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

Huang, Xiaojun; Ji, Shanyu

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

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.

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.

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

[Be] S. Bell; S. Krantz Ed. Compactness of families of holomorphic mappings up to the boundary (Lecture Notes in Math), Volume 1268, pp. 29-43 | Zbl

[BER1] M.S. Baouendi; P. Ebenfelt; L.P. Rothschild Parametrization of local biholomorphisms of real analytic hypersurfaces, Asian J. Math, Volume Vol 1 (1997), pp. 1-16 | MR | Zbl

[BER2] M. S. Baouendi; P. Ebenfelt; L. Rothschild Real Submanifolds in Complex Spaces and Their Mappings, Princeton Univ. Mathematics Series, 47, Princeton University, New Jersey, 1999 | MR | Zbl

[BER3] M.S. Baouendi; P. Ebenfelt; L.P. Rothschild Local geometric properties of real submanifolds in complex spaces, Bull. AMS, Volume 37 (2000), pp. 309-336 | DOI | MR | Zbl

[Bo] S. Bochner Analytic and meromorphic continuation by means of Green's formula, Ann. of Math, Volume 44 (1943), pp. 652-673 | DOI | MR | Zbl

[BS] D. Jr Burns; S. Shnider Projective connections in CR geometry, Manuscripta Math, Volume 33 (1980), pp. 1-26 | DOI | MR | Zbl

[BT] R. Bott; L. W. Tu Differential Forms in Algebraic Topology, Graduate Texts in Mathematics, Springer-Verlag, 1982 | MR | Zbl

[Ch] S.-S. Chern On the projective structure of a real hypersurface in $C_n+1$, Math. Scand, Volume 36 (1975), pp. 74-82 | MR | Zbl

[CJ1] S.-S. Chern; S. Ji Projective geometry and Riemann's mapping problem, Math Ann, Volume 302 (1995), pp. 581-600 | DOI | MR | Zbl

[CJ2] S.-S. Chern; S. Ji On the Riemann mapping theorem, Ann. of Math, Volume 144 (1996), pp. 421-439 | DOI | MR | Zbl

[CM] S. S. Chern; J. K. Moser Real hypersurfaces in complex manifolds, Acta Math, Volume 133 (1974), pp. 219-271 | DOI | MR | Zbl

[ES] S. Eilenberg; N. Steenrod Foundations of algebraic topology, Princeton Univ. Press, Princeton, N.J., 1952 | MR | Zbl

[Fa] J. Faran Segre families and real hypersurfaces, Invent. Math, Volume 60 (1980), pp. 135-172 | DOI | MR | Zbl

[Ga] R. Gardner The method of equivalence and its applications, CBMS-NSF (regional conference series in applied mathematics) (1989) | Zbl

[H1] X. Huang 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] X. Huang Geometric Analysis in Several Complex Variables (August, 1994) (Ph. D. Thesis, Washington University)

[H3] X. Huang; edited by L. Yang and S. T. Yau 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] X. Huang; S. Ji 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] X. Huang; S. Ji; S.S.T. Yau 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] J. Milnor On the Betti numbers of real varieties. Proc. Amer. Math. Soc, Volume 15 (1964), pp. 275-280 | MR | Zbl

[Pi] S. Pinchuk On holomorphic maps or real-analytic hypersurfaces, Mat. Sb., Nov. Ser., Volume 105 (1978), pp. 574-593 | MR

[V] A.G. Vitushkin Holomorphic mappings and geometry of hypersurfaces, Several Complex Variables I (Encyclopaedia of Mathematical Sciences), Volume Vol. 7 (1985), pp. 159-214 | Zbl

[We] S.M. Webster On the mapping problem for algebraic real hypersurfaces, Invent. Math, Volume 43 (1977), pp. 53-68 | DOI | MR | Zbl