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 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 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 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 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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