Additional Explanatory Notes on the Analytic Proof of the Finite
  Generation of the Canonical Ring
Additional Explanatory Notes on the Analytic Proof
of the Finite Generation of the Canonical Ring
Yum-Tong Siu 1
Introduction. This set of notes is put together to provide some addi-
tional explanatory material on the analytic proof of the finite generation of
the canonical ring for use by the participants in the Workshop on Minimal
and Canonical Models in Algebraic Geometry in April 16–20, 2007 at the
Mathematical Sciences Research Institute at Berkeley, California.
In late March 2007 before the Workshop Mihai Paun of Strasbourg came
to Harvard for two weeks to pose to me some questions which he and other
people have about the analytic proof of the finite generation of the canoni-
cal ring which I posted in October 2006 on the arXiv.org server [Siu 2006].
Besides orally answering his questions I also wrote up some notes for him
to give him more precise details. This set of notes is compiled by putting
together the notes which I wrote up to answer his questions. This set of
notes consists of two parts.
Part I is about how to apply the general nonvanishing theorem to prove
the precise achievement of stable vanishing order in codimension one. Part
II gives the argument for the precise achievement of the stable vanishing
order for higher codimension. For both Part I and Part II the new powerful
tool in the analytic proof is the use of the second case of the dichotomy of
the modified restriction of the curvature current of the metric of minimum
singularity for the canonical line bundle.
Let V = Vk be an irreducible subvariety of codimension k in the compact
complex algebraic manifold X of finite type whose canonical ring is to be
proved to be finitely generated such that V is an embedded branch of the
stable base point set. A modified restriction ΘV to V of the curvature current
of the canonical line bundle is given as follows. For some ην ≥ 0 and Vν
irreducible of codimension 1 in Vν−1 (k < ν < 0) with V0 = X and η0 = 0,
let Θ0 be the curvature current ΘKX on X of of the metric of minimum
singularity for the canonical line bundle KX and inductively
Θν = (Θν−1 − ην−1Vν−1) |Vν .
1Partially supported by a grant from the National Science Foundation.
http://arxiv.org/abs/0704.1940v1
Then Θk is the modified restriction ΘV to V of the curvature current of
KX for the sequence Vk−1 ⊂ · · · ⊂ V1 ⊂ X of nested subvarieties. For the
canonical decomposition
γj [Yj] +R
of ΘV on V there are two cases. The first case is either J = ∞ or R 6= 0
and the second case is J < ∞ and R = 0, where Yj is a subvariety of
codimension one in V and R is a closed positive current on V whose Lelong
number is zero outside a countable union of subvarieties of codimension at
least two in V . For Part I on how to apply the general nonvanishing theorem
to prove the precise achievement of stable vanishing order in codimension
one, the result of the use of this new powerful tool is that the second case
of the dichotomy must always eventually occur at some positive dimensional
V , because if the first case occurs all the time down to dimension zero then
there is some improvement in the stable vanishing at some point of V in the
multi-directions defined by the sequence of nested subvarieties. The second
case of the dichotomy gives a explicitly constructed section on V (unique
up to a nonzero constant factor) which belongs to the multiplier ideal sheaf
corresponding to ΘV .
In Part I of these additional explanatory notes, in order to make the
arguments of the general nonvanishing theorem more transparent, I treat
separately the two cases of the dichotomy. We use the process of using
the techniques for the Fuijta conjedture by constructing singular metrics
successively to cut down on the dimension of (the projection of) the zero-set
of the multiplier ideal sheaf until we end up with the inevitable second case
of the dichotomy and in the second case of the dichotomy we use the sections
explicitly constructed from the special form of the canonical decomposition of
the modified restriction of the curvature current. In extending the explicitly
constructed sections all the way back to the ambient manifoldX we introduce
the technique of constrained minimum center of log canonical singularity.
In Part II of these additional explanatory notes about the argument for
the precise achievement of the stable vanishing order for higher codimension,
we give its illustration in the low dimensional cases of complex surfaces and
complex threefolds. The illustration in the low dimensional cases avoids the
encumberment of complicated notations in the case of general dimension and
at the same time contains the essence of the argument of the general case.
For the case of a threefoldX in Part II, the key case of higher codimension
is that of an irreducible curve C which is an embedded curve in the stable base
point set. The important ingredient of using the second case of the dichotomy
is the following. For simplicity assume that there is no codimension-one base
point set.
Suppose s1, · · · , sk are pluricanonical sections on X such that C is cut
out by the family Sσ of surfaces defined by pluricanonical sections sσ =
j=1 σjsj parametrized by σ = (σ1, · · · , σk) ∈ Ck − {0}. Suppose s is a
pluricanonical section whose restriction to some Sσ achieves the restriction
of the stable vanishing order ασ on Sσ at some point Pσ of C. Then, after
taking away the restriction of the stable vanishing order ασ of s|Sσ across
C, the resulting section
(sC,σ)
ασ on Sσ (where sC,σ is the canonical section
of C in Sσ), when restricted to C, is the section (up to a nonzero constant
factor) explicitly constructed from the second case of the dichotomy for C in
Sσ. This means that at every point of C outside the finite zero-set Zσ of the
resulting section
(sC,σ)
ασ on C, the section
(sC,σ)
ασ achieves the restriction
of the stable vanishing order ασ on Sσ.
Suppose for every parameter σ there is a parameter τ and some point Pσ,τ
in C such that the restriction of the pluricanonical section sτ to Sσ achieves
the restriction of the stable vanishing order on Sσ at the point Pσ,τ of C. Let
Z be the finite subset of C such that as a point P varies in C outside Z the
Artinian subscheme defined by s1, · · · , sk in the normal direction of C at P
does not change. In other words, every point Q ∈ C − Z admits an open
neighborhood U in C − Z and a holomorphic family of local nonsingular
complex surfaces MP in X parametrized by P ∈ U such that each MP
intersecting C normally at P and
dimC OMP
OMP (sj|MP )
is independent of P ∈ U . From the above discussion about s being the
explicitly constructed section and about the achievement of the restriction
to Sσ of stable vanishing order at points of C outside Zσ it follows that the
stable vanishing order is achieved at points of C − Z.
