The Fujita Conjecture and
the Extension Theorem of Ohsawa-Takegoshi
Yum-Tong Siu
1
§1. Introduction and Statement of Results
§2. Multiplier Ideal Sheaves and the Induction Argumet
§3. Semicontinuity of Multiplier Ideal Sheaves
§4. Proof of the Extension Theorem of Ohsawa-Takegoshi
§5. Alternative to the Use of the Extension Theorem of Ohsawa-Takegoshi
§6. Difficulty in Improving the Quadratic Bound to the Conjectured Linear
Bound
§7. Remarks on Very Ampleness
§1. Introduction and Statement of Results
Let L be an ample line bundle over a compact complex manifold X of complex dimension n. We discuss here the most recent result of myself and Angehrn
[AS94] on the conjecture of Fujita [F87] on freeness. Fujita’s conjecture states
that (n + 1)L + KX is free. The conjecture of Fujita has a second part on very
ampleness which states that (n + 2)L + KX is very ample. We will confine ourselves to the freeness part of the Fujita conjecture. The case n = 1 is well-known
and the case n = 2 was proved by Reider [R88]. Ein-Lazarsfeld [EL93] proved
the freeness part for n = 3. My result with Angehrn [AS94] is the following.
1
Main Theorem. Let κ be a positive number. If (Ld · W ) d ≥ 12 n(n + 2r − 1) + κ
for any irreducible subvariety W of dimension 1 ≤ d ≤ n in X, then the global
holomorphic sections of L + KX over X separate any set of r distinct points
P1 , · · ·, Pr of X. In other words, the restriction map Γ(X, L) → ⊕rν=1 OX /mPν
is surjective, where mPν is the maximum ideal at Pν .
Corollary. mL + KX is free for m ≥ 21 (n2 + n + 2).
To avoid distracting technical details, we will discuss here the proof of the
Corollary to the Main Theorem instead of the Main Theorem itself. Besides
the routine part of the proof, there is one important new ingredient in the
proof. The routine part of the proof uses the well-known techniques of the
theorem of Riemann-Roch and the vanishing theorem of Nadel for multiplier
ideal sheaves. The most important ingredient in the proof is the new technique
of the semi-continuity of the multiplier ideal sheaf which is a consequence of the
extension theorem of Ohsawa-Takegoshi for L2 holomorphic functions [OT87].
This new technique of the semi-continuity of the multiplier ideal sheaf solves the
difficulty that has been, until now, the only obstacle to obtaining, for general
dimension, an effective bound m so that mL + KX is free. Recently a preprint
1 Partially
supported by a grant from the National Science Foundation
1
of Tsuji [T94] also gave a proof of the Corollary to the Main Theorem, but
in that preprint there was no discussion of that obstacle and no method was
introduced to handle it. After the circulation of the preprint of [AS94], a later
version of [T94] incorported the technique from [AS94] of the semicontinuity
of multiplier ideal sheaves and presented a procedure to improve the quadratic
bound of m to the conjectured linear bound m ≥ n + 1. However, Ein and
Lazarsfeld pointed out that such a procedure to improve the quadratic bound
already encountered insurmountable difficulties even in the case of dimension 3
and a divisor occurring in the first step with quadratic conic singularities. This
example of Ein and Lazarsfeld of a surface with quadratic conic singularities in
dimension 3 will be given in §6 where we will discuss of the difficulty involved in
improving the quadratic bound of m to the linear bound m ≥ n + 1 conjectured
by the freeness part of the Fujita conjecture.
The extension theorem of Ohsawa-Takegoshi needed in the new technique requires heavy analysis. After learning of this new technique of the semi-continuity
of the multiplier ideal sheaf, Kollar [K94] came up with an algebraic proof of the
semi-continuity of the multiplier ideal sheaf by using the inversion of adjunction
(instead of the the extension theorem of Ohsawa-Takegoshi for L2 holomorphic
functions). Kollar’s algebraic proof enabled him to generalize the Corollary to
the Main Theorem to the case where X may have log terminal singularities.
The known proof of the the extension theorem of Ohsawa-Takegoshi for L2
holomorphic functions uses a long series of commutation identities in Kähler
geometry and some specially contructed complete Kähler metrics. The usual
L2 estimates of ∂ is not sufficient to give a proof of it. We present here a simpler
and clearer proof of the the theorem of Ohsawa-Takegoshi and explains the new
methods needed and why the usual L2 estimates of ∂ is insufficient. We also
explain, in terms of vanishing theorems in an analytic setting, Kollar’s algebraic
proof of the semi-continuity of the multiplier ideal sheaf by the inversion of
adjunction. I am indebted to Kawamata for patiently explaining to me the
meanings of a host of related terms, Kawamata log terminal, log canonical, etc.
in order for me to understand Kollar’s algebraic proof.
§2. Multiplier Ideal Sheaves and the Induction Argument
Fix P0 in X. We now explain the induction argument needed to produce a
singular metric for mL so that the vanishing theorem for its multiplier ideal sheaf
would give the existence of a global holomorphic section of mL + KX over X
which is nonzero at P0 . In order to make the explanation easier to understand,
we do the first couple of steps before we give the induction statement.
First we fix some terminology and introduce Nadel’s vanishing theorem for
multiplier ideal sheaves ([N89],[D93]). For a positive rational number α, by a
multivalued holomorphic section s of αL we mean that, for some positive integer
p, pα is a positive integer and sp is a global holomorphic section of pαL. We say
that the vanishing order of s at P0 is q if the vanishing order of sp at P0 is pq.
2
For a finite number of multivalued holomorphic sections s1 , · · · , sk of L we have
a (possibly) singular metric which, with respect to some local trivialization of
Pk
L, is locally given by ( ν=1 |sν |2 )−1 .
