ガロア第一論文と乗数イデアル他関連資料スレ5

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710: 2023/07/16(日)06:08 ID:Gig56QD8(1/17) AAS
References
[A] Asserda, S., The Levi problem on projective manifolds, Math. Z. 219 (1995),
no. 4, 631-636.
[B-P] B locki, Z. and Pflug, P., Hyperconvexity and Bergman completeness, Nagoya
Math. J., 151 (1998), 221-225.
[C] Chen, B.-Y., Bergman completeness of hyperconvex manifolds, Nagoya Math.
J. 175 (2004), 165-170.
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711: 2023/07/16(日)06:11 ID:Gig56QD8(2/17) AAS
[M-2] ——, On quotients of bounded homogeneous domains by unipotent discrete
groups, arXiv:2202.10283v2 [math.CV]
[Mi] Miyazawa, K., A remark on the paper “On the infinite dimensionality of the
middle L
2
cohomology of complex domains”, preprint (in Japanese)
[Oh-1] Ohsawa, T., Vanishing theorems on complete K¨ahler manifolds, Publ. Res.
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712: 2023/07/16(日)06:13 ID:Gig56QD8(3/17) AAS
[R] Rosay, J.-P., Extension of holomorphic bundles to the disc (and Serre’s problem
on Stein bundles), Ann. Inst. Fourier, Grenoble 57 (2007), 517-523.
[S] Siu, Y.-T., Holomorphic fiber bundles whose fibers are bounded Stein domains
with zero first Betti number, Math. Ann. 219 (1976), 171-192.
[St] Stehl´e, J.-L., Fonctions plurisousharmoniques et convexit´e holomorphe de certains
fibr´es analytiques, Lecture Notes in Math., Vol. 474, Springer, Berlin,
1975, pp. 155-179.
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713: 2023/07/16(日)08:25 ID:Gig56QD8(4/17) AAS
NOTES ON THE BUNDLE OF COEURE AND LOEB ´

Abstract
A weak holomorphic-convexity property of a fiber bundle constructed
by Coeur´e and Loeb will be proved after an observation that it admits a
complete K¨ahler metric. Other remarks on their geometric and function
theoretic properties will be presented, too. In particular, it will be shown that there exists a
locally pseudoconvex branched Riemann domain
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714: 2023/07/16(日)08:28 ID:Gig56QD8(5/17) AAS
Introduction
Given a complex manifold M and a holomorphic vector bundle E →
M, the notion of E-convexity of M was introduced by Grauert [G-2]
by generalzing holomorphic convexity. Then, for locally pseudoconvex
smooth bounded domains Ω ⋐ M, Pinney [P] showed that Ω is Econvex in a suitably weakened
sense if E is a line bundle euqipped
with a fiber metric with positive curvature. Here the E-convexity of
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715: 2023/07/16(日)08:30 ID:Gig56QD8(6/17) AAS
The E-convexity in this sense was verified by Asserda [A] assuming
the compactness of M and quite recently by [Oh-3,5] in other situations
including the cases where ∂Ω is a proper analytic set of M, under the
assumtion of the curvature positivity of E|∂Ω
Compared to the classical holomorphically convex cases, not so much
more has been known on E-convex domains. For instance, the precise
boundary behavior of the bundle-valued Bergman kernels is not known
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716: 2023/07/16(日)08:33 ID:Gig56QD8(7/17) AAS
More recently, [T] was extended in [Oh-6] as the holomorphic convexity of weakly 1-complete manifolds
whose canonical bundle is negative at infinity (i.e. on the complement of a compact subset of
the manifold). Another extension of [T] was obtained in [Oh-4,7] asserting a similar conclusion
under certan regularity or curvature assumptions of ∂Ω.
In the latter extension, the conclusion is weaker than the genuine holomorhic convexity,
because it only says that the domain is properly mapped onto a locally closed analytic set
in some C^N.

717: 2023/07/16(日)08:35 ID:Gig56QD8(8/17) AAS
This seems to suggest that it may be worthwhile to study locally pseudoconvex domains in
complex manifolds by focusing on the finer structures of function spaces.
Since the well-known Coeur\'e-Loeb's counterexample to the Serre problem is
a locally pseudoconvex domain of similar type which actually arises in nature,
we would like to study here some of its function theoretic properties from this viewpoint.

718: 2023/07/16(日)08:38 ID:Gig56QD8(9/17) AAS
Let us recall that the Serre problem asks whether or not
analytic fiber bundles over Stein manifolds with Stein fibers are Stein.
Both positive and negative answers are known to contain significant contents.
In particular, counterexamples sometimes share interesting function-theoretic
properties with Stein manifolds, such as the Oka's principle (cf. [R]).

After recalling Coeur\'e-Loeb's example, we shall prove the following.