This use of the sections, explicitly constructed from the curvature current
in the second case of the dichotomy, in order to come up with a finite sub-
set Z of C can be regarded as the higher-codimensional way of constructing
sections of vector bundles explicitly from the second case of the dichotomy of
the modified restriction of the curvature current. Note that the codimension
one case for a surface is invoked for us to get the rationality of the restric-
tion of the stable vanishing order to a surface and its achievement on the
surface at some point of C. This part of the hard work already done for the
codimension one case is used here. This vector bundle section on C explicitly
constructed from the second case of the dichotomy of the modified restric-
tion of the curvature current is obtained, so to speak, by piecing together the
line bundle sections on C explicitly constructed by using the surface slices S
from the second case of the dichotomy of the modified restriction to the sur-
face slice S of the curvature current. This is the reason why the usual worry
about interminable blowups to transform into hypersurfaces newly appearing
higher-codimensional base point sets is not a problem here.
Before presenting Part I and Part II, we would like to make two remarks.
The first remark is about the finite generation of the canonical ring without
the assumption of finite type. In analysis, when KX is not big, adding a line
bundle E to KX to have a big bundle KX + E and working with KX + E
instead of KX would only mean a standard modification of the argument for
the finite generation of the canonical ring for the case of general type. (This
is actually used in the form of KX − γY , with γ being the stable vanishing
order across the hypersurface Y , in the proof of the finite generation of
the canonical ring for the case of general type.) However, getting rid of E
afterwards needs the justification of a rather involved limiting process. Such
a limiting process is of the same nature as the limiting process which is used
to extend the proof of the deformational invariance of the plurigenera for the
case of general type [Siu 1998] to the general algebraic case [Siu 2002].
For the finite generation of the canonical ring, when KX is big, we intro-
duce the curvature ΘKX of the metric of minimum singularity constructed
from pluricanonical sections. When KX is not big, we use an ample line
bundle A on X and the limit curvature current
Θ̃KX := lim
ΘKX+ 1mA
to replace ΘKX . All the diophantine approximation arguments deal with the
modified restriction of the canonical decomposition of Θ̃KX . This makes it
possible to go ahead with the limiting process.
In the case of the deformational invariance of the plurigenera the limiting
process is handled by abandoning the metric of minimum singularity and
replacing it by another metric with maximum allowable singularity which
is still good enough for the finiteness of the L2 norm of the section to be
extended, because there is a need to bound the dimension of the space of
L2 holomorphic m-canonical sections with the bound independent of m or
at least growing more slowly than m. It is not yet clear what the analogous
procedure is for the problem of the finite generation of the canonical ring.
The second remark is to answer the question why one does not simply
take a pluricanonical section and get the finite generation of the canonical
ring of its divisor and then use extension of pluricanonical sections from that
divisor to the entire ambient space. This procedure is explained in §8 of
the posted notes of my analytic proof [Siu 2006]. The extension technique
for pluricanonical sections in the problem of deformational invariance of the
plurigenera was developed with that particular application in mind [Siu 1998,
Siu 2006]. Though the pluricanonical extension technique for deformational
invariance of the plurigenera is for extension from the fiber at a point to
the open unit disk, in analysis it does not matter whether one deals with a
compact manifold or a Stein manifold if the Stein manifold is a Zariski open
subset and one keeps track of the L2 bounds.
However, the situation of the singularities in the pluricanonical divisor
poses technical problems in analysis. In my article in the proceedings of a
2001 conference in Hanoi [Siu 2004] I addressed the problem of pluricanonical
extension from a singular divisor. There are technical problems concerning
finite L2 bounds. The difficulty in handling such technical problems concern-
ing finite L2 bounds is the same as getting the comparability of the metric
of minimum singularity for KX and that from an appropriate truncated fi-
nite sum in the application of Skoda’s theorem of ideal generation [Skoda
1972]. Moreover, the Ohsawa-Takegoshi-type extension theorem used in the
pluricanonical extension is closely related to Skoda’s theorem of ideal gen-
eration (see, for example, Ohsawa’s article in the Festschrift for Grauert’s
70th birthday [Ohsawa 2002]) and the constant in Skoda’s theorem is more
precise. That is why we use the approach by Skoda’s theorem instead of the
essentially equivalent approach of pluricanonical extension.
It seems that at this point any approach to the problem of finite gener-
ation of the canonical ring has to rely, as starting point, on the technique
of pluricanonical extension. Any difference in the different approaches are
more a matter of technical handling of the singularities of the pluricanonical
divisor.
In the definition of a stable metric in [Siu 2006, (6.1)] there are some
typographical errors and inaccuracies. The correct definition is as follows.
A metric e−κ of a line bundle L on a compact Kähler manifold M is said to
be stable if κ is locally plurisubharmonic and there exists some ε > 0 with
the following property. If U is an open neighborhood of a point P ∈ M ,
and ϕ and ψ are plurisubharmonic functions on U such that the total mass,
with respect to the Kähler form of M , of the sum of the two closed positive
(1, 1)-currents Θϕ and Θψ is less than ǫ and if κ − ϕ is plurisubharmonic,
then there exists an open neighborhood U ′ of P inM such that the multiplier
ideal sheaf Iκ+ψ of the metric e−κ−ψ on U ′ is equal to the multiplier ideal
sheaf Iκ−ϕ of the metric e−κ+ϕ on U ′.
PART I
How to Apply Nonvanishing Theorem to Precisely Achieve
Stable Vanishing Order in Codimension One
The proof of the precise achievement of stable vanishing order in codi-
mension one has the following ingredients.
