On semipositivity theorems

0.21 MB·

by  Osamu Fujino

ON SEMIPOSITIVITY THEOREMS
OSAMU FUJINO AND TARO FUJISAWA
7 Abstract. WegeneralizetheFujita–Zucker–Kawamatasemipositivitytheoremfromthe
1 analytic viewpoint.
0
2

n Contents
a
J
1. Introduction 1
8
2. Preliminaries 3

] 3. Nefness 5
G
4. Proof of Theorem 1.1 6
A
5. Proof of Theorem 1.5 12
.
h References 12
t
a
m
[

1 1. Introduction
v
9 Themainpurposeofthispaperistogeneralizethewell-knownFujita–Zukcer–Kawamata
3
semipositivity theorem (see [Kaw1, 4. Semi-positivity], [Kaw2, Theorem 2], [FF, Section
0 §
2 5], and [FFS, Theorem 3]) from the analytic viewpoint.
0
. Theorem 1.1. Let X be a complex manifold and let X X be a Zariski open set such
1 0
⊂
0 that D = X X is a normal crossing divisor on X. Let V be a polarizable variation of
0 0
7 R-Hodge stru\cture over X with unipotent monodromies around D. Let Fb be the canonical
1 0
: extension of the lowest piece of the Hodge filtration. Let Fb L be a quotient line bundle
v →
of Fb. Then the Hodge metric of Fb induces a singular hermitian metric h on L such that
i
X
√ 1Θ (L) 0 and the Lelong number of h is zero everywhere.
h
r − ≥
a
As a direct consequence of Theorem 1.1, we have:
Corollary 1.2 (cf. [Kaw3]). Let X be a complex manifold and let X X be a Zariski
0
⊂
open set such that D = X X is a normal crossing divisor on X. Let V be a polarizable
0 0
\
variation of R-Hodge structure over X with unipotent monodromies around D. Let Fb be
0
the canonical extension of the lowest piece of the Hodge filtration. Then O (1) has a
PX(Fb)
singular hermitian metric h such that √ 1Θ (O (1)) 0 and that the Lelong number
− h PX(Fb) ≥
of h is zero everywhere. Therefore, Fb is nef in the usual sense when X is projective.
Remark 1.3. There exists a quite short published proof of Corollary 1.2 (see the proof
of [Kaw3, Theorem 1.1]). However, we have been unable to follow it. We also note that
the arguments in [Kaw1, 4. Semi-positivity] contain various troubles. For the details, see
§
[FFS, 4.6. Remarks].
Date: 2016/12/21,version 0.08.
2010 Mathematics Subject Classification. Primary 14D07; Secondary 14C30, 32J25, 32U25.
Key words and phrases. variationofHodgestructure,singularhermitianmetric,nefness, Higgsbundle,
system of Hodge bundles.
1
2 OSAMU FUJINO ANDTAROFUJISAWA
Remark 1.4. When X is projective and V is geometric in Corollary 1.2, the nefness of Fb
0
has already played important roles in the Iitaka program and the minimal model program
for higher-dimensional complex algebraic varieties.
More generally, we can prove:
Theorem 1.5. Let X be a complex manifold and let X X be a Zariski open set such
0
⊂
that D = X X is a normal crossing divisor on X. Let V be a polarizable variation of
0 0
\
R-Hodge structure over X with unipotent monodromies around D. If M is a holomorphic
0
line subbundle of the associated system of Hodge bundles Gr•F V = pGrpF V which is
contained in the kernel of the Higgs field
L
θ : Gr V Ω1 (logD) Gr V ,
•F → X ⊗OX •F
then the Hodge metric induces a singular hermitian metric h on its dual M such that
∨
√ 1Θ (M ) 0 and that the Lelong number of h is zero everywhere.
h ∨
− ≥
For the details of the Higgs field θ : Gr•F V → Ω1X(logD) ⊗OX Gr•F V in Theorem 1.5,
see Definition 2.7 below.
