Mappings of real algebraic hypersurfaces

Abstract

A holomorphic function h(Z) defined in a neighborhood of a point Z0 E CN is algebraic if there are polynomials qj(Z), j = 0, ..., k, not all identically zero, such that qk(Z)h(Z)k + ... + qo(Z) _ 0. A holomorphic mapping is algebraic if all its components are algebraic. A smooth real hypersurface M in CN is algebraic if it is contained in the zero set of a nontrivial real-valued polynomial in Z and Z. We assume throughout this paper that N > 1. In [WI], Webster proved the following celebrated result: If M and M' are algebraic hypersurfaces in CN with nondegenerate Levi forms and if H is a biholomorphism defined in an open neighborhood of M and mapping M into M', then H is algebraic. Previously, it had long been known that if M and M are open subsets of spheres, then the components of H are in fact rational functions (Poincare [P], Tanaka [T]). In this paper we go a step further and give a complete characterization of algebraic hypersurfaces in CN such that any holomorphic mapping with nonvanishing Jacobian determinant between two such hypersurfaces must be algebraic. Our main result is stated in Theorem 1 below. ,N By a germ at p0 of a holomorphic vector field in C , we shall mean a