On the Lie group structure of automorphism groups
type: Item
instance of: scholarly article
title: On the Lie group structure of automorphism groups
zbMATH Open document ID: 1375.32039
author: Yoshikazu Nagata
published in: Comptes Rendus. Mathématique. Académie des Sciences, Paris
publication date: 2017-07-31T00:00:00Z
review text: The author gives a sufficient condition for complex manifolds such that the automorphism groups become Lie groups. A classical result of H. Cartan says that the automorphism groups of bounded domains in \(\mathbb C^n\) are Lie groups. A key fact for its proof is that the isometry group of any Hermitian metric is a Lie group with respect to the compact-open topology. Using a similar approach, the author shows that for a Stein manifold \(M\), if \(M\neq Z\cup D\), then \(\text{Aut}(M)\) has a Lie group structure. Here, \(Z\) denotes the zero locus of the Bergman kernel and \(D\) denotes the degenerating locus of the Bergman pseudometric \(ds^2_{M\backslash Z}\). The author also gives a sufficient condition for \(M\neq Z\cup D\). In particular, the author discusses its connection with the notion of the ``core'' of a Stein domain.
Mathematics Subject Classification ID: 32M05
Mathematics Subject Classification ID: 32Q28
zbMATH DE Number: 6754872
zbMATH Keywords: Lie groups
zbMATH Keywords: automorphism groups
zbMATH Keywords: Bergman kernel
MaRDI profile type: Publication
full work available at URL: j.crma.2017.06.007
OpenAlex ID: W2732045749
Wikidata QID: Q115358180
cites work: Mappings of Partially Analytic Spaces
cites work: The Bergman metric on a Stein manifold with a bounded plurisubharmonic function
cites work: On the dimension of the Bergman space for some unbounded domains
cites work: Infinite-dimensionality of the automorphism groups of homogeneous Stein manifolds
cites work: Geometry of Bounded Domains
cites work: Transformation groups in differential geometry.
cites work: Imbedding of Holomorphically Complete Complex Spaces
cites work: Q5628476
cites work: On complete Kähler domains with \(C^ 1-\)boundary
cites work: \(L_h^2\)-functions in unbounded balanced domains
DOI: 10.1016/J.CRMA.2017.06.007
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description: scientific article; zbMATH DE number 6754872