Cartan-Chern-Moser Theory on Algebraic Hypersurfaces and Automorphism Group of Algebraic Domains

Abstract

§0. Introduction It is known that for a projective compact Riemann surface S, the number of the elements in its automorphism group Aut(S) is finite (when g(S) > 1), which is moreover bounded by a certain constant depending only on the degree of the equations defining S. It would be interesting to find an analogue of this fact for a bounded strongly pseudoconvex domain D ⊂ C defined by a real polynomial. Motivated by this problem, we shall prove in this paper, that for a strongly pseudoconvex domain D defined by a real polynomial of degree k0, the Lie group Aut(D) can be identified with a constructible Nash algebraic smooth variety in the CR structure bundle Ỹ of ∂D̃, and the sum of its Betti numbers is bounded by a certain constant Cn,k0 depending only on n and k0. 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. Since the domain under consideration is strongly pseudoconvex, the study of its automorphism group can be pushed to that for the CR automorphism group of its boundary. Applying the CR equivalence theory, this can be reduced to investigating a certain variety in its structure bundle defined by certain curvature equations. The precise estimate of the total Betti number of the just mentioned variety is then obtained by making use of a result of Milnor [M] and the specific construction of its defining equations. Now, we give our main result, whose statement requires some terminology to be explained in §1− §3:

Cite this paper

@inproceedings{Huang2004CartanChernMoserTO, title={Cartan-Chern-Moser Theory on Algebraic Hypersurfaces and Automorphism Group of Algebraic Domains}, author={Xiaojun Huang and Shanyu Ji}, year={2004} }