On the convergence of normalizations of real analytic surfaces near hyperbolic complex tangents
type: Item
instance of: scholarly article
title: On the convergence of normalizations of real analytic surfaces near hyperbolic complex tangents
zbMATH Open document ID: 0826.32012
DOI: 10.1007/BF02564504
author: Xianghong Gong
published in: Commentarii Mathematici Helvetici
publication date: 1995-12-04T00:00:00Z
full work available at URL: https://eudml.org/doc/140354
review text: If \(M \subset \mathbb{C}^2\) is a real analytic surface with a nondegenerate complex tangent at a point (= the origin), being thus locally given by \(z_2 = z_1 \overline {z}_1 + \gamma z^2_1 + \gamma \overline {z}^2_1 + H(z_1, \overline {z}_1)\) (\(H\) is a convergent power series in \(z_1\) and \(\overline {z}_1\) starting with third order terms and \(\gamma\) is the Bishop invariant for the hyperbolic case, \({1\over 2} < \gamma < \infty\)) which is formally equivalent to a quadric \(Q_\gamma : z_2 = z_1 \overline {z}_1 + \gamma z^2_1 + \gamma \overline {z}^2_1\), the author shows that \(M\) is actually equivalent to \(Q\) through biholomorphic mappings provided that \(\gamma \lambda^2 - \lambda + \gamma = 0\) for \(\lambda\) either a root of unity or satisfying a condition called the diophantine condition. Also the author deals with holomorphic flatness of a real analytic surface in \(\mathbb{C}^2\) obtaining a result which indicates that the holomorphic flatness (transforming by holomorphic mappings a manifold into a real hyperplane) is not equivalent to the convergence of normalizations for surfaces near hyperbolic complex tangents.
Mathematics Subject Classification ID: 32V40
Mathematics Subject Classification ID: 32B15
Mathematics Subject Classification ID: 53C99
zbMATH DE Number: 745324
zbMATH Keywords: real analytic surface in \(\mathbb{C}^ 2\)
zbMATH Keywords: holomorphic flatness
zbMATH Keywords: convergence of normalizations
zbMATH Keywords: hyperbolic complex tangents
reviewed by: Eugen Pascu
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 745324