A converse to the Oka principle over certain complex Banach manifolds
type: Item
instance of: scholarly article
title: A converse to the Oka principle over certain complex Banach manifolds
zbMATH Open document ID: 1190.32014
author: Imre Patyi
published in: Bulletin des Sciences Mathématiques
publication date: 2009-08-20T00:00:00Z
review text: Conditions that can be viewed as a converse to the Grauert-Oka principle and that ensure that a complex manifold be Stein have been given by \textit{J. Kajiwara} [Mem. Fac. Sci., Kyushu Univ., Ser. A. 33, 83--93 (1979; Zbl 0404.32007)]. A slightly time later \textit{J. Leiterer} [Math. Nachr. 89, 181--183 (1979; Zbl 0445.32012)] characterized Steinness, in a similar spirit, when holomorphic properties are determined by topological conditions. For example, he showed that an open subset \(D \subset M\) of a Stein manifold is itself Stein if and only if \(H^1(D,{\mathcal O})=0\) and every holomorphic vector bundle that is topologically trivial over \(D\) is holomorphically trivial over \(D\). The purpose of the present paper is to give analogues of Leiterer's conditions for certain complex Banach manifolds. An essential ingredient is the notion of plurisubharmonic domination due to \textit{L. Lempert} [J. Am. Math. Soc. 17, No. 2, 361--372 (2004; Zbl 1042.32013)].
Mathematics Subject Classification ID: 32L10
Mathematics Subject Classification ID: 32Q28
Mathematics Subject Classification ID: 46G20
zbMATH DE Number: 5597497
zbMATH Keywords: pseudoconvex domains
zbMATH Keywords: holomorphic Banach vector bundles
zbMATH Keywords: complex Banach manifolds
zbMATH Keywords: Oka principle
reviewed by: Bruce Gilligan
MaRDI profile type: Publication
full work available at URL: j.bulsci.2008.10.004
OpenAlex ID: W2016503673
cites work: EQUIVALENCE OF STEINNESS AND VALIDITY OF OKA'S PRINCIPLE FOR SUBDOMAINS WITH CONTINUOUS BOUNDARIES OF A STEIN MANIFOLD
cites work: CHARAKTERISIERUNG DER STEINSCHEN TEILGEBIETEN DURCH OKASCHES PRINZIP IN ZWEI-DIMENSIONALER STEINSCHER MANNIGFALTIGKEIT
cites work: Q5693078
cites work: Equivalence of Steinness and Validity of Oka's principle for subdomains fo Stein manifolds
cites work: The Dolbeault complex in infinite dimensions. III: Sheaf cohomology in Banach spaces
cites work: Plurisubharmonic domination
cites work: Vanishing cohomology for holomorphic vector bundles in a Banach setting
cites work: Analytic sheaves in Banach spaces☆
cites work: Q3711155
cites work: Q5664409
cites work: On holomorphic Banach vector bundles over Banach spaces
cites work: An embedding theorem for pseudoconvex domains in Banach spaces
DOI: 10.1016/J.BULSCI.2008.10.004
description: scientific article; zbMATH DE number 5597497