Canonical singular hermitian metrics on relative canonical bundles
Canonical singular hermitian metrics
on relative canonical bundles
Hajime TSUJI
November 5, 2007
Abstract
We introduce a new class of canonical AZD’s (called the supercanonical
AZD’s) on the canonical bundles of smooth projective varieties with pseu-
doeffective canonical classes. We study the variation of the supercanonical
AZD ĥcan under projective deformations and give a new proof of the in-
variance of plurigenera. This paper is a continuation of [T5].
MSC: 14J15,14J40, 32J18
Contents
1 Introduction 1
1.1 Canonical AZD hcan . . . . . . . . . . . . . . . . . . . . . . . . . 2
1.2 Supercanonical AZD ĥcan . . . . . . . . . . . . . . . . . . . . . . 3
1.3 Variation of the supercanonical AZD ĥcan . . . . . . . . . . . . . 5
2 Proof of Theorem 1.7 6
2.1 Upper estimate of K̂Am . . . . . . . . . . . . . . . . . . . . . . . . 6
2.2 Lower estimate of K̂Am . . . . . . . . . . . . . . . . . . . . . . . . 7
2.3 Independence of ĥcan,A from hA . . . . . . . . . . . . . . . . . . 9
2.4 Completion of the proof of Theorem 1.7 . . . . . . . . . . . . . . 10
2.5 Comparison of hcan and ĥcan . . . . . . . . . . . . . . . . . . . . 10
3 Variation of ĥcan under projective deformations 11
3.1 Construction of ĥcan on a family . . . . . . . . . . . . . . . . . . 12
3.2 Semipositivity of the curvature current of ĥm,A . . . . . . . . . . 13
3.3 Uniqueness of ĥcan,A for singular hA’s . . . . . . . . . . . . . . . 16
3.4 Case dimS > 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
3.5 Completion of the proof of Theorem 1.10 . . . . . . . . . . . . . 17
4 Appendix 20
1 Introduction
Let X be a smooth projective variety and let KX be the canonical bundle of X .
In algebraic geometry, the canonical ring R(X,KX) := ⊕∞m=0Γ(X,OX(mKX))
is one of the main object to study.
http://arxiv.org/abs/0704.0566v5
Let X be a smooth projective variety such that KX is pseudoeffective. The
purposes of this article are twofold. The first purpose is to construct a singular
hermitian metric ĥcan on KX such that
1. ĥcan is uniquely determined by X .
ĥcan
is semipositive in the sense of current.
3. H0(X,OX(mKX)⊗I(ĥmcan)) ≃ H0(X,OX(mKX)) holds for every m ≧ 0,
where I(ĥmcan) denotes the multiplier ideal sheaf of ĥmcan as is defined in [N].
And the second purpose is to study the behavior of ĥcan on projective families.
We may summerize the 2nd and the 3rd conditions by introducing the following
notion.
Definition 1.1 (AZD)([T1, T2]) Let M be a compact complex manifold and
let L be a holomorphic line bundle on M . A singular hermitian metric h on L
is said to be an analytic Zariski decomposition (AZD in short), if the followings
hold.
1. Θh is a closed positive current.
2. For every m ≥ 0, the natural inclusion
H0(M,OM (mL)⊗ I(hm)) → H0(M,OM (mL))
is an isomorphim.
Remark 1.2 A line bundle L on a projective manifold X admits an AZD, if
and only if L is pseudoeffective ([D-P-S, Theorem 1.5]). �
In this sense, the first purpose of this article is to construct an AZD on KX
depending only on X , when KX is pseudoeffective (by Remark 1.2 this is the
minimum requirement for the existence of an AZD).
The main motivation to construct such a singular hermitian metric is to
study the canonical ring in terms of it. This is indeed possible. For example,
we obtain the invariance of plurigenera under smooth projective deformations
(cf. Corollary 1.12).
In fact the hermitian metric constructed here is useful in many other con-
texts. Other applications and a generalization to subKLT pairs will be treated
in the forthcoming papers ([T6]).
I would like to express thanks to Professor Bo Berndtsson who pointed out
an error in the previous version.
1.1 Canonical AZD hcan
If we assume the stronger assumption that X has nonnegative Kodaira dimen-
sion, we have already konwn how to construct a canonical AZD for KX . Let us
review the construction in [T5].
Theorem 1.3 ([T5]) Let X be a smooth projective variety with nonnegative
Kodaira dimension. We set for every point x ∈ X
Km(x) := sup{| σ |
m (x);σ ∈ Γ(X,OX(mKX)), |
(σ ∧ σ̄) 1m |= 1}
K∞(x) := lim sup
Km(x).
hcan := the lower envelope of K
is an AZD on KX. �
Remark 1.4 By the ring structure of R(X,KX), we see that
lim sup
Km(x) = sup
Km(x)
holds. �
Remark 1.5 Since h∞ depends only on X, the volume
h−1can
is an invariant of X. �
Apparently this construction is very canonical, i.e., hcan depends only on the
complex structure of X . We call hcan the canonical AZD of KX . But this
construction works only if we know that the Kodaira dimension of X is nonneg-
ative apriori. This is the main defect of hcan. For example, hcan is useless to
solve the abundance conjecture.
1.2 Supercanonical AZD ĥcan
To avoid the defect of hcan we introduce the new AZD ĥcan. Let us use the
following terminology.