Theorem 1. There exists a logarithmically convex bounded Reinhardt domain F
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719: 2023/07/16(日)17:32 ID:Gig56QD8(10/17) AAS
In [C-D] it was remarked that the total space of the above bundle
can be realized as a branched Riemann domain over C^3
. In Theorem
1, F can be chosen in such a way that it admits a fixed point free automorphism generating an infinite cyclic subgroup Γ of AutF such thatthe bundle is C∗ ×ρ F for an isomorphism ρ between the fundamentalgroup of C∗ and Γ. As a by-product of Theorem 1 we shall show that F/Γ, which is a fortiori non-Stein, can be realized as a branched locally pseudoconvex Riemann domain over C^2.

720: 2023/07/16(日)17:33 ID:Gig56QD8(11/17) AAS
Coeure-Loeb’s bundle ´
Let Ω be a bounded Stein domain in C
n
, let AutΩ be the group of
biholomorphic self-maps of Ω and let Γ be an infinite cyclic properly
discontinuous subgroup of AutΩ generated (automatically) by a fixed
point free element. It is known that the quotient manifold Ω/Γ is not
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721: 2023/07/16(日)17:36 ID:Gig56QD8(12/17) AAS
Therefore, in particular, an analytic fiber bundle over the punctured disc
D∗:= {z ∈ C; 0 < |z| < 1} whose fiber is biholomorphic to a bounded
homogeneous domain Ω0 is Stein if it arises as the infinite cyclic quotient of
the product D × Ω0 associated to a nontrivial homomorphism
ρ : π_1(D∗) → AutΩ0. A theorem of Siu [S] says more generally that
analytic fiber bundles over Stein manifolds with fibers equivalent to a
bounded pseudoconvex domains in C^n with zero first Betti number is
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722: 2023/07/16(日)17:54 ID:Gig56QD8(13/17) AAS
If Ω is homogeneous, the Bergman metric on Ω defined as
∂¯∂ log K_Ω(z, z)　from the Bergman kernel KΩ(z, w) of Ω is invariant under the action of AutΩ,
so that it is a complete K¨ahler metric on Ω. Furthermore, the function log K_Ω(z, z) has bounded
gradient with respect to∂¯∂ log K_Ω(z, z) and limz→∂Ω log K_Ω(z, z) = ∞ (cf. [K-Oh]).
Therefore Ω is hyperconvex in the sense of Stehl´e [St], i.e. Ω admits a strictly
plurisubharmonic bounded exhaustion function. So the Steinness of
analytic Ω bundles over Stein manifolds follows also from Stehl´e’s theorem in [St].

723: 2023/07/16(日)19:08 ID:Gig56QD8(14/17) AAS
On the other hand, Coeur’e and Loeb [C-L] constructed a bounded
pseudoconvex Reinhardt domain F in C^2
satisfying the following property;
For the bounded domain Ω = {z ∈ C; |ζ| < 1}×F in C^3, there exists
an element σ ∈ AutF such that the element ˆσ ∈ AutΩ defined by
σˆ(ζ, z) := ((2i − 1)ζ + 1)/(−ζ + 1 + 2i), σ(z))
generates an infinite cyclic group Γ := {σˆµ; µ ∈ Z} by which Ω has a
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724: 2023/07/16(日)19:09 ID:Gig56QD8(15/17) AAS
Therefore, one will have a non-Stein fiber bundle over D
∗ with fiber F
in this way. By the construction, this fiber bundle is naturally extended
to a bundle over C
∗
. By Stehl´e’s theorem F is not hyperconvex. In
fact, the Bergman metric on F is not complete. ( It is due to [B-P], [H]
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725: 2023/07/16(日)19:11 ID:Gig56QD8(16/17) AAS
Construction of the domain F
We put H = {z ∈ C; Imz > 0}, T =
( 1+√
5
2
1−
√
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726: 2023/07/16(日)21:12 ID:Gig56QD8(17/17) AAS
It is easy to see by a direct computation that the set $\{\sigma^k_A(v_1,v_2); k\in \mathbb{Z}\}$ has no accumulation points in $F$, so that $\hat{F}$:=$F/\{\sigma^k_A; k\mathbb{Z}\}$ is a complex manifold. $\hat{F}$ is non-Stein since so is $\hat{\Omega}$. Since $F$ has a $\sigma_A$-invariant complete K\"ahler metric, $\hat{F}$ has also a complete K\"ahler metric. Combining this with the invariance of $du_1\wedge du_2$, the $\sigma_A$-invariance of the Bergman kernel function of $F$ follows. Thus $\hat{F}$ is a complete K\"ahler manifold with trivial canonical bundle which is positive. Therefore, by the $L^2$ method one can conclude that $\hat{F}$ is holomorphically separable.

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