The vanishing theorem of Nadel states the following. Suppose there is a
(possibly√singular) metric of L locally given by e−ϕ such that the curvature
current −1∂∂ϕ dominates a positive definite smooth (1, 1)-form on X. Let
I be the multiplier ideal sheaf (for the metric) which is defined as the ideal
sheaf consisting of all holomorphic function germs f such that |f |2 e−ϕ is locally
integrable. Then H k (X, I(L + KX )) vanishes for k ≥ 1. The algebraic version
of Nadel’s theorem is the theorem of Kawamata-Viehweg [Ka82,V82], which for
our purpose could be used instead of Nadel’s theorem.
Let ǫ denote some sufficiently small positive rational number. We will use ǫ
as a generic notation for a sufficiently small positive rational number so that even
after a small change we keep the same notation ǫ. For the first step we choose by
the theorem of Riemann-Roch a multivalued holomorphic section s of (n + ǫ)L
which vanish to order at least n at P0 . There exists a positive rational number α
(as one can see for example by resolving the singularity of the divisor of s) such
that the zero-set X̃1 of the multiplier ideal sheaf for the metric |s|−2α contains
P0 but the zero-set of the multiplier ideal sheaf for |s|−2β does not contain P0
for any β < α. Since L is ample, we can find a finite number of multivalued
(1)
(1)
holomorphic sections t1 , · · · , tk1 of L over X whose common zero-set is one
branch X1 of X̃1 containing P0 . Now we choose two suitable sufficiently small
(1)
(1)
positive numbers σ, τ so that, with sj = sα−σ (tj )τ (1 ≤ j ≤ k1 ), the zero-set
Pk 1
(1)
|sj |2 )−1 is X1 , and
of the multiplier ideal sheaf for the metric h1 := ( j=1
that, for any γ < 1, the zero-set of the multiplier ideal sheaf for the metric hγ1
does not contain P0 . The metric h1 is a metric for (n + ǫ1 )L, where ǫ1 is a
sufficiently small positive rational number obtained by slightly changing ǫ.
Let d1 be the dimension of X1 at P0 . The induction process ends if d1 = 0.
So we assume that d1 > 0. Now we do the second step which will construct a
metric h2 for (d1 + n + ǫ2 )L (for some sufficiently small positive number ǫ2 ) so
that precisely one branch with dimension < d1 of the zero-set of the multiplier
ideal sheaf for h2 contains P0 . The obstacle mentioned in the introduction is the
difficulty encountered in the application of the theorem of Riemann-Roch when
X1 is singular at P0 . Let us first set aside this obstacle by considering the case
where P0 is a regular point of X1 . Then by the theorem of Riemann-Roch we
can find a multivalued holomorphic section s of (d1 +ǫ)L over X1 which vanishes
to order at least d1 at P0 . We can extend s to a multivalued holomorphic section
ŝ of (d1 + ǫ)L over all of X. However, the vanishing order of ŝ along the normal
direction of X1 at P0 may be a very small positive rational number. To make
up for the very small vanishing order along the normal direction of X1 at P0 , for
some sufficiently small η we use the metric h̃2 := h11−η |ŝ|−2 so that the zero-set
of its multiplier ideal sheaf contains P0 . Since η > 0, by the choice of h1 we
know that near P0 the zero-set X̃2 of the multiplier ideal sheaf of the metric h̃2
3
must be contained in the zero-set of ŝ.
Now we look at the case when X1 is singular at P0 . Because we have no
control over the multiplicity of X1 at P0 and over the nature of the singularity
there, we are unable to say that we need only (d1 + ǫ)L and not mL for some
large m not effectively determinable in order to get a multivalued holomorphic
section s over X1 to construct our singular metric with the desired property
for its multiplied ideal sheaf. It is until now an insurmountable obstacle. To
overcome this obstacle, we take a local complex curve ∆ in X1 passing through
P0 so that ∆ intersects the singular set of X1 only at P0 . By using the theorem
of Riemann-Roch with ∆ as the parameter space, we can find a multivalued
holomorphic section sP of (d1 + ǫ)L over X1 depending holomorphically on
P ∈ ∆ such that for P ∈ ∆ − P0 the vanishing order of sP at P is at least d1 .
We extend each sP to a multivalued holomorphic section ŝP of (d1 + ǫ)L over
all of X so that ŝP is holomorphic in P ∈ ∆. Then we get a family of singular
metrics (h˜2 )P (P ∈ ∆) so that the zero-set of its multiplier ideal sheaf contains
P for P ∈ ∆ − P0 . Here is the key step of the new technique to overcome
the obstacle. The holomorphic family of multiplier ideal sheaves parametrized
by P ∈ ∆ enjoys the following semicontinuity property. From the fact that P
belongs to the zero-set of the multiplier ideal sheaf of (h˜2 )P for P ∈ ∆ − P0 ,
it follows that P0 belongs to the zero-set of the multiplier ideal sheaf of (h˜2 )P0 .
We will discuss this semicontinuity property of multiplier ideal sheaves in a later
section. With the obstacle overcome, we continue our discussion of the second
step in the induction argument with ŝ = ŝP0 .
At this point, it is still too early to conclude that the dimension of X̃2 at P0
is < d1 , because the extra vanishing order from ŝ may add to some of the pole
order of h1 which earlier was not high enough to contribute to X1 and thus may
prevent us from concluding that X̂2 is contained in X1 . One way to avoid this is
2 −1
to use locally the metric h11−η (h−ρ
for some sufficiently small positive
1 + |ŝ| )
rational number ρ (and we may have to replace η by a smaller positive rational
number). To order to get a globally defined metric we have to multiply h−ρ
1 by
a factor so that after the multiplication the sum of h−ρ
and |ŝ|2 are globally
1
meaningful. To do this, we let θ1 , · · · , θℓ be multivalued holomorphic sections of
Pℓ
Pk 1
(1)
|sj |2ρ )( i=1 |θi |2δ ),
L without common zeroes and we replace h−ρ
by ( j=1
1
where δ = d1 + ǫ − (n + ǫ1 )ρ. So we get a globally defined metric for (d1 + n +
ǫ)L (after slightly changing ǫ) so that the zero-set of its multiplier ideal sheaf
contains P0 and has dimension < d1 at P0 . As in the first step where we use
(1)
(1)
the multivalued holomorphic sections t1 , · · · , tk1 , before we go on to the next
step we modify the metric so that precisely one branch of the zero-set of the
multiplier ideal sheaf for the new metric of (d1 +n+ǫ)L contains P0 . After these
two steps it is clear how we should go on in our argument to get the following
induction statement on ν for some integers 0 = dr < dr−1 < · · · < d1 < d0 = n.