(a) The techniques for the Fujita conjecture which consists of
(i) constructing singular metrics with curvature current of strict pos-
itive lower bound whose multiplier ideal sheaf has high vanishing
order at a prescribed point,
(ii) blowing up the zero-set of the multiplier ideal sheaf of the new
singular metric,
(iii) repeating the procedure so that the zero-set of the multiplier ideal
sheaf of the new singular metric on a blow-up space is projected
down to lower and lower dimension in the original manifold, until
one gets to the case of a singular point in the original manifold,
(iv) extending a section defined on the single point to over all of
the original manifold by the vanishing theorem of Kawamata-
Viehweg-Nadel.
Because of the need of adding one canonical line bundle in the ap-
plication of the vanishing theorem of Kawamata-Viehweg-Nadel, the
singular metric is for a multiple of the line bundle in question minus
the canonical line bundle.
(b) Introducing a dichotomy depending on the canonical decomposition of
some curvature current so that
(i) in the first case of the dichotomy the technique of the Fujita con-
jecture for the construction of singular metric can be carried out,
(ii) in the second case of the dichotomy a section can be explicitly
constructed which can be extended to all of the original manifold.
Once we get to the second case of the dichotomy there is no need
to use the technique of the Fujita conjecture to construct any more
singular metric. The process is stopped and complete. For the precise
achievement of stable vanishing order, the second case of the dichotomy
must arise, otherwise the stable vanishing order can be improved, which
contradicts the definition of a stable vanishing order. The explicitly
constructed section is rather rigid in the sense that there is no choice
and the section comes from the curvature current in a rather unique
way. This uniqueness and rigidity of the explicitly constructed section
will be a key ingredient in the proof of the precise achievement of the
stable vanishing order in higher codimension.
(c) The new technique of constrained minimum center of log canonical sin-
gularity, whose motivation and precise application will be explained
below. This new technique is needed, because of the undesirable addi-
tional vanishing order in the process of constructing singular metrics,
which will be explained in detail below.
(d) Use of diophantine approximation to handle irrational coefficients. Since
the use of diophantine approximation has already been described in
detail in the posted notes of my proof of the finite generation of the
canonical ring [Siu 2006], we will not discuss diophantine approxima-
tion in these additional explanatory notes and we just assume that the
relevant coefficients are known to be rational numbers.
As the first step we will consider the first case of the dichotomy and describe
how to construct an appropriate singular metric. Then we will give the
motivation for constrained minimum center of log canonical singularity by
reviewing the goal and the strategy of the standard technique for the Fujita
conjecture. Finally we consider the second case of the dichotomy where a
rather rigid section is explicitly constructed.
Proposition(Construction of Metric with Multiplier Ideal Sheaf Vanishing
to High Order at a Prescribed Point for the First Case of the Dichotomy of
the Curvature Current). Let M be a compact complex projective algebraic
manifold of complex dimension n. Let L be a holomorphic line bundle on M
with a (possibly singular) metric e−ϕ along its fibers whose curvature current
Θϕ is a closed positive (1, 1)-current. Let
τj [Vj] +R
be the canonical decomposition of Θϕ, where J ∈ N∪{0,∞} and the Lelong
number of R is zero outside a countable union Z of subvarieties of codi-
mension at least two in M and Vj is an irreducible hypersurface in M and
τj > 0. Assume that either J = ∞ or R 6= 0 (that is, one is in the first
case of the dichotomy for the curvature current Θϕ). Assume that for some
positive integer p0 there is a (possibly singular) metric e
−χ along the fibers of
p0L−KM which is stable and whose curvature current Θχ is a closed positive
(1, 1)-current which dominates some strictly positive smooth (1, 1)-form on
M . Let P0 be a point of M such that the Lelong number of Θϕ is zero at
P0. Let q ∈ N. Then for some sufficiently divisible m ∈ N the line bundle
(m+ p0)L − KM admits a metric e−χ̃ whose curvature current dominates
some strictly positive smooth (1, 1)-form on M such that its multiplier ideal
sheaf Iχ̃ is contained in the maximum ideal of M at P0 raised to the q-th
power. Moreover, if M is a hypersurface in some compact complex algebraic
manifold X of general type so that L is the restriction of some line bundle
L̃ on X and KM = KX +M and the metric e
−χ is defined by a convergent
infinite sum of multi-valued holomorphic sections of p0L̃−KX −M over X ,
then the metric e−χ̃ can be chosen to be defined also by a convergent infinite
sum of multi-valued holomorphic sections of (p + p0) L̃−KX −M over X .
Proof. The idea of the proof is to use the techniques for Fujita’s conjecture
(see, for example, [Angehrn-Siu 1995]).
Slicing by an Ample Divisor. Let A be a very ample line bundle over M such
that A−KM is ample. Let hA be a smooth metric of A whose curvature form
ωA is positive on M . We assume that A is chosen to be sufficiently ample so
that for each point P ∈M the proper transform of A in the manifold obtained
from M by blowing up P is still very ample. This technical assumption will
enable us to choose a generic element of Γ (M, A) vanishing at P0 which is
not a zero-divisor of a prescribed coherent ideal sheaf.
Let p and k be positive integers and we will impose more conditions on p
and k later. Let s1 be a generic element of Γ (M, A) vanishing at P0 so that
the short exact sequence
0 → Ipϕ+χ ((p+ p0)L−KM + kA)
θs1−→ Ipϕ+χ ((p+ p0)L−KM + (k + 1)A)
→ (Ipϕ+χ /s1Ipϕ+χ ) ((p+ p0)L−KM + (k + 1)A) → 0
is exact, where θs1 is defined by multiplication by s1. Let M1 be the zero-set
of s1 and
OM1 = (OM /s1OM ) |M1,
which we can assume to be regular with ideal sheaf equal to s1OM because s1
is generic element of Γ (M,A) vanishing at P0. By choosing s1 generically we
can also assume that I(pϕ+χ)|M1 = Ipϕ+χ /s1Ipϕ+χ and I(pϕ)|M1 = Ipϕ /s1Ipϕ
and Iχ|M1 = Iχ /s1Iχ . We use χ (·, ·) to denote the arithmetic genus which
means
χ (·, ·) =
(−1)ν dimCHν (·, ·) .
