Sur la classification des germes d'applications holomorphes contractantes. (On the classification of germs of contracting holomorphic maps)
type: Item
instance of: scholarly article
title: Sur la classification des germes d'applications holomorphes contractantes. (On the classification of germs of contracting holomorphic maps)
zbMATH Open document ID: 0677.32004
DOI: 10.1007/BF01450082
published in: Mathematische Annalen
publication date: 1988-01-01T00:00:00Z
full work available at URL: https://eudml.org/doc/164390
review text: The author considers the problem of determining normal forms of germs of holomorphic maps F: (\({\mathbb{C}}^ 2,0)\to ({\mathbb{C}}^ 2,0)\) which satisfy the following conditions: (1) All the eigenvalues of the Jacobian matrix dF(0) have modulus strictly less than 1. (2) \(F=\Pi \cdot \sigma\), where \(\pi\) : \(B^{\Pi}\to B\) is a modification at the center 0 of the unit ball \(B\subset {\mathbb{C}}^ 2\), and \(\sigma\) : \(\bar B\to B^{\Pi}\) is a holomorphic embedding such that \(\sigma (0)\in \Pi^{-1}(0).\) If dF(0) is nonsingular, the normal forms are well-known. If dF(0) is singular, F defines a generalized Inoue-Hirzebruch surface. For each f, the author defines two kinds of invariants, a(F) and \(\sigma_ n(F)\). Both are given by the continued fraction associated to the self- intersection numbers of the curves on the surface. By these invariants, he characterizes the germs of holomorphic maps which define Inoue- Hirzebruch surfaces in the strict sense, and gives an algorithm of calculating their normal forms.
Mathematics Subject Classification ID: 32J15
Mathematics Subject Classification ID: 14J25
zbMATH DE Number: 4109109
zbMATH Keywords: holomorphic map
zbMATH Keywords: normal form
zbMATH Keywords: contraction
zbMATH Keywords: Inoue-Hirzebruch surface
author: Georges Dloussky
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 4109109