Definition 1.6 Let (L, hL) be a singular hermitian line bundle on a complex
manifold X. (L, hL) is said to be pseudoeffective, if the curvature current of hL
is semipositive. �
Let X be a smooth projective n-fold such that the canonical bundle KX is
pseudoeffective. Let A be a sufficiently ample line bundle such that for every
pseudoeffective singular hermitian line bundle (L, hL) on X , OX(A+L)⊗I(hL)
and OX(KX+A+L)⊗I(hL) are globally generated. Such an ample line bundle
A extists by L2-estimates. Let hA be a a C
∞ hermitian metric on A with
strictly positive curvature 1. Let us fix a C∞ volume form dV on X . By the
L2-extension theorem ([O]) we may and do assume that A is sufficiently ample
1Later we shall also consider the case that hA is any C
∞ hermitian metric (without posi-
tivity of curvature) or a singular hermitian metric on A.
so that for every x ∈ X and for every pseudoeffective singular hermitian line
bundle (L, hL), there exists a bounded interpolation operator
Ix : A
2(x, (A+ L)x, hA · hL, δx) → A2(X,A+ L, hA · hL, dV )
such that the operator norm of Ix is bounded by a positive constant independent
of x and (L, hL), where A
2(X,A + L, hA · hL, dV ) denotes the Hilbert space
defined by
A2(X,A+L, hA·hL, dV ) := {σ ∈ Γ(X,OX(A+L)⊗I(hL)) |
| σ |2 ·hA·hL·dV < +∞}
with the L2 inner product
(σ, σ′) :=
σ · σ̄′ · hA · hL · dV
and A2(x, (A+L)x, hA ·hL, δx) is defined similarly, where δx is the Dirac measure
supported at x. We note that if hL(x) = +∞, then A2(x, (A+L)x, hA ·hL, δx) =
0. For every x ∈ X we set
K̂Am(x) := sup{| σ |
m (x) | σ ∈ Γ(X,OX(A+mKX)), |
A · (σ ∧ σ̄)
m |= 1}.
Here | σ | 2m is not a function on X , but the supremum is takan as a section of
the real line bundle |A | 2m ⊗ |KX |2 in the obvious manner2. Then h
A · K̂Am is a
continuous semipositive (n, n) form on X . Under the above notations, we have
the following theorem.
Theorem 1.7 We set
K̂A∞ := lim sup
A · K̂
ĥcan,A := the lower envelope of K̂
Then ĥcan,A is an AZD of KX . And we define
ĥcan := the lower envelope of inf
ĥcan,A,
where inf means the pointwise infimum and A runs all the ample line bundles
on X. Then ĥcan is a well defined AZD
3 depending only on X. �
Definition 1.8 (Supercanonical AZD) We call ĥcan in Theorem 1.7 the
supercanonical AZD of KX . And we call the semipositive (n, n) form ĥ
can the
supercanonical volume form on X. �
Remark 1.9 Here “super” means that corresponding volume form ĥ−1can satisfies
the inequality :
ĥ−1can ≧ h
if X has nonnegative Kodaria dimension (cf. Theorem 2.9). �
In the statement of Theorem 1.7, one may think that ĥcan,A may dependent of
the choice of the metric hA. But later we prove that ĥcan,A is independent of
the choice of hA (cf. Theorem 2.7).
2We have abused the notations |A|, |KX| here. These notations are similar to the notations
of corresponding linear systems. But I think there is no fear of confusion.
3I believe that ĥcan,A is already independent of the sufficiently ample line bundle A.
1.3 Variation of the supercanonical AZD ĥcan
Let f : X −→ S be an algebraic fiber space, i.e., X,S are smooth projective
varieties and f is a projective morphism with connected fibers. Suppose that
for a general fiber Xs := f
−1(s), KXs is pseudoeffective
4. In this case we may
define a singular hermitian metric ĥcan on KX/S similarly as above. Then ĥcan
have a nice properties on f : X −→ S as follows.
Theorem 1.10 Let f : X −→ S be an algebraic fiber space such that for a
general fiber Xs, KXs is pseudoeffective. We set S
◦ be the maximal nonempty
Zariski open subset of S such that f is smooth over S◦ and X◦ = f−1(S◦).
Then there exists a unique singular hermitian metric ĥcan on KX/S such that
1. ĥcan has semipositive curvature in the sense of current.
2. ĥcan |Xs is an AZD of KXs for every s ∈ S◦.
3. There exists the union F of at most countable union of proper subvarieties
of S such that for every s ∈ S \F ,
ĥcan|Xs ≦ ĥcan,s
holds, where ĥcan,s denotes the supercanonical AZD of KXs .
4. There exists a subset G of measure 0 in S◦, such that for every s ∈ S◦ \G,
ĥcan |Xs = ĥcan,s holds.
Remark 1.11 Even for s ∈ G, ĥcan|Xs is an AZD of KXs by 2. I do not know
whether F or G really exists in some cases. �
By Theorem 1.10 and the L2-extension theorem ([O-T, p.200, Theorem]), we
obtain the following corollary immediately.
Corollary 1.12 ([S1, S2, T3]) Let f : X −→ S be a smooth projective family
over a complex manifold S. Then plurigenera Pm(Xs) := dimH
0(Xs,OXs(mKXs))
is a locally constant function on S �
The following corollary is immediate consequence of Theorem 1.10, since the
supercanonical AZD is always has minimal singularities (cf. Definition 2.2 and
Remark 2.8).
Corollary 1.13 Let f : X −→ Y be an algebraic fiber space. Suppose that
KX and KY are pseudoeffective. Let ĥcan be the canonical singular hermitian
metric on KX/Y constructed as in Theorem 1.10. Let ĥcan,X , ĥcan,Y be the
supercanonical AZD’s of KX and KY respectively. Then there exists a positive
constant C such that
ĥcan,X ≦ C · ĥcan · f∗ĥcan,Y
holds on X. �
4This condition is equivalent to the one that for some regular fiber Xs, KXs is pseudoef-
fective. This is well known. For the proof, see Lemma 3.7 below for example.
Cororally 1.13 is very close to Iitaka’s conjecture which asserts that
Kod(X) ≧ Kod(Y ) + Kod(F )
holds for any algebraic fiber space f : X −→ Y , where F is a general fiber of
f : X −→ Y and Kod(M) denotes the Kodaira dimension of a compact complex
manifold M .