From the long cohomology exact sequence of the above short exact sequence
we obtain
χ (M, Ipϕ+χ ((p+ p0)L−KM + (k + 1)A)) =
χ (M, Ipϕ+χ ((p+ p0)L−KM + kA))+χ
M1, I(pϕ+χ)|M1 ((p+ p0)L−KM + (k + 1)A) |M1
Since A − KM is ample and 2A − KM1 = A − KM is also ample, when we
assume k ≥ 1, by the theorem of Kawamata-Viehweg-Nadel
Hν (M, Ipϕ+χ ((p+ p0)L−KM + kA)) = 0 for ν ≥ 1,
M1, I(pϕ+χ)|M1 (((p+ p0)L−KM + (k + 1)A) |M1)
= 0 for ν ≥ 1
so that
dimC Γ (M, Ipϕ+χ ((p+ p0)L−KM + (k + 1)A))
= dimC Γ (M, Ipϕ+χ ((p+ p0)L−KM + kA))
+ dimC Γ
M1, I(pϕ+χ)|M1 (((p+ p0)L−KM + (k + 1)A) |M1)
≥ dimC Γ
M1, I(pϕ+χ)|M1 (((p+ p0)L−KM + (k + 1)A) |M1)
Slicing by Ample Divisors Down to a Curve. Instead of one single element
s ∈ Γ (M,A), we can choose generically
s1, · · · , sn−1 ∈ Γ (M, A)
all vanishing at P0 so that inductively for 1 ≤ ν ≤ n−1 the common zero-set
Mν of s1, · · · , sν with the structure sheaf
OMν :=
is regular and we end up with the inequality
dimC Γ (M, Ipϕ+χ ((p+ p0)L−KM + (k + n− 1)A))
≥ dimC Γ
Mn−1, I(pϕ+χ)|Mn−1
((p+ p0)L−KM + (k + n− 1)A) |Mn−1
Since Mn−1 is a curve, all coherent ideal sheaves on it are principal and are
locally free and they come from holomorphic line bundles.
We would like to remark also that this particular step of slicing by n− 1
ample divisors to get down to a curve roughly corresponds to the step in
Shokurov’s proof of his non-vanishing theorem [Shokurov 1985] where he
takes the product of his numerically effective divisor in his n-dimensional
manifold with the (n−1)-th power of a numerically effective big line bundle.
Application of the Theorem of Riemann-Roch to a Curve and Comparing
Contributions from the Curvature Current and the Multiplier Ideal Sheaves.
Let b be the Chern class of the line bundle onMn−1 defined by the multiplier
ideal sheaf Iχ|Mn−1 of the restriction to Mn−1 of the metric e
−χ. Let c be the
nonnegative number
R ∧ (ωA)n−1 .
(†) dimC Γ (M, Ipϕ+χ ((p+ p0)L−KM + (k + n− 1)A))
≥ dimC Γ
Mn−1, I(pϕ+χ|Mn−1)
((p+ p0)L−KM + (k + n− 1)A) |Mn−1
≥ 1− genus (Mn−1) + b+ (k + n− 1)An−1Mn−1
(pτj − ⌊pτj⌋)Vj · An−1 + p
where the last identity is from the theorem of Riemann-Roch applied to the
regular curve Mn−1 and the locally free sheaf
I(pϕ+χ|Mn−1)
((p+ p0)L−KM + (k + n− 1)A) |Mn−1
on Mn−1. From the assumption that J = ∞ or R 6= 0, we conclude that
the right-hand side of (†) goes to ∞ as p goes to ∞ through an appropriate
sequence, where for the case of J = ∞ and R = 0 a diophantine argument
has to be used whereas for the case R 6= 0 we simply need to use c > 0.
Construction of Sections with Extra Vanishing Order from Dimension Count-
ing and Construction of Metrics by Canceling Contributions from Ample Di-
visors by Using the General Type Property. For any ℓ ∈ N the number of
terms in a polynomial of degree ℓ in d variables is
. Take a positive
integer N and we will impose more condition on N later. By the behavior
of the right-hand side of (†) as p→ ∞, there exists p ∈ Z such that
dimC Γ (M, Ipϕ+χ ((p+ p0)L−KM + (k + n− 1)A)) ≥ 1+
n+N (k + n− 1) q
and we can find some non identically zero element s of
Γ (M, Ipϕ+χ ((p + p0)L−KM + (k + n− 1)A))
which vanishes to order at least N (k + n− 1) q at P0 so that s
N(k+n−1) is a
multi-valued holomorphic section of the Q-line-bundle p
N(k+n−1)
A over
M which vanishes to order at least q at P0. We assume that N is chosen
so large that the curvature current Θχ dominates
ωA. Let p̂ to be the
round-up of p
N(k+n−1)
and δp = p̂− pN(k+n−1) . We introduce the metric
e−χ̃ :=
e−χ−δpϕ
N |s|
of (p+ p0)L − KM so that the multiplier ideal of Iχ̃ at P0 is contained in
(mM,P0)
. Q.E.D.
Remark on Application of the Proposition on Construction of Metric for
the First Case of the Dichotomy. The above application is applied in the
following manner. Let X be the the compact complex algebraic manifold of
finite type whose canonical ring is to be proved to be finitely generated. Let
Y be a hypersurface in X across which the stable vanishing order is γ > 0.
Let P0 be a generic point of Y . We start out withM = Y and L = KX−γY .
After we get the new metric e−χ̃, we use an interpolation between e−χ̃ and
e−pϕ−χ and a slight modification to get to a minimum center of log canonical
singularity which, after blow-up, projects down to a proper subvariety of M
containing P0. Then we replace X by its blow-up and replace M by the new
minimum center of log canonical singularity and replace L by its pullback
to the blowup of X . We continue doing this until we inevitably come to the
second case of dichotomy eventually as explained in the Introduction.
