{{\bf L},\infty}(\tilde{f}) = \sum_{p \in \reallineinfinity} \xi_{p,\infty}(\tilde{f})$ \item $\xi_{\reallineinfinity}(\tilde{f})= \sum_{\scriptstyle p \in \reallineinfinity \atop \scriptstyle c \in {\bf R}} \xi_{p,c}(\tilde{f})$ \end{itemize} We will do Morse theory on the function $\tilde{f}: M \to {\bf R} \cup \{ \infty \} $. Suppose $ c_1 < c_2 \dots < c_r $ in ${\bf R}$ are the critical values of $\tilde{f}$ restricted to the inverse image of ${\bf R}$. Choose $ \epsilon > 0 $ so that $ c_i + \epsilon < c_{i+1} - \epsilon $ for $1 \leq i < r$. Choose $A > 0$ so that $-A < c_1$ and $c_r < A$. Since a level set of $\tilde{f}$ corresponding to a regular value is a union of circles, \begin{equation} \label{eqn2} \chi(M) = \chi( \{ \tilde{f} \leq -A \} \cup \{ \tilde{f} \geq A \} ) + \sum_i \chi( \{ c_i - \epsilon \leq \tilde{f} \leq c_i + \epsilon \} ) \end{equation} where $\chi$ denotes Euler characteristic. The set $\{ \tilde{f} \leq -A \} \cup \{ \tilde{f} \geq A \}$ is homotopy equivalent to the set $ \tilde{f}^{-1}(\infty) $. This is a connected set, and is homotopy equivalent to a join of circles. These circles are the exceptional sets where $\tilde{f} = \infty$ together with the proper transform of ${\bf L}$. Thus $$ \chi( \{ \tilde{f} \leq -A \} \cup \{ \tilde{f} \geq A \} ) = -\xi_{\reallineinfinity,\infty}(\tilde{f}) $$ Next, $M$ is a connected sum of copies of ${\bf P}^2$, so $$\chi(M) = 1 - ( \xi_{\reallineinfinity}(\tilde{f}) + \xi_{\reallineinfinity, \infty}(\tilde{f}) + \xi_{{\bf L},nc}(\tilde{f}) ) $$ At a critical value $c_i$, $\chi( \{ c_i - \epsilon \leq \tilde{f} \leq c_i + \epsilon \} )$ is the sum of the indices of the corresponding critical points, by Morse theory. The sum of all these indices can be split into the parts coming from critical points in the finite plane and the line at infinity. Using this fact and the two equations above changes Equation (\ref{eqn2}) to $$ 1 - \xi_{\reallineinfinity}(\tilde{f}) - \xi_{\reallineinfinity, \infty} (\tilde{f}) - \xi_{{\bf L},nc}(\tilde{f}) = - \xi_{\reallineinfinity,\infty}(\tilde{f}) + i + i_{{\bf L}}(\tilde{f})$$ which proves (1). \medskip \noindent Proof of (2): (See Figures \ref{std-crpt-l} and \ref{std-crpt-res-l}.) Choose $\epsilon > 0$ so that $c$ is the only critical value in $(c-\epsilon, c+\epsilon)$. Let $C'$ be the (closed) exterior of the circle $C$ in the plane. Let $N'$ be the connected component of $ \{(x,y) \in {\bf R}^2: c-\epsilon \leq f(x,y) \leq c+\epsilon \} \cap C' $ containing $p$ in its closure. Choose the circle $C$ large enough so that each boundary component of $N'$ consists of an arc of $f = c \pm \epsilon$ followed by an arc of $C$ followed by an arc of $f = c \pm \epsilon$. \begin{figure} \postscript{std-crpt-l.eps}{0.7} \caption{The region $N'$ (bounded by dotted lines) for the polynomial $y(xy-1)$ at $p = [1,0,0]$ and $c =0$.} \label{std-crpt-l} \end{figure} \begin{figure} \postscript{std-crpt-res-l.eps}{0.7} \caption{The region $N$ (bounded by dotted lines)} \label{std-crpt-res-l} \end{figure} Let $$N = \overline{\pi^{-1}(N')} \subset M$$ We assume that $N$ is connected; if it is disconnected the proof is similar. We need a variant of the Poincar\'{e}-Hopf Theorem for vector fields on a manifold, or more properly, a variant of Morse theory on manifolds with boundary. (See, for instance, \cite{Milnor-TDV}, p. 35). For an oriented manifold $X$ with boundary, the Euler characteristic $\chi(X)$ is given by $$\chi(X) = \sum \{\mbox{indices of internal critical points}\}+ \{\mbox{index on boundary}\} $$ where the index of the vector field on the boundary is measured with respect to the outward pointing normal vector. This result is true for a gradient vector field on a nonorientable two-manifold $X$ without boundary, provided that the index is defined to be $+1$ at a local extremum and $-1$ at a saddle, or more generally defined at an arbitrary critical point using the result of Arnold \cite{Arnold-78} that the index of a polynomial $f(x,y)$ at a point $p$ is $1-r$, where $r$ is the number of real branches at $p$ of the curve $f(x,y) = f(p)$. If $X$ has a boundary with an orientable collar neighborhood, then the result is still true, provided that the index on the boundary is measured according as Figure \ref{u-sign}. Finally, the form we will use for $N$ is $$\chi(N) = 1+ \sum \{\mbox{indices of internal critical points}\}+ \{\mbox{index on boundary}\} $$ The term $+1$ comes from the fact that $N$ has four corners (see Figure \ref{std-crpt-res-l}). Choose a Riemannian metric on $N$ so that it agrees, on the boundary components of $N$ consisting of arcs of $C$, with the standard metric on the plane. We apply the above result to the gradient vector field of $\tilde{f}$. In the interior of $N$ there are the exceptional sets with $\tilde{f} = c$ and those critical points of $\tilde{f}$ which have critical value $c$. Since $N$ retracts to the exceptional sets contained in it, $$\chi(N) = 1 - \xi_{p,c}(\tilde{f})$$ The index of the internal critical points of $\tilde{f}$ is $i_{p,c}(\tilde{f})$. Finally, the index of the gradient vector field on the boundary of $N$ is $i_{p,c}$. Combining these facts proves (2). \medskip \noindent Proof of (3): (See Figure \ref{large-level-curves}.) If $|f| = A$ is compact for $A \gg 0$, then Proposition \ref{geom-prop-cpt} and Proposition \ref{index-prop} prove the result. Hence we can suppose that $|f| = A$ is not compact. If $\gamma$ is an end of the curve of tangencies and $c(\gamma) = \infty$, then $\gamma$ intersects $|f| = A$ for $A >> 0$. Let $I$ be a connected component of $|f| = A$ in ${\bf R}^2$. Each $\gamma$ which meets $I$ has $c(\gamma) = \infty$. Since $I$ begins and ends outside $C$ by Proposition \ref{geom-prop-noncpt} part (1), clearly the sum of the $ i(\gamma)$ over all $\gamma$ meeting $I$ is $-1/2$. By Proposition \ref{geom-prop-noncpt} part (2), there is a two-to-one correspondence between connected components of $|f| = A$ and exceptional sets where $\tilde{f}$ is nonconstant. This proves (3). \end{xproof} \begin{figure} \postscript{std-2-large-l.eps}{0.7} \caption{The level curves of $f(x,y) = y(x^2y-1)$ in ${\bf P}^2$} \label{large-level-curves} \end{figure} \begin{example} Consider the resolution $\tilde{f}$ of the polynomial $y(xy-1)$ shown in Figure \ref{std-crpt-res-g}. Over $p = [1,0,0]$, $\tilde{f}$ has two saddle points with critical value $0$, so $i_{[1,0,0], 0}(\tilde{f}) = -2$. There is one exceptional set $E$ over $[1,0,0]$ with $\tilde{f} | E = 0$, so $\xi_{[1,0,0], 0}(\tilde{f}) = 1$, and one exceptional set where $\tilde{f}$ is nonconstant, so so $\xi_{[1,0,0], nc}(\tilde{f}) = 1$. Over $p = [0,1,0]$, there is just one exceptional set where $\tilde{f}$ is nonconstant, so $\xi_{[0,1,0], nc}(\tilde{f}) = 1$. \end{example} Recall that $f_d$ is the homogeneous term of highest degree of the polynomial $f$, and that $d_{{\bf R}}$ is the real degree of $f$ as defined in the Introduction. \begin{corollary} \label{corollary-dr} If $f_d$ has no repeated real linear factors, then $i = 1-d_{{\bf R}}$. Also, $i_{p,c}= 0$ for $p \in \reallineinfinity$ and $c \in {\bf R}$, and $i_{\reallineinfinity, \infty} = -d_{{\bf R}}$. \end{corollary} \begin{xproof} This ``geometrically obvious'' result follows from Proposition \ref{res-index-formula}, since over each point where the level curves of $f$ intersect $\reallineinfinity$ the resolution is as in Figure \ref{linear-res-fig}. \end{xproof} \begin{figure} \postscript{linear-res.eps}{0.7} \caption{Resolution of a point where $d_p = 1$} \label{linear-res-fig} \end{figure} Let $h(x,y)$ be a real polynomial whose homogeneous term of highest degree is a product of irreducible real quadratic factors (ie, $d_{{\bf R}} = 0$). Instead of using a family of concentric circles (the level curves of the polynomial $x^2+y^2$) to define the $i_{p,c}$ , we could use the level curves of the polynomial $h(x,y)$. We let $i^h_{p,c}$ be the decomposition of the index defined this way. \begin{proposition} \label{generic-h} Let $f(x,y)$ be a real polynomial with isolated critical points. If $h(x,y)$ is a real polynomial with $d_{{\bf R}} = 0$, then $i^h_{p,c} = i_{p,c}$ for all $p \in \reallineinfinity$ and $c \in {\bf R}$. \end{proposition} \begin{xproof} Let $e$ be the degree of $h$. We have that $i_{p,c} = i^{h'}_{p,c}$ for $h' = (x^2+y^2)^{e/2}$. Choose a family $h^s$ of polynomials with $h^0 = h$ and $h^1 = h'$, and so that $h^s$ is a polynomial in $x$, $y$ and $s$ with degree $e$ in $x$ and $y$ and the homogeneous term of highest degree $(h^s)_e$ is a product of irreducible real quadratic factors. The curve of tangencies for $h^s$ is $f_xh^s_y - f_yh^s_x=0$. Choose a resolution $\tilde{f} : M \to {\bf R}$ of $f$, with $\pi: M \to {\bf P}^2$. The curve of tangencies for $h^s$ lifts to $M$. Let $q^s$ be an intersection point of the proper transform of the lifted curve of tangencies with the exceptional set $\pi^{-1}(\reallineinfinity)$. If $\pi(q^s) = p$ and $\tilde{f}(q^s)=c$, then $q^s$ contributes to $i^{h^s}_{p,c}$. The point $q^s$ varies continuously with $s$. Clearly $\pi(q^s)$ is independent of $s$. We will show that $\tilde{f}(q^s)$ is also independent of $s$. If $q^s$ does not move as a function of $s$, then this is true. Suppose $q^s$ moves. Fix an $s$ and call it $s_0$. Suppose that $q^{s_0}$ is contained in an exceptional set $E$. We will show that $\tilde{f} | E$ is constant. The function $\tilde{f} | E$ is rational. By moving $s_0$ a little, we may assume that $q^{s_0}$ is not an intersection point of $\pi^{-1}(L)$ and that $q^{s_0}$ is not a critical point of $\tilde{f} | E$. For each $s$, the level sets of $\tilde{h^s}$ form a system of regular neighborhoods of $\pi^{-1}(L)$. The level curves of $\tilde{h^s}$ and $\tilde{f}$ are tangent along the curve of tangencies for $h^s$. Thus the level curves of $\tilde{f}$ become tangent to $E$ in a neighborhood of $s_0$. Hence $\tilde{f}$ is constant in a neighborhood of $s_0$, and hence on $E$. \end{xproof} The proposition is obviously not true for $i_{p,\infty}$ with $p \in \reallineinfinity'$. Lastly, is there a polynomial $f$ with a resolution $\tilde{f}$ and a point $q$ in the exceptional set such that $\tilde{f}$ has a local extremum at $q$? \section{A formula for $i$ in terms of a Morsification} This section contains a formula (Proposition \ref{morsification-formula}) for computing the index $i$ of a polynomial and the $i_{p,c}$ from Section 3 in terms of a Morsification. The proof is straight-forward, and the results will not be used later. Some examples and a conjecture are included. \begin{definition} A {\em deformation} of a real polynomial $f(x,y)$ of degree $d$ with isolated critical points is a real polynomial $h(x,y,s)$ of degree $d$ in $x$ and $y$ with $h(x,y,0) = f(x,y)$. We let $f^s(x,y) = h(x,y,s)$. A deformation is a {\em Morsification} if for small $s \neq 0$, $(f^s)_d$ (the homogeneous term of degree $d$ of $f^s$) has no repeated real linear factors and the critical points of $f^s$ in ${\bf R}^2$ are nondegenerate. \end{definition} It is easy to show that a polynomial of degree $d$ has a Morsification, and that the set of Morsifications is a dense open subset of the set of polynomials of degree $\leq d$. If $f^s$ is a Morsification of $f$ and $p \in \reallineinfinity$ is in the closure of some level set of $f$ and $c \in \real \cup \{\infinity \} $, we let \begin{itemize} \item $\tilde{d}_p(f^s)$ be the number of real linear factors in $(f^s)_d$ which are deformations of the factor corresponding to $p$ in $f_d$. (The number $\tilde{d}_p(f^s)$ is also the number of points on ${\bf L}$ through which the level sets of the Morsification pass and which go to $p$ as $s \to 0$.) \item $i^{\infty}_{p,c}(f^s)$ be the index of the critical points of $f^s$ which go to $p$ and whose critical value goes to $c$ as $s \to 0$ \item $i^{\infty}_{p}(f^s) = \sum_{c \in \real \cup \{\infinity \} } i^{\infty}_{p,c}(f^s)$ \item $i^{\infty}(f^s) = \sum_{p \in \reallineinfinity} i^{\infty}_{p}(f^s)$ \end{itemize} Note that these invariants depend not just on the expression for $f^s$ but also on the sign of $s$. \begin{proposition} \label{morsification-formula} Let $f^s$ be a Morsification of $f$. \begin{enumerate} \item $i = 1- d_{{\bf R}}(f^s) - i^{\infty}(f^s)$ \item $i_p = 1-\tilde{d}_p(f^s) -i^{\infty}_p(f^s)$ for $p \in \reallineinfinity''$ (ie, for $p$ in the closure of some level set of $f$). \end{enumerate} \end{proposition} \begin{xproof} (1): We have $i(f^s) = i^{\infty}(f^s)+i$ by summing the indices of the critical points of $f^s$, and $i(f^s) = 1 - d_{{\bf R}}(f^s)$ by Corollary \ref{corollary-dr}. \noindent (2): We may assume that $p = [1,0,0]$. Choose a four-sided region on the $x > 0$ side of the plane containing the critical points of $f^s$ on that side which go to $p$ as $s \to 0$, and such that the left side of the region is a segment of the circle $C$ containing the points of tangency $q$ which approach $p$, the top and bottom are level sets of $f^s$, and whose right side is a segment of a larger circle $C'$. Orient the boundary of this region counter-clockwise. Choose a similar region on the left side of the plane. The index of $grad \, f^s$ about these two regions is clearly $1 - i_p - \tilde{d}_p(f^s)$. It is also the sum of the indices of the critical points in the interior of the regions, which is $i^{\infty}_p(f^s)$. \end{xproof} There is no obvious formula for $i_{p,c}$; see Conjecture \ref{ipc-conjecture} at the end of this section. \begin{example} Let $f(x,y) = y^2-x$ and $p = [1,0,0]$. Here $i_{p,c} =0$ for $c \in \real \cup \{\infinity \} $. The Morsification $f^s(x,y) = y^2 +sx^2 -x$ has a critical point at $(1/2s,0)$ with critical value $-1/4s$. If $s>0$ then $\tilde{d}_{p}(f^s) = 0$ and the critical point is a mininum, so $i^{\infty}_{p,0}(f^s) = 1$. If $s<0$ then $\tilde{d}_{p}(f^s) = 2$ and the critical point is a saddle, so $i^{\infty}_{[p,0}(f^s) = -1$. \end{example} \begin{example} Let $f(x,y) = y(xy-1)$ and $p = [1,0,0]$. Here $i_{p,0} = 1$ and $i_{p,c} = 0$ for $c \neq 1$. Define a Morsification by $f^s(x,y) = (y-sx)(xy-1)$. (This deformation simply tilts the line in the zero locus of $f$.) We have $\tilde{d}_{p}(f^s) = 2$. For $s > 0$, $f^s$ has two real nondegenerate critical points: \medskip \begin{tabular}{|l|l|l|} \hline critical point & type & critical value \\ \hline $(+ 1/ \sqrt{s}, + \sqrt{s})$ & saddle & $0$ \\ \hline $(- 1/ \sqrt{s}, - \sqrt{s})$ & saddle & $0$ \\ \hline \end{tabular} \medskip \noindent Thus $i^{\infty}_{p,0}(f^s) = -2$. \end{example} \begin{example} Let $f(x,y) = x(y^2-1)=x(y+1)(y-1)$ and $p = [1,0,0]$. Here $i_{p,\infty} = -1$ and $i_{p,c} = 0$ for $c \in {\bf R}$. We give two Morsifications. The first is $f^s(x,y) = x(y-sx+1))(y+sx-1)$. We have $\tilde{d}_p(f^s) = 2$. The function $f^s$ has critical points: \medskip \begin{tabular}{|l|l|l|} \hline critical point & type & critical value \\ \hline $(0,\pm 1)$ & saddle & $0$ \\ \hline $(1/s,0)$ & saddle & $0$ \\ \hline $(1/(3s), 0)$ & minimum ($s>0$) & $-4/(27s)$ \\ & maximum ($s<0$) & \\ \hline \end{tabular} \medskip \noindent Thus $i^{\infty}_{p,0} = -1$ and $i^{\infty}_{p,\infty} = 1$. The second Morsification is $f^s(x,y) = x(y^2 -1) + sx^3$. For $s > 0$, $\tilde{d}_p(f^s) = 0$ and $f^s$ has critical points \medskip \begin{tabular}{|l|l|l|} \hline critical point & type & critical value \\ \hline $(0,\pm 1)$ & saddle & $0$ \\ \hline $(1/\sqrt{3s},0)$ & mimimum & $-2/3\sqrt{3s}$ \\ \hline $(-1/\sqrt{3s}, 0)$ & maximum & $2/3\sqrt{3s}$ \\ \hline \end{tabular} \medskip \noindent Thus $i^{\infty}_{p,\infty} = 2$. (If $s < 0$, the only real critical points of $f^s$ are at $(0, \pm 1)$, and $\tilde{d}_p(f^s) = 2$.) \end{example} There is no obvious formula for $i_{p,c}$ in terms of a deformation, as can be seen in the case of $x(y^2-1)$ above. However, the following conjecture seem reasonable: \begin{conjecture} \label{ipc-conjecture} If $f^s$ is a deformation of $f$ and $p \in \reallineinfinity''$, then $i_{p,c} \leq -i^{\infty}_{p,c}(f^s)$ for $c \in {\bf R}$, and $i_{p,\infty} \leq 1 - i^{\infty}_{p,\infty}(f^s)$. \end{conjecture} For a deformation $f^s$ of $f$ it is easy to find bounds on the number of local maxima, minima and saddles near a point $p \in \reallineinfinity$. It would be interesting to see what possible combinations of these can occur, similar to the investigation in \cite{REU} or \cite{Shustin-96}. \section{Bounds on $i$} This section contains the main results of this paper, the bounds on the index $i$ of the gradient vector field of a real polynomial. The main tool is a bound on $i_p$ (Lemma \ref{i-d-local-estimate}). This, together with the interpretation of $i_{\reallineinfinity, \infty}$ in terms of a resolution (Proposition \ref{res-index-formula}) and some lemmas using techniques from Section 4 give the first main result (Theorem \ref{dr-theorem}). We next give a refinement (Lemma \ref{lemma-2}) of Lemma \ref{i-d-local-estimate}. This and a number of technical details gives the second main result (Theorem \ref{max-theorem}). As remarked in the Introduction, these is still a large gap between the upper bounds and known examples. Let $f(x,y)$ be a real polynomial of degree $d$ with isolated critical points. For $p \in \reallineinfinity$ and $c \in \real \cup \{\infinity \} $, recall that $$i_{p,c} = \sum i(\gamma)$$ where the sum is over ends $\gamma$ of the curve of tangencies with $p(\gamma) = p$ and $c(\gamma) = c$. We let $$i_{p,c}^{abs} = \sum |i(\gamma)| $$ and $$i^{abs}_p = \sum_{c \in \real \cup \{\infinity \} } i^{abs}_{p,c}$$ These invariants can be computed from a resolution of $f$, and in particular are integers, although this is not evident from the proof of \ref{res-index-formula}. (The invariants $i_{p,c}^{N}$ and so forth later in this section can also be computed from a resolution.) Recall that $\reallineinfinity'$ and $\reallineinfinity''$ were introduced in Section 4, that $\reallineinfinity'' \subset \reallineinfinity'$ and that $l''$ is the number of elements in $\reallineinfinity''$. If $p = [a,b,0] \in \reallineinfinity$, we let $d_p$ be the multiplicity of the factor $(bx-ay)$ in $f_d$. The next result is the local analogue of the estimate $|i| \leq d-1$ from the Introduction. This estimate, like the global one, is proved by relating the index to an algebraic intersection number. \begin{lemma} \label{i-d-local-estimate} If $p \in \reallineinfinity'$ (ie, if $d_p > 0$), then $$i_p^{abs} \leq d_p - 1 $$ \end{lemma} This follows from the next two results. We let $\Gamma$ be the projective completion of the curve of tangencies, and $\Gamma_{\bf C}$ be its complexification. We use $(A,B)_p$ to denote the intersection number of the curves $A$ and $B$ at $p$. \begin{lemma} \label{i-d-local-estimate-1} $i^{abs}_p \leq (\Gamma_{\bf C} \cdot \complexlineinfinity)_p$ \end{lemma} \begin{xproof} The number $i^{abs}_p$ is at most one half the number of ends $\gamma$ of the curve of tangencies with $p(\gamma) = p$. This number is the number of real branches of the completion of the curve of tangencies at $p$, which is at most the number of branches of $\Gamma_{\bf C}$ at $p$. This number is at most $(\Gamma_{\bf C} \cdot \complexlineinfinity)_p$, since no component of the curve of tangencies is contained in $\complexlineinfinity$. \end{xproof} \begin{lemma} \label{i-d-local-estimate-2} $(\Gamma_{\bf C} \cdot \complexlineinfinity)_p = d_p -1$ \end{lemma} \begin{xproof} Without loss of generality we may assume that $p = [1,0,0]$. We have that $f = y^{d_p}h(x,y) +{\mbox { \{terms of lower order\} } }$ where $d_p \geq 1$, $h(x,y)$ is homogeneous of degree $d-d_p$, and $y$ does not divide $h(x,y)$. Changing coordinates to $p$ and computing as in \cite[III.3]{Fulton} shows that $(\Gamma_{\bf C} \cdot \complexlineinfinity)_p = d_p - 1$ \end{xproof} Lemma \ref{i-d-local-estimate} is sharp: The polynomial of Example \ref{many-max-min-ex} at $p = [1,0,0]$ has $d_p = k$ and $i^{abs}_{p} = i_{p,0} = k-1$. Another example is provided by the polynomial $x(y+1)(y+2) \dots (y+k)$ at $p = [1,0,0]$, which has $d_p = k$ and $i^{abs}_{p} = -i_{p,0} = k-1$. Recall that the real degree of the polynomial is $$d_{{\bf R}} = \sum_{p \in \reallineinfinity} d_p $$ We let $$\tilde{d}_{{\bf R}} = \sum_{p \in \reallineinfinity''} d_p $$ This is the sum of the $d_p$'s over those $p$ in the line at infinity which are in the closure of some level set of the polynomial $f$. Note that \begin{equation} \label{string} l'' \leq \tilde{d}_{{\bf R}} \leq d_{{\bf R}} \leq d \end{equation} \begin{proposition} \label{i-upper-bound} If $f(x,y)$ is a real polynomial with isolated critical points, then $$i \leq 1 + \tilde{d}_{{\bf R}} - 2l''$$ \end{proposition} \begin{xproof} Let $\tilde{f}$ be a resolution of $f$. We have $$i = 1 + \sum_{\scriptstyle p \in \reallineinfinity \atop \scriptstyle c \in {\bf R}} i_{p,c} + i_{{\bf L}, \infty}$$ $$\leq 1 + \sum_{\scriptstyle p \in \reallineinfinity'' \atop \scriptstyle c \in {\bf R}} i_{p,c}^{abs} - \xi_{{\bf L}, nc}(\tilde{f})$$ $$\leq 1 +\sum_{p \in \reallineinfinity''} (d_p -1 ) - \xi_{{\bf L}, nc}(\tilde{f})$$ $$ \leq 1 + \tilde{d}_{{\bf R}} - l'' - \xi_{{\bf L}, nc}(\tilde{f})$$ The first line follows from Proposition \ref{geometric-index-formula}. The second follows since $i_{\reallineinfinity, \infty} = - \xi_{{\bf L}, nc}(\tilde{f})$ by Part (3) of Proposition \ref{res-index-formula}, $i_{p,c} \leq i_{p,c}^{abs}$ by definition, and $i_{p,c}^{abs} = 0$ for $p \in \reallineinfinity - \reallineinfinity''$ by Lemma \ref{lemma-i-p-c}. The third line follows from Lemma \ref{i-d-local-estimate}. The result follows from Corollary \ref{l-xi}. \end{xproof} To get a lower bound on the index, we need to compactify the plane ${\bf R}^2$ by the circle $$ {\bf S} = \{ (a,b,0) \in ({\bf R}^3 - 0) / {\bf R}^+ \} $$ The projection map $${\bf S} \to \reallineinfinity$$ which takes $(a,b,0)$ to $[a,b,0]$ will be denoted by $q \mapsto |q|$. If $\gamma$ is a end of the curve of tangencies, we let $q(\gamma) \in {\bf S}$ be the endpoint of the closure of $\gamma$ in ${\bf S}$. For example, for $y-(xy-1)^2$ there are ends $\gamma$ and $\gamma'$ with $q(\gamma) = (1,0,0)$, $q(\gamma') = (-1,0,0)$, $c(\gamma) = c(\gamma') = 0$ and $i(\gamma) = + 1/2$, $i(\gamma') = - 1/2$. The two cancel out to give $i_{[1,0,0],0} = 0$. Let $${\bf S}'' = \{ q \in {\bf S} : \mbox{ \ There is a $t \in {\bf R}$ such that $q$ is in the closure of $f(x,y) = t$} \} $$ This is a finite set of points; we let $s''$ denote the number of points in this set. Since the fibers of the projection map $${\bf S}'' \to \reallineinfinity''$$ consist of one or two points, $$l'' \leq s'' \leq 2l''$$ Thus the string of inequalities (\ref{string}) becomes \begin{equation} \label{string-2} 0 \leq \frac{1}{2} l'' \leq \frac{1}{2} s'' \leq l'' \leq \tilde{d}_{{\bf R}} \leq d_{{\bf R}} \leq d \end{equation} The next two lemmas are preparation for proving Proposition \ref{i-lower-bound}. \begin{lemma} $$\biggl | \sum_{q(\gamma) \in {\bf S}''} i(\gamma)\biggr | \leq \tilde{d}_{{\bf R}} - l''$$ \end{lemma} \begin{xproof} $$\biggl | \sum_{q(\gamma) \in {\bf S}''} i(\gamma) \biggr | \leq \sum_{q(\gamma) \in {\bf S}''} |i(\gamma)| $$ $$\leq \sum_{p \in \reallineinfinity''} \sum_{| q(\gamma)| = p} |i(\gamma)| $$ $$\leq \sum_{p \in \reallineinfinity''} (d_p - 1)$$ $$= \tilde{d}_{{\bf R}} - l''$$ \end{xproof} \begin{lemma} $$ \biggl| \sum_{q(\gamma) \in {\bf S} - {\bf S}''} i(\gamma) \biggr| \leq \frac{1}{2} s''$$ \end{lemma} \begin{xproof} If ${\bf S}''$ is empty ($s'' = 0$), then all the level sets of $f$ are compact and the left-hand sum is zero by Proposition \ref{index-prop}. Now suppose that ${\bf S}''$ is not empty. Fix a connected component $V$ of ${\bf S} - {\bf S}''$. Let $A \gg 0$ and let $I(V)$ be the connected component of $|f| = A$ which goes to $V$ as $A$ goes to infinity. By Propositions \ref{geom-prop-cpt} and \ref{geom-prop-noncpt}, $I(V)$ is not compact and hence has its endpoints on ${\bf S}$. We have (see the proof of Part (3) of Proposition \ref{res-index-formula}.) that $$\sum i(\gamma) = - \frac{1}{2}$$ where the sum is over all ends $\gamma$ of the curve of tangencies which intersect $I(V)$. If an end $\gamma$ has $q(\gamma) \in V$, then $\gamma$ intersects $I(V)$, and no other connected component of $|f| = A$. (Since the closure of no level curves of $f$ pass through the endpoint of $\gamma$, the function $f$ approaches infinity monotonely on $\gamma$. (Lemma \ref{lemma-cgamma})) However, some of ends $\gamma$ of the curve of tangencies may intersect $I(V)$ but have their endpoints on the endpoints of $V$. (I know of no examples of polynomials with this property, though.) The sum of the $i(\gamma)$ which intersect $I(V)$ and have endpoint a chosen endpoint of $V$ is 0 or $+ 1/2$. The sum over both endpoints of $V$ is thus 0, $+1/2$ or 1. Hence $$\sum_{q(\gamma) \in V} i(\gamma) = - \frac{1}{2},\ 0 \mbox{ or } +\frac{1}{2} $$ Since ${\bf S} - {\bf S}''$ has $s''$ connected components, this proves the lemma. \end{xproof} \begin{proposition} \label{i-lower-bound} If $f(x,y)$ is a real polynomial with isolated critical points, then $$i \geq 1 - d_{{\bf R}}$$ \end{proposition} \begin{xproof} We have $$i = 1 + \sum_{\gamma} i(\gamma)$$ $$= 1 + \sum_{q(\gamma) \in {\bf S}''} i(\gamma) + \sum_{q(\gamma) \in {\bf S} - {\bf S}''} i(\gamma) $$ $$\geq 1 + l'' - \tilde{d}_{{\bf R}} -\frac{1}{2}s''$$ $$\geq 1 - \tilde{d}_{{\bf R}}$$ $$\geq 1 - d_{{\bf R}}$$ The third line follows from the two lemmas above, and the fourth and fifth by the string of inequalities (\ref{string-2}). \end{xproof} The following is our first main result. \begin{theorem} \label{dr-theorem} Let $f(x,y)$ be a real polynomial of real degree $d_{{\bf R}}$ with isolated critical points, and let $i$ be the index of $grad \, f$ around a large circle containing the critical points. If all the level sets of $f$ are compact, then $i = 1$. Otherwise $$|i| \leq d_{{\bf R}} - 1$$ \end{theorem} \begin{xproof} If the level sets are compact then the result is obvious (see Proposition \ref{index-prop}.) If some level sets are not compact, then $l'' >0$. The upper bound follows from Proposition \ref{i-upper-bound}, and the lower bound from Proposition \ref{i-lower-bound}. \end{xproof} As remarked in the Introduction and Corollary \ref{corollary-dr}, it is easy to find ``generic'' polynomials which realize the lower bound of this theorem. The upper bound appears too high; the estimate of Proposition \ref{i-upper-bound} is somewhat better. Finally, the result seems somewhat obvious and one could hope for a better proof. We now further decompose $i_{p,c}$ and its refinements defined above. For $p \in \reallineinfinity$ and $c \in \real \cup \{\infinity \} $, recall that $i_{p,c} = \sum i(\gamma)$, summed over all ends $\gamma$ of the curve of tangencies with $p(\gamma) = p$ and $c(\gamma) = c$. We let $i^T_{p,c}$ (respectively, $i^N_{p,c}$) be the sum of the $i(\gamma)$'s such that the corresponding curve $\gamma$ is tangent (respectively, not tangent) to $\reallineinfinity$ at $p$. Thus $$i_{p,c} = i^N_{p,c} + i^T_{p,c} $$ We similarly decompose $i_{p,c}^{abs}$. As before, these numbers are all integers. For example, the polynomial $y(xy-1)$ of Example \ref{std-crpt-ex} has $i_{[1,0,0],0} = i^{N}_{[1,0,0],0} = 1$, and the polynomial in Example \ref{two-parabola-ex} has $i_{[1,0,0],0} = i^{T}_{[1,0,0],0} = 1$. We also let $$i^{N, abs}_p = \sum_{c \in \real \cup \{\infinity \} } i^{N, abs}_{p,c}$$ and define $i^{T, abs}_p$ similarly. The following lemma is a refinement of Lemma \ref{i-d-local-estimate}. \begin{lemma} \label{lemma-2} If $p \in \reallineinfinity'$, then $$i^{N,abs}_{p} + 2 i^{T,abs}_{p} \leq d_p-1$$ \end{lemma} \begin{xproof} We let $\Gamma^T$ (respectively, $\Gamma^N$) be the product of the branches of the curve of tangencies $\Gamma$ tangent (respectively, not tangent) to $\reallineinfinity$ at $p$, so that $\Gamma = \Gamma^T \Gamma^N$ near $p$. As in Lemma \ref{i-d-local-estimate-1}, we have that \begin{equation} \label{eqn-n} i^{N,abs}_p \leq (\Gamma^N_{\bf C} \cdot \complexlineinfinity )_p \end{equation} Similarly \begin{equation} \label{eqn-t} i^{T,abs}_p \leq \frac{1}{2}(\Gamma^T_{\bf C} \cdot \complexlineinfinity )_p \end{equation} since these branches are tangent to $\reallineinfinity$ at $p$. Thus $$i^{N,abs}_p + 2i^{T,abs}_p \leq (\Gamma^N_{\bf C} \cdot \complexlineinfinity )_p + (\Gamma^T_{\bf C} \cdot \complexlineinfinity )_p = (\Gamma_{\bf C} \cdot \complexlineinfinity )_p = d_p -1$$ as before. \end{xproof} Some stronger local estimates are probably true. In fact, let $f = f^Nf^T$ at $p$, where $f^T$ (respectively, $f^N$) are the branches of $f = t$ tangent (respectively, not tangent) to $\reallineinfinity$ at $p$ (which is independent of $t \in {\bf R}$), and let $d_p^T$ (respectively, $d_p^N$) be the intersection number of $f^T = t$ (respectively, $f^N = t$) with $\reallineinfinity$ at $p$. Thus $$d_p = d_p^N + d_p^T$$ It seems reasonable to expect that $i^{N,abs}_{p} \leq d^N_p - 1$ and $i^{T,abs}_{p} \leq (1/2) d^T_p - 1$ for $p \in \reallineinfinity$, and that these estimates are sharp. We need one more technical lemma: \begin{lemma} \label{lemma-n-t} Fix $p \in \reallineinfinity$ and let $\tilde{f}$ be a resolution of $f$. If there are $c, c' \in {\bf R}$ such that $i^{N,abs}_{p,c} > 0$ and $i^{T,abs}_{p,c'}>0$, then $\xi_{p,nc}(\tilde{f}) \geq 2$ (ie, there are at least two exceptional sets over $p$ on which $\tilde{f}$ is not constant). \end{lemma} \begin{xproof} There are ends $\gamma$ and $\gamma'$ of the curve of tangencies with $p(\gamma) = p(\gamma') = p$, $c(\gamma) = c$ and $c(\gamma') = c'$, and with $\gamma$ (respectively $\gamma'$) tangent (respectively, not tangent) to $\reallineinfinity$ at $p$. By the proof of Lemma \ref{lemma-i-p-c}, the closure of the level curve $f = c$ (respectively, $f = c'$) intersects an exceptional set $E$ (respectively, $E'$) over $p$. By the proof of Lemma \ref{geom-lemma-1}, there is at least one exceptional set over $p$ where $\tilde{f}$ is not constant. In fact, there are at least two such exceptional sets: Since $\gamma$ and $\gamma'$ have distinct tangents at $p$, the limit of $f$ must be infinite on all but a finite number of tangent directions between these by \cite[Proposition 1.3]{REU}, so $E$ and $E'$ are distinct and the chain of exceptional sets connecting $E$ and $E'$ must have at least one member $E_0$ with $f | E_0 = \infty$. Thus there must be an exceptional set in the chain connecting $E$ and $E_0$ where $\tilde{f}$ is nonconstant, and similarly between $E'$ and $E_0$. \end{xproof} \begin{theorem} \label{max-theorem} If $f(x,y)$ is a real polynomial of degree $d$ with isolated critical points, and $i$ is the index of $grad \, f$ around a large circle containing the critical points, then $$ i \leq max \{ 1,d-3 \}$$ \end{theorem} The difficult part of this proof is the case when $l''=0$, i.e. when the closures of the level curves of the polynomial intersect the line at infinity at just one point. \begin{xproof} If $l'' = 0$ then $i = 1$ by Lemma \ref{index-prop}. If $l'' \geq 2$ then $i \leq d_{{\bf R}} -3$ by Proposition \ref{i-upper-bound}. Thus we must treat the case $l'' = 1$. We may suppose without loss of generality that $p = [1,0,0]$ is the only point where the real level curves of $f$ intersect $\reallineinfinity$. The point $p$ will remain fixed for the rest of the proof. Let $\tilde{f}$ be a resolution of $f$. From Proposition \ref{geometric-index-formula}, Lemma \ref{lemma-i-p-c} and part (3) of Proposition \ref{res-index-formula} we have that \begin{equation} \label{equation-l-1} i = 1 + \sum_{c \in {\bf R}} i_{p,c} - \xi_{p,nc}(\tilde{f}) \end{equation} Since $\xi_{\reallineinfinity, nc}(\tilde{f}) \geq l'' = 1$ by Corollary \ref{l-xi}, we also have the weaker form of this equation: $$i \leq \sum_{c \in {\bf R}} i_{p,c}$$ Suppose $d_p < d$. We have that $d_p \leq d-2$ since the roots of $f_d$ other than $p$ are complex and hence occur in conjugate pairs. Thus $$i \leq \sum_{c \in {\bf R}} i_{p,c} \leq \sum_{c \in {\bf R}} i^{abs}_{p,c} \leq i^{abs}_p \leq d_p -1 \leq d-3$$ where the fourth inequality follows from Lemma \ref{i-d-local-estimate}. Thus we may assume that $d_p = d$, so that $$f(x,y) = \pm y^d + h(x,y)$$ where $h$ has degree $e < d$. If $h$ is a function of $x$ alone, then from far away $f(x,y)$ looks like $\pm y^d \pm x^e$, which has $i = 0$ or $\pm 1$. Thus we may assume that $h$ is a nonconstant function of both $x$ and $y$. The rest of the proof is divided into three cases: \noindent Case 1: Suppose $\sum_{c \in {\bf R}} i^N_{p,c} = 0$. Then $$i \leq \sum_{c \in {\bf R}} i_{p,c} = \sum_{c \in {\bf R}} i^T_{p,c} \leq \sum_{c \in {\bf R}} i^{T,abs}_{p,c} \leq i^{T,abs}_p$$ $$\leq \frac{1}{2}(d_p-1) \leq max \, \{1, d-3 \} $$ where the fourth inequality follows from Lemma \ref{lemma-2}. \noindent Case 2: Suppose that $\sum_{c \in {\bf R}} i_{p,c}^T = 0$. We have that $$i \leq \sum_{c \in {\bf R}} i_{p,c} = \sum_{c \in {\bf R}} i_{p,c}^N \leq \sum_{c \in {\bf R}} i_{p,c}^{N,abs} \leq i^{N,abs}_p \leq ( \Gamma^N_{\bf C} \cdot \complexlineinfinity ) _p$$ where the last inequality is Equation (\ref{eqn-n}). Since $f(x,y) = \pm y^d + h(x,y)$ where $h$ is a nonconstant function of both $x$ and $y$ of degree less than $d$, a computation shows that $z$ divides the term of lowest degree in the curve of tangencies localized at $p$. Hence $\Gamma^T$ is nonempty, so $( \Gamma^T_{\bf C} \cdot \complexlineinfinity ) _p \geq 2$. Thus $$( \Gamma^N_{\bf C} \cdot \complexlineinfinity ) _p = ( \Gamma_{\bf C} \cdot \complexlineinfinity )_p - ( \Gamma^T_{\bf C} \cdot \complexlineinfinity )_p$$ $$ \leq (d_p - 1) - 2 $$ $$ = d_p - 3$$ $$ = d-3$$ \noindent Case 3: Suppose that $i^{N, abs}_{p,c} > 0$ and $i^{T, abs}_{p,c'}>0$ for some $c, c' \in {\bf R}$. We have by Equation (\ref{equation-l-1}) and Lemma \ref{lemma-n-t} that $$i \leq \sum_{c \in {\bf R}} i_{p,c} - 1$$ $$ \leq \sum_{c \in {\bf R}} i_{p,c}^{abs} - 1$$ $$ = \sum_{c \in {\bf R}} i_{p,c}^{N,abs} + 2 \biggl( \sum_{c \in {\bf R}} i_{p,c}^{T,abs} \biggr) - \sum_{c \in {\bf R}} i_{p,c}^{T,abs} - 1$$ $$ \leq (d-1) -1-1 = d-3 $$ where the last inequality follows from Lemma \ref{lemma-2}. \end{xproof} \section{Vanishing cycles} Suppose $p \in \reallineinfinity$ and $c \in {\bf R} \cup \{ \infty \}$. In this section we relate the term $i_{p,c}$ in the decomposition of the index $i$ of a real polynomial $f(x,y)$ to the number of vanishing cycles $\nu_{p,c}$ of the corresponding complex polynomial at $(p,c)$. This number is defined to be the ``jump'' in the Milnor number of the family $f(x,y)=t$ of complex polynomials at $p$ when $t=c$. (For a detailed discussion of this notion, in particular the case $c = \infty$, the reader is referred to \cite{Durfee-P96}.) Recall that $i^{abs}_{p,c}$ was defined in the last section, and that $i_{p,c} \leq i^{abs}_{p,c}$. \begin{proposition} Let $f(x,y)$ be a real polynomial with isolated critical points. If $p \in \reallineinfinity'$ (ie, $p$ is a zero of $f_d$) and $c \in \real \cup \{\infinity \} $, then $$i_{p,c}^{abs} \leq \nu_{p,c} $$ \end{proposition} \begin{xproof} Suppose $c \in {\bf R}$. (The proof for $c = \infty$ is similar.) As in the proof of Lemma \ref{i-d-local-estimate-1}, The number $i^{abs}_{p,c}$ is at most one half the number of ends $\gamma$ of the curve of tangencies with $p(\gamma) = p$ and $c(\gamma)= c$. Since $f$ is either strictly increasing or decreasing on each end $\gamma$ (Lemma \ref{lemma-cgamma}), we may assume without loss of generality (replace $f$ by $-f$), that the number of ends $\gamma$ with $p(\gamma) = p$, $c(\gamma)= c$ and $f|\gamma < c$ is at most the number of ends $\gamma$ with $p(\gamma) = p$, $c(\gamma)= c$ and $f|\gamma > c$. Hence $i^{abs}_{p,c}$ is at most the number $v$ of ends $\gamma$ with $p(\gamma) = p$, $c(\gamma)= c$ and $f|\gamma > c$. Since $f$ is strictly decreasing to $c$ along $\gamma$, $v$ is the number of intersection points in ${\bf R}^2$ of the curves $xf_y - yf_x = 0$ (the curve of tangencies) and $f = c + \epsilon$ which approach $p$ as $\epsilon \downarrow 0$. If we assume without loss of generality that $p = [1,0,0]$, we may replace the curve $xf_y - yf_x = 0$ by the curve $f_y = 0$. Thus $v$ is at most the number of intersection points of the complex curves $f_y = 0$ and $f = c + \epsilon$ which approach $p$ as $\epsilon \downarrow 0$. This number is well-known to be $\nu_{p,c}$ (see for example \cite[2.13]{Durfee-P96}). \end{xproof} The inequality of the proposition can be strict; for example the polynomial $y(x^ay-1)$ at $p = [1,0,0]$ and $c=0$ has $i_{p,c} = 1$ and $\nu_{p,c} = a+1$. \bibliographystyle{alpha} \newcommand{\etalchar}[1]{$^{#1}$}

9307

alg-geom/9307002

en

https://arxiv.org/abs/alg-geom/9307002

alg-geom/9307002

Robert Friedman

Robert Friedman

Vector bundles and $SO(3)$ invariants for elliptic surfaces I

19 pages, AMS-TeX

This paper is the first in a series of three devoted to the smooth classification of simply connected elliptic surfaces. The method is to compute some coefficients of Donaldson polynomials of $SO(3)$ invariants whose second Stiefel-Whitney class is transverse to the unique primitive class $\kappa$ such that a positive multiple of $\kappa$ is the class of a general fiber on the surface. In this paper, we collect preliminary results on elliptic surfaces and vector bundles and give the general outline of the argument.