In this paper all the varieties are defined over C. And we frequently use the
classical result that the supremum of a family of plurisubharmonic functions
locally uniformly bounded from above is again plurisubharmonic, if we take the
uppersemicontinuous envelope of the supremum ([L, p.26, Theorem 5]). For
simpliciy, we denote the upper(resp. lower)semicontinuous envelope simply by
the upper(resp. lower) envelope. We note that this adjustment occurs only
on the set of measure 0. In this paper all the singular hermitian metrics are
supposed to be lowersemicontinuous.
There are other applications of the supercanonical AZD. Also it is imme-
diate to generalize it to the log category and another generalization involving
hermitian line bundles with semipositive curvature is also possible. These will
be discussed in the forthcoming papers.
2 Proof of Theorem 1.7
In this section we shall prove Theorem 1.7. We shall use the same notations as
in Section 1.2. The upper estimate of K̂Am is almost the same as in [T5], but the
lower estimate of K̂Am requires the L
2 extension theorem ([O-T, O]).
2.1 Upper estimate of K̂Am
Let X be as in Theorem 1.7 and let n denote dimX and let x ∈ X be an
arbitrary point. Let (U, z1, · · · , zn) be a coordinate neighbourhood ofX which is
biholomorphic to the unit open polydisk ∆n such that z1(x) = · · · = zn(x) = 0.
Let σ ∈ Γ(X,OX(A+mKX)). Taking U sufficiently small, we may assume
that (z1, · · · , zn) is a holomorphic local coodinate on a neighbourhood of the
closure of U and there exists a local holomorphic frame eA of A on a neighbour-
hood of the clousure of U . Then there exists a bounded holomorphic function
fU on U such that
σ = fU · eA · (dz1 ∧ · · · ∧ dzn)⊗m
holds. Suppose that
A · (σ ∧ σ̄)
m |= 1
holds. Then we see that
| fU (z) |
m dµ(z) ≦ (inf
hA(eA, eA))
hA(eA eA)
m | fU |2 dµ(z)
≦ (inf
hA(eA, eA))
hold, where dµ(z) denotes the standard Lebesgue measure on the coordinate.
Hence by the submeanvalue property of plurisubharmonic functions,
A · | σ |
m (x) ≦ { hA(eA, eA)(x)
infU hA(eA, eA)
m · π−n· |dz1 ∧ · · · ∧ dzn |2 (x)
holds. Let us fix a C∞ volume form dV on X . Since X is compact and every
line bundle on a contractible Stein manifold is trivial, we have the following
lemma.
Lemma 2.1 There exists a positive constant C independent of the line bundle
A and the C∞ metric hA such that
lim sup
A · K̂
m ≦ C · dV
holds on X. �
2.2 Lower estimate of K̂Am
Let hX be any C
∞ hermitian metric on KX . Let h0 be an AZD of KX defined
by the lower envelope of :
inf{h(x) | h is a singular hermitian metric on KX with Θh ≧ 0,h ≧ hX}.
Then by the classical theorem of Lelong ([L, p.26, Theorem 5]) it is easy to
verify that h0 is an AZD of KX (cf. [D-P-S, Theorem 1.5]). h0 is of minimal
singularities in the following sense.
Definition 2.2 Let L be a pseudoeffective line bundle on a smooth projective
variety X. An AZD h on L is said to be an AZD of minimal singularities,
if for any AZD h′ on L, there exists a positive constant C such that
h ≦ C · h′
holds. �
Let us compare h0 and ĥcan.
By the L2-extension theorem ([O]), we have the following lemma.
Lemma 2.3 There exists a positive constant C independent of m such that
K(A+mKX , hA · hm−10 ) ≧ C · (hA · hm0 )−1
holds, where K(A+mKX , hA · hm−10 ) is the (diagonal part of) Bergman kernel
of A+mKX with respect to the L
2-inner product:
(σ, σ′) := (
σ ∧ σ̄′ · hA · hm−10 ,
where we have considered σ, σ′ as A+ (m− 1)KX valued canonical forms. �
Proof of Lemma 2.3. By the extremal property of the Bergman kernel (see
for example [Kr, p.46, Proposition 1.4.16]) we have that
K(A+mKX , hA·hm−10 )(x) = sup{|σ(x) |
2| σ ∈ Γ(X,OX(A+mKX)⊗I(hm−10 )), ‖σ‖= 1},
holds for every x ∈ X , where ‖σ‖= (σ, σ) 12 . Let x be a point such that h0 is not
+∞ at x. Let dV be an arbitrary C∞ volume form on X as in Section 1.2. Then
by the L2-extension theorem ([O, O-T]) and the sufficiently ampleness of A (see
Section 1.2), we may extend any τx ∈ (A+mKX)x with hA·hm−10 ·dV −1(τx, τx) =
1 to a global section τ ∈ Γ(X,OX(A+mKX)⊗ I(hm−10 )) such that
‖τ ‖≦ C0,
where C0 is a positive constant independent of x and m. Let C1 be a positive
constant such that
h0 ≧ C1 · dV −1
holds on X . By (1), we obtain the lemma by taking C = C−10 · C1. �
Let σ ∈ Γ(X,OX(A+mKX)⊗ I(hm−10 )) such that
σ ∧ σ̄ · hA · hm−10 = 1
| σ |2 (x) = K(A+mKX , hA · hm−10 )(x)
hold, i.e., σ is a peak section at x. Then by the Hölder inequality we have that
A · (σ ∧ σ̄)
m | ≦ (
hA · hm0 · | σ |2 ·h−10 )
m · (
h−10 )
h−10 )
hold. Hence we have the inequality:
K̂Am(x) ≧ K(A+mKX , hA · hm−10 )(x)
m · (
h−10 )
m (2)
holds. Now we shall consider the limit
lim sup
A ·K(A+mKX , hA · h
Let us recall the following result.