Remark on the Second Case of the Dichotomy. Suppose
τj [Vj]
with J < ∞. Then we can explicitly construct a section s0 of pL over
M . The reason why a minimal center of log canonical singularity is used
in the techniques for the Fujita conjecture is to make sure that when we
take the subspace defined by the multiplier ideal sheaf, the subspace has a
reduced structure. In our case we have to introduce the notion of constrained
minimal center of log canonical singularity so that the center is not completely
contained in the zero-set of s0. For that we have to pay the price that
the subspace defined by the multiplier ideal sheaf may have an unreduced
structure, but the set where nonzero nilpotent elements of its structure sheaf
occurs is contained in the zero-set of s0. By raising s0 to a sufficiently high
power, we can handle the unreduced structure and get the extension of a
sufficiently high power of s0 to the ambient manifoldX by using the vanishing
theorem from the metric of pL−KM . We are going to elaborate on this by
reviewing the goal of the techniques for the Fujita conjecture and the use of
minimal center of log canonical singularity and also how we are naturally and
by necessity led to the concept of constrained minimal center of log canonical
singularity.
Main Idea of the Techniques for the Fujita Conjecture. For the discussion
about the main idea of the technique for the Fujita conjecture, we forget the
above meaning of X and L and for the time being use the symbols X and
L in another context. We will so indicate when we later return to the above
meaning of X and L. The goal of the technique for the Fujita conjecture is
to find global sections to generate some positive power mL of a line bundle
L over a compact complex algebraic manifold X .
For a more general setting, the goal is to find global sections to globally
generate J (mL) over X for some given coherent ideal sheaf J at points
outside some given subvariety Z of X . The problem of proving the finite
generation of the canonical ring by verifying the precise achievement of stable
vanishing orders actually involves this more general setting. There the even
more complicated situation of supremum norm is used instead of just the L2
norm. However, for the sake of simplicity in our discussion of the main idea
of the techniques of the Fujita conjecture, we stick with the simpler goal of
find global sections to generate some positive power mL of a line bundle L
over a compact complex algebraic manifold X .
The main idea of the technique is to focus on the subvariety where the
global generation fails. We take a basis of s1, · · · , sk ∈ Γ (X,mL) and let Y
be their common zero-set so that global generation precisely fails at points of
Y . The main idea of the technique is simply to focus on Y if Y is nonempty.
We seek elements of Γ (Y,mL|Y ) which are not identically zero and then
extend them to elements of Γ (X,mL), which would then contradict the def-
inition of Y . Hopefully the extension of elements of Γ (Y,mL|Y ) to elements
of Γ (X,mL) could be done by the vanishing of some appropriate first co-
homology group. Usually this first cohomology comes from the vanishing
theorem of Kawamata-Viehweg-Nadel. We seek a metric e−ϕ of mL−KX so
(i) the zero-set of its multiplier ideal sheaf Iϕ is Y , and
(ii) the curvature current of e−ϕ dominates some strictly positive smooth
(1, 1)-form on X .
In such a case we have
1 (X, Iϕ (mL)) = 0
from the vanishing theorem of Kawamata-Viehweg-Nadel and the map
Γ (X,mL) → Γ (Y, (OX /Iϕ ) (mL))
is surjective. The next step is to come up with some element of
Γ (Y, (OX /Iϕ ) (mL))
which induces a non identically zero element of Γ (Y,mL|Y ).
It is at this point that the question of a possibly unreduced complex
structure OX /Iϕ arises. It means that the structure sheaf OX /Iϕ for Y
may have nonzero nilpotent elements. This is the case when Iϕ is a proper
subsheaf of the full ideal sheaf IY of Y and is not equal to IY . An element of
IY which is not in Iϕ would yield a nonzero nilpotent element of the structure
sheaf OX /Iϕ for Y . When we have an unreduced structure OX /Iϕ for Y ,
it is more difficult to produce some element of
Γ (Y, (OX /Iϕ ) (mL))
which induces a non identically zero element of Γ (Y,mL|Y ). To handle the
problem of unreduced structure sheaf, the technique is to use minimum cen-
ters of log canonical singularity whose role we are going to explain.
Minimum Center of Log Canonical Singularity. The idea is to seek a metric
e−ψ for mL −K which is less singular than e−ϕ so that the multiplier ideal
sheaf Iψ of e−ψ contains the multiplier ideal sheaf Iϕ of e−ϕ. This procedure
usually involves the interpolation of two metrics and a slight modification
of the result of the interpolation. We seek to make the metric of e−ψ for
mL −K to be as least singular as possible, with just enough singularity to
make the multiplier ideal sheaf Iψ of e−ψ not equal to OX . Let Y ′ be the
support of OX /Iψ . This kind of least or minimum singularity for the choice
of e−ψ gives us a reduced complex structure OX /Iψ for Y ′. The reduced
complex subspace (Y ′,OX /Iψ ) is called a minimum center of log canonical
singularity. (Usually for this technique of minimum center of log canonical
singularity one requires, in addition, that the proper transform of Y ′ in some
appropriate blow-up X̃ of X to be a nonsingular hypersurface in X̃ .) Now
one replaces Y by Y ′ and uses the vanishing of
H1 (X, Iψ (mL)) = 0
and the surjectivity of the map
Γ (X,mL) → Γ (Y ′, (OX /Iψ ) (mL))
to reduce the problem to the construction of nonzero element of
Γ (Y ′, (OX /Iψ ) (mL)) = Γ (Y ′, mL|Y ′) .
Constrained Minimum Center of Log Canonical Singularity. For our case
at hand for the finite generation of the canonical ring one modification has
to be added in the application of the technique of minimum center of log
canonical singularity. This modification necessitates the introduction of a
new concept which we give the name constrained minimum center of log
canonical singularity just to make it easier to refer to. Let us now describe
the situation. In the second case of the dichotomy of the curvature current,
there is some non identically zero element s0 of Γ (Y,mL|Y ) which is explicitly
constructed from the canonical decomposition on Y of a modified restriction
of the curvature current.