alg-geom

\section{1. Introduction.} Beginning with Donaldson's seminal paper on the failure of the $h$-cobordism theorem in dimension 4 [4], the techniques of gauge theory have proved to be highly successful in analyzing the smooth structure of simply connected elliptic surfaces. Recall that a simply connected elliptic surface $S$ is specified up to deformation type by its geometric genus $p_g(S)$ and by two relatively prime integers $1\leq m_1\leq m_2$, the multiplicities of its multiple fibers. Here, if $p_g(S) = 0$, a surface $S$ such that $m_1 =1$ is rational, and thus all surfaces $S$ with $p_g(S) = 0$ and $m_1=1$ are deformation equivalent and in particular diffeomorphic. Moreover, if $p_g(S) = 1$ and $m_1 = m_2 =1$, then $S$ is a $K3$ surface. In all other cases, $S$ is a surface with Kodaira dimension one. Our goal in this series of three papers is to prove the following result: \theorem{} Two possibly blown up simply connected elliptic surfaces are diffeomorphic if and only if they are deformation equivalent. More precisely, suppose that $S$ and $S'$ are relatively minimal simply connected elliptic surfaces. Suppose that $S$ has multiple fibers of multiplicities $m_1$ and $m_2$, with $1\leq m_1 \leq m_2$, and that $S'$ has multiple fibers of multiplicities $m_1'$ and $m_2'$, with $1\leq m_1' \leq m_2'$. Let $\tilde S$ be a blowup of $S$ at $r$ points and $\tilde S'$ a blowup of $S'$ at $r'$ points. Suppose that $\tilde S$ and $\tilde S '$ are diffeomorphic. Then $r=r'$ and $p_g(S) = p_g(S')$, and moreover: \roster \item"{(i)}" If $p_g(S)> 0$, then $m_1=m_1'$ and $m_2=m_2'$. \item"{(ii)}" If $p_g(S) = 0$, then $S$ is rational, i.e\. $m_1=1$, if and only if $S'$ is rational if and only if $m_1'=1$. If $S$ and $S'$ are not rational, then $m_1=m_1'$ and $m_2=m_2'$. \endroster \endstatement \smallskip There is also a routine generalization to the case of a finite cyclic fundamental group. The statements in the theorem that $r=r'$ and $p_g(S) = p_g(S')$ are easy consequences of the fact that $\tilde S$ and $\tilde S'$ are homotopy equivalent, and the main point is to determine the multiplicities. Before discussing the proof of the theorem in more detail, we shall review some of the history of the classification of simply connected elliptic surfaces: \theorem{1.1 [8]} There is a function $f(m_1, m_2)$ defined on pairs of relatively prime positive integers $(m_1, m_2)$ such that $f$ is symmetric and finite-to-one provided that neither $m_1$ nor $m_2$ is $1$, with the following property: Let $S$ and $S'$ be two simply connected surfaces with $p_g(S) = 0$. Denote the multiplicities of the multiple fibers of $S$ by $m_1, m_2$ and the multiplicities for $S'$ by $m_1', m_2'$. If $S$ and $S'$ are diffeomorphic, then $f(m_1, m_2) = f(m_1', m_2')$. Moreover, let $\tilde S$ and $\tilde S'$ be blowups of $S$ and $S'$ at $r$ points. Then: \roster \item"{(i)}" Every diffeomorphism $\psi \: \tilde S \to \tilde S'$ pulls back the cohomology class of an exceptional curve on $\tilde S'$ to $\pm$ the cohomology class of an exceptional curve on $\tilde S$. \item"{(ii)}" Every diffeomorphism $\psi \: \tilde S \to \tilde S'$ pulls back the cohomology class of a general fiber on $\tilde S'$ to a rational multiple of the cohomology class of a general fiber on $\tilde S$. \item"{(iii)}" If $\tilde S$ and $\tilde S'$ are diffeomorphic, then $f(m_1, m_2) = f(m_1', m_2')$. \qed \endroster \endstatement The function $f(m_1, m_2)$ was then determined by S\. Bauer [1] (the case $m_1 = 2$ is also in [8]): \theorem{1.2} In the above notation, $$f(m_1, m_2) = \frac{(m_1^2-1)(m_2^2-1)}3 -1.\qed$$ \endstatement For the case $p_g(S) > 0$, there is the following result [9]: \theorem{1.3} Let $S$ and $S'$ be two simply connected surfaces with $p_g(S) > 0$. Denote the multiplicities of the multiple fibers of $S$ by $m_1, m_2$ and the multiplicities for $S'$ by $m_1', m_2'$. If $S$ and $S'$ are diffeomorphic, then $m_1m_2 = m_1'm_2'$. Moreover, let $\tilde S$ and $\tilde S'$ be blowups of $S$ and $S'$ at $r$ points. Then: \roster \item"{(i)}" Every diffeomorphism $\psi \: \tilde S \to \tilde S'$ pulls back the cohomology class of an exceptional curve on $\tilde S'$ to $\pm$ the cohomology class of an exceptional curve on $\tilde S$. \item"{(ii)}" Except possibly for $p_g(S) =1$, every diffeomorphism $\psi \: \tilde S \to \tilde S'$ pulls back the cohomology class of a general fiber on $\tilde S'$ to a rational multiple of the cohomology class of a general fiber on $\tilde S$. \item"{(iii)}" If $\tilde S$ and $\tilde S'$ are diffeomorphic, then $m_1m_2 = m_1'm_2'$. \qed \endroster \endstatement The crux of the argument involves calculating a coefficient of a suitable Donaldson polynomial invariant $\gamma _c(S)$. In fact, it is shown in [9] that $\gamma _c(S)$ can be written as a polynomial in the intersection form $q_S$ and the primitive class $\kappa$ such that the class of a general fiber $[f]$ of $S$ is equal to $m_1m_2\kappa$, and that, for $c$ sufficiently large, the first nonzero coefficient of this polynomial is given as follows: let $n = 2c-2p_g(S) -1$ and $d = 4c - 3p_g(S) -3$. If $\gamma _c(S) = \sum _{i=0}^{[d/2]}a_iq_S^i\kappa ^{d-2i}$, then $a_i = 0$ for $i>n$ and $$a_n = \frac{d!}{2^nn!}(m_1m_2)^{p_g(S)}.$$ The proof of this statement involves showing that the moduli space of stable vector bundles $V$ with $c_1(V) = 0$ and $c_2(V) = c$ fibers holomorphically over a Zariski open subset of a projective space, and that the fiber consists of $m_1m_2$ copies of a complex torus. It is natural to wonder if the techniques of [9] can be pushed to determine some of the remaining terms. However, it seems to be difficult to use the vector bundle methods used in [9] to make the necessary calculations, even in the case of no multiple fibers. Thus, it is natural to look for other techniques to complete the $C^\infty$ classification of elliptic surfaces. Using a detailed analysis of certain moduli spaces of vector bundles, Morgan and O'Grady [15] together with Bauer [2] were able to calculate the coefficient $a_{n-1}$ in case $p_g(S) = 1$ and $c=3$. The calculation is long and involved for the following reason: the moduli spaces are nonreduced, not necessarily of the correct dimension, and (in the case of trivial determinant) the integer $c$ is not in the ``stable range." The final answer is that, up to a universal combinatorial factor, $a_{n-1} = m_1m_2(2m_1^2m_2^2 - m_1^2 - m_2^2)$. {}From this and from the knowledge of $m_1m_2$, it is easy to determine the unordered pair $\{m_1, m_2\}$. In addition, the calculation shows that the class of a fiber of $S$ is preserved up to rational multiples in case $p_g(S) = 1$ as well (the possible exception in (ii) of Theorem 1.3 above), provided that not both of $m_1$ and $m_2$ are 1. In the proof of the main theorem, we shall use the following results. Aside from standard techniques in the theory of vector bundles, and the gauge theory results that are described in the book [9], we use only the results of this series of papers and of [9] to handle the case $p_g(S) >0$. In case $p_g(S) = 0$, we use the results in this series and in [8], as well as the calculation of Bauer described in Theorem 1.2 in case $m_1m_2 \equiv 0\mod 2$. In case $m_1m_2\equiv 1 \mod 2$, our proof does not depend on Bauer's results. Next we outline the strategy of the argument. Following a well-established principle [6], [10], we shall work with $SO(3)$-invariants instead of $SU(2)$-invariants since these are often much easier to calculate. Moreover, in case $b_2^+=1$ a good choice of an $SO(3)$ invariant can simplify the problem that the invariant depends on the choice of a certain chamber. Thus we must choose a class $w\in H^2(S; \Zee/2\Zee)$ to be the second Stiefel-Whitney class of a principal $SO(3)$ bundle, although it will usually be more convenient ot work with a lift of $w$ to $\Delta \in H^2(S; \Zee)$. One possible choice of a lift of $w$ would be the class $\kappa$, the primitive generator of $\Zee ^+\cdot [f]$, or perhaps $c_1(S)$, or even $[f]$. All of these classes are rational multiples of $[f]$, and they do not simplify the problem. Instead we shall consider the case where $\Delta$ is transverse to $f$, more specifically where $\Delta \cdot \kappa = 1$. Of course, we shall need to choose $\Delta$ to be the class of a holomorphic divisor as well in order to be able to apply algebraic geometry. As we shall see in Section 2, we can always make the necessary choices and the final calculation will show that the answer does not depend on the choices made. Note that $\Delta$ is well-defined up to a multiple of $\kappa$, and that the choices $\Delta$ and $\Delta -\kappa$ correspond to different choices for $w^2\equiv p\mod 4$. Finally, as we shall show in Section 3, in case $b_2^+(S) =1$ or equivalently $p_g(S)=0$, there is a special chamber $\Cal C(w,p)$ which is natural in an appropriate sense under diffeomorphisms. With this choice of $\Delta$, the study of the relevant vector bundles divides into two very different cases, depending on whether $m_1m_2\equiv 0\mod 2$ or $m_1m_2\equiv 1\mod 2$. In this paper we shall collect results which are needed for both cases and show how the main theorems follow from the calculations in Parts II and III. In Part II of this series, we shall consider the case where $\Delta \cdot \kappa = 1$ and $m_1m_2\equiv 0\mod 2$. In this case, $m_1$, say, is even. Since $\Delta \cdot f =m_1m_2 \Delta \cdot \kappa$, a vector bundle $V$ with $c_1(V) = \Delta$ has even degree on a general fiber $f$. At first glance, then, it seems as if we are again in the situation of [7] and [9] and that there is no new information to be gained from the Donaldson polynomial. However, it turns out that the asymmetry between $m_1$ and $m_2$ appears in the moduli space as well. In this case, the moduli space again fibers holomorphically over a Zariski open subset of a projective space. But the fibers now consist of just $m_2$ copies of a complex torus. We then have, by an analysis that closely parallels [9], the following result: \theorem{1.4} Let $w$ and $p$ be as above, and set $d = -p - 3(p_g(S) + 1)$ and $n = (d-p_g(S))/2$. Suppose that $\gamma _{w,p}(S)$ is the Donaldson polynomial for the $SO(3)$-bundle $P$ over $S$ with $w_2(P) = w$ and $p_1(P) = p$ where if $p_g(S) =0$ this polynomial is associated to the chamber $\Cal C(w,p)$ defined in \rom{(3.6)} below. Then, assuming that $m_1$ is even and writing $\gamma _{w, p}(S)= \sum _{i=0}^{[d/2]}a_iq_S^i\kappa _S^{d-2i}$, we have, for all $p$ such that $-p\geq 2(4p_g(S) + 2)$, $a_i= 0$ for $i>n$ and $$a_n = \frac{d!}{2^nn!}(m_1m_2)^{p_g(S)}m_2.$$ \endstatement \medskip In particular, the leading coefficient contains an ``extra" factor of $m_2$. Using this and either [9] in case $p_g(S) > 0$ or [1] in case $p_g(S) = 0$, we may then determine $\{m_1, m_2\}$. Note that in case $p_g(S) = 0$ and one of $m_1$, $m_2$ is 1, then it cannot be $m_1$ since $m_1$ is even. Thus $m_2 =1$ and the leading coefficient does not determine $m_1$ (as well it cannot). Finally, in Part III we shall discuss the case where $\Delta \cdot \kappa = 1$ and $m_1m_2\equiv 1\mod 2$. If $m_1m_2 \equiv 1 \mod 2$, then vector bundles $V$ with $c_1(V ) = \Delta$ have odd degree when restricted to a general fiber, and the general methods for studying vector bundles on elliptic surfaces described in [7] and [9] Chapter 7 do not apply. Thus we must develop new techniques for studying such bundles, and this is the subject of Part III of this series. Fortunately, it turns out that this moduli problem is in many ways much simpler to study than the case of even degree on the general fiber. For example, as long as the expected dimension is nonnegative, for a suitable choice of ample line bundle the moduli space is always nonempty, irreducible, and smooth of the expected dimension. Moreover a Zariski open subset of the moduli space is independent of the multiplicities, and from this one can show easily that the leading coefficient of the Donaldson polynomial for the corresponding $SO(3)$-bundle is (up to the usual combinatorial factors) equal to 1. At first glance, this rather disappointing result suggests that no new information can easily be gleaned from the Donaldson polynomial. However, this suggestion is misleading: in some sense, the structure of the moduli space allows the contribution of the multiple fibers to be localized around the multiple fibers, enabling us to calculate the next two coefficients in the Donaldson polynomial. By contrast, in the case of trivial determinant, the moduli space for a surface with two multiple fibers of multiplicities $m_1$ and $m_2$ looks roughly like a branched cover of the corresponding moduli space for a surface without multiple fibers. A further simplification is that we can work with moduli spaces of small dimension, for example dimension two or four. Using the vector bundle results, we shall show: \theorem{1.5} Let $S$ be a simply connected elliptic surface with two multiple fibers $m_1$ and $m_2$, with $m_1m_2 \equiv 1\mod 2$. Let $w \in H^2(S; \Zee/2\Zee)$ satisfy $w\cdot \kappa = 1$. Suppose that $\gamma _{w,p}(S)$ is the Donaldson polynomial for the $SO(3)$-bundle $P$ over $S$ with $w_2(P) = w$ and $p_1(P) = p$ where if $p_g(S) =0$ this polynomial is associated to the chamber $\Cal C(w,p)$ defined in \rom{(3.6)} below. \roster \item"{(i)}" Suppose $w$ and $p$ are chosen so that the expected complex dimension of the moduli space $-p -3(p_g(S) +1)$ is $2$. Then for all $\Sigma \in H_2(S; \Zee)$, $$\gamma _{w,p}(S)(\Sigma, \Sigma) = \Sigma ^2 + ((m_1^2m_2^2)(p_g(S) + 1)- m_1^2-m_2^2) (\Sigma \cdot \kappa)^2.$$ \item"{(ii)}" Suppose $w$ and $p$ are chosen so that the expected complex dimension of the moduli space $-p -3(p_g(S) +1)$ is $4$. Then for all $\Sigma \in H_2(S; \Zee)$, $$\gamma _{w,p}(S)(\Sigma, \Sigma ,\Sigma, \Sigma ) = 3(\Sigma ^2)^2 + 6C_1(\Sigma ^2 )(\Sigma \cdot \kappa)^2 + (3C_1^2 - 2C_2)(\Sigma \cdot \kappa)^4, $$ where $$\align C_1&= (m_1^2m_2^2)(p_g(S) + 1)-m_1^2-m_2^2;\\ C_2 &= (m_1^4m_2^4) (p_g(S) + 1) -m_1^4-m_2^4. \endalign$$ Here $C_1$ is the second coefficient of the degree two polynomial. \endroster \endstatement \medskip Note that the final answer has the following self-checking features. First, it is a polynomial in $q_S$ and $\kappa _S$. If $p_g(S) = 1$ and $m_1 = m_2 = 1$, so that $S$ is a $K3$ surface, then the term $(\Sigma \cdot \kappa)$ does not appear. This is in agreement with the general result that $\gamma _{w,p}(S)$ is a multiple of a power of $q_S$ alone. If $p_g(S) = 0$ and $m_1 =1$, then the answer is independent of $m_2$, since in this case all of the surfaces $S$ for various choices of $m_2$ are diffeomorphic. In fact, we shall turn this remark around and use the knowledge of $\gamma _{w,p}(S)$ for $p_g =0$, $m_2 =1$ and $m_1$ arbitrary, to determine $\gamma _{w,p}(S)$ in general. The techniques used to prove Theorem 1.5 should be capable of further generalization. For example, these methods should give in principle (that is, up to the knowledge of the multiplication table for divisors in $\operatorname{Hilb}^nS$), the full polynomial invariant in case $m_1$ and $m_2$ are odd. One might make a conjectural formula for $\gamma _{w,p}(S)$ in general along the lines suggested by Kronheimer and Mrowka in [13]. In our case the formula should conjecturally read as follows: let $\gamma _t(\Sigma)$ be the Donaldson polynomial $\gamma _{w,p}(S)(\Sigma , \dots , \Sigma)$ for $w = \Delta \mod 2$ or $w= \Delta -\kappa \mod 2$ and $p$ chosen so that $w^2 \equiv p \mod 4$ and $-p - 3\chi (\scrO _S) =2t$, so that the complex dimension of the moduli space is $2t$. It follows from Proposition 2.1 below that $\gamma _t$ depends only on $t$. Then the natural analogue of the conjectures in [13] is the conjecture that $$\sum _{t\geq 0}\frac{\gamma _t(\Sigma)}{(2t)!} = \exp \fracwithdelims(){q_S}2\frac{\bigl(\cosh ( m_1m_2(\kappa \cdot \Sigma))\bigr)^{p_g+1}}{\cosh (m_1(\kappa \cdot \Sigma)) \cosh (m_2(\kappa \cdot \Sigma))}.$$ It essentially follows from Theorem 1.5 that this formula is correct through the first three terms, including the case $p_g =0$ where the quotient is not given by a finite sum of exponentials, and it is likely that a further extension of the methods in Part III and the knowledge of the multiplication in $\operatorname{Hilb}^nS$ can establish the general formula. Finally we should add that many of these techniques have applications to the $SU(2)$ case. Morgan and Mrowka [14] have independently determined the second coefficient $a_{n-1}$ for all $S$ such that $p_g(S) \geq 1$, for the case of the $SU(2)$-invariant $\gamma _c(S)$. The answer is that, up to combinatorial factors, $$a_{n-1} = (m_1m_2)^{p_g(S)}((m_1^2m_2^2)(p_g(S) + 1)-m_1^2-m_2^2).$$ {}From this, it is again easy to see that the diffeomorphism type of $S$ determines the unordered pair $\{m_1, m_2\}$ in case $p_g(S) \geq 1$. The proof of this formula uses the knowledge of $a_{n-1}$ for the case of $p_g(S)=1$, together with the gauge theory gluing techniques developed by Mrowka in [16], to determine the coefficient $a_{n-1}$ for $p_g(S) > 1$. Let us finally describe the contents of this paper. In Section 2 we discuss the possible choices for $w=w_2(P)$ up to diffeomorphisms of $S$ and show that there is always a generic elliptic surface for which $w$ is the mod 2 reduction of a holomorphic divisor. There is also a discussion of certain elliptic surfaces which can be constructed from $S$. In Section 3 we introduce a class of ample line bundles which we shall use to define stability and which are well-adapted to the geometry of $S$. Section 4 explains the meaning of stability of a vector bundle $V$ with respect to such a line bundle: stability is equivalent to the assumption that the restriction of $V$ to almost every fiber is semistable. Finally, in Section 5, we show how the main results concerning Donaldson polynomials lead to $C^\infty$ classification results. \section{2. Preliminaries on elliptic surfaces.} Let $S$ be a simply connected elliptic surface with at most two multiple fibers of multiplicities $m_1 \leq m_2$. Here we shall allow $m_1$ or both $m_1$ and $m_2$ to be one. Let $[f]$ denote the class in homology of a smooth nonmultiple fiber of $S$. There is a unique homology class $\kappa _S = \kappa$ such that $[f] = m_1m_2\kappa$, and $\kappa$ is primitive [9]. Let $P$ be a principal $SO(3)$-bundle over $S$ with $w_2(P) = w$ and $p_1(P) = p$. Note that $w^2 \equiv p \mod 4$. We shall be concerned with bundles $P$ such that $w\cdot \kappa\bmod 2 = 1$. In this section, we shall show that, modulo diffeomorphism, the choice of $w$ is not essential. Indeed, we shall prove that, given a class $w$, there is a diffeomorphism $\psi \: S \to S'$, where $S'$ is again a simply connected elliptic surface with two multiple fibers of multiplicities $m_1$ and $m_2$, such that $\psi ^*\kappa _{S'} = \kappa _S$ and such that there exists a holomorphic divisor $\Delta$ with $w=\psi ^*[\Delta] \mod 2$. Thus we may always assume that $w$ is the reduction of a $(1,1)$ class. We begin with an arithmetic result, which is not in fact needed in what follows but which helps to clarify the role of the choice of $w$ modulo diffeomorphisms. In the arguments below, we shall sometimes blur the distinction between $H_2(S)$ and $H^2(S)$ using the canonical identification between these two groups. \proposition{2.1} Let $S$ be a simply connected elliptic surface. \roster \item"{(i)}" Suppose that $m_1m_2 \equiv 1\mod 2$, and let $a\in \Zee/4\Zee$. Then the group of orientation-preserving diffeomorphisms $\psi \: S \to S$ such that $\psi _*([f]) = [f]$ acts transitively on the set of $w\in H^2(S; \Zee/2\Zee)$ such that $w\cdot \kappa =1$ and $w^2 \equiv a \mod 4$. \item"{(ii)}" If $m_1m_2 \equiv 0\mod 2$ and $a\in \Zee/4\Zee$, then there are at most three orbits of the set $$\{\, w\in H^2(S; \Zee /2\Zee): w\cdot \kappa =1 \text{ and } w^2\equiv a \mod 4\,\}$$ under the group of diffeomorphisms of $S$ which fix $\kappa$. \endroster \endstatement \proof Let $L$ be the image of $H_2(S - \pi ^{-1}(D))$ in $H_2(S)$, where $D$ is a small disk in $\Pee ^1$ which we may assume contains the multiple fibers and no other singular fiber. Thus $L\subseteq (\kappa^\perp)$, and in fact $L$ has index $m_1m_2$ in $(\kappa ^\perp)$. Let $\varphi$ be an automorphism of the lattice $H_2(S; \Zee)$ fixing $\kappa$. Thus by restriction $\varphi$ induces an automorphism of $(\kappa ^\perp)$. The method of proof of Theorem 6.5 of Chapter 2 of [9] shows that there is a diffeomorphism $\psi$, automatically orientation-preserving, inducing $\varphi$ provided that $\varphi(L) \subseteq L$ and that $\varphi$ has real spinor norm one. Clearly we may write $L = \Zee [f] \oplus W$, where $W$ is an even unimodular lattice. Moreover $(\kappa ^\perp) = \Zee \cdot\kappa \oplus W$, with the inclusion $L\subseteq (\kappa ^\perp)$ the natural inclusion given by $[f] = m_1m_2\kappa$. If $W^\perp$ denotes the orthogonal complement of $W$ in $H_2(S; \Zee)$, then $W^\perp = \operatorname{span}\{\kappa, x\}$ for some class $x$ with $x\cdot \kappa = 1$. Given $a\bmod 4$, we can always assume after replacing $x$ by $x+\kappa$ that $x^2 \equiv a \mod 4$. Now it is easy to describe all automorphisms of $H_2(S; \Zee)$ fixing $\kappa$: choosing an isometry $\tau$ of $W$, $\varphi $ is given by $$\align \varphi (\kappa ) &= \kappa;\\ \varphi (\alpha) &= \tau(\alpha) +\ell(\alpha)\kappa, \quad\alpha \in W;\\ \varphi (x) &= x + c\kappa + \beta. \endalign$$ Here $\ell$ is an arbitrary homomorphism $W\to \Zee$ and $\beta$ is the unique element of the unimodular lattice $W$ such that $-\beta \cdot \alpha = \ell (\tau (\alpha))$ for all $\alpha \in W$. Furthermore $c=-\beta ^2/2$. It is clear that every choice of $\tau$ and $\ell$ (or equivalently $\beta$) produces an automorphism $\varphi$, and that $\varphi(L) = L$ if and only if $m_1m_2$ divides $\ell$ or equivalently $\beta$. If $x'$ is another class such that $x'\cdot \kappa \equiv 1\mod 2$ and $(x')^2 \equiv x^2 \mod 4$, we can write $x' = nx + b\kappa + \beta$, where $\beta \in W$. Since we only care about $x'\bmod 2$, we may assume that $n=1$. Note that $2b + \beta ^2 \equiv 0\mod 4$ and thus $b \equiv \beta ^2/2 \mod 2$. First assume that $m_1m_2$ is odd. Then since $\beta \equiv m_1m_2\beta \mod 2$, we may assume that $\beta $ is divisible by $m_1m_2$. Choosing $\tau = \operatorname{Id}$ and $\ell$, $c$ in the definition of $\varphi$ as specified by $\beta$ gives $\varphi$ such that $\varphi (x) \equiv x' \mod 2$. As $\varphi$ is unipotent, it is easy to see that $\varphi$ has spinor norm one, i.e\. that $\varphi$ is in the same connected component of the group of automorphisms of the quadratic form of $H_2(S; \Ar)$ as the identity. Thus there is a diffeomorphism $\psi$ realizing $\varphi$. Next suppose that $2|m_1m_2$. In fact in this case the class $x$ defined above is fixed $\mod 2$ by every isometry $\varphi$ as above which satisfies $\varphi (L) = L$: Since $m_1m_2|\beta$ and $c\cong \beta ^2/2\mod 2$, it follows that $\varphi (x) \equiv x \mod 2$. Now let $x'$ be a class with $x'\cdot \kappa \equiv 1\mod 2$ and $(x')^2 \equiv x^2 \mod 4$. We may assume that $x'\neq x$. First consider the case where $x' = x + b\kappa + \alpha$ and $b\equiv 0 \mod 2$. Thus we may replace $x'$ by $x+\alpha$. By assumption $\alpha ^2 \equiv 0 \mod 4$. We may assume that $\alpha $ is primitive (otherwise $\alpha \equiv 0\mod 2$ or $\alpha$ is congruent to a primitive nonzero element mod 2). Replacing $\alpha $ by $\alpha + 2\beta$, where $\beta \in W^\perp$, replaces $\alpha ^2$ by $\alpha ^2 + 4(\alpha \cdot \beta) + 4\beta ^2$. Since $\alpha$ is primitive, it is easy to see that there is a choice of $\beta$ so that $(\alpha + 2\beta)^2 = 0$. Thus we may assume that $\alpha$ is primitive and that $\alpha ^2=0$. The group $SO(W)$ includes into the automorphism group of $L$, and every element of $SO(W)^*$, the set all elements of $SO(W)$ with spinor norm one, is realized by a diffeomorphism. Moreover an easy exercise shows that $SO(W)^*$ acts transitively on the set of primitive $\alpha \in W$ with $\alpha ^2=0$. Thus the set of all possible $x+\alpha$, with $\alpha \neq 0$, is contained in a single orbit under the diffeomorphism group. In case $x' = x + \kappa + \alpha$ with $\alpha ^2 \equiv 2\mod 4$, an argument similar to that given above shows that we may assume that $\alpha ^2 = 2$ and that every two classes $x_1 = x + \kappa + \alpha_1$ and $x_2 = x + \kappa + \alpha_2$ with $\alpha _i^2 = 2$ are conjugate under the group of diffeomorphisms of $S$ which fix $\kappa$. Thus there are at most three orbits in this case. \endproof The following result is really only needed in the case where $m_1m_2$ is even, since in case $m_1m_2$ is odd we can appeal to (i) of (2.1) above. \proposition{2.2} Let $S$ be a simply connected elliptic surface and $w$ be a class in $H^2(S;\Zee/2\Zee)$ with $w\cdot \kappa = 1$. Then after replacing $S$ with a deformation equivalent elliptic surface, we may assume that there is a divisor $\Delta$ on $S$ with $\Delta \cdot \kappa = 1$, and such that all singular fibers of $S$ are irreducible rational curves with a singular ordinary double point, i.e\. $S$ is nodal. \endstatement \proof Fix a nodal simply connected elliptic surface with a section $B$ such that $p_g(B) = p_g(S)$. Using [9], $S$ is deformation equivalent through elliptic surfaces to a logarithmic transform of $B$ at two smooth fibers, where the multiplicities of the logarithmic transforms are $m_1$ and $m_2$. Fix one such logarithmic transform $S_0$, and let $\psi\: S \to S_0$ be a diffeomorphism preserving the class of the fiber. Using this diffeomorphism, we shall identify $S$ and $S_0$. Let $\Delta $ be an element in $H^2(S; \Zee)$ whose mod 2 reduction is $w$ and such that $\Delta \cdot \kappa = 1$. We shall show that, by further modifying the complex structure on $S$, we may assume that $\Delta$ is of type $(1,1)$. Given $\Delta$, we have the image $i_* ([\Delta]) \in H^2(S;\scrO _S)$, where $i_*$ is the map induced on sheaf cohomology by the inclusion $\Zee \subset \scrO _S$. The set of all complex structures of an elliptic surface on $S$ for which the associated Jacobian surface is $B$ and which are locally isomorphic to $S$ is a principal homogeneous space over $H^1(\Pee ^1; \Cal B)$, where $\Cal B$ is the sheaf of local holomorphic cross sections of $B$ ([9] Chapter 1 Theorem 6.7). Moreover there is a surjective map $H^2(S; \scrO _S) \cong H^2(B; \scrO _B)\to H^1(\Pee ^1; \Cal B)$ ([9], Chapter 1, Lemma 5.11). Thus given a cohomology class $\eta \in H^2(S; \scrO _S)$, we can form the associated surface $S^\eta$ and consider the element $i_* ^\eta([\Delta]) \in H^2(S^\eta;\scrO _{S^\eta}) \cong H^2(S;\scrO _S)$. \lemma{2.3} In the above notation, $i_* ^\eta([\Delta]) = i_* ([\Delta]) + m_1m_2\eta$. \endstatement \proof This presumably could be proved by a rather involved direct calculation. For another argument, note that the map $H^2(S; \scrO_S) \to H^2(S; \scrO_S)$ defined by $$\eta \mapsto i_* ^\eta([\Delta]) - i_* ([\Delta]) - m_1m_2\eta$$ is holomorphic, since it arises from a variation of Hodge structure. The argument of Lemma 6.13 in Chapter 1 of [9] shows that $i_* ^\eta([\Delta]) - i_* ([\Delta]) - m_1m_2\eta$ lies in the countable (not necessarily discrete) subgroup $H^1(\Pee ^1; R^1\pi _*\Zee)$ of $H^2(S; \scrO_S)= H^1(\Pee ^1; R^1\pi _*\scrO _S)$. This is only possible if the image of the map is contained in a single point, and since the image contains the origin the map is identically zero. \endproof Returning to the proof of (2.2), since $H^2(S;\scrO _S)$ is divisible, there is a choice of $\eta$ so that $i_* ^\eta([\Delta]) = 0$. For the corresponding complex structure, $[\Delta]$ is then a $(1,1)$ class. \endproof Finally we shall describe a way to associate new elliptic surfaces to $S$ which generalizes the construction of the Jacobian surface. Suppose that $S$ is an elliptic surface over $\Pee ^1$. Let $\eta = \operatorname{Spec}k$ be the generic point of $\Pee ^1$, where $k=k(\Pee ^1)$ is the function field of the base curve, let $\bar \eta= \operatorname{Spec}\bar k$, where $\bar k =\overline{k(\Pee ^1)}$ is the algebraic closure of $k$, and let $S_\eta$ and $S_{\bar \eta}$ be the restrictions of $S$ to $\eta$ and $\bar \eta$. Thus $S_\eta$ is a curve of genus one over $k$. Given an algebraic elliptic surface with a section $\pi \:B\to \Pee ^1$, it has an associated Weil-Chatelet group $WC(B)$ [3], which classifies all algebraic elliptic surfaces $S$ whose Jacobian surface is $B$. As above we let $B_\eta$ be the elliptic curve over $k$ defined by the generic fiber of $B$. By definition $WC(B)$ is the Galois cohomology group $H^1(G, B_\eta(\bar k))$, where $G = \operatorname{Gal}(\bar k/k)$ and $B_\eta(\bar k)$ is the group of points of the elliptic curve $B_\eta$ defined over $\bar k$. There is an exact sequence $$0 \to {\cyr Sh} (B) \to WC(B) \to \bigoplus _{t\in C}H_1(\pi ^{-1}(t); \Bbb Q/\Zee) \to 0.$$ The subgroup ${\cyr Sh} (B)$ corresponds to elliptic surfaces without multiple fibers whose Jacobian surface is isomorphic to $B$, and the quotient describes the possible local forms for the multiple fibers. Thus if $\xi\in WC(B)$ corresponds to the surface $S$, then $S$ has a multiple fiber of multiplicity $m$ at $t\in \Pee ^1$ if and only if the projection of $\xi$ to $H_1(\pi ^{-1}(t); \Bbb Q/\Zee)$ has order $m$. The surface $S$ is specified by an element $\xi$ of $WC(J(S))$, where $J(S)$ is the Jacobian surface associated to $S$. Let us recall the recipe for $\xi$ [18]: we have the curve $S_\eta$ and its Jacobian $J(S_\eta )$ defined over $k$. The curve $S_\eta$ is a principal homogeneous space over $J(S_\eta )$, and thus defines a class $\xi \in WC(J(S))$, by the following rule: let $\sigma$ be a point of $S_{\bar \eta}$. Given $g\in \operatorname{Gal}(\bar k/k)$, the divisor $g(\sigma) - \sigma$ has degree zero on $S_{\bar \eta}$ and so defines an element of $J(S_{\bar \eta})$, which is easily checked to be a 1-cocycle. The induced cohomology class is $\xi$. For every integer $d$ there is an algebraic elliptic surface $J^d(S)$, whose restriction to the generic fiber $\eta$ is the Picard scheme of divisors of degree $d$ on the curve $S_\eta$. Thus $J^0(S) = J(S)$ and $J^1(S) = S$. We claim that, if $S$ corresponds to the class $\xi \in WC(J(S))$, then $J^d(S)$ corresponds to the class $d\xi$. Indeed, using the above notation, if $\sigma$ defines a point of $S_{\bar \eta}$, then $d\sigma$ is a point of $J^d(S_{\bar \eta})$. Thus, the corresponding cohomology class is represented by $d(g(\sigma) - \sigma)$ and so is equal to $d\xi$. In particular, if $S$ has a multiple fiber of multiplicity $m$ at $t$, then $J^d(S)$ has a multiple fiber of multiplicity $m/\gcd (m, d)$. Of course if $m|d$ then the multiplicity is one. Finally note that $J(S)$ is the Jacobian surface of $J^d(S)$ for every $d$ and that $p_g(J^d(S)) = p_g(S)$. Ideally we would like there to be a Poincar\'e line bundle $\Cal P_d$ over $S\times _{\Pee ^1}J^d(S)$ such that the restriction of $\Cal P_d$ to the slice $S\times _{\Pee ^1}\{\lambda\}$ is the line bundle of degree $d$ on the fiber of $S$ over $\pi (\lambda)$ corresponding to $\lambda$. In general this is too much to ask. However such a bundle exists locally around every smooth nonmultiple fiber: if $X$ is the inverse image in $S$ of a small disk $D$ in $\Pee ^1$ such that all fibers on $X$ are smooth and nonmultiple, and $X_d$ is the corresponding preimage in $J^d(S)$, then there is a Poincar\'e line bundle over $X\times _DX_d$. There is also an analogous statement where we replace a small classical open set in $\Pee ^1$ with an \'etale open set. The proof for this result is essentially contained in the proof of Theorem 1.3 of Chapter 7 in [9]. Another construction is given in Section 7 of Part III of this series. \section{3. Suitable line bundles.} Suppose that we are given a class $w\in H^2(S; \Zee/2\Zee)$ with $w\cdot \kappa \bmod 2 = 1$ and an integer $p$ with $w^2 \equiv p \mod 4$. Choose once and for all a complex structure on $S$ for which there is a divisor $\Delta $ with $w= \Delta \bmod 2$. Let $c$ be the integer $(\Delta ^2 - p)/4$. The principal $SO(3)$-bundle $P$ over $S$ with $w_2(P) = w$ and $p_1(P) = p$ lifts uniquely to a principal $U(2)$-bundle $P'$ over $S$ with $c_1(P') = \Delta$ and $c_2(P') = c$. Moreover, by Donaldson's theorem, if $g$ is a Hodge metric on $S$ corresponding to the ample line bundle $L$, we can identify the moduli space of gauge equivalence classes of $g$-anti-self-dual connections on $P$ with the moduli space of $L$-stable rank two vector bundles $V$ over $S$ with $c_1(V) = \Delta$ and $c_2(V) = c$. We shall also have to make a choice of the ample line bundle $L$. If $p_g(S) > 0$, then the resulting Donaldson polynomial invariant does not depend on the choice of $L$, whereas if $p_g(S) = 0$, then the invariant depends on the chamber containing $c_1(L)$ [11], [12]. We then make the following definition [17]: \definition{Definition 3.1} A {\sl wall of type $(\Delta, c)$} is a class $\zeta \in H^2(S; \Zee)$ such that $\zeta \equiv \Delta \mod 2$ and $$\Delta ^2 - 4c \leq \zeta ^2 <0.$$ In particular there are no such walls unless $\Delta ^2 - 4c <0$. Clearly this definition depends only on $\Delta \bmod 2 = w$ and $p= \Delta ^2 - 4c$, and we shall also refer to walls of type $(w, p)$. Now suppose that $p_g(S) = 0$, i.e\. that $b_2^+(S) = 1$. Let $$\Omega _S = \{ x\in H^2(S; \Ar): x^2 >0\}.$$ Let $W^\zeta = \Omega _S \cap (\zeta)^\perp$. A {\sl chamber of type $(\Delta, c)$} (or of type $(w, p)$) for $S$ is a connected component of the set $$\Omega _S - \bigcup\{W^\zeta: \zeta {\text{ is a wall of type $(\Delta, c')$, $c'\leq c$}}\, \}.$$ \enddefinition For the purposes of algebraic geometry, walls of type $(\Delta, c)$ arise as follows: let $L$ be an ample line bundle and let $V$ be a rank two bundle over $S$ with $c_1(V) = \Delta$ and $c_2(V) = c$ which is strictly $L$-semistable. Let $\scrO _S(F)$ be a destabilizing sub-line bundle. Thus there is an exact sequence $$0 \to \scrO _S(F) \to V \to \scrO _S(-F+\Delta )\otimes I_Z \to 0,$$ where $I_Z$ is the ideal sheaf of a codimension two local complete intersection subscheme. Thus $$c_2(V) = c = -F^2 + F \cdot \Delta + \ell (Z).$$ Since $\ell(Z)$ is nonnegative, we can rewrite this as $$ -F^2 + F \cdot \Delta \leq c.$$ Moreover $$(2F -\Delta)^2 = -4(-F^2 + F \cdot \Delta) + \Delta ^2 \leq \Delta ^2 -4c,$$ so that we can rewrite the last condition by $$ \Delta ^2 - 4c \leq (2F -\Delta)^2.$$ Using the fact that $L\cdot F = (L\cdot \Delta)/2$, we have $$L\cdot (2F -\Delta) = 0,$$ and so by the Hodge index theorem $(2F -\Delta)^2 \leq 0$, with equality holding if and only if $2F -\Delta= 0$ (recall that $S$ is simply connected). This case cannot arise for us since $\Delta \cdot \kappa = 1$ and thus $\Delta$ is primitive. In particular $\zeta = 2F-\Delta$ is a wall of type $(\Delta, c)$. Of course, it is also the cohomology class of a divisor, and thus has type $(1,1)$. It then follows easily that, if $L_1$ and $L_2$ are two ample line bundles such that $c_1(L_1)$ and $c_1(L_2)$ lie in the interior of the same chamber of type $(\Delta, c)$, then a rank two vector bundle $V$ with $c_1(V) = \Delta$ and $c_2(V)=c$ is $L_1$-stable if and only if it is $L_2$-stable. With this said, we can make the following definition: \definition{Definition 3.2} Let $c$ be an integer, and set $w = \Delta \bmod 2$ and $p = \Delta ^2 - 4c$. An ample line bundle $L$ is {\sl $(\Delta, c)$-suitable\/} or {\sl $(w,p)$-suitable\/} if, for all walls $\zeta$ of type $(\Delta, c)$ which are the classes of divisors on $S$, we have $\operatorname{sign} f\cdot \zeta = \operatorname{sign} L\cdot \zeta$. \enddefinition \medskip \noindent {\bf Remark.} 1) Suppose that $\zeta$ is a $(1,1)$ class satisfying $\zeta ^2 \geq 0$. It follows from the Hodge index theorem that if $\zeta \cdot f>0$, then $\zeta \cdot L >0$ as well. Thus we can drop the requirement that $\zeta ^2 <0$. 2) In our case $\zeta \equiv \Delta \bmod 2$ and thus $\zeta \cdot \kappa \equiv 1 \mod 2$. It follows that $\zeta \cdot \kappa \neq 0$ and thus that $\zeta \cdot f \neq 0$. Thus the condition $\zeta \cdot f \neq 0$ (which was included as part of the definition in [9]) is always satisfied in our case. 3) In case $b_2^+(S)=1$, $L$ is $(\Delta, c)$-suitable if and only if the class $\kappa$ lies in the closure of the chamber containing $c_1(L)$. \medskip \lemma{3.3} For every $c$, $(\Delta, c)$-suitable ample line bundles exist. \endstatement \proof Let $L_0$ an ample line bundle. For $n\geq 0$, let $L_n = L_0 \otimes \scrO _S(nf)$. It follows from the Nakai-Moishezon criterion that $L_n$ is ample as well. We claim that if $n> -p (L_0 \cdot f)/2$, then $L_n$ is $(\Delta, c)$-suitable. To see this, let $\zeta = 2F - \Delta$ be a wall of type $(\Delta, c)$ with $$\Delta ^2 - 4c \leq \zeta^2 <0.$$ We may assume that $a = \zeta \cdot f >0$, and must show that $\zeta \cdot L_n > 0$ as well. The class $ac_1(L_0) - (L_0\cdot f) \zeta$ is perpendicular to $f$. Since $f^2=0$, we may apply the Hodge index theorem to conclude: $$0 \geq ac_1(L_0) - (L_0\cdot f) \zeta = a^2L_0^2 -2a(L_0\cdot f) (L_0 \cdot \zeta) + (L_0\cdot f)^2\zeta ^2.$$ Using the fact that $\zeta ^2 \geq\Delta ^2 - 4c = p $, we find that $$L_0 \cdot \zeta \geq \frac{a(L_0^2)}{2(L_0\cdot f)} + \frac{\zeta ^2}{2a}(L_0 \cdot f) > \frac{p}{2a}(L_0 \cdot f).$$ Thus $$\align L_n \cdot \zeta &= (L_0 \cdot \zeta) + n(\zeta \cdot f) >\frac{p}{2a}(L_0 \cdot f) -\frac{pa}{2}(L_0\cdot f) \\ &= -\frac{p }{2}(L_0\cdot f)\Bigl(a-\frac{1}{a}\Bigr) \geq 0. \endalign$$ Thus $L_n$ is $(\Delta, c)$-suitable. \endproof In the case where $b_2^+ (S) =1$, we have the following interpretation of $(\Delta, c)$-suitability. \lemma{3.4} Suppose that $p_g(S) = 0$. If $L_1$ and $L_2$ are both $(\Delta, c)$-suitable, then $c_1(L_1)$ and $c_1(L_2)$ lie in the same chamber of type $(\Delta, c)$. Thus there is a unique chamber $\Cal C(w,p)$ of type $(\Delta, c)$ which contains the first Chern classes of $(\Delta, c)$-suitable ample line bundles. Conversely, if $L$ is ample and $c_1(L) \in \Cal C(w,p)$, then $L$ is $(\Delta, c)$-suitable. \endstatement \proof Let $L_1$ and $L_2$ be $(\Delta, c)$-suitable. Since $p_g(S) =0$, every cohomology class is of type $(1,1)$. Thus if $\zeta$ is a wall of type $(\Delta, c)$, then $$\operatorname{sign} L_1\cdot \zeta =\operatorname{sign} f\cdot \zeta = \operatorname{sign} L_2\cdot \zeta.$$ This exactly implies that $c_1(L_1)$ and $c_1(L_2)$ are not separated by any wall $(\zeta )^\perp$. Conversely suppose that $c_1(L) \in \Cal C(w,p)$, where $\Cal C(w,p)$ is the unique chamber containing the first Chern classes of $(\Delta, c)$-suitable ample line bundles. This means in particular that $L\cdot \zeta \neq 0$ for every $\zeta$ of type $(\Delta, c)$. The proof of (3.4) shows that $c_1(L) + N[f] \in \Cal C(w,p)$ for all sufficiently large $N$. Thus, for all $\zeta$ of type $(\Delta, c)$, $$\operatorname{sign} L\cdot \zeta = \operatorname{sign} L\cdot \zeta + N\operatorname{sign} f\cdot \zeta.$$ Since $f\cdot \zeta \neq 0$, $\operatorname{sign} L\cdot \zeta + N\operatorname{sign} f\cdot \zeta = \operatorname{sign} f\cdot \zeta$ for all $N\gg 0$. Thus $\operatorname{sign} f\cdot \zeta = \operatorname{sign} L\cdot \zeta$, and $L$ is $(\Delta, c)$-suitable. \endproof \lemma{3.5} Suppose that $p_g(S) = 0$. The chamber $\Cal C(w,p)$ is the unique chamber of type $(w,p)$ which contains $\kappa$ in its closure. Thus every diffeomorphism $\psi$ of $S$ satisfies $\psi ^*\Cal C(w,p) = \pm \Cal C(\psi ^*w,p)$. More generally, if $S$ and $S'$ are two elliptic surfaces with $p_g=0$ and $\psi \: S \to S'$ is a diffeomorphism, then $\psi ^*\Cal C(w,p) = \pm \Cal C(\psi ^*w,p)$. \endstatement \proof Let $\Cal C_1$ and $\Cal C_2$ be two distinct chambers which contain $\kappa$ in their closures. Let $\zeta$ be a wall separating $\Cal C_1$ and $\Cal C_2$. We may assume that $\zeta \cdot x > 0 $ for all $x\in \Cal C_1$ and $\zeta \cdot x < 0$ for all $x \in \Cal C_2$. Thus $0\leq\zeta \cdot \kappa \leq 0$, so that $\zeta \cdot \kappa = 0$. However this contradicts the fact that $\zeta \cdot \kappa \neq 0$. Thus there is at most one chamber containing $\kappa$ in its closure. We have seen in the proof of Lemma 3.3 that, for all ample line bundles $L$ and integers $N\gg 0$, $c_1(L) + N\kappa \in \Cal C(w,p)$. Thus $\kappa + (1/N)c_1(L) \in \Cal C(w,p)$. It follows that $\kappa$ indeed lies in the closure of $\Cal C(w,p)$, so that $\Cal C(w,p)$ is the unique chamber with this property. To see the final statement, we use [8] to see that every diffeomorphism $\psi$ of $S$ satusfies $\psi ^*\kappa = \pm \kappa$. Thus $\pm \kappa$ lies in the closure of $\psi ^*\Cal C(w,p)$. Clearly, if $\Cal C$ is a chamber of type $(w,p)$, then $\psi ^*\Cal C$ is a chamber of type $(\psi ^*w,p)$. It follows that $\psi ^*\Cal C(w,p) = \pm \Cal C(\psi ^*w,p)$. The statement about two different surfaces is proved similarly. \endproof \definition{Definition 3.6} The chamber described in Lemma 3.4 will be called the {\sl suitable chamber\/} of type $(\Delta, c)$ or of type $(w,p)$ or the {\sl $(\Delta, c)$-suitable\/} or {\sl $(w, p)$-suitable chamber\/}. \enddefinition \section{4. The geometric meaning of suitability.} The goal of this section is to describe the meaning of $(\Delta, c)$-suitability. Given the bundle $V$ on $S$, it defines by restriction a bundle $V|f$ on each fiber $f$. Our main result says essentially that $V$ is stable for one, or equivalently all, $(\Delta, c)$-suitable line bundles $L$ if and only if $V|f$ is semistable for almost all $f$. It will be more convenient to use the language of schemes to state this result. As in Section 2, let $k(\Pee ^1)$ denote the function field of $\Pee ^1$ and let $\overline {k(\Pee ^1)}$ be the algebraic closure of $k(\Pee ^1)$. Set $\eta = \operatorname{Spec}k(\Pee ^1)$ and $\bar\eta = \operatorname{Spec} \overline{k(\Pee ^1)}$. Thus $\eta$ is the generic point of $\Pee ^1$. Let $S_\eta= S\times _{\Pee ^1}\eta$ be the generic fiber of $\pi$ and let $S_{\bar \eta} = S\times _{\Pee ^1}\bar\eta$. Here $S_\eta$ is a curve of genus one over the field $k(\Pee ^1)$ and $S_{\bar\eta}$ is the curve over $\overline{k(\Pee ^1)}$ defined by extending scalars. Let $V_\eta$ and $V_{\bar \eta}$ be the vector bundles over $S_\eta$ and $S_{\bar\eta}$ respectively obtained by restricting $V$. We can then define stability and semistability for $V_{\bar \eta}$ and $V_\eta$; for $V_\eta$, a destabilizing subbundle must also be defined over $k(\Pee ^1)$. Trivially, if $V_\eta$ is unstable (resp\. not stable) then $V_{\bar \eta}$ is unstable (resp\. not stable). Thus if $V_{\bar \eta}$ is stable, then $V_\eta$ is stable as well. \lemma{4.1} $V_\eta$ is semistable if and only if $V_{\bar \eta}$ is semistable. \endstatement \proof We have seen that, if $V_\eta$ is not semistable, then $V_{\bar \eta}$ is not semistable. Conversely suppose that $V_{\bar \eta}$ is not semistable. Then there is a canonically defined maximal destabilizing line subbundle of $V_{\bar \eta}$, which thus is fixed under by every element of $\operatorname{Gal}(\overline {k(\Pee ^1)}/k(\Pee ^1)$. By standard descent theory this line subbundle must then be defined over $k(\Pee ^1)$. Thus $V_\eta$ is not semistable. \endproof \noindent {\bf Remark.} If $V_{\bar \eta}$ is strictly semistable, it is typically the case that $V_\eta$ is actually stable. \medskip \lemma{4.2} In case $\Delta \cdot \kappa = 1$, the bundle $V_\eta$ is semistable if and only if it is stable. \endstatement \proof First assume that $m_1m_1 \equiv 1\mod 2$. In this case $V_\eta$ has odd fiber degree, and so there are no strictly semistable bundles over $\overline{k(\Pee ^1)}$. Hence, if $V_\eta$ is semistable, then by (4.1) $V_{\bar \eta}$ is semistable and therefore stable. Thus $V_\eta$ is stable by the remarks preceding (4.1). In case $m_1m_1 \equiv 0\mod 2$, suppose that $V_\eta$ is strictly semistable. Thus there is a line bundle on $S_\eta$ of degree $m_1m_2/2$. There would thus exist a divisor $D$ on $S$ with $D\cdot f = m_1m_2/2$. Since $f = m_1m_2\kappa$, this possibility cannot occur. Thus $V_\eta$ is stable. \endproof Here then is the theorem of this section: \theorem{4.3} Let $V$ be a rank two vector bundle on $S$ with $c_1(V) = \Delta$ and $c_2(V) = c$ and let $L$ be a $(\Delta, c)$-suitable ample line bundle. Then $V$ is $L$-stable if and only if the restriction $V_\eta$ of $V$ to the generic fiber $S_\eta$ is stable. \endstatement \proof First suppose that $V$ is $L$-stable. Let $F_\eta$ be a subbundle of $V_\eta$ of rank one. Then there is a divisor $F$ on $S$ such that $\scrO _S(F)$ restricts to $F_\eta$ and an inclusion $\scrO _S(F) \to V$. Hence there is an effective divisor $D$ and an inclusion $\scrO_S(F+D) \to V$ and the cokernel is torsion free. Since $F_\eta$ is a subbundle of $V_\eta$, the divisor $D$ cannot have positive intersection number with $f$. As $D$ is effective it is supported in the fibers of $\pi$ and so $F$ and $F+D$ have the same restriction to the generic fiber. We may thus replace $F$ by $F+D$. Then $V/\scrO _S(F)$ is torsion free. Hence there is an exact sequence $$0 \to \scrO _S(F) \to V \to \scrO _S(\Delta - F)\otimes I_Z \to 0,$$ where $Z$ is a codimension two subscheme of $S$. Thus $$\Delta ^2 - 4c \leq (2F-\Delta )^2.$$ Since $V$ is $L$-stable, $L\cdot (2F-\Delta ) <0$. It follows from Definition 3.2 and 2) of the remark following it that $f\cdot (2F-\Delta ) <0$ as well. Thus $\deg F_\eta <\deg V_\eta /2$, which says that $V_\eta$ is stable. Conversely suppose that $V_\eta$ is stable. Let $\scrO _S(F)$ be a sub-line bundle of $V$, where we may assume that $V/\scrO _S(F)$ is torsion free. Reversing the argument above shows that $f\cdot (2F-\Delta ) <0$ and therefore that $L\cdot (2F-\Delta )<0$ as well. Thus $V$ is $L$-stable. \endproof \corollary{4.4} Let $V$ be a rank two vector bundle on $S$ with $c_1(V) = \Delta$ and $c_2(V) = c$. Then the following are equivalent: \roster \item"{(i)}" There exists a $(\Delta, c)$-suitable ample line bundle $L$ such that $V$ is $L$-stable. \item"{(ii)}" $V$ is $L$-stable for every $(\Delta, c)$-suitable ample line bundle $L$. \item"{(iii)}" $V_\eta$ is stable. \item"{(iv)}" $V_{\bar \eta}$ is semistable. \item"{(v)}" The restriction $V|\pi^{-1}(t)$ is semistable for almost all $t \in \Pee ^1$. \item"{(vi)}" There exists a $t\in \Pee^1$ such that $\pi^{-1}(t)$ is smooth and the restriction $V|\pi^{-1}(t)$ is semistable. \endroster \endstatement \proof By (4.1) and (4.2), (iii) and (iv) are equivalent, and by (4.3) (i) $\implies$ (iii) $\implies$ (ii). The implication (ii) $\implies$ (i) is trivial. The implication (iv) $\implies$ (v) follows from the openness of semistability in the Zariski topology in the sense of schemes, and the implication (v) $\implies$ (vi) is trivial. To see that (vi) $\implies$ (iv), suppose that $V_{\bar \eta}$ is not semistable. Then a destabilizing sub-line bundle extends to give a sub-line bundle over the pullback of $S$ to some finite base change of $\Pee^1$. Thus $V|\pi^{-1}(t)$ is unstable for every $t\in \Pee^1$ such that $\pi^{-1}(t)$ is smooth, and so (vi) $\implies$ (iv). \endproof \noindent {\bf Remark.} In case $\Delta \cdot \kappa \equiv 0\mod 2$, there can exist strictly semistable bundles on $S_\eta$ of degree $\Delta \cdot f$. There are examples of rank two bundles $V$ on $S$ with $c_1(V) = \Delta$ and $V_\eta$ strictly semistable such that $V$ is either stable, strictly semistable, or unstable (cf\. [8]). \section{5. Donaldson polynomials and the main theorems.} As above we let $S$ denote a simply connected elliptic surface with $p_g(S) \geq 0$. Fix $w = \Delta \bmod 2$ and let $p$ be an integer satisfying $w^2 \equiv p \bmod 4$. For $p_g(S) >0$, there is the Donaldson polynomial $\gamma _{w,p}(S)$ corresponding to the $SO(3)$-bundle $P$ with invariants $w$ and $p$. Here for simplicity we shall always choose the orientation on the moduli space which agrees with the natural complex orientation. The polynomial $\gamma_{w,p}(S)$ is invariant up to sign under self-diffeomorphisms $\psi$ of $S$ such that $\psi ^*w=w$. If $p_g(S) = 0$, then we have the distinguished chamber $\Cal C(w,p)$ which contains $\kappa$ in its closure. We shall then use $\gamma_{w,p}(S)$ to denote the Donaldson polynomial for $S$ with respect to the chamber $\Cal C(w,p)$, again with the orientation chosen to be the complex orientation. Since $\psi ^*\Cal C(w,p) = \pm \Cal C(\psi ^*w,p)$, the invariant $\gamma_{w,p}(S)$ is again natural up to sign under orientation-preserving self-diffeomorphisms which fix $w$. Of course, there are only finitely many choices for $w$, so that there is a subgroup of finite index in the full group of diffeomorphisms fixing $[f]$ which will also fix $w$. \lemma{5.1} For every choice of $w$ and $p$, $\gamma_{w,p}(S)$ lies in $\Bbb Q[q_S, \kappa _S]$. Moreover, if for some choice of $w$ and $p$, $\gamma_{w,p}(S)$ does not lie in $\Bbb Q[q_S]$, then every diffeomorphism $\psi$ from $S$ to another simply connected elliptic surface $S'$ satisfies $\psi ^*\kappa _{S'} = \pm \kappa _S$. \endstatement \proof The set of automorphisms of $H_2(S; \Zee)$ of the form $\psi _*$, where $\psi$ is a diffeomorphism satisfying $\psi _*([f]) = [f]$, $\psi ^*w = w$, and $\psi ^* \gamma_{w,p}(S) = \gamma_{w,p}(S)$, is a subgroup of finite index in the group of all isometries of $H_2(S;\Zee)$ preserving $[f]$, by [8] Part I Theorem 6 and [9] Chapter 2 Theorem 6.5. Thus by [9] Chapter 6 Theorem 2.12, $\gamma_{w,p}(S) \in \Bbb Q[q_S, \kappa _S]$. Moreover $\kappa$ is the unique such class. The last statement of the lemma is then clear. \endproof Next let us discuss the effect of blowing up. Suppose that $\rho\: \tilde S\to S$ is the $r$-fold blowup of $S$, and let the exceptional classes in $H_2(\tilde S)$ be denoted by $e_1, \dots, e_r$. Likewise let $S'$ be another simply connected elliptic surface and let $\rho'\:\tilde S'\to S'$ be the $r$-fold blowup of $S'$, with exceptional classes $e_1', \dots, e_r'$. If $\psi \: \tilde S \to \tilde S'$ is a diffeomorphism, then $\psi ^*e_i' = \pm e_j$ for a uniquely determined $j$, by [8] Part I Theorem 7 and [9] Chapter 6 Corollary 3.8. It follows that, if $w'\in H^2(\tilde S'; \Zee/2\Zee)$ is of the form $(\rho ')^*w_0'$ for some $w_0' \in H^2(S'; \Zee/2\Zee)$, then there is a $w_0 \in H^2(S; \Zee/2\Zee)$, such that $\psi ^*w' = \rho ^*w_0$. Finally, we shall need the following extension of [9] Chapter 6 Theorem 3.1: \proposition{5.2} Let $w_0 \in H^2(S; \Zee/2\Zee)$ and let $\rho \:\tilde S\to S$ be the $r$-fold blowup of $S$. If $b_2^+(S) =1$, assume moreover that $\gamma _{\rho ^*w_0, p}$ is defined with respect to some chamber $\Cal D$. Let $\Cal C$ be a chamber of type $(w_0, p)$ on $H^2(S; \Bbb R)$ such that $\Cal D$ contains $\rho ^*\Cal C$ in its closure. Then $$\gamma _{\rho ^*w_0, p}|\rho ^*H_2(S; \Zee) = \gamma _{w_0, p},$$ where if $b_2^+(S)=1$, the polynomial $\gamma _{w_0, p}$ is defined with respect to the chamber $\Cal C$. \endstatement \medskip Here, in case $p_g(S) = 0$, the chamber $\Cal C$ does not in general determine a unique chamber $\Cal D$ on $\tilde S$. However the conclusion of the proposition implies in particular that the value of $\gamma _{\rho ^*w_0, p}$ on classes in $\rho ^*H_2(S; \Zee)$ is independent of the chamber for $\tilde S$ of type $(\rho ^*w_0, p)$ which contains $\Cal C$ in its closure. This result follows from standard gauge theory techniques [5]. It can also be proved in our case via algebraic geometry, using the blowup formulas for instance in [8]. Since it does not appear with an explicit proof in the literature, we shall outline a proof in the only case that concerns us, where the chamber $\Cal C$ contains the first Chern class of an ample line bundle. We shall just write down the argument in the most interesting case, where $p_g(S) = 0$. We shall also assume that the moduli space has the expected dimension, although the arguments given here can easily be extended to handle the general case. By induction we may assume that $\rho \: \tilde S \to S$ is the blowup of $S$ at a single point $p$. Let $E$ be the exceptional curve and $e$ be its cohomology class. We shall usually identify $H^2(S)$ with its image in $H^2(\tilde S)$ under $\rho ^*$. Let $\Cal D$ be a chamber for $\tilde S$ of type $(\rho ^*w_0, p)$ containing $\Cal C$ in its closure and let $\zeta$ be a wall for $\Cal D$. Then $\zeta = \zeta ' + ae$, where $\zeta ' \in H^2(S; \Zee)$ and $a\in \Zee$ (in fact $2|a$ since $\zeta \equiv \Delta \mod 2$). After possibly reflecting in $e$, which is realized by an orientation-preserving diffeomorphism $r_e$ of $\tilde S$, we may assume that $a\geq 0$: Indeed, $r_e^*$ switches the two possible chambers corresponding to $\pm a$, and so, if $\gamma _1$ and $\gamma _2$ are the two invariants corresponding to the two choices of chambers, then $r_e^*\gamma _1 = \gamma _2$. Since $r_e^*|H_2(S; \Zee)$ is the identity, it suffices to prove the result for either chamber. So we can assume that $a\geq 0$. Since $\Cal C$ is in the closure of $\Cal D$, if $x\in \Cal C$, then $x\cdot \zeta '= x\cdot \zeta \geq 0$. Conversely, if we start with an ample line bundle $L$ on $S$ such that $c_1(L) \in \Cal C$, then for all $N\gg 0$, $Nc_1(L) - e$ is the first Chern class of an ample line bundle $L_N$ on $\tilde S$. Moreover $(Nc_1(L) - e)\cdot \zeta \geq N(c_1(L) \cdot \zeta ') \geq 0$. It follows from this that $c_1(L_N)$ lies in $\Cal D$, and if $c_1(L)$ is in the interior of $\Cal C$ then $c_1(L_N)$ is in the interior of $\Cal D$. Consider rank two vector bundles $\tilde V$ on $\tilde S$ with $c_1(\tilde V) = \rho ^*\Delta$ and $c_2(\tilde V) = c$. Set $V = (\rho _*\tilde V)\spcheck{}\spcheck$. Then $V$ is a rank two vector bundle on $S$ with $c_1(V) = \Delta$ and $c_2(V) \leq c_2(\tilde V)$, where equality holds if and only if $\tilde V = \rho ^*V$. The arguments of the proof of Theorem 5.5 in Part II of [8], which essentially just depend on the determinant of $\tilde V$ being a pullback, show the following. There is a constant $N_0$, depending only on $L$ and $c$, such that, for all $N\geq N_0$, if $\tilde V$ is $L_N$-stable then $V$ is $L$-semistable, and conversely if $V$ is $L$-stable then $\tilde V$ is $L_N$-stable. Moreover the map $V\mapsto \rho ^*V$ defines an open immersion of schemes from the moduli space of $L$-stable rank two vector bundles on $S$ with $c_1=\Delta$ and $c_2=c$ to the corresponding moduli space for $\tilde S$ and $c_1 = \rho ^*\Delta$. To evaluate the Donaldson polynomial on $\tilde S$ on a collection of classes of the form $\rho ^*\alpha$, represent $\alpha$ by a smoothly embedded Riemann surface $C$ on $S$ which does not pass through $p$, the center of the blowup, and choose a theta characteristic on $C$. This choice leads to a divisor $D_C$ on the moduli space. By definition $\tilde V$ lies in $D _C$ if and only if the Dirac operator coupled to the ASD connection induced on $C$ has a kernel. From this it is clear that $\tilde V$ lies in $D _C$ if and only if $V$ lies in the corresponding divisor on the moduli space for $S$ of bundles with $c_1 = \Delta$ and $c_2 = c_2(V)\leq c$. An easy counting argument then shows that, if $d$ is the dimension of the moduli space for $\tilde S$ and we choose $C_1, \dots, C_d$ in general position and general theta characteristics on $C_i$, then $\tilde V$ lies in the intersection $D _{C_1} \cap \dots \cap D_{C_d}$ if and only if $\tilde V = \rho ^*V$ and $V$ lies in the corresponding intersection for the moduli space of $S$. As $\#(D _{C_1} \cap \dots \cap D_{C_d})$ calculates the value $\gamma _{\rho ^*w_0, p}([C_1], \dots, [C_d])$, it is then clear that $$\gamma _{\rho ^*w_0, p}|\rho ^*H_2(S; \Zee) = \gamma _{w_0, p}. \qed $$ \medskip Assuming Theorems 1.4 and 1.5, we can now state the main results of this series of papers. \theorem{5.3} Suppose that $S$ and $S'$ are two simply connected elliptic surfaces with $p_g(S) = p_g(S') =1$. Suppose that neither $S$ nor $S'$ is a $K3$ surface. Let $\tilde S$ and $\tilde S'$ be two blowups of $S$ and $S'$, and let $\psi \: \tilde S \to \tilde S'$ be a diffeomorphism. Identify $H^2(S; \Zee)$ with its image in $H^2(\tilde S; \Zee)$ under the natural map, and similarly for $H^2(S'; \Zee)$. Then $\psi ^*\kappa _{S'} = \pm \kappa _S$. \endstatement \proof Arguing as in Corollary 3.6 of Chapter 6 of [9], we see that it suffices to show that some $\gamma _{w,p}(S)$ actually involves $\kappa _S$. If $m_1m_2\equiv 1\mod 2$, then the coefficient of $\kappa _S^2$ in $\gamma _{w,p}(S)$, for the choice of $p$ given in Theorem 1.5 (i) corresponding to the two-dimensional moduli space, is $2m_1^2m_2^2 -m_1^2 -m_2^2$. This number is zero if $m_1 = m_2 =1$, in which case $S$ is a $K3$ surface. Otherwise $2m_1^2m_2^2 -m_1^2 -m_2^2 >0$. Thus the coefficient of $\kappa _S^2$ is nonzero. If $m_1m_2\equiv 0\mod 2$, then the expected dimension of the moduli space is $4c -\Delta ^2 - 6 \equiv \Delta ^2 \mod 2$. Moreover $\Delta \cdot \kappa _S = 1$ and $K_S = (2m_1m_2 - m_1 - m_2)\kappa$. Since exactly one of $m_1, m_2$ is even, $\Delta \cdot K_S \equiv 1 \mod 2$. Thus by the Wu formula $\Delta ^2 \equiv 1\mod 2$, and the dimension of the moduli space is odd. It follows that every nonzero invariant must involve $\kappa_S$. Since nonzero invariants exist by Theorem 1.4 or more generally by Donaldson's theorem on the nonvanishing of the invariants, there are choices of $p$ for which $\gamma _{w,p}(S)$ actually involves $\kappa _S$. \endproof \theorem{5.4} Suppose that $S$ and $S'$ are two elliptic surfaces with $p_g(S) = p_g(S') \geq 1$ with finite cyclic fundamental group and multiple fibers of multiplicities $\{m_1,m_2\}$ and $\{m_1', m_2'\}$, respectively, that $\tilde S$ and $\tilde S'$ are two blowups of $S$ and $S'$, and that $\tilde S$ and $\tilde S'$ are diffeomorphic. Then $\{m_1,m_2\}=\{m_1', m_2'\}$. Hence $S$ and $S'$ are deformation equivalent. \endstatement \proof As in [9] we can reduce to the simply connected case. Using Theorem 1.3, we know that $m_1m_2 = m_1'm_2'$ and that, if $\psi \: \tilde S \to \tilde S'$ is a diffeomorphism, then $\psi _*(H_2(S; \Zee)) = H_2(S'; \Zee)$ under the identification of $H_2(S; \Zee)$ with a subspace of $H_2(\tilde S;\Zee)$ and likewise for $S'$. Fix a class $w\in H^2(S'; \Zee)$ with $w\cdot \kappa_{S'}=1$. Thus $$\gamma _{\psi ^*w,p}(\tilde S)|H_2(S; \Zee) = \gamma _{w,p}(\tilde S')|H_2(S'; \Zee)$$ under the natural identifications. Moreover $\psi ^*w\cdot \kappa _S = w\cdot \kappa _{S'} =1$. Thus the Donaldson polynomial invariants for the minimal surfaces $S$ and $S'$ for the values $\psi ^*w$ and $w$ respectively are equal. If $m_1m_2 = m_1'm_2'\equiv 1\mod 2$, then $$(p_g(S) + 1)m_1^2m_2^2 -m_1^2 -m_2^2 = (p_g(S) + 1)(m_1')^2(m_2')^2 -(m_1')^2 -(m_2')^2.$$ Thus $m_1m_2 = m_1'm_2'$ and $m_1^2 +m_2^2= (m_1')^2 +(m_2')^2$. It follows that $(m_1 + m_2)^2 = (m_1' + m_2')^2$ and so $m_1 + m_2 = m_1' + m_2'$. We can thus determine the elementary symmetric functions of $m_1$ and $m_2$ from the diffeomorphism type, and hence the unordered pair. If $m_1m_2 = m_1'm_2'\equiv 0\mod 2$, then, assuming that $2|m_1$, it follows from Theorem 1.4 that we can determine $m_1m_2$ and $m_2$. Thus, we can determine $m_1$ as well. \endproof \theorem{5.5} Suppose that $S$ and $S'$ are two nonrational elliptic surfaces with finite cyclic fundamental group and with $p_g(S) = p_g(S') =0$ with multiple fibers of multiplicities $\{m_1,m_2\}$ and $\{m_1', m_2'\}$, respectively, that $\tilde S$ and $\tilde S'$ are two blowups of $S$ and $S'$, and that $\tilde S$ and $\tilde S'$ are diffeomorphic. Suppose further that $m_1m_2 \equiv 0 \mod 2$. Then $m_1'm_2' \equiv 0\mod 2$, and $\{m_1,m_2\}=\{m_1', m_2'\}$. \endstatement \proof We may again reduce to the simply connected case. Note that every diffeomorphism $\psi$ from $\tilde S$ to $\tilde S'$ sends the subspace $H^2(S';\Zee)$ to $H^2(S; \Zee)$. Choose a class $w\in H^2(S'; \Zee/2\Zee)$ with $w\cdot \kappa _{S'} = 1$. We must have $\psi ^*\Cal C(w,p)= \pm \Cal C(\psi ^*w,p)$, by Lemma 3.5. As in the preceding argument, we are immediately reduced to comparing the Donaldson invariants for the surfaces $S$ and $S'$. Let us first show that $m_1'm_2' \equiv 0\mod 2$. First note that the two-dimensional invariant corresponds to $-p=5> 4=2(4p_g(S) +2)$. Thus we are in the stable range and can apply Theorem 1.4 to conclude that the leading coefficient of $\gamma _{w, -5}(S) = m_2$. Since $S$ is not rational $m_2>1$. But if $m_1'm_2'\equiv 1\mod 2$, then by (i) of Theorem 1.5 the leading coefficient of $\gamma _{w, -5}(S')$ is 1, a contradiction. Hence $m_1'm_2'\equiv 0\mod 2$ and the leading coefficient of $\gamma _{w,-5}(S')$ is just $m_2'$. It follows that $m_2 = m_2'$ and, by Bauer's result (Theorem 1.2), that $(m_1^2-1)(m_2^2 -1) = ((m_1')^2-1)((m_2')^2 -1)$. Thus $m_1 = m_1'$ as well. \endproof \theorem{5.6} Suppose that $S$ and $S'$ are two nonrational elliptic surfaces with finite cyclic fundamental group and with $p_g(S) = p_g(S') =0$ with multiple fibers of multiplicities $\{m_1,m_2\}$ and $\{m_1', m_2'\}$, respectively, that $\tilde S$ and $\tilde S'$ are two blowups of $S$ and $S'$, and that $\tilde S$ and $\tilde S'$ are diffeomorphic. Suppose further that $m_1m_2 \equiv 1 \mod 2$. Then $m_1'm_2' \equiv 1\mod 2$, and $\{m_1,m_2\}=\{m_1', m_2'\}$. \endstatement \proof As before we pass to the simply connected case. If $m_1'm_2' \equiv 0\mod 2$, then by (5.5) $m_1m_2 \equiv 0\mod 2$ as well, a contradiction. Thus $m_1'm_2' \equiv 1\mod 2$. Using (i) and (ii) of Theorem 1.5, we see that the Donaldson polynomials determine the quantities $$\align A &= m_1^2m_2^2 - m_1^2 - m_2^2+1=(m_1^2-1)(m_2^2-1);\\ B &= m_1^4m_2^4 - m_1^4 - m_2^4+1=(m_1^4-1)(m_2^4-1). \endalign$$ We must show that $A$ and $B$ determine $\{m_1, m_2\}$ provided that both $m_1$ and $m_2$ are greater than one. This is just a matter of elementary algebra: let $\sigma _1= m_1^2 + m_2^2$ and $\sigma _2 = m_1^2m_2^2$. Then $\sigma _1$ and $\sigma _2$ are the elementary symmetric functions in $m_1^2$ and $m_2^2$ and thus determine $\{m_1^2, m_2^2\}$. As $m_1$ and $m_2$ are positive the knowledge of $\{m_1^2, m_2^2\}$ determines $\{m_1, m_2\}$. To read off $\sigma _1$ and $\sigma _2$ from $A$ and $B$, note that if $A\neq 0$ then $$\frac{B}{A} = (m_1^2+1)(m_2^2+1) = m_1^2m_2^2 + m_1^2 + m_2^2+1.$$ Thus $2\sigma _2 = B/A + A-2$ and $2\sigma _1 = B/A -A$. provided that $A\neq 0$. Now $A= 0$ if $m_1$ or $m_2$ is one, and otherwise $A\geq 1$. Thus, provided neither of $m_1$ or $m_2$ is one, $A$ and $B$ determine $\sigma _2$ and $\sigma _1$. \endproof \Refs \widestnumber\no{99} \ref \no 1\by S. Bauer\paper Some nonreduced moduli of bundles and Donaldson invariants for Dolgachev surfaces\jour J. reine angew. Math.\vol 424\yr 1992\pages 149--180\endref \ref \no 2\bysame \paper Diffeomorphism classification of elliptic surfaces with $p_g=1$\toappear \endref \ref \no 3\by I. Dolgachev \paper Algebraic surfaces with $q = p_g = 0$ \inbook in Algebraic Surfaces \bookinfo C.I.M.E. Cortona 1977 \publ Liguori \publaddr Napoli \yr 1981 \pages 97--215 \endref \ref \no 4\by S. K. Donaldson \paper Irrationality and the $h$-cobordism conjecture \jour J. Differential Geometry \vol 26 \yr 1987 \pages 141--168 \endref \ref \no 5\by S. K. Donaldson and P. B. Kronheimer \book The Geometry of Four-Manifolds \publ Clarendon \publaddr Oxford \yr 1990 \endref \ref \no 6\by R. Fintushel and R. Stern \paper $SO(3)$-connections and the topology of 4-manifolds \jour J. Differential Geometry \vol 20 \yr 1984 \pages 523--539\endref \ref \no 7\by R. Friedman \paper Rank two vector bundles over regular elliptic surfaces \jour Inventiones Math. \vol 96 \yr 1989 \pages 283--332 \endref \ref \no 8\by R. Friedman and J. W. Morgan \paper On the diffeomorphism types of certain algebraic surfaces I \jour J. Differential Geometry \vol 27 \yr 1988 \pages 297--369 \moreref \paper II \jour J. Differential Geometry \vol 27 \yr 1988 \pages 371--398 \endref \ref \no 9\by R. Friedman and J. W. Morgan \book Smooth 4-manifolds and complex surfaces \toappear \endref \ref \no 10\by D. Kotschick\paper On manifolds homeomorphic to $\Bbb C P^2\# 8\overline{\Bbb C P}^2$\jour Inventiones Math.\vol 95\yr 1989\pages 591--600\endref \ref \no 11\bysame \paper $SO(3)$-invariants for 4-manifolds with $b_2^+=1$ \jour Proc. London Math. Soc. \vol 63 \yr 1991 \pages 426--448 \endref \ref \no 12\by D. Kotschick and J. W. Morgan \paper $SO(3)$-invariants for 4-manifolds with $b_2^+=1$ II\toappear\endref \ref \no 13\by P. Kronheimer and T. Mrowka \paper Recurrence relations and asymptotics for four-manifold invariants \toappear\endref \ref \no 14\by J. W. Morgan and T. Mrowka \paper On the diffeomorphism classification of regular elliptic surfaces \toappear\endref \ref \no 15\by J. W. Morgan and K. O'Grady \paper The smooth classification of fake $K3$'s and similar surfaces \toappear \endref \ref \no 16\by T. Mrowka \paper A local Mayer-Vietoris principle for Yang-Mills moduli spaces \paperinfo Berkeley PhD. thesis \yr 1989\endref \ref \no 17\by Z. Qin \paper Equivalence classes of polarizations and moduli spaces of sheaves \jour J. Differential Geometry \vol 37 \yr 1993 \pages 397--415 \endref \ref \no 18\by J.-P. Serre \book Cohomologie Galoisienne \bookinfo Lecture Notes in Mathematics {\bf 5} \publ Springer Verlag \publaddr Berlin Heidelberg New York \yr 1973 \endref \endRefs \enddocument

9307

alg-geom/9307003

en

https://arxiv.org/abs/alg-geom/9307003

alg-geom/9307003

Robert Friedman

Robert Friedman

Vector bundles and $SO(3)$ invariants for elliptic surfaces II: The case of even fiber degree

18 pages, AMS-TeX

This paper is the second in a series of three devoted to the smooth classification of simply connected elliptic surfaces. In this paper, we study the case where one of the multiple fibers has even multiplicity, and describe the moduli space of stable rank two vector bundles with the appropriate first Chern class needed to calculate Donaldson polynomials. The analysis is in many ways parallel to the analysis in the case of vector bundles of trivial determinant, but the asymmetry between the multiplicities also appears in the moduli space. In this way, the first coefficient of an appropriate Donaldson polynomial determines one of the multiplicities.