(ν)
(ν)
There exist a finite number of multivalued holomorphic sections s1 , · · · , skν
Pν−1
of (ǫν + λ=0 dλ )L (for some sufficiently small positive rational number ǫν ) so
4
that precisely one branch of dimension dν of the zero-set of the multiplier ideal
Pν−1
Pk ν
(ν)
|sj |2 )−1 of (ǫν + λ=0 dλ )L contains P0 .
sheaf of the metric hν := ( j=1
Finally we take any smooth metric of L with positive definite curvature form
and with it, for any m ≥ 1 + (1 + 2 + · · · + n), we construct from hr a singular
metric of mL whose multiplier ideal sheaf has isolated zero at P0 . From Nadel’s
vanishing theorem it follows that there exists a global holomorphic section of
mL + KX which is nonzero at P0 , finishing the proof of the Corollary to the
Main Theorem.
§3. Semicontinuity of Multiplier Ideal Sheaves
We now state and prove the semicontinuity property of multiplier ideal
sheaves.
Lemma on the semicontinuity of multiplier ideal sheaves. Let X be a compact
complex manifold of complex dimension n and L be an ample line bundle over
X. Let P0 be a point of X and U ′ be a local holomorphic curve in X passing
through P0 with P0 as the only singularity and U be the open unit disk in C and
σ : U → U ′ be the normalization of U ′ so that σ(0) = P0 . Let β be a positive
rational number. Let s1 , · · ·, sk be multivalued holomorphic sections of pr1∗ (βL)
over X×U . Suppose that for almost all u ∈ U −0 (in the sense that the statement
is true up to a subset of measure zero) the point σ(u) × u belongs to the zeroPk
set of the multiplier ideal sheaf of the singular metric ( ν=1 |sν |2 )−1 |X × u
P
k
of βL = pr1∗ (βL)|X × u (i.e., the function ( ν=1 |sν |2 )−1 (·, u) is not locally
integrable at σ(u)). Then P0 × 0 belongs to the zero-set of the multiplier ideal
Pk
sheaf of the singular metric ( ν=1 |sν |2 )−1 |X × 0 of βL = pr1∗ (βL)|X × 0. (i.e.,
Pk
the function ( ν=1 |sν |2 )−1 (·, 0) is not locally integrable at P0 ).
The proof depends on the following extension theorem of Ohsawa-Takegoshi.
Theorem of Ohsawa-Takegoshi. Let Ω be a bounded smooth pseudoconvex domain in Cn+1 with coordinates z1 , · · ·, zn , w. Let H be defined by w = 0. Let
ϕ be a smooth plurisubharmonic function on Ω. There exists a constant CΩ
depending
only on Ω such that for any holomorphic function f on Ω ∩ H with
R
2 −ϕ
|f
|
e
< ∞Rthere exists a holomorphic
function F on Ω extending f with
H∩Ω
R
the property that Ω |F |2 e−ϕ ≤ CΩ H∩Ω |f |2 e−ϕ . Moreover, CΩ can be chosen
1 1/2
2
to be 64
, where A is any positive number with Ω ⊂ {|w| < A}.
9 πA (1 + 4e )
Proof of the lemma on the semicontinuity of multiplier ideal sheaves. Assume
the
Pkcontrary. Then for some open neighborhood D of P0 in X the function
( ν=1 |sν |2 )−1 (·, 0) is integrable on D. We can assume without loss of generality
that pr1∗ L|D ×U is holomorphically trivial and D is biholomorphic to a bounded
pseudoconvex domain in Cn . We apply Theorem (5.1) to the domain Ω = D×U
and the P
hyperplane H = Cn × 0. For the plurisubharmonic function we use
k
ϕ = log( ν=1 |sν |2 ) and for the function to be extended we use f ≡ 1. Let F
R
Pk
be the holomorphic function on D × U such that D×U |F |2 ( ν=1 |sν |2 )−1 < ∞
5
and F (·, 0) = f on D. There exist an open neighborhood D′ of P0 in D and
an open neighborhood W of 0 in U such that |F | is bounded from below on
D′ × W byR some positive
number. There is a set E of measure zero in W
Pk
such that D′ ×{u} ( ν=1 |sν |2 )−1 is finite for u ∈ W − E, contradicting the
Pk
assumption that ( ν=1 |sν |2 )−1 (·, u) is not locally integrable at σ(u) for almost
all u ∈ U − 0. Q.E.D.
§4. Proof of the Extension Theorem of Ohsawa-Takegoshi
Before we prove the theorem, we would like to make some remarks about its
proof.
First we recall the standard technique of using functional analysis and Hilbert
spaces to solve the ∂ equation. Denote ∂ by T and the ∂ in the next step of
the Dolbeault complex by S. Given g with Sg = 0 we would like to solve the
equation T u = g. The equation T u = g is equivalent to (v, T u) = (v, g) for all
v ∈ Ker S ∩ Dom T ∗ , which means (T ∗ v, u) = (v, g) for all v ∈ Ker S ∩ Dom T ∗ .
To get a solution u it suffices to prove that the map T ∗ v → (v, g) can be extended to a bounded linear functional, which means that |(v, g)| ≤ CkT ∗ vk for
all v ∈ Ker S ∩ Dom T ∗ . In that case we can solve the equation T u = g with
kuk ≤ C. We could also use the equaivalent inequality
|(v, g)|2 ≤ C 2 (kT ∗ vk2 + kSvk2 )
for all v ∈ Dom S ∩ Dom T ∗ . Suppose L is a semipositive operator. Suppose we
have the following inequality
|(Lv, v)| ≤ kT ∗ vk2 + kSvk2 .
