Compact varieties of surjective holomorphic endomorphisms
type: Item
instance of: scholarly article
title: Compact varieties of surjective holomorphic endomorphisms
zbMATH Open document ID: 0581.32013
DOI: 10.1007/BF01214749
author: Camilla Horst
published in: Mathematische Zeitschrift
publication date: 1985-01-01T00:00:00Z
full work available at URL: https://eudml.org/doc/173644
review text: Let X denote a normal compact connected reduced complex space. By Douady, the set \({\mathcal H}(X,X)\) of all holomorphic mappings \(X\to X,\) with the compact-open topology, carries a complex structure. Denote by \(Sur X\) a subset of \({\mathcal H}(X,X)\), which consists of all holomorphic surjections \(X\to X.\) The author shows that \(Sur X\) is open and closed in \({\mathcal H}(X,X)\), in particular it is analytic. The main result of the paper says that if Z is any compact connected reduced subspace of \(Sur X,\) then for every two surjections \(\sigma\), \(\tau\in Z\), there exists a biholomorphic automorphism \(\alpha\) of X such that \(\sigma =\alpha \circ \tau\). Such an \(\alpha\) can be described more exactly. An example shows, that the normality of X is essential. The author uses standard methods of complex analytic geometry and complex Lie groups.
Mathematics Subject Classification ID: 32C20
Mathematics Subject Classification ID: 32C25
Mathematics Subject Classification ID: 32M05
Mathematics Subject Classification ID: 32H99
Mathematics Subject Classification ID: 32J99
zbMATH DE Number: 3929422
zbMATH Keywords: weakly normal complex space
zbMATH Keywords: Stein factorization
zbMATH Keywords: complex analytic geometry
zbMATH Keywords: complex Lie groups
reviewed by: Krzysztof Dabrowski
MaRDI profile type: Publication
cites work: Groups on analytic manifolds
cites work: Le problème des modules pour les sous-espaces analytiques compacts d'un espace analytique donné
cites work: Über die Automorphismengruppen kompakter komplexer Räume
OpenAlex ID: W2083416720
description: scientific article; zbMATH DE number 3929422