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何卒、統合化に御協力お願いいたします Abstract. A theorem asserting the existence of proper holomorphic maps
with connected fibers to an open subset of C
N from a locally pseudoconvex
bounded domain in a complex manifold will be proved under the negativity
of the canonical bundle on the boundary. Related results of Takayama on
the holomorphic embeddability and holomorphic convexity of pseudoconvex
manifolds will be extended under similar curvature conditions. Abstract. A theorem asserting the existence of
proper holomorphic maps
with connected fibers to an open subset of C^N
from a locally pseudoconvex bounded domain
in a complex manifold will be proved under the
negativity of the canonical bundle on the
boundary. Related results of Takayama on
the holomorphic embeddability and holomorphic
convexity of pseudoconvex manifolds will be
extended under similar curvature conditions. This is a continuation of [Oh-5] where the following
was proved among other things.
Theorem 1.1. Let M be a complex manifold and let Ω be a proper
bounded domain in M with C^2-smooth pseudoconvex boundary
∂Ω. Assume that M admits a K¨ahler metric and the
canonical bundle K_M of M admits a fiber metric
whose curvature form is negative on a
neighborhood of ∂Ω. Then there exists a holomorphic
map with connected fibers from Ω to C^N for some
N ∈ ℕ which is proper onto the image.
The main purpose of the present article is to strengthen it
by removing the K¨ahlerness assumption (see §2).
For that, the proof of Theorem 0.1 given in [Oh-5]
by an application of the L^2 vanishing theorem on
complete K¨ahler manifolds will be replaced by an
argument which is more involved but also seems to be basic (see §1). More precisely, the proof is an application of the finite-dimensionality
of L^2 ¯∂-cohomology groups on M with coefficients in line bundles whose
curvature form is positive at infinity. Recall that the idea of exploiting
the finite-dimensionality for producing holomorphic sections originates
in a celebrated paper [G] of Grauert. Shortly speaking, it amounts to
finding infinitely many linearly independent C^∞ sections s1, s2, . . . of
the bundle in such a way that some nontrivial linear combination of
¯∂s1,
¯∂s2, . . . , say 膿N_{k=1} c_k¯∂sk(ck ∈ C), is equal to ¯∂u for some u which
is more regular than 膿N_{k=1} cksk. 訂正
¯∂s1,¯∂s2, . . . , say ΣN_{k=1} c_k¯∂sk(ck ∈ C), is equal to ¯∂u for some u which
is more regular than ΣN_{k=1} cksk. This works if one can attach mutually
different orders of singularities to sk for instance as in [G] where the
holomorphic convexity of strongly pseudoconvex domains was proved. Although such a method does not directly work for the weakly pseudoconvex
cases, the method of solving the ¯∂-equation with L^2
estimates is available to produce a nontrivial holomorphic section of the form
Σ^N_{k=1} cksk −u by appropriately estimating u. More precisely speaking,
instead of specifying singularities of sk, one finds a solution u which
has more zeros than Σ^N_{k=1} ck¯∂sk. For that, finite-dimensionality of the
L^2 cohomology with respect to singular fiber metrics would be useful. However, this part of analysis does not seem to be explored a lot. For
instance, the author does not know whether or not Nadel’s vanishing theorem
as in [Na] can be extended as a finiteness theorem with
coefficients in the multiplier ideal sheaves of singular fiber metrics under
an appropriate positivity assumption of the curvature current near infinity. So, instead of analyzing the L^2
cohomology with respect to singular
fiber metrics, we shall avoid the singularities by simply removing them
from the manifold and consider the L^2
cohomology of the complement, which turns out to have similar
finite-dimensionality property because
of the L^2 estimate on complete Hermitian manifolds. Such an argument
is restricted to the cases where the singularities of the fiber metic are
isolated. As a technique, it was first introduced in [D-Oh-3] to estimate
the Bergman distances. It is useful for other purposes and applied also
in [Oh-3,4,5,6], but will be repeated here for the sake of the reader’s convenience. Once one has infinitely many linearly independent holomorphic sections
of a line bundle L → M, one can find singular fiber metrics of L
by taking the reciprocal of the sum of squares of the moduli of local
trivializations of the sections. Very roughly speaking, this is the main
trick to derive the conclusion of Theorem 0.1 from K_M|∂Ω < 0. In fact,
for the bundles L with L|∂Ω > 0, the proof of
dim H^{n,0}(Ω, L^m) = ∞ for
m >> 1 will be given in detail here (see Theorem 1.4, Theorem 1.5 and
Theorem 1.6). The rest is acturally similar as in the case K_M < 0.