Then we can use the Schwarz inequality and get
|(v, g)|2 ≤ |(Lv, v)||(L−1 g, g)| ≤ |(L−1 g, g)|(kT ∗ vk2 + kSvk2 ),
where |(L−1 g, g)| means ∞ if g is not in the domain of L−1 . This means that
1
we could choose C to be equal to |(L−1 g, g)| 2 and conclude that the equation
1
T u = g can be solved with kuk ≤ |(L−1 g, g)| 2 . In the usual application L is
the operator defined byP
multiplication by a positive number. In our application
where
g
is
a
(0,
1)-form
gα dz α , the semipositive operator L is defined by Lg =
P P
α
α(
β γαβ gβ )dz where (γαβ ) is a semipositive matrix of scalar functions. An
equivalent way of achieving the same effect as using the semipositive operator
L is to use the (limit) Kähler metric defined by (γαβ ).
Since the constant CΩ is to be independent of ϕ, it suffices to prove the
theorem for a smooth ϕ, we can write ϕ as the limit of smooth plurisubharmonic
functions ϕRν and then for each
R ν get a holomorphic function Fν extending f
such that Ω |Fν |2 e−ϕν ≤ CΩ H∩Ω |f |2 e−ϕν and then we can pass to limit as
6
ν → ∞ to get F from Fν . By the same reasoning we can assume without loss
of generality that the boundary of Ω is smooth.
The natural approach clearly is the following. We extend f to some holomorphic function on Ω without any condition on the bound of the extension and
we denote the extension also by f . Then we take 0 < λ < 1 and take a cut-off
function χ so that χ(ξ) is identically 1 on ξ ≤ λ and ξ is supported in ξ ≤ 1.
2
1
1
Let χǫ (w) = χ( |w|
ǫ2 ). Consider vǫ = w (∂(χǫ f )) = w (∂χǫ )f . We will solve the
∂-equation ∂uǫ = vǫ and set F = χǫ f − wuǫ so that ∂F = 0 and F agrees with
f on w = 0. The difficulty is to keep track of the estimates.
Let Uǫ = Ω ∩ {χǫ 6= 0}. Then
Z
Uǫ
|∂χǫ |2 2 −ϕ
|f | e
|w|2
is of the order ǫ12 . To offset this order, we can introduce the weight function
2
log(|w|2 + ǫ2 ) so that ∂w ∂w log(|w|2 + ǫ2 ) = (|w|2ǫ+ǫ2 )2 can be used to define L
2
2 2
)
of ∂w ∂w log(|w|2 + ǫ2 ) which is of order
and L−1 gives the reciprocal (|w| ǫ+ǫ
2
2
ǫ precisely cancelling the unwanted order ǫ12 . However, the use of the weight
function log(|w|2 + ǫ2 ) necessitates the introduction of the metric |w|21+ǫ2 which
contributes back the unwanted factor ǫ12 . The ideal situation is to be able to
2
get the contribution of the curvature term (|w|2ǫ+ǫ2 )2 without the contribution
of the metric |w|21+ǫ2 . Such an ideal situation is clearly impossible by the usual
Bochner-Kodaira type formula. Ohsawa-Takegoshi introduced a way of producing the curvature term without the contribution of the metric. In their proof
and also the later generalization by Manivel [M93] to the case of a general manifold, the actual underlying reason for the argument to work is obscured by a
long series of commutation identities in Kähler geometry and the introduction
of a specially contructed complete Kähler metric. Actually the key point for
the argument to work is the replacement, in the ∂-equation, of ∂ by ∂ composed with a scalar function on the right. The square of this scalar function
is broken down as the sum of two scalar functions. The first summand is used
in the usual Bochner-Kodaira formula to produce a term similar to the desired
curvature term. The second summand is used to insure that an inequality holds
so that the standard technique of functional analysis and Hilbert spaces can be
2
applied. The first summand used to produce the curvature term is log |w|A2 +ǫ2
2
so that the curvature term it produces is (|w|2ǫ+ǫ2 )2 . In the proof below the
√
2
salar function is η + γ with η = log |w|A2 +ǫ2 . In this easier way of looking at
the proof of Ohsawa-Takegoshi, there is no need to introduce any specially constructed complete Kähler metric and the estimate obtained is far sharper than
those produced in the original proofs of Ohsawa-Takegoshi and the subsequent
generalizationn by Manivel.
7
Proof of the extension theorem of Ohsawa-Takegoshi. For the proof we use the
notations introduced in the preceding remarks and most of the time use the summation convention of repeated indices. In order to separate the new arguments
in the approach of Ohsawa-Takegoshi from the standard basic estimates of the
Bochner-Kodaira formula, we divide the proof into two parts. Part I is simply
a reproduction of the usual Bochner-Kodaira formula with an introduction of a
scalar factor η. Part II consists of the new arguments.
∗
Part I. Let u be in the domain of ∂ on Ω. We consider the integration by parts
for
Z
Z
∗
∗
< η∂u, ∂u > e−ψ
< η∂ ψ u, ∂ ψ u > e−ψ +
=
Z
Ω
Ω
∗
< ∂(η∂ ψ u), u > e−ψ +
+
Z
Z
Ω
∗
Ω
< ∂ ψ (η∂u), u > e−ψ
η(∂α uβ − ∂β uα )(∂α ρ)uβ e−ψ .
