For a strongly pseudoconvex domain defined by a real polynomial of degree , we prove that the Lie group can be identified with a constructible Nash algebraic smooth variety in the CR structure bundle of , and that the sum of its Betti numbers is bounded by a certain constant depending only on and . In case 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 est un domaine fortement pseudo-convexe de , défini par un polynôme réel de degré , nous montrons que le groupe de Lie s’identifie à une variété algébrique de Nash constructible du CR fibré de , et que la somme de ses nombres de Betti est bornée par une constante , dépendant seulement de et de . Lorsque 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.
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
Huang, Xiaojun 1; Ji, Shanyu 2
@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}, pages = {1793--1831}, publisher = {Association des Annales de l{\textquoteright}institut Fourier}, volume = {52}, number = {6}, year = {2002}, doi = {10.5802/aif.1935}, zbl = {1023.32024}, mrnumber = {1954325}, language = {en}, url = {https://aif.centre-mersenne.org/articles/10.5802/aif.1935/} }
TY - JOUR AU - Huang, Xiaojun AU - Ji, Shanyu TI - Cartan-Chern-Moser theory on algebraic hypersurfaces and an application to the study of automorphism groups of algebraic domains JO - Annales de l'Institut Fourier PY - 2002 SP - 1793 EP - 1831 VL - 52 IS - 6 PB - Association des Annales de l’institut Fourier UR - https://aif.centre-mersenne.org/articles/10.5802/aif.1935/ DO - 10.5802/aif.1935 LA - en ID - AIF_2002__52_6_1793_0 ER -
%0 Journal Article %A Huang, Xiaojun %A Ji, Shanyu %T Cartan-Chern-Moser theory on algebraic hypersurfaces and an application to the study of automorphism groups of algebraic domains %J Annales de l'Institut Fourier %D 2002 %P 1793-1831 %V 52 %N 6 %I Association des Annales de l’institut Fourier %U https://aif.centre-mersenne.org/articles/10.5802/aif.1935/ %R 10.5802/aif.1935 %G en %F 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. https://aif.centre-mersenne.org/articles/10.5802/aif.1935/
[Be] Compactness of families of holomorphic mappings up to the boundary (Lecture Notes in Math), Volume 1268, pp. 29-43 | Zbl
[BER1] Parametrization of local biholomorphisms of real analytic hypersurfaces, Asian J. Math, Volume Vol 1 (1997), pp. 1-16 | MR | Zbl
[BER2] Real Submanifolds in Complex Spaces and Their Mappings, Princeton Univ. Mathematics Series, 47, Princeton University, New Jersey, 1999 | MR | Zbl
[BER3] Local geometric properties of real submanifolds in complex spaces, Bull. AMS, Volume 37 (2000), pp. 309-336 | DOI | MR | Zbl
[Bo] Analytic and meromorphic continuation by means of Green's formula, Ann. of Math, Volume 44 (1943), pp. 652-673 | DOI | MR | Zbl
[BS] Projective connections in CR geometry, Manuscripta Math, Volume 33 (1980), pp. 1-26 | DOI | MR | Zbl
[BT] Differential Forms in Algebraic Topology, Graduate Texts in Mathematics, Springer-Verlag, 1982 | MR | Zbl
[Ch] On the projective structure of a real hypersurface in , Math. Scand, Volume 36 (1975), pp. 74-82 | MR | Zbl
[CJ1] Projective geometry and Riemann's mapping problem, Math Ann, Volume 302 (1995), pp. 581-600 | DOI | MR | Zbl
[CJ2] On the Riemann mapping theorem, Ann. of Math, Volume 144 (1996), pp. 421-439 | DOI | MR | Zbl
[CM] Real hypersurfaces in complex manifolds, Acta Math, Volume 133 (1974), pp. 219-271 | DOI | MR | Zbl
[ES] Foundations of algebraic topology, Princeton Univ. Press, Princeton, N.J., 1952 | MR | Zbl
[Fa] Segre families and real hypersurfaces, Invent. Math, Volume 60 (1980), pp. 135-172 | DOI | MR | Zbl
[Ga] The method of equivalence and its applications, CBMS-NSF (regional conference series in applied mathematics) (1989) | Zbl
[H1] On the mapping problem for algebraic real hypersurfaces in the complex spaces of different dimensions, Ann. Inst. Fourier, Grenoble, Volume 44 (1994) no. 2, pp. 433-463 | DOI | Numdam | MR | Zbl
[H2] Geometric Analysis in Several Complex Variables (August, 1994) (Ph. D. Thesis, Washington University)
[H3] On some problems in several complex variables and Cauchy-Riemann Geometry, Proceedings of ICCM (AMS/IP Stud. Adv. Math), Volume 20 (2001), pp. 383-396 | Zbl
[HJ] Global holomorphic extension of a local map and a Riemann mapping Theorem for algebraic domains, Math. Res. Lett, Volume 5 (1998), pp. 247-260 | MR | Zbl
[HJY] An example of real analytic strongly pseudoconvex hypersurface which is not holomorphically equivalent to any algebraic hypersurfaces, Ark. Mat., Volume 39 (2001), pp. 75-93 | DOI | MR | Zbl
[M] On the Betti numbers of real varieties. Proc. Amer. Math. Soc, Volume 15 (1964), pp. 275-280 | MR | Zbl
[Pi] On holomorphic maps or real-analytic hypersurfaces, Mat. Sb., Nov. Ser., Volume 105 (1978), pp. 574-593 | MR
[V] Holomorphic mappings and geometry of hypersurfaces, Several Complex Variables I (Encyclopaedia of Mathematical Sciences), Volume Vol. 7 (1985), pp. 159-214 | Zbl
[We] On the mapping problem for algebraic real hypersurfaces, Invent. Math, Volume 43 (1977), pp. 53-68 | DOI | MR | Zbl
Cited by Sources: