Equivariant complex extensions of homogeneous spaces
type: Item
instance of: scholarly article
title: Equivariant complex extensions of homogeneous spaces
zbMATH Open document ID: 0807.32023
DOI: 10.1007/BF01263293
published in: Mathematical Notes
publication date: 1994-04-12T00:00:00Z
review text: Let \(M\) be a real analytic manifold of dimension \(n\). By its complex extension we mean a pair \((\Omega, \varphi)\), where \(\Omega\) is a complex manifold of complex dimension \(n\) and \(\varphi\) is a real analytic isomorphism of the manifold \(M\) onto a closed totally real submanifold \(\varphi (M) \subset \Omega\). It is known that if \(M\) is paracompact, then a complex extension always exists. Moreover, as \textit{H. Grauert} has proved [Ann. Math., II. Ser. 68, No. 2, 460-472 (1958; Zbl 0108.078)], we can take a Stein manifold for \(\Omega\). Assume that the analytic action of a real Lie group \(G\) is given on \(M\). We say that a complex extension \((\Omega, \varphi)\) is \(G\)-equivariant if the analytic action of the group \(G\) on \(\Omega\) is defined, where each transformation \(g : \Omega \to \Omega\) \((g \in G)\) is holomorphic and the mapping \(M @>> \sim > \varphi (M) \subset \Omega\) commutes with the action of \(G\). The goal of the present article is twofold. First, we prove that equivariant extensions do not always exist, even if \(M\) is a homogeneous space of the group \(G\). Second, for homogeneous spaces of linear groups with a compact stationary subgroup that have obvious equivalent complex extensions, we construct equivariant Stein extensions with proper actions.
Mathematics Subject Classification ID: 32M10
Mathematics Subject Classification ID: 32C05
zbMATH DE Number: 525686
zbMATH Keywords: real analytic manifold
zbMATH Keywords: complex extension
zbMATH Keywords: equivariant extensions
zbMATH Keywords: Stein extensions
author: Dmitri Akhiezer
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 525686