alg-geom

\section{Introduction.} Let $S$ be a simply connected elliptic surface with at most two multiple fibers. In this paper, the second in a series of three, we are concerned with describing moduli spaces of stable vector bundles $V$ over $S$ such that the restriction of $c_1(V)$ to a general fiber has the smallest possible nonzero degree, namely the product of the multiplicities, in the case where this product is even. We then apply this study toward a partial calculation of the corresponding Donaldson polynomial invariants of $S$. Our goal is the completion of the $C^\infty$ classification of such surfaces, and the general outline of this classification has been described in the introduction to Part I. Aside from quoting a few results from Part I, this paper can however be read independently. On the other hand, the methods of this paper draw heavily on the book [4], and many arguments which are very similar to arguments in [4] are sketched or simply omitted. Roughly speaking, the new ingredients in the proof consist of the algebraic geometry of certain elliptic surfaces associated to $S$, which have a single multiple fiber of multiplicity two and are birational to double covers of rational ruled surfaces. The vector bundle parts of the argument run more or less parallel to the arguments in [4], with a few new cases to analyze. The outline of this paper is as follows. In this paper, we shall only be concerned with elliptic surfaces $S$ over $\Pee ^1$ with multiple fibers of multiplicities $2m_1$ and $m_2$, where $m_2$ is odd, and such that there exists a divisor $\Delta$ on $S$ with $\Delta \cdot f = 2m_1m_2$, the minimum possible value, for a smooth fiber $f$. In this case, there is an associated surface $J^{m_1m_2}(S)$ defined in [3]. The surface $J^{m_1m_2}(S)$ fibers over $\Pee ^1$ and the fiber over a point $t$ lying under a smooth fiber $f$ of $S$ is $J^{m_1m_2}(f)$, the set of line bundles of degree $m_1m_2$ on the fiber $f$ of $S$. The surface $J^{m_1m_2}(S)$ has an involution defined by $\lambda \in J^{m_1m_2}(f) \mapsto \scrO _f(\Delta |f) \otimes \lambda ^{-1}$. The quotient of $J^{m_1m_2}(S)$ by this involution is birational to a rational ruled surface $\Bbb F_N$, and we describe the geometry of the double cover in detail. In Section 2, we describe the rough classification of stable bundles $V$ on $S$ with $c_1(V)=\Delta$. To each such bundle there is an associated bisection $C$ of $J^{m_1m_2}(S)$ which is invariant under the involution, and so defines a section of the quotient ruled surface. In Section 3, we show that for general bundles $V$, $V$ is determined up to finite ambiguity by the section of the ruled surface and the choice of a certain line bundle on the associated bisection $C$ of $J^{m_1m_2}(S)$. Thus, a Zariski open subset of the moduli space fibers over an open subset of the linear system of all sections on a certain rational ruled surface, and the fibers are a number of copies of the Jacobian $J(C)$ of the bisection $C$ of $J^{m_1m_2}(S)$. It is here that the asymmetry between $2m_1$ and $m_2$ becomes apparent: the number of connected components of the fiber is just $m_2$. The reasons for this are explained following Lemma 2.4. Finally, in Section 4 we calculate the leading coefficient of a Donaldson polynomial invariant and show that it contains an ``extra" factor of $m_2$. \section{1. Geometry of elliptic surfaces.} We fix notation for this paper. Let $\pi \: S\to \Pee ^1$ be an elliptic surface with two multiple fibers $F_{2m_1}$ and $F_{m_2}$ of multiplicities $2m_1$ and $m_2$, where $\gcd (2m_1, m_2)=1$. We shall further assume that the reductions of the multiple fibers are smooth and that all other singular fibers are reduced and irreducible with just one ordinary double point. In other words, $S$ is nodal in the terminology of [4]. Let $\kappa _S=\kappa \in H^2(S;\Zee)$ be the unique class such that $2m_1m_2\kappa = [f]$, where $f$ is a general fiber of $\pi$. Finally we shall assume that there is a $2m_1m_2$-section $\Delta$, i.e\. a divisor $\Delta$, not necessarily effective, with $\Delta \cdot f = 2m_1m_2$, or equivalently $\Delta \cdot \kappa = 1$. By [4], there always exist such nodal elliptic surfaces. In particular $S$ is algebraic. We shall be concerned with the associated elliptic surface $J^{m_1m_2}(S)$ defined in Section 2 of [3]. By the discussion in Section 2 of [3], $J^{m_1m_2}(S)$ has exactly one multiple fiber of multiplicity two, above the point of $\Pee ^1$ corresponding to $F_{2m_1}$. We shall denote this fiber by $F_2$. Moreover, given the $2m_1m_2$-section $\Delta$, there is an involution of $J^{m_1m_2}(S)$ defined on the generic fiber $S_\eta$ by $\lambda \mapsto \scrO_{S_{\eta}}(\Delta)\otimes \lambda ^{-1}$. Since $J^{m_1m_2}(S)$ is relatively minimal, this involution on the generic fiber extends to an involution on $J^{m_1m_2}(S)$, which we shall denote by $\iota$. The data of $J^{m_1m_2}(S)$ and the involution $\iota$ do not depend on $\Delta$, but only on the restriction of $\Delta$ to the generic fiber. A line bundle which has the same restriction as $\Delta$ to the generic fiber differs from $\Delta$ by a line bundle which is trivial on the generic fiber and thus is a multiple of $\kappa$. Up to twisting by a line bundle which is divisible by 2, the only possibilities are then $\Delta$ and $\Delta - \kappa$. Replacing $\Delta$ by $\Delta - \kappa$ replaces $\Delta ^2$ by $\Delta ^2 -2$ and thus changes $\Delta ^2 \mod 4$. Of course $\Delta ^2\equiv 1 \mod 2$ since $$\Delta \cdot K_S = 2m_1m_2(p_g+1) - 2m_1-m_2\equiv 1 \mod 2.$$ Let us determine the fixed point set of $\iota$. \lemma{1.1} The fixed point set of $\iota$ consists of a smooth $4$-section together with two isolated fixed points on $F_2$. \endstatement \proof The fixed point set of an involution on the smooth surface $J^{m_1m_2}(S)$ must consist of a smooth curve together with some isolated fixed points. For a smooth nonmultiple fiber $f$, there are four divisors $\lambda$ such that $2\lambda = f$. Thus there is a component of the fixed point set which is a 4-section of $J^{m_1m_2}(S)$. Every other curve component of the fixed point set has trivial restriction to the generic fiber and (since the curve component is smooth) cannot meet the 4-section. Since all fibers are irreducible, there can be no other component, and the remaining fixed points are isolated. Let us consider the possibilities for isolated fixed points away from $F_2$. In an analytic neighborhood of a nonmultiple fiber, there is a section of the elliptic fibration. Using this section to make the local identification of $J^{m_1m_2}(S)$ with $J^0(S)$, and since the group of local sections is divisible, it is easy to see that, after a translation, we can assume that $\iota$ corresponds to the involution $x\mapsto -x$. Direct inspection shows that this involution has no isolated fixed points, even at the nodal fibers. To handle the multiple fiber, the explicit description of a neighborhood $X$ of a multiple fiber shows that after a base change of order two, say $\tilde X \to X$, we can assume that there is a local section of $\tilde X$. The induced map from the central fiber of $\tilde X$ to the central fiber of $X$ corresponds to taking the quotient by a subgroup of order two. After a translation we can further assume that the pulled back involution on $\tilde X$ is given by $x\mapsto -x$. Since inverses commute with translations by a point of order two, the restriction of $\iota$ to $F_2$ again has four fixed points. Two of these lie on the 4-section (recall that a 4-section can meet $F_2$ in at most two distinct points) and the remaining two are isolated. \endproof Thus there are two isolated fixed points of $\iota$ on $F_2$. If we blow these up and then take the quotient, the result will be a smooth surface $\Bbb F$ mapping to $\Pee ^1$ whose fibers are smooth rational curves except over the point corresponding to $F_2$ ({\smc Fig. 1}). Over this point, the fiber is a curve $\frak d_1+2\frak e+\frak d_2$, where $\frak d_1$ and $\frak d_2$ are the images of the exceptional curves, $\frak e$ is the image of $F_2$, and we have $(\frak d_i)^2 = -2$ and $\frak e^2 = -1$, $\frak d_1\cdot \frak e =\frak d_2\cdot \frak e = 1$ and $\frak d_1\cdot \frak d_2 = 0$. In particular we may contract $\frak e$ and then either $\frak d_1$ or $\frak d_2$ to obtain a rational ruled surface $\Bbb F_N$. We shall fix notation so that $\frak d_2$ is the curve we contract and the resulting surface is $\Bbb F_N$. However, as we shall see, it is important to keep in mind the symmetry between $\frak d_1$ and $\frak d_2$. Contracting $\frak d_1$ instead corresponds to making an elementary modification of $\Bbb F_N$ and thus replacing it by $\Bbb F_{N\pm 1}$. As we shall see, the symmetry between $\frak d_1$ and $\frak d_2$ corresponds to the choice of either $\Delta$ or $\Delta - \kappa$. The branch divisor $B'$ on the blowup $\Bbb F$ of $\Bbb F_N$ is of the form $B+ \frak d_1+\frak d_2$, where $B$ is a smooth 4-section which does not meet $\frak d_1$ or $\frak d_2$ and hence $B\cdot \frak e = 2$. Thus if we use the basis $\{\sigma, f, \frak d_2, \frak e\}$ for $\operatorname{Pic}(\Bbb F)$, where $\sigma $ is the negative section of $\Bbb F_N$ and $f$ is the class of a fiber, viewed as curves on $\Bbb F$, it is easy to see that $$B = 4\sigma + (2k+1)f -4\frak e -2\frak d_2$$ for some odd integer $k$ and that $$B+\frak d_1+\frak d_2 = 4\sigma + (2k+2)f -6\frak e -2 \frak d_2,$$ which is indeed divisible by 2. Note that we cannot say {\it a priori\/} that $B$ is irreducible. However it cannot be a union of a 3-section and a section, since there are no sections of $J^{m_1m_2}(S)$. In particular $B\cdot \sigma \geq 0$. Thus if $\frak d_1$ is the proper transform of the fiber on $\Bbb F_N$, and we assume that the negative section $\sigma$ does not pass through the point of the original fiber that was blown up, so that $\sigma \cdot \frak d_2 = 0$, then $2k+1 \geq 4N$, or equivalently $$k\geq 2N.$$ The same conclusion holds by a similar argument if $\sigma$ does pass through the point that is blown up. It is now easy to reverse this procedure. Begin with $\Bbb F_N$ and blow up a point in a fiber. Then blow up the point of intersection of the exceptional curve with the proper transform of the fiber. The result is a a nonmiminal ruled surface $\Bbb F$ with a reducible fiber of the ruling of the form $\frak d_1+2\frak e + \frak d_2$, where $\frak d_1$ is the proper transform of the fiber, $\frak d_2$ is the proper transform of the first exceptional curve, and $\frak e$ is the second exceptional curve. Choose a smooth element $B$ in the linear system $|4\sigma +(2k+1)f - 4\frak e -2\frak d_2|$, if any exist, where $\sigma$ is the negative section of $\Bbb F_N$ and $f$ is a fiber. The double cover of $\Bbb F$ branched along $B+\frak d_1+\frak d_2$ is then an elliptic surface with a multiple fiber of multiplicity 2, bisections corresponding to the pullbacks of sections of $\Bbb F_N$, and an involution $\iota$. Let us calculate $p_g(J^{m_1m_2}(S)) = p_g(S)$ in terms of $\Bbb F$ and $B$. The canonical bundle of $\Bbb F_N$ is given by $K_{\Bbb F_N} = -2\sigma -(N+2)f$. Thus, recalling that $\frak d_2$ is the proper transform of the first exceptional curve and that $\frak e$ is the second, we have $$K_{\Bbb F} = -2\sigma -(N+2)f +\frak d_2 +2\frak e.$$ As for the branch locus $B+\frak d_1+\frak d_2$, we have $$B+\frak d_1+\frak d_2 = 2(2\sigma +(k+1)f -3\frak e -\frak d_2).$$ By standard formulas for double covers, $$H^0(J^{m_1m_2}(S); K_{J^{m_1m_2}(S)}) = H^0(\Bbb F; \scrO _{\Bbb F}(K_{\Bbb F} + 2\sigma +(k+1)f -3\frak e -\frak d_2)).$$ Now, using the calculations above, we have $$K_{\Bbb F} + 2\sigma +(k+1)f -3\frak e -\frak d_2 = (k-N-1)f - \frak e.$$ Recalling that $f$ is linearly equivalent to $\frak d_1+2\frak e + \frak d_2$, it is clear that $$h^0((k-N-1)f - \frak e) = \cases k-N-1, &\text{if $k-N\geq 2$}\\ 0, &\text{otherwise.} \endcases$$ Now if $k-N \leq 1$, since $k\geq 2N$ we must have $N\leq 1$. If $N=1$, then $k=2$ and so $k-N-1=0$ in this case as well. If $N=0$, then $k=0$. In this case $B= 4\sigma +f-4\frak e -2\frak d_2= 4\sigma +\frak d_1-2\frak e - \frak d_2$, and it is easy to see that the effective curve $\sigma - \frak d_2- \frak e$, which is the proper transform of the unique element of $|\sigma|$ passing through the point of the fiber which is blown up, satisfies $$ (\sigma - \frak d_2-\frak e)\cdot (4\sigma +f-4\frak e -2\frak d_2) =-1.$$ Thus $|B|$ has the fixed component $\sigma - \frak d_2-\frak e$, which is a section, and this case does not arise. So in all cases we have $h^0((k-N-1)f - \frak e) = k-N-1$. Thus we may summarize this discussion as follows: \lemma{1.2} With notation and conventions as above, $$p_g(S) =p_g(J^{m_1m_2}(S)) = k-N-1. \qed$$ \endstatement In particular, suppose that $p_g(S) = 0$. Since $k\geq 2N$, and the case $k=N = 0$ has been ruled out above, the only possibilities are $N = 0$, $k=1$ or $N=1$, $k=2$. Thus $\Bbb F$ is the blowup of either $\Bbb F_0$ or $\Bbb F_1$, and of course the two cases are elementary transformations of each other. Thus we may assume that $k=2$ and $N=1$ in this case. Moreover the negative section of $\Bbb F_1$ does not pass through the exceptional point in the blowup. \section{2. Classification of stable bundles.} We let $S$ be a nodal elliptic over $\Pee ^1$ with exactly two multiple fibers of multiplicities $2m_1$ and $m_2$ and let $\Delta$ be a divisor on $S$ with $\Delta \cdot f = 2m_1m_2$. Fix an integer $c$. In this section we shall study rank two vector bundles $V$ with $c_1(V) = \Delta$ and $c_2(V) = c$. We shall also let $w = \Delta \bmod 2$ and $p = \Delta ^2 -4c$. First recall the following standard definition from [3]: \definition{Definition 2.1} An ample line bundle $L$ on $S$ is {\sl $(\Delta, c)$-suitable\/} or {\sl $(w,p)$-suitable\/} if for all divisors $D$ on $S$ such that $-D^2 + D\cdot \Delta \leq c$, either $f\cdot(2D - \Delta) = 0$ or $$\operatorname{sign} f\cdot (2D - \Delta) = \operatorname{sign}L\cdot (2D - \Delta).$$ \enddefinition The following is Lemma 3.3 of [3]: \lemma{2.2} For all pairs $(\Delta, c)$, $(\Delta, c)$-suitable ample line bundles exist. \qed \endstatement With this said, here is the rough classification of rank two vector bundles $V$ with $c_1(V) = \Delta$ and $c_2(V) = c$ which are stable with respect to a $c$-suitable line bundle $L$. \theorem{2.3} Let $L$ be $(\Delta, c)$-suitable, and let $V$ be an $L$-stable rank two vector bundle with $c_1(V) = \Delta$ and $c_2(V) = c$. Then there exist: \roster \item"{(i)}" A smooth irreducible curve $C$ and a birational map $C \to \overline C \subseteq J^{m_1m_2}(S)$, where $\overline C$ is a bisection of $J^{m_1m_2}(S)$ invariant under the involution $\iota$; \item"{(ii)}" A divisor $D$ on $ T$, the minimal desingularization of the normalization of $C\times _{\Pee^1}S$, such that $D\cdot f = m_1m_2$, where $f$ is a general fiber of $T\to C$, and moreover $D$ has the same restriction to the generic fiber of $ T$ as the divisor induced by the section of $J^{m_1m_2}(T)$ corresponding to the map $C \to J^{m_1m_2}(S)$; \item"{(iii)}" A codimension two local complete intersection $Z$ and an exact sequence $$0 \to \scrO_{T}(D) \to \nu ^*V \to \scrO_{T}(\nu ^*\Delta -D)\otimes I_Z \to 0,$$ where $\nu \:T \to S$ is the natural degree two map. \endroster Moreover the bisection $\overline C$ and the double cover $T$ are uniquely determined by the bundle $V$, and $D$ is determined by the bundle $V$ and the choice of a map $C \to J^{m_1m_2}(S)$. Finally, every rank two vector bundle $V$ with $c_1(V) = \Delta$ and $c_2(V) = c$ satisfying \rom{(i)}--\rom{(iii)} above is stable with respect to every $(\Delta, c)$-suitable ample line bundle $L$. \endstatement \proof First suppose that $V$ is $L$-stable. It follows from Theorem 4.3 of [3] that the restriction of $V$ to the geometric fiber of $\pi$ is semistable. More precisely, let $\eta = \operatorname{Spec} k(\Pee^1)$ and let $\bar \eta = \operatorname{Spec} \overline{k(\Pee^1)}$, where $\overline{k(\Pee^1)}$ denotes the algebraic closure of $k(\Pee^1)$. Let $V_{\bar \eta}$ denote the pullback of $V$ to the curve $S_{\bar \eta}$ which is the geometric fiber of $\pi$. Then $V_{\bar \eta}$ is semistable. By the classification of rank two bundles on an elliptic curve, $V_{\bar \eta}= L_1 \oplus L_2$, where each $L_i$ is a line bundle over $S_{\bar \eta}$ of degree $m_1m_2$ and $L_1\otimes L_2$ corresponds to the restriction of $\Delta$ to $S_{\bar \eta}$. The Galois group $\operatorname{Gal}(\overline{k(\Pee^1)}/k(\Pee ^1)$ permutes the set $\{L_1, L_2\}$. This action cannot be trivial, since otherwise $L_i$ would be rational over $k(\Pee ^1)$ and then $S$ would have an $m_1m_2$-section. Thus the fixed field of the subgroup of $\operatorname{Gal}(\overline{k(\Pee^1)}/k(\Pee ^1)$ which operates trivially on $\{L_1, L_2\}$ defines a degree two extension of $k(\Pee^1)$, corresponding to a morphism $C\to \Pee ^1$. Setting $T $ to be the minimal resolution of the normalization of $C\times _{\Pee ^1}S$, there is a section of $J^{m_1m_2}(T )$ defined by $L_1$, say. The image of this section in $J^{m_1m_2}(S)$ is then the bisection $\overline C$. By construction $\overline C$ is invariant under the involution $\iota$. Let $\nu \:T \to S$ be the natural degree two map. The inclusion $L_1 \to V_{\bar \eta}$ induces a sub-line bundle $\scrO_{T}(D) \to \nu ^*V$, which we may assume to have torsion free cokernel. Since $L_2 \neq L_1$, it is clear that this sub-line bundle is unique. The quotient is then necessarily of the form $\scrO_{ T}(\nu ^*\Delta -D)\otimes I_Z$. It remains to prove that every $V$ satisfying the above description is indeed $L$-stable. It follows from (iii) that $ V_{\bar \eta}$ is an extension of two line bundles of degree $m_1m_2$ and is therefore semistable. Again using Theorem 4.3 of [3], $V$ is $L$-stable. \endproof Next we discuss the meaning of the scheme $Z$ and the bisection $\overline C$. The following is the analogue of Lemma 1.11 in [4] and is proved in exactly the same way: \lemma{2.4} Let $f$ be a smooth fiber of $\pi$ and let $g$ be a component of $\nu^{-1}(f)$. Then $\operatorname{Supp}Z\cap g \neq \emptyset$ if and only if $V|f$ is unstable. In particular, if $\nu$ is not branched over $f$, so that $\nu^{-1}(f) = g\cup g'$, then $\operatorname{Supp}Z\cap g \neq \emptyset$ if and only if $\operatorname{Supp}Z\cap g' \neq \emptyset$. \qed \endstatement Next we turn to the section $\overline C$. Since $\overline C$ is invariant under $\iota$, its proper transform on the blowup of $J^{m_1m_2}(S)$ at the two isolated fixed points of $\iota$ is the pullback of a section $A'$ of $\Bbb F$, which in turn induces a section $A$ of $\Bbb F_N$. We shall use throughout the notation and conventions of the previous section. Notice that the section $A'$ meets the reducible fiber either along $\frak d_1$ or $\frak d_2$, the two components of multiplicity one. Here $A'\cdot \frak d_1 =1$ and $A'\cdot \frak d_2=0$ if $A$ does not pass through the point of the corresponding fiber of $\Bbb F_N$ which was blown up, and $A' \cdot \frak d_2 =1$ and $A'\cdot \frak d_1=0$ in the remaining case. Since the branch locus of the map $J^{m_1m_2}(S) \to \Bbb F$ consists of $B+\frak d_1+\frak d_2$, we see that $A'$ always passes through the branch locus over the point corresponding to $F_2$. Of course, this is also clear from the picture on $J^{m_1m_2}(S)$: since $\overline C$ is a bisection, $\overline C \cdot f =2$ and therefore $\overline C \cdot F_2=1$. It follows that $\overline C$ is smooth at the point of intersection with $F_2$, that the intersection is transverse, and that the natural map $\overline C \to \Pee ^1$ is always branched at the point corresponding to $F_2$. A similar statement will hold for the map $C\to \Pee ^1$. This fact is the fundamental difference between the case studied in this paper and the case of trivial determinant studied in [2] and [4]. Since $A'$ always meets $\frak d_1$ or $\frak d_2$, it follows that the inverse image of $A'$ in the blowup of $J^{m_1m_2}(S)$ always meets exactly one of the two exceptional curves, and in fact meets it transversally at one point. Thus the inverse image of $A'$ is the proper transform of $\overline C$, and therefore $$(\overline C)^2 = 2(A')^2 + 1.$$ Let us consider the section $A$ of $\Bbb F_N$ in more detail. Either $A= \sigma$ or $A\in |\sigma + (N+s)f|$ for a uniquely specified nonnegative integer $f$. Moreover either $A$ does not pass through the point on $\Bbb F_N$ which is the image of the exceptional divisor, in which case $A'\cdot \frak d_2 =0$, or it does, in which case $A' \cdot \frak d_2=1$. The following lemma relates the odd integer $\Delta ^2 - 4c=p$ to the invariants of $V$: \lemma{2.5} With notation as above, denote by the exceptional point the point of $\Bbb F_N$ which is blown up under the morphism $\Bbb F \to \Bbb F_N$. Then, if we set $p = p_1(\ad V) = \Delta ^2 - 4c$, $$-p = \cases 4k-6N+1+2\ell(Z) +\delta, &\text{if $A=\sigma$ does not pass through}\\ {} &\text{the exceptional point;}\\ 4s+4k-2N+1+2\ell(Z) +\delta, &\text{if $A\in |\sigma +(N+s)f|$}\text{does not pass} \\ {} &\text{through the exceptional point;}\\ 4k-6N-1+2\ell(Z) +\delta, &\text{if $A=\sigma$ passes through}\\ {} &\text{the exceptional point;}\\ 4s+4k-2N-1+2\ell(Z) +\delta, &\text{if $A\in |\sigma +(N+s)f|$ passes through}\\ {}&\text{the exceptional point.} \endcases$$ Here $\delta$ is a nonnegative integer which is zero if the map $C\to \Pee ^1$ is not branched over any point corresponding to a singular nonmultiple fiber of $\pi \: S \to \Pee ^1$. \endstatement \proof Since $c_1^2(\nu ^*V) = 2c_1^2(V)=2\Delta ^2$ and $c_2(\nu ^*V) = 2c_2(V) = 2c$, it will suffice to work with $\nu ^*V$. Clearly $$c_2(\nu ^*V) = -D^2 + D\cdot \nu ^*\Delta + \ell (Z).$$ Thus $$2(4c - \Delta ^2) = -(2D- \nu ^*\Delta )^2 + 4\ell(Z).$$ Now we can write $$2D- \nu ^*\Delta = D-(\nu ^*\Delta -D),$$ where both $D$ and $\nu ^*\Delta -D$ naturally correspond to sections of $J^{m_1m_2}(T)$. In fact, if $\varphi\: J^{m_1m_2}(T) \to J^{m_1m_2}(S)$ is the obvious map, then the bisection $\overline C$ satisfies $\varphi ^*\overline C = C_1 + C_2$, where $C_1$ and $C_2$ are sections of $J^{m_1m_2}(T)$ corresponding to the divisors $D$ and $\nu ^*\Delta -D$ on $T$. An argument essentially identical to the proofs of Claim 1.17 and 1.18 in Chapter 7 of [4] shows that there is a nonnegative integer $\delta$ such that $$ -(2D- \nu ^*\Delta )^2 = -(C_1-C_2)^2 + 2\delta.$$ Moreover $\delta = 0$ if the map $C\to \Pee ^1$ is not branched over any point corresponding to a singular nonmultiple fiber of $\pi \: S \to \Pee ^1$. Thus we must calculate $(C_1-C_2)^2$. But, using the fact that $C_i$ is a section of $J^{m_1m_2}(T)$, we have $$(C_i)^2 = -2(1+p_g(S)) = -2(k-N).$$ Moreover $C_1+C_2 = \varphi ^*\overline C$. Thus $$\align (C_1-C_2)^2 &= 2(C_1)^2 + 2(C_2)^2 - (C_1+C_2)^2\\ &=-8(k-N) -2(\overline C)^2\\ &=-8(k-N) -4(A')^2 - 2. \endalign$$ Clearly we have $$(A')^2 = \cases -N, &\text{if $A=\sigma$ does not pass through the exceptional point;}\\ N+2s, &\text{if $A\in |\sigma +(N+s)f|$ does not pass through}\\ {} &\text{the exceptional point;}\\ -N-1, &\text{if $A=\sigma$ passes through the exceptional point;}\\ N+2s-1, &\text{if $A\in |\sigma +(N+s)f|$ passes through}\\ {}&\text{the exceptional point.} \endcases$$ Putting these formulas together gives the statement of the lemma. \endproof Using Lemma 2.5, the inequality $k\geq 2N$ and the fact that if $N=0$ then $k\geq 1$, whereas if $N=1$ and $k=2$ then the section $\sigma$ does not pass through the exceptional point, we can easily deduce the following slight strengthening of Bogomolov's inequality in our case: \corollary{2.6} We have the following inequality for $-p$: $$-p\geq \cases 4p_g(S) -2N +5 &\text{if $A=\sigma$ does not pass through the exceptional point;}\\ 4p_g(S) +2N +5 &\text{if $A\in |\sigma +(N+s)f|$ does not pass through}\\ {} &\text{the exceptional point;}\\ 4p_g(S) -2N +3 &\text{if $A=\sigma$ passes through the exceptional point;}\\ 4p_g(S) +2N +3 &\text{if $A\in |\sigma +(N+s)f|$ passes through}\\ {}&\text{the exceptional point.} \endcases$$ In all cases $-p \geq 2p_g(S) +1$, and if $p_g(S)=0$, then $-p\geq 3$. \qed \endstatement \section{3. A Zariski open subset of the moduli space.} Our goal in this section is to prove the following theorem: \theorem{3.1} Let $V$ be an $L$-stable rank two bundle on $S$. Suppose that \roster \item"{(i)}" The associated bisection $\overline C$ of $J^{m_1m_2}(S)$ is smooth, or equivalently that $\overline C = C$, and the image of $\overline C$ in $\Bbb F$ is not the proper transform of $\sigma$; \item"{(ii)}" The map $C \to \Pee ^1$ is not branched at any point corresponding to a singular fiber of $\pi$ or at the multiple fiber of odd multiplicity $m_2$; \item"{(iii)}" The scheme $Z$ on the associated double cover $T$ is empty, and thus there is an exact sequence $$0 \to \scrO _T(D) \to \nu ^*V \to \scrO _T(\nu ^*\Delta - D) \to 0.$$ \endroster Then $V = \nu _*\scrO _T(D+F)=\nu _*\scrO _T(\nu ^*\Delta -D)$. In particular $V$ is uniquely determined by the associated section $A$ of $\Bbb F_N$ and the divisor $D$ on $T$. Finally $V$ is a smooth point of its moduli space, which is of dimension $-p -3\chi (\scrO_S)$ at $V$. \endstatement \medskip It is clear that the conditions above are equivalent to assuming that $A'$ meets the branch locus $B$ transversally, and that no point of intersection lies over a point of $\Pee ^1$ corresponding to a singular nonmultiple fiber or to the multiple fiber of multiplicity $m_2$, and that $Z=\emptyset$. The proof of (3.1) will proceed along lines very similar to the proof of Theorem 1.12 in Chapter 7 of [4], and we shall simply sketch some of the details. Let $A$ be the section of $\Bbb F_N$ corresponding to $A'$. By assumption $A \neq \sigma$. Let $r$ be the nonnegative integer such that $A\in |\sigma +(N+r)f|$. If the section $A$ does not pass through the exceptional point of the blowup, then $$(A')\cdot (B+\frak d_1+\frak d_2) = (\sigma +(N+r)f)\cdot (4\sigma +(2k+2)f -6\frak e -2\frak d_2)= 4r+2k+2.$$ Of these points, one corresponds to the intersection $A'\cdot \frak d_1$, and so the branch divisor of the map $T \to S$ is $(4r+2k+1)f$, where $f$ is a general fiber of $\pi$. This divisor is even since $f$ is divisible by $2$, and we set $G =(4r+2k+1)f/2$. Likewise, if $A$ does pass through the exceptional point of the blowup, then $$(A')\cdot (B+\frak d_1+\frak d_2) = (\sigma +(N+r)f-\frak d_2-\frak e)\cdot (4\sigma +(2k+2)f -6\frak e -2\frak d_2)= 4r+2k.$$ In this case we set $G= (4r+2k-1)f/2$. Let $F$ be the branch divisor in $T$, so that $\nu ^*G\equiv F$. Thus $F = (4r+2k+1)f$ or $(4r+2k-1)f$. For future reference, let us also record the genus of $C$: \lemma{3.2} Let $C$ satisfy \rom{(i)} and \rom{(ii)} of \rom{(3.1)}. Then $$g(C) = \cases 2r+k, &\text{if $A$ does not pass through the exceptional point;}\\ 2r+k-1, &\text{if $A$ passes through the exceptional point.} \endcases$$ \endstatement \proof The map $C\to \Pee ^1$ is branched at $A'\cdot (B+\frak d_1+\frak d_2) = 4r+2k+2$ points if $A$ does not pass through the exceptional point, and $4r + 2k$ points otherwise. The lemma now follows from the Riemann-Hurwitz formula. \endproof Now $\det \nu _*\scrO_T(D) = \nu _*D - G$. Clearly $\nu _*D$ and $\Delta$ have the same restriction to the generic fiber. Arguing as in Chapter 7, (1.20) of [4], there is an injective map $$\nu _* \scrO _T(D+F) \to V.$$ Set $W = \nu _* \scrO _T(D+F)$. Our goal will be to show that $W = V$. Preliminary to this goal we shall analyze $W$ and the map $W\to V$. As a divisor class $\det W = \nu _*D-G +2G= \nu _* D + G$. In addition there is an effective divisor $E$ such that $(\det W)^{-1} \otimes \det V = \scrO _S(E)$. Thus $\det W = \Delta - E$ and $E$ has trivial restriction to the generic fiber, so that $E$ is a union of fibers (possibly including the reductions of the multiple fibers). Moreover $$\nu _*D = \Delta - G - E.$$ Set $E' = \nu ^*E$. We have $$D + \iota ^*D = \nu ^*\nu _*D = \nu ^*\Delta -F -E',$$ and therefore $$\iota ^*D = \nu ^*\Delta - D -F -E'.$$ We can thus write $$W = \nu _*\scrO _T(D+F) = \nu _* \scrO _T(\nu ^*\Delta -D -E').$$ Using the fact that there is a surjection from $\nu ^*W$ to $\scrO _T(\nu ^* \Delta -D -E')$, we conclude that there is an exact sequence $$0 \to \scrO _T(D) \to \nu ^*W \to \scrO _T(\nu ^* \Delta -D -E') \to 0.$$ Comparing this sequence to the defining exact sequence for $\nu ^*V$ and arguing as in (1.24) of Chapter 7 of [4], we may conclude: \lemma{3.3} Let $Q =V/W$. Then $\nu ^*Q \cong [\scrO _T/\scrO_T(-E')] \otimes \scrO_T(\nu ^*\Delta -D)$.\qed \endstatement \medskip Our goal now is to prove the following: \lemma{3.4} In the above notation, $E'=0$. Thus $Q=0$ and $V = W =\nu _* \scrO _T(D+F)$ where $\nu _* D = \Delta -G$. \endstatement \medskip We begin with the following construction. Let $e$ be a component of the support of $E'$, and write $E' = ae + E''$, where $E''$ is effective and disjoint from $e$ and $a>0$. If $e$ is not the multiple fiber of multiplicity $m_1$ on $T$, then either $\nu$ is unbranched over $e$ or $e$ is a smooth fiber. In either of these cases $\nu$ induces an isomorphism from $e$ to $\nu (e)=f$, and we shall identify $\nu (e)$ with $e$. In the remaining case $e=F_{m_1}$ is the multiple fiber of multiplicity $m_1$ and $\nu$ is an \'etale double cover. There is then the following analogue of (1.25) of Chapter 7 of [4]: \lemma{3.5} There is a subsheaf $Q_0$ of $\nu _*Q$ which is isomorphic to \roster \item"{(i)}" $\scrO _e(-(a-1)e+\nu ^*\Delta -D),$ viewed as a sheaf on $\nu (e) = f$, if $e\neq F_{m_1}$; \item"{(ii)}" A line bundle on $F_{2m_1}$ such that $$\nu ^*Q_0 \cong \scrO _{F_{m_1}}(-(a-1)F_{m_1}+\nu ^*\Delta -D),$$ in case $e=F_{m_1}$. \endroster \endstatement \proof The argument in case $e\neq F_{m_1}$ runs as in (1.25) of Chapter 7 of [4]. If $e = F_{m_1}$, then $\nu ^*Q$ contains the subsheaf $$Q_0' = [\scrO _T(-(a-1)e/\scrO_T(-ae)] \otimes \scrO_T(\nu ^*\Delta -D),$$ which is a line bundle on $F_{m_1}$. The vector bundle $\nu _*Q_0'$ is a rank two vector bundle on $F_{2m_1}$ with $\deg (\nu _*Q_0') = m_1m_2$. Consider the rank two vector bundle $\nu ^* \nu _*Q_0'$ on $F_{m_1}$. Its determinant is $Q_0'\otimes \iota^*Q_0' = (Q_0')^{\otimes 2}$ (recall that $\nu ^*\Delta -D$ is fixed under the involution) and there is a surjective map $\nu ^* \nu _*Q_0' \to Q_0'$. Thus there is an exact sequence $$0 \to Q_0' \to \nu ^* \nu _*Q_0' \to Q_0' \to 0.$$ It follows that $\nu ^* \nu _*Q_0'$ is semistable and is either $Q_0'\oplus Q_0'$ or the nontrivial extension of $Q_0'$ by $Q_0'$. On the other hand, $\nu _*Q_0'$ is either a direct sum of line bundles, say $Q_0 \oplus Q_1$ for two line bundles $Q_i$ or a nontrivial extension of a line bundle $Q_0$ by $Q_0$. In the first case we must have $\nu ^*Q_i = Q_0'$ and in the second $\nu ^*Q_0 = Q_0'$. In either case there is a subbundle $Q_0$ of $\nu _*Q$ as desired. \endproof To prove Lemma 3.4, we shall assume that $E'\neq 0$ and derive a contradiction. Again, the argument will be very similar to the argument given in [4]. It will suffice to show that $\dim \operatorname{Ext}^1(Q_0, W) \leq 1$. We also have that $$\operatorname{Ext}^1(Q_0, W) = H^0(W\otimes \scrO _S(e)\otimes Q_0^{-1}).$$ The case where $e$ does not lie over the branch locus of $C\to \Pee ^1$ follows exactly as in [4]. The case where $e$ is a smooth fiber in the branch locus also follows by these methods provided we can show that $R^1\rho _*\scrO _T(2D-\nu ^*\Delta )$ has length one at the point of $C$ corresponding to $e$. This is a local calculation, which we shall leave to the reader; it uses the fact that $A'$ meets the branch locus transversally and can in fact be deduced from the global argument in [4], proof of Lemma 1.19 of Chapter 7. There remains the new case, where $e = F_{m_1}$. In this case, the natural map $$W = \nu _* \scrO _T(\nu ^*\Delta -D -E')\to \nu _* \scrO _{F_{m_1}}(\nu ^*\Delta -D -aF_{m_1})$$ is surjective, as one can see from applying the surjective map $\nu _*$ to the exact sequence $$0 \to \scrO _T (\nu ^*\Delta -D -E'-F_{m_1})\to \scrO _T(\nu ^*\Delta -D -E')\to \scrO _{F_{m_1}}(\nu ^*\Delta -D -aF_{m_1})\to 0.$$ It follows that $W|F_{2m_1} = \nu _* \scrO _{F_{m_1}}(\nu ^*\Delta -D -aF_{m_1})$. Thus $$\gather H^0(W\otimes \scrO _S(F_{2m_1})\otimes Q_0^{-1}) = H^0(\nu _*\scrO _{F_{m_1}}(\nu ^*\Delta -D -aF_{m_1}) \otimes \scrO _S(F_{2m_1})\otimes Q_0^{-1}) \\ = H^0(\nu _*\big[\scrO _{F_{m_1}}(\nu ^*\Delta -D -aF_{m_1}) \otimes \nu ^*\scrO _S(F_{2m_1})\otimes \nu ^*Q_0^{-1}\bigr]) = H^0(\scrO _{F_{m_1}}), \endgather$$ where we have used the fact that $\nu ^*Q_0 = \scrO _{F_{m_1}}(\nu ^*\Delta -D -(a-1)F_{m_1})$. Hence $$\dim\operatorname{Ext}^1(Q_0, W) = h^0(W\otimes \scrO _S(F_{2m_1})\otimes Q_0^{-1}) = 1$$ as desired. \qed \medskip We see that we have proved all of Theorem 3.1 except the statement about the smoothness of the moduli space, which follows from: \lemma{3.6} Suppose that $V= \nu _*\scrO _T(D+F)$ as above. Then $V$ is good, in the terminology of \rom{[2]}. In other words, $H^2(S; \operatorname{ad} V) = 0$. \endstatement \proof It suffices to show that $\dim \operatorname{Hom}(V, V\otimes K_S) = h^0(K_S)$. Now $$\align \operatorname{Hom}(V, V\otimes K_S) &= \operatorname{Hom}(V, \nu _*\Big(\scrO _T(\nu ^*\Delta -D)\otimes \nu ^*K_S\Bigr)) \\ &= \operatorname{Hom}(\nu ^*V, \scrO _T(\nu ^*\Delta -D)\otimes \nu ^*K_S). \endalign$$ Using the defining exact sequence for $\nu ^*$, there is an exact sequence $$0 \to H^0(\nu ^*K_S) \to \operatorname{Hom}(\nu ^*V, \scrO _T(\nu ^*\Delta -D)\otimes \nu ^*K_S) \to H^0(\scrO _T(\nu ^*\Delta -2D)\otimes \nu ^*K_S).$$ Since $K_S$ is a rational multiple of the fiber and $\nu ^*\Delta -2D$ is nontrivial on the generic fiber, the term $H^0(\scrO _T(\nu ^*\Delta -2D)\otimes \nu ^*K_S)$ is zero. Thus $\dim \operatorname{Hom}(V, V\otimes K_S) = h^0(\nu ^*K_S)$. Using the isomorphism $H^0(\nu ^*K_S) \cong H^0(K_S) \oplus H^0(K_S(-G))$, it suffices to show that $ H^0(K_S(-G)) = 0$. Now $K_S= \scrO_S((k-N-2)f + (2m_1-1)F_{2m_1} + (m_2-1)F_{m_2})$. Also $G = (2r+k \pm 1/2)f$. Thus it suffices to observe that the linear system $$|(-N -2r -2\pm 1/2)f + (2m_1-1)F_{2m_1} + (m_2-1)F_{m_2})|$$ is empty. \endproof Next let us describe the subset of the moduli space consisting of bundles $V$ which satisfy the hypotheses of Theorem 3.1. We begin by reversing the procedure outlined above. Fix the section $A'$, which is generic in the sense of Theorem 3.1: it meets $B$ transversally and no point of intersection corresponds to a singular nonmultiple fiber or to the multiple fiber of odd multiplicity. The section $A'$ determines the bisection $C = \overline C$, and thus a double cover $\nu \: T \to S$ together with an elliptic fibration $\rho \: T\to C$ and a divisor $D_0$, well-defined on the generic fiber. Moreover by construction $\nu _*D_0$ and $\Delta$ have the same restriction on the generic fiber, and thus differ by a multiple of $\kappa$. It is easy to see that changing $D_0$ by a sum of fiber components on $T$ replaces $\nu _*D_0$ by an arbitrary even multiple of $\kappa$. Thus we may assume that we have $\nu _* D_0 = \Delta - G$ or $\nu _* D_0 = \Delta - \kappa - G$. It is an exercise in the formulas of the preceding section to see that we have $$\Delta ^2 \equiv 2(A')^2 + 1 \mod 4.$$ Additionally $$2(A')^2 + 1 \equiv \cases 2N+1 \mod 4, &\text{if $A$ does not pass through the exceptional point;}\\ 2N-1 \mod 4 , &\text{otherwise.} \endcases$$ Thus the symmetry between the possibility that $A$ does or does not pass through the exceptional point, which is essentially the choice of blowing $\Bbb F$ down to $\Bbb F_N$ or $\Bbb F_{N\pm 1}$, reflects the choice of $\Delta$ or $\Delta - \kappa$, which in turn reflects $\Delta ^2\bmod 2$, or equivalently the dimension of the moduli space mod 2. Having made one choice for a line bundle $D_0$ on the double cover $T$, where $D_0$ is specified on the generic fiber of $T$ and satisfies $\nu _*D_0 = \Delta - G$, or $\nu _*D_0 = \Delta - \kappa -G$, the possibilities for $D$ are given by the next lemma. \lemma{3.7} Given the double cover $T\to S$, the set of all $D$ whose restriction to the generic fiber equals $D_0$ and which satisfy $\nu _*D = \Delta - G$, or $\nu _*D = \Delta - \kappa -G$ is a principal homogeneous space over $\operatorname{Pic}^\tau T$, which in turn is an extension of the Jacobian $J(C)$ by a cyclic group of order $m_2$. Moreover this principal homogeneous space is nonempty for exactly one of the two choices for $\nu _*D$ above. \endstatement \proof By the remarks preceding the lemma, there exists a $D_0$ with $\nu _*D_0 = \Delta - G$, or $\nu _*D_0 = \Delta - \kappa -G$, and only one of these possibilities can hold. If $D$ has the same restriction to the generic fiber as $D_0$ and $\nu _*D = \nu _*D_0$, then $D-D_0$ has trivial restriction to the generic fiber and $\nu _*(D-D_0) = 0$. The first condition says that $D-D_0$ is of the form $\rho ^*\lambda \otimes \scrO_T(aF_{m_1} + bF'_{m_2} + cF''_{m_2})$, where $F_{m_1}$ is the multiple fiber of multiplicity $m_1$ lying above $F_{2m_1}$ and $F'_{m_2}$, $F_{m_2}''$ are the two multiple fibers of multiplicity $m_2$ lying over $F_{m_2}$. We may further assume that $0\leq a <m_1$ and that $0\leq b < m_2$, $0\leq c <m_2$. Here $\lambda$ is a line bundle of degree $d$ on $C$. Thus $$\nu _* (D-D_0) = df + 2aF_{2m_1} + (b+c)F_{m_2}.$$ It is easy to see that this divisor is trivial if and only if $d=0$, $a=0$, and $b\equiv -c\mod m_2$. Thus, there is a natural identification of the set of all $D$ (given the fixed divisor $D_0$) with $J(C) \times \Zee/m_2\Zee$. \endproof We now assume that $A$ is not the negative section of $\Bbb F_N$ and write $A\in |\sigma +(N+r)f|$ where $r\geq 0$. The dimension of the linear system $A'$ is then equal to $n$, where $$n = \cases N+2r +1, &\text{if $A$ does not pass through the exceptional point;}\\ N+ 2r, &\text{otherwise.} \endcases$$ We also let $g = g(C)$ be the genus of the bisection $C$, as given in Lemma 3.2. Note that $$n+g = \cases 4r +N + k + 1 &\text{if $A$ does not pass through the exceptional point;}\\ 4r + N + k -1 &\text{otherwise.} \endcases$$ In both cases, comparing this with the formula for $-p$ given in Lemma 2.5 and using the fact that $1+ p_g(S) = k-N$, we see that $$-p-3\chi (\scrO _S) = g+n.$$ Note finally that the moduli space will be nonempty provided that $-p \geq 4p_g(S) +2N+3$. Since $p_g(S) = k-N -1 \geq N-1$, the moduli space will be nonempty as long as $$-p \geq 6p_g(S) + 5.$$ Arguing as in Theorem 1.14 of Chapter 7 of [4], we obtain the following: \theorem{3.8} Let $p$ be an odd negative integer, and choose $w\in H^2(S; \Zee/2\Zee)$ such that $w = \Delta \bmod 2$ or $w= \Delta - \kappa \bmod 2$, and that $w^2 \equiv p\mod 4$. Let $L$ be a $(w,p)$-suitable ample line bundle on $S$. Let $\frak M = \frak M(S,L; w,p)$ denote the moduli space of $L$-stable rank two bundles on $S$ with $w_2(V) = w$ and $p_1(\ad V) = p$. Then for all $p$ such that $-p \geq 6p_g(S)+5$, $\frak M$ contains a nonempty Zariski open subset $M$ corresponding to vector bundles $V$ satisfying the hypotheses of Theorem \rom{3.1}. The set $M$ is smooth of dimension $-p-3\chi (\scrO _S)$. Moreover, there is a holomorphic map from $M$ to a Zariski open subset $U \subseteq \Pee ^n$ and the fibers are isomorphic to $m_2$ copies of a complex torus of dimension $g$. \qed \endstatement Let us finally consider the case where $p_g(S) =0$ and $-p=3$, the case of a moduli space of expected dimension zero. In this case we fix $N=1$ and $k=2$, and the negative section does not pass through the exceptional point. The Chern class calculations of Lemma 2.5 show that we must have $\ell (Z) = \delta =0$ and $A$ must be the negative section $\sigma$ of $\Bbb F_1$. Note that $\sigma \cdot (B+\frak d_1+\frak d_2) = 2$, and the intersection must be transverse since $\sigma$ cannot split into a union of two sections in the double cover. Thus $C= \overline C \to \Pee ^1$ is branched at two points, so that $C = \Pee ^1$ again. Assuming as we may that the multiple fiber of odd multiplicity does not correspong to a branch point, we see that the arguments of Theorem 3.1 go through to show that there are exactly $m_2$ vector bundles $V$ whose associated section is $\sigma$. (Here, in case the intersection point of $\sigma$ with $B$ corresponds to a singular nonmultiple fiber, we must use the more detailed analysis of [2] (5.12) and (5.13) to see that the section $\sigma$ and the line bundle on $T$ determine $V$.) Each of these is a smooth point of the moduli space, by a slight modification of the proof of Lemma 3.6 (in this case $G= f/2$ and $K_S = \scrO_S(-f + (2m_1-1)F_{2m_1} + (m_2-1)F_{m_2})$). Summarizing, then: \theorem{3.9} In case $p_g(S) = 0$, the moduli space corresponding to $-p = 3$ consists of $m_2$ reduced points. \qed \endstatement \section{4. Calculation of the leading coefficient.} Fix $w$ and $p$ with $w^2 \equiv p \mod 4$, and let $\frak M = \frak M(S,L; w , p)$ denote the moduli space of $L$-stable rank two bundles on $S$ with $w_2(V) = w$ and $p_1(\ad V) = p$. Let $d$ be the (complex) dimension of $\frak M$: $$d = \cases 4r +N + k + 1 &\text{if $A$ does not pass through the exceptional point}\\ 4r + N + k -1 &\text{otherwise,} \endcases$$ where the section $A\in |\sigma +(N+r)f|$. With this notation we see that $2n = d-p_g(S)$ and that $g= d-n$. Finally let us recall from (3.4) of [3] that in case $p_g(S)=0$ there is a unique chamber of type $(w,p)$ which contains $\kappa$ in its closure, called the {\sl $(w, p)$-suitable\/} chamber. We can now state the main result of this paper: \theorem{4.1} For $p_g(S) >0$, let $\gamma _{w, p}(S, \beta)$ be the Donaldson polynomial corresponding to the $SO(3)$-bundle $P$ over $S$ with $w_2(P) = \Delta \bmod 2$ and $p_1(P) = p$, and $\beta$ is a choice of orientation agreeing with the usual complex orientation for $\frak M$. If $p_g(S)=0$, let $\gamma _{w, p}(S, \beta)$ be the corresponding Donaldson polynomial for metrics whose associated self-dual harmonic $2$-form lies in the $(w,p)$-suitable chamber. Then, writing $\gamma _{w, p}(S, \beta)$ as a polynomial in $\kappa _S$ and $q_S$, say $\gamma _{w, p}(S, \beta)= \sum _{i=0}^{[d/2]}a_iq_S^i\kappa _S^{d-2i}$, we have, for all $p$ with $-p\geq 2(4p_g+2)$, $a_i= 0$ for $i>n$ and $$a_n = \frac{d!}{2^nn!}(2m_1m_2)^{p_g(S)}m_2.$$ \endstatement \medskip We first remark that the assumption that $-p\geq 2(4p_g+2)$ implies that $\frak M$ is nonempty and contains a smooth Zariski open subset as described in Theorem 3.8. Indeed $-p$ is an odd integer greater than $8p_g+4$, and so $-p \geq 8p_g+5 \geq 6p_g+5$. Thus we are in the range of Theorem 3.8. Let $X= X_{w,p}$ denote the Uhlenbeck compactification associated to $\frak M$ [1], [4]. The orientation of $\frak M$ induces a fundamental class of $X$. There is a $\mu$-map $H_2(S) \to H^2(\frak M)$, which roughly speaking is given by taking slant product with the class $-p_1(P)/4$, where $P$ is the universal $SO(3)$-bundle over $S\times \frak M$. If $P$ lifts to a holomorphic bundle $\Cal V$ over $S\times \frak M$, then $p_1(P) = c_1^2(\Cal V) - 4c_2(\Cal V)=p_1(\ad \Cal V)$. The classes $\mu (\alpha) \in H^2(\frak M)$ extend uniquely to classes in $H^2(X)$, which we shall also denote by $\mu (\alpha)$. The Donaldson polynomial is then defined by taking cup products of the $\mu (\alpha)$ and evaluating on the fundamental cycle of $X$. Arguing as in [4], it is a combinatorial exercise to deduce Theorem 4.1 from the following: \theorem{4.2} Let $\mu$ denote the $\mu$-map associated to the Uhlenbeck compactification $X$ of $\frak M$. Then, using the complex orientation to identify $H^{2d}($X$; \Zee)\cong \Zee$, we have, for all $p$ with $-p\geq 2(4p_g+2)$ and for all $\Sigma \in H_2(S)$, $$\mu (f)^m\cup \mu (\Sigma)^{d-m} = \cases 0, &\text{if $m>n$}\\ (d-n)!(2m_1m_2)^{d-n}m_2(\Sigma \cdot \kappa _S)^{d-n}, &\text{if $m=n$.} \endcases$$ \endstatement Here the factor $(2m_1m_2)^{p_g(S)}$ appearing in Theorem 4.1 arises from the fact that $\mu (f)^n = (2m_1m_2)^n\mu (\kappa)$ and that $d-2n = p_g(S)$. In order to prove Theorem 4.2, we shall introduce geometric divisors which will represent the cohomology class $\mu(f)$. Fix a smooth fiber $f$. Then there are exactly four line bundles $\theta$ of degree $m_1m_2$ on $f$ such that $\theta ^{\otimes 2} = \scrO _f(\Delta)$. Each line bundle $\theta$ corresponds to a point of intersection of $f$ with the branch divisor $B\subset \Bbb F$. If $V$ is a rank two vector bundle over $S$ with $c_1(V) = \Delta$, then $(V|f)\otimes (\theta)^{-1}$ is a vector bundle over $f$ with trivial determinant. Thus by the Riemann-Roch theorem $\chi (f; (V|f)\otimes (\theta)^{-1}) =0$. For each integer $c$, fix a $(\Delta, c)$-suitable line bundle $L$. Given an integer $b\leq c$, we define $\frak M_b$ to be the moduli space of $L$-stable rank two bundles $V$ with $c_1(V) = \Delta$ and $c_2(V) = b$. Given $f$ and $\theta$ as above, define the divisor $\Cal Z_b(f, \theta)$ as a set by: $$\Cal Z_b(f, \theta) = \{\, V \in \frak M_b: h^0(f; (V|f)\otimes (\theta)^{-1}) \neq 0\,\}.$$ A calculation with the Grothendieck-Riemann-Roch theorem as in the proof of Proposition 1.1 in Chapter 5 of [4] show that we can use the divisors $\Cal Z_b(f, \theta)$ to calculate the $\mu$-map. More precisely, since the $f_i$ are disjoint, suppose that, for all choices of $b>0$ the intersection $$\Cal Z_{c-b}(f_{i_1}, \theta _{i_1}) \cap \dots \cap \Cal Z_{c-b}(f_{i_{n-b}}, \theta _{i_{n-b}})= \emptyset,$$ that $\bigcap _{i=1}^n\Cal Z_c(f_i, \theta_i)=J$ is compact and is contained in the Zariski open subset $M$ of $\frak M$, and that the $\Cal Z_c(f_i, \theta _i)$ meet transversally along $J$. Here by (2.6), $\frak M_{c-b} = \emptyset$ if $b >(-p-3)/4$ and so it is enough to check the above for $0< b \leq (-p-3)/4$. In particular we must have $n>b$ for all $b\leq (-p-3)/4$. Then we can define $\mu ([\Sigma])| J$ and arguments along the lines of the proof of Theorem 1.12 in Chapter 5 of [4] show that $$\gamma _{w,p}(S)(f_1, \dots, f_n, [\Sigma], \dots, [\Sigma]) = (\mu ([\Sigma])|J)^{d-n}.$$ Thus, Theorem 4.2 is a consequence of the following: \theorem{4.3} Suppose that $-p\geq 2(4p_g+2)$. Let $f_1, \dots, f_t$ denote distinct general fibers of $\pi$, and for each $i$ choose $\theta _i$ a line bundle on $f_i$ with $\theta _i^2 = \scrO _{f_i}(\Delta)$. Then: \roster \item"{(i)}" For all $t\geq n$ and for all choices of $b>0$, the intersection $$\Cal Z_{c-b}(f_{i_1}, \theta _{i_1}) \cap \dots \cap \Cal Z_{c-b}(f_{i_{t-b}}, \theta _{i_{t-b}})= \emptyset.$$ \item"{(ii)}" If $t> n$, then $\bigcap _{i=1}^t\Cal Z_c(f_i, \theta_i)= \emptyset$, and moreover $\bigcap _{i=1}^n\Cal Z_c(f_i, \theta_i)$ is compact and is contained in the Zariski open subset $M$ of $\frak M$. The intersection is transverse and is a fiber $J$ of the map $M \to U\subseteq \Pee ^n$. \item"{(iii)}" If $\Sigma$ is a smooth curve in $S$, then the restriction of $\mu ([\Sigma])$ to each of the $m_2$ connected components of $J$ is equal to $(\Sigma \cdot f)\cdot [\Theta]$, where $\Theta$ is the theta divisor of the component. \endroster \endstatement \medskip We begin by determining the set-theoretic intersection of the $\Cal Z_c(f_i, \theta _i)$. Recall that we have associated to $V$ the section $A$ and the scheme $Z$ on the double cover $T$. The following is straightforward: \lemma{4.4} Let $V\in \frak M$. Then $V\in \Cal Z_c(f, \theta)$ is and only if either the section $A'$ of $\Bbb F$ corresponding to $V$ meets $B$ transversally at the point corresponding to $\theta$ or $\operatorname{Supp}Z \cap \nu ^{-1}(f) \neq \emptyset$. \qed \endstatement \medskip For the rest of the argument, we shall not worry about the case where the sections pass through the exceptional point, since this can be handled by symmetry. Arguing as in [4], Lemma 2.7 of Chapter 7, for a general choice of fibers $f_1, \dots, f_t$ and line bundles $\theta _i$ on $f_i$, and for all $s\leq r$, setting $H_i$ to be the hyperplane of sections in $|\sigma +(N+s)f|$ passing through the point corresponding to $\theta _i$, then $H_1\cap \dots \cap H_{N+2s +1} = \{A\}$ and the intersection of more than $N+2s +1$ of the $H_i$ is empty. In particular, this means that the $f_i$ are chosen so that the negative section $\sigma$ does not neet the branch divisor at any of the $f_i$. Thus if $V$ is a bundle whose associated section $A$ is the negative section $\sigma$ of $\Bbb F_N$ and $V$ lies in the intersection $\bigcap _{i=1}^t\Cal Z_c(f_i, \theta_i)$, then $\operatorname{Supp}Z$ meets the preimage in $T$ of $f_i$ for all $i$. A counting argument as in the proof of Proposition 2.9 of Chapter 7 in [4] then establishes (i) and (ii) of Theorem 4.3, at least set-theoretically, except possibly for those $V$ such that the corresponding section of $\Bbb F_N$ is the negative section. Likewise, arguments identical to the proof of (2.6) in Chapter 7 of [4] show that the intersection of the divisor $\Cal Z_c(f_i, \theta_i)$ with the Zariski open subset $M$ is reduced, so that the intersection in (ii) of Theorem 4.3 is transverse. Thus the only new case to consider is the possibility that the section associated to $V$ is the negative section. We shall briefly outline the argument in this case; it is here that we must assume that $-p$ is sufficiently large. In particular, recalling that $$n = \frac{d -p_g}2 = \frac{-p -4p_g-3}2,$$ and that we may assume that $b\leq (-p-3)/4$, the condition $-p\geq 2(4p_g+2)$ insures that $$\align n-b&\geq n+\frac{p+3}4 = -\frac{p}4 -\frac{8p_g+3}4 \\ &\geq \frac{8p_g+4}2 - \frac{8p_g+3}4 >0. \endalign$$ Thus the intersection in (i) is not over the empty collection of divisors $\Cal Z_{c-b}(f_i, \theta_i)$, i.e\. the intersection is always contained in a divisor $\Cal Z_{c-b}(f_i, \theta_i)$. Given $t$ general fibers $f_1, \dots, f_t$, we can assume that they do not correspond to intersections of the negative section $\sigma$ with $B+\frak d_1+\frak d_2$. Now let $V$ be a bundle whose associated section is $\sigma$. Suppose further that $c_2(V) = c-b$ and that $V$ lies in the intersection $\Cal Z_{c-b}(f_{i_1}, \theta _{i_1}) \cap \dots \cap \Cal Z_{c-b}(f_{i_{t-b}}, \theta _{i_{t-b}})$ for $t\geq n$. We claim that, if $0\leq b<(-p-3)/4$, then $-p \leq 4p_g +2N -2$. It clearly suffices to assume that $t=n$. Let $\nu \:T \to S$ be the double cover corresponding to $\sigma$. Then $\nu$ is not branched over the $f_i$. Writing $\nu ^*V$ as an extension $$0 \to \scrO _T(D) \to \nu ^*V \to \scrO _T(\nu ^*\Delta - D)\otimes I_Z \to 0,$$ we have by (2.5) that $-p =-p_1(\ad V) +4b = 4k-6N +1+ 2\ell (Z) + \delta+4b$. Moreover $\operatorname{Supp}Z$ must meet each of the two components of $\nu ^{-1}(f_i)$ for $i=1, \dots, t$. Thus $\ell(Z) \geq 2n-2b$. But we also have $$\align 2n &= d-p_g =-p - 4p_g -3\\ &= 4k-6N +1 + 2\ell (Z) + \delta -4(k-N-1) -3+4b\\ &=-2N +2 + 2\ell (Z) +\delta+4b. \endalign$$ Thus $2n \geq -2N +2 + 4n-4b +\delta+4b$ and so $2n\leq 2N-2$. This says that $$-p\leq 4p_g+2N-2.$$ On the other hand, $2(4p_g+2) =4p_g +4(k-N)\geq 4p_g + 4N>4p_g+2N-2$. Thus with our assumptions on $p$ there can be no bundle $V$ in the intersection with an associated section equal to $\sigma$. Taking $b>0$ establishes (i) in the case where the section associated to $V$ is $\sigma$. Taking $b=0$ shows that no bundle with associated section $\sigma$ lies in the intersection $\bigcap _{i=1}^t\Cal Z_c(f_i, \theta _i)$ if $t\geq n$. This establishes (i) and (ii). Finally, to prove (iii) of Theorem 4.3, we must determine $\mu ([\Sigma])|J$, where $J$ is a fiber of the map $M\to U$. The argument again closely parallels that of [4] Chapter 7. A fiber of $M\to U$ determines and is determined by a generic double cover $T\to S$. There is a divisor $D_0$ on $T$ such that every $V$ in the fiber is of the form $\nu _*\scrO _T(D_0+F)\otimes \rho ^*\lambda$, for a line bundle $\lambda \in \Pic ^0C$. Fix a smooth holomorphic multisection $\Sigma$ of $\pi$, transverse to the double cover $T\to S$. Let $\Sigma'$ be the inverse image of $\Sigma$ under $\nu$. There is a commutative diagram $$\CD \Sigma ' @>{\nu}>> \Sigma\\ @V{\rho}VV @VV{\pi}V\\ C @>{g}>> \Pee ^1. \endCD$$ Let $\Cal P$ be the Poincar\'e bundle over $\operatorname{Pic}^0C\times C$. Let $E$ be the divisor on $\Sigma '$ induced by $D_0 + F$. Then there is a universal bundle over $\operatorname{Pic}^0C\times \Sigma$ of the form $$(\Id \times \nu)_*\Bigl[\pi _2^*\scrO _{\Sigma '}(E)\otimes (\Id \times \rho ^*)\Cal P\Bigr].$$ Here $\pi _2\: \operatorname{Pic}^0C\times \Sigma' \to \Sigma '$ is the second projection. The Chern classes of $\Cal V_E$ only depend on the numerical equivalence class of $E$. Moreover, $p_1(\ad \Cal V_E) = p_1(\ad \Cal V_E \otimes q_2^*\lambda)$ for every line bundle $\lambda$ on $\Sigma$, where $q_2\: \operatorname{Pic}^0C\times \Sigma \to \Sigma $ is the projection. This has the effect of replacing $\scrO _{\Sigma '}(E)$ by $\scrO _{\Sigma '}(E)\otimes \nu ^*\lambda$. Replacing $\Sigma$ by $2\Sigma$, replaces $\deg E$ by $2\deg E$. Thus we can assume that $\deg E$ is even, and then after twisting by an appropriate $\lambda$ we can assume that $\deg E =0$. So as far as the Chern classes are concerned we may as well assume that $E=0$, and we need to calculate $$p_1(\ad (\Id \times \nu)_*(\Id \times \rho) ^*\Cal P).$$ Now since cohomology commutes with flat base change, we have $$(\Id \times \nu)_*(\Id \times \rho) ^*\Cal P= (\Id \times \pi)^*(\Id \times g)_*\Cal P.$$ Thus we need to find $$p_1(\ad (\Id \times \pi)^*(\Id \times g)_*\Cal P) = (\Id \times \pi)^*p_1(\ad (\Id \times g)_*\Cal P).$$ Let us first calculate $p_1(\ad (\Id \times g)_*\Cal P)$. A straightforward calculation using e.g\. Lemma 2.14 of Chapter 7 of [4] shows the following \lemma{4.5} Let $f\:X \to Y$ be a double cover with $X$ and $Y$ smooth, and let $\Upsilon$ be the line bundle on $Y$ defining the double cover, so that $\Upsilon ^{\otimes 2} = \Cal {O}_Y(B)$, where $B$ is the branch locus of $f$. If $D$ is a divisor on $X$, then $$p_1(\ad f_*\scrO _X(D)) = c_1(\Upsilon )^2 - (f_*D)^2 + 2f_*(D^2).\qed$$ \endstatement \medskip Applying this in our situation, with $X = \Pic ^0C\times C$ and $f=\Id \times g$, we see that $\Upsilon$ is the pullback of a divisor on $C$ and thus $c_1(\Upsilon )^2 = 0$. So we are left with calculating (in the sense of cycles) $(\Id \times g)_*\Cal P$ and $(\Id \times g)_*[\Cal P]^2$. Also, as in the proofs of (2.15) and (2.16) in Chapter 7 of [4], $(\Id \times g)_*\Cal P =0$ and $[\Cal P ]^2 = -2r_1^*[\Theta] \cup r_2^*x$, where $x$ is the class of a point in $C$ and $\Theta$ is the theta divisor on $\Pic ^0C$, and $r_1, r_2$ are the first and second projections on $\Pic ^0C\times C$. Thus $$\align 2(\Id \times g)_*[\Cal P]^2 &= 2(\Id \times g)_*(-2r_1^*[\Theta] \cup r_2^*x)\\ &= -4p_1^*[\Theta] \cup p_2^*y, \endalign$$ where $p_1$, $p_2$ are the first and second projections on $\Pic ^0C \times \Pee^1$ and $y$ is the class of a point on $\Pee ^1$. It follows that $-p_1(\ad \Cal V_E)/4 = q_1^*\Theta \cup q_2^*\pi ^*y$. Hence the slant product of $-p_1(\ad \Cal V_E)/4$ with $[\Sigma]$ is $(\deg \pi)\cdot[\Theta]$. Since $\deg \pi = (\Sigma \cdot f)$, we have now established (iii) of Theorem 4.3. This concludes the proof of Theorem 4.3 and thus of Theorem 4.1. \qed \Refs \widestnumber\no{9} \ref \no 1\by S. K. Donaldson and P. B. Kronheimer \book The Geometry of Four-Manifolds \publ Clarendon \publaddr Oxford \yr 1990 \endref \ref \no 2\by R. Friedman \paper Rank two vector bundles over regular elliptic surfaces \jour Inventiones Math. \vol 96 \yr 1989 \pages 283--332 \endref \ref \no 3\bysame \paper Vector bundles and $SO(3)$-invariants for elliptic surfaces I \toappear \endref \ref \no 4\by R. Friedman and J. W. Morgan \book Smooth 4-manifolds and complex surfaces \toappear \endref \endRefs \enddocument