As a direct easy consequence of Theorem 1.5, we obtain:
Corollary 1.6 ([Z] and [B, Theorem 1.8]). Let X be a complex manifold and let X X
0
⊂
be a Zariski open set such that D = X X is a normal crossing divisor on X. Let
0
\
V be a polarizable variation of R-Hodge structure over X with unipotent monodromies
0 0
around D. If A is a holomorphic subbundle of the associated system of Hodge bundles
Gr•F V = pGrpF V which is contained in the kernel of the Higgs field
L θ : Gr•F V → Ω1X(logD)⊗Gr•F V ,
then OPX(A∨)(1) has a singular hermitian metric h such that √−1Θh(OPX(A∨)(1)) ≥ 0 and
that the Lelong number of h is zero everywhere. Therefore, the dual vector bundle A is
∨
nef in the usual sense when X is projective.
Corollary 1.6 is an analytic version of [B, Theorem 1.8]. For some generalizations of [B,
Theorem 1.8] from the Hodge module theoretic viewpoint, see [PoS, Theorem 18.1] and
[PoW, Theorem A].
Remark 1.7. Let a be the integer such that Fa+1 ( Fa = V . Then, in Corollary
0 0 0
1.6, GraF V is a holomorphic subbundle of Gr•F V and is contained in the kernel of θ.
Therefore, we can use Corollary 1.6 for A = Gra V . By considering the dual Hodge
F
structure in Corollary 1.6 and putting A = Gra V , Corollary 1.6 is also a generalization
F
of the Fujita–Zucker–Kawamata semipositivity theorem (see, for example, [FF, Remark
3.15]). Of course, by considering the dual Hodge structure, Theorem 1.5 contains Theorem
1.1 as a special case.
Ourproofinthispaperheavilydependson[Ko], whichisbasedon[CKS], andDemailly’s
approximation result for quasi-plurisubharmonic functions on complex manifolds (see [D1]
and [D2]).
Remark 1.8 (Singular hermitian metrics on vector bundles). We note that our results
explained above are local analytic. Therefore, we can easily see that the Hodge metric
of Fb in Theorem 1.1 is a semipositively curved singular hermitian metric in the sense
of Pa˘un–Takayama (see [Pa˘T, Definition 2.3.1]). Moreover, in Corollary 1.6, the induced
metric on A is a seminegatively curved singular hermitian metric in the sense of Pa˘un–
Takayama (see [Pa˘T, Definition 2.3.1]). For the details of singular hermitian metrics on
vector bundles, see [Pa˘T].
ON SEMIPOSITIVITY THEOREMS 3
Acknowledgments. The first author was partially supported by JSPS KAKENHI Grant
Numbers JP2468002, JP16H03925, JP16H06337. He thanks Professor Dano Kim whose
question made him consider the problems discussed in this paper. Moreover, he pointed
out an ambiguity in a preliminary version of this paper. The authors thank Professor
Shin-ichi Matsumura for answering their questions and giving them some useful comments
on singular hermitian metrics on vector bundles.
2. Preliminaries
In this section, we collect some basic definitions and results.
2.1 (Singularhermitianmetrics, multiplier ideal sheaves, andsoon). Let usrecall someba-
sic definitions and facts about singular hermitian metrics and plurisubharmonic functions.
For the details, see [D2, (1.4), (3.12), (5.4), and so on].
Definition 2.2 (Singular hermitian metrics). Let L be a holomorphic line bundle on a
complex manifold X. A singular hermitian metric on L is a metric which is given in every
trivialization θ : L U C by
U
| ≃ ×
ξ = θ(ξ) e ϕ(x), x U, ξ L ,
− x
|| || | | ∈ ∈
where ϕ L1 (U) is an arbitrary function, called the weight of the metric with respect to
∈ loc
the trivialization θ. Note that L1 (U) is the space of locally integrable functions on U.
loc
Definition 2.3 ((Quasi-)plurisubharmonic functions). A function ϕ : U [ , )
→ −∞ ∞
defined on an open set U Cn is called plurisubharmonic if
⊂
(i) ϕ is upper semicontinuous, and
(ii) for every complex line L Cn, ϕ is subharmonic on U L, that is, for every
U L
a U and ξ Cn satisfy⊂ing ξ <| d∩(a,Uc) = inf a x x ∩ Uc , the function ϕ
∈ ∈ | | {| − || ∈ }
satisfies the mean inequality
1 2π
ϕ(a) ϕ(a+eiθξ)dθ.
