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

Xiaojun Huang[1]; Shanyu Ji[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 (2002)

  • Volume: 52, Issue: 6, page 1793-1831
  • ISSN: 0373-0956

Abstract

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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.

How to cite

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Huang, Xiaojun, and 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 52.6 (2002): 1793-1831. <http://eudml.org/doc/116028>.

@article{Huang2002,
abstract = {For a strongly pseudoconvex domain $D\subset \{\mathbb \{C\}\}^\{n+1\}$ defined by a real polynomial of degree $k_0$, we prove that the Lie group $\{\rm Aut\}(D)$ 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.},
affiliation = {Rutgers University, Department of Mathematics, New Brunswick NJ 08903 (USA); Houston University, Department of Mathematics, Houston TX 77204 (USA)},
author = {Huang, Xiaojun, Ji, Shanyu},
journal = {Annales de l’institut Fourier},
keywords = {real algebraic hypersurfaces; automorphism group; algebraic domains; Cartan-Chern-Moser theory; strongly pseudoconvex domain; Betti numbers},
language = {eng},
number = {6},
pages = {1793-1831},
publisher = {Association des Annales de l'Institut Fourier},
title = {Cartan-Chern-Moser theory on algebraic hypersurfaces and an application to the study of automorphism groups of algebraic domains},
url = {http://eudml.org/doc/116028},
volume = {52},
year = {2002},
}

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
PB - Association des Annales de l'Institut Fourier
VL - 52
IS - 6
SP - 1793
EP - 1831
AB - For a strongly pseudoconvex domain $D\subset {\mathbb {C}}^{n+1}$ defined by a real polynomial of degree $k_0$, we prove that the Lie group ${\rm Aut}(D)$ 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.
LA - eng
KW - real algebraic hypersurfaces; automorphism group; algebraic domains; Cartan-Chern-Moser theory; strongly pseudoconvex domain; Betti numbers
UR - http://eudml.org/doc/116028
ER -

References

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  1. M.S. Baouendi, P. Ebenfelt, L.P. Rothschild, Parametrization of local biholomorphisms of real analytic hypersurfaces, Asian J. Math Vol 1 (1997), 1-16 Zbl0943.32021MR1480988
  2. M. S. Baouendi, P. Ebenfelt, L. Rothschild, Real Submanifolds in Complex Spaces and Their Mappings, 47 (1999), Princeton University, New Jersey Zbl0944.32040MR1668103
  3. M.S. Baouendi, P. Ebenfelt, L.P. Rothschild, Local geometric properties of real submanifolds in complex spaces, Bull. AMS 37 (2000), 309-336 Zbl0955.32027MR1754643
  4. S. Bell, Compactness of families of holomorphic mappings up to the boundary, 1268, 29-43, Springer-Verlag Zbl0633.32020
  5. S. Bochner, Analytic and meromorphic continuation by means of Green's formula, Ann. of Math 44 (1943), 652-673 Zbl0060.24206MR9206
  6. R. Bott, L. W. Tu, Differential Forms in Algebraic Topology, (1982), Springer-Verlag Zbl0496.55001MR658304
  7. D. Jr Burns, S. Shnider, Projective connections in CR geometry, Manuscripta Math 33 (1980), 1-26 Zbl0478.32018MR596374
  8. S.-S. Chern, On the projective structure of a real hypersurface in C n + 1 , Math. Scand 36 (1975), 74-82 Zbl0305.53019MR379910
  9. S.-S. Chern, S. Ji, Projective geometry and Riemann's mapping problem, Math Ann 302 (1995), 581-600 Zbl0843.32013MR1339928
  10. S.-S. Chern, S. Ji, On the Riemann mapping theorem, Ann. of Math 144 (1996), 421-439 Zbl0872.32016MR1418903
  11. S. S. Chern, J. K. Moser, Real hypersurfaces in complex manifolds, Acta Math 133 (1974), 219-271 Zbl0302.32015MR425155
  12. S. Eilenberg, N. Steenrod, Foundations of algebraic topology, (1952), Princeton Univ. Press, Princeton, N.J. Zbl0047.41402MR50886
  13. J. Faran, Segre families and real hypersurfaces, Invent. Math 60 (1980), 135-172 Zbl0464.32011MR586425
  14. R. Gardner, The method of equivalence and its applications, CBMS-NSF (1989) Zbl0694.53027
  15. X. Huang, On the mapping problem for algebraic real hypersurfaces in the complex spaces of different dimensions, Ann. Inst. Fourier, Grenoble 44 (1994), 433-463 Zbl0803.32011MR1296739
  16. X. Huang, Geometric Analysis in Several Complex Variables, (August, 1994) 
  17. X. Huang, On some problems in several complex variables and Cauchy-Riemann Geometry, Proceedings of ICCM 20 (2001), 383-396 Zbl1048.32022
  18. X. Huang, S. Ji, Global holomorphic extension of a local map and a Riemann mapping Theorem for algebraic domains, Math. Res. Lett 5 (1998), 247-260 Zbl0912.32010MR1617897
  19. 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. 39 (2001), 75-93 Zbl1038.32034MR1821083
  20. J. Milnor, On the Betti numbers of real varieties. Proc. Amer. Math. Soc, 15 (1964), 275-280 Zbl0123.38302MR161339
  21. S. Pinchuk, On holomorphic maps or real-analytic hypersurfaces, Mat. Sb., Nov. Ser. 105 (1978), 574-593 Zbl0438.32009MR496595
  22. A.G. Vitushkin,, Holomorphic mappings and geometry of hypersurfaces, Several Complex Variables I Vol. 7 (1985), 159-214, Springer-Verlag, Berlin Zbl0781.32013
  23. S.M. Webster, On the mapping problem for algebraic real hypersurfaces, Invent. Math 43 (1977), 53-68 Zbl0348.32005MR463482

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