9307

alg-geom/9307001

en

https://arxiv.org/abs/alg-geom/9307001

alg-geom/9307001

Lisa Jeffrey

L.C. Jeffrey and F.C. Kirwan

Localization for nonabelian group actions

42 pages, LaTex version no. 2.09, Introduction and Section 8 have been rewritten in revised version

Suppose $X$ is a compact symplectic manifold acted on by a compact Lie group $K$ (which may be nonabelian) in a Hamiltonian fashion, with moment map $\mu: X \to {\rm Lie}(K)^*$ and Marsden-Weinstein reduction $\xred = \mu^{-1}(0)/K$. There is then a natural surjective map $\kappa_0$ from the equivariant cohomology $H^*_K(X) $ of $X$ to the cohomology $H^*(\xred)$. In this paper we prove a formula (Theorem 8.1, the residue formula) for the evaluation on the fundamental class of $\xred$ of any $\eta_0 \in H^*(\xred)$ whose degree is the dimension of $\xred$, provided that $0$ is a regular value of the moment map $\mu$ on $X$. This formula is given in terms of any class $\eta \in H^*_K(X)$ for which $\kappa_0(\eta ) = \eta_0$, and involves the restriction of $\eta$ to $K$-orbits $KF$ of components $F \subset X$ of the fixed point set of a chosen maximal torus $T \subset K$. Since $\kappa_0$ is