∂Ω
Here for the integration by parts for
∗
R
Ω
∗
∗
< η∂ ψ u, ∂ ψ u > e−ψ there is no boundary
term because u ∈ Dom ∂ . Though ∂u = 12 (∂α uβ − ∂β uα )dz α ∧ dz β , we do not
have a factor 21 in the boundary term because the last factor in that integral is
only uβ . More precisely,
< ∂u, ∂u >=
X
α<β
=
(∂α uβ − ∂α uβ )(∂α uβ − ∂α uβ )
1X
(∂α uβ − ∂α uβ )(∂α uβ − ∂α uβ )
2
α,β
=
X
(∂α uβ − ∂α uβ )(∂α uβ ).
α,β
The sign for the boundary term on the right-hand side is checked against the
usual divergence theorem
Z
Z
Z
(∂j uj ) · 1 = −
uj · (∂j 1) +
(∂j ρ)uj
Ω
Ω
∂Ω
where ρ is the defining function for Ω in the sense that Ω = {ρ < 0} and
|dρ| ≡ 1 on ∂Ω. Note that the rule for the boundary term is that we replace the
differentiation ∂uj of uj by ∂ρj .
R
Ω
On the right-hand side of the Bochner-Kodaira formula we have the term
< η∇u, ∇u > e−ψ which is transformed by
Z
Z
< η∇u, ∇u > e−ψ =
η∂α uβ ∂α uβ e−ψ
Ω
Ω
8
Z
ψ
−ψ
Z
η∂α uβ (∂α ρ)uβ e−ψ .
R
We use the boundary term to cancel with part of the boundary term ∂Ω η(∂α uβ −
R
R
∗
∗
∂β uα )(∂α ρ)uβ e−ψ of Ω < η∂ u, ∂ u > + Ω < η∂u, ∂u > and we are left with
R
the other half − ∂Ω η∂β uα (∂α ρ)uβ e−ψ of boundary term. We use the trick of
=−
e ∂α (e
Ω
−ψ
η∂α uβ )uβ e
+
∂Ω
∗
C.B. Morrey to handle it. Since u ∈ Dom ∂ , it follows that uα ∂α ρ = 0 on ∂Ω.
In other words, uα ∂α ρ = θρ for some smooth function θ. Differentiation yields
∂β uα ∂α ρ + uα ∂β ∂α ρ = ρ∂β θ + θ∂β ρ = θ∂β ρ
on ∂Ω and
uβ ∂β uα ∂α ρ + uβ uα ∂β ∂α ρ = uβ θ∂β ρ = 0.
Hence
−
Z
∂Ω
η∂β uα (∂α ρ)uβ e−ψ =
Z
∂Ω
ηuβ uα (∂β ∂α ρ)e−ψ .
We have the following error term E = E1 + E2 + E3 , where
Z
Z
∗
∗
E1 =
< ∂(η∂ ψ u), u > e−ψ −
< η∂∂ ψ u, u > e−ψ
Ω
Ω
Z
∗
(∂α η)(∂ ψ u)uα e−ψ ,
Ω
Z
Z
∗
∗
−ψ
E2 =
< ∂ ψ (η∂u), u > e −
< η∂ ψ ∂u, u > e−ψ
Ω
Ω
Z
= − (∂α η)(∂α uβ − ∂β uα )uβ e−ψ ,
Ω
Z
Z
E3 =
< ∇ψ (η∇u), u > e−ψ −
< η∇ψ ∇u, u > e−ψ
Ω
Ω
Z
= (∂α η)(∂α uβ )uβ e−ψ .
=
Ω
With this error term we need up with
Z
Z
∗
∗
−ψ
< η∂u, ∂u > e−ψ
< η∂ ψ u, ∂ ψ u > e +
=
Z
∂Ω
Ω
ηuβ uα (∂β ∂α ρ)e−ψ +
Z
Ω
Ω
< η∇u, ∇u > e−ψ +
Z
Ω
ηuβ uα (∂β ∂α ψ)e−ψ + E.
Part II. We now introduce the main new arguments which transform the error
term to yield a curvature term. We leave E1 alone and combine E2 and E3
together and transform the sum. We get
Z
E2 + E3 = (∂α η)(∂β uα )uβ e−ψ
Ω
9
=
Z
∂Ω
−ψ
(∂α η)(∂β ρ)uα uβ e
=−
Z
Ω
−
Z
Ω
−ψ
(∂β ∂α η)uα uβ e
(∂β ∂α η)uα uβ e−ψ +
where the vanishing of the boundary term
∗
R
Z
+
Z
∗
Ω
(∂α η)uα ∂ ψ ue−ψ
∗
Ω
∂Ω
(∂α η)uα (∂ ψ u)e−ψ ,
(∂α η)(∂β ρ)uα uβ e−ψ from uβ ∂β ρ =
0 on ∂Ω because u ∈ Dom ∂ . By putting together the three error terms, we get
Z
Z
∗
−ψ
E1 + E2 + E3 = − (∂β ∂α η)uα uβ e + 2Re (∂α η)uα (∂ ψ u)e−ψ .
Ω
Ω
Thus we have
Z
<
Ω
=
+
Z
Ω
Z
∂Ω
∗
∗
η∂ ψ u, ∂ ψ u
−ψ
>e
+
ηuβ uα (∂β ∂α ψ)e
−
Z
< η∂u, ∂u > e−ψ
Ω
ηuβ uα (∂β ∂α ρ)e−ψ +
−ψ
Z
Z
< η∇u, ∇u > e−ψ
Ω
−ψ
Ω
(∂β ∂α η)uα uβ e
+ 2Re
Z
Ω
∗
(∂α η)uα (∂ ψ u)e−ψ .
Recall that Ω ⊂ {|w| < A}. Let
η = log
γ=
Then
A2
,
|w|2 + ǫ2
1
.
|w|2 + ǫ2
ǫ2
,
(|w|2 + ǫ2 )2
−∂w ∂w η =
w
,
|w|2 + ǫ2
w
.
