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

Huang, Xiaojun; Ji, Shanyu

doi:10.5802/aif.1935

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 ¹ ; Ji, Shanyu ²

¹ Rutgers University, Department of Mathematics, New Brunswick NJ 08903 (USA)
² Houston University, Department of Mathematics, Houston TX 77204 (USA)

Annales de l'Institut Fourier, Tome 52 (2002) no. 6, pp. 1793-1831.

Résumé
Abstract

Si $D$ est un domaine fortement pseudo-convexe de $ℂ^{n + 1}$ , défini par un polynôme réel de degré $k_{0}$ , nous montrons que le groupe de Lie $Aut (D)$ 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.

For a strongly pseudoconvex domain $D \subset ℂ^{n + 1}$ defined by a real polynomial of degree $k_{0}$ , we prove that the Lie group $Aut (D)$ 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.

MR Zbl

DOI : 10.5802/aif.1935

Classification : 32V40, 14P15, 32E99, 32H02, 32T15
Keywords: real algebraic hypersurfaces, automorphism group, algebraic domains, Cartan-Chern-Moser theory, strongly pseudoconvex domain, Betti numbers
Mot clés : hypersurfaces algébriques réelles, groupe d'automorphismes, domaines algébriques, théorie de Cartan-Chern-Moser, domaine fortement pseudoconvexe, nombres de Betti

Affiliations des auteurs :

Huang, Xiaojun ¹ ; Ji, Shanyu ²

¹ Rutgers University, Department of Mathematics, New Brunswick NJ 08903 (USA)
² Houston University, Department of Mathematics, Houston TX 77204 (USA)

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     author = {Huang, Xiaojun and Ji, Shanyu},
     title = {Cartan-Chern-Moser theory on algebraic hypersurfaces and an application to the study of automorphism groups of algebraic domains},
     journal = {Annales de l'Institut Fourier},
     pages = {1793--1831},
     publisher = {Association des Annales de l{\textquoteright}institut Fourier},
     volume = {52},
     number = {6},
     year = {2002},
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     mrnumber = {1954325},
     language = {en},
     url = {https://aif.centre-mersenne.org/articles/10.5802/aif.1935/}
}

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Huang, Xiaojun; Ji, Shanyu. Cartan-Chern-Moser theory on algebraic hypersurfaces and an application to the study of automorphism groups of algebraic domains. Annales de l'Institut Fourier, Tome 52 (2002) no. 6, pp. 1793-1831. doi : 10.5802/aif.1935. https://aif.centre-mersenne.org/articles/10.5802/aif.1935/

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