Lemma 2.4 ([D, p.376, Proposition 3.1])
lim sup
A ·K(A+mKX , hA · h
m = h−10
holds. �
Remark 2.5 In ([D, p.376, Proposition 3.1], Demailly only considered the local
version of Lemma 2.4. But the same proof works in our case by the sufficiently
ampleness of A. This kind of localization principle for Bergman kernels is quite
standard. �
In fact the L2-extension theorem ([O-T, O]) implies the inequality
lim sup
A ·K(A+mKX , hA · h
m ≧ h−10
and the converse inequality is elementary. See [D] for details and applications.
Hence letting m tend to infinity in (2), by Lemma 2.4, we have the following
lemma.
Lemma 2.6
lim sup
A · K̂
m ≧ (
h−10 )
−1 · h−10
holds. �
By Lemmas 2.1 and 2.6, we see that
K̂A∞ := lim sup
A · K̂
exists as a bounded semipositive (n, n) form on X . We set
ĥcan,A := the lower envelope of (K
2.3 Independence of ĥcan,A from hA
In the above construction, ĥcan,A depends on the choice of the C
∞ hermitian
metric hA apriori. But actually ĥcan,A is independent of the choice of hA.
Let h′A be another C
∞-hermitian metric on A. We define
(K̂Am)
′ := sup{| σ | 2m ; σ ∈ Γ(X,OX(A+mKX)), |
(h′A)
m · (σ ∧ σ̄) 1m |= 1}.
We note that the ratio hA/h
A is a positive C
∞-function on X and
m = 1
uniformly on X . Since the definitions of K̂Am and (K̂
′ use the extremal prop-
erties, we see easily that for every positive number ε, there exists a positive
integer N such that for every m ≧ N
(1− ε)(K̂Am)′ ≦ K̂Am ≦ (1 + ε)(K̂Am)′
holds on X . Hence we obtain the following uniqueness theorem.
Theorem 2.7 K̂A∞ = lim supm→∞ h
A · K̂Am is independent of the choice of the
C∞ hermitian metric hA. Hence hcan,A is independent of the choice of the C
hermitian metric hA. �
2.4 Completion of the proof of Theorem 1.7
Let h0 be an AZD of KX constructed as in Section 2.1. Then by Lemma 2.6 we
see that
ĥcan,A ≦ (
h−10 ) · h0
holds. Hence we see
I(ĥmcan,A) ⊇ I(hm0 )
holds for every m ≧ 1. This implies that
H0(X,OX(mKX)⊗I(hm0 )) ⊆ H0(X,OX(mKX)⊗I(ĥmcan,A)) ⊆ H0(X,OX(mKX))
hold, hence
H0(X,OX(mKX)⊗ I(ĥmcan,A)) ≃ H0(X,OX(mKX))
holds for every m ≧ 1. And by the construction and the classical theorem of
Lelong ([L, p.26, Theorem 5]) stated in Section 1.3, ĥcan,A has semipositive
curvature in the sense of current. Hence ĥcan,A is an AZD of KX and depends
only on X and A by Lemma 2.7.
Let us consider
K̂∞ := sup
K̂∞,A
where sup means the pointwise supremum and A runs all the sufficiently am-
ple line bundle on X . Then Lemma 2.1, we see that K̂∞ is a well defined
semipositive (n, n) form on X . We set
ĥcan := the lower envelope of K̂
Then by the construction, ĥcan ≦ ĥcan,A for every ample line bundle A. Since ĥA
is an AZD ofKX , ĥcan is also an AZD ofKX indeed (again by [L, p.26, Theorem
5]). Since ĥcan,A depends only on X and A, ĥcan is uniquely determined by X .
This completes the proof of Theorem 1.7. �
Remark 2.8 As one see Section 2.2, we see that ĥcan is an AZD of KX of
minimal singularities (cf. Definition 2.2). �
2.5 Comparison of hcan and ĥcan
Suppose that X has nonnegative Kodaira dimension. Then by Theorem 1.3, we
can define the canonical AZD hcan on KX . We shall compare hcan and ĥcan.
Theorem 2.9
ĥcan,A ≦ hcan
holds on X. In particular
ĥcan ≦ hcan
holds on X �
Proof of Theorem 2.9. If X has negative Kodaira dimension, then the right
hand side is infinity. Hence the ineuqality is trivial.
Suppose thatX has nonnegative Kodaira dimension. Let σ ∈ Γ(X,OX(mKX))
be an element such that
(σ ∧ σ̄)
m |= 1
Let x ∈ X be an arbitrary point on X . Since OX(A) is globally generated by
the definition of A, there exists an element τ ∈ Γ(X,OX(A)) such that τ(x) 6= 0
and hA(τ, τ) ≦ 1 on X . Then we see that
hA(τ, τ)
m · (σ ∧ σ̄) 1m ≦ 1
holds. This implies that
K̂Am(x) ≧|τ(x) |
m ·Km(x)
holds at x. Noting τ(x) 6= 0,letting m tend to infinity, we see that
K̂A∞(x) ≧ K∞(x)
holds. Since x is arbitrary, this completes the proof of Theorem 2.9. �
Remark 2.10 The equality hcan = ĥcan implies the abundance of KX . �
By the same proof we obtain the following comparison theorem (without
assuming X has nonnegative Kodaira dimension).
Theorem 2.11 Let A,B a sufficiently ample line bundle on X. Suppose that
B −A is globally generated, then
ĥcan,B ≦ ĥcan,A
holds. �
Remark 2.12 Theorem 2.11 implies that
ĥcan = lim
ĥcan,ℓA
holds for any ample line bundle A on X. �
3 Variation of ĥcan under projective deforma-
tions
In this section we shall prove Theorem 1.10. The main ingredient of the proof
is the variation of Hodge structure.