∂w η = − 2
|w| + ǫ2
∂w η = −
The function η is introduced to give us the desired the curvature term while γ
is introduced to help us take care of the error term by Schwarz inequality. We
use the estimate
Z
Z
|w|
∗
∗
|2 Re
(∂α η)uα (∂ ψ u)e−ψ | ≤ 2
|u||∂ ψ u|e−ψ
2 + ǫ2
|w|
Ω
Ω
Z
Z
|w|2
1
∗
≤
|u|2 e−ψ +
|∂ ψ u|2 e−ψ .
2
2
2
2
Ω |w| + ǫ
Ω |w| + ǫ
10
Assume that η(∂α ∂β ψ)uα uβ ≥
Z
< (η +
Ω
∗
γ)∂ ψ u,
≥
Z
Ω
|w|2
2
|w|2 +ǫ2 |u|
∗
∂ψ u
−ψ
>e
+
and that Ω is pseudoconvex. Then
Z
< (η + γ)∂u, ∂u > e−ψ
Ω
(∂β ∂α log(|w|2 + ǫ2 ))uα uβ e−ψ .
√
We now consider the operator T defined by T u = ∂( η + γu) and the operator
√
S defined by Su = η + γ∂u. Then ST = 0 and
Z
kT ∗ uk2 + kSuk2 ≥ (∂β ∂α log(|w|2 + ǫ2 ))uα uβ e−ψ .
Ω
√
Let Uǫ = Ω ∩ {χǫ 6= 0} ⊂ Ω ∩ {|w|2 < λǫ2 }. We can solve ∂( η + γuǫ ) = vǫ
with
Z
Z
(|w|2 + ǫ2 )2 |∂χǫ |2 2
e−ψ
|uǫ |2 e−ψ ≤
|f | .
ǫ2
|w|2
Uǫ
Ω
2
(1+λ)2 π R
|f |2 e−ψ ,
The limit of the right-hand side as ǫ → 0 becomes 2 λ (1−λ)
2
Ω∩H
because
|∂χǫ |2 = |χ′ (
1
1 2 λǫ2
λ
|w|2 2 |w|2
)| 4 |dw|2 ≤ (
) 4 |dw|2 =
|dw|2
2
2
ǫ
ǫ
1−λ ǫ
(1 − λ) ǫ2
and the integral of |dw|2 over {|w|2 < λǫ2 } is 2πλǫ2 . Recall that F = χǫ f −
√
w η + γuǫ . Observe that the L2 norm of χǫ f vanishes as ǫ → 0, because f is
2
L and the support of χǫ f approaches a set of measure zero. The L2 norm of
√
w η + γuǫ is dominated by
Z
Z
√
√
λ2 (1 + λ)2
|f |2 e−ψ .
sup |w η + γ|
|uǫ |2 e−ψ ≤ sup |w η + γ|2π
2
(1
−
λ)
Ω
Ω
Ω∩H
Ω
Finally we take a plurisubharmonic function σ such that (∂α ∂β σ)uα uβ ≥ |u|2
and use ψ = ϕ + σ. Let
supΩ e−σ
B=
infΩ e−σ
The constant CΩ can be taken to be any positive number which dominates
B(1 + sup |w|2 log
Ω
A2 1/2 λ2 (1 + λ)2
) 2
|w|2
(1 − λ)2
for some 0 < λ < 1.
We now choose σ and λ to get a good explicit numerical value for the constant
2
(1+λ)2
which is the same as the critical
CΩ . We first find the critical value of λ(1−λ)
2
11
value 94 for log λ + log(1 + λ) − log(1 − λ) achieved at λ = 13 where its derivative
vanishes. For the choice of σ we consider first the case where Ω is contained in
{|w| < 12 } and for such a case we choose σ = log(|w|2 + ( 12 )2 ). Then ∂w ∂ w σ =
1
supΩ |w|2 +(
1 )2
supΩ e−σ
( 21 )2
2
=
≤ 2 Finally we
1 2 > 1 on Ω. We have B =
2
1
|w| +( 2 )
infΩ e−σ
infΩ |w|2 +(
1 )2
estimate supΩ |w|2 log
1
4|w|2 )
2
2
by finding its critical value as a function of |w|2
1
1 1/2
1
achieved at |w| = 4e
. Thus we can choose CΩ to be 16
which is 4e
9 π(1 + 4e )
1
when Ω ⊂ {|w| < 2 }. In general, we do a change of scale in the variable w and
1 1/2
2
. Thus the theorem of
conclude that we can choose CΩ to be 64
9 πA (1 + 4e )
Ohsawa-Takegoshi is proved.
§5. Alternative to the Use of the Extension Theorem of OhsawaTakegoshi
We now present, in an analytic setting, the replacement by Kollar of the use
of the extension theorem of Ohsawa-Takegoshi by the inversion of adjunction.
The extension theorem of Ohsawa-Takegoshi was used only to show that, for
local holomorphic functions Fj (z1 , · · · , zn , w) at 0 (1 ≤ j ≤ k) and for any
is locally integrable at 0 as a function
positive rational number α, if Pk 1
2α
(
on C , then Pk
n
(
1
|Fj |)2α
j=1
|fj |)
is locally integrable at 0 as a function on Cn+1 , where
j=1
fj (z1 , · · · , zn ) = Fj (z1 , · · · , zn , 0). Since the techniques of the proof can already
be seen in the case of k = 1, we present here only the case k = 1.
Lemma on the inheritance of non-integrability by restriction. Let F (z1 , · · · , zn , w)
be a holomorphic function defined on some Stein open neighborhood U × W of
0 in Cn × C and let α be a positive rational number. Let f (z1 , · · · , zn ) =
F (z1 , · · · , zn , 0). If |f 1|2α is locally integrable at 0 as a function on Cn , then
1
n+1
.
|F |2α is locally integrable at 0 as a function on C
Proof. By shrinking U and W we can assume without loss of generality that
1
|F |2α is integrable on U × W . Let D = α div F and S = U × 0 ⊂ U × W be
Q-divisors in U × W . We use D|S to denote α div f . Let π : Ũ → U × W
be a resolution of singularities for the Q-divisor D + S obtained by successive
monoidal transformations with nonsingular centers so that for some collection
{Eµ }ℓµ=0 of nonsingular hypersurfaces in Ũ in normal crossing the following
three conditions hold.P
ℓ
(i) π ∗ (D + S) = E0 + µ=1 aµ Eµ with aµ being nonnegative rational numbers,.