≤ 2π
Z0
Let X be an n-dimensional complex manifold. A function ϕ : X [ , ) is said to
→ −∞ ∞
be plurisubharmonic if there exists an open cover X = U such that ϕ is plurisub-
i I i |Ui
harmonic on U ( Cn) for every i. A quasi-plurisubhar∈monic function is a function ϕ
i
⊂ S
which is locally equal to the sum of a plurisubharmonic function and of a smooth function.
Let ϕ be a quasi-plurisubharmonic function on a complex manifold X. Then the multi-
plier ideal sheaf J(ϕ) O is defined by
X
⊂
Γ(U,J(ϕ)) = f O (U) f 2e 2ϕ L1 (U)
{ ∈ X || | − ∈ loc }
for every open set U X. It is well known that J(ϕ) is a coherent ideal sheaf on X.
⊂
Definition 2.4 (Lelong numbers). Let ϕ be a quasi-plurisubharmonic function on U (
⊂
Cn). The Lelong number ν(ϕ,x) of ϕ at x U is defined as follows:
∈
ϕ(z)
ν(ϕ,x) = liminf .
z x log z x
→ | − |
It is well known that ν(ϕ,x) 0.
≥
In this paper, we will implicitly use the following easy lemma repeatedly.
Lemma 2.5. Let L be a holomorphic line bundle on a complex manifold X. Let h = ge 2ϕ
−
be a singular hermitian metric on L, where g is a smooth hermitian metric on L and ϕ is
a locally integrable function on X. We assume that √ 1Θ (L) 0. Then there exists a
h
− ≥
quasi-plurisubharmonic function ψ on X such that ϕ coincides with ψ almost everywhere.
4 OSAMU FUJINO ANDTAROFUJISAWA
In this situation, we put J(h) = J(ψ). Moreover, we simply say the Lelong number of
h to denote the Lelong number of ψ if there is no risk of confusion.
2.6 (Systems of Hodge bundles, Higgs fields, curvatures, and so on). Let us recall the
definition of systems of Hodge bundles.
Definition 2.7 (Systems of Hodge bundles). Let V = (V ,F ) be a polarizable variation
0 0 0
of R-Hodge structure on a complex manifold X , where V is a local system of finite-
0 0
dimensional R-vector spaces on X and Fp is the Hodge filtration. Then we obtain a
0 { 0}
Higgs bundle (E ,θ ) on X by setting
0 0 0
E0 = Gr•F0V0 = F0p/F0p+1
p
M
where V = V O . Note that θ is induced by the Griffiths transversality
0 0 ⊗ X0 0
∇ : F0p → Ω1X0 ⊗OX0 F0p−1.
More precisely, induces
∇
θ0p : F0p/F0p+1 → Ω1X0 ⊗OX0 F0p−1/F0p
for every p. Then
(cid:0) (cid:1)
θ = θp : E Ω1 E .
0 0 0 → X0 ⊗OX0 0
p
M
The pair (E ,θ ) is usually called the system of Hodge bundles associated to V = (V ,F )
0 0 0 0 0
and θ is called the Higgs field of (E ,θ ).
0 0 0
We further assume that X is a Zariski open set of a complex manifold X such that
0
D = X X is a normal crossing divisor on X and that the local monodromy of V around
0 0
\
D is unipotent. Then, by [S], we can extend (E ,θ ) to (E,θ) on X, where
0 0
E = Gr•F V = Fp/Fp+1
p
M
and
θ : E Ω1 (logD) E.
→ X ⊗OX
Note that V is the canonical extension of V and Fp is the canonical extension of Fp for
0 0
every p.
We need the following important calculations of curvatures by Griffiths.
