Strongly pseudoconvex domains as subvarieties of complex manifolds

Abstract

In this paper we obtain existence and approximation results for closed complex subvarieties that are normalized by strongly pseudoconvex Stein domains. Our sufficient condition for the existence of such subvarieties in a complex manifold $X$ is expressed in terms of the Morse indices and the number of positive Levi eigenvalues of an exhaustion function on $X$. Examples show that our conditions cannot be weakened in general. We obtain optimal results for subvarieties of this type in complements of compact complex submanifolds with Griffiths positive normal bundle; in the projective case these generalize classical theorems of Remmert, Bishop and Narasimhan concerning proper holomorphic maps and embeddings to ${\Bbb C}^n ={\Bbb P}^n \backslash {\Bbb P}^{n-1}$..