The section s0 may have a nonempty zero-set W in Y . If we just use
the technique of minimum center of log canonical singularity without any
modification, we may end up with a minimum center of log canonical sin-
gularity Y ′ which is completely contained inside the zero-set W of s0. In
such a case the extension of s0|Y ′ is useless, because the restriction s0|Y ′ of
s0 to Y
′ is identically zero on Y ′. So we need to introduce a modification to
the technique of minimum center of log canonical singularity. In the proce-
dure of using a metric e−ψ with least singularity to get the minimum center
of log canonical singularity, we introduce the additional condition that the
support of OX /Iψ is not contained entirely in the zero-set W of s0. With
this additional condition we can no longer require that the structure sheaf
OX /Iψ of Y ′ is reduced, but we can require that the set E of points where
the structure sheaf OX /Iψ of Y ′ fails to be reduced is entirely contained
in W . We call Y ′, which is obtained from this procedure of the additional
condition, a constrained minimum center of log canonical singularity.
The key point about the use of a constrained minimum center of log
canonical singularity is the following. Though the restriction s|Y ′ to Y ′ is
only holomorphic on the reduced structure of Y ′, yet since s0 vanishes on E
we can take a sufficiently high power sN of s so that sN |Y ′ is holomorphic
on the unreduced structure OX /Iψ of Y ′. We now extend sN |Y ′ to X . Of
course, we have to replace m by Nm.
Proposition (Global Generation of the Pluricanonical Bundle at Points
of Zero Stable Vanishing Order). Let X be a compact complex algebraic
manifold of complex dimension n. Let e−ϕ be the metric of KX of minimum
singularity and let Θϕ be its curvature current. Then there exist a positive
integer m0 such that the common zero-set W of a C-basis of Γ (X,m0KX) is
precisely the set of points where the Lelong number of Θϕ is positive.
Proof. We use the technique for the Fujita conjecture and constrained min-
imum centers of log canonical singularity. We use W as the set of constraint
for the constrained minimum center of log canonical singularity. We will not
go into further details here, because a similar but harder situation will be
handled in the proof of precisely achieving stable vanishing order γ > 0 for
codimension one in the case of a hypersurface Y whose coefficient in Θϕ is γ.
The only difference is that here the number γ is replaced by 0 and we do the
argument in the ambient space X instead of in the hypersurface Y taking
the place of X .
We now finish the use of the temporary meaning of X and L and return
to the earlier meaning of X and L.
Proposition(Extension of Explicitly Constructed Section to Ambient Man-
ifold by Constrained Minimum Center of Log Canonical Singularity). Let
X be a compact complex algebraic manifold of complex dimension n. Let
e−ϕ be the metric of KX of minimum singularity and let Θϕ be its curvature
current. Let M be a nonsingular hypersurface in X such that the stable
vanishing order η for M is a positive rational number. Let
(Θϕ − ηM) |M =
be the canonical decomposition of the closed positive (1, 1)-current (Θϕ − ηM) |M
on M with J < ∞ and each γj being rational. Then the stable vanishing
order η for M is precisely achieved.
Proof. By the previous Proposition we find a positive integer m0 such that
the common zero-set of a C-basis s1, · · · , sk of Γ (X,m0KX) is precisely the
set of points where the Lelong number of Θϕ is positive. If any of the elements
of Γ (X,m0KX) precisely achieves the stable vanishing order η forM , we are
already done. By replacing m0 by another sufficiently large integer, we can
make the vanishing order across M of
j=1 |sj|
m0 to be as close to η as
prescribed (though still > η). On the other hand, by raising
j=1 |sj|
m0 to a
positive integral power afterwards and using interpolation, we can adjust the
stable vanishing order across M to η plus any positive prescribed number.
Let L = KX − ηM . We can thus use s1, · · · , sk and interpolation of
metrics and their slight modifications to construct a metric e−χ ofmL−KX =
(m− 1)L− (η + 1)M of strictly positive curvature current so that
(i) the zero-set of its multiplier ideal sheaf Iχ is contained in the set of
points where the Lelong number of Θϕ is positive, and
(ii) the generic vanishing order of the its multiplier ideal sheaf Iχ across
M is precisely 1.
We are able to fulfill Condition(i), because we can construct the metric e−χ
by using the (m−1)-th power
j=1 |sj|
m0 and since the common zero-set of
a C-basis s1, · · · , sk of Γ (X,m0KX) is precisely the set of points where the
Lelong number of Θϕ is positive. We are able to fulfill Condition(ii), because
we have the additional order η+1 acrossM to spare when we use the (m−1)-
th power
j=1 |sj|
m0 and require only the higher generic vanishing order of
mη + 1 = (m − 1)η + (η + 1) across M instead of the order (m − 1)η from
the stable vanishing order η across M .
The vanishing theorem of Kawamata-Viehweg-Nadel gives
H1 (X, Iχ (mL)) = 0.
To get elements of Γ (X,mL), we need to use elements of Γ (X, (OX /Iχ ) (mL)).
At points of Y the additional vanishing order of the multiplier ideal sheaf Iχ
of e−χ occurs only at points where the Lelong number of Θϕ is positive. When
we construct a constrained minimum centerM ′ of log canonical singularity in
M by interpolation (with the subvariety ∪Jj=1Yj as the set of constraint), the
complex structure of M ′ is reduced outside ∪Jj=1Yj. Now we can explicitly
construct the section
of the line bundle NL|M on the reduced structure ofM for some appropriately
chosen positive integerN , where sYj is the canonical section of Yj onM . Since
s0 vanishes on sYj , by replacing N by a large positive integral multiple, we
can assume that the restriction of s0 toM
′ can be extended to a holomorphic
section over any unreduced structure of M ′ which is reduced outside ∪Jj=1Yj
and, moreover, can be extended to an element of Γ (X, (OX /Iχ ) (NL)). Note
that the support of OX /Iχ may be more than just Y , but its intersection
with Y is contained in the zero-set ∪Jj=1Yj of s0 so that the last extension
to an element of Γ (X, (OX /Iχ ) (NL)) is possible for a sufficiently large N .
