An equivariant version of Grauert's Oka principle
type: Item
instance of: scholarly article
title: An equivariant version of Grauert's Oka principle
zbMATH Open document ID: 0837.32004
DOI: 10.1007/BF01245185
author: P. Heinzner
author: Frank Kutzschebauch
published in: Inventiones Mathematicae
publication date: 1996-05-13T00:00:00Z
full work available at URL: https://eudml.org/doc/144259
review text: An equivariant version of Grauert's Oka principle for a compact Lie group of holomorphic transformations of a Stein space is proved in this paper. Let \(K\) be a compact Lie group and \(X\) be a Stein \(K\)-space. Fix a Kempf- Ness subset \(R\) of \(X\) with respect to a \(K\)-invariant strictly plurisubharmonic exhaustion function. Let \(X//K\) be the categorial quotient. The major object in this version is \(Q(R)\) which is defined to be the sheaf of groups over \(X//K\). The main results are the following three theorems. Connectedness Theorem: The group \(Q(R) (X//K)\) is arcwise connected; Runge Theorem: If \(U\) is Runge in \(X//K\), then the image of \(Q(R) (X//K)\) in \(Q(R) (U)\) is dense; Vanishing Theorem: The first cohomology group \(H^1 (X//K, {\mathcal Q} (R)) = 0\). There are also corresponding theorems for analytic cubes of dimension \(d\). These theorems have many consequences. For example, it is proved here that if \(K\) is a compact Lie group of holomorphic transformations of a reduced Stein space \(X\), then every topological complex \(K\)-vector bundle over \(X\) is \(K\)-equivariantly isomorphic to a holomorphic one, and two holomorphic ones are \(K\)-equivariantly holomorphically isomorphic if and only if they are \(K\)-equivariantly topologically isomorphic.
reviewed by: Na Jisheng
Mathematics Subject Classification ID: 32C15
Mathematics Subject Classification ID: 32E10
zbMATH DE Number: 783991
zbMATH Keywords: \(K\)-equivariantly holomorphically isomorphic
zbMATH Keywords: Oka principle
zbMATH Keywords: Stein space
zbMATH Keywords: sheaf
zbMATH Keywords: topologically isomorphic
MaRDI profile type: Publication
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OpenAlex ID: W2071046881
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description: scientific article; zbMATH DE number 783991