3.1 Construction of ĥcan on a family
Let f : X −→ S be an algebraic fiber space as in Theorem 1.10.
The construction of ĥcan can be performed simultaeneously on the family
as follows. The same construction works for flat projective family with only
canonical singularities. But for simplicity we shall work on smooth category.
Let S◦ be the maximal nonempty Zariski open subset of S such that f is
smooth over S◦ and let us set X◦ := f−1(S◦).
Hereafter we shall assume that dimS = 1. The general case of Theorem 1.10
easily follows from just by cutting down S to curves. Let A be a sufficiently
ample line bundle on X such that for every pseudoeffective singular hermitian
line bundle (L, hL), OX(A+L)⊗I(hL) and OX(KX+A+L)⊗I(hL) are globally
generated and OXs(A+L |Xs)⊗I(hL |Xs) and OXs(KXs+A+L |Xs)⊗I(hL |Xs)
are globally generated for every s ∈ S◦ as long as hL|Xs is well defined.
Let us assume that there exists a smooth member D of | 2A | such that D
does not contain any fiber over S◦. Let σD a holomorphic section of 2A with
divisor D. We consider the singular hermitian metric
hA :=
| σD |
on A. We set
Em := f∗OX(A+mKX/S).
Since we have assumed that dimS = 1, Em is a vector bundle for every m ≧ 1.
We denote the fiber of the vector bundle over s ∈ S by Em,s. Then we shall
define the sequence of 1
A-valued relative volume forms by
K̂Am,s := sup{|σ |
m ;σ ∈ Em,s, |
A · (σ ∧ σ̄)
m |= 1}
for every s ∈ S◦. This fiberwise construction is different from that in Section
1.2 in the following two points :
1. We use the singular metric hA |Xs instead of a C∞ hermitian metric on
A |Xs.
2. We use Em,s instead of Γ(Xs,OXs(A|Xs +mKXs)).
We note that the 2nd difference occurs only over at most countable union of
proper analytic subsets in S◦. Since hA is singular, at some point s ∈ S◦ and
for some positive integer m0, K̂
might be identically 0 on Xs. But for any
s ∈ S◦ we find a positive integer m0 such that for every m ≧ m0, we have
A |Xs) = OXs holds for every m ≧ m0. Hence even in this case we see that
K̂Am,s is not identically 0 for every sufficiently large m.
We define the relative |A | 2m valued volume form K̂Am by
K̂Am|Xs := K̂Am,s(s ∈ S)
and a relative volume form K̂A∞ by
K̂A∞|Xs := lim sup
A · K̂
m,s(s ∈ S).
Of course the above construction of K̂Am,s(s ∈ S◦) works also for C∞ hermitian
metric instead of the singular hA as above. The reason why we use the singular
hA is that we shall use the variation of Hodge structure to prove the plurisub-
harmonic variation of log K̂Am,s.We may use a C
∞ metric with strictly positive
curvature on A, instead of the singular hA as above, if we use the plurisubhar-
monicity properties of Bergman kernels ([Ber, Theorem 1.2]) instead of Theorem
3.1. See Theorem 4.1 below.
We define singular hermitian metrics on A+mKX/S by
ĥm,A := the lower envelope of (K̂
Let us fix a C∞ hermitian metric hA,0 on A and we set
ĥcan,A := the lower envelope of lim inf
A,0 · ĥm,A.
Cleary ĥcan,A does not depend on the choice of hA,0 (in this sense, the presence
of hA,0 is rather auxilary). Then we define
ĥcan := the lower envelope of inf
ĥcan,A,
where A runs all the ample line bundle on X . At this moment, ĥcan is defined
only on KX/S |X◦. The extension of ĥcan to the singular hermitian metric on
the whole KX/S will be discussed later.
3.2 Semipositivity of the curvature current of ĥm,A
To prove the semipositivity of the curvature of ĥm,A, the following theorem is
essential.
Theorem 3.1 ([Ka3, p.174,Theorem 1.1] see also [F, Ka1]) φ : M −→ C be a
projective morphism with connected fibers from a smooth projective variety M
onto a smooth curve C. Let KM/C be the relative canonical bundle. We set
F := φ∗OM (KM/C)) and let C◦ denote the nonempty maximal Zariski open
subset of C such that φ is smooth over C◦. Let hM/C be the hermitian metric
on F | C◦ by
hM/C(σ, σ
′) := (
σ ∧ σ′,
where n = dimM − 1. Let π : P(F ∗) −→ C be the projective bundle associated
with F ∗ and Let L −→ P(F ∗) be the tautological line bundle. Let hL denote the
hermitian metric on L | π−1(S◦) induced by hM/C .
Then hL has semipositive curvature on π
−1(S◦) and hL extends to the sin-
gular hermitian metric on L with semipositive curvature current. �
We define the pseudonorm ‖σ‖ 1
of σ ∈ Em,s by
‖σ‖ 1
A · (σ ∧ σ̄)
m |m2 .
We set Em = f∗OX(A+mKX/S) and let Lm be the tautological line bundle
on P(E∗m), where E
m denotes the dual of Em. By Theorem 3.1 and the branched
covering trick, we obtain the following essential lemma.
