On the group of holomorphic automorphisms of \(\mathbb{C}{}^ n\)
type: Item
instance of: scholarly article
title: On the group of holomorphic automorphisms of \(\mathbb{C}{}^ n\)
zbMATH Open document ID: 0770.32015
DOI: 10.1007/BF01231337
author: Erik Andersén
author: László Lempert
published in: Inventiones Mathematicae
publication date: 1993-05-16T00:00:00Z
full work available at URL: https://eudml.org/doc/144055
review text: The paper deals with the problem of the structure of the group of holomorphic automorphisms \(\text{Aut}(\mathbb{C}^ n)\) of \(\mathbb{C}^ n\) (with the topology of locally uniform convergence). The authors consider two classes of holomorphic automorphisms of \(\mathbb{C}^ n\): 1. shears --- of the form \((z_ 1,\dots,z_ n)\mapsto(z_ 1,\dots,z_{n-1}\), \(f(z_ 1,\dots,z_{n-1})+z_ n)\), 2. overshears --- of the form \((z_ 1,\dots,z_ n)\mapsto(z_ 1,\dots,z_{n-1}, f(z_ 1,\dots,z_{n-1})+h(z_ 1,\dots,z_{n- 1})z_ n)\), where \(f, h\) are holomorphic functions on \(\mathbb{C}^{n-1}\), \(h\) nowhere 0. Let \(\text{Aut}_ 1(\mathbb{C}^ n)\) be the group of holomorphic automorphisms of \(\mathbb{C}^ n\) with Jacobi determinant 1. Denote by \(G(\mathbb{C}^ n)\) (resp. \(G_ 1(\mathbb{C}^ n))\) the subgroup of \(\text{Aut}(\mathbb{C}^ n)\) (resp. \(\text{Aut}_ 1(\mathbb{C}^ n))\) generated by overshears (resp. shears) i.e. all finite compositions of overshears (resp. shears). The first author in Complex Variables, Theory Appl. 14, No. 1-4, 223-235 (1990; Zbl 0705.58008) proved that for \(n\geq 2\) \(G_ 1(\mathbb{C}^ n)\) is dense in \(\text{Aut}_ 1(\mathbb{C}^ n)\) and that \(G_ 1(\mathbb{C}^ 2)\neq\text{Aut}_ 1(\mathbb{C}^ 2)\). In the paper the authors push further these results. Namely, they prove: 1. \(G_ 1(\mathbb{C}^ n)\neq\text{Aut}_ 1(\mathbb{C}^ n)\), 2. \(G(\mathbb{C}^ n)\neq\text{Aut}(\mathbb{C}^ n)\), 3. \(G(\mathbb{C}^ n)\) is dense in \(\text{Aut}(\mathbb{C}^ n)\).
Mathematics Subject Classification ID: 32M05
Mathematics Subject Classification ID: 32H02
zbMATH DE Number: 167399
zbMATH Keywords: holomorphic automorphism
zbMATH Keywords: shears
zbMATH Keywords: overshears
zbMATH Keywords: Runge property
reviewed by: Tadeusz Krasiński
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 167399