Lemma 2.8. We use the same notation as in Definition 2.7. Let Fb be the lowest piece of
0
the Hodge filtration. Let q be the metric of Fb induced by the Hodge metric. Let Θ (Fb)
0 0 q0 0
be the curvature form of (Fb,q ). Then we have
0 0
Θ (Fb)+(θb) θb = 0
q0 0 0 ∗ ∧ 0
where (θb) is the adjoint of θb with respect to the Hodge metric (see, for example, [GT] and
0 ∗ 0
[S, (7.18) Lemma]). Let L be a quotient line bundle of Fb. Then we have the following
0 0
short exact sequence of locally free sheaves.
0 S Fb L 0.
→ 0 → 0 → 0 →
Let A be the second fundamental form of the subbundle S Fb. Let h be the induced
0 ⊂ 0 0
metric of L . Then we obtain
0
√ 1Θ (L ) = √ 1Θ (Fb) +√ 1A A
− h0 0 − q0 0 |L0 − ∧ ∗
= √ 1(θb) θb +√ 1A A .
− − 0 ∗ ∧ 0|L0 − ∧ ∗
Note that A is the adjoint of A with respect to q . Therefore, the curvature form of
∗ 0
(L ,h ) is a semipositive smooth (1,1)-form on X .
0 0 0
ON SEMIPOSITIVITY THEOREMS 5
Lemma 2.9. We use the same notation as in Definition 2.7. Let q be the Hodge metric
0
on the system of Hodge bundles (E ,θ ) induced by the original Hodge metric. Let Θ (E )
0 0 q0 0
be the curvature form of (E ,q ). Then we have
0 0
Θ (E )+θ θ +θ θ = 0
q0 0 0 ∧ 0∗ 0∗ ∧ 0
where θ is the adjoint of θ with respect to q (see, for example, [GT] and [S, (7.18)
0∗ 0 0
Lemma]). Therefore, we have
√ 1Θ (E ) = √ 1θ θ √ 1θ θ .
− q0 0 − − 0 ∧ 0∗ − − 0∗ ∧ 0
Let L be a line subbundle of E which is contained in the kernel of θ and let h be the
0 0 0 0
induced metric on L . Then
0
√ 1Θ (L ) = √ 1Θ (E ) +√ 1A A
− h0 0 − q0 0 |L0 − ∗ ∧
= √ 1θ θ √ 1θ θ +√ 1A A
− − 0 ∧ 0∗|L0 − − 0∗ ∧ 0|L0 − ∗ ∧
= √ 1θ θ +√ 1A A
− − 0 ∧ 0∗|L0 − ∗ ∧
where A is the second fundamental form of the line subbundle L E and A is the
0 0 ∗
⊂
adjoint of A with respect to q . Therefore, the curvature of (L ,h ) is a seminegative
0 0 0
smooth (1,1)-form on X .
0
3. Nefness
Let us start with the definition of nef line bundles on projective varieties.
Definition 3.1 (Nef line bundles). A line bundle L on a projective variety X is nef if
L C 0 for every curve C on X.
· ≥
In this paper, we need the notion of nef locally free sheaves (or vector bundles) on
projective varieties, which is a generalization of Definition 3.1.
Definition 3.2 (Nef locallyfree sheaves). A locallyfree sheaf (or vector bundle) E of finite
rank on a projective variety X is nef if the following equivalent conditions are satisfied:
(i) E = 0 or O (1) is nef on P (E).
PX(E) X
(ii) For every map from a smooth projective curve f : C X, every quotient line
→
bundle of f E has nonnegative degree.
∗
Anef locallyfreesheaf inDefinition3.2was originallycalleda(numerically)semipositive
sheaf in the literature.
Let us recall the definition of nef line bundles in the sense of Demailly (see [D2, (6.11)
Definition]).
Definition 3.3 (Nef line bundles in the sense of Demailly). A holomorphic line bundle L
on a compact complex manifold X is said to be nef if for every ε > 0 there is a smooth
hermitian metric h on L such that √ 1Θ (L) εω, where ω is a fixed hermitian
ε − hε ≥ −
metric on X.