alg-geom

\section{Introduction} Suppose $X$ is a compact oriented manifold acted on by a compact connected Lie group $K$ of dimension $s$; one may then define the equivariant cohomology $\hk(X)$. Throughout this paper we shall consider only cohomology with complex coefficients. If $X$ is a symplectic manifold with symplectic form $\om$ and the action of $K$ is Hamiltonian (in other words, there is a moment map $\mu: X \to \lieks$), then we may form the symplectic quotient $\xred = \mu^{-1}(0)/K$. The restriction map $i_0: X \to \mu^{-1}(0)$ gives a ring homomorphism $i_0^*: \hk(X) \to \hk (\mu^{-1}(0) ) $. Using Morse theory and the gradient flow of the function $|\mu|^2: X \to {\Bbb R }$, it is proved in \cite{Ki1} that the map $i_0^*$ is surjective. Suppose in addition that $0$ is a {\em regular value} of the moment map $\mu$. This assumption is equivalent to the assumption that the stabilizer $K_x$ of $x$ under the action of $K$ on $X$ is {\em finite } for every $x \in \mu^{-1} (0)$, and it implies that $\xred$ is an orbifold, or $V$-manifold, which inherits a symplectic form $\omega_0$ from the symplectic form $\omega$ on $X$. In this situation there is a canonical isomorphism $\pi_0^*: H^*(\mu^{-1}(0)/K) \to \hk (\zloc)$.\footnote{This isomorphism is induced by the map $\pi_0: \zloc \times_K EK \to \zloc/K$. Recall that we are only considering cohomology with complex coefficients.} Hence we have a surjective ring homomorphism \begin{equation} \label{0.1} \kappa_0 = (\pi_0^*)^{-1} \circ i_0^*: \hk(X) \to H^*(\xred). \end{equation} Henceforth, if $\eta \in \hk(X)$ we shall denote $\kappa_0(\eta)$ by $\eta_0$. Previous work \cite{Brion,Ki2} on determining the ring structure of $\hk(X)$ has presented methods which in some situations permit the direct determination of the kernel of the map $\kappa_0$, and hence of generators and relations in $H^*(\xred)$ in terms of generators and relations in $\hk(X)$. (Note that the generators of $\hk(X)$ give generators of $H^*(\xred)$ via the surjective map $\kappa_0$, and also that generators of $H^*(BK)$ together with extensions to $\hk(X)$ of generators of $H^*(X)$ give generators of $\hk(X)$ because the spectral sequence of the fibration $X \times_K EK \to BK$ degenerates \cite{Ki1}.) Here we present an alternative approach to determining the ring structure of $H^*(\xred)$ when $0$ is a regular value of $\mu$, which complements the results obtained by directly studying the kernel of $\kappa_0$. Our approach is based on the observation that since $H^*(\xred) $ satisfies Poincar\'e duality, a class $\eta \in \hk(X)$ is in the kernel of $\kappa_0$ if and only if for all $\zeta \in \hk(X)$ we have \begin{equation} \eta_0 \zeta_0 [\xred] = (\eta \zeta)_0 [\xred] = 0 . \end{equation} Hence to determine the kernel of $\kappa_0$ (in other words the relations in the ring $H^*(\xred)$) given the ring structure of $\hk(X)$, it suffices to know the {\em intersection pairings}, in other words the evaluations on the fundamental class $[\xred]$ of all possible classes $ \xi_0 = \kappa_0 (\xi)$. In principle the intersection pairings thus determine generators and relations for the cohomology ring $H^*(\xred)$, given generators and relations for $\hk(X)$. There is a natural pushforward map $\pisk: \hk(X) \to \hk $ $ = \hk({\rm pt} )$ $ \cong S(\lieks)^K$, where we have identified $\hk$ with the space of $K$-invariant polynomials on the Lie algebra $\liek$. This map can be thought of as integration over $X$ and will sometimes be denoted by $\int_X$. If $T$ is a compact {\em abelian} group (i.e. a torus) and $\zeta \in \hht(X)$, there is a formula\footnote{Atiyah and Bott \cite{abmm} give a cohomological proof of this formula, which was first proved by Berline and Vergne \cite{BV1}.} (the {\em abelian localization theorem}) for $\pist \zeta$ in terms of the restriction of $\zeta$ to the components of the fixed point set for the action of $T$. In particular, for a general compact Lie group $K$ with maximal torus $T$ there is a canonical map $\tau_X: \hk(X) \to \hht(X)$, and we may apply the abelian localization theorem to $\tau_X (\zeta)$ where $\zeta \in \hk(X)$. In terms of the components $F$ of the fixed point set of $T$ on $X$, we obtain a formula (the residue formula, Theorem \ref{t8.1}) for the evaluation of a class $\eta_0 \in H^*(\xred) $ on the fundamental class $[\xred]$, when $\eta_0$ comes from a class $\eta \in \hk(X)$. There are two main ingredients in the proof of Theorem \ref{t8.1}. One is the abelian localization theorem \cite{abmm,BV1}, while the other is an equivariant normal form for $\om$ in a neighbourhood of $\zloc$, given in \cite{STP} as a consequence of the coisotropic embedding theorem. The result is the following: \noindent{\bf Theorem 8.1} {\em Let $\eta \in \hk(X)$ induce $\eta_0 \in H^*(\xred)$. Then we have $$ \eta_0 e^{i\omega_0} [\xred] = \frac{(-1)^{n_+} }{(2 \pi)^{s-l} |W| \,{\rm vol}\, (T) } \treso \Biggl ( \: \nusym^2 (\psi) \sum_{F \in {\mbox{$\cal F$}}} e^{i \mu_T(F) (\psi) } \int_F \frac{i_F^* (\eta(\psi) e^{i \omega} ) }{e_F(\mar \psi) } [d \psi] \Biggr ). $$ In this formula, $n_+$ is the number of positive roots of $K$, and $\nusym(\psi) = \prod_{\gamma > 0} \gamma(\psi)$ is the product of the positive roots, while ${\mbox{$\cal F$}}$ is the set of components of the fixed point set of the maximal torus $T$ on $X$. If $F \in {\mbox{$\cal F$}}$ then $i_F$ is the inclusion of $F$ in $ X$ and $e_F$ is the equivariant Euler class of the normal bundle to $F$ in $X$.} \noindent Here, via the {\em Cartan model}, the class $\tau_X ( \eta )\in \hht(X)$ has been identified with a family of differential forms $\eta (\psi)$ on $X$ parametrized by $\psi \in \liet$. The definition of the residue map $\treso$ (whose domain is a suitable class of meromorphic differential forms on $\liet \otimes {\Bbb C }$) will be given in Section 8 (Definition \ref{d8.5n}). It is a linear map, but in order to apply it to the individual terms in the statement of Theorem 8.1 some choices must be made. The choices do not affect the residue of the whole sum. When $\liet$ has dimension one the formula becomes $$ \eta_0 e^{i\omega_0} [\xred] = -\frac{1}{2} {\rm Res}_0 \Biggl ( \psi^2 \: \sum_{F \in {\mbox{$\cal F$}}_+} e^{i \mu_T(F) (\psi) } \int_F \frac{i_F^* (\eta(\psi) e^{i \omega} ) }{e_F(\mar \psi) } \Biggr ), $$ where ${\rm Res}_0$ denotes the coefficient of $1/\psi$, and ${\mbox{$\cal F$}}_+$ is the subset of the fixed point set of $T = U(1)$ consisting of those components $F$ of the $T$ fixed point set for which $\mu_T(F) > 0 $. We note that if $\dim \eta_0 = \dim \xred$ then the left hand side of the equation in Theorem \ref{t8.1} is just $\eta_0 [\xred]$. More generally one may obtain a formula for $\eta_0[\xred]$ by replacing the symplectic form $\omega$ by $\delta \omega$ (where $\delta > 0 $ is a small parameter), and taking the limit as $\delta \to 0$. This has the effect of replacing the moment map $\mu$ by $\delta \mu$. In the limit $\delta \to 0$, the Residue Formula (Theorem \ref{t8.1}) becomes a sum of terms corresponding to the components $F$ of the fixed point set, where the term corresponding to $F$ is (up to a constant) the residue (in the sense of Section 8) of $\nusym^2(\psi) \int_F {i_F^* (\eta(\psi) ) }/{e_F(\mar \psi) } $, and the only role played by the symplectic form and the moment map is in determining which $F$ give a nonzero contribution to the residue of the sum and the signs with which individual terms enter. Results for the case when $K = S^1$, which are related to our Theorem \ref{t8.1}, may be found in the papers of Kalkman \cite{Kalk} and Wu \cite{wu}. Witten in Section 2 of \cite{tdg} gives a related result, the {\em nonabelian localization theorem}, which also interprets evaluations $\eta_0[\xred]$ of classes on the fundamental class $[\xred]$ in terms of appropriate data on $X$. For $\epsilon > 0$ and $\zeta \in \hk(X)$, he defines\footnote{The normalization of the measure in $\ie(\zeta)$ will be described at the beginning of Section 3. As above, $\pis: \hk(X) \to \hk \cong S(\lieks)^K$ is the natural pushforward map, where $S(\lieks)^K$ is the space of $K$-invariant polynomials on $\liek$. } \begin{equation} \label{1.1} \ie(\zeta) = \frac{1}{(2 \pi i)^s \,{\rm vol}\, K} \intk [d \phi] e^{- \epsilon \inpr{\phi, \phi}/2 } \pisk \zeta (\phi) \end{equation} (where $\inpr{\cdot,\cdot} $ is a fixed invariant inner product on $\liek$, which we shall use throughout to identify $\lieks$ with $\liek$) and expresses it as a sum of local contributions. Witten's theorem tells us that just as $\pisk \zeta$ would have contributions from the components of the fixed point set of $K$ if $K $ were abelian, the quantity $\ie(\zeta)$ (if $K$ is not necessarily abelian) reduces to a sum of integrals localized around the critical set of the function $\rho = |\mu|^2,$ i.e. the set of points $x$ where $(d |\mu|^2)_x = 0$. (Of course $d |\mu|^2 = 2\inpr{\mu, d \mu}$, so the fixed point set of the $K$ action, where $d \mu = 0$, is a subset of the critical set of $d|\mu|^2$.) More precisely the critical set of $\rho = |\mu|^2$ can be expressed as a disjoint union of closed subsets $C_\beta$ of $X$ indexed by a finite subset ${\mbox{$\cal B$}}$ of the Lie algebra $\liet$ of the maximal torus $T$ of $K$ which is explicitly known in terms of the moment map $\mu_T$ for the action of $T$ on $X$ \cite{Ki1}. If $\beta \in {\mbox{$\cal B$}}$ then the critical subset $C_\beta$ is of the form $C_\beta = K (Z_\beta \cap \mu^{-1}(\beta))$ where $Z_\beta$ is a union of connected components of the fixed point set of the subtorus of $T$ generated by $\beta$. The subset $\mu^{-1}(0)$ on which $\rho = |\mu|^2$ takes its minimum value is $C_0$. There is a natural map\footnote{This map is induced by the projection ${\rm pr}:\zloc \to {\rm pt}.$} ${\rm pr}^*: \hk \to \hk(\zloc)$ so that the distinguished class $f(\phi) = -\inpr{\phi, \phi}/2$ in $H^4_K$ gives rise to a distinguished class $\Theta \in H^4(\zloc/K) \cong H^4_K(\zloc).$ Witten's result can then be expressed in the form \begin{theorem} \label{t1.1} $$ \ie(\zeta) = \zeta_0e^{\epsilon \Theta} [\xred] + \sum_{\beta \in {\mbox{$\cal B$}} - \{0\} } \int_{U_\beta} {\zeta'}_\beta. $$ Here, the $U_\beta$ are open neighbourhoods in $X$ of the nonminimal critical subsets $C_\beta$ of the function $\rho$. The $\zeb$ are certain differential forms on $U_\beta$ obtained from $\zeta$. \end{theorem} In the special case $\zeta = \eta \exp \iins \bar{\om} $ (where $\bar{\om}(\phi) = \om + \evab{\phi}{ \mu} $ is the standard extension of the symplectic form $\om$ to an element of $H^2_K(X)$, and $\eta$ has polynomial dependence\footnote{The equivariant cohomology $\hk(X) $ is defined to consist of classes which have polynomial dependence on the generators of $\hk$, but we shall also make use of formal classes such as $\exp \iins \bom$ which are formal power series in these generators.} on the generators of $\hk$), Witten's results give us the following estimate on the growth of the terms $\int_{U_\beta} \zeb$ as $\epsilon \to 0$: \begin{theorem} \label{t1.2} Suppose $\zeta = \eta \exp i \bar{\om}$ for some $\eta \in \hk(X)$. If $\beta \in {\mbox{$\cal B$}} - \{0\}$ then $\int_{U_\beta} \zeb = e^{- \rho_\beta/{2 \epsilon} } \: h_\beta(\epsilon)$, where $\rho_\beta = |\beta|^2$ is the value of $|\mu|^2$ on the critical set $C_\beta$ and $|h_\beta(\epsilon)|$ is bounded by a polynomial in $\epsilon^{-1}$. \end{theorem} Thus one should think of $\epsilon > 0$ as a small parameter, and one may use the asymptotics of the integral $\ie$ over $X$ to calculate the intersection pairings $\eta_0 e^{\epsilon \Theta} e^{i\om_0} [\xred],$ since the terms in Theorem \ref{t1.2} corresponding to the other critical subsets of $\rho$ vanish exponentially fast as $\epsilon \to 0$. Notice that when $\zeta = \exp i\bom$, the vanishing of $\mu$ on $\zloc$ means that $\zeta_0 = \exp i\om_0$, where $\omega_0$ is the symplectic form induced by $\omega$ on $\xred = \mu^{-1}(0)/K$. In this paper we shall give a proof of a variant of Theorems \ref{t1.1} and \ref{t1.2}, for the case $\zeta = \eta \exp i \bom $ where $\eta \in \hk(X)$. Before outlining our proof, it will be useful to briefly recall Witten's argument. Witten introduces a $K$-invariant 1-form $\lambda$ on $X$, and shows that $\ie(\zeta) = \ie(\zeta \exp s D \lambda)$, where $D $ is the differential in equivariant cohomology and $s \in {\Bbb R }^+$. He then does the integral over $\phi \in \liek$ and shows that in the limit as $s \to \infty$, this integral vanishes over any region of $X$ where $\lambda(\va) \ne 0$ for at least one of the vector fields $\va, j = 1, \dots, s$ given by the infinitesimal action of a basis of $\liek$ on $X$ indexed by $j$. Thus, after integrating over $\phi \in \liek$, the limit as $s \to \infty$ of $\ie(\zeta)$ reduces to a sum of contributions from sets where $\lambda(\va) = 0$ for all the $\va$. In our case, when $X$ is a symplectic manifold and the action of $K$ is Hamiltonian, Witten chooses $\lambda(Y) = d|\mu|^2(JY)$, where $J$ is a $K$-invariant almost complex structure on $X$. Thus $\lambda(\va)(x) = 0$ for all $j$ if and only if $(d|\mu|^2)_x = 0$, so $\ie(\zeta)$ reduces to a sum of contributions from the critical sets of $\rho = |\mu|^2$. Further, he obtains the contribution from $ \zloc$ as $e^{\epsilon \Theta} \zeta_0 [\xred] $. If $\zeta = \eta e^{\iins \bom} $ he also obtains the estimates in Theorem \ref{t1.2} on the contributions from the neighbourhoods $U_\beta$. In general, the contributions to the localization theorem depend on the choice of $\lambda$. In the symplectic case, with $\lambda = J d |\mu|^2$, the contribution from $\mu^{-1}(0)$ is canonical but the contributions from the other critical sets $C_\beta$ depend in principle on the choice of $J$. Further, the properties of these other terms are difficult to study. Ideally they should reduce to integrals over the critical sets $C_\beta$, and indeed when proving Theorem \ref{t1.2} Witten makes the assumption (before (2.52)) that the $C_\beta$ are nondegenerate critical manifolds in the sense of Bott \cite{bnd}. In general the $C_\beta$ are not manifolds; and even when they are manifolds, they are not necessarily nondegenerate. They satisfy only a weaker condition called {\em minimal degeneracy} \cite{Ki1}.\footnote{However, minimal degeneracy may be sufficient for Witten's argument.} We shall treat the integrals over neighbourhoods of the $C_\beta$ in a future paper. In the case when $X$ is a symplectic manifold and $\zeta = \eta \exp \bom$ for any $\eta \in \hk(X)$, we have been able to use our methods to prove a variant of Theorems \ref{t1.1} and \ref{t1.2} (see Theorems \ref{t4.1}, \ref{t4.3} and \ref{t7.1} below) which bypasses these analytical difficulties and reduces the result to fairly well known results on Hamiltonian group actions on symplectic manifolds. We assume that $0$ is a regular value of $\mu$, or equivalently that $K$ acts on $\mu^{-1}(0)$ with finite stabilizers.\footnote{Witten assumes that $K$ acts freely on $\mu^{-1}(0)$.} By treating the pushforward $\pisk \zeta$ as a function on $\liek$, we may use the abelian localization formula \cite{abmm,BV1} for the pushforward in equivariant cohomology of torus actions to find an explicit expression for $\pisk \zeta$ as a function on $\liek$. Thus, analytical problems relating to integrals over neighbourhoods of $C_\beta$ are circumvented, and localization reduces to studying the image of the moment map and the pushforward of the symplectic or Liouville measure under the moment map.\footnote{If $K$ is abelian, this pushforward measure is equal (at $\phi \in \liek$) to Lebesgue measure multiplied by a function which gives the symplectic volume of the reduced space $\mu^{-1}(\phi)/K$; this function is sometimes called the Duistermaat-Heckman polynomial \cite{DH}.} Seen in this light, the nonabelian localization theorem is a consequence of the same results that underlie the residue formula: the abelian localization formula for torus actions \cite{abmm,BV1} and the normal form for $\om$ in a neighbourhood of $\zloc$. We now summarize the key steps in our proof. Having replaced integrals over $X$ by integrals over $\liek$, we observe that in turn these may be replaced by integrals over the Lie algebra $\liet$ of the maximal torus. Then, applying properties of the Fourier transform, we rewrite $\ie$ as the integral over $\liets$ of a Gaussian $\gtsoe(y) $ $\sim e^{- |y|^2/(2 \epsilon) } $ multiplied by a function $Q = D_\nusym R$ where $R$ is piecewise polynomial and $D_\nusym$ is a differential operator on $\liets$: \begin{equation} \label{0.2}\ie =i^{-s} \tintt \widetilde{\gsoe} (y) Q(y), \end{equation} where $s$ is the dimension of $K$. The function $Q$ is obtained by combining the abelian localization theorem (Theorem \ref{t2.1}) with a result (Proposition \ref{p3.5}) on Fourier transforms of a certain class of functions which arise in the formula for the pushforward. The function $Q$ is smooth in a neighbourhood of the origin when $0$ is a regular value of $\mu$: thus there is a polynomial $Q_0 = D_\nusym R_0 $ which is equal to $Q$ near $0$. It turns out that the cohomological expression $\eeth[\xred]$ is obtained as the integral over $\liets$ of a Gaussian multiplied not by $Q$ but by the polynomial $Q_0$: \begin{equation} \eeth [\xred] = i^{-s} \tintt \widetilde{\gsoe} (y) Q_0(y). \end{equation} This result follows from a normal form for $\om$ near $\zloc$.\footnote{This normal form is a key tool in the original proof \cite{DH} of the Duistermaat-Heckman theorem; this theorem motivated the proof by Atiyah and Bott \cite{abmm} of the abelian localization theorem.} To obtain our analogue of Witten's estimate (Theorem \ref{t1.2}) for the asymptotics of $\ie - \eeth [\xred]$ as $\epsilon \to 0$, we then write \begin{equation} \ie - \eeth [\xred] = i^{-s} \tintt \widetilde{\gsoe}(y) D_\nusym (R - R_0) (y). \end{equation} Here, $R - R_0$ is piecewise polynomial and supported {\em away} from $0$. By studying the minimum distances from $0$ in the support of $R - R_0$ we obtain an estimate (Theorem \ref{t4.1}) similar to Witten's estimate (Theorem \ref{t1.2}). In our estimate, the terms in the sum are indexed by the set ${\mbox{$\cal B$}} - \{0\}$; however, our estimate is weaker than Witten's estimate since some of the subsets $C_\beta$ indexed by $\beta \in {\mbox{$\cal B$}} - \{0\}$ (which a priori contribute to our sum\footnote{In a future paper we hope to prove that the nonzero contributions to our estimate (Theorem \ref{t4.1}) come only from those $|\beta|^2$ which are nonzero critical values of $|\mu|^2$.} ) may be empty in which case $\rho_\beta = |\beta|^2$ may not be a critical value of $|\mu|^2$. To summarize, the following related quantities appear in this paper: \begin{enumerate} \item The cohomological quantity $\eta_0 e^{i \om_0} [\xred]$. \item The integral ${\mbox{$\cal I$}}^\epsilon $ (\ref{1.1}) coming from the pushforward of an equivariant cohomology class $\eta \in \hk(X)$ to $\hk$. \item Sums of terms of the form $$\int_F \frac{i_F^* \eta}{e_F}$$ where $F$ is a connected component of the fixed point set of the maximal torus $T$ acting on $X$. Such sums appear after mapping $\eta \in \hk(X)$ into $\hht(X) $ and then applying the abelian localization theorem. \end{enumerate} Witten's work relates (1) and (2), while our Theorem \ref{t8.1} relates (1) and (3). This paper is organized as follows. Section 2 contains background material on equivariant cohomology and the abelian localization formula. In Section 3 we collect a number of preliminary results which we use in Section 4 to reduce our integral $\ie$ to an integral over $\liets$ of a piecewise polynomial function multiplied by a Gaussian. Section 4 also contains the statement of two of our main results, Theorems \ref{t4.1} and \ref{t4.3}; Theorem \ref{t4.3} is proved in Section 5, and Theorem \ref{t4.1} in Section 6. In Section 7, Theorems \ref{t4.1} and \ref{t4.3} (which are for the case $\zeta = \exp \bom$) are extended to the case $\zeta = \eta \exp \bom $ for $\eta \in \hk(X)$: the result is Theorem \ref{t7.1}. Finally, in Section 8 we prove the residue formula (Theorem \ref{t8.1}) for the evaluation of cohomology classes from $\hk(X)$ on the fundamental class of $\xred$, and in Section 9 we apply it when $K = SU(2)$ to specific examples. This formula may be related to an unpublished formula due to Donaldson. In future papers we shall treat the case when $\xred$ is singular using intersection homology; we shall also apply the nonabelian localization formula to moduli spaces of bundles over Riemann surfaces regarded as finite dimensional symplectic quotients, in singular as well as nonsingular cases. \noindent{\em Acknowledgement:} We are most grateful to H. Duistermaat and M. Vergne for helpful suggestions and careful readings of the paper. In addition, one of us (L.C.J.) wishes to thank E. Lerman and E. Prato for explaining their work, and also to thank E. Witten for discussions about the nonabelian localization formula while the paper \cite{tdg} was being written. { \setcounter{equation}{0} } \section{Equivariant cohomology and pushforwards} In this section we recall the localization formula for torus actions (Theorem \ref{t2.1}) and express it in a form convenient for our later use (Lemma \ref{l2.2}). Let $X$ be a compact manifold equipped with the action of a compact Lie group $K$ of dimension $s$ with maximal torus $T$ of dimension $l$. We denote the Lie algebras of $K$ and $T$ by $\liek$ and $\liet$ respectively, and the Weyl group by $W$. We assume an invariant inner product $\inpr{\cdot, \cdot} $ on $\liek$ has been chosen (for example, the Killing form): we shall use this to identify $\liek$ with its dual. The orthocomplement of $\liet$ in $\liek$ will be denoted $\lietp$. Throughout this paper all cohomology groups are assumed to have coefficients in the field ${\Bbb C }$. The $K$-equivariant cohomology of a point is $\hk = H^*(BK)$, and similarly the $T$-equivariant cohomology is $\hht = H^*(BT)$. We identify $\hk $ with $S(\liek^*)^K$, the $K$-invariant polynomial functions on $\liek$, and $\hht$ with $S(\liets)$. Hence we have a bijective map (obtained from the restriction from $\lieks$ to $\liets$) which identifies $\hk$ with the subset of $\hht$ fixed by the action of the Weyl group $W$: \begin{equation} \label{2.1} \hk \cong S(\lieks)^K \cong S(\liets)^W \subseteq S(\liets) \cong \hht \end{equation} This natural map $\hk \to \hht$ will be denoted $\tau$ or $\tau_X$. We shall use the symbol $\phi$ to denote a point in $\liek$, and $\psi$ to denote a point in $\liet$. For $f \in \hk$ we shall write $f = f (\phi)$ as a function of $\phi$. The $K$-equivariant cohomology of $X$ is the cohomology of a certain chain complex (see, for instance, Chapter 7 of \cite{BGV} or Section 5 of \cite{MQ}; the construction is due to Cartan \cite{cartan}) which can be expressed as \begin{equation} \label{2.0} \Om^*_K(X) = \Bigl ( S(\lieks) \otimes \Om^*(X) \Bigr )^K \end{equation} (where $\Om^*(X)$ denotes differential forms on $X$). An element in $\Om^*_K(X)$ may be thought of as a $K$-equivariant polynomial function from $\liek$ to $\Om^*(X)$. For $ \alpha \in \Om^*(X)$ and $f \in S(\lieks)$, we write $(\alpha \otimes f) (\phi) = f(\phi) \alpha$. In this notation, the differential $D$ on the complex $\Om^*_K(X)$ is then defined by\footnote{This definition and the definition (\ref{2.0''}) of the extension $\bom(\phi) = \omega + \mu(\phi)$ of the symplectic form $\omega$ to an equivariant cohomology class are different from the conventions used by Witten \cite{tdg}: a factor $\phi$ appears in the our definitions where $i \phi$ appears in Witten's definition. In other words Witten's definition is $D(\alpha \otimes f) (\phi) = f(\phi) (d \alpha - i \iota_{ \tilde{\phi} } \alpha)$ and $\bom(\phi) = \omega + i \mu(\phi)$. Witten makes this substitution so that the oscillatory integral $ \int_X \exp (\om + i \evab{\phi}{ \mu} ) $ will appear as the integral of an equivariant cohomology class.} \begin{equation} \label{2.0'} D(\alpha \otimes f) (\phi) = f(\phi) (d \alpha - \iota_{ \tilde{\phi} } \alpha) = \, f (\phi) d \alpha - \sum_{j= 1}^s \phi_j f(\phi) \: \iota_{\va} \alpha . \end{equation} Here, $\tilde{\phi}$ is the vector field on $X$ given by the action of $\phi \in \liek$, and $\iota_{\tilde{\phi} }$ is the interior product with the vector field $\tilde{\phi}$. We have introduced an orthonormal basis $\{ \hat{e}^j, \: j = 1, \dots, s \} $ for $\liek$, and the $\phi_j$ $ \in \lieks$ are simply the coordinate functions $\phi_j = \inpr{\hat{e}^j, \phi}, $ while the $\va$ are the vector fields on $X$ generated by the action of $\hat{e}^j$. The $\phi_j $ are assigned degree $2$, so that the differential $D$ increases degrees by $1$. One may define the pushforward $\pis^K: \hk(X) \to \hk$, which corresponds to integration over the fibre of the map $X \times_K EK \to BK $ (see Section 2 of \cite{abmm}). The pushforward satisfies $\pis^T = \tau \circ \pisk$. Because of this identification, we shall usually simply write $\pis$ for $\pis^K$ or $\pis^T$. A localization formula for $\pist$ was given by Berline and Vergne in \cite{BV1}; a more topological proof of this formula is given in Section 3 of \cite{abmm}. \begin{theorem} \label{t2.1} {\bf \cite{BV1} } If $\si \in \hht(X)$ and $\psi \in \liet$ then $$(\pist \si) (\psi) = \sum_{F \in {\mbox{$\cal F$}}} \int_F \frac{i_F^* \si(\psi) }{e_F( \psi) }. $$ Here we sum over the set ${\mbox{$\cal F$}}$ of components $F$ of the fixed point set of $T$, and $e_F $ is the $T$-equivariant Euler class of the normal bundle of $F$; this Euler class is an element of $H^*_T(F) \cong H^*(F) \otimes \hht$, as is $i_F^* \si$. The map $i_F: F \to X$ is the inclusion map. The right hand side of the above expression is to be interpreted as a rational function of $\psi$. \end{theorem} We shall now prove a lemma about the image of the pushforward, which will be applied in Section 4. \begin{lemma}\label{l2.2} If $\si \in \hht(X) $ then $(\pist \si)(\psi)$ is a sum of terms \begin{equation} \label{2.m0} (\pist \si)(\psi) = \sum_{F \in {\mbox{$\cal F$}}, \;\alpha \in {\mbox{$\cal A$}}_F}\tau_{F,\alpha} \end{equation} such that each term $\tau_{F,\alpha}$ is of the form \begin{equation} \label{2.m1} \tau_{F,\alpha} = \frac{\int_F c_{F, \alpha}(\psi) }{ \efo(\mar \psi) \prod_j \beta_{F,j}(\mar \psi)^{n_{F,j}(\alpha)} } \end{equation} for some component $F$ of the fixed point set of the $T$ action. Here, the $\beta_{F,j}$ are the weights of the $T$ action on the normal bundle $\nu_F$, and $\efo (\mar \psi) = \prod_j \beta_{F,j}(\mar \psi)$ is the product of all the weights, while $n_{F,j}(\alpha)$ are some nonnegative integers. The class $c_{F,\alpha}$ is in $H^*(F) \otimes H^*_T$, and is equal to $i_F^* \si \in H^* (F) \otimes H^*_T$ times some characteristic classes of subbundles of $\nu_F$. \end{lemma} \Proof The normal bundle $\nu_F$ to $F$ decomposes as a direct sum of weight spaces $\nu_F = \oplus_{j = 1}^r \nu_F^{(j)}$, on each of which $T$ acts with weight $\bfj$. All these weights must be nonzero. By passing to a split manifold if necessary (see section 21 of \cite{BT}), we may assume without loss of generality that the subbundle on which $T$ acts with a given weight decomposes into a direct sum of $T$-invariant real subbundles of rank $2$. In other words, we may assume that the $ \nu_F^{(j)}$ are rank $2$ real bundles, and the $T$ action enables one to identify them in a standard way with complex line bundles. Then the equivariant Euler class $e_F(\psi)$ is given for $\psi \in \liet$ by \begin{equation} \label{2.3} e_F(\psi) = \prod_{j = 1}^r \Bigl (c_1(\nu_F^{(j)}) + \beta_{F,j}(\psi) \Bigr ). \end{equation} Thus we have $$ \frac{1}{e_F (\mar \psi)} = \frac{1}{\efo(\mar \psi) } \, \prod_j (1 + \frac{c_1 (\nu_F^{(j)} )}{\beta_{F,j}(\mar \psi) } )^{-1} $$ \begin{equation} \label{2.4} = \frac{1}{\efo(\mar \psi) } \: \prod_j \sum_{r_j \ge 0 } \: \, (-1)^{r_j} \Bigl ( \frac{c_1 (\nu_F^{(j)} )} {\beta_{F,j}(\mar \psi) } \Bigr )^{r_j} . \end{equation} Here, $c_1(\nu_F^{(j)} ) \in H^2(F)$, so that $c_1(\nu_F^{(j)} )/\beta_{F,j}(\mar \psi)$ is {\em nilpotent} and the inverse makes sense in $H^*(F) \otimes {\Bbb C }(\psi_1, \dots, \psi_l)$, where ${\Bbb C } (\psi_1, \dots, \psi_l)$ denotes the complex valued rational functions on $\liet$. $\square$ Let us now assume that $X$ is a symplectic manifold and the action of $K$ is Hamiltonian with moment map $\mu: X \to \lieks$. Denote by $\mu_T $ the moment map for the action of $T$ given by the composition of $\mu$ with the restriction map $\lieks \to \liets$. We shall be interested in one particular (formal) equivariant cohomology class $\si $, defined by \begin{equation} \label{2.0''}\si(\phi) = \exp \iins \bom(\phi), \phantom{bbbbb} \bom(\phi) = \omega + \mu( \phi ) . \end{equation} For this class the localization formula gives \begin{equation} \label{2.2} (\pist \si) (\psi) = \sum_F \rf(\psi), \phantom{bbbbb} \rf (\psi) = \int_F \frac{ e^{i \mu_T(F)(\psi ) } e^\om } {e_F(\mar \psi) }. \end{equation} (This formula does not require the fixed point set of $T$ to consist of isolated fixed points.) \noindent{\em Remark:} For any $\eta \in \hk(X) $ the function $\pisk \eta \in \hk$ is a polynomial on $\liek$, and in particular is smooth. However, $\sigma = e^{\iins \bom} $ does not have polynomial dependence on $\phi$. Although it is not immediately obvious from the formula (\ref{2.2}), the function $\pisk (\eta e^{\iins \bom}) $ is still a { smooth} function on $\liek$ (for any $\eta \in \hk(X)$ represented by an element $\tilde{\eta} \in \Om^*_K (X)$): this follows from its description as $$ \pisk (\eta e^{\iins \bom}) (\phi) = \int_{x \in X} e^{i \om} \tilde{\eta} (\phi) e^{i \mu(x) (\phi) }. $$ { \setcounter{equation}{0} } \section{Preliminaries} This section contains results which will be applied in the next section to reduce the integral $\ie$ to an integral over $\liets$ of a Gaussian multiplied by a piecewise polynomial function. The first, Lemma \ref{l3.1}, reduces integrals over $\liek$ to integrals over $\liet$. Lemma \ref{l3.2} enables us to replace the $L^2$ inner product of two functions by the $L^2$ inner product of their Fourier transforms. Lemma \ref{l3.3} relates Fourier transforms on $\liek$ to Fourier transforms on $\liet$. Finally Proposition \ref{p3.5} describes certain functions whose Fourier transforms are the terms appearing in the localization formula (\ref{2.4}). We would like to study a certain integral that arises out of equivariant cohomology: \begin{equation} \label{3.1} {\mbox{$\cal I$}}^\epsilon = \frac{1}{(2 \pi i )^s \,{\rm vol}\, K} \int_{\phi \in \liek} [d\phi] \, e^{- \epsilon \inpr {\phi, \phi } /2 } \int_X \si(\mar \phi). \end{equation} Here, $\si \in \Om^*_K(X) $ (see (\ref{2.0})); we are mainly interested in the class $\si$ defined by (\ref{2.0''}). Also, $\epsilon > 0 $ and we shall consider the behaviour of ${\mbox{$\cal I$}}^\epsilon$ as $\epsilon \to 0^+$. The measure $[d \phi]$ is a measure on $\liek$ which corresponds to a choice of invariant metric on $\liek$ (for instance, the metric given by the Killing form): such a metric induces a volume form on $K$, and $ \,{\rm vol}\, K$ is the integral of this volume form over $K$. Thus $[d \phi]/ \,{\rm vol}\, K$ is independent of the choice of metric on $\liek$. The metric also gives a measure $[d \psi] $ on $\liet$ and a volume form on $T$: it is implicit in our notation that the measures on $T$ and $\liet$ come from the same invariant metric as those on $K$ and $\liek$. It will be convenient to recast integrals over $\liek$ in terms of integrals over $\liet$. For this we use a function $\nusym: \liet \to {\Bbb R }$, satisfying $\nusym(w \psi) = (\det w) \nusym(\psi)$ for all elements $w$ of the Weyl group $ W$, and defined by \begin{equation} \label{3.2} \nusym(\psi) = \prod_{\gamma > 0} \gamma(\psi), \end{equation} where $\gamma$ runs over the positive roots. Using the inner product to identify $\liet$ with $\liets$, $\nusym$ also defines a function $\liets \to {\Bbb R }$. We have \begin{lemma} \label{l3.1}[Weyl Integration Formula] If $f: \liek \to {\Bbb R }$ is $K$-invariant, then $$\int_{\phi \in \liek} f (\phi) [d \phi] = \ck^{-1} \intt f (\psi) \nusym(\psi)^2 [d \psi], $$ where $s$ and $l$ are the dimensions of $K$ and $T$, and $ \ck = { \wn \,{\rm vol}\, T}/ \,{\rm vol}\, K $. \end{lemma} \Proof There is an orthonormal basis $\{X_\gamma, Y_\gamma | $ $\gamma $ a positive root$\}$ for $\lietp$ such that $$[X_\gamma, \psi] = \gamma(\psi) Y_\gamma, $$ $$ [Y_\gamma, \psi] = - \gamma(\psi) X_\gamma$$ for all $\psi \in \liet$. The Riemannian volume form of the coadjoint orbit through $\psi \in \liet \cong \liets$ (with the metric on the orbit pulled back from the metric on $\lieks$ induced by the inner product $\inpr{\cdot, \cdot}$) evaluated on the tangent vectors $[X_\gamma, \psi]$ and $[ Y_\gamma, \psi]$ is thus $ \prod_{\gamma> 0} \gamma(\psi)^2$, while the volume form of the homogeneous space $K/T$ (induced by the chosen metric on $\liek$) evaluated on the tangent vectors corresponding to $X_\gamma, Y_\gamma \in \liek $ is $1$. Hence the Riemannian volume of the orbit through $\psi \in \liet$ is $ \nusym(\psi)^2$ times the volume of the homogeneous space $K/T$. $\square$ \newcommand{\lamax}{\Lambda^{\rm max} } \newcommand{\dist}{{\mbox{$\cal D$}}'} It will be convenient also to work with the Fourier transform. Given $f: \liek \to {\Bbb R }$ we define $F_K f: \lieks \to {\Bbb R }$, $F_T f: \liets \to {\Bbb R }$ by \begin{equation} \label{3.3} (F_K f) (z) = \frac{1}{(2 \pi)^{s/2} } \intk f (\phi) e^{-i \evab{\phi}{z} } \, [d \phi], \end{equation} \begin{equation} \label{3.4} (F_T f) (y) = \frac{1}{(2 \pi)^{l/2} } \intt f (\psi) e^{-i \evab{\psi}{y} } \, [d \psi]. \end{equation} More invariantly, the Fourier transform is defined on a vector space $V$ of dimension $n$ with dual space $V^*$ as a map $F: \Omega^{\rm max}(V) \to \Omega^{\rm max} (V^*)$ where $\Omega^{\rm max} (V) = \Lambda^{\rm max} (V^*) \otimes \dist (V)$, $\Lambda^{\rm max}(V^*) $ is the top exterior power of $V^*$ and $\dist (V) $ are the tempered distributions on $V$ (see \cite{hor}). Indeed for $z \in V^*$, $f \in \dist(V) $ and $u \in \lamax(V^*)$ we define \begin{equation} \Bigl ( F (u \otimes f )\Bigr ) (z) = \frac{v}{(2 \pi)^{n/2}} \int_{\phi \in V} f(\phi) e^{- i z(\phi)} u , \end{equation} where the element $v\in \lamax (V) $ satisfies $u(v) = 1 $ under the natural pairing $\lamax(V^*) \cong \Bigl (\lamax (V) \Bigr)^*.$ (The normalization has been chosen so that $$ \tilde{f} (\phi) = F (F \tilde{f}) (- \phi) $$ for any $\tilde{f} \in \Omega^{\rm max}(V)$.) For notational convenience we shall often ignore this subtlety and identify $\liek, \liet$ with $\lieks, \liets$ under the invariant inner product $\inpr{\cdot, \cdot}$: we shall also suppress the exterior powers of $\liek$ and $\liet$ and write $\fk: \dist(\liek) \to \dist (\lieks)$, $\ft: \dist(\liet) \to \dist(\liets)$. Further, although we shall work with functions whose definition depends on the choice of the element $[d \phi] \in \lamax(\lieks)$ (associated to the inner product), some of our end results do not depend on this choice\footnote{The statement of Theorem \ref{t8.1}, for instance, does not depend on the invariant inner product $\inpr{\cdot, \cdot}$.} and the use of such functions is just a notational convenience. A fundamental property of the Fourier transform is that it preserves the $L^2$ inner product: \begin{lemma} \label{l3.2} {\bf [Parseval's Theorem]}(\cite{hor}, Section 7.1) If $f: \liek \to {\Bbb C }$ is a tempered distribution and $g: \liek \to {\Bbb C }$ is a Schwartz function then $F_K f: \liek \to {\Bbb C }$ is also a tempered distribution and $F_K g: \liek \to {\Bbb C }$ a Schwartz function, and we have $$ \intk \overline{g(\phi)} f (\phi) [d \phi] = \tintk \overline{(F_K g)(z) } (F_K f)(z) \, [d z], $$ $$ \intt \overline{g(\psi)} f (\psi) [d \psi] = \tintt \overline{(F_Tg)(y) } (F_T f)(y) \, [d y]. $$ \end{lemma} We note also that if $g_\epsilon: \liek \to {\Bbb R }$ is the Gaussian defined by $g_\epsilon(\phi) = e^{-\epsilon \inpr{\phi, \phi}/2 } $, then \begin{equation} \label{3.5} (F_K g_\epsilon) (z) = \frac{1}{\epsilon^{s/2} } e^{- \inpr{z,z} /{2 \epsilon} } = \frac{1}{\epsilon^{s/2} } \gsoe(z), \phantom{bbbbb} (F_T g_\epsilon ) (y) = \frac{1}{\epsilon^{l/2} } e^{- \inpr{y,y} /{2 \epsilon} } = \frac{1}{\epsilon^{l/2} } \gsoe(y). \end{equation} We have also that \begin{lemma} \label{l3.2'} The symplectic volume form $d \Om^S_\phi$ at a point $\phi$ in the orbit $K \cdot \psi$ through $\psi \in \liek$ is related to the Riemannian volume form $d \Om^R_\phi$ (induced by the metric on $\liek$) by $$d \Om^R_\phi = \nusym(\psi) d \Om^S_\phi.$$ \end{lemma} \Proof The symplectic form is $K$-invariant and is given at the point $\psi \in \liet_+$ in the orbit $K \cdot \psi$ (for $\xi, \eta \in \lietp$ giving rise to tangent vectors $[\xi,\psi], $ $[\eta, \psi]$ to the orbit) by \begin{equation} \label{3.7} \om([\xi,\psi], [ \eta,\psi]) = \inpr{[ \xi,\psi], \eta } = \inpr{\psi, [ \eta,\xi] }. \end{equation} In the notation of Lemma \ref{l3.1}, the symplectic volume form evaluated on the tangent vectors $[ X_\gamma,\psi], $ $[ Y_\gamma,\psi]$ is given by $\prod_{\gamma > 0} \om ([ X_\gamma,\psi], [ Y_\gamma,\psi] ).$ But $\om ([X_\gamma,\psi], [ Y_\gamma,\psi] ) = \inpr{[ X_\gamma,\psi], Y_\gamma} $ $ = \gamma(\psi)$, from which (comparing with the proof of Lemma \ref{l3.1}) the Lemma follows. $\square$ We shall use this Lemma to prove the following Lemma relating Fourier transforms on $\liek$ to those on $\liet$: \begin{lemma} \label{l3.3} Let $f \in \dist(\liek)$ be $K$-invariant, and let $\nusym$ be defined by (\ref{3.2}). Then $$\ft(\nusym f) = \nusym \fk(f)$$ as distributions on $\liet$. \end{lemma} \Proof $$(\fk f) (z) = \frac{1}{(2 \pi)^{s/2} } \intk e^{-i \evab{\phi}{z} } \, f(\phi) [d \phi] $$ \begin{equation} \label{3.6} = \frac{1} {(2 \pi)^{s/2} } \int_{\psi \in \liet_+} \int [d \psi] f(\psi) \: \int_{\phi \in K \cdot \psi} e^{- i \evab{\phi}{z} } d \Om^R_\phi \end{equation} (where $\liet_+$ denotes the fundamental Weyl chamber). We have from Lemma \ref{l3.2'} that $d \Om^R_\phi = \nusym(\psi) d \Om^S_\phi$. Thus we have \begin{equation} \label{3.8}(F_K f)(z) = (2 \pi)^{-s/2} \int_{\psi \in \liet_+} [d \psi] f (\psi) \nusym(\psi) \int_{\phi \in K \cdot \psi} e^{- i \evab{\phi}{ z} } d \Om^S_\phi. \end{equation} Now the integral over the coadjoint orbit may be computed by the Duistermaat-Heckman theorem \cite{DH} applied to the action of $T$ on the orbit $K \cdot \psi$ (or equivalently as a consequence of the abelian localization theorem, Theorem \ref{t2.1}): we have (see e.g. \cite{BGV} Theorem 7.24) \begin{equation} \label{3.9} \int_{\phi \in K \cdot \psi} e^{- i \evab{\phi}{ z} } \, d \Om^S_\phi = \frac{(2 \pi)^{(s-l)/2} } {\nusym(z) } \sum_{w \in W} e^{- i \evab{w \psi}{ z} } (\det w). \end{equation} This is a well known formula due originally to Harish-Chandra (\cite{Harish}, Lemma 15). (Notice that the $z$ in (\ref{3.9}) plays the role of the $\psi$ in Theorem \ref{t2.1}, while the $\psi$ in (\ref{3.9}) specifies the orbit.) Now since $\nusym(w \psi) = (\det w) \nusym(\psi), $ we may replace the integral over $\liet_+$ by an integral over $\liet$: in other words we have \begin{equation} \label{3.10} \nusym(z) (F_K f) (z) = (2 \pi)^{- l/2} \int_{\psi \in \liet} [d \psi] \, f(\psi) \nusym(\psi) \eminevb{\psi}{ z} = F_T(\nusym f) (z). \phantom{bbbbb} \square \end{equation} \newcommand{\ims}{i^{-s} } Applying this to the Gaussian $\gse(\psi) = e^{-\epsilon \inpr{\psi, \psi}/2 } $ we have \begin{corollary} \label{c3.4} $$ F_T(\gse \nusym) (y) = \frac{1}{\epsilon^{s/2} } \nusym(y) e^{- \inpr{y,y} /{2 \epsilon} } = \frac{1}{\epsilon^{s/2} } \nusym(y) \gsoe(y). $$ \end{corollary} We shall also need the following result which occurs in the work of Guillemin, Lerman, Prato and Sternberg concerning Fourier transforms of a class of functions on $\liets$. This result will be applied to functions appearing in the abelian localization formula (\ref{2.4}). \begin{prop} \label{p3.5} {\bf (a)} (see \cite{JGP}, section 3.2 of \cite{GLS}, and \cite{GP}) Define $h(y) = H_{\bar{\beta} }(y + \tau)$ for some $\tau \in \liet$, where $H_{\bar{\beta} } (y) = \,{\rm vol}\, \{ (s_1, \dots, s_N): s_i \ge 0, \phantom{a} y = \sum_j s_j \beta_j \}$ for some $N$-tuple $\bar{\beta} = $ $\{ \beta_1, \dots, \beta_N \}$ , $\beta_j \in \liets$, such that the $\beta_j$ all lie in the interior of some half-space of $\liets$. Thus $H_{\bar{\beta} } $ is a piecewise polynomial function supported on the cone $C_\barb = \{ \sum_j s_j \beta_j \, | \, s_j \ge 0 \}. $ Then the Fourier transform of $h$ is given for $\psi$ in the complement of the hyperplanes $\{ \psi \in \liet| {\beta_j}({ \psi}) = 0 \}$ by the formula \begin{equation} \label{3.10'} F_T h(\psi) = \frac{\epinev{\tau}{ \psi} } {i^N \prod_{j = 1}^N \beta_j (\mar \psi) }. \end{equation} \noindent{\bf(b)} (see Section 2 of \cite{JGP} and (2.15) of \cite{GP}) The function $H_{\bar{\beta}} (y) $ is also given as $$H_{\bar{\beta}} (y) = H_{\beta_1} * H_{\beta_2} * \dots * H_{\beta_r}, $$ where for $\beta \in {\rm Hom}({\Bbb R }^l,{\Bbb R })$ we have $H_\beta = (i_\beta)_* dt$, i.e. $H_\beta$ is the pushforward of the Euclidean measure $dt$ on ${\Bbb R }^+$ under the map $i_\beta: {\Bbb R }^+ \to {\Bbb R }^l$ given by $i_\beta(t) = \beta t$. Here, $*$ denotes convolution. \noindent{\bf (c)} (see Section 2 of \cite{JGP}) The function $H_\barb $ satisfies the differential equation $$ \prod_{j = 1}^N \beta_j (\partial/\partial y) H_\barb(y) = \delta_0(y) $$ where $\delta_0$ is the Dirac delta distribution. \noindent{\bf (d)} (see Proposition 2.6 of \cite{JGP}) The function $H_\barb$ is smooth at any $y \in U_\barb$, where $U_\barb$ are the points in $\liets$ which are not in any cone spanned by a subset of $\{\beta_1, \dots, \beta_N \} $ containing fewer than $l$ elements. \end{prop} { \setcounter{equation}{0} } \section{Reduction to a piecewise polynomial function} In this section we apply the results stated in Section 3 to reduce $\ie$ to the form given in Proposition \ref{p4.2}, as the integral over $y \in \liets$ of a Gaussian $\gtsoe(y) = \gsoe(y)/((2 \pi)^s \wn \,{\rm vol}\, ( T) \epsilon^{s/2} ) $ times a function $Q(y)$ which is $D_\nusym R(y)$ where $D_\nusym$ is a differential operator and $R$ is a piecewise polynomial function. As a byproduct we obtain also a generalization of Theorem 2.16 of \cite{GP}: we may relax the hypothesis of \cite{GP} that the action of the maximal torus $T$ have isolated fixed points. Two of our main results related to Witten's work in \cite{tdg} are Theorem \ref{t4.1} and \ref{t4.3}: these are stated in this section. The proof of Theorem \ref{t4.3} will be given in Section 5. It tells us that the cohomological contribution $\eeth[\xred]$ given in Witten's Theorem \ref{t1.2} for the zero locus of the moment map is given by the integral over $y \in \liets$ of $\gtsoe(y)$ times a polynomial $Q_0(y)$ which is equal to $Q$ near $y = 0$. Theorem \ref{t4.1} is our version of the asymptotic estimates given in Witten's Theorem \ref{t1.1}, and will be proved in Section 6 below. Theorem \ref{t7.1} in Section 7 extends Theorems \ref{t4.1} and \ref{t4.3} to more general equivariant cohomology classes. We assume throughout the rest of the paper that $K$ acts on $X$ in a Hamiltonian fashion, and that $0$ is a regular value of the moment map $\mu$ for the $K$ action. This is equivalent to the assumption that $K$ acts on $\mu^{-1}(0)$ with finite stabilizers, and it implies that $\mu^{-1}(0)$ is a smooth manifold. Under these hypotheses, the space $\xred = \mu^{-1}(0)/K$ is a $V$-manifold or orbifold (see \cite{kaw}) and $P = \mu^{-1}(0) \to \mu^{-1}(0)/K $ is a $V$-bundle: we have a class $\Theta \in H^4 (\xred) $ which represents the class $-\inpr{\phi, \phi}/2 \in H^4_K(\mu^{-1}(0) ) $ $\cong H^4(\zloc/K)$, and which is a four-dimensional characteristic class of the bundle $P \to \xred$. In \cite{Ki1} it is proved that the set of critical points of the function $\rho = |\mu|^2: X \to {\Bbb R }$ is a disjoint union of closed subsets $C_\beta$ in $X$ indexed by a finite subset ${\mbox{$\cal B$}}$ of $\liet$. In fact if $\liet_+$ is a fixed positive Weyl chamber for $K$ in $\liet$ then $\beta \in {\mbox{$\cal B$}}$ if and only if $\beta \in \liet_+$ and $\beta$ is the closest point to $0$ of the convex hull in $\liet$ of some nonempty subset of the finite set $\{ \mu_T(F): F \in {\mbox{$\cal F$}}\}$, i.e. the image under $\mu_T$ of the set of fixed points of $T$ in $X$. Moreover if $\beta \in {\mbox{$\cal B$}}$ then $$C_\beta = K(Z_\beta \cap \mu^{-1}(\beta))$$ where $Z_\beta$ is the union of those connected components of the set of critical points of the function $\mu_\beta$ defined by $\mu_\beta (x) = \eva{\mu(x)}{ \beta} $ on which $\mu_\beta$ takes the value $|\beta|^2$. Note that $C_0 = \mu^{-1}(0) $ and in general the value taken by the function $\rho = |\mu|^2$ on the critical set $C_\beta$ is just $|\beta|^2$. We shall prove the following version of Witten's nonabelian localization theorem, for the integral $\ie$ defined in (\ref{3.1}) with the class $\si = e^{\iins \bom} $ defined by (\ref{2.0''}): \begin{theorem} \label{t4.1} For each $\beta \in {\mbox{$\cal B$}}$ let $\rho_\beta = |\beta|^2$ (this is the critical value of the function $\rho = |\mu|^2: X \to {\Bbb R }$ on the critical set $C_\beta$ when this set is nonempty). Then there exist functions $h_\beta: {\Bbb R }^+ \to {\Bbb R } $ such that for some $N_\beta \ge 0$, $\epsilon^{N_\beta} h_\beta(\epsilon)$ remains bounded as $\epsilon \to 0^+$, and for which $$|\ie - e^{\epsilon \Theta} e^\om [\xred] | \le \sum_{\beta \in {\mbox{$\cal B$}} - \{0\} } e^{- \rho_\beta /{2 \epsilon} } \, h_\beta(\epsilon). $$ \end{theorem} \noindent{\em Remark:} The estimate given in Theorem \ref{t4.1} is {\em weaker} than Witten's estimate (Theorem 1.2) since $|\beta|^2 $ is not in fact a critical value of $|\mu|^2 $ when $C_\beta$ is empty. Nevertheless in many interesting cases all the $C_\beta$ are nonempty, and our estimate then coincides with Witten's. To prove this result, we shall rewrite $\ie$ using Lemma \ref{l3.1}: \begin{equation} \label{4.1} \ie = \frac{1}{(2 \pi i)^s \wn \,{\rm vol}\, (T) } \: \intt [d \psi] \, \Bigl ( \gse(\psi) \: \nusym(\psi) \Bigr ) \: \Bigl ( \: \nusym(\psi) (\pist \sigma ) (\psi) \Bigr ). \end{equation} Now we apply Lemma \ref{l3.2} to get \begin{equation} \label{4.2} \ie = \frac{1}{(2 \pi i)^s \wn \,{\rm vol}\, (T) } \tintt [dy] \Bigl ( F_T (g_\epsilon \nusym) (y) \Bigr ) \Bigl ( F_T (\nusym \, \pist \si ) (y) \Bigr ). \end{equation} Applying Corollary \ref{c3.4} we have \begin{equation} \label{4.3} \ie = \frac{1}{(2 \pi i)^s \wn \,{\rm vol}\, (T) \epsilon^{s/2} } \tintt [dy] \nusym(y) e^{- \inpr{y,y}/{2 \epsilon} } \, F_T ( \nusym \pist \si) (y). \end{equation} \newcommand{\bwd}{\lasub} \newcommand{\cf}{C_F} \newcommand{\cfl}{C_{F,\lasub } } \newcommand{\dcf}{\check{C}_{F,\lasub} } Following Guillemin, Lerman and Sternberg \cite{JGP}, we may use the abelian localization formula (Theorem \ref{t2.1}) to give a formula for $F_T ( \pist \si)$ where $\si = e^{\iins \bom}$, and from it obtain a formula for $F_T (\nusym \pist \si)$. In terms of the notation of Lemma \ref{l2.2}, we choose a component $\bwd$ of the set $\cap_{F,j} \Bigl \{ \psi \in \liet:$ $ \bfj(\psi) \ne 0 \Bigr \}, $ where $\beta_{F, j}$ are the weights of the action of $T$ on the normal bundle to a component $F$ of the fixed point set. Thus $\bwd$ is a cone in $\liet$. If we denote by $\cf = \cfl$ the component of $\cap_j \{ \psi \in \liet: $ $ \bfj (\psi) \ne 0 \} $ containing $\bwd$, then $\bwd = \cap_F \cfl$. Also, $\bfj \in \liets$ lies in the dual cone $\dcf$ of $\cfl$: indeed, this dual cone is simply the cone $\dcf = \{ \sum_j s_j \bfj: s_j \ge 0 \}. $ We then define $\sigma_\fj = {\rm sign} \bfj(\xi) $ for any $\xi \in \bwd$, and $\bfjw = \sigma_\fj \bfj$. Then we set $k_F(\alpha) = \sum_{j, \sigma_\fj = - 1} n_\fj(\alpha)$. We define a function $\hh: \liets \to {\Bbb C }$ by \begin{equation} \label{4.g} \hh(y) = \sum_{F \in {\mbox{$\cal F$}}} \sum_{\alpha \in {\mbox{$\cal A$}}_F} (-1)^{k_F(\alpha) } H_{\bar{\gamma_F}(\alpha) } (-y + \mu_T(F) ) \int_F (e^{i\om} \tilde{c}_{F,\alpha}) . \end{equation} Here as before ${\mbox{$\cal F$}}$ is the set of components $F$ of the fixed point set of $T$ and the ${\mbox{$\cal A$}}_F$ are the indexing sets which appeared in Lemma \ref{l2.2}. If $\alpha \in {\mbox{$\cal A$}}_F$ then $\bar{\gamma_F}(\alpha)$ consists of the elements $\bfjw$ where each $\bfjw$ appears with multiplicity $n_\fj(\alpha) $. Then $H_{\bar{\gamma_F}(\alpha) } $ is as defined in Proposition \ref{p3.5}. The $\tilde{c}_{F,\alpha} \in H^*(F) $ are related to the $c_{F,\alpha} $ in (\ref{2.m1}), in that \begin{equation} \label{4.004} c_{F, \alpha}(\psi) = e^{i\om} e^{i \eva{\mu_T(F)}{ \psi} } \tilde{c}_{F,\alpha}. \end{equation} We then have the following Theorem, which in the case when the action of $T$ has isolated fixed points is the main theorem of Section 3 of the paper \cite{JGP} of Guillemin, Lerman and Sternberg. For the most part our proof is a direct extension of the proof given in that paper; the major difference is in the use of the abelian localization theorem rather than stationary phase. \begin{theorem} \label{t4.1'} The (piecewise polynomial) function $\hh$ given in (\ref{4.g}) is identical to the distribution $ \ggh = F_T ( \pist e^{\iins \bom}) $. \end{theorem} \Proof We first apply the abelian localization formula (\ref{2.2}) to $\pist \si$. Then the formula obtained from Lemma \ref{l2.2} for $\pist \si$ is \begin{equation} \label{4.3p} \pist \si = \sum_{F \in {\mbox{$\cal F$}}} \rf, \phantom{bbbbb} \rf = \sum_{\alpha \in {\mbox{$\cal A$}}_F} \tau_{F, \alpha} , \end{equation} \begin{equation} \label{4.03p}\tau_{F, \alpha} = (-1)^{k_F(\alpha) } \frac{ \epinevb{\psi}{ \mu_T(F)} \int_F (e^{\iins\om} \tilde{c}_{F,\alpha} ) } {\prod_j \Bigl ( \bfjw (\mar \psi) \Bigr )^{{n}_\fj(\alpha)} } , \end{equation} where $\mu_T $, the moment map for the $T$ action, is simply the projection of $\mu$ onto $\liet$ and the $ \bfj $ are the weights of the $T$ action. Recall that the quantity $\bfjw$ is $\bfj$ if $\bfj (\xi) > 0 $ and $- \bfj$ if $\bfj (\xi) < 0 $. The class $\tilde{c}_{F, \alpha} \in H^*(F)$ is equal to some characteristic classes of subbundles of the normal bundle $\nu_F$; it is independent of $\psi$. Notice that each $\bfjw$ $\in \liets$ lies in the half space $\{ y \in \liets \, | y( \xi) > 0 \}$, for any $\xi \in \bwd$. The expression (\ref{4.03p}) is hence of the form appearing on the right hand side of (\ref{3.10'}), up to multiplication by a factor independent of $\psi$. The conclusion of the proof goes, as in Section 3 of \cite{JGP}, by applying a lemma about distributions (see Appendix A of \cite{GP}): \begin{lemma} \label{l4} Suppose $\gggh$ and $\hhh$ are two tempered distributions on ${\Bbb R }^l$ such that: \begin{enumerate} \item $F_T \gggh - F_T \hhh$ is supported on a finite union of hyperplanes. \item There is a half space $\{ y \, | \, \inpr{y,\zeta} > k_0 \}$ containing the support of $\hhh - \gggh$. \end{enumerate} Then $\gggh = \hhh$. \end{lemma} Here we apply the lemma to $\hhh$ as given in (\ref{4.g}) and $\gggh = F_T ( \pist e^{\iins \bom} ) $. The first hypothesis is satisfied because we know from Proposition \ref{p3.5} that $F_T \hh$ is given by the formula (\ref{4.3p}) on the complement of the hyperplanes $\{ \psi \, | \, \bfjw ( \psi) = 0 \}$; but this is just the formula for $F_T \ggh = \pist e^{\iins \bom}$. Further, $\hh$ is supported in a half space since all the weights $\bfjw$ satisfy $\bfjw(\xi) > 0$ for any $\xi \in \bwd$, while the support of $\ggh$ is contained in the compact set $\mu_T(X)$ (see Section 5 below). Therefore the support of $H-G$ is contained in a half space of the form $\{ y: \inpr{y,\xi} > k_0 \}$ for some $k_0$. This completes the proof of Theorem \ref{t4.1'}. $\square$ Define \begin{equation} \label{4.3pp} R(y) = F_T ( \pist \si) (y), \end{equation} where $\si = \exp \iins \bom$. Then (\ref{4.3}) and (\ref{4.g}) give us the following \begin{prop} \label{p4.2} The function $R$ is a piecewise polynomial function supported on cones each of which has apex at $\mu_T(F)$ for some component $F$ of the fixed point set of $T$. Let $Q$ be the distribution defined by $$ Q(y) = \nusym(y) D_\nusym R(y) $$ where the differential operator $D_\nusym$ is given by $$D_\nusym = \prod_{\gamma> 0} (i \gamma(\partial/\partial y) )$$ and $\gamma$ runs over the positive roots (cf. (\ref{3.2})). Then \begin{equation} \label{4.3ppp} \ie = \frac{1}{(2 \pi i )^s \wn \,{\rm vol}\, ( T) \epsilon^{s/2} } \tintt [dy] e^{- \inpr{y,y}/{2 \epsilon} } Q(y). \end{equation} \end{prop} \Proof It only remains to note that $\ft (\nusym \pist \si) = D_\nusym \ft (\pist \si), $ where $D_\nusym$ is defined above. $\square$ \noindent{\em Remark:} The formulas for $Q(y)$ obtained from (\ref{4.3p}) will in general be different for different choices of $\bwd$. In addition, certain formulas simplify if we impose the additional assumption that at any point $x$ in a component $F$ of the fixed point set of the $T$ action, the orthocomplement $\lietp$ of $\liet$ in $\liek$ injects into the tangent space $T_x X$ under the infinitesimal action of $K$: in other words, that the stabilizer ${\rm Stab} (x)$ of $x$ is such that ${\rm Stab}(x)/T$ is a finite group. Under this additional hypothesis, we may indeed prove a somewhat stronger result. Notice that by Lemma \ref{l2.2}, each term (corresponding to a component $F$ in the fixed point set of $T$) in the localization formula for $\pist \si$ has a factor $\efo(\mar \psi)$ in the denominator. Now for $x \in F$, the fibre $(\nu_F)_x$ over $x$ of the normal bundle to $F$ will contain $ \lietp \cdot x$, the image of $\lietp$ under the infinitesimal action of $K$. Under the additional assumption that $\lietp$ injects into $T_x X$, the set of weights for $\nu_F$ contains for each root $\gamma > 0$ either the root $\gamma$ or the root $- \gamma$: in other words, $\efo (\mar \psi)$ is divisible by $\nusym(\psi)$. Thus from Lemma \ref{l2.2} we obtain the formula for $\nusym(\psi) \pist \si$: \begin{equation} \label{4.3pinj} \nusym(\psi) \pist \si = \sum_{F \in {\mbox{$\cal F$}}, \alpha \in {\mbox{$\cal A$}}_F} \ttfa, \end{equation} \begin{equation} \label{4.03pinj} \ttfa = (-1)^{k_F(\alpha) } \frac{ \epinevb{\psi}{ \mu_T(F)} } {\prod_j \Bigl ( \bfjw (\mar \psi) \Bigr )^{\tilde{n}_\fj(\alpha)} } \int_F ( e^{i \om} \tilde{c}_{F,\alpha} ) . \end{equation} Here, the notation is as in (\ref{4.3p}) and (\ref{4.03p}) except that $\tilde{n}_\fj(\alpha) = n_\fj(\alpha) $ if $\bfj$ is not a root, while $\tilde{n}_\fj(\alpha) = n_\fj(\alpha) - 1$ if $\bfj$ is a root. We may then use the abelian localization formula to give a formula for $F_T (\nusym \pist \si)$ where $\si = e^{\iins \bom}$. As in (\ref{4.g}), we define a function $\hh^\nusym(y) $ for $y \in \liets$ by \begin{equation} \label{4.ginj} \hh^\nusym(y) = \sum_{F \in {\mbox{$\cal F$}}} \sum_{\alpha \in {\mbox{$\cal A$}}_F} (-1)^{k_F(\alpha) } H_{\hat{\gamma_F}(\alpha) } (-y +\mu_T(F) ) \int_F e^\omega \tilde{c}_{F,\alpha} . \end{equation} The notation is as in ({\ref{4.g}) except that each $\bfjw$ appears in $\hat{\gamma}_F(\alpha)$ with multiplicity $n_\fj(\alpha) $ if it is not a root and $n_\fj(\alpha) - 1 $ if it is a root. The function $H_{\hat{\gamma_F}(\alpha) } $ is then as defined in Proposition \ref{p3.5}. In the case when the action of $T$ has isolated fixed points, the theorem \ref{t4.1'inj} below is the main result (Theorem 2.16) of Part I of the paper \cite{GP} of Guillemin and Prato. For the most part our proof translates directly from the proof given in that paper, except that we use the abelian localization theorem in place of stationary phase. \begin{theorem} \label{t4.1'inj} Suppose that $\lietp$ injects into $T_x X$ for all fixed points $x$ of the action of $T$. Then the distribution $\hh^\nusym$ given in (\ref{4.ginj}) is identical to the distribution $ \ggh = F_T (\nusym \pist e^{\iins \bom}) $. \end{theorem} \Proof This theorem is proved in exactly the same way as Theorem \ref{t4.1'}. The conclusion of the proof goes, as in the proof of Theorem 2.16 of \cite{GP}, by applying Lemma \ref{l4} directly to $\hh^\nusym$ as given in (\ref{4.ginj}) and $\gggh = F_T (\nusym \pist e^{\iins \bom} ) $. $\square$ In particular we have the following \begin{prop} \label{p4.2inj} Suppose that $\lietp$ injects into $T_x X$ for all fixed points $x$ of the action of $T$. Define \begin{equation} \label{4.3ppinj} Q(y) = \nusym(y) F_T (\nusym \pist \si) (y), \end{equation} so that \begin{equation} \label{4.3pppinj} \ie = \frac{1}{(2 \pi i )^s \wn \,{\rm vol}\, ( T) \epsilon^{s/2} } \tintt [dy] e^{- \inpr{y,y}/{2 \epsilon} } Q(y). \end{equation} Then $Q$ is a piecewise polynomial function supported on cones each of which has apex at $\mu_T(F)$ for some component $F$ of the fixed point set of $T$. \end{prop} We now drop the hypothesis that $\lietp$ injects into $T_x X$ at the fixed points $x$ of the action of $T$, and return to the general situation described in Proposition \ref{p4.2}. It will follow from (\ref{5.6}) that $Q$ is smooth near $y = 0$: thus in particular there is a polynomial $Q_0$ which is equal to $Q$ near $y = 0$. Of course $Q_0 = D_\nusym R_0$ where $R_0$ is the polynomial which is equal to $\ft (\pist \si)$ near $y = 0$. In the next section we shall provide an alternative description of $Q_0$ and prove \begin{theorem} \label{t4.3} $$ \ie_0 \;\: {\stackrel{ {\rm def} }{=} } \;\: \frac{1}{ (2 \pi i)^s \wn \,{\rm vol}\, ( T) \epsilon^{s/2} } \tintt [dy] e^{- \inpr{y,y}/{2 \epsilon} } \: Q_0(y) = e^{\epsilon \Theta} e^{i \om_0} [\xred]. $$ \end{theorem} This tells us that the contribution to $\ie$ from $\mu^{-1}(0)/K$ is obtained by integrating $Q_0$ rather than $Q$ (weighted by the Gaussian $\tilde{g}_{\epsilon^{-1}}$ defined in the first paragraph of this Section) over $y \in \liets$. { \setcounter{equation}{0} } \section{The proof of Theorem 4.7} This section gives the proof of Theorem \ref{t4.3}, which identifies $\eeth [\xred]$ with the integral of a Gaussian $\gtsoe$ times a polynomial $Q_0$. The key step in the proof is the well known result Proposition \ref{p5.2} below, which gives a normal form for the symplectic form, the $K$ action and the moment map in a neighbourhood ${\mbox{$\cal O$}}$ of $\zloc$. We first recast the distribution $Q$ in terms of an integral over $X$ (Proposition \ref{p5.1}), so that $Q(y)$ is given by the integral over $X$ of a distribution supported where $\mu$ takes the value $y$. (This step occurs also in the proof \cite{DH} of the Duistermaat-Heckman theorem.) Hence, for sufficiently small $y$, this distribution is supported in ${\mbox{$\cal O$}}$ and we may do the integral over $X$ to obtain the value of the polynomial $Q_0 $ which is equal to $Q $ near $0$. This turns out to be given by an integral over $\xred$ involving the symplectic form and the curvature of a bundle over $\xred$ (see (\ref{5.6})). Finally, we multiply $Q_0$ by the Gaussian $\gtsoe$ and integrate over $\liet$ to see that the result is $\eeth [\xred]$. \begin{prop} \label{p5.1} $Q(y) = \nusym^2(y) (2 \pi)^{s/2} \int_{x \in X} e^{i \om} \delta (y - \mu(x) ) $, where $\delta$ denotes the (Dirac) delta distribution. \end{prop} \Proof We have by Lemma \ref{l3.3} that $$Q = \nusym F_T (\nusym \pist \si ) = \nusym^2 F_K(\pisk \si) ,$$ so that $$ Q(y) = \frac{\nusym^2 (y) }{(2 \pi)^{s/2} } \intk [d \phi] \, e^{-i \evab{\phi}{ y} } \int_{x \in X} e^{i \om} e^{ i \eva{\mu(x)}{ \phi} } \, $$ \begin{equation} \label{5.1} = \nusym^2 (y) (2 \pi)^{s/2} \int_X e^{i \om} \delta(\mu - y). \square \end{equation} We would like to study this for $|y| < h $ for sufficiently small $h > 0$. Now there is a neighbourhood of $\mu^{-1}(0)$ on which the symplectic form is given in a standard way related to the symplectic form $\om_0$ on $\xred$: this follows from the coisotropic embedding theorem (see sections 39-41 of \cite{STP}). \begin{prop} \label{p5.2} {\bf (Gotay\cite{gotay}, Guillemin-Sternberg \cite{STP}, Marle \cite{marle})} Assume $0$ is a regular value of $\mu$ (so that $\mu^{-1}(0) $ is a smooth manifold and $K$ acts on $\mu^{-1}(0) $ with finite stabilizers). Then there is a neighbourhood ${\mbox{$\cal O$}} \cong \mu^{-1}(0) \times \{ z \in \lieks, |z| \le h \} $ $\subseteq \mu^{-1}(0) \times \lieks$ of $\mu^{-1} (0)$ on which the symplectic form is given as follows. Let $P \;\: {\stackrel{ {\rm def} }{=} } \;\: \mu^{-1}(0) \stackrel{q}{\to} \xred $ be the orbifold principal $K$-bundle given by the projection map $q: \zloc \to \zloc/K$, and let $\theta$ $ \in \Om^1(P) \otimes \liek$ be a connection for it. Let $\omr$ denote the induced symplectic form on $\xred$, in other words $q^* \om_0 = i_0^* \om$. Then if we define a 1-form $\tau$ on ${\mbox{$\cal O$}}\subset P \times \lieks$ by $\tau_{p,z} = z(\theta)$ (for $p \in P$ and $z \in \lieks$), the symplectic form on ${\mbox{$\cal O$}}$ is given by \begin{equation} \label{5.2} \om = q^* \omr + d \tau. \end{equation} Further, the moment map on ${\mbox{$\cal O$}} $ is given by $\mu (p, z) = z$. \end{prop} \noindent{\em Proof of Theorem \ref{t4.3}:}~ We assume for simplicity of notation that $K$ acts freely on $\mu^{-1}(0)$, but all of the following may be transferred to the case when $K$ acts with finite stabilizers by introducing $V$-manifolds or orbifolds (see \cite{kaw}). In other words, we work locally on finite covers of subsets of $\mu^{-1}(0)$ and $\mu^{-1}(0)/K$, where the covering group is the stabilizer of the $K$ action at a point $x \in \mu^{-1}(0)$. When $|y| < h$ and $h$ is sufficiently small, the distribution $\delta(\mu(x) - y)$ is supported in ${\mbox{$\cal O$}}$, so we may compute $Q_0(y)$ from (\ref{5.1}) by restricting to ${\mbox{$\cal O$}}$. We have \begin{equation} \label{5.3} Q_0(y) = (2 \pi)^{s/2} \nusym^2(y) \int_{(p, z') \in P \times \lieks} e^{i \om} \delta(y - z') \end{equation} \begin{equation} \label{5.4} \phantom{a} = (2 \pi)^{s/2} \nusym^2(y) \int_{(p, z') \in P \times \lieks} \exp i(q^* \om_0 + \evab{d \theta}{ z'}) \exp i\evab{\theta}{ dz'} \delta(y - z') \end{equation} Now the term in $\exp i \evab{\theta}{ dz'} $ which contributes to the integral (\ref{5.4}) is $i^s \Om \,[dz'] $ where $[dz'] $ is the volume form on $\liek$ (since all factors $d z'_1 \dots {dz'}_l$ must appear in order to get a contribution to the integral). Here, $\Om = \prod_{j = 1}^s \theta^j$ (for $j$ indexing an orthonormal basis of $\liek$ and $\theta^j$ the corresponding components of the connection $\theta$) is a form integrating to $ \,{\rm vol}\, (K)$ over each fibre of $P \to \xred$. Doing the integral over $z' \in \lieks$, we get \begin{equation} \label{5.5} Q_0(y) = i^s \nusym^2(y) (2 \pi)^{s/2} \: \int_P \exp i (q^* \omr + \evab{d \theta + [\theta, \theta]/2 }{ y } ) \: \Om \end{equation} \begin{equation} \label{5.6} \phantom{bbbbb} = i^s \nusym^2 (y) (2 \pi)^{s/2} \: \int_{\zloc} \exp i (q^* \omr + \evab{F_\theta }{ y } ) \, \Om. \end{equation} Here, $F_\theta = d \theta + {\frac{1}{2} } [\theta, \theta]$ is the curvature associated to the connection $\theta$; we may introduce the term $[\theta, \theta]$ into the exponential in (\ref{5.5}) since the additional factors $\theta$ will give zero under the wedge product with $\Om$. Formula (\ref{5.6}) shows that $Q_0$ is a polynomial in $y$. Now we were interested in $$\ie_0 = \frac{1}{(2 \pi i )^s \wn \,{\rm vol}\, ( T) \epsilon^{s/2} } \tintt [dy] e^{- \inpr{y,y}/{2 \epsilon} } Q_0(y) $$ \begin{equation} \label{5.7} = \frac{1 } { (2 \pi)^{s/2} \wn \,{\rm vol}\, ( T) \epsilon^{s/2} } \tintt [dy] \nusym^2 (y) \: e^{- \inpr{y,y}/{2 \epsilon} } \: \int_{P} \exp (q^* \omr + \evab{F_\theta}{ y} ) \, \Om, \end{equation} \begin{equation} \label{5.9} = \frac{1}{ (2 \pi \epsilon)^{s/2} \,{\rm vol}\, (K) } \int_{z \in \lieks} \, [dz] \, e^{- \inpr{z,z}/{2\epsilon} } \: \int_{P} \exp (q^* \omr + \evab{F_\theta}{z} ) \, \Om, \end{equation} where the last step uses Lemma \ref{l3.1} and the fact that $\int_{P} \exp (q^* \omr + \evab{F_\theta}{ z} ) \Om$ is an invariant function of $z$. We now regard $F_\theta$ as a formal parameter and complete the square to do the integral over $z$: we have (identifying $z(F_\theta)$ with $\inpr{F_\theta, z}$ using the invariant inner product $\inpr{\cdot, \cdot}$) \begin{equation} \label{5.10} \int_{z \in \lieks} [dz] e^{- \inpr{z,z}/{2 \epsilon} } \: \exp \inpr{F_\theta, z} = (2 \pi \epsilon)^{s/2} \exp \epsilon \inpr{F_\theta, F_\theta}/2. \end{equation} But $\inpr{F_\theta, F_\theta}/2$ is just the class $\pi^* \Theta$ on $P$, for $\Theta \in H^4 (\xred)$. Hence we obtain (integrating over the fibre of $P \to \xred$ and using the fact that the integral of $\Om$ over the fibre is $ \,{\rm vol}\, (K)$) \begin{equation} \label{5.11} \ie_0 = \int_{\xred} \exp i\om_0 \: \exp \epsilon \Theta, \end{equation} completing the proof of Theorem \ref{t4.3}. $\square$ { \setcounter{equation}{0} } \section{The proof of Theorem 4.1 } In this section we complete the proof of Theorem \ref{t4.1}. This is done by observing that $\ie - \eeth[\xred]$ is of the form $\ims \int_{\liet} \gtsoe (Q - Q_0) = $ $\int_{\liet} \gtsoe D_\nusym ( R - R_0)$ = $\int_{\liet} (D_\nusym^* \gtsoe) (R-R_0)$, where $R - R_0$ is piecewise polynomial and supported away from $0$. (Here, $D_\nusym^* = (-1)^{(s-l)/2} D_\nusym.