We shall also generalize the following theorems of Takayama. Theorem 1.2. (cf. [T-1]) Weakly 1-complete manifolds with positive
line bundles are embeddable into CP^N
(N >> 1).
Theorem 1.3. (cf. [T-2]) Pseudoconvex manifolds with negative canonical bundles
are holomorphically convex. Let M be a complex manifold. We shall say that M is a C^k
pseudoconvex manifold if M is equipped with a C^k plurisubharmonic
exhaustion function, say φ. C^∞ (resp. C^0) pseudoconvex manifolds are
also called weakly 1-complete (resp. pseudoconvex) manifolds. The
sublevel sets {x; φ(x) < c} will be denoted by Mc.
Theorem 0.2 and Theorem 0.3 are respectively a generalization of
Kodaira’s embedding theorem and that of Grauert’s characterization
of Stein manifolds. Our intension here is to draw similar conclusions by assuming the
curvature conditions only on the complement of a compact subset of
the manifold in quetion Theorem 0.2 will be generalized as follows.
Theorem 1.4. Let (M, φ) be a connected and noncompact C^2
pseudoconvex manifold which admits a holomorphic Hermitian line bundle
whose curvature form is positive on M - Mc.
Then there exists a holomorphic embedding of M - Mc into CP^N which
extends to M meromorphically. Theorem 0.3 will be extended to
Theorem 1.5. A C^2 pseudoconvex manifold (M, φ) is holomorphically
convex if the canonical bundle is negative outside a compact set.
This extends Grauert’s theorem asserting that strongly 1-convex
manifold are holomorphically convex. The proofs will be done by combining the method of Takayama with
an L^2 variant of the Andreotti-Grauert theory [A-G] on complete Hermitian manifolds whose special form needed here will be recalled in§3.
In §4 we shall extend Theorem 0.4 for the domains Ω as in Theorem
0.1. Whether or not Ω in Theorem 0.1 is holomorphically convex is
still open. The proof of the desired improvement of Theorem 0.1 will rely on
the following. Theorem 2.1. (cf. [Oh-4, Theorem 0.3 and Theorem 4.1]) Let M be
a complex manifold, let Ω ⊊ M be a relatively compact pseudoconvex
domain with a C^2-smooth boundary and let B be a holomorphic line
bundle over M with a fiber metric h whose curvature form is positive
on a neighborhood of ∂Ω. Then there exists a positive integer m0 such
that for all m ≥ m0
dimH^{0,0}(Ω, B^m) = ∞ and that, for any compact
set K ⊂ Ω and for any positive number R, one can find a compact set
K˜ ⊂ Ω such that for any point x ∈ Ω -K˜ there exists an element s of
H^{0,0}(Ω, B^m) satisfying
sup_{K} |s|_h^m < 1 and |s(x)|_h^m > R. We shall give the proof of Theorem 1.1 in this section for the convenience of the reader, after recalling the basic L^2
estimates in a general setting. Let (M, g) be a complete Hermitian manifold of dimension n and let
(E, h) be a holomorphic Hermitian vector bundle over M.
