Finiteness Theorems for Products and Symmetric Products of Hyperbolic Riemann Surfaces

TL;DR

This paper proves that the set of dominant holomorphic maps from finite-type hyperbolic Riemann surface products to certain complex manifolds is finite, extending to products and symmetric products of these surfaces.

Contribution

It establishes finiteness theorems for dominant holomorphic mappings from products and symmetric products of hyperbolic Riemann surfaces to complex manifolds.

Findings

Finiteness of dominant holomorphic maps from product of hyperbolic Riemann surfaces.

Finiteness results extend to symmetric products of hyperbolic Riemann surfaces.

Applicable to mappings into complex manifolds with bounded simply-connected domains.

Abstract

We prove that if $X = X_1 \times \dots \times X_n$ 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.