$) The results of \cite{Ki1} establish that the distance of any point of Supp($Q - Q_0$) from $0$ is at least $|\beta|$ for some nonzero $\beta$ in the indexing set ${\mbox{$\cal B$}}$ defined in Section 4. Hence we obtain the estimates in Theorem \ref{t4.1}. In fact the function $R - R_0$ is known explicitly in terms of the values of $\mu_T(F)$ (where $F$ are the components of the fixed point set), the integrals over $F$ of characteristic classes of subbundles of the normal bundle $\nu_F$, and the weights of the action of $T$ on $\nu_F$ (see (\ref{2.4}) and Proposition\ \ref{p3.5}). The function $R - R_0$ is polynomial on polyhedral regions of $\liet$, so that the quantity $\ie - \eeth [\xred]$ can in principle be computed from the integral of a polynomial times a Gaussian over these polyhedral regions. We shall study these integrals in another paper and relate them to the cohomology of the higher strata in the stratification of $X$ according to the gradient flow of $|\mu|^2$ given in \cite{Ki1}. We now examine $\ie - \ie_0$ and prove Theorem \ref{t4.1}. Recall from section 4 that the indexing set ${\mbox{$\cal B$}}$ of the critical sets $C_\beta$ for the function $\rho = |\mu|^2$ is ${\mbox{$\cal B$}} = \liet_+ \cap W{\mbox{$\cal B$}}$ where $W {\mbox{$\cal B$}} = \{ w \beta: \; \beta \in {\mbox{$\cal B$}}, \phantom{a} w \in W \}$ is the set of closest points to $0$ of convex hulls of nonempty subsets of the set $\{ \mu_T (F): F \in {\mbox{$\cal F$}}\}$ of images under $\mu_T$ of the connected components of the fixed point set of $T$ in $X$. We shall refer to $\{ \mu_T (F): F \in {\mbox{$\cal F$}}\}$ as the set of {\em weights}\footnote{The motivation for this is that if $X$ is a nonsingular subvariety of complex projective space ${ \Bbb P}_n$ and $T$ acts on $X$ via a linear action on ${\Bbb C }^{n+1} $ then each $\mu_T(F)$ is a weight of this action when appropriate identifications are made.} associated to $X$ equipped with the action of $T$. Let ${\mbox{$\cal J$}}$ denote the locus $${\mbox{$\cal J$}} = \{ y \in \liets : \, Q \phantom{a} \mbox{is not smooth at} \phantom{a} y\}. $$ Then we have \begin{prop} ${\mbox{$\cal J$}} \subset \caljab, $ where $\caljab = \{ y \in \liets : \, \ft (\pis e^{\iins \bom}) $ is not smooth at $y \}. $ \end{prop} \Proof $Q = \nusym \ft (\nusym \pis e^{\iins \bom})$ $ = \nusym D_\nusym \ft(\pis e^{\iins \bom})$. Hence if $\ft (\pis e^{\iins \bom})$ is smooth at $y$ then so is $Q$. $\square$ Now it follows from \cite{JGP} (Section 5) that $\caljab = \cup_{\gamma \in \Gamma} \mu_T (V_\gamma)$ where $V_\gamma$ is a component of the fixed point set of a one parameter subgroup $T_\gamma$ of $T$ and $\Gamma$ indexes all such one parameter subgroups and components of their fixed point sets. Let $$D = \cap \{ D_\beta: \; \beta \in W{\mbox{$\cal B$}} - \{0\} \}$$ where $D_\beta$ denotes the open half-space $$ D_\beta = \{ y \in \liets: \; y(\beta) < |\beta|^2 \}. $$ Note that if $\beta \in {\mbox{$\cal B$}} - \{0\}$ then $D_\beta$ contains $0$ and its boundary is the hyperplane $$H_\beta = \{ y \in \liets: \; y(\beta) = |\beta|^2 \}.$$ \begin{lemma} \label{l6.2}The support of $Q - Q_0$ is contained in the complement of $D$ (or equivalently $Q = Q_0$ on $D$). \end{lemma} \Proof Suppose $V_\gamma$ is a component of the fixed point set of a one parameter subgroup $T_\gamma$. By the Atiyah-Guillemin-Sternberg convexity theorem \cite{aam,gsconv}, $\mu_T(V_\gamma)$ is the convex hull of some subset of the weights. Hence the closest point to $0$ in $\mu_T(V_\gamma)$ is in $W {\mbox{$\cal B$}}$. Now either this closest point is $0$, or else the closest point is a point $\beta \in W{\mbox{$\cal B$}} - \{0\}$ (in which case $\mu_T(V_\gamma) \subset \liet - D_\beta \subset \liet - D$). Now if $x$ is a point in Supp($Q - Q_0$) then the ray from $0$ to $x$ must pass through at least one point in ${\mbox{$\cal J$}}$: hence it suffices to prove ${\mbox{$\cal J$}} \subset \liet - D$ since $D$ is the intersection of a number of open half spaces all of which contain $0$. But ${\mbox{$\cal J$}} \subset \cup_\gamma \mu_T (V_\gamma)$ and every point in $\mu_T(V_\gamma)$ is in $\liet - D$ unless $0$ is in $\mu_T(V_\gamma)$. Moreover if $x \in \mu_T(V_\gamma) \cap {\mbox{$\cal J$}}$ and $0 \in \mu_T(V_\gamma)$, we may consider the function $Q - Q_0$ restricted to a small neighbourhood of the ray from $0$ through $x$. This ray lies in the hyperplane $\tilde{H}_\gamma$ which is the orthocomplement in $\liet$ to the Lie algebra $\liet_\gamma$ of $T_\gamma$. (Since the component of $\mu_T$ in the direction of $\liet_\gamma$ is constant along $V_\gamma$ and since $0 \in \mu_T(V_\gamma)$, it follows that $\mu_T(V_\gamma)$ is contained in $\tilde{H}_\gamma$.) Since $Q - Q_0$ is identically zero near $0$ but not near $x$, the ray from $0$ to $x$ must contain a point $x'$ in ${\mbox{$\cal J$}}$ which is contained in some $\mu_T(V_{\gamma'})$ with $t_\gamma \ne t_{\gamma'}. $ If $0 \notin \mu_T(V_{\gamma'})$ then $\mu_T(V_{\gamma'} ) \subset \liet - D$ and so $x \in \liet - D$. If $0 \in \mu_T(V_{\gamma'}) $ then we simply repeat the argument, considering the restriction of $Q - Q_0$ to a neigbourhood of $\tilde{H}_\gamma \cap \tilde{H}_{\gamma'}$. Since $0 \notin {\mbox{$\cal J$}}$, after finitely many repetitions of this argument we get the required result. Hence the Lemma is proved. $\square$ To complete the proof of Theorem \ref{t4.3}, we then use Lemma \ref{l6.2} to express $\ie - \ie_0 $ as \begin{equation} \label{6.3} \ie - \ie_0 = \frac{1}{(2 \pi i)^s |W| \,{\rm vol}\, T \epsilon^{s/2} } \int_{\liets - D} [dy] (Q - Q_0) e^{- |y|^2/{2 \epsilon} } . \end{equation} Denote by $C$ the set $\{ y \in \liets - D \: : \: |Q(y) - Q_0(y) | \le 1 \}. $ Then $$(2 \pi)^s |W| \,{\rm vol}\, (T) \epsilon^{s/2} |\ie - \ie_0| \le \int_C [dy]\stg + \int_{\liets - D}[dy] |Q - Q_0|^2 \stg. $$ If $b $ is the minimum value of $|\beta | $ over all $\beta \in {\mbox{$\cal B$}} - \{0\}$, then $$\int_C [dy]\stg \le \int_{|y| \ge b } [dy] \stg \le e^{- b^2/{2 \epsilon} } q(\epsilon), $$ where $q(\epsilon) $ is a polynomial in $\epsilon^ {\frac{1}{2} } $. Further, denote by $p$ the function $|Q - Q_0|^2$. Then \begin{equation} \label{6.4}\int_{\liets - D} [dy] p(y) \stg \le \sum_{\beta \in W{\mbox{$\cal B$}} - \{0\}} \int_{y \in D_\beta}[dy] \stg p(y). \end{equation} For each $\beta \in W {\mbox{$\cal B$}} - \{0\} $ one can now decompose $y \in \liets$ into $y = w_0 \hatb + w, w \in \beta^\perp$, $w_0 \in {\Bbb R }$ (where $\hatb = \beta/|\beta|$). Hence each of the integrals (\ref{6.4}) is of the form $$ \int_{w_0 \ge |\beta| } \int_{w \in \beta^\perp} e^{- w_0^2/{2 \epsilon} } e^{ - |w|^2/{2 \epsilon} } p(w_0, w), $$ and this is clearly bounded by $e^{- |\beta|^2/{2 \epsilon} } $ times a polynomial in $\epsilon$. This completes the proof of Theorem 4.1. $\square$ { \setcounter{equation}{0} } \section{Extension of Theorems 4.1 and 4.7 to other classes} In this section we extend Theorems \ref{t4.1} and \ref{t4.3} to equivariant cohomology classes of the form $\zeta = \eta e^{\iins \bom} $ where $\eta \in \hk(X)$. More precisely we shall show the following \begin{theorem} \label{t7.1} Suppose $\eta \in \hk(X)$ and suppose that $i_0^* \eta \in \hk (\zloc)$ is represented by $\eta_0 \in H^*(\zloc/K)$ (where $i_0: \zloc \to X$ is the inclusion map). Then {\bf(a)} We have that $$ \eta_0 e^{\epsilon \Theta} e^{i \om_0} [\xred] = \frac{1}{(2 \pi i )^s |W| \,{\rm vol}\, (T) \epsilon^{s/2} } \tintt [dy] e^{- |y|^2/{2 \epsilon} } \, Q_0^\eta (y), $$ where $Q^\eta(y) = \nusym(y) \ft (\nusym \pist (\eta \exp \iins \bom) ) $ and $Q_0^\eta(y) $ is a polynomial which is equal to $Q^\eta(y) $ near $y = 0$. {\bf (b)} Let $ \rho_\beta = |\beta|^2$ be the value of the function $|\mu|^2: X \to {\Bbb R }$ on the critical set $C_\beta$. Then there exist functions $h_\beta: {\Bbb R }^+ \to {\Bbb R } $ such that for some $N_\beta \ge 0$, $\epsilon^{N_\beta} h_\beta(\epsilon)$ remains bounded as $\epsilon \to 0^+$, and for which $$\Bigl | \, \frac{1}{(2 \pi i )^s \,{\rm vol}\, (K) } \intk [d \phi] e^{-\epsilon |\phi|^2/2} \pisk \eta e^{\iins \bom} - \eta_0 e^{\epsilon \Theta} e^{i \om_0} [\xred] \, \Bigr | \le \sum_{\beta \in {\mbox{$\cal B$}} - \{0\} } e^{- \rho_\beta /{2 \epsilon} } \, h_\beta(\epsilon). $$ {\bf (c)} Suppose $\eta$ is represented by $\tilde{\eta} = \sum_J \eta_J \phi^J $ $ \in \Om^*_K(X)$ for $\eta_J \in \Om^*(X)$. Then $Q^\eta$ is of the form $Q^\eta (y) = \sum_J D_J R_J(y)$, where $D_J$ are differential operators on $\liets$ and $R_J$ are piecewise polynomial functions on $\liets$. \end{theorem} \noindent{\em Proof of } {\bf(a)}: We examine the function $ \qeta(y) $ near $y = 0$. We have from Lemma \ref{l3.3} and the paragraph before Theorem \ref{t2.1} that $$ \nusym \ft (\nusym \pist (\eta e^{\iins \bom} ) ) (y) = \Bigl (\nusym^2 \fk \pisk (\eta e^{\iins \bom} ) \Bigr ) (y)$$ $$ = \frac{\nusym^2 (y)}{(2 \pi)^{s/2} } \intk [d \phi] \int_{x \in X } e^{- i \eva{y}{ \phi} } e^{i \om} e^{ i \eva{\mu(x)}{ \phi} } \eta (\phi). $$ Now since $\eta (\phi) = \sum_I \phi^I \eta_I$ for $ \eta_I \in \Om^*(X)$ (where the $I$ are multi-indices), we may define for any $x \in X$ a distribution ${\mbox{$\cal S$}}_x$ with values in $\Lambda^* T^*_x X$ as follows: for any $y \in \liets$ we have $${\mbox{$\cal S$}}_x(y) = \intk [d \phi] \eminev{y}{ \phi } e^{i \om} \epinev{\mu(x)}{ \phi} \sum_I \eta_I \phi^I $$ \begin{equation} = \sum_I (i \partial/\partial y)^I \intk [d \phi] \epinev{ (\mu(x) - y)}{ \phi} e^{i \om} \eta_I \end{equation} $$ = (2 \pi)^s \sum_I (i \partial/\partial y)^I \delta(\mu(x) - y) e^{i \om} \eta_I. $$ Thus the distribution ${\mbox{$\cal S$}}_x(y)$, viewed as a distribution $ {\mbox{$\cal S$}} (x,y) $ on $X \times \liets$, is supported on $\{ (x , y) \in X \times \liets \, | \, \mu(x) = y \}. $ Hence for sufficiently small $y$, ${\mbox{$\cal S$}}(x,y) $ (viewed now as a function of $x$) is supported on $ {\mbox{$\cal O$}}$ (in the notation of Section 5) and we find that \begin{equation} \qeta(y) = \frac{\nusym^2(y)}{\htps} \int_{x \in X} {\mbox{$\cal S$}}(x, y) \end{equation} $$ = \frac{\nusym^2(y)}{ (2 \pi)^{s/2} } \intk [d \phi] \int_{P \times \lieks} \epinev{(\mu(x) - y)}{ \phi} e^{i \om} \eta(\phi) . $$ Now consider the restriction of $\eta$ to $\hk(P \times \lieks) \cong \hk(P)$ (where $P = \mu^{-1}(0)$ as in Section 5). Recall that there exists $\eta_0 \in \Om^*(P/K)$ such that $\eta - \pi^* q^*\eta_0 = D \gamma$ for some $\gamma$, where $D$ is the equivariant cohomology differential on $P \times \lieks$ and $\pi: P \times \lieks \to P \times \{0\}$ and $q: P \to \to P/K$ are the projection maps.\footnote{This is because the map $i \circ \pi: P \times \lieks \to P \times \lieks$ is homotopic to the identity by a homotopy through equivariant maps, where $i: P \times \{ 0 \} \to P \times \lieks$ is the inclusion map. Hence $i$ induces an isomorphism $i^*: \hk (P \times \lieks) \to \hk(P \times \{0\}). $} We then have that \begin{equation} \label{7.d3} Q^\eta (y) - \frac{\nusym^2(y)}{(2 \pi)^{s/2} } \intk [d \phi] \int_{x \in P \times \lieks} \epinev{(\mu(x) - y)}{\phi} e^{i \om} \pi^* q^*\eta_0 \end{equation} $$ \phantom{a} \phantom{a} = \frac{\nusym^2(y)}{(2 \pi)^{s/2} } \intk [d \phi] \int_{x \in P \times \lieks} \epinev{(\mu(x) - y)}{ \phi} e^{i \om} D \gamma \;\: {\stackrel{ {\rm def} }{=} } \;\: \frac{\nusym^2(y)}{(2 \pi)^{s/2}} \bigtriangleup. $$ But also $$ \epinev{(\mu(x) - y)}{ \phi} e^{i \om} D \gamma = D ( \epinev{(\mu(x) - y)}{ \phi} e^{i \om} \gamma ) $$ (since $D \phi_j = 0 $ and $D \bom = 0$). Hence $\bigtriangleup = \intk [ d \phi] \int_{P \times \lieks} d (\epinev{(\mu(x) - y)}{ \phi} e^{i \om} \gamma) $ (since for any differential form $\Psi$, the term $\iota_{\tilde{\phi}} \Psi$ in $D \Psi$ cannot contain differential forms of top degree in $x$). Using Stokes' Theorem and replacing $P \times \lieks$ by $P \times B(\lieks) $ where $B(\lieks)$ is a large ball in $\lieks$ with boundary $S(\lieks)$, we have that $$ \bigtriangleup = \intk [d \phi] \int_{P \times S(\lieks) } \epinev{(\mu(x) - y)}{ \phi} e^{i \om} \sum_I \gamma_I \phi^I$$ for $\gamma_I \in \Om^*(P \times \lieks)$. Thus we have $ \bigtriangleup = \sum_I (i \partial/\partial y)^I S_I(y) $ where $$ S_I (y) = \intk [d \phi] \int_{x \in P \times S(\lieks) } \epinev{(\mu(x) - y)}{ \phi} e^{i \om} \gamma_I. $$ Now we do the integral over $\phi$ to obtain $$ S_I(y) = (2 \pi)^{s} \int_{x \in P \times S(\lieks)} \delta (\mu(x) - y) e^{i \om} \gamma_I. $$ This is zero since the delta distribution is supported off $S(\lieks)$ (recall that $\mu(p, z) = z$ for $(p, z) \in P \times \lieks$). Hence we have that $\bigtriangleup = 0$, and so \begin{equation} \qeta(y) = \frac{\nusym^2(y)}{(2 \pi)^{s/2} } \intk [d \phi] \int_{(p, z) \in P \times \lieks} \epinev{(z - y)}{ \phi} e^{i \om} \pi^* \eta_0, \end{equation} and the rest of the proof is exactly the same as the proof of Theorem \ref{t4.3} which was for the case $\eta_0 = 1$. In particular the analogue of (\ref{5.6}) is \begin{equation} \label{5.6ext} Q^\eta_0(y) = i^s \nusym^2 (y) (2 \pi)^{s/2} \: \int_{\zloc} \eta_0 \exp (q^*\omr + \evab{F_\theta}{ y } ) \, \Om \, ; \end{equation} this equation shows that $Q_0^\eta$ is a polynomial (and in particular smooth) in $y$. \noindent{\em Proof of }{\bf(b):} This is a direct extension of the proof of Theorem \ref{t4.1}, with $Q$ and $Q_0$ replaced by $\qeta$ and $Q^\eta_0$. \noindent{\em Proof of }{\bf (c):} Since $\eta = \sum_J \eta_J \phi^J$, the abelian localization formula for $\pis (\eta e^{\iins \bom})$ yields \begin{equation} \label{7.6n} \nusym(\phi) \pis (\eta e^{\iins \bom}) (\phi) = \nusym(\phi) \sum_{F \in {\mbox{$\cal F$}}} \rfe(\phi), \phantom{bbbbb} \rfe(\phi) = \int_F \frac{i_F^* \eta( \phi) e^{i \bom(\phi)} }{e_F(\phi)}, \end{equation} $$ \phantom{bbbbb} = \sum_J \phi^J e^ {i \eva{\mu_T(F)}{ \phi} } \int_F \frac{i_F^* \eta_J e^{i \om} }{e_F(\mar \phi) } $$ $$ \phantom{a} = \nusym(\phi) \sum_J \phi^J \sum_{F \in {\mbox{$\cal F$}}, \alpha \in {\mbox{$\cal A$}}_F} \tfaj(\phi) , $$ \begin{equation} \label{7.06}\mbox{where} \phantom{a} \phantom{a} \tfaj(\phi) = (-1)^{k_F(\alpha) } \int_F i_F^* \eta_J e^{i \om} \tilde{c}_{F,\alpha} \frac{ \epinev{ \mu_T(F)}{ \phi } } {\prod_j \Bigl ( \bfjw (\mar \phi) \Bigr )^{{n}_\fj(\alpha)} } \end{equation} (and the $\tilde{c}_{F,\alpha}$ are as in (\ref{4.004})). Here, as in Section 4, we may form a distribution \begin{equation} \label{7.07}H(y) = D_\nusym \tilde{H}, \phantom{a} \tilde{H} = \sum_J (i \partial/\partial y)^J \sum_{F \in {\mbox{$\cal F$}}, \alpha \in {\mbox{$\cal A$}}_F} (-1)^{k_F(\alpha)} \Bigl ( \int_F i_F^* \eta_J e^{i \om} \tilde{c}_{F,\alpha} \Bigr ) H_{\bar{\gamma}_F(\alpha) } ( \mu_T(F) - y ), \end{equation} where the piecewise polynomial function $H_{\bar{\beta} }$ is as in Proposition \ref{p3.5}. Thus $\tilde{H}$ is of the form $\tilde{H}(y) = \sum_J D_J R_J (y)$ where the $R_J(y)$ are piecewise polynomial and the $D_J$ are differential operators. We also define the distribution \begin{equation} \label{7.07'} G(y) = \ft( \nusym \pis \eta e^{\iins \bom}) . \end{equation} Thus $\nusym(y) G(y) = Q^\eta (y)$. Then we may show that the distributions $G$ and $H$ are identical. For we apply Lemma \ref{l4} as before. We find by Proposition \ref{p3.5}(a) that $(\ft G)(\psi) $ and $(\ft H)(\psi) $ are identical off the hyperplanes $\bfj( \psi) = 0$, so the first hypothesis of Lemma \ref{l4} is satisfied. Moreover, $H$ is supported in a half space and \begin{equation} G(y) = \nusym(y) \fk (\pis \sum_J \eta_J \phi^J e^{\iins \bom} ) (y) \end{equation} $$ = (2 \pi)^s \nusym(y) \sum_J (i \partial/\partial y)^J \int_{x \in X} \eta_J \delta(\mu(x) - y) e^{i \om}, $$ so $G$ is supported in the compact set $\mu(X) \cap \liets$ and hence $H - G$ is supported in a half space. Thus the second hypothesis of Lemma \ref{l4} is also satisfied and we may conclude that $G = H$, which completes the proof. $\square$ { \setcounter{equation}{0} } \section{Relations in the cohomology ring of symplectic quotients} In this section we shall prove a formula for the evaluation of cohomology classes from $\hk(X) $ on the fundamental class of $\xred$, and apply it to study the cohomology ring $H^*(\xred)$ in two examples. \newcommand{\gamp}{\Gamma(P) } \newcommand{\npl}{ {n_+} } We shall prove the following theorem: \begin{theorem}\label{t8.1} Let $\eta \in \hk(X)$ induce $\eta_0 \in H^*(\xred)$. Then we have \begin{equation} \label{8.00} \eta_0 e^{i \om_0} [\xred] = \frac{(-1)^\npl }{ (2 \pi)^{s-l} |W| \,{\rm vol}\, (T) } \treso \Biggl ( \: \nusym^2 (\psi) \sum_{F \in {\mbox{$\cal F$}}} \rfe(\psi) [d \psi] \Biggr ), \end{equation} where $$ \phantom{a} \rfe(\psi) = e^{i \mu_T(F) (\psi) } \int_F \frac{i_F^* (\eta(\psi) e^{i \omega}) }{e_F(\mar \psi) } . $$ \end{theorem} Here $s$ and $l$ are the dimensions of $K$ and its maximal torus $T$, and $\nusym(\psi) = \prod_{\gamma > 0 } \gamma(\psi)$ where $\gamma$ runs over the positive roots of $K$; the number of positive roots $(s - l)/2$ is denoted by $\npl$. If $F \in {\mbox{$\cal F$}}$ (where ${\mbox{$\cal F$}}$ denotes the set of components of the $T$ fixed point set) then $i_F: F \to X$ is the inclusion of $F$ in $X$ and $e_F $ is the equivariant Euler class of the normal bundle to $F$ in $X$. The quantity $\treso (\Omega)$ will be defined below (Definition \ref{d8.5n}). The definition of $\treso (\Omega)$ will depend on the choice of a cone $\lasub$, a test function $\testf$, and a ray in $\liets$ specified by a parameter $ \zray \in \liets$ -- but in fact if the form $\Omega$ is sufficiently well behaved (as is the case in (\ref{8.00})) then the quantity $\treso (\Omega) $ will turn out to be independent of these choices. (See Propositions \ref{p8.6n}, \ref{p8.7n} and \ref{p8.8n}.) In the case of $K = SU(2) $ the result of Theorem \ref{t8.1} is as follows: \begin{corollary} \label{c8.2} Let $K = SU(2)$, and let $\eta \in \hk(X)$ induce $\eta_0 \in H^*(\xred)$. Then the cohomology class $\eta_0 e^{i \om_0} $ evaluated on the fundamental class of $\xred$ is given by the following formula: $$ \eta_0 e^{i \om_0} [\xred] = -\frac{ 1}{2} {\rm Res}_0 \Biggl ( \psi^2 \sum_{F \in {\mbox{$\cal F$}}_+ } \rfe(\psi) \Biggr ), \phantom{a} {\rm where} \phantom{a} \rfe (\psi) = e^{i \mu_T(F)(\psi) } \int_F \frac{i_F^* \eta(\psi) e^{i \omega} }{e_F( \psi) }. $$ Here, ${\rm Res_0}$ denotes the coefficient of $1/\psi$, and ${\mbox{$\cal F$}}_+$ is the subset of the fixed point set of $T = U(1)$ consisting of those components $F$ of the $T$ fixed point set for which $\mu_T(F) > 0 $. \end{corollary} An important special case (cf. Witten \cite{tdg}, Section 2.4) is as follows: \begin{corollary} \label{c8.00} Let $\eta \in \hk(X)$ induce $\eta_0 \in H^*(\xred)$, and let $\Theta \in H^4(\xred)$ be induced by the polynomial function $- \inpr{\phi, \phi}/2$ of $\phi$, regarded as an element of $H^4_K$. Then if $\epsilon > 0$ we have \begin{equation} \label{8.010} \eta_0 e^{\epsilon \Theta} e^{i \om_0} [\xred] = \frac{(-1)^\npl }{(2 \pi)^{(s-l)} |W| \,{\rm vol}\, (T) } \sum_{m \ge 0} \frac{1}{m!} \treso \Biggl ( (- \epsilon |\psi|^2/2)^m \: \nusym^2 (\psi) \sum_{F \in {\mbox{$\cal F$}}} \rfe(\psi) [d \psi] \Biggr ), \end{equation} $$ \phantom{bbbbb} {\rm where} \phantom{bbbbb} \rfe (\psi) = e^{i \mu_T(F) (\psi) } \int_F \frac{i_F^* (\eta(\psi) e^{i \om}) }{e_F(\mar \psi) } . $$ \end{corollary} We now give a general definition (Definition \ref{d8.5n}) of $\ress h [d \psi]$ when $h $ is a meromorphic function on an open subset of $\liet \otimes {\Bbb C }$ satisfying certain growth conditions at infinity. For the restricted class of meromorphic forms of the form $e^{i \lambda(\psi)} [d \psi]/\prod_j \beta_j (\psi), $ Definition \ref{d8.5n} implies the existence of an explicit procedure for computing these residues by successive contour integrations, which is outlined in Proposition \ref{p8.4}. Our definition of the residue will be based on the following well-known result (\cite{hor}, Theorem 7.4.2 and Remark following Theorem 7.4.3): \begin{prop} \label{cty} (i) Suppose $u$ is a distribution on $\liets$. Then the set $\gu = \{ \xi \in \liet: \: e^{(\cdot, \xi) } u $ is a tempered distribution$\}$ is convex. (Here, $(\cdot, \cdot)$ denotes the pairing between $\liet$ and $\liets$.) \noindent (ii) If the interior $\guo$ of $\gu$ is nonempty, then there is an analytic function $\hat{u} $ in $\liet + i \guo$ such that the Fourier transform of $e^{(\cdot, \xi) } u $ is $\hat{u} (\cdot + i \xi)$ for all $\xi \in \guo$. \noindent (iii) For every compact subset $M$ of $\guo$ there is an estimate \begin{equation} \label{bds} | \hat{u} (\zeta) | \le C (1 + |\zeta|)^N, \phantom{a} {\rm Im} (\zeta) \in M. \end{equation} \noindent (iv) Conversely if $\Gamma$ is an open convex set in $\liet$ and $h$ is a holomorphic function on $\liet + i \Gamma$ with bounds of the form (\ref{bds}) for every compact $M \subset \Gamma$, then there is a distribution $u$ on $\liets$ such that $e^{(\cdot, \xi)} u $ is a tempered distribution and has Fourier transform $h (\cdot + i \xi)$ for all $\xi \in \Gamma$. \noindent (v) Finally, if $u$ itself is tempered then the Fourier transform $\hat{u}$ is the limit (in the space ${\mbox{$\cal S$}}'$ of tempered distributions) of the distribution $ \psi \mapsto \hat{u} (\psi + i t \theta)$ as $t \to 0^+$, for any $\theta \in \guo$. \end{prop} \begin{definition} \label{d8.5n} Let $\lasub$ be a (proper) cone in $\liet$. Let $h$ be a holomorphic function on $\liet - i {\rm Int} (\Lambda) \subseteq \liet \otimes {\Bbb C } $ such that for any compact subset $M$ of $\liet - i {\rm Int} (\lasub) $ there is an integer $N \ge 0 $ and a constant $C$ such that $|h (\zeta)| \le C (1 + |\zeta|)^N$ for all $\zeta \in M$. Let $\testf: \lieks \to {\Bbb R }$ be a smooth invariant function with compact support and strictly positive in some neighbourhood of $0$, and let $\htestf = \fk \testf: $ $\liek \to {\Bbb C }$ be its Fourier transform. Let $\htestfe(\phi) = \htestf (\epsilon \phi)$ so that $(\fk \htestfe)(z) = \testfe (z) = \epsilon^{-s} \testf (z/\epsilon). $ Assume $\htestf(0) = 1. $ Then we define \begin{equation} \label{8.1n} \reslch (h [d \psi]) = \lim_{\epsilon \to 0^+} \frac{1}{(2 \pi i )^l } \int_{\psi \in \liet - i \xi} \htestf (\epsilon \psi) h (\psi) [ d\psi] \end{equation} where $\xi$ is any element of $\Lambda$. \end{definition} By the Paley-Wiener theorem (\cite{hor}, Theorem 7.3.1), for any fixed $\xi \in \liek$ the function $\htestf (\psi - i \xi) = \fk(e^{- \inpr{\xi, \cdot} } \testf)(\psi) $ is rapidly decreasing since $\fk \htestf = \testf$ is smooth and compactly supported. Hence the integral (\ref{8.1n}) converges. Now the function $\htestf$ extends to a holomorphic function on $\liek \otimes {\Bbb C }$ and in particular on $\liet \otimes {\Bbb C }$ (Proposition \ref{cty}), and by assumption $h$ extends to a holomorphic function on $\liet - i \, {\rm Int} (\lasub)$; hence, by Cauchy's theorem, the integral (\ref{8.1n}) is independent of $\xi \in {\rm Int} (\lasub)$. The independence of $\reslch (\Omega)$ of the choices of $\lasub$ and $\testf$ when $\Omega$ is sufficiently well behaved is established by the next results. \begin{prop} \label{p8.6n} Let $h: \liek \to {\Bbb C }$ be a $K$-invariant function. Assume that $\fk h$ is compactly supported; it then follows that $h: \liek \to {\Bbb C }$ is smooth (\cite{hor}, Lemma 7.1.3) and extends to a holomorphic function on $\liek \otimes {\Bbb C }$ (Proposition \ref{cty}). Then $\reslch ( h [d \psi] ) $ is independent of the cone $\lasub$. \end{prop} \Proof As above, define $\testfe (z) = \epsilon^{-s} \testf (z/\epsilon),$ so that $\htestfe (\phi) = \htestf(\epsilon \phi). $ The function $h$ extends in particular to a holomorphic function on $\liet \otimes {\Bbb C }$, and by the remarks after Definition \ref{d8.5n}, the function $\htestfe$ also extends to a holomorphic function on $\liet \otimes {\Bbb C }$. Hence Cauchy's theorem shows that for any choice of the cone $\lasub$, \begin{equation} \label{8.4n} \reslch ( h(\psi) \dps ) = \lim_{\epsilon \to 0^+} \frac{1}{(2 \pi i)^l} \intt \htestf (\epsilon\psi) h(\psi) [d \psi]. \phantom{bbbbb} \square \end{equation} \noindent{\em Remark:} If $h$ is as in the statement of Proposition \ref{p8.6n}, then $\nusym^2 h$ also satisfies the hypotheses of the Proposition, so $\reslch(\nusym^2 h [d \psi] ) $ is also independent of the cone $\lasub$. The following is a consequence of Proposition \ref{cty}: \begin{prop} \label{p8.7n} Let $u: \liets \to {\Bbb C }$ be a distribution, and assume the set $\gu$ defined in Proposition \ref{cty} contains $- {\rm Int} (\Lambda)$. Thus $h = \ft u$ is a holomorphic function on $\liet - i {\rm Int} \Lambda$, satisfying the hypotheses in Definition \ref{d8.5n}. Suppose in addition that $u$ is smooth at $0$. Then $\reslch (h \dps) $ is independent of the test function $\chi$, and equals $i^{-l} u(0)/(2 \pi)^{l/2} $. \end{prop} We shall be dealing with functions $h: \liek \to {\Bbb C }$ whose Fourier transforms are smooth at $0$ but which are sums of other functions not all of whose Fourier transforms need be smooth at $0$. For this reason we must introduce a small generic parameter $\zray \in \liets$ where all the functions in this sum are smooth. More precisely we make the following \begin{definition} \label{d8.7n} Let $\lasub, $ $\testf$ and $h$ be as in Definition \ref{d8.5n}. Let $\zray \in \liets$ be such that the distribution $\ft h$ is smooth on the ray $t \zray$ for $t \in (0, \delta)$ for some $\delta > 0 $, and suppose $(\ft h)(t \rho) $ tends to a well defined limit as $t \to 0^+$. Then we define \begin{equation} \label{8.6n} \resrlch (h \dps) = \lim_{t \to 0^+} \reslch \Bigl ( h (\psi ) e^{i t \zray (\psi) } \dps \Bigr ). \end{equation} \end{definition} Under these hypotheses, $\resrlch ( h \dps) $ is independent of $\testf$ (by Proposition \ref{p8.7n}), but it may depend on the ray $\{ t \zray: t \in {\Bbb R }^+ \}. $ However we have by Proposition \ref{p8.7n} \begin{prop} \label{p8.8n} Suppose $\ft h $ is smooth at $0$. Then the quantity $\resrlch ( h \dps)$ satisfies $\resrlch (h \dps) = \reslch ( h \dps) $. \end{prop} \noindent{\em Remark:} In the light of Propositions \ref{p8.6n}, \ref{p8.7n} and \ref{p8.8n}, it makes sense to write ${\rm Res} (\Omega) $ for $\resrlch (\Omega) $ when $\Omega = \nusym^2 h \dps $ for a $K$-invariant function $h: \liek \to {\Bbb C }$ for which $\fk h$ is compactly supported and $\ft (\nusym^2 h)$ is smooth at $0$. In the proof of Theorem \ref{t8.1} which we are about to give, we shall check the validity of these hypotheses for the form $\Omega$ which appears in the statement of the Theorem. We now give the proof of Theorem \ref{t8.1}. We shall first show \begin{prop} \label{p7.6} {\noindent (a)} The distribution $\fk(\pisk \eta e^{i \bom} ) (\zp) $ defined for $\zp \in \liek$ is represented by a smooth function for $\zp$ in a sufficiently small neighbourhood of $0$. {\noindent (b) } We have \begin{equation} \label{7.001} \eta_0 e^{i \om_0} [\xred] = \frac{1}{(2 \pi)^{s/2}i^s \,{\rm vol}\, (K) } F_K (\pis \eta e^{\iins \bom}) (0), \end{equation} \begin{equation} \label{7.002} \phantom{ \eta_0 e^{i \om} [\xred]} = \frac{(2 \pi)^{l/2} }{(2 \pi)^{s} |W| \,{\rm vol}\, (T)i^s } F_T (\nusym^2 \pis \eta e^{\iins \bom}) (0). \end{equation} \end{prop} \noindent{\em Proof of (a):} To evaluate $\fk(\pis \eta e^{i \bom} ) (0)$, we introduce a test function $\testf: \lieks \to {\Bbb R }^+$ which is smooth and of compact support, and for which $(\fk \testf) (0) = 1/(2 \pi)^{s/2}. $ We define $\testfe(z) = \epsilon^{-s} \testf(z/\epsilon) $; as $\epsilon \to 0 $, the functions $\testfe $ tend to the Dirac delta distribution on $\lieks$ (in the space ${\mbox{$\cal D$}}'$ of distributions on $\lieks$). Then we have \begin{equation} (\fk \testfe) (\phi) = (\fk \testf) (\epsilon \phi). \end{equation} Now to evaluate $\fk (\pis \eta e^{i \bom} ) (\zp)$, we integrate it against the sequence of test functions $\testfe$: \begin{equation} \fk ( \pis \eta e^{i \bom} ) (\zp) = \lim_{\epsilon \to 0^+} i^s \je (\zp) \end{equation} where \begin{equation} i^s \je (\zp)= \tintk [d z] \fk (\pis \eta e^{i \bom} ) (z) \chi_\epsilon (z - \zp) \end{equation} $$ \phantom{bbbbb} = \intk [d \phi] \int_{x \in X} \eta(\phi) e^{i \om} e^{i (\mu(x) - \zp) (\phi) } \htestfe(\phi) $$ (by Parseval's Theorem). Now because $\testfe$ is smooth and of compact support, the Paley-Wiener Theorem (Theorem 7.3.1 of \cite{hor}) implies that $\htestfe$ is rapidly decreasing. So we may use Fubini's theorem to interchange the order of integration and get \begin{equation} \label{7.t1} i^s \je(\zp) = \int_{x \in X } e^{i \om} \int_{ \phi \in \liek} [d \phi] \eta(\phi) \htestfe(\phi) e^{i (\mu(x) - \zp) (\phi)} \end{equation} \begin{equation} \label{7.t2} \phantom{bbbbb} = (2 \pi)^{s/2} \int_{x \in X} e^{i \om} \eta( - i \partial/\partial z) \testfe(z)|_{z = \mu(x) - \zp}. \end{equation} As $\epsilon \to 0$, $\testfe$ is supported on an arbitrarily small neighbourhood of $\mu^{-1}(0)$. Thus, because of Proposition \ref{p5.2}, the integral may be replaced by an integral over $P \times \lieks$: \begin{equation} i^s \je(z') = (2 \pi)^{s/2} \int_{(p,z) \in P \times \lieks} e^{i \om} \eta( - i \frac{ \partial }{\partial z}) \testfe(z - \zp) \end{equation} Now by the same argument as given in the proof of Theorem \ref{t7.1}(a), there is $\eta_0 \in \Omega^*(P/K) $ such that \begin{equation} \eta - \pi^* \eta_0 = D \gamma \phantom{a} \mbox{for some $\gamma$, } \end{equation} where $D$ is the equivariant cohomology differential on $P \times \lieks$ and $\pi: P \times \lieks \to P \to P/K$ is the projection map. By the argument given after (\ref{7.d3}), we have \begin{equation} i^s \je(\zp) - (2 \pi)^{s/2} \int_{(p,z) \in P \times \lieks} e^{i \om} (\pi^* \eta_0) \testfe(z - \zp) \end{equation} $$ \phantom{bbbbb} = \int_{(p,z) \in P \times \lieks} e^{i \om} \int_{ \phi \in \liek} [d \phi] D \gamma(\phi) \htestfe(\phi) e^{i (z - \zp)( \phi) } $$ $$ = \int_{ \phi \in \liek} [d \phi] \htestfe( \phi) \int_{ (p,z) \in P \times \lieks } e^{i \om} D \gamma (\phi) e^{i (z - \zp) (\phi) } \;\: {\stackrel{ {\rm def} }{=} } \;\: \deleps. $$ But \begin{equation} e^{i \om} e^{i (z-\zp)(\phi) } D \gamma = D (e^{i \om} e^{i (z - \zp) (\phi) } \gamma(\phi) ). \end{equation} Hence \begin{equation} \deleps = \intk [d \phi] \htestfe (\phi) \int_{(p,z) \in P \times \lieks} d ( e^{i \om} e^{i (z - \zp) (\phi) } \gamma(\phi) ) \end{equation} (since for any differential form $\Psi$, the term $\iota_{\tilde{\phi}} \Psi$ in $D \Psi$ cannot contain differential forms of top degree in $x$). Replacing $P \times \lieks$ by $P \times B_R(\lieks)$ where $B_R (\lieks)$ is a large ball in $\lieks$ with boundary $S_R (\lieks)$, we have that \begin{equation} \deleps = \lim_{R \to \infty} \intk [d \phi] \htestfe( \phi) \int_{(p,z) \in P \times B_R (\lieks) } d \Bigl ( e^{i \om} e^{i (z - \zp) (\phi) } \gamma( \phi) \Bigr ) . \end{equation} We then interchange the order of integration to get \begin{equation} \deleps = \lim_{R \to \infty} \int_{(p,z) \in P \times B_R (\lieks) } d \left ( e^{i \om} \intk [d \phi] \htestfe ( \phi) e^{i (z - \zp) (\phi) } \gamma( \phi) \right ) \end{equation} $$ \phantom{bbbbb} = \lim_{R \to \infty} \int_{(p,z) \in P \times B_R(\lieks) } d \Bigl (e^{i \om} \gamma( \frac{\partial}{\partial z}) \testfe (z - \zp) \Bigr ) . $$ $$ \phantom{bbbbb} = \lim_{R \to \infty} \int_{(p,z) \in P \times S_R(\lieks) } \Bigl (e^{i \om} \gamma( \frac{\partial}{\partial z}) \testfe (z - \zp) \Bigr ) \phantom{a} \mbox{by Stokes' Theorem}. $$ This limit equals $0$ for sufficiently small $\zp$ since $\testfe$ is compactly supported on $\lieks$. Hence $\deleps = 0 $. Finally we obtain using the expression for $\omega$ given in Proposition \ref{p5.2} \begin{equation} i^s \je(\zp) = (2 \pi)^{s/2} \int_{(p,z) \in P \times \lieks} e^{i \om} (\pi^* \eta_0) \testfe (z - \zp), \end{equation} \begin{equation} \phantom{bbbbb} = ( 2 \pi)^{s/2} \int_{ (p,z) \in P \times \lieks} e^{i \pi^* \om_0} (\pi^* \eta_0) e^{i \theta(dz)} e^{i d \theta(z) } \testfe (z - \zp). \end{equation} Thus we have as in Section 5 \begin{equation} \label{8.26} \lim_{\epsilon \to 0 } \je (\zp) = ( 2 \pi)^{s/2} \int_{(p,z) \in P \times \lieks} e^{i \pi^* \om_0} (\pi^* \eta_0) \Om [dz] e^{i F_\theta (z)} \delta(z - \zp) \end{equation} $$ = ( 2\pi)^{s/2} \int_P e^{i \pi^* \om_0} (\pi^* \eta_0) e^{i F_\theta(\zp) } \Om $$ where $\Omega$ is the differential form introduced after (\ref{5.4}). This shows in particular that $(\fk \rbare)(\zp)$ (where $\rbare (\phi) = \pis (\eta e^{i \bom})( \phi) $) is a polynomial in $\zp$ for small $\zp$, and hence is smooth in $\zp$ for $\zp$ sufficiently close to $0$. This completes the proof of (a). When $\zp = 0 $, equation (\ref{8.26}) becomes $$\lim_{\epsilon \to 0} \je (0) = i^{-s} F_K (\pis \eta e^{\iins \bom})(0) = {(2 \pi)^{s/2} \,{\rm vol}\, (K) } \eta_0 e^{i \om_0} [\xred], $$ which proves (\ref{7.001}). Using Lemma \ref{l3.1} again, we have for any $K$-invariant function $f$ on $\liek$ that \begin{equation} \label{7.003} \frac{(2 \pi)^{s/2} }{ \,{\rm vol}\, (K) } (F_K f) (0) = \frac{( 2 \pi)^{l/2} }{|W| \,{\rm vol}\, (T) } F_T(f \nusym^2) (0). \end{equation} Combining this with (\ref{7.001}) we obtain $$ \eta_0 e^{i \om_0}[\xred] = \frac{(2 \pi)^{l/2} }{(2 \pi)^{s} |W| \,{\rm vol}\, (T) i^s } F_T (\nusym^2 \pis \eta e^{\iins \bom}) (0), $$ which is (\ref{7.002}). $\square$ Theorem \ref{t8.1} follows from Proposition \ref{p7.6} and Proposition \ref{p8.7n} by applying Theorem \ref{t2.1} to decompose $r^{{\eta}} = \pis (\eta e^{i \bom} ) $ as a sum of meromorphic functions $\rfe$ on $\liet \otimes {\Bbb C } $ corresponding to the components $F$ of the fixed point set of $T$. We now complete the proof of this theorem. \noindent{\em Proof of Theorem \ref{t8.1}:} As in (\ref{7.6n}), the abelian localization formula yields $ r^{{\eta} } (\psi) = \sum_{F \in {\mbox{$\cal F$}}} \rfe (\psi), $ where $$ \rfe(\psi) = e^{i \mu_T(F) (\psi) } \int_F \frac{i_F^* \eta(\psi) e^{i \om} } {e_F(\psi) }. $$ Now the distribution $(\fk r^{{\eta} } ) (\phi) $ is represented by a smooth function near $0$ (Proposition \ref{p7.6} (a)); also, the distribution $\ft(\nusym^2 r^{{\eta} } ) $ $ = D_{\nusym} \ft (\nusym \rbare) $ is represented by a smooth function near $0$ since $\ft (\nusym r^{{\eta} } ) = \nusym \fk r^{{\eta} } $ (Lemma \ref{l3.3}) and $\fk r^{{\eta} } $ is smooth near $0$ (Proposition \ref{p7.6}(a)). We choose $ \zray \in \liets$ so that the distribution $\ft \rfe$ is smooth along the ray $t \zray, t \in (0, \delta) $, for all $F$ and sufficiently small $\delta > 0 $, and that this distribution tends to a well defined limit as $t \to 0^+$: this is possible because the $\rfe (\psi)$ are sums of terms of the form $e^{i \mu_T(F) (\psi) } / \prod_j \bfj(\psi)^{n_j} $ so their Fourier transforms $\ft \rfe$ are piecewise polynomial functions of the form $H_\barb (y) $ (see Proposition \ref{p3.5}). These functions are smooth on the set $U_\barb$ consisting of all points $y$ where $y$ is not in the cone spanned by any subset of the $\bfjw$ containing less than $l$ elements. Thus $$\lim_{t \to 0^+} (2 \pi)^{-l/2} i^{-l} \ft \rfe(t \zray) = \resrlch (\rfe \dps) $$ by Definition \ref{d8.5n} and Proposition \ref{p8.7n}. It follows that \begin{equation} \label{8.7n} (2 \pi)^{-l/2} i^{-l} \ft (\nusym^2 \rbare) (0) = \resrlch (\nusym^2 \rbare \dps ) \end{equation} \begin{equation} \label{8.8n} \phantom{bbbbb} = \sum_{F \in {\mbox{$\cal F$}}} \resrlch (\nusym^2 \rfe). \end{equation} The residue in (\ref{8.7n}) is independent of $\testf$, $\lasub$ and $\zray$ by Propositions \ref{p8.6n}, \ref{p8.7n} and \ref{p8.8n}. The residues in (\ref{8.8n}) are independent of $\testf$ by Proposition \ref{p8.7n}, but they may depend on $\zray$ and $\lasub$. To conclude the proof of Theorem \ref{t8.1} we note that (\ref{7.002}) gives \begin{equation} \eta_0 e^{i \om} [\xred] = \frac{(2 \pi)^{l/2} }{(2 \pi)^{s} |W| \,{\rm vol}\, (T)i^s } F_T (\nusym^2 \pis \eta e^{\iins \bom}) (0), \end{equation} $$ = \frac{i^l }{(2 \pi)^{s-l} |W| \,{\rm vol}\, (T)i^s } \reso(\rbare(\psi) \nusym^2(\psi) [ d\psi]) \phantom{a} \mbox{(by Proposition \ref{p8.7n})} $$ $$ = \frac{-1)^\npl }{ (2 \pi)^{s-l} |W| \,{\rm vol}\, (T) } \reso \Biggl ( \sum_{F \in {\mbox{$\cal F$}}} \rfe(\psi) \nusym^2 (\psi) [d \psi] \Biggr ) $$ as claimed. $\square$ \noindent{\em Proof of Corollary 8.2:} In a normalization where $ \,{\rm vol}\, (T) = 1$, the factor $\nusym(\psi) $ becomes $2 \pi \psi$. This gives $$ \frac{1}{(2 \pi)^{s-l}}\treso \Biggl ( \: \nusym^2 (\psi) \sum_{F \in {\mbox{$\cal F$}}} \rfe(\psi) \dps \Biggr ) = \reslch ( \psi^2 \sum_{F \in {\mbox{$\cal F$}}} \rfe(\psi) [d \psi] ). $$ Each term $\rfe (\psi) $ is a sum of terms of the form $\tau_\alpha (\psi) = c_{\alpha} \psi^{-n_\alpha} e^{i \mu(F) \psi} $ for some constants $c_\alpha$ and integers $n_\alpha$. By (\ref{8.1n}), the residue is given by \begin{equation} \label{8.001} \reslch (h \dps) = \lim_{ \epsilon \to 0^+ } \frac{1} {2 \pi i} \int_{ \psi \in {\Bbb R } - i \xi} \htestf (\epsilon \psi) h(\psi) \dps, \end{equation} where we choose $\xi$ to be in the cone $\lasub = {\Bbb R }^+$. Proposition \ref{cty} (i), (ii) shows that the function $\htestf: {\Bbb R } \to {\Bbb C }$ extends to an entire function on ${\Bbb C }$. We may now decompose the integral in (\ref{8.001}) into terms corresponding to the $\tau_\alpha$. If $n_\alpha > 0 $, we complete each such integral over ${\Bbb R } - i \xi$ to a contour integral by adding a semicircular curve at infinity, which is in the upper half plane if $\mu(F) > 0 $ and in the lower half plane if $\mu(F) < 0 $. This choice of contour is made so that the function $\tau_\alpha (\psi) $ is bounded on the added contours, so the added semicircular curves do not contribute to the integral. Since only the contours corresponding to values of $F$ for which $\mu(F) > 0 $ enclose the pole at $0$, application of Cauchy's residue formula now gives the result. A similar argument establishes that the terms $\tau_\alpha $ for which $n_\alpha \le 0 $ contribute $0$ to the sum.\footnote{The formula obtained by choosing $\lasub = {\Bbb R }^-$ is in fact equivalent to the formula we have obtained using the choice $\lasub = {\Bbb R }^+$. This can be seen directly from the Weyl invariance of the function $\rbare$, where the action of the Weyl group takes $\psi$ to $- \psi$ and so converts terms with $\mu(F) > 0 $ to terms with $\mu(F) < 0 $.} $\square$ \noindent{\em Remarks:} (a) The quantity $\resrlch (\nusym^2 \rfe \dps ) $ depends on the cone $\lasub$ for each $F$; however, it follows from Proposition \ref{p8.6n} that the sum $\sum_{F \in {\mbox{$\cal F$}}} \resrlch ( \nusym^2 \rfe \dps ) $ is independent of $\lasub$. } \noindent (b) Let ${\mbox{$\cal F$}}_\lasub$ be the set of those $F \in {\mbox{$\cal F$}}$ for which $\mu_T(F)$ lies in the cone $\dcf = \{ \sum_j s_j \bfjw: s_j \ge 0 \} $ (defined in Section 4) spanned by the $\bfjw$. Then by Proposition \ref{p8.4} (iii), $\resrlch ( \nusym^2 \rfe ) = 0 $ if $F \notin {\mbox{$\cal F$}}_\lasub$, so in fact $\resrlch (\nusym^2 \rbare \dps) = \resrlch \sum_{F \in {\mbox{$\cal F$}}_\lasub} ( \nusym^2 \rfe \dps ) $. \noindent (c) Finally, it follows from Proposition \ref{p8.4} that if we replace the symplectic form $\omega$ and the moment map $\mu$ by $\delta \omega$ and $\delta \mu$ (where $\delta > 0 $) and then let $\delta $ tend to $0$, we obtain an expression where $\mu$ and $\omega $ appear only in determining the set ${\mbox{$\cal F$}}_\lasub$ indexing terms which yield a nonzero contribution. We now restrict ourselves to the special case when $\Omega$ is of the form $$\Omega_\lambda(\psi) = e^{i \lambda(\psi)} [d \psi]/\prod_{j = 1}^N \beta_j (\psi).$$ If $\beta_j \in \lasub $ then the distribution $\resrlch (\Omega_\lambda) $ is just (up to multiplication by a constant) the piecewise polynomial function $H_\barb(\lambda)$ from Proposition \ref{p3.5}. We shall now give a Proposition which gives a list of properties satisfied by the residues $\reso (\Oma{\lambda})$: these properties in fact characterize the residues uniquely and enable one to compute them. \newcommand{\pee}{P} \begin{prop} \label{p8.4} Let $\xi \in \liet$ and suppose $\beta_1, \dots, \beta_N \in \liets$ are all in the dual cone of a cone $\lasub \in \liet$. Denote by $\pee: \liet \to {\Bbb R } $ the function $\pee(\psi) = \prod_j \beta_j(\psi)$. Suppose $\lambda \in U_\barb \subset \liets$ (see Proposition \ref{p3.5}), and define $\Oma{\lambda}(\psi) = e^{i \lambda(\psi)}[d \psi]/\pee(\psi) $. Then we have \begin{description} \item[(i)] $\reso (\psi_k \psi^J \Oma{\lambda}) = (- i \partial/\partial \lambda_k) \reso (\psi^J \Oma{\lambda}). $ \item[(ii)] $(2 \pi i)^l \reso (\Oma{\lambda}) = i^N H_\barb (\lambda)$, where $H_\barb$ is the distribution given in Proposition \ref{p3.5}. (Recall we have assumed that $\beta_j $ is in the dual cone of $ \lasub $ for all $j$.) \item[(iii)] $\reso (\Oma{\lambda}) = 0 $ unless $\lambda$ is in the cone $C_\barb$ spanned by the $\beta_j$. \item[(iv)] $$\lim_{s \to 0^+} \reso \Bigl ( \Oma{s \lambda} \psi^J \Bigr ) = 0$$ unless $ N - |J| = l$. \item[(v)] $$\lim_{s \to 0^+} \reso \Bigl ( \Oma{s \lambda} \psi^J \Bigr ) = 0 $$ if the monomials $\beta_j$ do not span $\liets$. \item[(vi)] If $\beta_1, \dots, \beta_l$ span $\liets$ and $\lambda = \sum_j \lambda^j \beta_j$ with all $\lambda^j > 0$, then $$ \lim_{s \to 0^+} \reso \Bigl ( \frac{ e^{i s \lambda(\psi) } [d \psi] }{\beta_1(\psi) \dots \beta_l(\psi) } \Bigr ) = \frac{1}{\det \barb } , $$ where $\det \barb $ is the determinant of the $l $ by $l$ matrix whose columns are the coordinates of $\beta_1, \dots, \beta_l$ written in terms of any orthonormal basis of $\liet$. \item[(vii)] $$ \reso \Bigl ( \frac{e^{i \lambda(\psi) } \psi^J } {\pee(\psi)} [d \psi] \Bigr ) = \sum_{m \ge 0} \lim_{s \to 0^+} \reso \Bigl ( \frac{ (i \lambda(\psi))^m e^{i s \lambda(\psi) } \psi^J } {m! \pee(\psi)} [d \psi]\Bigr ). $$ \end{description} \end{prop} \noindent{\em Remark:} The limits in Proposition \ref{p8.4} are not part of the definition of the residue map: rather these limits are described in order to specify a procedure for computing the piecewise polynomial function $H_\barb (\lambda) = (2 \pi i)^l i^{-N} \reso (\Om_\lambda). $ (See the example below.) Proposition \ref{p8.4} (ii) identifies $H_\barb (\lambda)$ with an integral over $\liet$, which may be completed to an appropriate contour integral: the choice of contour is determined by the value of $\lambda$, and requires $\lambda$ to be a nonzero point in $U_\barb$. Proposition \ref{p8.4} (vii) says that one may compute $H_\barb$ by expanding the numerator $e^{i \lambda(\psi)}$ in a power series, but only provided one keeps a factor $e^{is \lambda(\psi)}$ in the integrand (for small $s > 0$) in order to specify the contour. The limits in Proposition \ref{p8.4} (iv)-(vii) exist because according to Proposition \ref{p8.4} (i) and (ii), they specify limits of derivatives of polynomials on subdomains of $U_\barb$, as one approaches the point $0$ in the boundary of $U_\barb$ along the fixed direction $s \lambda$ as $s \to 0$ in ${\Bbb R }^+$. \noindent{\em Proof of (i):} This follows directly from Definition \ref{d8.5n}. \noindent{\em Proof of (ii):} This follows because $(2 \pi i)^l i^{- N} \reso(\Oma{\lambda})$ is the fundamental solution $E(\lambda)$ of the differential equation $P(\partial/\partial \lambda)E(\lambda) = \delta_0$ with support in a half space containing the $\beta_j$. (See \cite{abg} Theorem 4.1 or \cite{hor2} Theorem 12.5.1.) But this fundamental solution is given by $H_\barb$ (see Proposition \ref{p3.5}(c)). \noindent{\em Proof of (iii):} This is an immediate consequence of (i) and (ii), in view of Proposition \ref{p3.5}(a).\footnote{Alternatively there is the following direct argument, which was pointed out to us by J.J. Duistermaat. We recall that $$ \reso (\Omega_\lambda) = \frac{1}{(2 \pi i)^l} \intt \frac{ e^{i \lambda(\psi - i \xi) }}{P(\psi - i \xi) } \, [d \psi], $$ which is defined and independent of $\xi$ for $\xi \in (C_\barb)^* $ (see the discussion after Definition \ref{d8.5n}). Hence we may replace $\xi $ by $t \xi $ for any $t \in {\Bbb R }^+$: \begin{equation} \label{resrec} \reso (\Omega_\lambda) = \frac{1}{(2 \pi i)^l} \intt \frac{ e^{i \lambda(\psi - i t\xi) }}{P(\psi - i t \xi) } \, [d \psi]. \end{equation} Taking the limit as $t \to \infty$, we see that $\reso (\Omega_\lambda) = 0 $ if $ \lambda(\xi) < 0 $, because of the factor $e^{ t \lambda(\xi) } $ that appears in the numerator of (\ref{resrec}). Since this holds for all $\xi \in (C_\barb)^*$, $\reso (\Omega_\lambda)$ is only nonzero when $\lambda(\xi) \ge 0 $ for all $\xi \in (C_\barb)^*$, in other words when $\lambda \in C_\barb$.} \noindent{\em Proof of (iv):} By (i), $$ \limpl \reso \Bigl ( \Oma{s \lambda_0} \psi^J \Bigr ) = \limpl (- i \partial/\partial \lambda)^J H_\barb (\lambda)|_{\lambda = s\lambda_0}, $$ but $H_\barb$ is a homogeneous piecewise polynomial function of degree $N-l$, so the conclusion holds for $|J| > N - l.$ If $|J| < N - l$, we find that $(\partial/\partial \lambda)^J H_\barb(\lambda)$ is homogeneous of order $N - l - |J|$ (on any open subset of $\liets$ where $H_\barb$ is smooth). Hence it is of order $s^{N - l - |J|} $ at $\lambda = s \lambda_0$ as $s \to 0^+$, and the conclusion also holds in this case. \noindent{\em Proof of (v):} By (ii), we know that $\reso \Bigl ( \Oma{\lambda} \Bigr ) = 0$ for $\lambda$ in a neighbourhood of $s \lambda_0$ (since $s \lambda_0$ is not in the support of $H_\barb$). Applying (i), $\reso \Bigl ( \Oma{\lambda} \psi^J \Bigr ) $ must also be zero. \noindent{\em Proof of (vi):} If $\beta = \{\beta_1, \dots, \beta_l \} $ span $\liets$, and $\lambda = \sum_j \lambda^j \beta_j$ with all $\lambda^j > 0$, we have $$\reso \Bigl ( \Oma{\lambda} \Bigr ) = \frac{1}{(2 \pi i)^l} \intt \frac{ [d \psi] e^{i \sum_j \lambda^j (\psi_j - i \vt_j) } } {\prod_{k = 1}^l (\psi_k - i \vt_k ) } , $$ where the $\vt_k = \beta_k (\vt) > 0$. But $ [d \psi] = d \psi_1 \dots d \psi_l/(\det \barb)$, where $\psi_j = \beta_j(\psi)$. Since the integrals over $\psi_1, \dots, \psi_l$ may be completed to integrals over semicircular contours $C_+(R) $ in the upper half plane, for each of which the contour integral may be evaluated by the Residue Theorem to give the contribution $ 2 \pi i$, we obtain the result. \noindent{\em Proof of (vii):} We have $$ \int_{\psi + i \vt \in {\Bbb R }^l} \frac{ e^{i \lambda(\psi)} }{P(i\psi)} [d \psi] = (2 \pi i)^l i^{-N} \reso(\Oma{\lambda}) = H_\barb (\lambda). $$ Also, by (i), $$\sum_{m_1, \dots, m_l \ge 0} \frac{(2 \pi i)^l i^{-N} }{m_1!\dots m_l! } \limpl \reso \Bigl ( (i \lambda^1\psi_1)^{m_1} \dots (i \lambda^l\psi_l)^{m_l} \Oma{s \lambda_0} \Bigr ) $$ $$ = \sum_{m_1, \dots, m_l \ge 0} \frac{(2 \pi i)^l i^{-N} (\lambda^1)^{m_1} \dots (\lambda^l)^{m_l}} {m_1!\dots m_l! } \limpl ( \partial/\partial \lambda^1)^{m_1} \dots ( \partial/\partial \lambda^l)^{m_l} \reso (\Oma{\lambda} ) |_{\lambda = s \lambda_0} $$ $$ = \sum_{m_1, \dots, m_l \ge 0} \frac{ (\lambda^1)^{m_1} \dots (\lambda^l)^{m_l}} {m_1!\dots m_l! } \limpl ( \partial/\partial \lambda^1)^{m_1} \dots ( \partial/\partial \lambda^l)^{m_l} H_\barb(\lambda) |_{\lambda = s \lambda_0} $$ which is equal to $H_\barb(\lambda)$ since $H_\barb$ is a polynomial on certain conical subregions of $\liets$, and there is such a subregion containing the ray $\lambda = s \lambda_0$.$\square$ \noindent{\underline{ Example.}} In the following simple example, the explicit formula for $H_\barb$ follows immediately from the definition of $H_\barb$ in Proposition \ref{p3.5}(a). The example is included to show how this result may alternatively be derived by successive contour integrations. Suppose $l = 2$ and $N = 3$, and $\beta_1(\psi) = \psi_1$, $\beta_2(\psi) = \psi_2$, $\beta_3(\psi) = \psi_1 + \psi_2$. We compute \begin{equation} \label{10.6}{\mbox{$\cal R$}} = \reso \Bigl ( \frac{e^{i \lambda(\psi)} } {\psi_1 \psi_2 (\psi_1 + \psi_2)} \Bigr ) \end{equation} where $\lambda(\psi) = \lambda^1 \psi_1 + \lambda^2 \psi_2. $ We assume $\lambda^1, \lambda^2 > 0$, and $\vt_1, \vt_2 > 0 $. The quantity (\ref{10.6}) is given by \begin{equation} {\mbox{$\cal R$}} = \frac{1}{(2 \pi i)^2} \int_{\psi_2 + i \vt_2 \in {\Bbb R }} d \psi_2 \frac{e^{i \lambda^2 \psi_2}}{\psi_2} \int_{\psi_1+ i \vt_1\in {\Bbb R }} \frac{e^{i \lambda^1 \psi_1}}{(\psi_1 + \psi_2)\psi_1} . \end{equation} (This integral in fact gives the Duistermaat-Heckman polynomial $F_T (\pist e^{\iins \bom}) (\lambda)$ near $\mu_T(F) $ where $F$ is a fixed point of the action of $T$ on $X$, when $X$ is a coadjoint orbit of $SU(3)$ and $T\cong (S^1)^2$ is the maximal torus : see \cite{JGP}.) We compute this by first integrating over $\psi_1$: since $\lambda^1 > 0 $, the integral may be completed to a contour integral over a semicircular contour in the upper half plane. We obtain contributions from the two residues $\psi_1 = 0 $ and $\psi_1 = - \psi_2$. Hence we have \begin{equation} {\mbox{$\cal R$}} = \frac{1}{(2 \pi i)} \int_{\psi_2 + i \vt_2 \in {\Bbb R }} d \psi_2 \frac{e^{i \lambda^2 \psi_2}}{\psi_2^2} -\frac{1}{(2 \pi i)} \int_{\psi_2 + i \vt_2 \in {\Bbb R }} d \psi_2 \frac{e^{i (\lambda^2- \lambda^1) \psi_2}}{\psi_2^2} . \end{equation} Since $\lambda^2 > 0 $, the first of these integrals may be completed to a contour integral over a semicircular contour in the upper half plane, and the residue at $0$ yields the value $i \lambda^2$. If $\lambda^2 - \lambda^1 > 0 $, the second integral likewise yields $- i (\lambda^2 - \lambda^1)$. However if $\lambda^2 - \lambda^1 < 0 $ the second integral is instead equal to a contour integral over a semicircular contour in the {\em lower} half plane, which does not enclose the pole at $0$, and hence the second integral gives $0$. Thus we have \begin{equation} \label{9.009}(2 \pi i)^2 \reso(\Oma{\lambda}) = \cases{i \lambda^2, & $\lambda^1 > \lambda^2$ \cr i \lambda^1, & $\lambda^2 > \lambda^1$. \cr } \end{equation} According to (iv) and (vii), the quantity ${\mbox{$\cal R$}}$ is also given by \begin{equation} {\mbox{$\cal R$}} = \limpl \int_{\psi+ i \vt\in {\Bbb R }^2} \frac{ e^{i s (\lambda^1 \psi_1 + \lambda^2 \psi_2)} (i \lambda^1 \psi_1 + i \lambda^2 \psi_2)[d \psi]}{\psi_1 \psi_2 (\psi_1 + \psi_2)} \end{equation} $$ = \limpl (i \lambda^1) \int_{\psi+ i \vt\in {\Bbb R }^2} \frac{ e^{i s [\lambda^1 (\psi_1 + \psi_2) + (\lambda^2 - \lambda^1)\psi_2]} [d \psi] }{(\psi_1 + \psi_2) \psi_2} + \limpl (i \lambda^2) \int_{\psi+ i \vt\in {\Bbb R }^2} \frac{ e^{i s [(\lambda^1 - \lambda^2)\psi_1 + \lambda^2 (\psi_1 + \psi_2)]} [d \psi] }{\psi_1 (\psi_1 + \psi_2) } . $$ This clearly gives the result (\ref{9.009}). { \setcounter{equation}{0} } \section{Examples} In this section we shall show in the case $K = SU(2)$ how Corollary \ref{c8.2} may be used to prove relations in the cohomology ring $H^*(\xred)$ for two specific $X$. These $X$ are the examples treated at the end of Section 6 of \cite{Ki2}. There, all the relations in the cohomology ring are determined, which is equivalent to exhibiting all the vanishing intersection pairings. We shall show how the results of the present paper may be used to show these are indeed vanishing intersection pairings, although we shall not be able to rederive the result that there are no others. \noindent\underline{\em Example 1: $X = ({\Bbb P}_1)^N, $ $N$ odd.} Consider the action of $K = SU(2)$ on the space $X = ( {\Bbb P } _1)^N$ of ordered $N$-tuples of points on the complex projective line $ {\Bbb P } _1$, defined by the $N$th tensor power of the standard representation of $K$ on ${\Bbb C }^2$. Equivalently when $ {\Bbb P } _1$ is identified with the unit sphere $S^2$ in ${\Bbb R }^3$ then $K$ acts on $X = (S^2)^N$ by rotations of the sphere. When the dual of the Lie algebra of $K$ is identified suitably with ${\Bbb R }^3$ then the moment map $\mu$ is given (up to a constant scalar factor depending on the conventions used) by $$\mu(x_1, \dots, x_N) = x_1 + \dots + x_N$$ for $x_1 , \dots, x_N \in S^2$. We assume that $0$ is a regular value for $\mu$; this happens if and only if there is no $N$-tuple in $\mu^{-1}(0)$ containing a pair of antipodal points in $S^2$ each with multiplicity $N/2$, and so $0$ is a regular value if and only if $N$ is odd. In order to apply Corollary \ref{c8.2} we note that the fixed points of the action of the standard maximal torus $T$ of $K$ are the $N$-tuples $(x_1, \dots, x_N)$ of points in $ {\Bbb P } _1$ such that each $x_j$ is either $0$ or $\infty$. We shall index these by sequences $n = (n_1, \dots, n_N)$ where $n_j = + 1$ if $x_j = 0$ and $n_j = - 1$ if $x_j = \infty$. Denote by $e_n$ the fixed point indexed by $n$. Then $$ \mu_T(e_n) = \sum_{j = 1}^N n_j $$ and the weights of the action of $T$ at $e_n$ are just $\{ n_1, \dots, n_N\}$. Hence the sign of the product of weights at $e_n$ is $\prod_j n_j$ and its absolute value is $1$. The cohomology ring $H^*(X)$ has $N$ generators $\xi_1, \dots, \xi_N$ say, of degree two, satisfying $\xi_j^2 = 0$ for $1 \le j \le N$. The equivariant cohomology ring $H^*_T(X)$ with respect to the torus $T$ has generators $\xi_1, \dots, \xi_N$ and $\alpha$ of degree two subject to the relations $$(\xi_j)^2 = \alpha^2$$ for $1 \le j \le N$. The Weyl group action sends $\alpha$ to $-\alpha$ so $\hk(X)$ has generators $\xi_1, \dots, \xi_n, \alpha^2$ subject to the same relations. According to the last example of section 6 of \cite{Ki2}, the kernel of the map $\hk(X) \to H^*(\xred)$ is spanned by elements of the form \begin{equation} \label{8.beta} (1/\alpha) \Bigl ( q( \xi_1, \dots, \xi_N, \alpha) \prod_{i \in Q} (\xi_i + \alpha) - q( \xi_1, \dots, \xi_N, -\alpha) \prod_{i \in Q} (\xi_i - \alpha) \Bigr ) \end{equation} for some $Q \subset \{ 1, \dots, N\} $ containing at least $(N+1)/2$ elements and some polynomial $q$ in $N + 1$ variables with complex coefficients. We can use Corollary \ref{c8.2} to give an alternative proof that the evaluation against the fundamental class $[\xred]$ of the image in $H^*(\xred)$ of any element of this form of degree $N-3$ is zero. This amounts to showing that \begin{equation} {\rm Res}_0 \frac{1}{\psi^{N-1} } \sum_{\stackrel{\indd_j = \pm 1,}{ \sum_j \indd_j > 0} } \Bigl ( \prod_j \indd_j \Bigr ) \Bigl \{ q(\psi \indd_1, \dots, \psi \indd_N, \psi) \prod_{i \in Q} \psi(\indd_i + 1) - q(\psi \indd_1, \dots, \psi \indd_N, -\psi) \prod_{i \in Q} \psi(\indd_i - 1) \Bigr \} \end{equation} is zero when $q$ is homogeneous of degree $N - 2 - |Q|$. In other words it amounts to showing that $\dell = 0$ for any $\dell$ of the form \begin{equation} \dell = \sum_{\stackrel{\indd_j = \pm 1,}{ \sum_j \indd_j > 0}} \Bigl ( \prod_j \indd_j \Bigr ) \Bigl \{ q( \indd_1, \dots, \indd_N, 1) \prod_{i \in Q} (\indd_i + 1) - q( \indd_1, \dots, \indd_N, -1) \prod_{i \in Q} (\indd_i - 1) \Bigr \} \end{equation} where $q$ is homogeneous of degree $N - 2 - |Q|$. Let us assume without loss of generality that $q (\xi_1, \dots, \xi_N, \alpha) = \prod_i \xi_i^{r_i} \alpha^p$ where $p + \sum_i r_i + |Q| = N-2$. Thus $p - 1 = |Q| + \sum_i r_i $ (mod $2$) since $N$ is odd. Hence we have that \begin{equation} \dell = \sum_{\stackrel{\indd_j = \pm 1,} {\sum_j \indd_j > 0} } \prod_j (\indd_j^{r_j + 1} ) \Bigl \{ \prod_{i \in Q} (\indd_i + 1) + (-1)^{|Q|} \prod_k (-1)^{r_k} \prod_{i \in Q} (\indd_i - 1) \: \Bigr \} \end{equation} \begin{equation} \phantom{a} = \sum_{\stackrel{\indd_j = \pm 1,} {\sum_j \indd_j > 0} } \prod_j (\indd_j^{r_j + 1} ) \prod_{k \in Q} (\indd_k + 1) - \sum_{\stackrel{\Indd_j = \pm 1,}{ \sum_j \Indd_j < 0} } \prod_j (\Indd_j^{r_j + 1} ) \prod_{k \in Q} (\Indd_k + 1) \end{equation} where we have introduced $\Indd_j = - \indd_j$. Now the second sum vanishes, for if $\Indd_j = 1$ for all $j \in Q$ then we must have $\sum_j \Indd_j > 0$ since $|Q|> N/2$. Hence we are reduced to proving the vanishing of $$ \sum_{ \indd\in \Gamma} \prod_j \indd_j^{r_j + 1} $$ where $$\Gamma = \{ \indd \: | \: \sum_j \indd_j > 0, \; \indd_j = 1 \; \mbox{for $j \in Q$} \}. $$ Hence we have to prove the vanishing of $$\dell = \sum_{\indd_j = \pm 1, j \notin Q} \prod_{j \in S} n_j$$ where $S$ is the set $\{ j \notin Q \, | \, r_j = 0 \pmod{2} \}. $ The sum thus vanishes by cancellation in pairs provided $S$ is nonempty. However if $S$ were empty then $r_j = 1 \pmod{2}$ for all $j \notin Q$, so $r_j \ge 1$ for all $j \notin Q$, which is impossible since $\sum_j r_j + |Q| \le N-2$. This proves the desired result. \noindent\underline{\em Example 2: $X = {\Bbb P } _N$, $N$ odd.} A closely related example is given by the action of $K = SU(2)$ on the complex projective space $X = {\Bbb P}^N $ defined by the $N$th symmetric power of the standard representation of $K$ on ${\Bbb C }^2$. Equivalently we can identify $X$ with the space of of {\em unordered} $N$-tuples of points in the complex projective line $ {\Bbb P } _1$ or the sphere $S^2$, and then $K$ acts by rotations as in example $1$. We take the symplectic form $\om$ on $X$ to be the Fubini-Study form on $ {\Bbb P } _N$. The moment map is given by the composition of the restriction map ${\bf u}(N+1)^* \to \lieks$ with the map $\mu: {\Bbb P } _N \to {\bf u}(N+1)^* $ defined for $a \in {\bf u}(N+1)$ by $$ \inpr{\mu(x) , a} = (2 \pi i |x^*|^2 )^{-1} \bar{x^*}^t a x^*, $$ where $x^* = (x^*_0, \dots, x^*_N)$ is any point in ${\Bbb C }^{N+1}$ lying over the point $x \in {\Bbb P } _N$. The restriction of this moment map to $\liets$ is $\mu_T (x) = (2 \pi i |x^*|^2 )^{-1} ) \sum_{j = 0}^N (N - 2j)|x^*_j|^2 $. Again we assume that $N$ is odd in order to ensure that $0$ is a regular value of the moment map $\mu$. Again the fixed points of the action of $T$ are the $N$-tuples of points in $ {\Bbb P } _1$ consisting entirely of copies of $0$ and $\infty$. Equivalently they are the points $e_0 = [1, 0, \dots, 0]$, $e_1 = [0, 1, \dots, 0]$, $\dots, e_N = [0, \dots, 0, 1]$ of $ {\Bbb P } _N$. The image of $e_k$ under $\mu_T$ is $\mu_T(e_k) = N - 2 k = \mu_k$ say. Since $\diag (t, t^{-1} ) \in T$ acts on $ {\Bbb P } _N$ by sending $[x_0, \dots, x_j, \dots, x_N]$ to $[t^{-N} x_0, \dots, t^{2j-N} x_j, \dots, t^N x_N]$ the weights at $e_k$ are $$\{ 2 (j - k): \: 0 \le j \le N, \; j \ne k \}$$ The number of negative weights at $e_k$ is equal to $k$ (modulo $2$), and the absolute value of the product of weights at $e_k$ is $v_k = \prod_{j \ne k } |j - k| = 2^N k! (N-k)! $ The cohomology ring $H^*( {\Bbb P } _N)$ is generated by $\xi$ of degree two subject to the relation $\xi^{N+1} = 0$. The equivariant cohomology ring $H^*_T ( {\Bbb P } _N)$ is generated by $\xi$ and $\alpha$ of degree two subject to the relation $\prod_{0 \le j \le N} (\xi - (2j-N) \alpha) = 0, $ and the equivariant cohomology ring $\hk( {\Bbb P } _N)$ is generated by $\xi$ and $\alpha^2$ subject to the same relation. According to section 6 of \cite{Ki2} the kernel of the natural map $\hk(X) \to H^*(\xred)$ is generated as an ideal in $\hk(X) $ by $P_+(\xi, \alpha)$ and $P_-(\xi, \alpha)/\alpha$ where $$P(\xi, \alpha) = \prod_{k > N/2} (\xi + \mu_k \alpha) $$ and $$P_\pm(\xi, \alpha) = P (\xi, \alpha) \pm P(\xi, - \alpha). $$ (Note that $P_+ (\xi, \alpha)$ and $P_-(\xi, \alpha)/\alpha$ are actually polynomials in $\xi$ and $\alpha^2$.) We would like to check that the evaluation against the fundamental class $[\xred]$ of the image of $R_+(\xi, \alpha^2) P_+(\xi, \alpha) $ and $R_-(\xi, \alpha^2) P_-(\xi, \alpha)/\alpha$ in $H^*(\xred)$ is zero for any $R_\pm(\xi, \alpha^2)$ $\in \hk(X)$ of the appropriate degree. Now we have from the abelian fixed point formula that for any $S(\gnx, \gna^2)$, $$\Pi^+_* (S(\gnx, \gna^2) ) = \sum_{k < N/2} (-1)^k \frac{S(\mu_k \psi, \psi^2) }{\veee_k } \psi^{-N}. $$ (Here, if $\zeta \in \hk(X)$, the notation $\Pi^+(\zeta)$ means the portion of the abelian formula (\ref{2.1}) for $\pis(\zeta)$ corresponding to fixed points $F$ for which $\mu_T(F) > 0$.) To evaluate this on the fundamental class of $\xred$ we must then find the term of degree $-1$ in $\nusym^2(\psi) \Pi^+_* (S(\gnx, \gna^2 ) )(\psi) $, or in other words the term of degree $N - 3$ in $\sum_{k < N/2}(-1)^k S(\mu_k \psi, \psi^2) /(\veee_k ) $. Having found the term of degree $N-3$ in $\psi$, we evaluate it at $\psi = 1$ to get the residue. In the case when $S(\mu_k \psi, \psi^2) = R_+(\mu_k \psi, \psi^2) P_+(\mu_k \psi, \psi) $ or $S(\mu_k \psi, \psi^2) = R_-(\mu_k \psi, \psi^2) P_-(\mu_k \psi, \psi)/\psi $ is homogeneous of degree $N - 3$ in $\psi$, we need to show that $$ \sum_{k = 0 }^{(N-1)/2} \: (-1)^k \frac{1}{\veee_k } \Bigl ( \, R_\pm (\mu_k, 1) \prod_{j \ge N/2} (\mu_k + \mu_j) \pm R_\pm(\mu_k, -1) \prod_{j \ge N/2} (\mu_k - \mu_j) \: \Bigr ) = 0. $$ Since $\mu_k = - \mu_{N-k} $, we have $\prod_{j \ge N/2} (\mu_k + \mu_j ) = 0$ for all $k < N/2$, so we just have to prove the vanishing of $$\sum_{k \le N/2} \: (-1)^k \frac{1}{ \veee_k } R(\mu_k) \prod_{j \ge N/2} (\mu_k - \mu_j) $$ for every polynomial $R$ of degree at most $(N-1)/2 - 2$, or without loss of generality the vanishing of $$ \sum_{k \le N/2} (\mu_k)^s \frac{(-1)^k}{ \prod_{l \ne k} |l - k| } \prod_{j \ge N/2} (\mu_k - \mu_j), $$ where $s \le (N-1)/2 - 2$. Since $ \prod_{l \ne k} |l - k| = k! (N-k)!$ and $\mu_k - \mu_j = 2(j - k), $ we have that $$ \prod_{j \ge N/2} |\mu_k - \mu_j | = 2^{(N+1)/2} \frac{(N - k)!} {( (N-1)/2 - k )! } $$ Define $r = (N-1)/2$. We need to show the vanishing of $$\sum_{k = 0}^r (N-2k)^s \frac{(-1)^k (N-k)!}{k! (N-k)! (r - k)! } $$ for $s \le r - 2$. It then suffices to show the vanishing of $$\sum_{k = 0}^r k^s (-1)^k \colvec{r}{k} , \phantom{bbbbb} \mbox{ $s \le r - 2$.}$$ This follows since one may expand $(1- e^\lambda)^r$ as a power series in $e^\lambda$ using the binomial theorem: we have $$ (1 - e^\lambda)^r = \sum_{j= 0}^r (-1)^j \colvec{r}{j} \exp j \lambda \: = \sum_{s \ge 0} \lambda^s/s! \sum_{j= 0}^r (-1)^j j^s \colvec{r}{j} , $$ but the terms in this expansion corresponding to $s < r$ must vanish since $1 - e^\lambda = \lambda h(\lambda)$ for some function $h$ of $\lambda$ which is analytic at $\lambda = 0$.