Let C^{p,q}(M, E) denote the space of E-valued C^∞ (p, q)-forms on M
and letC^{p,q}_0(M, E) = {u ∈ C^{p,q}(M, E); suppu is compact}. Given a C^2
function φ : M → R, let L^{p,q}_{(2),φ}(M, E) (= L^{p,q}_{(2),g,φ}(M, E))
be the space of E-valued square integrable measurable (p, q)-forms on
M with respect to g and he^{−φ}
. The definition of L^{p,q}_{(2),φ}(M, E) will be
naturally extended for continuous metrics and continuous weights. Recall that L^{p,q}_{(2),φ}(M, E) is identified with the completion of
C^{p,q}_0(M, E)
with respect to the L^2 norm
||u||φ := (∫_Me^{−φ}|u|^2_{g,h}dVg)1/2.
Here dVg := 1/n!ω^n
for the fundamental form ω = ω_g of g. More explicitly, when E is given by a system of transition functions eαβ with
respect to a trivializing covering {Uα} of M and h is given as a system
of C∞ positive definite Hermitian matrix valued functions hα on Uα satisfying hα =t
eβαhβeβα on Uα ∩ Uβ, |u|2
g,hdVg is defined by tuαhα ∧ ∗uα,
where u = {uα} with uα = eαβuβ on Uα ∩ Uβ and ∗ stands for the
Hodge’s star operator with respect to g. We put ∗¯u = ∗u so that
tuαhα ∧ ∗uα =tuαhα ∧ ∗¯uα Let us denote by ¯∂ (resp. ∂) the complex exterior derivative of
type (0, 1) (resp. (1, 0)). Then the correspondence uα 7→ ¯∂uα defines
a linear differential operator ¯∂ : C
p,q(M, E) → C
p,q+1(M, E). The
Chern connection Dh is defined to be ¯∂ + ∂h, where ∂h is defined by
uα 7→ h
−1
α ∂(hαuα). Since ¯∂
2 = ∂
2
h = ∂
¯∂ + ¯∂∂ = 0, there exists a
E
∗ ⊗ E-valued (1, 1)-form Θh such that D2
hu = Θh ∧u holds for all u ∈
C
p,q(M, E). Θh is called the curvature form of h. Note that Θhe−φ =
Θh+IdE ⊗∂
¯∂φ. Θh is said to be positive (resp. semipositive) at x ∈ M
if Θh =
馬
j,k=1 Θjk¯dzj ∧ dzk in terms of a local coordinate (z1, . . . , zn)
LEVI PROBLEM UNDER THE NEGATIVITY 5
around x and (Θjk¯(x))j,k = (Θµ
νjk¯
(x))j,k,µ,ν is positive (semipositive) in
the sense (of Nakano) that the quadratic form
(
µ
hµκ¯Θ
µ
νjk¯
)(x)ξ
νj ξ
κk
is positive definite (resp. positive semidefinite). Θ > 0 (resp. ≥0) for an E^∗ ⊗ E-valued (1,1)-form Θ will mean the positivity (resp.
semipositivity) in this sense. Whenever there is no fear of confusion, as well as the Levi form ∂¯∂φ
of φ, Θ_h will be identified with a Hermitian form along the fibers of
E ⊗ TM, where TM stands for the holomorphic tangent bundle of M. By an abuse of notation, ¯∂ (resp. ∂he−φ ) will also stand for the maximal closed extension of ¯∂|C
p,q
0
(M,E)
(resp. ∂he−φ |C
p,q
0
(M,E)
) as a closed
operator from L
p,q
(2),φ
(M, E) to L
p,q+1
(2),φ
(M, E) (resp. L
p+1,q
(2),φ
(M, E)). The
adjoint of ¯∂ (resp. ∂he−φ ) will be denoted by ¯∂
∗ = ¯∂
∗
g,he−φ (resp. ∂
∗
he−φ ).
We recall that ∂
∗
he−φ = −∗¯∂∗¯ holds as a differential operator acting on
C
p,q(M, E), so that ∂
∗
he−φ will be also denoted by ∂
∗
. By Dom¯∂ (resp.