P
ℓ
(ii) KŨ − π ∗ KU ×W = µ=1 bµ Eµ with bµ being nonnegative integers.
(iii) the restriction of π to E0 is a modification of S so that from (i) and (ii) it
follows that
∗
π ∗ (D|S) − KE0 + (π|E0 )∗ KS
= π (D + S|S) − π ∗ (S|S) − (KŨ + E0 )|E0 + π ∗ (KU ×W + S)|S
12
=
X
µ∈J
(aµ − bµ )(Eµ ∩ E0 ),
where J is the set of all 1 ≤ µ ≤ ℓ with Eµ ∩ E0 6= ∅.
Note that one needs to consider π ∗ (D + S) instead of π ∗ D in (i) in order
for the formula for KE0 − (π|E0 )∗ KS in (iii) to hold, because of the identity
KE0 = (KŨ + E0 )|E0 and the corresponding identity KS = (KU ×W + S)|S,
compatible with the map π, for the trivial line bundles KS , KU ×W |S, S|S .
Since |f 1|2α is integrable on U × W , it follows that aµ − bµ < 1 for µ ∈ J. We
can conclude that |F1|2α is locally integrable at every point of S if aµ − bµ < 1
for 1 ≤ µ ≤ ℓ whenever π(Eµ ) ∩ S 6= ∅. Thus to finish the proof it suffices to
show that there is no 1 ≤ µ ≤ ℓ with Eµ ∩ E0 = ∅ and π(Eµ ) ∩ S 6= ∅. Suppose
there is such an index µ. We are going to derive a contradiction.
Since π : Ũ → U × W is obtained by a finite number of monoidal transformations, we can find sufficiently small positive rational numbers δµ (1 ≤ µ ≤ ℓ)
Pℓ
such that the Q-bundle − µ=1 δµ Eµ admits a Hermitian metric h0 along its
fibers whose curvature form is positive definite at every point of Ũ . Moreover, δµ
(1 ≤ µ ≤ ℓ) are chosen so small that aµ − bµ + δµ < 1 whenever aµ − bµ < 1. Let
Pℓ
s0 be the canonical section of the Q-bundle µ=1 δµ Eµ so that the divisor of s0
Pℓ
is µ=1 δµ Eµ . Let ω be the holomorphic (n + 1)-form π ∗ (dz1 ∧ · · · ∧ dzn ∧ wdw)
on Ũ . Let h be the (singular) Hermitian metric of along the fibers of −KŨ
defined by h0 |s0 |−2 |ω|2 |F |−2α . The multiplier ideal sheaf of h is the ideal sheaf
Pℓ
defined by the Q-divisor E0 + µ=1 ⌊aµ −bµ +δµ ⌋Eµ , where ⌊aµ −bµ +δµ ⌋ is the
rounddown (i.e. the integral part) of aµ − bµ + δµ . One concludes the vanishPℓ
ing of H 1 (Ũ , E0 + µ=1 ⌊aµ − bµ + δµ ⌋Eµ ). Though the vanishing theorem for
multiplier ideal sheaves is usually for compact manifolds, a simple modification
of its proof also gives its extension to the case of a complex manifold admitting
a proper holomorphic map onto a Stein manifold. Since aµ − bµ < 1 for every
Pℓ
Eµ ∩ E0 = ∅, the support of the divisor µ=1 ⌊aµ − bµ + δµ ⌋Eµ is disjoint from
E0 . There exists a holomorphic function
PℓG over Ũ which is identically 0 on E0
and is identically 1 on the support of µ=1 ⌊aµ − bµ + δµ ⌋Eµ ) in Ũ . The holomorphic function G on Ũ descends to a holomorphic function g on U × W which
is at the same time identically 0 on S due to itsPidentically zero value on E0 and
ℓ
identically 1 on the image of the support of µ=1 ⌊aµ − bµ + δµ ⌋Eµ , yielding
Pℓ
a contradiction, because the image of the support of µ=1 ⌊aµ − bµ + δµ ⌋Eµ
intersects S. Q.E.D.
§6. Difficulty in Improving the Quadratic Bound to the Conjectured
Linear Bound
We would like to discuss the difficulty in improving the quadratic bound to
the linear bound conjectured by the Fujita conjecture on freeness. By using
13
the theorem of Riemann-Roch for any 0 < ǫ < 1 we could get a multivalued
holomorphic section s of L over X vanishing at P0 to order at least 1 − ǫ. We
choose a positive number q (as small as possible) such that |s|−2q is locally
non-integrable at P0 and let X1 be the subvariety of X consisting of all the
points of X where |s|−2q is locally non-integrable. To be able to use iterative
notations, we let a(X) = q, V (X) = X1 , b(X) = dim P0 X1 and inductively define
V (ν) (X) = V (V (ν−1) (X)) with V (0) (X) = X. For this inductive definition
we have to handle the case of X being singular at P0 . In such a case, as
discussed above, we consider a 1-parameter holomorphic family of multivalued
holomorphic sectons sP with the holomorphic parameter P being a point in a
local holomorphic curve ∆ in X containing P0 so that s is equal to sP when
P = P0 and that X is regular at P for P 6= P0 . The vanishing order of s at P0
is replaced by the lim sup at P0 of the vanishing order of sP at P ∈ ∆ − {P0 }
as P approaches P0 . The property of local non-integrability of |s|−2q at P0
is replaced by the existence of a sequence of Pj ∈ ∆ − {P0 } of ∆ with limit
point P0 such that |sPj |−2q is not locally integrable at every Pj . Let ℓ be the
smallest ν such that b(V (ν) (X)) = 0. With such notations we have the freeness
Pℓ
of mL + KX for m > ν=0 a(V (ν) (X)).
