純粋・応用数学・数学隣接分野（含むガロア理論）13

[過去ﾛｸﾞ] 純粋・応用数学・数学隣接分野（含むガロア理論）13 (1002ﾚｽ)
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963: 2023/07/31(月)09:41 ID:jznoxopE(13/50) AAS
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.

964: 2023/07/31(月)09:43 ID:jznoxopE(14/50) AAS
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.

965: 2023/07/31(月)09:45 ID:jznoxopE(15/50) AAS
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.

966: 2023/07/31(月)09:58 ID:jznoxopE(16/50) AAS
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.

967: 2023/07/31(月)09:59 ID:jznoxopE(17/50) AAS
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

968: 2023/07/31(月)10:01 ID:jznoxopE(18/50) AAS
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.

969: 2023/07/31(月)10:02 ID:jznoxopE(19/50) AAS
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.

970: 2023/07/31(月)10:05 ID:jznoxopE(20/50) AAS
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.

971: 2023/07/31(月)10:05 ID:jznoxopE(21/50) AAS
The proof of the desired improvement of Theorem 0.1 will rely on
the following.

972: 2023/07/31(月)10:10 ID:jznoxopE(22/50) AAS
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
省1

973: 2023/07/31(月)10:12 ID:jznoxopE(23/50) AAS
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.

974: 2023/07/31(月)10:13 ID:jznoxopE(24/50) AAS
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}.

975: 2023/07/31(月)10:15 ID:jznoxopE(25/50) AAS
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}
.

976: 2023/07/31(月)10:17 ID:jznoxopE(26/50) AAS
The definition of L^{p,q}_{(2),φ}(M, E) will be
naturally extended for continuous metrics and continuous weights.

977: 2023/07/31(月)10:27 ID:jznoxopE(27/50) AAS
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.

978: 2023/07/31(月)10:29 ID:jznoxopE(28/50) AAS
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

979: 2023/07/31(月)10:30 ID:jznoxopE(29/50) AAS
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 = ∂
省28

980: 2023/07/31(月)10:32 ID:jznoxopE(30/50) AAS
Θ > 0 (resp. ≥0) for an E^∗ ⊗ E-valued (1,1)-form Θ will mean the positivity (resp.
semipositivity) in this sense.

981: 2023/07/31(月)10:32 ID:jznoxopE(31/50) AAS
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.

982: 2023/07/31(月)10:33 ID:jznoxopE(32/50) AAS
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
省29

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