We can easily check:
Lemma 3.4. If X is projective in Definition 3.3, then L is nef in the sense of Demailly
if and only if L is nef in the usual sense.
Proof. It is an easy exercise. For the details, see [D2, (6.10) Proposition]. (cid:3)
The following proposition is the main result of this section, which is more or less known
to the experts. We write the proof for the reader’s convenience.
6 OSAMU FUJINO ANDTAROFUJISAWA
Proposition 3.5. Let X be a compact complex manifold and let L be a holomorphic line
bundle equipped with a singular hermitian metric h. Assume that √ 1Θ (L) 0 and
h
− ≥
the Lelong number of h is zero everywhere. Then L is a nef line bundle in the sense of
Definition 3.3.
First, we give a quick proof of Proposition 3.5 when X is projective. It is an easy
application of the Nadel vanishing theorem and the Castelnuovo–Mumford regularity.
Proof of Proposition 3.5 when X is projective. Let A be an ample line bundle on X such
that A isbasepoint-free. BySkoda’stheorem(see[D2, (5.6)Lemma]), wehaveJ(hm) =
| |
O for every positive integer m, where J(hm) is the multiplier ideal sheaf of hm. Here,
X
we used the fact that the Lelong number of h is zero everywhere. By the Nadel vanishing
theorem,
Hi(X,ω L m A n+1 i) = 0
X ⊗ ⊗ −
⊗ ⊗
for every 0 < i n = dimX and every positive integer m. By the Castelnuovo–Mumford
≤
regularity, ω L m A n+1 is generated by global sections for every positive integer
X ⊗ ⊗
⊗ ⊗
m. We take a curve C on X. Then C (ω L m A n+1) 0 for every positive integer
X ⊗ ⊗
m. This means that C L 0. Ther·efore,⊗L is n⊗ef in the u≥sual sense. (cid:3)
· ≥
Next, we prove Proposition 3.5 when X is not necessarily projective. The proof depends
on Demailly’s approximation theorem for quasi-plurisubharmonic functions on complex
manifolds (see [D1]).
Proof of Proposition 3.5: general case. Let ω be a hermitian metric on X and let ε be any
positive real number. We fix a smooth hermitian metric g on L. Then we can write
h = ge 2ϕ, where ϕ is an integrable function on X. Since √ 1Θ (L) 0, we see that
− h
− ≥
1
√ 1∂∂ϕ √ 1Θ (L) =: γ.
g
− ≥ −2 −
By Lemma 2.5, we may assume that ϕ is quasi-plurisubharmonic. Note that γ is a smooth
(1,1)-form on X. By [D1, Proposition 3.7] (see also [D2, (13.12) Theorem]), we can
construct a quasi-plurisubharmonic function ψ on X such that
ε
1
√ 1∂∂ψ γ εω.
ε
− ≥ − 2
Since the Lelong number of h is zero everywhere on X, we can make ψ smooth. For the
ε
details, see [D1, Proposition 3.7] (see also [D2, (13.12) Theorem]). We put h = ge 2ψε.
ε −
Then h is a smooth hermitian metric on L such that √ 1Θ (L) εω. This means
ε − hε ≥ − (cid:3)
that L is a nef line bundle in the sense of Definition 3.3.
4. Proof of Theorem 1.1
In this section, we will prove Theorem 1.1 and Corollary 1.2. The arguments below
heavily depend on [Ko, Section 5]. Therefore, we strongly recommend the reader to see
[Ko, Section 5], especially [Ko, Definition 5.3], before reading this section.
4.1. We put ∆ = z C z < a and ∆ = ∆ 0 . On ∆n, we fix coordinates
a { ∈ || | } ∗a a \ { } a
z , ,z .
1 n
···
Let us quickly recall the definition of nearly boundedness and almost boundedness due
to Koll´ar for the reader’s convenience.
Definition 4.2 (see [Ko, Definition 5.3 (vi) and (vii)]). On (∆ )n with 0 < a < e 1, we
∗a −
define the Poincar´e metric by declaring the coframe
dz dz¯
i i
,
z log z z¯ log z
(cid:26) i | i| i | i|(cid:27)
ON SEMIPOSITIVITY THEOREMS 7
to be unitary. This defines a frame of every Ωk which we will refer to as the Poincar´e
frame.