9307

alg-geom/9307010

en

https://arxiv.org/abs/alg-geom/9307010

alg-geom/9307010

Victor Batyrev

Victor V. Batyrev and Duco van Straten

Generalized Hypergeometric Functions and Rational Curves on Calabi-Yau Complete Intersections in Toric Varieties

45 pages, Latex 2.09

Commun. Math. Phys. 168 (1995) 493

10.1007/BF02101841

We formulate general conjectures about the relationship between the A-model connection on the cohomology of a $d$-dimensional Calabi-Yau complete intersection $V$ of $r$ hypersurfaces $V_1, \ldots, V_r$ in a toric variety ${\bf P}_{\Sigma}$ and the system of differential operators annihilating the special hypergeometric function $\Phi_0$ depending on the fan $\Sigma$. In this context, the Mirror Symmetry phenomenon can be interpreted as the following twofold characterization of the series $\Phi_0$. First, $\Phi_0$ is defined by intersection numbers of rational curves in ${\bf P}_{\Sigma}$ with the hypersurfaces $V_i$ and their toric degenerations. Second, $\Phi_0$ is the power expansion near a boundary point of the moduli space of the monodromy invariant period of the holomorphic differential $d$-form on an another Calabi-Yau $d$-fold $V'$ which is called Mirror of $V$. Using the generalized hypergeometric series, we propose a general construction for Mirrors $V'$ of $V$ and canonical $q$-coordinates on the moduli spaces of Calabi-Yau manifolds.