Dom¯∂
∗
) we shall denote the domain of ¯∂ (resp. ¯∂
∗
). We put
H
p,q
(2),φ
(M, E)(= H
p,q
(2),g,φ
(M, E)) =
Ker (
¯∂ : L
p,q
(2),φ
(M, E) → L
p,q+1
(2),φ
(M, E)
)
Im (
¯∂ : L
p,q−1
(2),φ
(M, E) → L
p,q
(2),φ
(M, E)
)
and
H p,q
φ
(M, E) = Ker ¯∂ ∩ Ker ¯∂
∗ ∩ L
p,q
(2),φ
(M, E). Let Λ = Λg denote the adjoint of the exterior multiplication by ω.
Then Nakano’s formula
(2.2) ¯∂
¯∂
∗ + ¯∂
∗ ¯∂ − ∂h∂
∗ − ∂
∗
∂h =
√
−1(ΘhΛ − ΛΘh)
holds if dω = 0. Here Θh also stands for the exterior multiplication by
Θh from the left hand side. Hence, for any open set Ω ⊂ M such that
dω|Ω = 0 and for any u ∈ C
n,q
0
(Ω, E), one has
(2.3) k
¯∂uk
2
φ + k
¯∂
∗uk
2
φ ≥ (
√
−1(Θh + IdE ⊗ ∂
¯∂φ)Λu, u)φ.
Here (u, w)φ stands for the inner product of u and v with respect to
(g, he−φ
). Here (u, w)φ stands for the inner product of u and v with respect to
(g, he−φ
). The following direct consequence of (1.3) is important for
our purpose. Proposition 2.1. Let M, E, g, h and φ be as above. Assume that there
exists a compact set K ⊂ M such that dωg = 0 holds on M \ K. Then
there exist a compact set K′
containing K and a constant C such that
K′ and C do not depend on the choice of φ and
(
√
−1(Θh+IdE⊗∂
¯∂φ)Λu, u)φ ≤ C
(
k
¯∂uk
2
φ + k
¯∂
∗uk
2
φ +
∫
K′
e
−φ
|u|
2
g,hdVg
)
holds for any u ∈ C
n,q
0
(M, E) (q ≥ 0). From Proposition 1.1 one infers Proposition 2.2. Let (M, E, g, h, φ, K) and (K′
, C) be as above. Assume moreover that one can find a constant C0 > 0 such that C0(Θh +
IdE ⊗∂
¯∂φ)−IdE ⊗g ≥ 0 holds on M \K. Then there exists a constant
C
′ depending only on C, K′ and C0 such that
kuk
2
φ ≤ C
′
(
k
¯∂uk
2
φ + k
¯∂
∗uk
2
φ +
∫
K′
e
−φ
|u|
2
g,hdVg
)
holds for any u ∈ C
n,q
0
(M, E) (q ≥ 1). By a theorem of Gaffney, the estimate in Proposition 1.2 implies the
following.
