The Grauert-Oka principle in a Banach space
type: Item
instance of: scholarly article
title: The Grauert-Oka principle in a Banach space
zbMATH Open document ID: 1460.32028
author: Imre Patyi
published in: Journal of Mathematical Analysis and Applications
publication date: 2021-02-25T00:00:00Z
review text: The main result here is the following. Theorem: Let \(M\) be a Stein Banach manifold modeled on a separable Banach space \(X\) with the bounded approximation property, let \(\Gamma \to M\) be a holomorphic Banach Lie group bundle over \(M,\) and let \({\mathcal O}^\Gamma \to M\)\ (resp. \({\mathcal C}^\Gamma\)) \(\to M\) be the sheaf of germs of holomorphic (resp. continuous) sections of \(\Gamma.\) Then the natural map \(H^1(M, {\mathcal O}^\Gamma) \to H^1(M, {\mathcal C}^\Gamma)\) of first cohomology sets is bijective. The proof of this and related results are non-trivial, relying on a classical paper of \textit{H. Cartan} [Sympos. Int. Topologia Algebraica 97--121 (1958; Zbl 0121.30503)] and several papers of L. Lempert. It is worth noting that this paper begins with a short, but helpful, introduction as well as a somewhat apt quotation of Schiller.
Mathematics Subject Classification ID: 32K12
Mathematics Subject Classification ID: 46G20
zbMATH DE Number: 7315378
zbMATH Keywords: complex Banach manifold
zbMATH Keywords: holomorphic embedding
zbMATH Keywords: sheaf cohomology
zbMATH Keywords: vector bundle
zbMATH Keywords: principal bundle
reviewed by: Richard M. Aron
MaRDI profile type: Publication
full work available at URL: j.jmaa.2020.124711
OpenAlex ID: W3094036490
cites work: On analytic fiber bundles. I: Holomorphic fiber bundles with infinite dimensional fibers
cites work: Q5732807
cites work: Stein Manifolds and Holomorphic Mappings
cites work: Cohomologie non abélienne et espaces fibrés
cites work: Q3685657
cites work: Oka's principle in pseudoconvex domains
cites work: Approximationssaetze für holomorphe Funktionen mit Werten in komplexen Räumen
cites work: Holomorphe Funktionen mit Werten in komplexen Lieschen Gruppen
cites work: Analytische Faserungen über holomorph-vollständigen Räumen
cites work: The Oka-Grauert principle without induction over the base dimension
cites work: On bases, finite dimensional decompositions and weaker structures in Banach spaces
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: Q5419023
cites work: On the Oka principle in a Banach space. I
cites work: On the Oka principle in a Banach space. II
cites work: On holomorphic Banach vector bundles over Banach spaces
cites work: On holomorphic domination. I
cites work: Plurisubharmonic domination in Banach spaces
cites work: On complex Banach manifolds similar to Stein manifolds
cites work: On holomorphic domination. II.
cites work: Projections in certain Banach spaces
DOI: 10.1016/J.JMAA.2020.124711
description: scientific article; zbMATH DE number 7315378