A function f defined on a dense Zariski open set of ∆n is called nearly bounded on ∆n
a a
if f is smooth on (∆ )n and there are C > 0, k > 0 and ε > 0 such that for every ordering
∗a
of the coordinate functions z , ,z at least one of the following conditions is satisfied
1 n
···
for every z z (∆ )n z z .
∈ { ∈ ∗a || 1| ≤ ··· ≤ | n|}
(a): f C,
| | ≤
(b ): z exp( z ε) and f C( log z )k for some 2 m n.
m 1 m − m
| | ≤ −| | | | ≤ − | | ≤ ≤
A form η defined on a dense Zariski open set of ∆n is called nearly bounded on ∆n if the
a a
coefficient functions are nearly bounded on ∆n when we write η in terms of the Poincar´e
a
frame. If η and η are nearly bounded on the same ∆n, then η η is nearly bounded on
1 2 a 1∧ 2
∆n.
a
A form η defined on a dense Zariski open set of ∆n is called almost bounded on ∆n
a a
if there is a proper bimeromorphic map p : W ∆n such that W is smooth and every
→ a
w W has a neighborhood where p η is nearly bounded.
∗
∈
Remark 4.3. The definition of nearly boundedness and almost boundedness in Definition
4.2 is slightly different from Koll´ar’s original one (see [Ko, Definition 5.3 (vii)]).
4.4 (Proof of Theorem 1.1). We fix a smooth hermitian metric g on L. The Hodge metric
induces a smooth hermitian metric h on L . Then we can write
0 |X0
h = ge 2ϕ0
0 −
for some smooth function ϕ on X . We take an arbitrary point P of X. Then, by [CKS,
0 0
Theorem 5.21] and [Kas], we can take a local coordinate (z , ,z ) centered at P such
1 n
···
that
ϕ (z) log(C( log z )a1 ( log z )an)
0 1 n
− ≤ − | | ··· − | |
for z (∆ )n for some 0 < a < e 1, where a , ,a are some positive integers and C is a
∈ ∗a − 1 ··· n
large positive real number (cf. [VZ, Claim 7.8]). By considering the dual line bundle L ,
∨
we may further assume that
ϕ (z) log(C( log z )a1 ( log z )an)
0 1 n
≤ − | | ··· − | |
for z (∆ )n. Let ϕ be the smallest upper semicontinuous function that extends ϕ to X.
∈ ∗a 0
More explicitly,
ϕ(z) = lim sup ϕ (w),
0
ε→0w∈∆nε∩X0
where ∆n is a polydisk on X centered at z X. Then, by Lemma 4.6, we obtain:
ε ∈
Lemma 4.5. ϕ is locally integrable on X.
Proof of Lemma 4.5. Let P be an arbitrary point of X. In a small open neighborhood of
P, we have
0 ϕ (z) log(C( log z )a1 ( log z )an)
1 n
≤ ± ≤ − | | ··· − | |
where ϕ = max ϕ,0 and ϕ = ϕ ϕ. By Lemma 4.6 below, we obtain that ϕ is
+ +
locally integrable {on X}. − − (cid:3)
We have already used:
Lemma 4.6. We have
a
rlog( logr)dr <
− ∞
Z0
for 0 < a < e 1.
−
8 OSAMU FUJINO ANDTAROFUJISAWA
Proof of Lemma 4.6. We put t = logr. Then we can easily check
−
a
rlog( logr)dr = ∞ e 2t(logt)dt ∞ te 2tdt ∞ e tdt = a <
− − −
− ≤ ≤ ∞
Z0 Z loga Z loga Z loga
− − −
(cid:3)
by direct calculations.
We put
h = ge 2ϕ.
−
Then h is a singular hermitian metric on L in the sense of Definition 2.2. The following
lemma is essentially contained in [Ko, Proposition 5.7 and Proposition 5.15].