Lemma 3.2 ([Ka1, p.63, Lemma 7 and p.64, Lemma 8]) Let
σ ∈ Γ(X,OX(A + mKX/S)). Then ‖ ‖ defines a singular hermitian metric
with semipositive curvature on Lm. �
Proof of Lemma 3.2. If there were no A, the lemma is completely the same as
[Ka1, p.63, Lemma 7 and p.64, Lemma 8]. In our case, we use the Kawamata’s
trick to reduce the logarithmic case to the non logarithmic case. Since this trick
has been used repeatedly by Kawamata himself (see [Ka2, Ka3] for example),
the following argument has no originality. We consider the multivalued relative
log canonical form
Then there exists a finite cyclic covering
µ : Y −→ X
such that µ∗( σ√
m is a (single valued) relative canonical form on Y 5. Here
the branch locus of µ may be much larger than the union of D∪(σ). But it does
not matter. The branch covering is used to reduce the log canonical case to the
canonical case. Let π : Ỹ −→ Y be an equivariant resolution of singularities
and let
f̃ : Ỹ −→ S
be the resulting family. We shall denote the composition µ ◦ π : Ỹ −→ X by
µ̃. Let U be a Zariski open subset of Sσ such that f̃ is smooth. We note that
the Galois group action is isometric on f̃∗OỸ (KỸ /S) with respect to the natural
L2-inner product on f̃∗OỸ (KỸ /S). Therefore by Theorem 3.1, we see that ‖ ‖
defines a singular hermitian metric on Lm with semipositive curvature on a
nonempty Zariski open of P(E∗m). Again by Theorem 3.1 the singular hermitian
metric extends to the whole P(E∗m) preserving semipositive curvature property.
We also present an alternative proof indicated by Bo Berndtsson at the workshop
at MSRI in April, 2007.
Alternative proof of Lemma 3.2(cf. [B-P, Section 6]).
We use the eqality
| σ | 2m= | σ |
| σ |2m−1m
and view
| σ |2m−1m
as a singular hermitian metric on (m− 1)KX/S +A. Then by [T4, Therem 5.4]
or [B-P], we see that
A · (σ ∧ σ̄)
m | 2m
defines a singular hermitian metric with semipositive curvature current on Lm.
The rest of the proof is identical as the previous one. �
5If we use a C∞ hermitian metric instead of the above hA, we also construct a cyclic
covering µ : Y −→ X such that 1
µ∗L is a genuine line bundle on Y and µ∗σ
m is a 1
valued canonical form on Y .
Remark 3.3 The metric hA can be replaced by a C
∞-hermitian metric with
semipositive curvature in the second proof. �
Corollary 3.4 (see also [B-P, Section 6]) The curvature Θ
ĥm,A
−1∂∂̄ log K̂m,A
is semipositive everywhere on X◦. �
Proof. Let x ∈ Xs(s ∈ S◦) and let Ω be a holomorphic local generatorof KX/S
and let eA be a holomorphic local generator of A on a neighbourhood U of x
in X◦. Viewing ξ(y) := (e−1A · Ω−m)(y) as an element of the dual of Em,f(y) by
σ ∈ Em,f(y) 7→ σ(y) · (e−1A · Ω−m)(y)(y ∈ U),
log(K̂m,A(y)· | eA |−
m · | Ω |−2 (y)) (y ∈ U)
is plurisubharmonic function on U , since
| ξ(y) |
m ·K̂m,A(y) = sup{
| ξ(y) · σ(y) | 2m
‖ [σ][ξ(y)] ‖
; σ ∈ Em,f(y), [σ][ξ(y)] 6= 0}
holds, where [σ][ξ(y)] denotes the class of σ ∈ Em,f(y) in the fiber Lm,[ξ(y)] at
[ξ(y)] ∈ P(E∗m). �
Now let us consider the behavior of ĥm,A along X\X◦. Since the problem is
local, we may and do assume S is a unit open disk ∆ in C for the time being.
For every local holomorphic section σ of Em the function
A · (σ ∧ σ̄)
is of algebraic growth along S \S◦. More precisely for s0 ∈ S \S◦ as in [Ka1,
p.59 and p. 66] there exist positive numbers C,α, β such that
A · (σ ∧ σ̄)
m |≦ C· |s− s0|−α · | log(s− s0) |β (3)
holds. Moreover as [Ka1, p.66] for a nonvanishing holomorphic section σ of Em
around p ∈ S \S◦, the pseudonorm
‖σ‖ 1
A (σ ∧ σ̄)
has a positive lower bound around every p ∈ S. This implies that ĥm,A is
bounded from below by a smooth metric along the boundary X \X◦. By the
above estimate, ĥm,A is of algebraic growth along the fiber on X \X◦ by its
definition and ĥm,A extends to a singular hermitian metric on
A+KX/S with
semipositive curvature on the whole X .
Now we set
ĥcan,A := the lower envelope of lim inf
A,0 · ĥm,A,
where hA,0 be a C
∞ metric on A (with strictly positive curvature) as in the last
subsetion 6.
To extend ĥcan,A across S \S◦, we use the following useful lemma.
6One may use hA instead of hA,0 here. But the corresponding limits may be different
along D, although the difference is negligible by taking the lower envelopes.
Lemma 3.5 ([B-T, Corollary 7.3]) Let {uj} be a sequence of plurisubharmonic
functions locally bounded above on the bounded open set Ω in Cm. Suppose
further
lim sup
is not identically −∞ on any component of Ω. Then there exists a plurisubhar-
monic function u on Ω such that the set of points
{x ∈ Ω | u(x) 6= (lim sup
uj)(x)}
is pluripolar. �
Since ĥm,A extends to a singular hermitian metric on
A + KX/S with
semipositive curvature current on the whole X and
ĥcan,A := the lower envelope of lim inf
A,0 · ĥm,A
exists as a singular hermitian metric on KX/S on X
◦ = f−1(S◦), we see that
ĥcan,A extends as a singular hermitian metric with semipositive curvature cur-
rent on the whole X by Lemma 3.5.
Repeating the same argument we see that ĥcan is a well defined singular her-
mitian metric with semipositive curvature current on KX/S |X◦ and it extends
to a singular hermitian metric on KX/S with semipositive curvature current on
the whole X .
