On the dimension of a group acting transitively on hypersurfaces in \(\mathbb{C}^3\)
type: Item
instance of: scholarly article
title: On the dimension of a group acting transitively on hypersurfaces in \(\mathbb{C}^3\)
zbMATH Open document ID: 0946.32013
DOI: 10.1007/BF02465145
author: A. V. Loboda
publication date: 2000-10-08T00:00:00Z
review text: Let \(M\) be a real-analytic CR-hypersurface in \({\mathbb C}^3\) with non degenerate Levi form. Assume that \(M\) is nonspherical (i.e. not locally isomorphic to a quadric). Let \(\xi\in M\). It is known that the group of germs of holomorphic automorphisms of \({\mathbb C}^3\) mapping \(M\) into \(M\) and stabilizing \(\xi\) is a finite-dimensional real Lie group which is linear in appropriately chosen local coordinates. The article under review establishes sharp upper bounds for the dimension of this group. It is of real dimension three if \(M\) is isomorphic to \(\{\Im(z_3)=(z_1\bar z_2 + z_2\bar z_1) + |z_1|^2 \}\) and at most two otherwise. As a consequence one obtains \(\dim_{\mathbb R}Aut(M)\leq 8\) for a homogeneous such \(M\).
reviewed by: Jörg Winkelmann
Mathematics Subject Classification ID: 32M17
Mathematics Subject Classification ID: 32V40
zbMATH DE Number: 1407616
zbMATH Keywords: CR-hypersurface
zbMATH Keywords: homogeneous
zbMATH Keywords: automorphism group
MaRDI profile type: Publication
cites work: Sur la géométrie pseudo-conforme des hypersurfaces de l'espace de deux variables complexes
cites work: Real hypersurfaces in complex manifolds
cites work: Deformations of strictly pseudoconvex domains
cites work: On the Moser normal form at a non-umbilic point
cites work: The classification of three-dimensional homogeneous complex manifold
full work available at URL: https://doi.org/10.1007/bf02465145
OpenAlex ID: W1999821576
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published in: Functional Analysis and Its Applications
description: scientific article; zbMATH DE number 1407616