Lectures on the automorphism groups of Kobayashi-hyperbolic manifolds
type: Item
instance of: scholarly article
title: Lectures on the automorphism groups of Kobayashi-hyperbolic manifolds
zbMATH Open document ID: 1196.32017
published in: Lecture Notes in Mathematics
publication date: 2007-02-23T00:00:00Z
review text: Let \(M\) be a hyperbolic complex manifold of dimension \(n\geq 2\). The group \(\Aut(M)\) of holomorphic automorphisms of \(M\) is a real Lie group acting properly on \(M\). In this situation, in fact, \(\Aut(M)\) acts on \(M\) as a group of isometries with respect to the Kobayashi distance. Denote by \(d(M)\) the real dimension of \(\Aut(M)\). The goal of this book is to present a self-contained exposition of the classification of hyperbolic manifolds with ``large'' automorphism group obtained by the author and collaborators in several papers [see for example, \textit{A. V. Isaev} and \textit{S. G. Krantz}, J. Reine Angew. Math. 534, 187--194 (2001; Zbl 0974.32018); \textit{A. V. Isaev}, J. Geom. Anal. 15, No.~2, 239--259 (2005; Zbl 1095.32007); Geom. Funct. Anal. 17, No.~1, 192--219 (2007; Zbl 1120.32014); Geom. Topol. 12, No.~2, 643--711 (2008; Zbl 1143.32016)]. The manifolds under consideration are precisely the \(n\)-dimensional hyperbolic manifolds \(M\) satisfying the condition \[ n^2-1\leq d(M)\leq n^2+2n. \] The upper bound arises from a result of \textit{W. Kaup} [Invent. Math 3, 43--70 (1967; Zbl 0157.13401)] stating that, if \(d(M)>n^2\), then \(M\) is necessarily homogeneous and \(d(M)\leq n^2+2n\). Moreover, \(d(M)=n^2+2n\) if and only if \(M\) is biholomorphic to \(B^n\), the unit ball in \({\mathbb C}^n\). The lower bound \(n^2-1\) is suggested by the existence in \({\mathbb C}^2\) of an infinite family of pairwise non-equivalent Reinhardt domains, all with automorphism group of dimension \(2\). Such a family cannot be described by any formula, thus excluding the possibility of classifying 2-dimensional manifolds \(M\) with \(d(M)=2\) and in general \(n\)-dimensional manifolds \(M\) with \(d(M)=n^2-2\), for all values of \(n\). One should also observe that, for \(M\) not the unit ball, actually \(d(M)\leq n^2+2\) holds. By the properness of the action, the isotropy subgroup of an arbitrary point in \(M\) can in fact be identified with a closed subgr…
reviewed by: Laura Geatti
Mathematics Subject Classification ID: 32Q45
Mathematics Subject Classification ID: 32M05
Mathematics Subject Classification ID: 32-02
zbMATH DE Number: 5129479
zbMATH Keywords: Kobayashi-hyperbolic manifolds
zbMATH Keywords: group of holomorphic automorphisms
zbMATH Keywords: non-homogeneous hyperbolic manifolds
zbMATH Keywords: CR-manifolds
zbMATH Keywords: CR-automorphisms
author: A. V. Isaev
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 5129479