3.3 Uniqueness of ĥcan,A for singular hA’s
In the above construction, we use a singular hermitian metric hA on A instead of
a C∞ hermitian metric. We note that hA is singular along the divisor D. Hence
the resulting metric may be a little bit different from the original construction
apriori. But actually Theorem 2.7 still holds. Our metric hA is defined as
as above. Let h′A be a C
∞ hermitian metric on A. Let us fix an arbitrary point
s ∈ S◦. Let us fix a Kähler metric on X and let Uε be the ε neighbourhood
of D with respect to the metric. By the upper estimate Lemma 2.1, we see
that although hA is singular along D, there exists a positive integer m0 and a
positive constnat C depending only on s such that for every m ≧ m0 and any
σ ∈ Em,s with
‖ σ ‖ 1
A · (σ ∧ σ̄)
2 = 1,
Uε∩Xs
A · (σ ∧ σ̄)
m |≦ C · ε
holds. This means that there is no mass concentration around the neighbour-
hood of D ∩ Xs. We note that on Xs \Uε the ratio (hA/h′A)
m converges uni-
formly to 1 as m tends to infinity. Hence by the definitions of K̂Am,s and (K̂
we see that for every s ∈ S◦ and δ > 0, there exists a positive integer m1 such
that for every m ≧ m1
(1− δ)(K̂Am,s)′ ≦ K̂Am,s ≦ (1 + δ)(K̂Am,s)′
holds on Xs. Hence we have the following lemma.
Lemma 3.6 K̂A∞,s is same as the one defined by a C
∞ hermitian metric on A
for every s ∈ S◦. �
3.4 Case dimS > 1
In Sections 3.1,3.2, we have assumed that dimS = 1. In the case of dimS > 1
the same proof works similarly. But there are several minor differences.
First there may not exist D ∈| 2A | which does not contain any fibers, hence
the restriction of hA may not be well defined on some fibers in this case. But
this can be taken care by Lemma 3.6. Namely ĥcan is independent of the choice
of D. Hence replacing hA by a C
∞ hermitian metric, we see that K̂A∞ is defined
on all fibers over S◦.
Second in this case Em = f∗OX(A+mKX/S) may not be locally free on S◦.
If Em.s is not locally free at s0 ∈ S◦, then K̂A∞ may be discontinuous at s0. But
J := {s ∈ S◦ | Em is not locally free at s for some m ≧ 1}
is at most a countable union of proper subvarieties of S◦ and
ĥcan,A := the lower envelope of
is a well defined singular hermitian metric with semipositive curvature current
on X◦, i.e., the construction is indifferent to the thin set J . Hence we may
construct ĥcan on X
◦ in this case. The extension of ĥcan as a singular hermitian
metric onKX/S with semipositive curvature current can be accomplished just by
slicing S by curves. Hence we complete the proof of the assertion 1 in Theorem
1.10.
3.5 Completion of the proof of Theorem 1.10
To complete the proof of Theorem 1.10, we need to show that ĥcan defines an
AZD for KXs for every s ∈ S. To show this fact, we modify the construction of
K̂Am. Here we do not assume dimS = 1.
Let us fix s ∈ S◦ and let h0,s be an AZD constructed as in Section 2.2. Let
U be a neighbourhood of s ∈ S◦ in S◦ which is biholomorphic to an open ball in
k(k := dimS). By the L2-extension theorem ([O-T, O]), we have the following
lemma.
Lemma 3.7 Every element of Γ(Xs,OXs(A | Xs+mKXs)⊗I(hm−10,s )) extends
to an element of Γ(f−1(U),OX(A+mKX)) for every positive integer m. �
Proof of Lemma 3.7. We prove the lemma by induction on m. If m =
1, then the L2-extension theorem ([O-T, O]) implies that every element of
Γ(Xs,OXs(A +KXs)) extends to an element of Γ(f−1(U),OX(A +KX)). Let
{σ(m−1)1,s , · · · , σ
(m−1)
N(m−1)} be a basis of Γ(Xs,OXs(A | Xs+(m−1)KXs)⊗I(h̃
0,s ))
for some m ≧ 2. Suppose that we have already constructed holomorphic exten-
sions
{σ̃(m−1)1,s , · · · , σ̃
(m−1)
N(m−1),s} ⊂ Γ(f
−1(U),OX(A+ (m− 1)KX))
of {σ(m−1)1,s , · · · , σ
(m−1)
N(m−1),s} to f
−1(U). We define the singular hermitian metric
Hm−1 on (A+ (m− 1)KX) | f−1(U) by
Hm−1 :=
1∑N(m−1)
j=1 | σ̃
(m−1)
j,s |2
We note that by the choice of A, OXs(A |Xs + mKXs) ⊗ I(hm−10,s ) is globally
generated. Hence we see that
I(hm0,s) ⊆ I(hm−10,s ) ⊆ I(Hm−1|Xs)
hold on Xs. Apparently Hm−1 has a semipositive curvature current. Hence
by the L2-extension theorem ([O-T, p.200, Theorem]), we may extend every
element of
Γ(Xs,OXs(A+mKXs)⊗ I(hm−10,s ))
extends to an element of
Γ(f−1(U),OX(A+mKX)⊗ I(Hm−1)).
This completes the proof of Lemma 3.7 by induction. �
Let hA,0 be a C
∞ hermitian metric on A with strictly positive curvature as
in the end of the last subsection. We define the sequence of {K̃Am,s} by
K̃Am,s := sup{| σ |
m ; σ ∈ Γ(Xs,OXs(A | Xs+mKXs)⊗I(hm−10,s )), |
A,0·(σ∧σ̄)
m |= 1}.
By Lemma 3.7, we obtain the following lemma immediately.
Lemma 3.8
lim sup
A,0 · K̃
m,s ≦ K̂
holds. �
Proof. We set
K̂A,0m,s = sup{| σ |
m ; σ ∈ Em,s, |
A,0 · (σ ∧ σ̄)
m |= 1}.