Lemma 4.7. Let P be an arbitrary point of X. Then ∂ϕ and ∂¯∂ϕ are almost bounded
0 0
in a neighborhood of P X. More precisely, there exists ∆n on X centered at P for some
0 < a < e 1 such that ϕ∈, ∂ϕ , and ∂¯∂ϕ are smooth on (∆a)n and that ∂ϕ and ∂¯∂ϕ are
− 0 0 0 ∗a 0 0
almost bounded on ∆n.
a
Proof of Lemma 4.7. We consider the following short exact sequence:
0 S Fb L 0.
→ → → →
We fix smooth hermitian metrics g ,g and g on S,Fb, and L, respectively. We assume
1 2
that g = g and that g is the orthogonal complement of g in g . Let h and h
1 2 S 1 2 1 2
|
be the induced Hodge metrics on S = S and Fb, respectively. By applying the
0 |X0 0
calculations in [Ko, Section 5] to detS and detFb, we obtain deth = detg e ϕ1 and
1 1 −
deth = detg e ϕ2 on X such that ∂ϕ ,∂¯∂ϕ ,∂ϕ , and ∂¯∂ϕ are almost b·ounded in
2 2 − 0 1 1 2 2
·
a neighborhood of P. For the details, see [Ko, Proposition 5.7 and Proposition 5.15].
¯
By construction, ϕ = ϕ + ϕ . Therefore, we can see that ∂ϕ and ∂∂ϕ are almost
0 1 2 0 0
− (cid:3)
bounded.
We prepare an easy lemma.
Lemma 4.8. We assume 0 < a < e 1. We have
−
a log( logr)
− dr < .
logr ∞
Z0 −
Proof of Lemma 4.8. We put t = logr. Then r = e t. We have
−
−
a log( logr) loga logt
− dr = − ( e t)dt
−
logr t −
Z0 − Z∞
logt
= ∞ e tdt
−
t
Z loga
−
∞ e tdt = a < .
−
≤ ∞
Z loga
−
(cid:3)
This is what we wanted.
The following lemma is missing in [Ko, Section 5]. This is because it is sufficient to
¯
consider the asymptotic behaviours of ∂ϕ and ∂∂ϕ for the purpose of [Ko, Section 5].
0 0
Lemma 4.9. Let η be a smooth (2n 1)-form on ∆n with compact support. We put
− a
S = z ∆n z εi for every i and z = εi0 for some i
~ε { ∈ a || i| ≥ | i0| 0}
where ~ε = (ε1, ,εn) with εi > 0 for every i. Then there is a sequence ~ε with ~ε 0
k k
··· { } ց
such that
lim ϕη = 0.
k
→∞ZS~εk
ON SEMIPOSITIVITY THEOREMS 9
Proof of Lemma 4.9. We put
S = z ∆n z = ε .
ε,1 { ∈ a || 1| }
Then it is sufficient to prove that
lim ϕη = 0
k
→∞ZSεk,1
for some sequence ε with ε 0. Without loss of generality, we may assume that η is
k k
{ } ց
a real (2n 1)-form by considering η+η¯ and η η¯ . Let us consider the real 1-form
−
− 2 2√ 1
−
1 dz dz¯
1 1
ω = + .
(2( log z1 )2)1/2 (cid:18) z1 z¯1 (cid:19)
− | |
This form is orthogonal to the foliation S and has length one everywhere by the Poincar´e
ε,1
metric. We consider the vector field
1 ∂ ∂
v = z (log z )2 +z¯ (log z )2 ,
(2( log z1 )2)1/2 (cid:18) 1 | 1| ∂z1 1 | 1| ∂z¯1(cid:19)
− | |
which is dual to ω. We fix ε with 0 < ε < a < e 1. We consider the flow f on ∆ ∆n 1
− t ∗a × a−
corresponding to v. We can explicitly write
−
f : [0, ) S ∆ ∆n 1
t ∞ × ε,1 → ∗a × a−
by
w 1
(t,(w,z , ,z )) exp exp t+log( logε) ,z , ,z .
