Finiteness Theorems for Products and Symmetric Products of Hyperbolic Riemann Surfaces

Abstract

We prove that if $X=X_1\times \cdots \times X_n$ is a product of hyperbolic Riemann surfaces of finite type and $Y=\Omega /\Gamma$ is a complex manifold, where $\Omega$ is a bounded simply-connected domain in ${\mathbb{C}}^m$, then the space of dominant holomorphic mappings from $X$ to $Y$ is a finite set. As corollaries, we obtain the finiteness of the space of dominant holomorphic mappings into products and symmetric products of hyperbolic Riemann surfaces.

This paper has been withdrawn as Lemma 11 is not true