Since s0 is nonzero at some point P0 of M
′, the stable vanishing order across
M is precisely achieved by
s̃0 (sM)
∈ Γ (X,NqKX)
at the point P0 of M . Q.E.D.
PART II
Illustration in Low Dimension of the Argument of
Precise Achievement of Stable Vanishing Order
for Higher Codimension
We now illustrate the argument of the precise achievement of the stable
vanishing order for higher codimension by using the low dimensional cases
of complex surfaces and complex threefolds. First we consider the case of
surfaces. For surfaces codimension two means isolated points. For isolated
points for any dimension there is a simple direct argument, which is given
in the following proposition. After we present the case of threefolds, we will
remark on how this simple direct argument can be interpreted in the context
of the argument for any dimension.
Proposition (Precise Achievement of Stable Vanishing Order at a Finite
Set). The stable vanishing order is automatically precisely achieved every-
where when it is precisely achieved outside a finite set of a compact complex
algebraic manifold X .
Proof. Let X be a compact complex algebraic manifold of general type.
Let e−ϕ be the metric of minimum singularity for the canonical line bundle
KX of X . Suppose it has been proved that the stable vanishing order is
precisely achieved outside a finite number of points P1, · · · , Pk of X by using
σ1, · · · , σℓ ∈ Γ (X,m0KX) .
We are going to show that this finite number of points must be the empty
set, otherwise there is a contradiction. Let
e−ψ =
j=1 |σj |
We take p ∈ N sufficiently large to magnify the discrepancy of the vanishing
orders of e−pm0ϕ and e−pψ so that the support of the quotient
Ipm0ϕ /Ipψ
of the multiplier ideal sheaves Ipm0ϕ and Ipψ of e−pm0ϕ and e−pψ respectively
is the finite set {P1, · · · , Pk}. We now apply a slight modification to Ipψ to
get a metric with strictly positive curvature current. Let e−θ be a metric
of KX with strictly positive curvature current. Since e
−ϕ is the metric of
minimum singularity for KX , it follows that when ε > 0 is sufficiently small
(which we assume to be the case) the multiplier ideal sheaf I(p−ε)ψ+εm0θ of
the metric e−((p−ε)ψ+εm0θ) agrees with the multiplier ideal sheaves Ipm0ϕ of
e−pm0ϕ on X − {P1, · · · , Pk} and the support of the quotient
Ipm0ϕ
I(p−ε)ψ+εm0θ
of the multiplier ideal sheaves Ipm0ϕ and Ipψ of e−pm0ϕ and e−((p−ε)ψ+εm0θ)
respectively is the finite set {P1, · · · , Pk}. By the vanishing theorem of
Kawamata-Viehweg-Nadel, we have
H1 (X, Ipψ ((pm0 + 1)KX)) = 0.
Then the map
Γ (X, Ipm0ϕ ((pm0 + 1)KX)) → Γ
Ipm0ϕ
I(p−ε)ψ+εm0θ
((pm0 + 1)KX)
is surjective. Note that for this we do not need the vanishing of the coho-
mology group
H1 (X, Ipm0ϕ ((pm0 + 1)KX)) .
Since
Ipm0ϕ
I(p−ε)ψ+εm0θ
((pm0 + 1)KX)
Ipm0ϕ
I(p−ε)ψ+εm0θ
it follows from Nakayama’s lemma and the surjectivity of
Γ (X, Ipm0ϕ ((pm0 + 1)KX)) →
Ipm0ϕ
I(p−ε)ψ+εm0θ
that the map
Γ (X, Ipm0ϕ ((pm0 + 1)KX)) →
(Ipm0ϕ)Pj
is surjective. This actually gives a contradiction, because the stable vanishing
order of (pm0 + 1)KX should be given by e
−(pm0+1)ϕ instead of by e−pm0ϕ.
Q.E.D.
Remarks. (a) With this proposition, to get the finite generation of the canon-
ical ring for the case of a compact complex algebraic surface of general type
it suffices to show that the stable vanishing order is precisely achieved at
codimension one.
(b) For the analytic proof of the finite generation of the canonical ring, when
we get down to the point of having already verified the precise achievement
of the stable vanishing order outside a finite set of points, we do not need to
blow up the points to reduce the argument to the case of a hypersurface in the
new blown-up manifold. Therefore the difficulty does not exist, in the case of
a surface, of blowing up a point P0 to get a curve C1 and then locating some
bad point P1 (where the precise achievement of stable vanishing order fails)
in the curve C1 and then blowing up P1 to get a curve C2 and locating some
bad point P2 (where the precise achievement of stable vanishing order fails)
in the curve C2 and then blowing up P2 and possibly finally ending up with
an unending infinite sequence of bad points, each in a tower of successively
blown-up surfaces.
Higher Codimension Argument for Threefold Case. Let X be a complex
complex algebraic threefold of general type. Let e−ϕ be the metric of mini-
mum singularity for the canonical line bundle KX of X . Suppose it has been
proved that the stable vanishing order is precisely achieved outside a curve
Cj (where each Cj is irreducible) by
s1, · · · , sk ∈ Γ (X,m0KX) .
Note that here we have used the Proposition given above to rule out the
possibility that, besides at the curve C, the stable vanishing order may fail
to be precisely achieved at a finite set of points in X − C.
On X we have the canonical decomposition of
ΘKX =
γjYj +R
of the curvature current of the metric of minimum singularity for KX . We
are skipping the diophantine argument which is explained in detail in [Siu
2006] and consider as verified the rationality of each γj. By replacing m0 by
an appropriate integral multiple, we can assume without loss of generality
that each m0γj is a positive integer. Let s
)m0γj
, where sYj is
the canonical section of Yj. Let L = KX −
j=1 γjYj. Since we can replace
σ1, · · · , σℓ by
, · · · , sk
∈ Γ (X,m0L)
and the essence of the rest of the argument that is to follows does not change,
for notational simplicity we are going to assume that all γj = 0 so that
L = KX . Also, for notational simplicity we are going to assume that the
curve C is irreducible.