2 n 2 n
··· 7→ ε − √2 − ···
(cid:18) (cid:18) (cid:18) (cid:19)(cid:19) (cid:19)
Therefore, by using the flow f , we can parametrize z C 0 < z ε ∆n 1 by
t { ∈ | | | ≤ } × a−
[0, ) S . If we write
ε,1
∞ ×
ω ϕη = f(z)dV,
∧
where dV is the standard volume form of Cn, then we put
(ω ϕη)+ = max f(z),0 dV
∧ { }
and
(ω ϕη) = (ω ϕη)+ ω ϕη.
−
∧ ∧ − ∧
We can easily see that
(ω ϕη) <
±
∧ ∞
Z∆na
by Lemma 4.6 and Lemma 4.8. Therefore, we obtain
(4.1) (ω ϕη) < .
±
∧ ∞
Z[0, ) Sε,1
∞ ×
The image of {t}×Sε,1 in ∆na is Sε′,1 for some 0 < ε′ ≤ ε. We note that ω is orthogonal
to Sε′,1 and unitary. More explicitly, we can directly check
f ω = dt.
t∗ −
Therefore, the above integral (4.1) transforms to
(ϕη) dt < .
±
∞
Z[0,∞) ZSε(t),1 !
Note that (ϕη) is defined by
±
f (ω ϕη) = dt (ϕη) .
t∗ ∧ ± − ∧ ±
10 OSAMU FUJINO ANDTAROFUJISAWA
This can happen only if
(ϕη) 0
±
→
ZSε(tk),1
for some sequence t with t . This implies that we can take a sequence ε with
k k k
{ } ր ∞ { }
ε 0 such that
k
ց
lim ϕη = 0.
k
→∞ZSεk,1
Therefore, we have a desired sequence ~ε . (cid:3)
k
{ }
Remark 4.10. The real 1-form ω and the corresponding flow f in the proof of Lemma
t
4.9 are different from the 1-form ω and the flow v in the proof of [Ko, Proposition 5.16],
t
respectively.
By combining the proof of [Ko, Proposition 5.16] and the proof of Lemma 4.9, we have:
Lemma 4.11. Let η be a nearly bounded (2n 1)-form on ∆n with compact support. Then
− a
there exists a sequence ε~ with ε~ 0 such that
′k ′k
{ } ց
lim η = 0.
ε~′kց0ZSε~′k
We leave the details of Lemma 4.11 as an exercise for the reader (see the proof of [Ko,
Proposition 5.16] and the proof of Lemma 4.9).
By Lemma 4.7 and Lemma 4.9, we have the following lemma.
Lemma 4.12. Let η be a smooth (2n 2)-form on ∆n with compact support. We further
assume that ∂ϕ and ∂¯∂ϕ are nearly−bounded on ∆n.aThen
0 0 a
¯ ¯
ϕ∂∂η = ∂∂ϕ η.
0
∧
Z∆na Z∆na
Note that the right hand side is an improper integral. Therefore, we obtain
¯ ¯
∂∂ϕ η = ∂∂ϕ η,
0
∧ ∧
Z∆na Z∆na
where we take ∂∂¯ of ϕ as a current.
Proof of Lemma 4.12. We put
V = z ∆n z εi for every i
~εk { ∈ a || i| ≥ k }
where ~ε = (ε1, ,εn) with εi > 0 for every i. Then
k k ··· k k
¯ ¯
ϕ∂∂η = lim ϕ ∂∂η
0
Z∆na ~εkց0ZV~εk
¯ ¯
= lim d(ϕ ∂η) lim ∂ϕ ∂η
0 0
~εkց0ZV~εk −ε~′kց0ZVε~′k ∧
¯ ¯
= lim ϕ ∂η + lim d(∂ϕ η) lim ∂∂ϕ η
0 0 0
~εkց0ZS~εk ε~′kց0ZVε~′k ∧ −ε~′kց0ZVε~′k ∧
¯
= lim ∂∂ϕ η
0
ε~′kց0ZVε~′k ∧
¯
= ∂∂ϕ η.
0
∧
Z∆na