Cartan-Chern-Moser theory on algebraic hypersurfaces and an application to the study of automorphism groups of algebraic domains

[Théorie de Cartan-Chern-Moser sur les hypersurfaces algébriques et une application à l'étude des groupes d'automorphismes des domaines algébriques]

Huang, Xiaojun ; Ji, Shanyu

Annales de l'Institut Fourier, Tome 52 (2002), p. 1793-1831 / Harvested from Numdam

Résumé

Si $D$ est un domaine fortement pseudo-convexe de ${\mathbb {C}}^{n+1}$, défini par un polynôme réel de degré $k_0$, nous montrons que le groupe de Lie ${\rm Aut}(D)$ s’identifie à une variété algébrique de Nash constructible du CR fibré $Y$ de $\partial D$, et que la somme de ses nombres de Betti est bornée par une constante $C_{n,k_0}$, dépendant seulement de $n$ et de $k_0$. Lorsque $D$ est simplement connexe, nous donnons une borne explicite, mais plus grossière, en fonction de la dimension et du degré du polynôme. Notre approche consiste à adapter la théorie de Cartan-Chern-Moser aux hypersurfaces algébriques.

Abstract

For a strongly pseudoconvex domain $D\subset {\mathbb {C}}^{n+1}$ defined by a real polynomial of degree $k_0$, we prove that the Lie group ${\rm Aut}(D)$ can be identified with a constructible Nash algebraic smooth variety in the CR structure bundle $Y$ of $\partial D$, and that the sum of its Betti numbers is bounded by a certain constant $C_{n,k_0}$ depending only on $n$ and $k_0$. In case $D$ is simply connected, we further give an explicit but quite rough bound in terms of the dimension and the degree of the defining polynomial. Our approach is to adapt the Cartan-Chern-Moser theory to the algebraic hypersurfaces.

Détail

Publié le : 2002-01-01

MR 1954325 | Zbl 1023.32024

DOI : https://doi.org/10.5802/aif.1935
Classification:  32V40,  14P15,  32E99,  32H02,  32T15
Mots clés: hypersurfaces algébriques réelles, groupe d'automorphismes, domaines algébriques, théorie de Cartan-Chern-Moser, domaine fortement pseudoconvexe, nombres de Betti

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