Proposition 2.3. In the situation of Proposition 1.2,
kuk
2
φ ≤ C
′
(
k
¯∂uk
2
φ + k
¯∂
∗uk
2
φ +
∫
K′
e
−φ
|u|
2
g,hdVg
)
holds for all u ∈ L
n,q
(2),φ
(M, E) ∩ Dom¯∂ ∩ Dom¯∂
∗
(q ≥ 1). Recall that the following was proved in [H] by a basic argument of
functional analysis. Theorem 2.2. (Theorem 1.1.2 and Theorem 1.1.3 in [H]) Let H1 and
H2 be Hilbert spaces and let T : H1 → H2 be a densely defined closed
operator. Let H3 be another Hilbert space and let S : H2 → H3 be a
densely defined closed operator such that ST = 0. Then a necessary
and sufficient condition for the ranges RT , RS of T, S both to be closed
is that there exists a constant C such that
(2.4) kgkH2 ≤ C(kT
∗
gkH1 +kSgkH3
); g ∈ DT ∗ ∩DS, g⊥(NT ∗ ∩NS),
where DT ∗ and DS denote the domains of T
∗ and S, respectively, and
NT ∗ = KerT
∗ and NS = KerS. Moreover, if one can select a strongly
convergent subsequence from every sequence gk ∈ DT ∗ ∩DS with kgkkH2
bounded and T
∗
gk → 0 in H1, Sgk → 0 in H3, then NS/RT
∼= NT ∗ ∩NS
holds and NT ∗ ∩ NS is finite dimensional. Theorem 2.3. In the situation of Proposition 1.2, dimH
n,q
(2),φ
(M, E) <
∞ and H n,q
φ
(M, E) ∼= H
n,q
(2),φ
(M, E) hold for all q ≥ 1. It is an easy exercise to deduce from Theorem 1.3 that every strongly
pseudoconvex manifold is holomorphically convex (cf. [G] or [H]). We
are going to extend this application to the domains with weaker pseudoconvexity. For any Hermitian metric g on M, a C
2
function ψ : M → R is called
g-psh (g-plurisubharmonic) if g + ∂
¯∂ψ ≥ 0 holds everywhere.
Then Theorem 1.3 can be restated as follows. Theorem 2.4. Let (M, g) be an n-dimensional complete Hermitian
manifold and let (E, h) be a Hermitian holomorphic vector bundle over
M. Assume that there exists a compact set K ⊂ M such that
Θh − IdE ⊗ g ≥ 0 and dωg = 0 hold on M \ K. Then, for any g-psh
function φ on M and for any ε ∈ (0, 1),
dim H
n,q
(2),εφ
(M, E) < ∞ and H n,q
εφ (M, E) ∼= H
n,q
(2),εφ
(M, E)
for q ≥ 1 §2 Infinite dimensionality and bundle convexity theorems
By applying Theorem 1.4, we shall show at first the following.
Theorem 2.5. Let (M, E, g, h) be as in Theorem 1.4 and let xµ (µ =
1, 2, . . .) be a sequence of points in M without accumulation points.
Assume that there exists a (1 − ε)g-psh function φ on M \ {xµ}
∞
µ=1 for
some ε ∈ (0, 1) such that e
−φ
is not integrable on any neighborhood of
xµ for any µ. Then
dim H
n,0
(M, E) = ∞. Proof. We put M′ = M \{xµ}
∞
µ=1 and let ψ be a bounded C
∞ ε
2
g-psh
function on M′
such that g
′
:= g + ∂
¯∂ψ is a complete metric on M′
.
Take sµ ∈ C
n,0
(M, E) (µ ∈ N) in such a way that |sµ(xν)|g,h = δµν and
∫
M′ e
−φ
|
¯∂sµ|
2
g,hdVg < ∞. Since ∫
M′ e
−φ−ψ
|
¯∂sµ|
2
g
′
,hdVg
′ ≤
∫
M′ e
−φ−ψ
|
¯∂sµ|
2
g,hdVg
and dim H
n,1
(2),g′
,φ
(M′
, E) < ∞ by Theorem 1.4, one can find a nontrivial finite linear combination of ¯∂sµ, say v =
把µ
¯∂sµ, which is in the
range of L
n,0
(2),φ
(M′
, E)
∂¯
−→ L
n,1
(2),g′
,φ
(M′
, E). Then take u ∈ L
n,0
(2),φ
(M′
, E)
satisfying ¯∂u = v and put s =
把µsµ − u. Clearly s extends to a
nonzero element of Hn,0
(M, E) which is zero at xµ except for finitely
many µ. Hence, one can find mutually disjoint finite subsets Σν 6=
ϕ (ν = 1, 2, . . .) of N and nonzero holomorphic sections sν of E such
that sν(xµ) = 0 if µ /∈ Σν, so that dim Hn,0
(M, E) = ∞ This observation will be basic for the proofs of Theorems 0.4 and
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