Then by the definition of K̃Am,s and Lemma 3.7 we have that
K̃Am,s ≦ K̂
m,s (4)
holds on Xs. On the other hand by Lemma 3.6, we see that
lim sup
A,0 · K̂
m,s = lim sup
A,0 · K̂
m,s = K̂∞,s (5)
hold. Hence combining (4) and (5), we complete the proof of Lemma 3.8. �
We set
h̃m,A,s := (K̃
We have the following lemma.
Lemma 3.9 If we define
K̃A∞,s := lim sup
A,0 · K̃
h̃∞,A,s := the lower envelope of K̃
∞.A,s,
h̃∞,A,s is an AZD of KXs . �
Proof. Let h0,s be an AZD of KXs as above. We note that
OXs(A |Xs + mKXs) ⊗ I(hm−10,s ) is globally generated by the definition of A.
Then by the definition of K̃Am,s,
I(hm0,s) ⊆ I(h̃mm,A,s)
holds for every m ≧ 1. Hence by repeating the arugument in Section 2.2, similar
to Lemma 2.6, we have that
h̃∞,A,s ≦ (
h−10,s) · h0,s
holds. Hence h̃∞,A,s is an AZD of KXs . �
Since by the construction and Lemma 3.6
ĥcan,s ≦ h̃∞,A,s
holds on s, we see that ĥcan |Xs is an AZD of KXs . Since s ∈ S◦ is arbitrary,
we see that ĥcan |Xs is an AZD of KXs for every s ∈ S◦. This completes
the proof of the assertion 2 in Theorem 1.10. We have already seen that the
singular hermitian metric ĥcan has semipositive curvature in the sense of current
(cf. Section 3.2 expecially Corollary 3.4). We note that there exists the union
F of at most countable union of proper subvarieties of S such that for every
s ∈ S◦ \F
E(ℓ)m,s = Γ(Xs,OX(ℓA+mKXs))
holds for every ℓ,m ≧ 1. Then by the construction and Theorem 2.11(see
Remark 2.12)7 for every s ∈ S◦ \F ,
ĥcan|Xs ≦ ĥcan,s
holds, where ĥcan,s is the supercanonical AZD of KXs . This completes the proof
of the assertion 3 in Theorem 1.10.
7Theorem 2.11 is used because some ample line bundle on the fiber may not extends to an
ample line bundle on X in general.
We shall define the singular hermitian metric Ĥcan on KX/S|X◦ by
Ĥcan|Xs := ĥcan,s (s ∈ S◦).
Then by the construction of ĥcan there exists a subset Z of measure 0 in X
such that
Ĥcan|X◦ \Z = ĥcan|X◦ \Z
holds. Let us set
G := {s ∈ S◦ | Xs ∩ Z is not of measure 0 in Xs}.
Then since Z is of measure 0, G is of measure 0 in S◦. For s ∈ S \G, by the
definition of the supercanonical AZD ĥcan,s of KXs , we see that
ĥcan|Xs = ĥcan,s
holds. This completes the proof of Theorem 1.10. �
Remark 3.10 As above we have used the singular hermitian metric hA to prove
Theorem 1.10 and then go back to the case of a C∞ metric by the uniqueness
result (Lemma 3.6). This kind of interaction between singular and smooth met-
rics have been seen in the convergence of the currents associated with random
sections of a positive line bundle to the 1-st Chern form of the positive line
bundle (see [S-Z]). My first plan of the proof of Theorem 1.10 was to use the
random sections to go to the smooth case from the singular case. Although I
cannot justify it, it seems to be interesting to pursue this direction. �
4 Appendix
The following theorem is a generalization of Theorem 3.1.
Theorem 4.1 φ : M −→ C be a projective morphism with connected fibers
from a smooth projective variety M onto a smooth curve C. Let KM/C be the
relative canonical bundle. Let (L, hL) be a pseudoeffective singular hermitian
line bundle on M Let m be a positive integer. We set F := φ∗OM (mKM/C +L)
and let C◦ denote the nonempty maximal Zariski open subset of C such that
φ is smooth over C◦. Let π : P(F ∗) −→ C be the projective bundle associated
with F ∗ and Let H −→ P(F ∗) be the tautological line bundle. Let hH denote
the singular hermitian metric on H | π−1(S◦) defined by
hH(σ, σ) := {(
L · (σ ∧ σ)
m }m2 ,
where n = dimM − 1. Then hH has semipositive curvature on π−1(S◦) and
hH extends to the singular hermitian metric on H with semipositive curvature
current. �
Proof. The proof is a minor modification of the proof of Lemma 3.2. Let σ
be a local holomorphic section of H on π−1(S◦). We consider the multivalued
L-valued canonical form m
σ and uniformize it by taking a suitable cyclic
Galois covering
µ : Y −→ X
as in Lemma 3.2. Then applying [Ber, Theorem 1.2] or [T4, Theorem 5.4] (see
also [B-P]) on Y , as in Lemma 3.2, we see that hH defines a singular hermitian
metric on the tautological line bundle on P(E∗m). Hence we see that hH has
semipositive curvature on π−1(S◦). The extension of hH to the whole H is also
follows from [T4, Theorem 5.4]. This completes the proof of Theorem 4.1. �
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Author’s address
Hajime Tsuji
Department of Mathematics
Sophia University
7-1 Kioicho, Chiyoda-ku 102-8554
Japan
http://arxiv.org/abs/math/0606023
http://arxiv.org/abs/math/0703729
	Introduction
	Canonical AZD hcan
	Supercanonical AZD can
	Variation of the supercanonical AZD can
	Proof of Theorem 1.7
	Upper estimate of mA
	Lower estimate of mA
	Independence of can,A from hA
	Completion of the proof of Theorem 1.7
	Comparison of hcan and can
	Variation of can under projective deformations
	Construction of can on a family
	Semipositivity of the curvature current of m,A
	Uniqueness of can,A for singular hA's
	Case dimS > 1
	Completion of the proof of Theorem 1.10
	Appendix