To get a small m for the freeness of mL + KX , we would like to have a
small a(V (ν) (X)) and at the same time a rapid decrease of b(V (ν) (X)) as a
function of ν. The reason why we have to use a quadratic bound is that we may
encounter the worst situation of a(X) = dim P0 X and b(X) = dim P0 X − 1 and
the corresponding worst situation at every subsequent step. So we end up with
m = 1 + (1 + 2 + · · · + n) for the freeness of mL + KX if the worst situation is
assumed in every step. One way to try to get a small m closer to the conjectured
linear bound is to use at any given step all the vanishing orders unused in the
previous steps, so to speak. However, even such savings of the unused vanishing
orders are in general not enough to give us the conjectured linear bound by this
method.
The following example from Ein and Lazarsfeld illustrates in the case n = 3
how a possible situation prevents the construction, by this method, of a singular metric for (n + ǫ)L whose multiplier ideal sheaf has an isolated zero at a
prescribed point. Again ǫ denotes a generic sufficiently small positive number.
When one uses a multivalued holomorphic section s of (1 + ǫ)L to get vanishing
order at least 1 at the point P0 , the divisor of s is locally 21 times the divisor
P3
of j=1 zj2 . So in order for the zero-set of the multiplier ideal sheaf to contain
P0 we have to use the metric |s2 |−2 . The zero set of the multiplier ideal sheaf
P3
locally is the zero-set X1 of j=1 zj2 . For the next step we have to find a section
of L over X1 vanishing at P0 . By the theorem of Riemann-Roch,
the best one
√
2(1
+
ǫ)L over X1
can do is to get a multivalued holomorphic section
s
of
1
√
whose divisor is locally z1 + z2 . The number 2 comes from the formula of
Riemann-Roch when one tries to get a section vanishing at two points in the
regular part of X1 . By blowing up the single point P0 , near the inverse image of
14
P0 , the pullbacks of the divisors of s2 and s1 are expressed in terms of regular
surfaces in normal crossing. One can see that, in order for the zero-set of the
multiplier ideal sheaf of the singular metric constructed from s2(1−ǫ) and sα
1 to
contain P0 for some sufficiently small ǫ,√one must have α ≥ 1. So for these two
steps one is already forced to use 2 + 2 + ǫ times L. It turns out that the
Fujita conjecture on freeness is true for n = 3 as was proved in [EL93] or from
the √
method of the semicontinuity of multiplier ideal sheaves, simply because
2P+ 2 < 3 + 1. In this example of a surface singularity given by the divisor of
3
2
j=1 zj , for the final step we need only use up an additional ǫL because some
very small additional pole order at P0 in the singular metric would already
produce an isolated zero at P0 for the singular metric.
In the higher dimensional case, analogus but far more complicated situations
of singularities occur, which makes it impossible to simply refine this method
to improve the quadratic bound to the linear bound conjectured by the Fujita
conjecture.
§7. Remarks on Very Ampleness
One can get very ampleness from freeness by using the following lemma
which is proved by simply using the the global holomorphic sections of the free
ample line bundle to get a holomorphic map into a complex projective space.
The map must have finite fibers by the ampleness of the line bundles. Then
one uses the pullbacks of suitable hyperplane sections of the projective space to
construct a singular metric with the desired multiplier ideal sheaf to get very
ampleness.
Lemma to conclude very ampleness from freeness. Let L be an ample line bundle
over a compact complex manifold X of complex dimension n such that L is free.
Let A be an ample line bundle. Then (n + 1)L + A + KX is very ample.
By using this lemma and the Corollary to the Main Theorem we conclude
that, for any ample line bundle L over a compact complex manifold X of complex
dimension n, the line bundle (n + 1)(mL + KX ) + L + KX is very ample for
m ≥ 21 (n2 +n+2). In particular, for a compact complex manifold X with ample
KX , the line bundle mKX is very ample for m ≥ 12 (n3 + 2n2 + 5n + 8).
References.
[AS94] Angehrn, U., Siu, Y.-T.: Effective freeness and separation of points for
adjoint bundles. Invent. math., to appear.
[D93] Demailly, J.-P.: A numerical criterion for very ample line bundles. J. Diff.
Geom., 37, 323-374 (1993).
[EL93] Ein, L. and Lazarsfeld, R.: Global generation of pluricanonical and
adjoint linear series on smooth projective threefolds, J. of the A.M.S., 6, 875903 (1993).
15
[F87] Fujita, T.: On polarized manifolds whose adjoint bundles are not semipositive. In: Algebraic Geometry, Sendai, Advanced Studies in Pure Math., 10,
167-178 (1987).
[Ka82] Kawamata, Y.: A generalization of Kodaira-Ramanujam’s vanishing theorem, Math. Ann., 261, 43-46 (1982).
[K94] Kollár, J.: private e-mail communication, October, 1994.
[M93] Manivel, L.: Un thórème de prolongement L2 de sections holomorphes
d’un fibré hermitien. Math. Zeitschr. 212, 107-122 (1993).
[N89] Nadel, A.: Multiplier ideal sheaves and the existence of Kähler-Einstein
metrics of positive scalar curvature, Proc. Natl. Acad. Sci. USA, 86, 7299-7300
(1989), and Ann. of Math., 132, 549-596 (1989).
[OT87] Ohsawa, T., Takegoshi, K.: On the extension of L2 holomorphic functions, Math. Z., 195, 197-204 (1987).
[R88] Reider, I.: Vector bundles of rank 2 and linear systems on algebraic
surfaces, Ann. of Math., 127, 309-316 (1988).
[T94] Tsuji, H., Global generations of adjoint bundles. Preprint 1994.
[V82] Viehweg, E.: Vanishing theorems, J. reine und angew. Math., 335, 1-8
(1982).
Department of Mathematics, Harvard University, Cambridge, MA 02138, U.S.A.
16