Cartan-Chern-Moser theory on algebraic hypersurfaces and an application to the study of automorphism groups of algebraic domains
Annales de l'Institut Fourier, Volume 52 (2002) no. 6, p. 1793-1831

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 $\mathrm{Aut}\left(D\right)$ 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.

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 $\mathrm{Aut}\left(D\right)$ 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.

DOI : https://doi.org/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
@article{AIF_2002__52_6_1793_0,
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},
publisher = {Association des Annales de l'institut Fourier},
volume = {52},
number = {6},
year = {2002},
pages = {1793-1831},
doi = {10.5802/aif.1935},
zbl = {1023.32024},
mrnumber = {1954325},
language = {en},
url = {http://www.numdam.org/item/AIF_2002__52_6_1793_0}
}

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, Volume 52 (2002) no. 6, pp. 1793-1831. doi : 10.5802/aif.1935. http://www.numdam.org/item/AIF_2002__52_6_1793_0/

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