On holomorphic homogeneity of real hypersurfaces of general position in \(\mathbb{C}^3\)
type: Item
instance of: scholarly article
title: On holomorphic homogeneity of real hypersurfaces of general position in \(\mathbb{C}^3\)
zbMATH Open document ID: 1396.32019
DOI: 10.1134/S0081543817060025
author: A. V. Atanov
author: A. V. Loboda
author: V. I. Sukovykh
published in: Proceedings of the Steklov Institute of Mathematics
publication date: 2018-02-27T00:00:00Z
review text: The article deals with the equivalence problem for homogeneous real-analytic hypersurfaces in \(\mathbb C^3\). A solution has been provided in the Levi degenerate case by \textit{G. Fels} and \textit{W. Kaup} [Acta Math. 201, No. 1, 1--82 (2008; Zbl 1171.32023)]. In the strictly pseudoconvex case, which is the setting of the reviewed paper, the second author in [Zbl 1023.32025; Zbl 1023.32025] gave results when there is a large isotropy group and another instance in [Zbl 1060.32021)]. For coordinates \((z_1,z_2,w)\) of \(\mathbb C^3\) write \(z=(z_1,z_2) \in \mathbb C^2\) and \(w=u+iv\). According to the result of \textit{S. S. Chern} and \textit{J. K. Moser} [Usp. Mat. Nauk 38, No. 2(230), 149--193 (1983; Zbl 0519.32014)] any strictly pseudoconvex real-analytic hypersurface in \(\mathbb C^3\) is biholomorphically equivalent to a hypersurface given by \[ \begin{aligned} v = |z_1|^2 + |z_2|^2 + \sum_{k,\ell \geq 2,m\geq 0} N_{k\ell m}(z,\bar z) u^m, \end{aligned} \] where \(N_{k\ell m}\) is a homogeneous polynomial of degree \(k\) in \(z\) and degree \(\ell\) in \(\bar z\) and the polynomials \(N_{22m}, N_{32m}\) and \(N_{33m}\) satisfy certain trace conditions. It was shown in [\textit{V. V. Ezhov} et al., Math. Notes 66, 3--4, 513--515 (1999); translation from Mat. Zametki 66, 4, 624--626 (1999; Zbl 1395.32020)] that after a change of coordinates preserving the Chern-Moser normal form, \(N_{220}\) can be assumed to be one of two specific polynomials. In the paper under review one case is considered and the corresponding hypersurface is referred to as being of ``general position''. The reader is referred to the paper for a detailed description. It was shown by the second author [Sb. Math. 192, No. 12, 1741--1761 (2001; Zbl 1029.32021); translation from Mat. Sb. 192, No. 12, 3--24 (2001)] that in the case of a homogeneous hypersurface, the above given Chern-Moser normal form is uniquely determined by the polynomials \(N_{220},N_{221},N_{222},N_{320},N_{321},N_{330}\)…
reviewed by: Michael Reiter
Mathematics Subject Classification ID: 32V40
zbMATH DE Number: 6844172
zbMATH Keywords: homogeneity
zbMATH Keywords: strict pseudoconvexity
zbMATH Keywords: Chern-Moser normal form
zbMATH Keywords: real hypersurface
zbMATH Keywords: biholomorphism
describes a project that uses: LieAlg
MaRDI profile type: Publication
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full work available at URL: s0081543817060025
OpenAlex ID: W2768365105
description: scientific article; zbMATH DE number 6844172