alg-geom

\section{Introduction} \noindent In this paper we consider complex projective smooth algebraic varieties $V$ of dimension $d$ whose canonical bundles ${\cal K}_V$ are trivial, i.e. ${\cal K}_V \cong {\cal O}_V$, and the Hodge numbers $h^{p,0}(V)$ are zero unless $p=0$, or $p=d$. These varieties are called {\em $d$-dimensional Calabi-Yau varieties}, or {\em Calabi-Yau $d$-folds}. For each dimension $d \geq 3$, there are many of examples of topologically different Calabi-Yau $d$-folds which can be constructed from hypersurfaces and complete intersections in weighted projective spaces \cite{cand0,cand01,cand02,kim}. Physicists have discovered a fascinating phenomen for Calabi-Yau manifolds, so called {\em mirror symmetry} \cite{dixon,greene,lerche,lynker}. Using the mirror symmetry, Candelas et al. in \cite{cand2} have computed the coefficients of the $q$-expansion of the Yukawa coupling for Calabi-Yau hypersurfaces of degree $5$ in ${\bf P}^4$. The method of Candelas et al. was applied to Calabi-Yau $3$-folds in weighted projective spaces \cite{font,morrison.picard,klemm1} and complete intersections in weighted and ordinary projective spaces \cite{klemm2,lib.teit}. The $q$-expansions for Yukawa couplings have been calculated also for Calabi-Yau hypersurfaces of dimension $d > 3$ in projective spaces \cite{greene1}. The interest of algebraic geometers to Yukawa couplings is explained by the conjectural relationship between the coefficients of the $q$-expansion of the Yukawa couplings and the intersection theory on the moduli spaces of rational curves on Calabi-Yau $d$-folds \cite{greene1,katz1}. For small values of degrees, this relationship was verified in \cite{katz2}. However, the main problem which remains unsolved is to find a general rigorous mathematical explanation of this relationship for rational curves of arbitrary degrees on Calabi-Yau $d$-folds. The purpose of this paper is to show that the mirror symmetry and the calculation of the Yukawa couplings for $d$-dimensional Calabi-Yau complete intersections in toric varieties bases essentially on the theory of special generalized hypergeometric functions. We remark that these hypergeometric functions satisfy the hypergeometric differential system considered by Gelfand, Kapranov and Zelevinsky in \cite{gel1}. We propose also a general method for computing the normalized canonical $q$-coordinates. The paper is organized as follows: In Section 2, we give a review of the calculation of Candelas et al. in \cite{cand2} of the coefficients $\Gamma_d$ of the $q$-expansion of the normalized Yukawa coupling \[ K^{(3)}_q = 5 + \sum_{d \geq 1} \Gamma_d \frac{q^d}{1 -q^d}. \] The coefficients $\Gamma_d = n_d d^3$ conjecturaly coincide with the Gromov-Witten invariants (introduced by D. Morrison in \cite{morrison.hodge}) for rational curves on quintic hypersurfaces in ${\bf P}^4$. Our review is greatly influenced by the work of D. Morrison \cite{morrison.mirror,morrison.picard}, but we want to emphasize on the fact that the computation of the prediction for the number of rational curves on quintic $3$-folds bases essentially on the properties of the special generalized hypergeometric series \[ \Phi_0(z) = \sum_{ n \geq 0} \frac{(5n)!}{(n!)^5}z^n \] which admits a combinatorial definition in terms of curves on ${\bf P}^4$. In Section 3, we explain a Hodge-theoretic framework for mirror symmetry and the ideas due to P. Deligne \cite{deligne1} and D. Morrison \cite{morrison.comp,morrison.hodge} The key-point of this framework is the existence of a new-type nilpotent connection on cohomology of Calabi-Yau manifolds. Following a suggestion of D. Morrison, we call it {\em A-model connection} (see also \cite{witten2}). The mirror symmetry identifies the A-model connection on the cohomology of a Calabi-Yau $d$-fold $V$ with the classical Gau\ss -Manin connection on cohomology of its mirror manifold $V'$. Section 4 contains a review of the standard computational technique based on the recurrent relations satisfied by coefficients of formal solutions of Picard-Fuchs equations. We use this techique later in explicit calculations of $q$-expansions for Yukawa couplings for some examples of Calabi-Yau complete intersections in toric varieties. Section 5 is devoted to complete intersections in ordinary projective spaces. Using explicit description of the series $\Phi_0(z)$ for Calabi-Yau complete intersections in projective spaces, we calculate the $d$-point Yukawa coupling and propose the explicit construction for mirrors of such Calabi-Yau $d$-folds for arbitrary dimension $d$. In Section 6, we give a general definition of special generalized hypergeometric functions and establish the relationships between these functions and combinatorial properties of rational curves on toric varieties containing Calabi-Yau complete intersections. It is easy to see that these generalized hypergeometric functions form a special subclass of the generalized hypergeometric functions with {\em resonance} parameters considered by Gelfand et al. in \cite{gel1}. We formulate general conjectures about the differential systems and canonial $q$-coordinates defined by the generalized hypergeometric series corresponding to Calabi-Yau complete intersections in toric varieties. Using a combinatorial interpretation of Calabi-Yau complete intersections in toric varieties due to Yu. I. Manin, we propose an explicit construction of mirrors. In Section 7, we consider in more details the example Calabi-Yau hypersurfaces $V$ of degree $(3,3)$ in ${\bf P}^2 \times {\bf P}^2$. We use this example for the illustration of the general computational method we used in Section 8, where we calculate the $q$-expansions of Yukawa couplings for some Calabi-Yau complete intersections in products of projective spaces. The computations in this section were done by the second author using an universal computer program based on Maple. {\bf Acknowledgements. } This work benefited greatly from conversations with F. Beukers, P. Candelas, B. Greene, D. Morrison, Yu.I. Manin, R. Plesser, R. Schimmrigk, Yu. Tschinkel and A. Todorov. We are very grateful to D. Morrison whose numerous remarks concerning a preliminary version of the paper helped us to give precise references on his work, especially on the forthcomming papers \cite{greene1,morrison.hodge} We would like to express our thaks for hospitality to the Mathematical Sciences Research Institute where this work was supported in part by the National Science Foundation (DMS-9022140), and the DFG (Forschungsschwerpunkt Komplexe Mannigfaltigkeiten). \section{Quintics in ${\bf P}^4$} \noindent In this section we give a review of the (conjectural) computation of the Gromov-Witten invariants $\Gamma_d$ and predictions $n_d$ for numbers of rational curves of degree $d$ on quintics $V$ in ${\bf P}^5$ due to P. Candelas, X. de la Ossa, P.S. Green, and L. Parkes \cite{cand2}. The main ingedients of this computations were considered in papers of D. Morrison \cite{morrison.mirror,morrison.picard}. The purpose of this review is to stress that this computation needs knowing only properties of the special generalized hypergeometric function $\Phi_0(z)$. We begin with the computational algorithm for computing the coefficients in the $q$-expansion of the Yukawa coupling and the predictions for number of rational curves. \newpage \subsection{The coefficients in the $q$-expansion of the Yukawa coupling} \noindent Consider the series \[ \Phi_0(z) = \sum_{n \geq 0} \frac{(5n)!}{(n!)^5}z^n. \] {\bf Step 1.} If we put $a_n = {\displaystyle \frac{(5n)!}{(n!)^5}}$, then the numbers $a_n$ satisfy the recurrent relation \[ (n+1)^4 a_{n+1} = 5(5n+1)(5n+2)(5n+3)(5n+4) a_n. \] This immediatelly implies that the series $\Phi_0(z)$ is the solution to the differential equation \[ {\cal D} \Phi(z) =0 ,\] where \[ {\cal D} = \Theta^4 - 5z(5 \Theta +1) (5 \Theta +2)(5 \Theta +3)(5 \Theta +4),\; \Theta = z \frac{\partial}{\partial z}. \] One can rewrite the differential operator ${\cal D}$ in powers of $\Theta$ as follows: \[ {\cal D} = A_4(z)\Theta^4 + A_3(z)\Theta^3 + \cdots + A_0(z). \] We denote by $C_i(z)$ the rational function $A_i(z)/A_4(z)$ $( i =0,\ldots, 3)$. \bigskip {\bf Step 2.} Following \cite{morrison.mirror}, define the normalized Yukawa $3$-differential as \[ {\cal W}_{3} = K_z^{(3)} (\frac{d z}{z})^{\otimes 3}, \] where $K_z^{(3)} = W_3(z)/ \Phi_0^2(z)$ is the $3$-point coupling function such that $W_3(z)$ satisfies the differential equation \[ \Theta W_3(z) = - \frac{1}{2}C_3(z) W_3(z) ({\rm \ref{coup-eq}}) \] and the normalizing condition $W_3(0) = 5$. One easily obtains \[ {\cal W}_{3} = \frac{5}{(1 - 5^5z)\Phi_0^2(z)} (\frac{d z}{z})^{\otimes 3}. \] \medskip {\bf Step 3.} The equation ${\cal D}\Phi =0$ is a Picard-Fuchs differential equation with maximal unipotent monodromy (in the sense of Morrison \cite{morrison.mirror}) at $z =0$. Therefore, there exists a unique solution $\Phi_1(z)$ to ${\cal D}\Phi =0$ such that $\Phi_1(z) = (\log z)\Phi_0(z) + \Psi(z)$, where $\Psi(z)$ is regular at $z =0$ and $\Psi(0) =0$. We define the new local coordinate $q = q(z)$ near the point $z =0$ as \[ q(z) = \exp \left( \frac{\Phi_1(z)}{\Phi_0(z)} \right) = z \exp \left( \frac{\Psi(z)}{\Phi_0(z)} \right). \] The, we rewrite the normalized Yukawa $3$-differential ${\cal W}_{3}$ in the coordinate $q$ as \[ {\cal W}_{3} = K_q^{(3)} (\frac{d q}{q})^{\otimes 3}. \] The function $K_q^{(3)}$ is called the Yukawa $3$-point coupling. This function has the power expansion \[ K_q^{(3)} = 5 + \sum_{d \geq 1}^{\infty} \frac{n_d d^3 q^d}{1 - q^d}, \] where $\Gamma_d = n_d d^3$ are conjectured to be the Gromov-Witten invariants of rational curves of degree $d$ on a quintic $3$-fold in ${\bf P}^4$ \cite{katz2,morrison.hodge}. The numbers $n_d$ are predictions for numbers of rational curves of degree $d$ on quintic $3$-folds. \bigskip It is important to remark that in the above algorithm for calculation of the numbers $n_d$ one needs to know only properties of the series $\Phi_0(z)$ and the normalization condition $W_3(0) = {\rm deg}\, V = 5$ for $W_3(z)$, i.e., one does not need to know anything about mirrors of quintics. \subsection{Philosophy of mirrors and the series $\Phi_0(z)$} \noindent The central role in the computation of Candelas et al. in \cite{cand2} is played by the orbifold construction of mirrors for quintics in ${\bf P}^4$ \cite{greene}. In \cite{batyrev.dual}, this construction of mirrors was generalized for hypersurfaces in toric Fano varieties with Gorenstein singularities. In the above algorithm, we have shown that one can forget about mirrors. However, the philosophy of mirrors proves to be very helpful. For quintic $3$-folds this philosophy appears as the following twofold interpretation of the series $\Phi_0(z)$. \medskip {\bf The first interpretation:} We compute the coefficients $a_n$ of the power series $\Phi_0(z)$ using combinatorial properties of curves $C \subset {\bf P}^4$ of degree $n$. Notice that any such a curve $C$ meets a generic quintic $V$ at $5n$ distinct points $ p_1,$ $\ldots,$ $p_{5n}$. There exists a degeneration of $V$ into a union of $5$ hyperplanes $H_1 \cup \cdots \cup H_5$. Every such a hyperplane $H_i$ intersects $C$ at $n$ points $p_{i_1},$ $\ldots,$ $p_{i_n}$ which can be considered as deformations of a subset of $n$ points from the set $\{ p_1,$ $\ldots,$ $ p_{5n} \}$. It remains to remark that there exists exactly $(5n)!/(n!)^5$ ways to divide $\{ p_1,$ $\ldots,$ $p_{5n} \}$ into $5$ copies of $n$-element disjoint subsets. \medskip {\bf The second interpretation: } We find the coefficients $a_n$ from an integral representation of $\Phi_0(z)$. Let ${\bf T} \cong ({\bf C}^*)^4$ be the $4$-dimensional algebraic torus with coordinate functions $X_1$, $X_2$, $X_3$, $X_4$. Take the Laurent polynomial \[f(u,X) = 1 - (u_1X_1 + u_2X_2 + u_3X_3 + u_4 X_4 + u_5(X_1 X_2 X_3 X_4)^{-1}) \] in variables $X_1, X_2, X_3, X_4$, where the coefficients $u_1, \ldots, u_5$ are considered as independent parameters. Let $z = u_1 u_2 u_3 u_4 u_5$. \begin{prop} \[ \Phi_0(u_1 \cdots u_5) = \Phi_0(z) = \frac{1}{(2 \pi \sqrt{-1})^4} \int_{\mid X_i \mid =1} \frac{1}{f(u,X)} \frac{dX_1}{X_1} \wedge \frac{dX_2}{X_2} \wedge \frac{dX_3}{X_3} \wedge \frac{dX_4}{X_4}. \] \end{prop} {\bf Proof. } One has \[ \frac{1}{f(u,X)} = \sum_{n \geq 0} (u_1X_1 + u_2X_2 + u_3X_3 + u_4 X_4 + u_5(X_1 X_2 X_3 X_4)^{-1})^n \] \[ = \sum_{m \in {\bf Z}^4} c_m(u) X^m. \] It is straightforward to see that $c_0(u) = \Phi_0(u_1 \cdots u_5)$. Now the statement follows from the Cauchy residue formula. \hfill $\Box$ \medskip The second interpretation of $\Phi_0(z)$ implicitly uses mirrors of quintics, since zeros of $f(u,X)$ define the affine Calabi-Yau $3$-fold $Z_f$ in ${\bf T}$ whose smooth Calabi-Yau compactification is mirror symmetric with respect to quintic $3$-folds (see \cite{batyrev.dual}). Moreover, the holomorphic $3$-form $\omega$ on $Z_f$ such that $\omega(z)$ extends regularly to a smooth compactification of $Z_f$ depends only on $z$, i.e., only on the product $u_1 \cdots u_5$. This $3$-form can be written as \[ \omega(z) = \frac{1}{(2\pi \sqrt{-1})^4}{\rm Res} \frac{1}{f(u,X)} \frac{dX_1}{X_1} \wedge \frac{dX_2}{X_2} \wedge \frac{dX_3}{X_3} \wedge \frac{dX_4}{X_4}. \] This shows that $\Phi_0(z)$ is exactly the monodromy invariant period of the $3$-form $\omega(z)$ near $z =0$. \begin{prop} The differential $3$-form $\omega(z)$ satisfies the same Picard-Fuchs differential equation ${\cal D} \Phi = 0$ as for the series $\Phi_0(z)$. In particular, all periods of $\omega(z)$ satisfy the Picard-Fuch differential equation with the operator \[ \Theta^4 - 5z(5 \Theta +1) (5 \Theta +2)(5 \Theta +3)(5 \Theta +4). \] \end{prop} {\bf Proof.} In order to prove the statement, it is sufficient to check that \[ \left({\cal D} \frac{1}{f(u,X)} \right) \frac{dX_1}{X_1} \wedge \frac{dX_2}{X_2} \wedge \frac{dX_3}{X_3} \wedge \frac{dX_4}{X_4} \] is a differential of a rational $3$-form on ${\bf T} \setminus Z_f$ . The latter follows from standard arguments using reduction by the Jacobian ideal $J_f$ (see \cite{batyrev.var}). \hfill $\Box$ \subsection{A-model connection} \noindent The Yukawa coupling can be described by a nilpotent connection $\nabla_A$ on the cohomology of quintic $3$-fold $V$ \[ \nabla_A \; : \; H^*(V, {\bf C}) \rightarrow H^*(V, {\bf C}) \otimes {\bf C} \langle \frac{dz}{z} \rangle. \] This connection is homogeneous of degree $2$, i.e., \[ \nabla_A\; : \; H^i (V, {\bf C}) \rightarrow H^{ i +2}(V, {\bf C})\otimes {\bf C} \langle \frac{dz}{z} \rangle, \] and hence $\nabla_A$ vanishes on $H^3(V, {\bf C})$. By this reason, we consider only the cohomology subring \[ H^{2*}(V, {\bf Z}) = \oplus_{i =0}^3 H^{2i}(V, {\bf Z}) \subset H^*(V, {\bf Z})\] of even-dimensional classes on a quintic $3$-fold $V$ (${\rm rk}\,H^{2i} (V, {\bf Z}) =1$). Let $\eta_i$ be the positive generator of $H^{2i} (V, {\bf Z})$. Then, in the basis $\eta_0, \eta_1, \eta_2, \eta_3$, the multiplication by $\eta_1$ is the endomorphism of $H^{2*}(V, {\bf Z})$ having as matrix \[ \Lambda = \left( \begin{array}{cccc} 0 & 1 & 0 & 0 \\ 0 & 0 & 5 & 0 \\ 0 & 0 & 0 & 1 \\ 0 & 0 & 0 & 0 \end{array} \right). \] Following \cite{cere} and \cite{morrison.hodge}, we define the $1$-parameter connection on $H^{2*}(V, {\bf C}) \otimes {\bf C} [[q]]$ considered as a trivial bundle over ${\rm Spec}\, {\bf C}[[q]]$ as follows \[ \left( \begin{array}{c} \nabla_A \eta_0 \\ \nabla_A \eta_1 \\ \nabla_A \eta_2 \\ \nabla_A \eta_3 \end{array} \right) = \left( \begin{array}{cccc} 0 & \frac{dq}{q} & 0 & 0 \\ 0 & 0 & K_q^{(3)}\frac{dq}{q} & 0 \\ 0 & 0 & 0 & \frac{dq}{q} \\ 0 & 0 & 0 & 0 \end{array} \right) \left( \begin{array}{c} \eta_0 \\ \eta_1 \\ \eta_2 \\ \eta_3 \end{array} \right). \] \begin{equation} K(q) = \left( \begin{array}{cccc} 0 & \frac{dq}{q} & 0 & 0 \\ 0 & 0 & K_q^{(3)}\frac{dq}{q} & 0 \\ 0 & 0 & 0 & \frac{dq}{q} \\ 0 & 0 & 0 & 0 \end{array} \right) \label{matrix} \end{equation} can be considered as the deformation of the matrix $\Lambda$ such that \[ \Lambda = {\rm Res}\mid_{q =0}\, K(q). \] \bigskip The mirror philosophy shows that the matrix (\ref{matrix}) can be identified with the matrix of the classical Gau\ss -Manin connection on the $4$-dimensional cohomology space $H^3(\hat{Z}_f, {\bf C})$ in a special symplectic basis. We notice that the quotients ${ F}^i/{ F}^{i+1}$ of the Hodge filtration \[ H^3(\hat{Z}_f, {\bf C}) = { F}^0 \supset { F}^1 \supset {F}^2 \supset { F}^3 \supset {F}^4 = 0 \] are $1$-dimensional. There is also the monodromy filtration on the homology $H_3(\hat{Z}_f, {\bf Z})$ \[ 0 = { W}_{-4} \subset { W}_{-3} \subset { W}_{-2} \subset { W}_{-1} \subset { W}_{0} = H_3(\hat{Z}_f, {\bf Z}) \] such that ${ W}_i/{ W}_{i-1}$ are also $1$-dimensional. We choose the symplectic basis $\gamma_0,\gamma_1, \gamma_2,\gamma_3$ in $H_3(\hat{Z}_f, {\bf Z})$ in such a way that $\{ \gamma_0, \ldots, \gamma_i \}$ form a ${\bf Z}$-basis of ${W}_i$. We choose also the basis $\omega_0, \omega_1, \omega_2, \omega_3$ of $H^3(\hat{Z}_f, {\bf C})$ such that $\{ \omega_0, \ldots, \omega_i \}$ form a ${\bf C}$-basis of ${F}^{3-i}$ and \[ p_{ij} = \int_{\gamma_j} \omega_i = \delta_{ij}\;\; {\rm for}\;i \geq j. \] So the period matrix $\Pi = (p_{ij})$ has the form \cite{greene1,morrison.hodge} \[ \Pi = \left( \begin{array}{cccc} 1 & p_{12} & p_{13} & p_{14} \\ 0 & 1 & p_{23} & p_{34} \\ 0 & 0 & 1 & p_{34} \\ 0 & 0 & 0 & 1 \end{array} \right). \] Notice that all coefficients $p_{ij}$ $(i < j)$ are multivalued functions of $z$ near $z = 0$. Applying the Griffiths transversality property, we obtain that the Gau\ss--Manin connection in the $z$-coordinate has the form \[ \left( \begin{array}{c} \nabla \omega_0 \\ \nabla \omega_1 \\ \nabla \omega_2 \\ \nabla \omega_3 \end{array} \right) = \left( \begin{array}{cccc} 0 & (\Theta p_{12})\frac{dz}{z} & 0 & 0 \\ 0 & 0 & (\Theta p_{23})\frac{dz}{z} & 0 \\ 0 & 0 & 0 & (\Theta p_{34})\frac{dz}{z} \\ 0 & 0 & 0 & 0 \end{array} \right) \left( \begin{array}{c} \omega_0 \\ \omega_1 \\ \omega_2 \\ \omega_3 \end{array} \right), \] where $\Theta p_{i,i+1}$ are single valued functions. Then the Yukawa $3$-differential is simply the tensor product \[ {\cal W}_3 = K_z^{(3)} (\frac{dz}{z})^{\otimes 3} = (\Theta p_{12})\frac{dz}{z} \otimes (\Theta p_{23})\frac{dz}{z} \otimes (\Theta p_{34})\frac{dz}{z}. \] \medskip By Griffiths transversality, one has $\omega_0 \wedge \omega_2 = 0$, i.e. we can assume that $p_{23} = p_{12}$. The differential form $w_0$ can be defined as $\omega / \Phi_0(z)$. Moreover, $p_{12} = \Phi_1(z) / \Phi_0(z)$. In the new coordinate $q$, we have $p_{12} = \log q$. Then the Gau\ss -Manin connection can be rewritten as \[ \left( \begin{array}{c} \nabla \omega_0 \\ \nabla \omega_1 \\ \nabla \omega_2 \\ \nabla \omega_3 \end{array} \right) = \left( \begin{array}{cccc} 0 & \frac{dq}{q} & 0 & 0 \\ 0 & 0 & K_q^{(3)}\frac{dq}{q} & 0 \\ 0 & 0 & 0 & \frac{dq}{q} \\ 0 & 0 & 0 & 0 \end{array} \right) \left( \begin{array}{c} \omega_0 \\ \omega_1 \\ \omega_2 \\ \omega_3 \end{array} \right). \] The role played by the $1$-form $dq/q$ in the connection matrix was first observed by P. Deligne \cite{deligne1}. \subsection{The $q$-coordinate and the Yukawa coupling} Since the coordinate $q$ was defined intrinsically as the ratio $\Phi_1(z)/\Phi_0(z)$ of two solutions of the differential equation ${\cal D} \Phi =0$, it is natural to ask about the form of the differential operator ${\cal D}$ in the new coordinate $q$. Denote by $\Xi$ the differential operator ${\displaystyle q \frac{\partial}{\partial q}}$. \begin{prop} The differential $3$-form $\omega_0$ satisfies the Picard-Fuchs differential equation with the differential operator \[ \Xi^4 + c_3(q) \Xi^3 + c_2(q) \Xi^2, \] where \[ c_3(q) = - 2\frac{\Xi K_q^{(3)}}{K_q^{(3)}}, \; c_2(q) = \frac{\Xi K_q^{(3)}}{(K_q^{(3)})^2} - \frac{\Xi^2 K_q^{(3)}}{K_q^{(3)}}. \] \end{prop} {\bf Proof. } By properties of the nilpotent connection, one has \[ \Xi \omega_0 = \omega_1, \; \Xi \omega_1 = K_q^{(3)} \omega_2, \; \Xi \omega_2 = \omega_3, \; \Xi \omega_3 = 0. \] So \[ \Xi^4 \omega_0 = \Xi^2 K_q^{(3)} \omega_2 = \Xi( (\Xi K_q^{(3)}) \omega_2 + K_q^{(3)} \omega_3 ) \] \[ = (\Xi^2 K_q^{(3)}) \omega_2 + 2 (\Xi K_q^{(3)}) \omega_3. \] On the other hand, \[ \omega_2 = \frac{1}{K_q^{(3)}} \Xi^2 \omega_0, \] \[ \omega_3 = \Xi(\frac{1}{K_q^{(3)}} \Xi^2 \omega_0) = - \frac{\Xi K_q^{(3)}}{(K_q^{(3)})^2} \Xi^2 \omega + \frac{1}{K_q^{(3)}}\Xi^3 \omega_0. \] \hfill $\Box$ \begin{rem} {\rm The differential equation for $\omega_0$ can be written also as \[ \Xi^2 (K_q^{(3)})^{-1} \Xi^2 \omega_0 = 0. \] In this form this equation first arose in \cite{ferrara.louis}. } \end{rem} The differential operator ${\cal D}$ which annihilates the function $\Phi_0(z)$ defines the connection in the basis $\omega$, $\Theta \omega$, $\Theta^2 \omega$, $\Theta^3 \omega$ of $H^3(\hat{Z}_f, {\bf C})$ : \[ \left( \begin{array}{c} \nabla\, \omega \\ \nabla\, \Theta \omega \\ \nabla\, \Theta^2 \omega \\ \nabla\, \Theta^3 \omega \end{array} \right) = \left( \begin{array}{cccc} 0 & \frac{dz}{z} & 0 & 0 \\ 0 & 0 & \frac{dz}{z} & 0 \\ 0 & 0 & 0 & \frac{dz}{z} \\ -C_0(z)\frac{dz}{z} & -C_1(z)\frac{dz}{z} & -C_2(z)\frac{dz}{z} & -C_3(z)\frac{dz}{z} \end{array} \right) \left( \begin{array}{c} \omega \\ \Theta \omega \\ \Theta^2 \omega \\ \Theta^3 \omega \end{array} \right). \] The basis $\omega$, $\Theta \omega$, $\Theta^2 \omega$, $\Theta^3 \omega$ is also compatible with the Hodge filtration in $H^3(\hat{Z}_f, {\bf C})$. Thus there exist a matrix \[ R = \left( \begin{array}{cccc} r_{11} & r_{12} & r_{13} & r_{14} \\ 0 & r_{22} & r_{23} & r_{34} \\ 0 & 0 & r_{33} & r_{34} \\ 0 & 0 & 0 & r_{44} \end{array} \right) \] such that \[ \left( \begin{array}{c} \omega \\ \Theta \omega \\ \Theta^2 \omega \\ \Theta^3 \omega \end{array} \right) = R \left( \begin{array}{c} \omega_0 \\ \omega_1 \\ \omega_2 \\ \omega_3 \end{array} \right). \] It is easy to see that \[ r_{11} = \Phi_0(z), \; r_{22} = \Phi_0(z) (\Theta p_{12}), \; r_{33} = \Phi_0(z) (\Theta p_{12})(\Theta p_{23}), \; \] \[ r_{44} = \Phi_0(z) (\Theta p_{12})(\Theta p_{23})(\Theta p_{34}). \] \bigskip \section{Quantum variations of Hodge structure on Calabi-Yau manifolds} \subsection{A-model connection and rational curves} \noindent A general appoach to the definition of a new connection on cohomology of algebraic and symplectic manifolds $V$ was proposed by Witten \cite{witten}. The construction of Witten bases on the interpretation of third partial derivatives \[ \frac{\partial^3 }{\partial z_i \partial z_j \partial z_k} P(z) \] of a function $P(z)$ on the cohomology space $H^*(V, {\bf C})$ as structure constants of a commutative associative algebra. The function $P(z)$ is defined via the intersection theory on the moduli spaces of mappings of Riemann surfaces $S$ to $V$. Using Poincare duality, one obtains the structure coefficients of the connection on $H^*(V, {\bf C})$. We consider a specialization of the general construction to the case when $V$ is a Calabi-Yau $3$-fold. We put $n = {\rm dim}\, H^2 (V, {\bf C}) = {\rm dim}\, H^4(V, {\bf C})$. Let $\eta_0$ be a generator of $H^0(V, {\bf Z})$, $\eta_1, \ldots, \eta_n$ a ${\bf Z}$-basis of $H^2(V, {\bf Z})$, $\zeta_1, \ldots, \zeta_n$ the dual ${\bf Z}$-basis of $H^4(V, {\bf Z})$ $( \langle \eta_i, \zeta_j \rangle = \delta_{ij})$, and $\zeta_0$ the dual to $\eta_0$ generator of $H^6(V, {\bf Z})$. We can always assume that the cohomology classes $\eta_1, \ldots, \eta_n$ are contained in the closed K\"ahler cone of $V$. \begin{opr} {\rm Let $R = {\bf C} [[ q_1, \ldots, q_n ]]$ be the ring of formal power series in $n$ independent variables. We denote by $H(V)$ the scalar extension \[ (\oplus_{i =0}^3 H^{2i}(V, {\bf C})) \otimes_{\bf C} R. \] We consider a flat nilpotent holomorphic connection \[ \nabla_A\; : H(V) \rightarrow H(V) \otimes \Omega^1_R(\log q) \] defined by the following formulas \cite{cere,morrison.hodge} \[ \nabla_A \eta_0 = \sum_{i =1}^n \eta_i \otimes \frac{dq_i}{q_i}; \] \[ \nabla_A \eta_k = \sum_{i =1}^n \sum_{j =1}^n K_{ijk} \; \zeta_j \otimes \frac{d q_i}{q_i},\; k =1, \ldots, n; \] \[ \nabla_A \zeta_j = \zeta_0 \; \frac{d q_j}{q_j} , \; j =1, \ldots, n; \] \[ \nabla_A \zeta_0 = 0. \] The coefficients $K_{ijk}$ are power series in $q_1, \ldots, q_n$ defined by rational curves $C$ on $V$, i.e., morphisms $f \; :\; {\bf P}^1 \rightarrow V$ as follows \[ K_{ijk} = \langle \eta_i, \eta_j, \eta_k \rangle + \sum_{\scriptstyle \begin{array}{c} {\scriptstyle C \subset V} \\ {\scriptstyle \lbrack C \rbrack \neq 0} \end{array} } n_{\lbrack C \rbrack} \langle C, \eta_i \rangle \langle C, \eta_j \rangle \langle C, \eta_k \rangle \frac{q^{\lbrack C \rbrack} } { 1 - q^{\lbrack C \rbrack}}, \] where $q^{\lbrack C \rbrack} = q_1^{c_1} \cdots q_n^{c_n}$ $( c_i = \langle C, \eta_i \rangle)$. The integer \[ \Gamma_{\lbrack C \rbrack}(\eta_i,\eta_j,\eta_k) = n_{\lbrack C \rbrack} \langle C, \eta_i \rangle \langle C, \eta_j \rangle \langle C, \eta_k \rangle \] is called {\em the Gromov-Witten invariant} \cite{katz2,morrison.hodge} {\em of the class $\lbrack C \rbrack$.} If the classes $\eta_i$, $\eta_j$, and $\eta_k$ are represented by effective divisors $D_i$, $D_j$, and $D_k$ on $V$, then $\Gamma_{\lbrack C \rbrack}(\eta_i,\eta_j,\eta_k)$ is the number of pseudo holomorphic immersions $\imath \; : \; {\bf P}^1 \rightarrow V$ such that $\lbrack \imath({\bf P}^1) \rbrack = \lbrack C \rbrack$ and $\imath(0) \in D_i$, $\imath(1) \in D_j$, $\imath(\infty) \in D_k$ for sufficiently general almost complex structure on $V$. In particular, the number $n_{\lbrack C \rbrack}$ which predicts the number of rational curves $C \subset V$ with the fixed class $\lbrack C \rbrack$ is always non-negative. The connection $\nabla_A$ will be called the {\em A-model connection}. The connection $\nabla_A$ defines on $H(V)$ a variation of Hodge structure of type $(1,n,n,1)$. We call this variation {\em the quantum variation of Hodge structure on $V$. } } \end{opr} \begin{rem} {\rm The Picard-Fuch differential system satisfied by $\eta_0$ was considered in details in \cite{cere}.} \end{rem} One immediatelly obtains: \begin{prop} Let $\eta = l_1 \eta_1 + \cdots + l_n \eta_n \in H^2(V, {\bf Z})$ be a class of an ample divisor on $V$. Define the $1$-parameter connection with the new coordinate $q$ by putting $q_1 =q^{l_1}\, \ldots, q_n = q^{l_n}$. Then the connection $\nabla_A$ on $H(V)$ induces the connection \[ \nabla_q\; : \; (\oplus_{i =0}^3 H^{2i}(V, {\bf C}[[q]])) \rightarrow (\oplus_{i =0}^3 H^{2i}(V, {\bf C}[[q]])) \otimes_{\bf C} \Omega^1_{{\bf C}[[q]]}(\log q). \] In particular, the residue of the connection operator $\nabla_q$ at $q = 0$ is the Lefschetz operator $L_{\eta} \;: H^{2i}(V, {\bf C}) \rightarrow H^{2i +2}(V, {\bf C})$, and \[ \langle (\nabla_q)^3 \eta_0, \zeta_0 \rangle = \left( \langle \eta, \eta, \eta \rangle + \sum_{d > 0} n_d \frac{d^3 q^d}{1 - q^d} \right) \left( \frac{dq}{q} \right)^{\otimes 3} \] where \[ n_d = \sum_{ \langle C, \eta \rangle = d } n_{\lbrack C \rbrack}. \] \end{prop} \begin{coro} The connection $\nabla_q$ defines a Picard-Fuchs differential operator of order $4$ annihilating $\eta_0$. \end{coro} \subsection{The Gau\ss -Manin connection for mirrors} \noindent Let $W$ be a Calabi-Yau $3$-fold such that ${\rm dim }\,H^3(W, {\bf C}) = 2n +2$. Assume that we are given a variation $W_z$ of complex structure on $W$ near a boundary point $p$ of the $n$-dimensional moduli space ${\cal M}_W$ of complex structures on $W$ in holomorphic coordinates $z_1, \ldots, z_n$ near $p$ such that $p = (0,\ldots, 0)$. \begin{opr} {\rm The family $W_z$ is said to have the {\em maximal unipotent monodromy at $z = 0$} if the weight filtration \[ 0 = W_{-1} \subset W_0 \subset W_1 \subset W_{2} \subset W_3 = H^3(W_z, {\bf C}) \] defined by $N$ is orthogonal to the Hodge filtration $\{ F^i \}$, i.e., \[ H^3(W_z, {\bf C}) = W^{\perp}_{i} \oplus F^{3-i} \; i =0, \ldots , 3.\]} \end{opr} (This is essentially the same definition given in \cite{morrison.comp,morrison.hodge}.) Choose a symplectic basis \[ \gamma_0, \gamma_1 , \ldots, \gamma_n, \delta_1, \ldots \delta_n, \delta_0 \] of $H_3(W_z, {\bf Z})$ in such a way that $\gamma_0$ generates $W_0$, $\gamma_0, \gamma_1, \ldots , \gamma_n$ is a ${\bf Z}$-basis of $W_1$, \[ \gamma_0, \gamma_1 , \ldots, \gamma_n, \delta_1, \ldots \delta_n \] is a ${\bf Z}$-basis of $W_2$. Then we choose a symplectic basis in $H^3(W_z, {\bf C})$ : \[ \omega_0, \omega_1, \ldots, \omega_n, \nu_1, \ldots, \nu_n, \nu_0 \] such that $\omega_0$ generates $F^3$, $\omega_0, \omega_1, \ldots, \omega_n$ is the basis of $F^2$, $\omega_0, \omega_1, \ldots, \omega_n, \nu_1, \ldots, \nu_n$ is the basis of $F^1$ such that \[ \langle \omega_i, \gamma_i \rangle = \langle \nu_i, \delta_i \rangle = 1,\; i =0,1, \ldots, n \; \] \[ \langle \omega_i, \gamma_0 \rangle = \langle \nu_j, \gamma_0 \rangle = \langle \nu_i, \gamma_j \rangle = \langle \nu_0, \gamma_j \rangle = \langle \nu_0, \delta_j \rangle = 0, \; i =1, \ldots, n, \; j =0, \ldots, n. \] The choice of the basis of $H^3(W_z, {\bf C})$ defines the splitting into the direct sum \[ H^3(W_z, {\bf C}) = H^{3,0} \oplus H^{2,1} \oplus H^{1,2} \oplus H^{0,3} \] such that all direct summand acquire {\em canonical integral structures}. By Griffiths transversality property, the Gau\ss -Manin connection $\nabla$ sends $H^{3-i,i}$ to $H^{3-i-1, i+1} \otimes \Omega^1(\log z)$. It is an observation of Deligne \cite{deligne1} that the weight and Hodge filtrations define a variation of mixed Hodge structure (VMHS). Two Calabi-Yau $3$-folds $V$ and $W$ are called mirror symmetric if the quantum variation of Hodge structure for $V$ is isomorphic to the classical VMHS for $W$. In this case the $q$-coordinates near $p$ up to constants are defined by the formula \cite{morrison.comp} \[ q_i = \exp (2\pi \sqrt{-1}) \int_{\gamma_i} \omega_0. \] \bigskip \section{Picard-Fuchs equations} \noindent In this section we recall standard facts about Picard-Fuch differential equations which we use in computations of Yukawa $d$-point functions and predictions for numbers of rational curves on Calabi-Yau manifolds. \subsection{Recurrent relations and differential equations} \noindent Let $a_n$ $n =0,1,2, \ldots$ be an infinite sequence of complex numbers. For our purposes, it will be more convenient to define $a_n$ for all integers $n \in {\bf Z}$ by putting $a_n =0$ for $n < 0$. We define the generating function for the sequence $\{ a_i \}$ as the formal power series \[ \Phi(z) = \sum_{i \geq 0}^{\infty} a_i z^i \in {\bf C}[[ z ]]. \] Consider two differential operators acting on ${\bf C} [[ z ]]$: \[ \Theta \; :\; f \mapsto z \frac{\partial}{\partial z} f, \] \[ z \; : \; f \mapsto z \cdot f \] satisfying the relation \begin{equation} \lbrack \Theta,z \rbrack = \Theta \circ z - z \circ \Theta = z. \label{commut} \end{equation} These operators generate the algebra ${\bf D} ={\bf C} \lbrack z, \Theta \rbrack$ of "logarithmic" differential operators which are polynomials in non-commuting operators $\Theta$ and $z$ \bigskip Fix a positive integer $d$. Assume that there exist $m+1$ $(m \geq 1)$ polynomials \[ P_{0}(y), \ldots, P_m(y) \in {\bf C}\lbrack y \rbrack \] of degree $d+1$ such that that for every $n \in {\bf Z}$ the numbers $\{a_i \}$ satisfy the recurrent relation: \begin{equation} P_0(n)a_{n} + P_1(n+1)a_{n +1} + \cdots + P_m(n+m) a_{n+m} = 0. \label{recurrent} \end{equation} (Here we consider $y$ as a new complex variable having no connection to our previous variable $z$.) Then $\Phi(z)$ is a formal solution of the linear differential equation \[ {\cal D} \Phi(z) = 0\] with the differential operator \begin{equation} {\cal D} = z^m P_0(\Theta) + z^{m-1}P_1(\Theta) + \cdots + P_m(\Theta). \label{diff1} \end{equation} This differential equation of order $d+1$ can be rewritten in powers of $\Theta$ as \begin{equation} {\cal D} = A_{d+1}(z)\Theta^{d+1} + \cdots + A_1(z) \Theta + A_0(z), \label{diff2} \end{equation} where $A_i$ are some polynomials in $z$. It is easy to check the following: \begin{prop} A power series $\Phi(z)$ is a formal solution to a differential equation ${\cal D}\Phi(z) = 0$ of order $d+1$ for some element ${\cal D} \in {\bf D}$ if and only if the coefficients $\{ a_i \}$ satisfy a recurrent relation as in {\rm (\ref{recurrent})} for some polynomials $ P_{0}(y), \ldots, P_m(y)$ of degree $d+1$. \end{prop} \subsection{Picard-Fuchs operators} \noindent Recall that a differential operator ${\cal D}$ as in (\ref{diff2}) is called {\em a Picard-Fuchs operator at point $z =0$} if $A_{d+1}(0) \neq 0$. Moreover, the condition that solutions of the Picard-Fuchs equatios ${\cal D}\Phi$ have {\em maximal unipotent monodromy at $z =0$} \cite{morrison.picard} is equivalent to $A_i(0) = 0$ for $ i = 0, \ldots, d$. These properties of the operator ${\cal D}$ can be reformulated in terms of the properties of the polynomial $P_m(y)$ in (\ref{recurrent}) as follows: \medskip {\em A differential operator ${\cal D}$ is a Picard-Fuchs operator if and only if the polynomial $P_m(y)$ has degree $d+1$, i.e., its leading coefficient is nonzero. Moreover, solutions of the equations ${\cal D}$ have maximal unipotent monodromy at $z =0$ if and only if the polynomial $P_m(y)$ equals $cy^k$ for some nonzero constant $c$. } \medskip Picard-Fuchs operators having the maximal unipotent monodromy at $z =0$ will be objects of our main interest. Therefore, we introduce the following definition: \begin{opr} {\em A Picard-Fuchs operator ${\cal D}$ with the maximal unipotent monodromy will be called {\em a MU-operator}. We will always assume that the corresponding polynomial $P_m(y)$ in (\ref{recurrent}) for any $MU$-operator ${\cal D}$ is $y^k$, i.e., $c =1$.} \label{MU} \end{opr} The fundamental property of $MU$-operators is the following one: \begin{theo} If ${\cal D}$ is $MU$-operator, then the subspace in ${\bf C}[[z]]$ of solutions of the linear differential equation \[ {\cal D}\Phi(z) = 0 \] has dimension $1$. Moreover, every solution is defined uniquely by the value $\Phi(0) = a_0$. \end{theo} {\bf Proof. } If we have chosen a value of $a_0$, the all coefficients $a_i$ for $i \geq 0$ are uniquely defined from the reccurent relation (\ref{recurrent}). ( We remind, that we put $a_i =0$ for $ i < 0$.) $\Box$ \begin{opr} {\rm Let ${\cal D}$ be a $MU$-operator. Then the power series solution $\Phi_0(z)$ of the equation ${\cal D}\Phi(z) = 0$ normalized by the condition $\Phi_0(0) =1$ will be called the {\em socle-solution}. } \end{opr} \subsection{Logarithmic solutions and the $q$-coordinate} \noindent Let ${\cal D}$ be a $MU$-operator of order ${d+1}$. Putting $C_i(z) = A_i(z)/A_{d+1}(z)$ we can define another differential operator \[ {\cal P} = {\cal P}(\Theta) = \sum_{i =0}^{{d+1}} C_i(z)\Theta^i \] which is also a $MU$-operator of order ${d+1}$, where $C_i(z)$ are rational functions in $z$, and $C_{d+1}(z) \equiv 1$. Assume that we have a formal regular solution \[ \Phi(z) = \sum_{i =0}^{\infty} a_n z^n. \] Consider a formal polynomial extension \[M_{z} = {\bf C} [[ z ]] \lbrack \log z \rbrack, \] where $\log z$ is considered as a new transcendent variable. We can define the structure of a left ${\bf D}$-module on $M_{z}$ putting by definition $\Theta \log z = 1$. In fact, $M_{z}$ will be a module over the larger algebra ${\bf D}_z$ containing the new operator $Log\,z$ such that \[ z \circ(\Theta \circ Logz\,) = (\Theta \circ Log\,z) \circ z = 1, \] \[ \Theta \circ Log\,z - Log\,z \circ \Theta =1, \] and $Log\,z$ acts on $M_z$ by multiplication on $\log z$. \begin{prop} Let ${\cal P} = \sum_{i =0}^{{d+1}} C_i(z)\Theta^i$ be any operator in ${\bf D}$. Then \[ \lbrack {\cal P}, Log\, z \rbrack = \sum_{i =1}^{{d+1}} iC_i(z)\Theta^{i-1} = {\cal P}_{\Theta}', \] where ${\cal P}_{\Theta}'$ is a formal derivative of ${\cal P}$ with respect to $\Theta$. \end{prop} {\bf Proof.} The statement follows from relation \[ \Theta^i \circ Log\, z - Log\, z \circ \Theta^i = i \Theta^{i-1} \] which can be proved by induction. \begin{flushright}$\Box$\end{flushright} \bigskip Assume that we want to find a element $\Phi_1(z)$ in $M_z$ such that ${\cal P} \Phi_1(z) = 0$ and $\Phi_1(z)$ has form \[ \Phi_1(z) = \log z \cdot \Phi_0(z) + \Psi(z) \] where $\Psi(z)$ is an element of ${\bf C} [[ z ]]$, and $\Psi(0) =0$. \begin{prop} The element $\Psi(z)$ satisfies the linear non-homogeneous differential equation \begin{equation} {\cal P}_{\Theta}' \Phi_0(z) + {\cal P} \Psi(z) = 0, \label{exp-diff} \end{equation} or, formally, \[ \Psi(z) = - {\cal P}^{-1} {\cal P}_{\Theta}'\Phi_0(z) = \partial_{\Theta} \log {\cal P} \cdot \Phi_0(z). \] \label{solut} \end{prop} {\bf Proof.} Since $\Phi_0$ and $\Phi_1$ are solutions, we obtain \[ 0 = {\cal P}\Phi_1 = {\cal P} \log z \Phi_0 + {\cal P} \Psi = \] \[ = ( Log \,z \circ{\cal P} + \lbrack {\cal P}, Log\, z \rbrack)\Phi_0 + {\cal P}\Psi = \lbrack {\cal P}, Log\, z \rbrack \circ \Phi_0(z) + {\cal P}\Psi = {\cal P}_{\Theta}' \Phi_0 + {\cal P}\Psi_0 = 0.\; \Box \] \begin{prop} If $\Phi_0(z)$ is the socle solution, then the function $\Psi(z)$ is uniquely defined by the equations (\ref{exp-diff}) and the condition $\Psi(0) = 0$ as an element of ${\bf C}[[z]]$. \label{log-q} \end{prop} {\bf Proof. } Let $\Psi(z) = \sum_{i = - \infty }^{+ \infty} b_i z^i$ be an element of $z{\bf C}[[z ]]$, i.e., $b_i = 0$ for $i \leq 0$. By \ref{solut}, for any $n \in {\bf Z}$, the coefficient by $z^n$ in ${\cal P}\Psi(z)$ is \[ P_m(n)b_n + P_{m-1}(n-1)b_{n-1} + \cdots + P_0(n -m)b_{n-m}. \] On the other hand, the coefficient by $z^n$ in ${\cal P}_{\Theta}'\Phi_0(z)$ is \[ P_m'(n)a_n + P_{m-1}'(n-1)a_{n-1} + \cdots + P_0'(n -m)a_{n-m}. \] Thus, we obtain the recurrent linear non-homogeneous relation \begin{equation} P_m(n)b_n + \sum_{i =1}^{m} P_{m -i}(n-i) b_{n -i} + \sum_{i =0}^{m} P_{m -i}'(n-i) a_{n -i} = 0. \label{rec-psi} \end{equation} Since $P_m(n) = n^{d+1} \neq 0$ for $n \geq 1$, one can find all coefficients $b_i$ $(i \geq 1)$ using (\ref{rec-psi}). For instance, we obtain \[ b_1 = -(d+1)a_1 - P_{m-1}'(0)a_0 \] \[ 2^{d+1}b_2 = -(d+1) 2^d a_2 - P_{m-1}(1)b_1 - P_{m-1}'(1) a_1 - P_{m-2}'(0)a_0; \; ...\; {\rm etc.} \] \begin{coro} Let ${\cal P}$ be a $MU$-operator then the quotient $\Psi/\Phi$ of the solutions of the linear system \[ {\cal P}\Phi =0,\;\; {\cal P}_{\Theta}' \Phi + {\cal P}\Psi,\;\; \Psi(0) = 0 \] is a function depending only on ${\cal P}$. \end{coro} We come now to the most important definition: \begin{opr} {\rm The element \[ q = \exp \left( \frac{\Phi_1(z)}{\Phi_0(z)} \right) = z \exp \left( \frac{\Psi(z)}{\Phi_0(z)} \right) \] is called the {\em q-parameter} for the $MU$-operator ${\cal P}$.} \end{opr} \subsection{Generalized hypergeometric functions and $2$-term recurrent relations} \noindent Since the number $m +1$ of terms in a recurrent relation (\ref{recurrent}) is at least $2$, $2$-term recurrencies are the simplest ones. Any such a relations is defined by two polymomials $P_0(y)$ and $P_1(y)$ of degree ${d+1}$: \begin{equation} P_0(n)a_{n} = P_1(n+1)a_{n +1}. \label{rec-2} \end{equation} Without loss of generality we again assume that the leading coefficient of $P_1(y)$ is $1$. \begin{opr} {\em Denote by \[ G_{d+1}(\alpha,\beta;w) = G_{d+1} \left( \begin{array}{c} \alpha_1, \ldots, \alpha_{d+1} \\ \beta_1, \ldots, \beta_{d+1} \end{array}; w \right) \] the series \[ \sum_{n \geq 0} \frac{ \prod_{i =1}^{d+1} \Gamma(\alpha_i)} {\prod_{i =1}^{d+1} \Gamma(\beta_i)} \times \left( \frac{ \prod_{i =1}^{d+1} \Gamma(\beta_i +n )} { \prod_{i =1}^{d+1} \Gamma(\alpha_i +n)} \right) w^n. \] which is the generalized hypergeometric function with parameters $\alpha_1,$ $\dots,$ $\alpha_{d+1}$, $\beta_1,$ $\ldots,$ $\beta_{d+1}$. (This is a slight modification of the well-known generalized hypergeometric function $\,_{d+1}F_{{d}}$ (see \cite{hyp.geom,slater}).)} \end{opr} \begin{prop} Assume that \[ P_1(y) = \prod_{i =1} (y + \alpha_i); \] \[ P_0(y) = \lambda \prod_{i =1} (y + \beta_i) \] then the function $G_{d+1}(\alpha,\beta; \lambda z)$ is a formal solution of the differential equation \[ {\cal P}\Phi = (P_1(\Theta) - z P_0(\Theta))\Phi = 0. \] \end{prop} Consider now the case when ${\cal D}$ is a $MU$-operator, i.e., $P_1(y) = y^{d+1}$, and the recurrent relation has the form \[ (n+1)^{d+1} a_{n+1} = P_{0}(n) a_n. \] Then for the power series $\Psi(z) = \sum_{i \geq 1} b_i z^i$ which is the solution to \[ {\cal D}_{\Theta}' \Phi_0(z) + {\cal P} \Psi(z) = 0, \] where \[ \Phi_0(z) = \sum_{i = 0}^{\infty} a_i z^i, \] is a regular solution to ${\cal P}\Psi =0$, the coefficients $\{ b_i \}$ satisfy the recurrent relation \[ n^{d+1}b_n = P_1(n-1)b_{n-1} + P_1'(n-1)a_{n-1} - (d+1)n^d a_n. \] \subsection{$d$-point Yukawa functions} \noindent Let $\pi\; :\; V_z \rightarrow S$ be a $1$-parameter family of Calabi-Yau $d$-folds, where $S = {\rm Spec}\, {\bf C}[[z]]$. Let $T$ be the corresponding monodromy transformation acting on $H_d(W_z, {\bf C})$, $T_u$ the unipotent part part of $T$, $N = Log\, T_u$. \begin{opr} {\rm The family $V_z$ is said to have the {\em maximal unipotent monodromy at $z = 0$} if the weight filtration \[ 0 = W_{-1} \subset W_0 \subset W_1 \subset \cdots \subset W_{d-1} \subset W_d = H^d(V_z, {\bf C}) \] defined by $N$ is orthogonal to the Hodge filtration $\{ F^i \}$, i.e., \[ H^d(V_z, {\bf C}) = W^{\perp}_{i} \oplus F^{d-i} \; i =0, \ldots , d.\]} \end{opr} (This is similar to definitions given in \cite{greene1,morrison.hodge}.) Assume that the family $V_z / S$ has the maximal unipotent monodromy at $z =0$ and ${\rm dim}\, F^i/F^{i+1} = 1$ for $ i = 0,\ldots, d$. Then the Jordan normal form of $N$ has exactly {\em one} sell of size $(d+1)\times (d+1)$. This means that there exists a $d$-cycle $\gamma \in H_d(V_z, {\bf Z})$ such that $\gamma, N\gamma, \ldots, N^d \gamma$ are linearly independent in $H_d(V_z, {\bf Z})$, and $N^d \gamma = \gamma_0$ is a monodromy invariant $d$-cycle. Take a $1$-parameter family $\omega(z)$ of holomorphic $d$-forms on $W_z$. It is well-known that the periods of $\omega(z)$ over the $d$-cycles in $H_d(V_z,{\bf C})$ satisfy a Picard-Fuchs differential equation of order $d+1$ defined by some differential $MU$-operator \begin{equation} {\cal P} = \Theta^{d+1} + C_d(z) \Theta^d + \cdots + C_0(z). \label{P-F} \end{equation} \begin{opr} {\rm Define the coupling functions $W_{k,l}(z)$ $(k,l \geq 0, \; k,l \in {\bf Z})$ as follows \[ W_{k,l} = \int_{V_z} \Theta^k \omega(z) \wedge \Theta^l \omega(z). \] (By definition, we put $\Theta^0 = 1$ to be the identity differential operator.) } \label{couplings} \end{opr} \begin{opr} {\rm \cite{morrison.mirror} The coupling function $W_{d,0}$ is called {\em unnormalized $d$-point Yukawa function}.} \end{opr} \begin{prop} The coupling functions $W_{k,l}(z)$ satisfy the properties {\rm (i)} $W_{k,l}(z) = (-1)^d W_{l,k}$; {\rm (ii)} $W_{k,l}(z) = 0$ for $k + l < d$; {\rm (iii)} $\Theta W_{k,l}(z) = W_{k+1,l}(z) + W_{k,l+1}(z)$. {\rm (iv)} $W_{d+k +1,0}(z) + C_d(z) W_{d+k,0}(z) + \cdots + C_0(z)W_{k,0} (z) = 0.$ \label{w-properties} \end{prop} {\bf Proof.} The statements follow immediatelly from the properties of the cup-product and from the Griffiths transversality property. \begin{theo} The $d$-pont Yukawa function $W_{d,0}(z)$ satisfies the linear differential equation of order one \begin{equation} \Theta W_{d,0}(z) + \frac{2}{d+1}C_d(z) W_{d,0} =0. \label{coup-eq} \end{equation} \end{theo} {\bf Proof. } By \ref{w-properties}(ii), we have \begin{equation} W_{d-i,i}(z) + W_{d-i-1,i+1}(z) =0\;\;{\rm for}\; i =0,1, \dots, d. \label{sum1} \end{equation} Therefore, $W_{d,0}(z) = (-1)^i W_{d-i,i}$. On the other hand, by \ref{w-properties}(iii), we have \begin{equation} \Theta W_{d-i,i} = W_{d-i+1,i}(z) + W_{d-i,i+1}(z) \;\;{\rm for}\; i =0,1, \dots, d. \label{sum2} \end{equation} It follows from (\ref{sum1}) and (\ref{sum2}) that \begin{equation} k \Theta W_{d,0}(z) = \sum_{i=0}^{k-1} (-1)^i W_{d-i,i}(z) = W_{d+1,0}(z) + (-1)^{k-1} W_{d- k + 1,k}(z). \label{sum3} \end{equation} {\sc Case I:} $d$ is odd. Since \[ W_{\frac{d+1}{2}, \frac{d+1}{2}}(z) = 0\;\; {\rm (\ref{w-properties}(i))},\] we obtain \[ \Theta W_{\frac{d+1}{2}, \frac{d-1}{2}}(z) = W_{\frac{d+3}{2}, \frac{d-1}{2}}(z). \] Using (\ref{sum1}) and (\ref{sum3}) for $k = (d-1)/2$, we obtain \[ \frac{(d+1)}{2} \Theta W_{d,0}(z) = W_{d+1,0}(z). \] By \ref{w-properties}(ii) and (iv), this implies the equation (\ref{coup-eq}) for $W_{d,0}(z)$. {\sc Case I:} $d$ is odd. One has \[ \Theta W_{\frac{d}{2}, \frac{d}{2}}(z) = W_{\frac{d+2}{2}, \frac{d}{2}}(z) + W_{\frac{d}{2}, \frac{d+2}{2}}(z) = 2 W_{\frac{d+2}{2}, \frac{d}{2}}(z). \] Using (\ref{sum1}) and (\ref{sum3}) for $k = d/2$, we obtain \[ (d+1)\Theta W_{d,0}(z) = 2 W_{d+1,0}(z). \] The latter again implies the same linear differential equation for $W_{d,0}(z)$. \begin{coro} \[ W_{d,0}(z) = c_0\exp \left( {-\frac{2}{d+1} \int_0^z C_d(v) \frac{dv}{v}}\right)\] for some nonzero constant $c_0 = W_{d,0}(0)$. \label{solution} \end{coro} \begin{exam} {\rm Assume that ${\cal P} = \Theta^{d+1} - zP_0(\Theta)$ be the $MU$-operator correspondng to a $2$-term recurrent relation $(n+1)^{d+1} a_{n+1} = P_0(n)a_n$, where $P_0(y) = \lambda y^{d+1} + \cdots$. Then the Yukawa $d$-point function $W_{d,0}(z)$ equals \[ W_{d,0}(z) = \frac{c_0}{1 - \lambda z}, \] i.e., $W_{d,0}(z)$ is a rational function in $z$. } \label{yukawa} \end{exam} \subsection{Multidimensional Picard-Fuchs differential systems with a symmetry group} \noindent So far we considered only the case of the $1$-parameter family of Calabi-Yau $d$-folds $V_z$ such that ${\rm dim}\, F^i/F^{i+1} =1$ for $ i =0, \ldots, d$. It is easy to see that the same methods can be applied to the case ${\rm dim}\, F^i/F^{i+1} \geq 1$ provided $V_z$ has a large authomorphisms group. \medskip \begin{prop} Let $V_z$ be a $1$-parameter family of Calabi-Yau $d$-folds with ${\rm dim}\, F^i/F^{i+1}$ $\geq 1$. Assume that there exists an action of a finite group $G$ on $V_z$ such that the $G$-invariant part $(F^i/F^{i+1})^G$ is $1$-dimensional for all $i =0, \ldots, d$. Then the holomorphic differential $d$-form $\omega(z)$ again satisfies the Picard-Fuchs differential equation of order $d +1 $. \label{symmetry} \end{prop} {\bf Proof. } The statement immediately follows from the fact that the cohomology classes of $\omega(z), \Theta \omega(z), \ldots, \Theta^d \omega(z)$ form the basis of the $G$-invariant subspace $H^d(V_z, {\bf C})^G \subset H^d(V_z, {\bf C})$. \hfill $\Box$. \bigskip \section{Calabi-Yau complete intersections in ${\bf P}^N$} \subsection{Rational curves and generalized hypergeometric series} \noindent Let $V$ be a complete intersection of $r$-hypersurfaces $V_1, \ldots, V_r$ of degrees $d_1, \ldots, d_r$ in ${\bf P}^{d+r}$. Then $V$ is a Calabi-Yau $d$-fold if and only if $d + r + 1 = d_1 + \ldots + d_r$. A rational curve $C$ of degree $n$ in ${\bf P}^{d+r}$ has $nd_i$ intersection points with a generic hypersurface $V_i$. On the other hand, there exists a degeneration of every divisor $V_{i}$ into the union of $d_i$ hyperplanes. Each of these hyperplanes has $n$ intersection points with $C$. This motivates the definition of the corresponding generalized hypergeometric series $\Phi_0(z)$ as \begin{equation} \sum_{i =0}^{\infty} \frac{(nd_1!)}{(n!)^{d_1}} \cdots \frac{(nd_r !)} {(n!)^{d_r}} z^n \label{proj-space} \end{equation} The coefficients \[ a_n = \frac{(nd_1!)\cdots (nd_r !)}{ (n!)^{d+r +1}} \] satisfy the recurrent relation \[ (n+1)^{d+1} a_{n+1} = P(n) a_n \] where $P(y)$ is the polynomial of degree $d +1$: \[ P(n) = \frac{(nd_1 + d_1)!}{(nd_1) !} \cdots \frac{(nd_r + d_r)!}{(n d_r) !}(n+1)^{-r} = \lambda n^{d+1} + \cdots . \] In particular, the leading coefficient of $P(y)$ is $\lambda = \prod_{ i=1}^{r} (d_i)^{d_i}$. \begin{exam} {\rm Let $V$ be a complete intersection of two cubics in ${\bf P}^5$. The corresponding generalized hypergeometric series is \[ \Phi_0(z) = \sum_{n \geq 0} \frac{(3n!)^2}{(n!)^6} z^n. \] This series was found in \cite{lib.teit} using the explicit construction of mirrors for $V$ by orbifolding the $1$-parameter family of special complete intersections of two cubics in ${\bf P}^5$: \[ Y_1^3 + Y_2^3 + Y_3^3 = 3 \psi Y_4Y_5Y_6; \] \[ Y_4^3 + Y_5^3 + Y_6^3 = 3 \psi Y_1Y_2Y_3, \] by an abelian group $G$ of order $81$. where $z = (3\psi)^{-6}$. \medskip We will give another interpretation of the construction of mirrors $V'$ for $V$ which immediatelly implies that $\Phi_0(z)$ is the monodromy invariant period for the regular differential $3$-form on $V'$. Let $Z_{f_1 f_2}$ be the complete intersection of two hypersurfaces in a $5$-dimensional algebraic torus ${\bf T} = {\rm Spec} [ X_1^{\pm 1}, \ldots, X_5^{\pm 1} ]$ defined by the Laurent polynomials \[ f_1(u,X) = 1 - (u_1 X_1 + u_2 X_2 + u_3 X_3), \; f_2(u,X) = 1 - (u_4 X_4 + u_5X_5 + u_6(X_1 \cdots X_5)^{-1} ). \] We define the differential $3$-form $\omega$ on $Z_{f_1 f_2}$ as the residue of the rational differential $5$-form on ${\bf T}$: \[ \omega = \frac{1}{(2 \pi \sqrt{-1})} {\rm Res} \frac{ 1}{f_1(u,X)f_2(u,X)} \frac{dX_1}{X_1} \wedge \cdots \wedge \frac{dX_5}{X_5}.\] Let $z = u_1 \cdots u_6$. By residue theorem, we obtain \[ \Phi_0(z) = \frac{1}{(2 \pi \sqrt{-1})} \int_{\mid X_i \mid =1} \frac{ 1}{f_1(u,X)f_2(u,X)} \frac{dX_1}{X_1} \wedge \cdots \wedge \frac{dX_5}{X_5}.\] In this interpretation, the mirrors for $V$ are smooth Calabi-Yau compactifications of of affine $3$-folds $Z_{f_1 f_2}$. The equivalence between the above two construction of mirrors for $V$ follows by the substitution \[ X_1 = Y_1^3/(Y_3Y_4 Y_5), \; X_2 = Y_2^3/(Y_3Y_4 Y_5), \; X_3 = Y_1^3/(Y_3Y_4 Y_5), \; \] \[ X_4 = Y_4^3/(Y_1Y_2 Y_3), \; X_5 = Y_5^3/(Y_1Y_2 Y_3), \;\] \[ u_1 = \cdots = u_6 = (3\psi)^{-1}. \] } \label{3-3} \end{exam} \begin{prop} The normalized Yukawa $d$-differential for Calabi-Yau complete intersections has the form \[ {\cal W}_{d} = \frac{d_1 \cdots d_r}{ (1 - \lambda z)\Phi_0^2(z)} \cdot (\frac{dz}{z})^{\otimes d}, \] where $d_1, \ldots, d_i$ are degrees of hypersurfaces. \end{prop} {\bf Proof. } The statement follows from \ref{yukawa} and the normalizing condition $d_1 \cdots d_r = W_d(0)$. \hfill $\Box$ \subsection{The construction of mirrors} \noindent Let $V$ be a $d$-dimensional Calabi-Yau complete intersection of $r$ hypersurfaces of degrees $d_1, \ldots, d_r$ in ${\bf P}^{d + r}$. We propose the explicit construction of candidats for mirrors with respect to $V$ as follows: Let $E = \{ v_1, \ldots, v_{d +r +1}\} $ be a generating set in the $(d+r)$-dimensional lattice $N \cong {\bf Z}^{d +r}$ such that there exist the relation \[ v_1 + \cdots + v_{d +r + 1} =0. \] We divide $E$ into a disjoint union of $r$ subsets $E_i \subset E$ such that ${\rm Card}\, E_i = d_i$. For $i =1, \ldots, r$, we define the Laurent polynomial $P_i(u, X)$ in variables $X_1, \ldots, X_{d+r}$ as \[ P_i(X) = 1 - (\sum_{v_j \in E_i} u_j X^{v_j}), \] where $u_1, \ldots, u_{d +r +1}$ are independent parameters. We denote by $V'$ a Calabi-Yau compactification of $d$-dimensional affine complete intersections $Z$ in ${\bf T} = {\rm Spec}\, [ X_1^{\pm 1}, \ldots, X_{d +r}^{\pm1}]$ defined by the polynomials $P_1(u,X), \ldots, P_r(u,X)$ with sufficiently general coefficients $u_i$. It is easy to see that up to an isomorphism the affine variaties $Z \subset {\bf T}$ depend only on $z = u_1 \cdots u_{d+r+1}$. Thus, we have obtained a $1$-parameter family of $d$-dimensional varieties $V'$. \begin{conj} The $1$-parameter family of $d$-dimensional varieties $V'$ yields the mirror family for $V$. \end{conj} This conjecture is motivated by the combinatorial interpretation proposed in \cite{batyrev.dual} of the well-known construction of mirrors for hypersurfaces of degree $d+2$ in ${\bf P}^{d+1}$ (see \cite{greene}). On the other hand, the conjecture is supported by the following property: \begin{prop} The hypergeometric series $\Phi_0(z)$ in {\rm (\ref{proj-space})} is the monodromy invariant period function of the holomorphic $d$-form $\omega$ on $V'$. \end{prop} {\bf Proof. } The statement follows from the equality \[\sum_{i =0}^{\infty} \frac{(nd_1!)}{(n!)^{d_1}} \cdots \frac{(nd_r !)} {(n!)^{d_r}} z^n = \frac{1}{(2\pi \sqrt{-1})^{d +r}} \int_{\mid X_j \mid =1} \frac{1}{P_{1}(X) \cdots P_{r}(X)} \frac{dX_1}{X_1} \wedge \cdots \wedge \frac{dX_{d+r}}{X_{d+r}}. \] \hfill $\Box$. \bigskip \section{Complete intersections in toric varieties} \subsection{The generalized hypergeometric series $\Phi_0$} \noindent Let $N$ be a free abelian group of rank $(d+r)$. Consider $r$ finite sets \[ E_i = \{ v_{i, 1}, \ldots, v_{i,k_i} \},\; i =1, \ldots, r \] consisting of elements $v_{i,j} \in N$. Let $E$ be the union $E_1 \cup \cdots \cup E_r$. We put $ k = {\rm Card}\, E = k_1 + \ldots + k_r$ and assume that $E$ generates the group $N$. Let $R(E)$ be the subgroup in ${\bf Z}^n$ consisting of all integral vectors $\lambda = \{ \lambda_{i,j} \}$ such that \[ \sum_{i=1}^r \sum_{j =1}^{k_i} \lambda_{i,j} v_{i,j} = 0. \] We denote by $R^+(E)$ submonoid in $R(E)$ consisiting of all $\lambda = \{ \lambda_{i,j} \} \in R(E)$ such that $\lambda_{i,j} \geq 0$. \begin{opr} {\rm Let $u_{i,j}$ be $k$ independent complex variables parametrized by $k$ integral vectors $v_{i,j}$. Define the power series $\Phi_0(u)$ as \[ \Phi_0(u) = \sum_{\lambda \in R^+(E)} \prod_{i =1}^r (\sum_{j =1}^{k_i} \lambda_{i,j})! \left( \prod_{j =1}^{k_i} \frac{u_{i,j}^{\lambda_{i,j}}}{(\lambda_{i,j})!}\right) .\]} \end{opr} Let $\lambda^{(1)}, \ldots, \lambda^{(t)}$ be a ${\bf Z}$-basis of the lattice $R(E)$ such that every element $\lambda \in R^+(E)$ is a non-negative integral linear combination of $\lambda^{(i)}$. We define new $r$ complex variables $z_1, \ldots, z_s$ as follows \[ z_s = \prod_{i =1}^t \prod_{j =1}^{k_i} u_{i,j}^{\lambda_{i,j}^{(s)}}; \; s =1 , \ldots, t. \] Thus, the series $\Phi_0(u)$ can be rewritten as the power series $\Phi_0(z)$ in $t$ variables $z_1, \ldots, z_t$. \medskip \begin{exam} {\rm Let $E = \{ v_1, \ldots, v_{d+1} \}$ be vectors generating $d$-dimensional lattice $N$ and satisfying the integral relation $v_1 + \cdots + v_{d+1} = 0$, i.e., the group $R(E)$ is generated by the vector $(1, \ldots, 1) \in {\bf Z}^{d+1}$. Then the corresponding generalized hypergeometric series is \[ \Phi_0(u) = \sum_{n \geq 0} \frac{(nd +n)!}{(n!)^{d+1}} (u_1 \cdots u_{d+1})^n = \sum_{n \geq 0} \frac{(nd +n)!}{(n!)^{d+1}} z^n = \Phi_0(z), \] where $z = u_1 \cdots u_{d+1}$. The integral representation of this series is the monodromy invariant period function for mirrors of hypersurfaces of degree $(d+1)$ in ${\bf P}^d$. } \end{exam} \begin{opr} {\rm Let ${\bf T}$ be a $(d +r)$-dimensional algebraic torus with the Laurent coordinates $X = (X_1, \ldots, X_{d+r})$. We define $r$ Laurent polynomials $P_{E_1}(X), \ldots, P_{E_r}(X)$ as follows \[ P_{E_i}(X) = 1 - \sum_{v_{i,j} \in E_i} u_{i,j} X^{v_{i,j}}. \] } \end{opr} \begin{prop} The series $\Phi_0(u)$ admits the following integral representation \[ \Phi_0(u) = \frac{1}{(2\pi \sqrt{-1})^{d +r}} \int_{\mid X_j \mid =1} \frac{1}{P_{E_1}(X) \cdots P_{E_r}(X)} \frac{dX_1}{X_1} \wedge \cdots \wedge \frac{dX_{d+r}}{X_{d+r}}. \] \label{integral} \end{prop} {\bf Proof. } The statement follows immediately from the residue formula. \hfill $\Box$ \subsection{Calabi-Yau complete intersections} \noindent Let ${\bf P}_{\Sigma}$ be a quasi-smooth $(d+r)$-dimensional projective toric variety defined by a $(d +r)$-dimensional simplicial fan $\Sigma$. Assume that there exist $r$ line bundles ${\cal L}_1, \ldots, {\cal L}_r$ such that each ${\cal L}_i$ is generated by global sections and the tensor product \[ {\cal L}_1 \otimes \cdots \otimes {\cal L}_r \] is isomorphic to the anticanonical bundle on ${\cal K}^{-1}$ on ${\bf P}_{\Sigma}$. If $V_i$ is the set of zeros of a generic global section of ${\cal L}_i$, then the complete intersection $V = V_1 \cap \cdots \cap V_r$ is a $d$-dimensional Calabi-Yau variety having only Gorenstein toroidal singularities which are analytically isomorphic to toric singularities of ${\bf P}_{\Sigma}$. \medskip Now let $E= \{v_1, \ldots, v_k\}$ be the set of all generators of $1$-dimensional cones in $\Sigma$. Denote by $D_j$ the toric divisor on ${\bf P}_{\Sigma}$ corresponding to $v_j$. Notice that \[ {\cal K}^{-1} = \otimes_{j=1}^k {\cal O}_{{\bf P}_{\Sigma}}(D_j). \] Following a suggestion of Yu. I. Manin, we assume that one can represent $E$ as a disjoint union \[ E = E_1 \cup \cdots \cup E_r \] such that the line bundle ${\cal L}_i$ is isomorphic to the tensor product \[ \otimes_{v_j \in E_i} {\cal O}_{{\bf P}_{\Sigma}}(D_j). \] The elements of the group $R(E)$ can be identified with the homology classes of $1$-dimensional algebraic cycles on ${\bf P}_{\Sigma}$. Moreover, one has the following property \begin{prop} Let $\lambda = (\lambda_1, \ldots, \lambda_k)$ be an arbitrary element of $R(E)$ representing the class of an algebraic $1$-cycle $C$. Then \[ \lambda_i = \langle D_i, C \rangle, \;i =1, \ldots, k. \] \end{prop} We can always choose a ${\bf Z}$-basis $\lambda^{(1)}, \ldots, \lambda^{(t)}$ of $R(E)$ such that every {\em effective} $1$-cycle on ${\bf P}_{\Sigma}$ is a non-negative linear combination of the elements $\lambda^{(1)}, \ldots, \lambda^{(t)}$. Since the submonoid $R^+(E)$ consists of classes of {\em nef-curves}, this implies that every element of $R^+(E)$ is also a non-negative linear combination of $\lambda^{(1)}, \ldots, \lambda^{(t)}$. This allows us to rewrite the series $\Phi_0(u)$ in $t$ algebraically independent variables $z_1\, \ldots, z_t$ $( t = {\rm rk}\, R(E))$. \begin{coro} The series $\Phi_0(z)$ can be interpreted via the intersection numbers of classes $\lbrack C \rbrack$ of curves $C$ on ${\bf P}_{\Sigma}$ as follows \[ \Phi_0(z) = \sum_{\lbrack C \rbrack \in R^+(E)} \frac{ (\langle V_1,C \rangle)! \cdots (\langle V_r, C \rangle)!} {\langle D_1, C \rangle ! \cdots \langle D_k, C \rangle !} z^{\lbrack C \rbrack}, \] where $z^{\lbrack C \rbrack} = z_1^{c_1} \ldots z_t^{c_t}$, $ \lbrack C \rbrack = c_1 \lambda^{(1)} + \cdots + c_t \lambda^{(t)}$. \end{coro} \subsection{General conjectures} \noindent Let $V$ be a $d$-dimensional Calabi-Yau complete intersection of hypersurfaces $V_1, \ldots, V_r$ in a $(d+r)$-dimensional quasi-smooth toric variety defined by a simplicial fan $\Sigma$. Choose a ${\bf Z}$-basis $\lambda^{(1)}, \ldots, \lambda^{(t)}$ in $R(E)$ such that the classes of all effective $1$-cycles have non-negative integral coordinates. We assume that the divisors $V_1, \ldots, V_r$ are numerically effective (in particular, they are not assumed to be necessary ample divisors). We assume also that the following conditions are satisfied: {\rm (i)} $V$ is smooth; {\rm (ii)} the restriction mapping ${\rm Pic}\, {\bf P}_{\Sigma} \rightarrow {\rm Pic}\, V$ is injective. \medskip In this situation, there exist two flat A-modle connections: the connection $\nabla_{AP}$ on $H^*({\bf P}_{\Sigma})$ and the connection $\nabla_{AV}$ on $H^*(V, {\bf C})$. Let $\tilde{H}^i$ be the image of $H^i({\bf P}_{\Sigma}, {\bf C})$ in $H^i(V, {\bf C})$. The connection $\nabla_{AP}$ defines the quantum variation on cohomology of toric variety ${\bf P}_{\Sigma}$. It follows from the result in \cite{batyrev.quant} the following. \begin{prop} The complex coordinates $z_1, \ldots, z_t$ on $\tilde{H}^2$ can be identified with flat coordinates with respect to $\nabla_{AP}$. \end{prop} \begin{conj} The generalized hypergeometric series $\Phi_0(z)$ as a function of $\nabla_{AP}$-flat $z$- coordinates on $\tilde{H}^2$ is a solution of the differential system ${\cal D}$ defined by the A-model connection $\nabla_{AV}$ on $\tilde{H}^2$ which defines the quantum variation of Hodge structures on the subring in $\bigoplus_{i = 0}^{d} H^{2i}(V, {\bf C})$ generated by restrictions of the classes in ${\rm Pic}\, {\bf P}_{\Sigma}$ to $V$. \label{conj.1} \end{conj} \begin{rem} {\rm One can check in many examples that the differential system ${\cal D}$ is already defined by the generalized hypergeometric series $\Phi_0(z)$. Probably there exists a general explanation of this fact.} \end{rem} \begin{conj} The $\nabla_{AV}$-flat coordinates $q_1, \ldots, q_t$ on $\tilde{H}^2$ are defined as \[ q_i = \exp (\Phi_i(z)/\Phi_0(z)), \; i = 1, \ldots, t, \] where $\Phi_i(z)$ is a logarithmic solution to the differential system ${\cal D}$ having the form \[ \Phi_i(z) = (\log z_i) \Phi_0(z) + \Psi_i(z), \; \Psi_i(0) = 0 \] for some regular at $z =0$ power series $\Psi_i(z)$. Moreover, all coefficients of the expansion of $\nabla_{AV}$-flat coordinates $q_i$ as power series of $\nabla_{AP}$-flat $z$-coordinates are {\bf integers}. \label{conj.2} \end{conj} \begin{rem} {\rm This conjecture establishes a general method for normalizing the logarithimic solutions defining the canonical $q$-coordinates for the differential system ${\cal D}$. There are two motivations for this conjecture. First, the conjecture is true for all already known examples of $q$-coordinates for Picard-Fuchs equations corresponding to Calabi-Yau complete intersections in products of weighted projective spaces (see examples in the remaining part of the paper). Second, the Lefschetz theorem and the calculation of the quantum cohomolgy ring of toric varieties \cite{batyrev.quant} imply the relation \[ q_i = z_i + O(\mid z \mid^2),\; ( i =1, \ldots, t). \] } \end{rem} \medskip \begin{conj} Assume that $V$ has dimension $3$. Let $K_{i,j,k}(z)$ be structure constants defining the A-model $\nabla_{AV}$ connection in the $z$-coordinates. Then \[ \Phi_0^2(z) K_{i,j,k}(z) \] is a rational function in $z$-coordinates. \label{conj.21} \end{conj} \begin{conj} The mirror Calabi-Yau varieties with respect to $V$ are Calabi-Yau compactifications of the complete intersection of the affine hypersurfaces in the $(d+r)$-di\-men\-sio\- nal algebraic torus ${\bf T}$ defined by the equations \[ P_{E_1}(X) = \cdots = P_{E_r}(X) = 0.\] \label{conj.3} \end{conj} \begin{rem} {\rm Recall that two Calabi-Yau $d$-folds $V$ and $V'$ are called mirror symmetric if $h^{p,d-p}(V) = h^{d-p,d-p}(V')$ and the superconformal field theories corresponding to $V$ and $V'$ are isomorphic. It was proposed in \cite{batyrev.dual} a general method for constructing pairs of mirror symmetric Calabi-Yau hypersurfaces in toric varieties based on the duality among so called {\em reflexive polyhedra} $\Delta$ and $\Delta^*$. However, the equality $h^{1,1}(\hat{Z}_f) = h^{d-1,1}(\hat{Z}_g)$ for the pair of Calabi-Yau $d$-folds $\hat{Z}_f$ and $\hat{Z}_g$ corresponding to the polyhedra $\Delta$ and $\Delta^*$ are not sufficient to prove the mirror duality between $\hat{Z}_f$ and $\hat{Z}_g$ in full strength. One needs to prove more: the isomorphism between the quantum cohomology of $\hat{Z}_f$ and $\hat{Z}_g$. Since the quantum cohomology are defined by the canonical form of the A-model connection $\nabla_A$ in $q$-coordinates, Conjecture \ref{conj.1} and Proposition \ref{integral} yield more evidence for validity of Conjecture \ref{conj.3}. We give below one example showing that Conjecture \ref{conj.3} agrees with an orbifold construction of mirrors for complete intersections in product of projective spaces inspired by superconformal field theories. } \end{rem} \begin{exam} {\rm Let $V$ be a Calabi-Yau complete intersection of two hypersurfaces of degrees $(3,0)$ and $(1,3)$ in the product ${\bf P}^3 \times {\bf P}^2$. It is known that the mirrors for $V$ can be obtained by orbifolding the complete intersection of two special hypersurfaces \[ S_1 T_1^3 + S_2 T_2^3 + S_3 T_3^3 = \phi S_4 T_1 T_2 T_3, \] \[ S_1^3 + S_2^3 + S_3^3 + S_4^3 = \psi S_1 S_2 S_3 \] by the group $G$ of order $27$, where $(S_1: S_2 : S_3 :S_4)$ and $(T_1 : T_2 : T_3)$ are the homogeneous coordinates on ${\bf P}^3$ and ${\bf P}^2$ respectively. On the other hand, the $5$-dimensional fan $\Sigma$ defining ${\bf P}^3 \times {\bf P}^2$ has $7$ generators $\{ v_1,$ $\ldots,$ $v_7 \}$ $= E$ satisfying the relations \[ v_1 + v_2 + v_3 + v_6 = v_4 + v_5 + v_7 = 0. \] We choose vectors $v_1, \ldots, v_5$ as the basis of the $5$-dimensional lattice $N$. The complete intersection $V$ is defined by dividing $E$ into two subsets $E_1 = \{v_1, v_2, v_3 \}$ and $E_2 =\{ v_4, v_5, v_6, v_7 \}$. The corresponding polynomials $P_{E_1}(X)$ and $P_{E_2}(X)$ are \[ P_{E_1} = 1 - (u_1 X_1 + u_2 X_2 + u_3 X_3), \] \[ P_{E_2} = 1 - (u_4 X_4 + u_5 X_5 + u_6(X_1 X_2 X_3)^{-1} + u_7(X_4 X_5)^{-1}). \] We obtain the equivalence between two construction of mirrors by putting $u_1 = u_2 = u_3 = \phi^{-1}$, $u_4 = u_5 = u_6 = u_7 = \psi^{-1}$, and \[ X_1 = \frac{S_1 T_1^2}{S_4 T_2 T_3}, \; X_2 = \frac{S_2 T_2^2}{S_4 T_1 T_3}, \; X_3 = \frac{S_3T_3^2}{S_4 T_1 T_2 }, \] \[ X_4 = \frac{S_1^2}{S_2S_3},\; X_5 = \frac{S_2^2}{S_1S_3}. \]} \end{exam} \bigskip \subsection{Calabi-Yau $3$-folds with $h^{1,1} = 1$} \noindent We consider below examples of the generalized hypergeometric series corresponding to smooth Calabi-Yau complete intersections $V$ of $r$ hypersurfaces in a toric variety ${\bf P}_{\Sigma}$ such that $h^{1,1}(V) = 1$. By Lefschetz theorem, $h^{1,1}({\bf P}_{\Sigma})$ must be also $1$. So $\Sigma$ is a $(r +3)$-dimensional fan with $(r +4)$ generators. There exists the unique primitive integral linear relation $\sum \lambda_i v_i = 0$ among the generators $\{v_i \}$ of $\Sigma$, i.e., ${\rm rk}\, R(E) =1$ where $E = \{ v_i \}$ is the whole set of generators of $\Sigma$ (${\rm Card}\, E = r + 4$). \medskip In all these examples the $MU$-operator ${\cal P}$ has form \[ {\cal P} = \Theta^4 - \mu z (\Theta + \alpha_1)(\Theta + \alpha_2) (\Theta + \alpha_3)(\Theta + \alpha_4), \] where the numbers $\alpha_1, \ldots, \alpha_4$ are positive rationals satisfying the relations \[ \alpha_1 + \alpha_4 = \alpha_2 + \alpha_3 = 1. \] The Yukawa $3$-differential in $z$-coordinate has form \[ {\cal W}_3 = \frac{W(0)}{(1 - \mu z)\Phi_0^2(z)} \left(\frac{dz}{z}\right)^{\otimes 3}. \] \begin{exam} {\sl Hypersurfaces in weighted projective spaces:} {\rm In this case we obtain Ca\-la\-bi-Yau hypersurfaces in the following weighted projective spaces ${\bf P}(\lambda_1, \cdots, \lambda_5)$ \begin{center} \begin{tabular}{|c|c|c|c|c|} \hline $(\lambda_1, \ldots, \lambda_5)$ & $\Phi_0(z)$ & $ W(0)$ & $\mu$ & $(\alpha_1, \alpha_2,\alpha_3, \alpha_4)$ \\ \hline $(1,1,1,1,1)$ & ${\displaystyle \sum_{n \geq 0} \frac{(5n)!}{(n!)^5}z^n}$ & 5 & $5^5$ & $(1/5,2/5,3/5,4/5)$ \\ \hline $(2,1,1,1,1)$ & ${\displaystyle \sum_{n \geq 0} \frac{(6n)!}{(n!)^4(2n!)}z^n}$ & $ 3 $ & $2^5 3^6$ & $(1/6,2/6,4/6,5/6)$ \\ \hline $(4,1,1,1,1)$ & ${\displaystyle \sum_{n \geq 0} \frac{(8n)!}{(n!)^4(4n!)}z^n}$ & $ 2 $ & $2^{18}$ & $(1/8,3/8,5/8,7/8)$ \\ \hline $(5,2,1,1,1)$ & ${\displaystyle \sum_{n \geq 0} \frac{(10n)!}{(n!)^3(2n!)(5n!)}z^n}$ & $ 1 $& $2^9 5^6$ & $(1/10,3/10,7/10,9/10)$ \\ \hline \end{tabular} \end{center} The $q$-expansion of the Yukawa $3$-point function and predictions $n_d$ for number of rational curves on these hypersurfaces were obtained in \cite{morrison.picard,klemm1,font}.} \end{exam} \medskip \begin{exam} {\sl Complete intersections in ordinary projective spaces:} {\rm Let $V_{d_1, \ldots, d_r}$ denotes the complete intersection of hypersurfaces of degrees $d_1, \ldots, d_r$. \begin{center} \begin{tabular}{|c|c|c|c|c|} \hline & $\Phi_0(z)$ & $W(0)$ & $\mu$ & $(\alpha_1, \alpha_2,\alpha_3, \alpha_4)$ \\ \hline $V_{3,3} \subset {\bf P}^5$ & ${\displaystyle \sum_{n \geq 0} \frac{((3n)!)^2}{(n!)^6}z^n}$ & $9$ & $3^6$ & $(1/3,1/3,2/3,2/3)$ \\ \hline $V_{2,4} \subset {\bf P}^5 $ & ${\displaystyle \sum_{n \geq 0} \frac{(2n)!(4n)!}{(n!)^6}z^n}$ & $8$ & $2^{10}$ & $(1/4,2/4,2/4,3/4)$ \\ \hline $V_{2,2,3} \subset {\bf P}^6$ & ${\displaystyle \sum_{n \geq 0} \frac{((2n)!)^2(3n)!}{(n!)^7}z^n}$ & $12$ & $2^4 3^3$ & $(1/3,1/2,1/2,2/3)$ \\ \hline $V_{2,2,2,2} \subset {\bf P}^8$ & ${\displaystyle \sum_{n \geq 0} \frac{((2n)!)^4}{(n!)^8}z^n}$ & $16$ & $2^8$ & $(1/4,1/4,1/4,1/4)$ \\ \hline \end{tabular} \end{center} These Calabi-Yau complete intersections in ordinary projective spaces were considered by Libgober and Teitelbaum \cite{lib.teit}. } \end{exam} \medskip \begin{exam} {\sl Complete intersections in weighted projective spaces:} {\rm \begin{center} \begin{tabular}{|c|c|c|c|c|} \hline & $\Phi_0(z)$ & $W(0)$ & $\mu$ & $(\alpha_1, \alpha_2,\alpha_3, \alpha_4)$ \\ \hline $V_{4,4} \in {\bf P}(1,1,1,1,2,2)$ & ${\displaystyle \sum_{n =0}^{\infty} \frac{(4n!)^2}{(n!)^4 (2n!)^2} z^n }$ & $4$ & $2^{12}$ & $(1/4,1/4,3/4,3/4)$ \\ \hline $V_{6,6} \in {\bf P}(1,1,2,2,3,3)$ & ${\displaystyle \sum_{n =0}^{\infty} \frac{(6n!)^2}{(n!)^2 (2n!)^2(3n!)^2}z^n }$ & $1$ & $2^8 3^6$ & $(1/6,1/6,5/6,5/6)$ \\ \hline $V_{3,4} \in {\bf P}(1,1,1,1,1,2)$ & ${\displaystyle \sum_{n =0}^{\infty} \frac{(4n!)(3n!)}{(n!)^5 (2n!)} z^n}$ & $ 6$ & $2^6 3^3$ & $(1/4,1/3,2/3,3/4)$ \\ \hline $V_{2,6} \in {\bf P}(1,1,1,1,1,3)$ & ${\displaystyle \sum_{n =0}^{\infty} \frac{(6n!)(2n!)}{(n!)^5 (3n!)} z^n }$ & $4$ & $2^8 3^3$ & $(1/6,1/2,1/2,5/6)$ \\ \hline $V_{4,6} \in {\bf P}(1,1,1,2,2,3)$ & ${\displaystyle \sum_{n =0}^{\infty} \frac{(6n!)(4n!)}{(n!)^3 (2n!)^2(3n!)} z^n}$ & $2$ & $2^{10}3^3$ & $(1/6,1/4,3/4,5/6)$ \\ \hline \end{tabular} \end{center} The coefficients of the Yukawa 3-point function $K_q^{(3)}$ for these five examples of Calabi-Yau $3$-folds $V$ having the Hodge number $h^{1,1}(V) =1$ were obtained by A. Klemm and S. Theisen \cite{klemm2}. } \end{exam} \bigskip \section{Calabi-Yau 3-folds in ${\bf P}^2 \times {\bf P}^2$} \subsection{The generalized hypergeometric series $\Phi_0$} \noindent Calabi-Yau $3$-folds $V$ in ${\bf P}^2 \times {\bf P}^2$ are hypersurfaces of degree $(3,3)$. The homology classes of rational curves on ${\bf P}^2 \times {\bf P}^2$ are parametrized by pairs of integers $(l_1, l_2)$. Let $\gamma_1$, $\gamma_2$ be the homology classes of $(1,0)$-curves and $(0,1)$-curves respectively. Then for any K\"ahler class $\eta$ we put \[ z_i = \exp (- \int_{\gamma_i} \eta ),\;\;(i =1,2) . \] The generalized hypergeometric series corresponding to the fan $\Sigma$ defining ${\bf P}^2 \times {\bf P}^2$ is \[ \Phi_0(z_1,z_2) = \sum_{l_1,l_2 \geq 0} \frac{(3l_1 + 3l_2)!}{(l_1!)^3 (l_2!)^3} z_1^{l_1}z_2^{l_2}. \] There are obvious two recurrent relations for the coefficients $a_{l_1,l_2}$ of the series \[ \Phi_0(z_1,z_2) = \sum_{l_1,l_2 \geq 0} a_{l_1,l_2} z_1^{l_1}z_2^{l_2} :\] \[ (l_1 + 1)^3 a_{l_1+1,l_2} = (3l_1 + 3l_2 +1)(3l_1 + 3l_2 +2) (3l_1 + 3l_2 +3)a_{l_1, l_2}; \] \[ (l_2 + 1)^3 a_{l_1,l_2+1} = (3l_1 + 3l_2 +1)(3l_1 + 3l_2 +2) (3l_1 + 3l_2 +3)a_{l_1, l_2}; \] Let \[ \Theta_1 = z_1 \frac{\partial}{\partial z_1}, \;\; \Theta_2 = z_2 \frac{\partial}{\partial z_2} \] Then the function $\Phi_0(z_1,z_2)$ satisfies the Picard-Fuchs differential system ${\cal D}$: \[ \left( \Theta_1^3 - z_1 (3 \Theta_1 + 3\Theta_2 + 1 ) ( 3\Theta_1 + 3\Theta_2 + 2 )( 3\Theta_1 + 3\Theta_2 + 3 ) \right) \Phi_0 = 0 ; \] \[ \left( \Theta_2^3 - z_2 ( 3 \Theta_1 + 3 \Theta_2 + 1 ) ( 3 \Theta_1 + 3\Theta_2 + 2 )( 3 \Theta_1 + 3\Theta_2 + 3 ) \right) \Phi_0 = 0. \] The differential system ${\cal D}$ has the maximal unipotent monodromy at $(z_1,z_2) =(0,0)$. There are two uniquely determined regular at $(0,0)$ functions $\Psi_1(z_1,z_2)$ and $\Psi_2(z_1,z_2)$ such that \[ (\log z_1) \Phi_0(z_1,z_2) + \Psi_1(z_1,z_2), \] \[ (\log z_2) \Phi_0(z_1,z_2) + \Psi_2(z_1,z_2) \] are solutions to ${\cal D}$, and $\Psi_1(0,0) = \Psi_2(0,0) =0$. If we put \[ \Psi_j(z_1,z_2) = \sum_{ \begin{array}{c} {\scriptstyle l_1,l_2 \geq 0}\\ {\scriptstyle (l_1, l_2) \neq (0,0)} \end{array}} b_{l_1,l_2}^{(j)} z_1^{l_1} z_2^{l_2}, \] then one finds the coefficients $b_{l_1,l_2}^{(j)}$ from the simple recurrent relations based on \ref{solut}. The $q$-coordinates $q_1$, $q_2$ defined by the formulas \[ q_1 = z_1 \exp (\Psi_1/\Phi_0), \;q_2 = z_2 \exp (\Psi_2/\Phi_0) \] are the power series with integral coefficients in $z_1, z_2$ of the form \[ q_j (z_1,z_2) = z_j \left( 1 + \sum_{ \begin{array}{c} {\scriptstyle l_1,l_2 \geq 0}\\ {\scriptstyle (l_1, l_2) \neq (0,0)} \end{array}} c_{l_1,l_2}^{(j)} z_1^{l_1} z_2^{l_2} \right), \; j =1,2. \] By symmetry, one has $c_{l_1,l_2}^{(1)} = c_{l_2,l_1}^{(2)}$. \subsection{Mirrors and the discriminant} \noindent Let $f$ be the Laurent polynomial \[ f(X,u) = 1 - u_1X_1 - u_2X_2 - u_3(X_1X_2)^{-1} - u_4 X_3 - u_5X_4 -u_6 (X_3 X_4)^{-1}. \] Let $\gamma_0$ be a generator of $H_4(({\bf C}^*)^4, {\bf Z})$, i.e., the cycle defined by the condition $\mid X_i \mid =1$ for $i =1,\ldots 4$. By the residue theorem, the integral \[ I(u) = \frac{1}{(2\pi \sqrt{-1})^4} \int_{\gamma_0} \frac{1}{f(X)}\frac{dX_1}{X_1}\wedge \frac{dX_2}{X_2}\wedge \frac{dX_3}{X_3}\wedge \frac{dX_4}{X_4}\] is the power series \[ I(u) = \sum_{k,m \geq 0} \frac{(3k + 3 m)!}{(k!)^3 (m!)^3} (u_1 u_2 u_3)^k (u_4 u_5 u_6)^m. \] Thus, putting $z_1 = u_1 u_2 u_3$; $z_2 = u_4 u_5 u_6$, we obtain exactly the generalized hypergeometric function $\Phi_0(z_1,z_2)$. It was proved in \cite{batyrev.dual} and \cite{batyrev.var} that the function $I(u)$ can be considered as the monodromy invariant period of the holomorphic differential $3$-form \[ \omega = \frac{1}{(2\pi \sqrt{-1})^4} {\rm Res}\; \frac{1}{f(X)}\frac{dX_1}{X_1}\wedge \frac{dX_2}{X_2}\wedge \frac{dX_3}{X_3}\wedge \frac{dX_4}{X_4}.\] for the family of Calabi-Yau 3-folds $\hat{Z}_f$ which are smooth compactifications of the affine hypersurfaces $Z_f$ in $({\bf C}^*)^4$ defined by Laurent polynomial $f$. One has $h^{1,1}(\hat{Z}_f)= 83$, $h^{2,1}(\hat{Z}_f) = 2$. The coordinates $z_1, z_2$ are natural coordinates on the moduli space of Calabi-Yau $3$-folds $\hat{Z}_f$. \bigskip The mirror construction helps to understand the the discriminant of the differential system ${\cal D}$ as a polynomial function in $z_1, z_2$. By definition \cite{gelfand}, the zeros of the discriminat are exactly those values of the coefficients $\{u_i \}$ of $f(X)$ such that the system \[ f(X) = X_1 \frac{\partial }{\partial X_1}f(X) = X_2 \frac{\partial }{\partial X_2}(X) = X_3 \frac{\partial }{\partial X_3}f(X) = X_4 \frac{\partial }{\partial X_4}f(X) = 0 \] has a solution in the toric variety ${\bf P}_{\Delta}$, where $\Delta$ is the Newton polyhedron of $f$. Since ${\bf P}_{\Delta}$ is isomorphic to the subvariety of ${\bf P}^6$ defined as \[ {\bf P}_{\Delta} = \{ ( Y_0 : \ldots : Y_6) \in {\bf P}^6 \mid Y_0^3 = Y_1 Y_2 Y_3, Y_0^3 = Y_4 Y_5 Y_6 \}, \] or equivalently, the system of the homogeneous equations \[ u_0 Y_0 + \cdots + u_6 Y_6 = u_1 Y_1 - u_3 Y_3 = u_2 Y_2 - u_3 Y_3 = u_4 Y_4 - u_6 Y_6 = u_5 Y_5 - u_6 Y_6 = 0; \] \[ Y_0^3 = Y_1 Y_2 Y_3 = Y_4 Y_5 Y_6 \] has a non-zero solution. If we put \[ A = u_3 Y_3,\; B = u_6 Y_6,\; C = u_0 Y_0 \] then the last system can be rewritten as \[ 3A + 3B + C = A^3 + z_1 C^3 = B^3 + z_2 C^3 = 0. \] So the discriminant the two-parameter family is the resultant of two binary homogeneous cubic equations in $A$ and $B$: \[ 27z_1 (A + B)^3 - A^3 = 0,\; 27z_2 (A + B)^3 - B^3 = 0. \] Put $27z_1 = x$, $27z_2 =y$. \begin{prop} The discriminant of the $2$-parameter family of Calabi-Yau $3$-folds $\hat{Z}_f$ is \[ {\rm Disc}\,f = 1 - (x+y) + 3(x^2 - 7xy + y^2) - (x^3 + 3x^2y + 3xy^2 + y^3). \] \end{prop} \subsection{The diagonal one-parameter subfamily} \noindent We consider the diagonal one-parameter subfamily of K\"ahler structures $\eta$ on $V$ which are invariant under the natural involution on $H^{1,1}(V)$, i.e., we assume that \[ \int_{\gamma_1} \eta = \int_{\gamma_2} \eta. \] This is equivalent to the substitution $z = z_1 = z_2$. \begin{rem} {\rm In this case we obtain the one-parameter family of mirrors \[ f_{\psi}(X) = X_1 + X_2 + (X_1 X_2)^{-1} + X_3 + X_4 + (X_3 X_4)^{-1} - {3}{\psi} = 0, \; {\psi}^3 = (27 z)^{-1} \] which is an analog to mirrors of quintic $3$-folds \cite{cand2}. } \end{rem} It is easy to check that the discriminant of $f_{\psi}(X)$ vanishes exactly when $\psi = \alpha + \beta$, where $\alpha^3 = \beta^3 =1$, i.e., $\psi^3 \in \{ 8, -1 \}$, or $z \in \{ -(3)^{-3}, (2\cdot3)^{-3} \}$. The monodromy invariant period function is \[ F_0(z) = \Phi_0(z,z) = \sum_{n \geq 0} \left( \sum_{k + m = n} \frac{(3n)!}{(k!)^3 (m!)^3} \right) z^n. \] It satisfies an ordinary Picard-Fuchs differential equation \[ {\cal D}\;: \; (\Theta^4 + \sum_{i =0}^3 C_i(z)\Theta^i )F(z) = 0,\;\; \Theta = z\frac{\partial}{\partial z}. \] We compute the Picard-Fuch differential equation ${\cal E}$ for $F_0(z)$ is from the recurrent formula for the coefficients \[ a_n = \sum_{k + m = n} \frac{(3n)!}{(k!)^3 (m!)^3} = \frac{(3n)!}{(n!)^3} ( \sum_{k =0}^n { n \choose k }^3) \] in the power expansion \[ F_0(z) = \sum_{n \geq 0} a_n z^n .\] \begin{prop} {\rm (\cite{stienstra}) } Let \[ b_n = \sum_{k =0}^n { n \choose k }^3. \] Then the numbers $b_n$ satisfy the recurrent relation \[ (n+1)^2 b_{n+1} = (7 n^2 + 7n + 2) b_{n} + 8 n^2 b_{n-1}. \] \end{prop} \begin{coro} The numbers $a_n$ satisfy the recurrent relation \[ (n+1)^4 a_{n+1} = 3( 7n^2 + 7n + 2) (3n +2) (3n +1) a_n + 72( 3n + 2)( 3n + 1)( 3n - 1)( 3n - 2)a_{n-1}. \] \end{coro} \begin{coro} The monodromy invariant period function $F_0(y)$ is annihilated by the differential operator ${\cal P}$ : \[ \Theta^4 - 3z (7 \Theta^2 + 7\Theta + 2) ( 3\Theta + 1) ( 3 \Theta + 2) - 72 z^2 ( 3\Theta + 5) (3 \Theta + 4) ( 3\Theta + 2)( 3\Theta + 1). \] \end{coro} The last operator can be rewritten also as \[ (1 - 216z)(1 + 27z) \Theta^4 - 54z( 7 + 432z) \Theta^3 - 3 z (10584 z + 95) \Theta^2 - 48z (351z + 2)\Theta - 12z - 2880z^2 .\] In particular, one has the coefficient \[ C_3(z) = \frac{-54z( 7 + 432z)}{(1 - 216z)(1 + 27z)}. \] The $z$-normalized Yukawa coupling $K_z^{(3)}$ is the solution to the differential equation \[ \frac{dK_z^{(3)}}{dz} = \frac{27( 7 + 432z)} {(1 - 216z)(1 + 27z)} K_z^{(3)}. \] Let $H$ be the cohomology class in $H^2(V, {\bf Z})$ such that $\langle H, \gamma_1 \rangle = \langle H, \gamma_2 \rangle = 1$. Since $H^3 = 18$, we obtain the normalization condition \[ K_{z}^{(3)}(0) = 18. \] Applying the general algorithm in \ref{log-q}, we find the $q$-expansion of the $z$-coordinate \[ z(q) = q - 48 q^2 - 18q^3 + 7976 q^4 - 1697115 q^5 + O(q^6), \] and the $q$-expansion of the $q$-normalized Yukawa coupling is \[ K_q^{(3)} = 18 + 378q + 69498 q^2 + 7724862 q^3 + 1030043898 q^4 + 132082090128 q^5 + O(q^6). \] We expect that \[ K_q^{(3)} = 18 + \sum_{d =1}^{\infty} \frac{n_d d^3 q^d}{1- q^d}. \] where $n_d$ are predictions for numbers rational curves of degree $d$ relative to the ample divisor of type $(1,1)$ on $V$. In paricular, one has $n_1 = 378.$ \subsection{Lines on a generic Calabi-Yau 3-fold in ${\bf P}^2 \times {\bf P}^2$} \noindent We show how to check the prediction for the number of lines on a generic Calabi-Yau 3-fold in ${\bf P}^2 \times {\bf P}^2$. First we formulate one lemma which will be useful in the sequel. \begin{lem} Let $M$ be a complete algebraic variety, ${\cal L}_1$ and ${\cal L}_2$ two invertible sheaves on $M$ such that the projectivizations ${\bf P}({\cal L}_i) = {\bf P} (H^0(M, {\cal L}_i))$ $(i =1,2)$ are nonempty. Define the morphism \[ p_{\lambda} \; : \; {\bf P}({\cal L}_1) \times {\bf P}({\cal L}_2) \rightarrow {\bf P}({\cal L}_1 \otimes {\cal L}_2) = {\bf P} ( H^0(M, {\cal L}_1 \otimes {\cal L}_2))\] by the natural mapping \[ \lambda\; : \; H^0(M, {\cal L}_1) \otimes H^0(M, {\cal L}_2) \rightarrow H^0(X, {\cal L}_1 \otimes {\cal L}_2). \] Then the pullback $p_{\lambda}^* {\cal O}(1)$ of the ample generator ${\cal O}(1)$ of the Picard group of ${\bf P}({\cal L}_1 \otimes {\cal L}_2)$ is isomorphic to ${\cal O}(1,1)$ on ${\bf P}({\cal L}_1) \times {\bf P}({\cal L}_2)$. \label{mapping} \end{lem} {\bf Proof.} The statement follows immediately from the fact that $\lambda$ is bilinear. \hfill $\Box$ \begin{prop} A generic Calabi-Yau hypersurface in ${\bf P}^2 \times {\bf P}^2$ contains $378$ lines relative to the ${\cal O}(1,1)$-polarization. \end{prop} {\bf Proof. } There are two possibilities for the type of lines: $(1,0)$ and $(0,1)$. By symmetry, it is sufficient to consider only $(1,0)$-lines whose projections on the second factor in ${\bf P}^2 \times {\bf P}^2$ are points. Let \[ \pi_2 \;: \; V \rightarrow {\bf P}^2 \] be the projection of $V$ on the second factor. Then for every point $p \in {\bf P}^2$ the fiber $\pi_2^{-1}(p)$ is a cubic in ${\bf P}^2 \times {p}$. We want to calculate the number of those fibers $\pi_2^{-1}(p)$ which are unions of a line $L$ and a conic $Q$ in ${\bf P}^2 \times {p}$. The space of the reducible cubics $L \cup Q$ is isomorphic to the image $\Lambda \subset {\bf P}^9 = {\bf P}({\cal O}_{{\bf P}^2}(3))$ of the morphism \[ {\bf P}( {\cal O}_{{\bf P}^2}(1)) \times {\bf P}( {\cal O}_{{\bf P}^2}(2))) = {\bf P}^2 \times {\bf P}^5 \rightarrow {\bf P}^9 = {\bf P}( {\cal O}_{{\bf P}^2}(3)). \] By \ref{mapping}, $\Lambda$ has codimension 2 and degree $21$. On the other hand, a generic Calabi-Yau hypersurface $V$ defines a generic Veronese embedding \[ \phi \; : \; {\bf P}^2 \hookrightarrow {\bf P}^9 = {\bf P}( {\cal O}_{{\bf P}^2}(3)), \; \; \phi(p) = \pi_2^{-1} (p). \] The degree of the image $\phi({\bf P}^2)$ is $9$. The number of $(1,0)$-lines is the intersection number of two subvarieties $\phi({\bf P}^2)$ and $\Lambda$ in ${\bf P}^9$, i.e., $9 \times 21 = 189$. Thus, the total amount of lines is $2 \times 189 = 378$. \hfill $\Box$ \bigskip \section{Further examples} \noindent In this section we consider more examples of Calabi-Yau $3$-folds $V$ obtained as complete intersections in product of projective spaces. In all these examples for simplicity we restrict ourselves to one-parameter subfamilies invariant under permutations of factors. The latter allows to apply the Picard-Fuchs operators of order $4$ to the calculation of predictions for numbers of rational curves on Calabi-Yau $3$-folds with $h^{1,1} > 1$. \bigskip \subsection{Calabi-Yau $3$-folds in ${\bf P}^1 \times {\bf P}^1 \times {\bf P}^1 \times {\bf P}^1$} \noindent We consider the diagonal subfamily of K\"ahler classes on Calabi-Yau hypersurfaces of degree $(1,1,1,1)$ in $({\bf P}^1)^4$. Repeating the same procedure as for hypersurfaces of degree $(3,3)$ in ${\bf P}^2 \times {\bf P}^2$, we obtain: \begin{center} \begin{tabular}{|c|c|} \hline & \\ $F_0(z)$ & $ {\displaystyle \sum_{n =0}^{\infty} \left( \sum_{ k_1 + k_2 + k_3 + k_4 = n } \frac{( 2k_1 + 2k_2 + 2k_3 + 2k_4 )! } { (k_1!)^2 (k_2!)^2 (k_3!)^2 (k_4!)^2 } \right) z^n }$ \\ & \\ \hline & \\ ${\cal P}$ & $\Theta^4 - 4 z (5\Theta^2 + 5 \Theta + 2) (2\Theta +1) + 64 z^{2} (2 \Theta +3) (2 \Theta +1) (2\Theta + 2 )^2 $ \\ & \\ \hline & \\ $ K_z^{(3)}$ & ${\displaystyle \frac{48}{(1 - 64z)(1 -16z)} }$ \\ & \\ \hline & \\ $ K_q^{(3)}$ & $ 48 + 192q + 7872 q^2 + 278400 q^3 + 9445056 q^4 + 315072192 q^5 + O(q^6)$ \\ & \\ \hline & \\ $ n_i$ & $ n_1 = 192,\; n_2 = 960, \; n_3 = 10304,\; n_4 = 147456,\; n_5 = 2520576$ \\ & \\ \hline \end{tabular} \end{center} \begin{prop} The number of lines on a generic Calabi-Yau hypersurface in ${\bf P}^1 \times {\bf P}^1 \times {\bf P}^1 \times {\bf P}^1$ relative to the $(1,1,1,1)$-polarization is equal to $192$. \end{prop} {\bf Proof. } Let $f$ be the polynomial of degree $(2,2,2,2)$ defining a Calabi-Yau hypersurface $V$ in $({\bf P}^1)^4$. If $V$ contains a $(0,0,0,1)$-curve whose projection on the product of first three ${\bf P}^1$ is a point $(p_1,p_2,p_3)$, then all three coefficients of the binary quadric obtained from $f$ by substitution of $(p_1,p_2,p_3)$ must vanish. Hence, the number of $(0,0,0,1)$ curves on $V$ equals the intersection number of $3$ hypersurfaces of degree $(2,2,2)$ in ${\bf P}^1 \times {\bf P}^1 \times {\bf P}^1$. This number is $48$. By symmetry, the total amount of lines on $V$ is $4 \times 48 = 192$. \hfill $\Box$ \begin{prop} The number of conics on a generic Calabi-Yau hypersurface in ${\bf P}^1 \times {\bf P}^1 \times {\bf P}^1 \times {\bf P}^1$ with respect to the $(1,1,1,1)$-polarization is equal to $960$. \end{prop} {\bf Proof.} By symmetry, it is sufficient to compute the number of rational curves of type $(0,0,1,1)$. Let $M$ be the product of first two ${\bf P}^1$ in $({\bf P}^1)^4$. Then we obtain the natural embedding \[ \phi \; : \; M \hookrightarrow {\bf P}^8 = {\bf P}({\cal O}_{{\bf P}^1 \times {\bf P}^1}(2,2)). \] On the other hand, the points on $M$ corresponding to projections of $(0,0,1,1)$-curves on $V$ are intersections of $\phi(M)$ with the $6$-dimensional subvariety $\Lambda \subset {\bf P}^8$ which is the image of the morphism \[ \phi' \; : \; {\bf P}({\cal O}_{{\bf P}^1 \times {\bf P}^1}(1,1))^2 = {\bf P}^3 \times {\bf P}^3 \rightarrow {\bf P}^8 = {\bf P}({\cal O}_{{\bf P}^1 \times {\bf P}^1}(2,2)). \] The image $\phi(M)$ has degree $8$. On the other hand, $\phi$ has degree two onto its image. Hence, the subvariety $\Lambda$ has degree $10$. Hence, we obtain $8 \times 20 = 160$ points on $M$. There are $6$ possibilities for the choice of the type of conics. Thus, the total amount of conics is $6 \times 160 = 960$. \hfill $\Box$ \bigskip \subsection{Complete intersections of three hypersurfaces in ${\bf P}^2 \times {\bf P}^2 \times {\bf P}^2$} \noindent We consider two examples of $3$-dimensional complete intersections with trivial canonical class in $({\bf P}^2)^3$. \medskip {\bf Calabi-Yau complete intersections of $3$ hypersurfaces of degree $(1,1,1)$:} \begin{center} \begin{tabular}{|c|c|} \hline & \\ $F_0(z)$ & $ {\displaystyle \sum_{n =0}^{\infty} \left( \sum_{ k + m + l = n } \frac{(( k+ m +l )!)^3 } { (k!)^3 (m!)^3 (l!)^3 } \right) z^n }$ \\ & \\ \hline & \\ ${\cal P}$ & $ 25 \Theta^4 - 15 z (5 + 30 \Theta + 72 \Theta^2 + 84 \Theta^3 + 51 \Theta^4)$ \\ & $ + 6z^2 (15 + 155 \Theta + 541 \Theta^2 + 828 \Theta^3 + 531 \Theta^4)$ \\ & $- 54 z^3 (1170 + 3795 \Theta + 4399 \Theta^2 + 2160 \Theta^3 + 423 \Theta^4)$ \\ & $ + 243z^4 (402 + 1586\Theta + 2270 \Theta^2 + 1386 \Theta^3 + 279 \Theta^4) - 59049 z^5 (\Theta + 1)^4 $ \\ & \\ \hline & \\ $ K_z^{(3)}$ & ${\displaystyle \frac{90 + 162 z}{(27z -1)(27z^2 +1)} }$ \\ & \\ \hline & \\ $ K_q^{(3)}$ & $90 + 108 q + 2916 q^2 + 57456 q^3 + 834084 q^4 + 13743108 q^5 + O(q^6)$ \\ & \\ \hline & \\ $n_i$ & $n_1 = 108,\; n_2 = 351, \; n_3 = 2124,\; n_4 = 12987,\; n_5 = 109944 $ \\ & \\ \hline \end{tabular} \end{center} \begin{prop} A generic complete intersection of 3 hyper\-sur\-faces of deg\-ree $(1,1,1)$ in ${\bf P}^2 \times {\bf P}^2 \times {\bf P}^2$ contains $108$ lines relative to the ${\cal O}(1,1,1)$-polarization. \end{prop} {\bf Proof.} Let $V$ be the complete intersection of three generic hypersurfaces $V_1$, $V_2$, $V_3$ in $M_1 \times M_2 \times M_3$, where $M_i \cong {\bf P}^2$. By symmetry, it is sufficient to consider lines having the class $(0,0,1)$ whose projections on $M_1 \times M_2$ are points. There is the morphism \[ \phi \; : \; M_1 \times M_2 \rightarrow {\bf P}^8 = {\bf P}(E) \] where $E$ is the space of all $3\times3$-matrices. By definition, $\phi$ maps a point $(p_1, p_2) \in M_1 \times M_2$ to the matrix of coefficents of there linear forms obtained from the substitution of $p_1$ and $p_2$ in the equations of $V_1, V_2$, and $V_3$. The morphism $\phi$ is the Segre embedding and its image has degree $6$. On the other hand, if a point $(p_1,p_2) \in M_1 \times M_2$ is a projection of a $(0,0,1)$-curve on $V$, then the image $\phi(p_1,p_2)$ must correspond to a matrix of rank $1$ in $E$. Thus, the number of $(0,0,1)$-curves equals $6\times 6 = 36$, the intersection number of two Segre subvarieties in ${\bf P}^8$. So the number of lines on $V$ is $3 \times 36 = 108$. \hfill $\Box$ \bigskip {\bf Abelian $3$-folds:} The complete intersection of three hyper\-sur\-fa\-ces of deg\-rees $(3,0,0)$, $(0,3,0)$, and $(0,0,3)$ are are abelian $3$-folds constructed by taking products of $3$ elliptic cubic curves in ${\bf P}^2$. Although abelian varieities are not Calabi-Yau manifolds from view point of algebraic geometers, these manifolds also present interest for physicists. \begin{center} \begin{tabular}{|c|c|} \hline & \\ $F_0(z)$ & $ {\displaystyle \sum_{ p + q + r = n } \left( \frac{((3p)!)^3 ((3q)!)^3 ((3r)!)^3}{(p!)^3 (q!)^3 (r!)^3} \right)z^n }$ \\ & \\ \hline & \\ ${\cal P}$ & $\Theta^4 - 3z (6 + 29\Theta + 56 \Theta^2 + 54 \Theta^3 + 27 \Theta^4) $ \\ & $ + 81 z^2(27 \Theta^2 + 54 \Theta + 40)(\Theta +1)^2 $ \\ & $ - 2187z^3 (3\Theta + 5)(3\Theta +4)(\Theta +2)(\Theta +1) $ \\ & \\ \hline & \\ $ K_q^{(3)}$ &162 \\ & \\ \hline \end{tabular} \end{center} Thus, we obtain that all Gromov-Witten invariants for the abelian $3$-folds are zero which agrees with the fact that there are no rational curves on abelian varieties. \bigskip \subsection{Calabi-Yau $3$-folds in ${\bf P}^3 \times {\bf P}^3$} \noindent {\bf Complete intersections of a hypersurface of degree $(2,2)$ and $2$ copies of hypersurfaces of degree $(1,1)$: } \newpage \begin{center} \begin{tabular}{|c|c|} \hline & \\ $F_0(z)$ & ${\displaystyle \sum_{n =0}^{\infty} \left( \sum_{ k + m =n } \frac{ (2 k + 2m)! ((k +m )!)^2 } {(k!)^4 (m!)^4 } \right) z^n }$ \\ & \\ \hline & \\ ${\cal P}$ & $ \Theta^4 - 4z(3\Theta + \Theta +1) (2\Theta +1)^2 - 4z^2(4\Theta + 5) (4\Theta + 6)(4\Theta + 2)(4\Theta + 3) $ \\ & \\ \hline & \\ $ K_z^{(3)}$ & ${\displaystyle \frac{40}{(1 + 16z)(1 - 64z)} }$ \\ & \\ \hline & \\ $ K_q^{(3)}$ & $ 40 + 160 q + 12640q^2 + 393280 q^3 + 17420640q^4 + 662416160 q^5 + O(q^6) $ \\ & \\ \hline & \\ $ n_i$ & $ n_1 = 160,\, n_2 = 1560, \, n_3 = 14560, \, n_4 = 272000, \, n_5 = 5299328 $ \\ & \\ \hline \end{tabular} \end{center} \bigskip \begin{prop} The number of lines on a generic complete intersections of a hypersurface of degree $(2,2)$ and $2$ copies of hypersurfaces of degree $(1,1)$ is equal to $160$. \label{3hyper} \end{prop} {\bf Proof. } Let $W = {\rm Gr}(2,4) \times {\bf P}^3$ be the $7$-dimensional variety parametrizing all $(1,0)$-lines on ${\bf P}^3 \times {\bf P}^3$. Let ${\cal E}$ be the tautological rank-$2$ locally free sheaf on ${\rm Gr}(2,4)$. We put $c_1({\cal E}) = c_1$, $c_2({\cal E}) = c_2$, and $h$ be the first Chern class of the ample generator $H$ of ${\rm Pic}({\bf P}^3)$. Let $S^2({\cal E})$ be the $2$-nd symmetric power of ${\cal E}$. By standard arguments, we obtain: \begin{lem} The Chern classes $c_1$, $c_2$ generate the cohomology ring of ${\rm Gr}(2,4)$. The elements $1,\,c_1,\,c_2,\,c_1^2,\,c_1c_2,\, c_1^2c_2$ form a ${\bf Z}$-basis of $H^*({\rm Gr}(2,4), {\bf Z})$, and one has the following \[ c_1^4 = c_2^2 = c_1^2,\; c_1^3 = 2c_1 c_2; \] \[ c_1(S^2({\cal E})) = 3c_1,\; c_2(S^2({\cal E})) = 2c_1^2 + 4 c_2, \; c_3(S^2({\cal E})) = 4c_1c_2. \] Moreover, for any invertible sheaf ${\cal L}$ on ${\bf P}^3$, one has \[ c_1(S^2({\cal E})\otimes{\cal L}) = 3c_1 + 3c_1({\cal L}),\; \] \[ c_2(S^2({\cal E})\otimes{\cal L}) = 2c_1^2 + 4 c_2^2 + 2c_1({\cal L})( 3c_1) + 3 c_1^2({\cal L}), \; \] \[ c_3(S^2({\cal E})\otimes{\cal L}) = 4c_1c_2 + c_1({\cal L})(2 c_1^2 + 4c_2) + c_1^2({\cal L})(3c_1) + c_1^3({\cal L}). \] \label{chern.cl1} \end{lem} Then the number of $(1,0)$-lines equals the following product in the cohomology ring of $W$: \[ c_2({\cal E}\otimes{\cal O}(H)) \cdot c_2({\cal E}\otimes{\cal O}(H)) \cdot c_3(S^2({\cal E})\otimes{\cal O}(2H)) \] \[ = (h^2 + c_1 h + c_2)^2 (8 h^3 + 3 c_1 \cdot 4h^2 + 2(c_1^2 + 2c_2)\cdot 2h + 4 c_1 c_2) \] \[ = (8 c_1^2 c_2 h^3 + 4(c_1^2 + 2c_2)^2 h^3 + 24 c_1^2 c_2 h^3 + 8 c_2^2 h^3) \] \[ = (8 + 4\times 10 + 24 + 8) c_1^2 c_2 h^3 = 80 c_1^2 c_2 h^3. \] Thus, the number of $(1,0)$-lines is $80$. By symmetry, the total amount of lines is $160$. \hfill $\Box$ \bigskip {\bf Complete intersections of hypersurfaces of degrees $(1,1)$, $(1,2)$ and $(2,1)$:} \begin{center} \begin{tabular}{|c|c|} \hline & \\ $F_0(z)$ & ${\displaystyle \sum_{n =0}^{\infty} \left( \sum_{ k + m =n } \frac{ (2 k + m)! (k + 2m)!(k +m )! } {(k!)^4 (m!)^4 } \right)z^n }$ \\ & \\ \hline & \\ ${\cal P}$ & $ 529 \Theta^4 - 23z( 92 + 621 \Theta + 1644 \Theta^2 + 2046 \Theta^3 + 921 \Theta^4) $ \\ & $ - z^2(221168 + 1033528 \Theta + 1772673 \Theta^2 + 1328584 \Theta^3 + 380851 \Theta^4) $ \\ & $ - 2z^3 (-27232 + 208932 \Theta + 1028791 \Theta^2 + 1310172 \Theta^3 + 475861 \Theta^4) $ \\ & $ - 68z^4 (-976 - 1664 \Theta + 5139 \Theta^2 + 14020 \Theta^3 + 8873 \Theta^4) $ \\ & $ + 6936z^5 (3\Theta + 4)(3 \Theta + 2)(\Theta +1)^2 $ \\ & \\ \hline & \\ $ K_z^{(3)}$ & ${\displaystyle \frac{46 + 68z}{(54z -1)(z^2 - 11 z -1)}}$ \\ & \\ \hline & \\ $ K_q^{(3)}$ & $46 + 160 q + 9416 q^2 + 251530 q^3 + 9120968 q^4 + 289172660 q^5 + O(q^6)$ \\ & \\ \hline & \\ $ n_i$ & $ n_1 = 160,\; n_2 = 1157, \; n_3 = 9310,\; n_4 = 142368,\; n_5 = 2313380 $ \\ & \\ \hline \end{tabular} \end{center} \bigskip \begin{prop} The number of lines on a generic complete intersections of hypersurfaces of degrees $(2,1)$, $(1,2)$, and $(1,1)$ is equal to $160$. \end{prop} {\bf Proof. } We use the same notations as in \ref{3hyper}. The number of $(1,0)$-lines equals the the following product in the cohomology ring of $W$: \[ c_2({\cal E}\otimes{\cal O}(H)) \cdot c_2({\cal E}\otimes{\cal O}(2H)) \cdot c_3(S^2({\cal E})\otimes{\cal O}(H)) \] \[ = (h^2 + c_1 h + c_2) \cdot (4 h^2 + 2 c_1 h + c_2) \cdot (h^3 + 3 c_1h^2 + 2( c_1^2 + 2 c_2) h + 4 c_1 c_2) \] \[ = (24 c_1^2 c_2 + 2 (5 c_2 + 2c_1^2)(2 c_2 + c_1^2) + 9 c_1^2 c_2 + c_2^2)h^3 \] \[ = (24 + 2 ( 5 + 10 + 4 + 4 ) + 9 + 1)c_1^2 c_2 h^3 = 80 c_1^2 c_2 h^3. \] Thus, the number of $(1,0)$-lines equals $80$. By symmetry, the total amount of lines is $160$. \hfill $\Box$ \bigskip {\bf Hypersurfaces in product of two Del Pezzo surfaces of degree $3$:} A Calabi-Yau hypersurface in product of two Del Pezzo surfaces of degree $3$ is a complete intersections of $(1,1)$, $(3,0)$ and $(0,3)$-hypersurfaces in ${\bf P}^3 \times {\bf P}^3$. \begin{center} \begin{tabular}{|c|c|} \hline & \\ $F_0(z)$ & ${\displaystyle \sum_{n =0}^{\infty} \left( \sum_{ k + m =n } \frac{ (3 k)! (3m)!(k +m )! } {(k!)^4 (m!)^4 } \right) z^n }$ \\ & \\ \hline & \\ ${\cal P}$ & $ \Theta^4 - 3z(4 + 23 \Theta + 53 \Theta^2 + 60 \Theta^3 + 48 \Theta^4)$ \\ & $ + 9z^2(304 + 1344\Theta + 2319\Theta^2 + 1980\Theta^3 + 873 \Theta^4)$ \\ & $ - 162z^3(800 + 3348 \Theta + 5259\Theta^2 + 3888\Theta^3 + 1269 \Theta^4)$ \\ & $ + 2916z^4(688 + 2952\Theta + 4653 \Theta^2 + 3240\Theta^3 + 891 \Theta^4 $\\ & $ - 1417176z^5 (3\Theta +4)(3\Theta + 2)(\Theta +1)^2 $ \\ & \\ \hline & \\ $ K_z^{(3)}$ & ${\displaystyle \frac{54 -972z}{(1 - 54z)(1 -27z)^2} }$ \\ & \\ \hline & \\ $ K_q^{(3)}$ & $54 + 162 q + 7290 q^2 + 119232 q^3 + 3045114 q^4 + 79845912 q^5 + O(q^6)$ \\ & \\ \hline & \\ $ n_i$ & $ n_1 = 162,\; n_2 = 891, \; n_3 = 4410,\; n_4 = 47466,\; n_5 = 638766 $ \\ & \\ \hline \end{tabular} \end{center} \begin{prop} Let $S_1$, $S_2$ be two Del Pezzo surfaces of degree $3$. Then the number of lines on a generic Calabi-Yau hypersurface $V$ in $S_1 \times S_2$ is $162$. \end{prop} {\bf Proof.} If $C$ is a line of type $(1,0)$ on $S_1 \times S_2$, then $\pi_1(C)$ is one of $27$ lines on $S_1$, and $\pi_2(C)$ is a point on $S_2$. Let ${\cal O}_{S_i}(-K)$ denotes the anticanonical bundle over $S_i$. Then the zero set of a generic global section $s$ of $\pi_1^*{\cal O}_{S_1}(-K) \otimes \pi_2^*{\cal O}_{S_2}(-K)$ defines a morphism \[ \phi\; : \; S_2 \rightarrow {\bf P}^3 = {\bf P}({\cal O}_{S_1}(-K)). \] On the other hand, for any line $L \in S_1$, one has the linear embedding \[ \phi'\; : \; {\bf P}({\cal O}_{S_1}(-K-L)) \cong {\bf P}^1 \hookrightarrow {\bf P}^3 = {\bf P}({\cal O}_{S_1}(-K)). \] The intersection number of ${\rm Im}\,\phi$ and ${\rm Im}\,\phi'$ in ${\bf P}^3$ equals $3$, i.e., one has exactly $3$ lines $C$ on a generic $V$ such that $\pi_1(C) = L$ and $\pi_2(C)$ is a point on $S_2$. Thus, there are $3 \times 27 = 81$ lines of type $(1,0)$ on $V$. By symmetry, the total amount of lines is $162$. \hfill $\Box$ \begin{prop} Let $S_1$, $S_2$ be two Del Pezzo surfaces of degree $3$. Then the number of conics on a generic Calabi-Yau hypersurface $V$ in $S_1 \times S_2$ is $891$. \end{prop} {\bf Proof.} If $C$ is a conic of type $(1,1)$ on $S_1 \times S_2$, then $L_1 = \pi_1(C)$ is one of $27$ lines on $S_1$, and $L_2 = \pi_2(C)$ is one of $27$ lines on $S_2$. On the other hand, for any pair of lines $L_1 \in S_1$, $L_2 \in S_2$, the intersection of the product $L_1 \times L_2 \subset S_1 \times S_2$ with $V$ is a conic of type $(1,1)$. So we obtain $27 \times 27 = 729$ conics of type $(1,1)$ on $V$. On the other hand, the number of $(2,0)$ and $(0,2)$ conics is obviously equals to the number of $(1,0)$ and $(0,1)$ lines. Thus, the total number of conics is equal to $729 + 162 = 891$. \hfill $\Box$ \bigskip \subsection{Calabi-Yau $3$-folds in ${\bf P}^4 \times {\bf P}^4$} \noindent {\bf Complete intersection of hypersuraces of degrees $(2,0)$, $(0,2)$, and $3$ copies of hypersurfaces of degree $(1,1)$ :} \begin{center} \begin{tabular}{|c|c|} \hline & \\ $F_0(z)$ & ${\displaystyle \sum_{n =0}^{\infty} \left( \sum_{ k + m = n } \frac{(( k+ m )!)^3 (2k)!(2m)!} { (k!)^5 (m!)^5 } \right) z^n }$ \\ & \\ \hline & \\ ${\cal P}$ & $25 \Theta^4 - 20z(5 + 30\Theta + 72 \Theta^2 + 84 \Theta^3 + 36 \Theta^4) $ \\ & $- 16 z^2 (- 35 - 70 \Theta + 71 \Theta^2 + 268 \Theta^3 + 181 \Theta^4) $ \\ & $+ 256 z^3 (\Theta +1)(165 + 375\Theta + 248\Theta^2 + 37\Theta^3) $ \\ & $+ 1024z^4 (59 + 232 \Theta + 331\Theta^2 + 198\Theta^3 + 39 \Theta^4) + 32768 z^5 (\Theta +1)^4 $ \\ & \\ \hline & \\ $ K_z^{(3)}$ & ${\displaystyle \frac{80 + 128 z}{(1 + 4z)(1 -4z)(1 -32z)}}$ \\ & \\ \hline & \\ $ K_q^{(3)}$ & $80 + 128q + 3776q^2 + 65792 q^3 + 1299136 q^4 + 23104128 q^5 + O(q^6)$ \\ & \\ \hline & \\ $ n_i$ & $n_1 = 128,\; n_2 = 456, \; n_3 = 2432,\; n_4 = 20240,\; n_5 = 184832 $ \\ & \\ \hline \end{tabular} \end{center} \begin{prop} The number of lines on a generic complete intersections of hypersuraces of degrees $(2,0)$, $(0,2)$, and $3$ copies of hypersurfaces of degree $(1,1)$ is equal to $128$. \label{5hyper} \end{prop} {\bf Proof. } Let $W = {\rm Gr}(2,5) \times {\bf P}^4$ be the $10$-dimensional variety parametrizing all $(1,0)$-lines on ${\bf P}^4 \times {\bf P}^4$. Let ${\cal E}$ be the tautological rank-$2$ locally free sheaf on ${\rm Gr}(2,5)$. We put $c_1({\cal E}) = c_1$, $c_2({\cal E}) = c_2$, and $h$ be the first Chern class of the ample generator $H$ of ${\rm Pic}({\bf P}^4)$. Let $S^2({\cal E}))$ be the $2$-nd symmetric power of ${\cal E}$. Again, by standard arguments, we obtain: \begin{lem} The Chern classes $c_1$, $c_2$ generate the cohomology ring of ${\rm Gr}(2,5)$. The elements $1,\,c_1,\,c_2,\,c_1^2,\,c_1c_2,\,c_1^3,\, c_1^2c_2, \, c_1^4, \, c_1c_2^2,\, c_2^3$ form a ${\bf Z}$-basis of $H^*({\rm Gr}(2,5), {\bf Z})$, and satisfy the following relations \[ c_1^4 c_2 = 2 c_1^2 c_2^2 = 2 c_2^3,\; c_1^5 = 5 c_1 c_2^2,\; c_1^6 = 5c_1^2 c_2^2 = 5c_2^3, \; c_1^3 c_2 = 2 c_1 c_2^2. \] \end{lem} Then the number of $(1,0)$-lines equals the following product in the cohomology ring of $W$: \[ c_1({\cal O}(H)\cdot (c_2(S^2({\cal E}))^3 \cdot c_3(S^2({\cal E})) \] \[ = (2h) \cdot (h^2 + c_1 h + c_2)^3 \cdot (4 c_1 c_2) = 64 c_1^2 c_2^2 h^4. \] Thus, the number of $(1,0)$-lines is $64$. By symmetry, the total amount of lines is $128$. \hfill $\Box$ \bigskip {\bf Complete intersection of $5$ hypersurfaces of degree $(1,1)$:} \begin{center} \begin{tabular}{|c|c|} \hline & \\ $F_0(z)$ & ${\displaystyle \sum_{n =0}^{\infty} \left( \sum_{ k + m = n } \frac{(( k+ m )!)^5 } { (k!)^5 (m!)^5 } \right) z^n }$ \\ & \\ \hline & \\ ${\cal P}$ & $49 \Theta^4 - 7z(14 + 91 \Theta + 234 \Theta^2 + 286 \Theta^3 + 155 \Theta^4)$ \\ & $ - z^2(15736 + 66094 \Theta + 102261 \Theta^2 + 680044 \Theta^3 + 16105 \Theta^4) $ \\ & $+ 8z^3 (476 + 3759 \Theta + 9071 \Theta^2 + 8589 \Theta^3 + 2625 \Theta^4)$ \\ & $ - 16 z^4 (184 + 806 \Theta + 1439 \Theta^2 + 1266 \Theta^3 + 465 \Theta^4) + 512 z^5 (\Theta +1)^4$ \\ & \\ \hline & \\ $ K_z^{(3)}$ & ${\displaystyle \frac{70 - 40z}{(32z -1)(z^2 - 11 z -1)} }$ \\ & \\ \hline & \\ $ K_q^{(3)}$ & $K_q(q) = 70 + 100 q + 5300 q^2 + 79750 q^3 + 1966900 q^4 + 37143850 q^5 + O(q^6)$ \\ & \\ \hline & \\ $ n_i$ & $n_1 = 100,\; n_2 = 650, \; n_3 = 2950,\; n_4 = 30650,\; n_5 = 297150$ \\ & \\ \hline \end{tabular} \end{center} \begin{prop} A generic complete intersection of generic $5$ hypersurfaces of degree $(1,1)$ in ${\bf P}^4 \times {\bf P}^4$ contains $100$ lines. \end{prop} {\bf Proof.} We give below two different proofs of the statement. {\bf I}: We keep the notation from the proof of \ref{5hyper}. Then the number of $(1,0)$-lines equals the following product in the cohomology ring of $W$: \[ (c_2(S^2({\cal E}))^5 = (h^2 + c_1 h + c_2)^5 \] \[ = (c_1 h + c_2)65 + 5 (c_1 h + c_2)64 h^2 + 10 (c_1 h + c_2)^3 h^4 = 5 c_1^4 c_2 h^4 + 5 { 4 \choose 2} c_1^2 c_2^2 h^4 + 10 c_2^3 h^4 \] \[ = ( 5 \times 2 + 5 \times 6 + 10) c_1 ^4 c_2 h^4 = 50 c_1^2 c_2^2 h^4. \] Thus, the number of $(1,0)$-lines is $50$. By symmetry, the total amount of lines is $100$. \medskip {\bf II}: Let $M = M_1 \times M_2$ where $M_i \cong {\bf P}^4$ ($i =1,2$). By symmetry, we consider only lines of type $(0,1)$ whose projections on $M_1$ are points. The substitution of a point $p \in M_1$ in the equations of the hypersurfaces $H_1, \ldots, H_5 \subset M$ gives $5$ linear forms $f_1, \ldots, f_5$ in homogeneous coordinates on $M_2$. A point $p \in M_1$ is a projection of a $(0,1)$-line on $H_1 \cap \cdots \cap H_5$ if the system of linear forms has rank $3$. The space of 5 copies of linear forms can be identified with the space $L$ of matrices $5\times5$. We are interested in the determinantal subvariety $D$ in ${\bf P}^{24}$ consisting of matrices of rank $\leq 3$. The subvariety $D$ has the codimension $4$, the ideal of $D$ is generated by all $4\times 4$ minors. Using the free graded resolution of the homogeneous coordinate ring of $D$ as a module over the homogeneous coordinate ring of ${\bf P}^{24}$, we can compute the degree of $D$ which is equal to $50$ (The Hilbert-Poincare series equals $( 1 + 4t + 10 t^2 + 20 t^3 + 10 t^4 + 4 t^5 + t^6)/(1 - t)^{21}$). On the other hand, the equatons of generic hypersurfaces $H_1, \ldots, H_5$ define a generic embedding \[ {\bf P}^4 \hookrightarrow {\bf P}^{24} \] of ${\bf P}^4$ as a linear subspace. So the number of lines of type $(0,1)$ on a generic complete intersection is $50$. Thus, the total number of lines is $100$. \hfill $\Box$ \bigskip {\bf Hypersurfaces in product of two Del Pezzo surfaces of degree $4$:} A Calabi-Yau hypersurface in the product of two Del Pezzo surfaces of degree $4$ is a complete intersection of $5$ hypersurfaces in ${\bf P}^4 \times {\bf P}^4$: two copies of type $(2,0)$, two copies of type $(0,2)$, and one copy of type $(1,1)$. \begin{center} \begin{tabular}{|c|c|} \hline & \\ $F_0(z)$ & ${\displaystyle \sum_{n =0}^{\infty} \left( \sum_{ k + m =n } \frac{ ((2 k)!)^2 ((2m)!)^2(k +m )! } {(k!)^5 (m!)^5 } \right)z^n }$ \\ & \\ \hline & \\ ${\cal P}$ & $ 9 \Theta^4 - 4z (6 + 33\Theta + 73 \Theta^2 + 80 \Theta^3 + 64 \Theta^4) $\\ & $ + 128z^2 (75 + 315\Theta + 527 \Theta^2 + 440 \Theta^3 + 194 \Theta^4)$ \\ & $ - 4096z^3 (66 + 261 \Theta + 397 \Theta^2 + 288 \Theta^3 + 94 \Theta^4)$ \\ & $ + 131072 z^4(19 + 77 \Theta + 117 \Theta^2 + 80 \Theta^3 + 22 \Theta^4) - 8388608 z^5 (\Theta + 1)^4 $ \\ & \\ \hline & \\ $ K_z^{(3)}$ & ${\displaystyle \frac{96 - 1024z}{(1 - 32z)(1 -16z)^2} }$ \\ & \\ \hline & \\ $ K_q^{(3)}$ & $96 + 128 q + 3456 q^2 + 38144q^3 + 572800 q^4 + 9344128 q^5 + O(q^6)$ \\ & \\ \hline & \\ $ n_i$ & $n_1 = 128,\; n_2 = 416, \; n_3 = 1408,\; n_4 = 8896,\; n_5 = 74752 $ \\ & \\ \hline \end{tabular} \end{center} \begin{prop} Let $S_1$, $S_2$ be two Del Pezzo surfaces of degree $4$. Then the number of lines on a generic Calabi-Yau hypersurface $V$ in $S_1 \times S_2$ is $128$. \end{prop} {\bf Proof.} If $C$ is a line of type $(1,0)$ on $S_1 \times S_2$, then $\pi_1(C)$ is one of $16$ lines on $S_1$, and $\pi_2(C)$ is a point on $S_2$. Let ${\cal O}_{S_i}(-K)$ denotes the anticanonical bundle over $S_i$. Then the zero set of a generic global section $s$ of $\pi_1^*{\cal O}_{S_1}(-K) \otimes \pi_2^*{\cal O}_{S_2}(-K)$ defines a morphism \[ \phi\; : \; S_2 \rightarrow {\bf P}^4 = {\bf P}({\cal O}_{S_1}(-K)). \] On the other hand, for any line $L \in S_1$, one has the linear embedding \[ \phi'\; : \; {\bf P}({\cal O}_{S_1}(-K-L)) \cong {\bf P}^2 \hookrightarrow {\bf P}^4 = {\bf P}({\cal O}_{S_1}(-K)). \] The intersection number of ${\rm Im}\,\phi$ and ${\rm Im}\,\phi'$ in ${\bf P}^3$ equals $4$, i.e., one has exactly $4$ lines $C$ on a generic $V$ such that $\pi_1(C) = L$ and $\pi_2(C)$ is a point on $S_2$. Thus, there are $4 \times 16 = 64$ lines of type $(1,0)$ on $V$. By symmetry, the total amount of lines is $128$. \hfill $\Box$ \begin{prop} Let $S_1$, $S_2$ be two Del Pezzo surfaces of degree $3$. Then the number of conics on a generic Calabi-Yau hypersurface $V$ in $S_1 \times S_2$ is $416$. \end{prop} {\bf Proof.} If $C$ is a conic of type $(1,1)$ on $S_1 \times S_2$, then $L_1 = \pi_1(C)$ is one of $16$ lines on $S_1$, and $L_2 = \pi_2(C)$ is one of $16$ lines on $S_2$. On the other hand, for any pair of lines $L_1 \in S_1$, $L_2 \in S_2$, the intersection of the product $L_1 \times L_2 \subset S_1 \times S_2$ with $V$ is a conic of type $(1,1)$. So we obtain $16 \times 16 = 256$ conics of type $(1,1)$ on $V$. In order to compute the number of $(2,0)$-conics, we notice that $S_1$ has exactly $10$ of conic bundle structures. Moreover, these conic bundle structures can be divided into $5$ pairs such that the union of degenerate fibers of each pair is the set of all $16$ lines on $S_1$. A generic global section $s$ of $\pi_1^*{\cal O}_{S_1}(-K) \otimes \pi_2^*{\cal O}_{S_2}(-K)$ defines the anticanonical embedding \[ \phi\; : \; S_2 \hookrightarrow {\bf P}^4 = {\bf P}({\cal O}_{S_1}(-K)). \] On the other hand, the points $p \in S_2$ such that $\phi(p)$ splits into the union of two conics $C_1 \cup C_2$ are exactly intersection points of $\phi(S_2)$ and the image of the embedding \[ \phi' \; : \; {\bf P}({\cal O}_{S_1}(C_1) \times {\cal O}_{S_1}(C_2) \cong {\bf P}^1 \times {\bf P}^1 \hookrightarrow {\bf P}^4 = {\bf P}({\cal O}_{S_1}(-K)). \] Since the image of $\phi'$ has degree $2$, we obtain $8$ points $p \in S_2$. Each such a point yields $2$ conics on $\pi^{-1}_2(p)$. Therefore, for each of $5$ pairs of conic bundle structures we have $16$ $(2,0)$-conics. Thus, the total number of conics is equal to $256 + 2 \times 80 = 416$. \hfill $\Box$ \bigskip