For σ = (σ1, · · · , σk) ∈ Ck − {0} let sσ =
j=1 σjsj and let Sσ be the
surface in X defined by sσ. We consider those σ for which Sσ is irreducible
across which sσ vanishes to order 1. By considering the blow-up of X by the
ideal generated by s1, · · · , sk and the precise achievement of stable vanishing
order for codimension one, after replacing m0 by a positive integral multiple
if necessary, we have the following situation.
(a) For σ = (σ1, · · · , σk) ∈ Ck − {0} for which Sσ is irreducible where sσ
vanishes to order 1, there exist
(i) some τ = (τ1, · · · , τk) ∈ Ck − {0} and
(ii) some finite subset Zτ,σ of C
such that the section sτ is not identically zero on Sσ and the multi-
valued section
sτ |Sσ
(sC,σ)
on a neighborhood of C − Zτ,σ in Sσ is nonzero at points of C − Zτ,σ,
where ησ is the stable vanishing order at C on the surface Sσ and sC,σ
is the canonical section of C in Sσ.
(b) One has the second case of the dichotomy for the curvature current
(ΘKX |Sσ − ησ [C])|C
on C.
(c) The multi-valued section
sτ |Sσ
(sC,σ)
on C is constructed from the second case of the dichotomy for the
curvature current
(ΘKX |Sσ − ησ [C])|C
on C.
(c) The stable vanishing order ησ for C on Sσ is achieved by sτ at points
of C − Zτ,σ.
Note that, because of Condition(b) and Condition(c) we can choose Zτ,σ in-
dependent of τ so that Zτ,σ = Zσ for some finite subset Zσ of C depending
only on σ. To finish the proof of the precise achievement of the stable vanish-
ing order for the threefold X , it suffices to show that there is a finite subset
Z of C such that every Zσ is contained in Z.
At a regular point P0 of C, there is some open neighborhood U of P0 in
X such that
(i) the pair (U,C) is biholomorphic to the pair (∆3, {(0, 0)} ×∆), where
∆ is the open unit disk in C, and
(ii) sj|U is represented by a holomorphic function fj (z1, z2, t) on ∆3 for
1 ≤ j ≤ k.
The existence of the finite subset Z of C with the property that Zτ,σ ⊂ Z
follows from Lemma 2 given below.
Lemma 1. Let f0, · · · , fk be holomorphic function germs on C2 at the origin
so that the origin is the common zero-set of any two of the holomorphic
function germs f0, · · · , fk. Let C0 be the complex curve germ at the origin
defined by f0 = 0, which is assumed to be irreducible and across which f0
vanishes to order 1. Then the following numbers are the same.
(i) The multiplicity of the ideal
j=0OC2,0fj at the origin.
(ii) The dimension over C of
OC2,0
OC2,0fj .
(iii) The Lelong number λ of
Θ̂ :=
∂∂̄ log
|fj |2
on C0 at the origin. Here the Lelong number λ means the Lelong
number of the pullback of the closed positive (1, 1)-current Θ̂ to the
normalization of C0.
(iv) The number η such that
j=1 |fj|
j=1 |zj |
is bounded between two positive numbers near the origin on C0.
Lemma 2. Let k ≥ 2 and let fj(z1, z2, t), for 1 ≤ j ≤ k, be holomorphic
functions on the tri-disk ∆3 with coordinates z1, z2, t. Assume that the com-
mon zero-set of any two of f1, · · · , fk is {(0, 0)} × ∆ = {z1 = z2 = 0}. For
any k-tuple (a1, · · · , ak) of complex numbers not all zero let Sa1,··· ,ak be the
zero-set of
j=1 ajfj . Then there exists a discrete subset Z of {(0, 0)} ×∆
with the following property. For any k-tuple (a1, · · · , ak) of complex numbers
not all zero there exists some nonnegative number γa1,··· ,ak such that
j=1 |fj |
|zj |2
)γa1,··· ,ak
is continuous nonzero on some neighborhood of (Sa1,··· ,ak ∩ {(0, 0)} ×∆)−Z
in Sa1,··· ,ak if Sa1,··· ,ak is irreducible and
j=1 ajfj vanishes to order 1 across
Sa1,··· ,ak .
Proof. Observe that for fixed t if a1 6= 0 then each of
j=1 |fj |
|a1f1 + · · ·+ akfk|2 +
j=2 |fj |
is bounded by a positive constant times the
other on some neighborhood of the origin in C2. Use Lemma 1. Q.E.D.
Remarks. (a) We would like to highlight the intuitive geometric reason for
the existence of a discrete set Z in C such that the “bad set” Zσ in C for
the restriction of the closed positive (1, 1)-current ΘKX restricted to Sσ is
contained in Z. The key point is that the surface Sσ is sliced out by a C-
linear combination of the pluricanonical sections s1, · · · , sk and that these
pluricanonical sections s1, · · · , sk have the property that the “bad set” Zσ of
C can be described by the extra vanishing of
j=1 |sj|
beyond their generic
vanishing order at points of C.
(b) We need to restrict ΘKX to Sσ, because we have to take away the vanish-
ing order of ΘKX at C and it is only for codimension one we can take away
the vanishing order. The vanishing order on different Sσ may be different.
(c) The “bad set” Zσ in C describes the points where the relative position
of the pair of two Artinian subschemes in the normal direction of C jumps.
One Artinian subscheme comes from the restriction of s1, · · · , sk to a local
surface T normal to C and the other one comes from s1, · · · , sk plus a C-linear
combination σ1s1+ · · ·+ σksk after restricting them to T . The main point is
that there is a finite subset Z of C such that Zσ ⊂ Z with Z independent of
the choice of σ.
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http://arxiv.org/abs/math/0610740
