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681: １３２人目の素数さん [] 2023/03/23(木) 13:40:44.38 ID:gtBUMZjM

>>673
>Levi problemも有名

まるほど
下記か、”The Levi problem for Cn  was affirmatively solved in 1953?1954 independently by K. Oka, H. Bremermann and F. Norguet, and Oka solved the problem in a more general formulation, concerned with domains spread over Cn ( cf. Covering domain) (see ?[6]).”
岡先生ね
とすると、>>675 シュタイン多様体 "X 内でコンパクトとなるようなものである。これはいわゆる、エフジェニオ・エリア・レヴィ（英語版）(Eugenio Elia Levi) (1911) にちなんで名付けられたレヴィ問題の解でもある[1]"
とも、対応しているね

不勉強で、初めて知りました

https://encyclopediaofmath.org/wiki/Levi_problem
Levi problem Encyclopedia of Mathematics

The problem of the geometric characterization of domains in a given analytic space that are Stein spaces (cf. Stein space); it was posed by E.E. Levi [1] for domains in the affine space Cn
in the following form. Let D
略
The Levi problem for Cn
was affirmatively solved in 1953?1954 independently by K. Oka, H. Bremermann and F. Norguet, and Oka solved the problem in a more general formulation, concerned with domains spread over Cn
( cf. Covering domain) (see ?[6]).

つづく

http://rio2016.5ch.net/test/read.cgi/math/1677671318/681

682: １３２人目の素数さん [] 2023/03/23(木) 13:41:19.56 ID:gtBUMZjM

>>681
つづき

Oka's result has been generalized to domains spread over any Stein manifold: If such a domain D
is a pseudo-convex manifold, then D
is a Stein manifold. The Levi problem has also been affirmatively solved in a number of other cases, for example, for non-compact domains spread over the projective space CPn
or over a Kahler manifold on which there exists a strictly plurisubharmonic function (see ), and for domains in a Kahler manifold with positive holomorphic bisectional curvature [7]. At the same time, examples of pseudo-convex manifolds and domains are known that are not Stein manifolds and not even holomorphically convex. A necessary and sufficient condition for a complex space to be a Stein space is that it is strongly pseudo-convex (see Pseudo-convex and pseudo-concave). Also, a strongly pseudo-convex domain in any complex space is holomorphically convex and is a proper modification of a Stein space (see , [4] and also Modification; Proper morphism).
(引用終り)
以上

http://rio2016.5ch.net/test/read.cgi/math/1677671318/682

683: １３２人目の素数さん [] 2023/03/23(木) 14:03:33.27 ID:gtBUMZjM

>>681 追加
検索ヒットしたので貼る
”The Levi problem was first solved by Oka”ね
YUM-TONG SIUは、例のSIUさんか
https://projecteuclid.org/journals/bulletin-of-the-american-mathematical-society/volume-84/issue-4/Pseudoconvexity-and-the-problem-of-Levi/bams/1183540919.pdf
BULLETIN OF THE
AMERICAN MATHEMATICAL SOCIETY
Volume 84, Number 4, July 1978
PSEUDOCONVEXTTY AND THE PROBLEM OF LEVI
BY YUM-TONG SIU

The Levi problem is a very old problem in the theory of several complex
variables and in its original form was solved long ago. However, over the
years various extensions and generalizations of the Levi problem were proposed
and investigated. Some of the more general forms of the Levi problem
still remain unsolved. In the past few years there has been a lot of activity in
this area. The purpose of this lecture is to give a survey of the developments
in the theory of several complex variables which arise from the Levi problem.
We will trace the developments from their historical roots and indicate the
key ideas used in the proofs of these results wherever this can be done
intelligibly without involving a lot of technical details. For the first couple of
sections of this survey practically no knowledge of the theory of several
complex variables is assumed on the part of the reader. However, as the
survey progresses, an increasing amount of knowledge of the theory of several
complex variables is assumed.

Table of Contents
1. Domains of holomorphy
2. The original Levi problem
3. Stein manifolds
4. Locally Stein open subsets
5. Increasing sequence of Stein open subsets
6. The Serre problem
7. Weakly pseudoconvex

P484
The Levi problem was first solved by Oka. He did the case n = 2 in [67]
and the general case in [68]. The case of a general n was also solved at the
same time independently by Bremermann [8] and Norguet [66].

http://rio2016.5ch.net/test/read.cgi/math/1677671318/683

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