Proper holomorphic mappings onto symmetric products of a Riemann surface

TL;DR

This paper demonstrates that proper holomorphic maps between symmetric products of non-compact Riemann surfaces are highly constrained, being determined by maps between the original surfaces, extending known results to a non-compact setting.

Contribution

It extends the understanding of proper holomorphic maps to symmetric products of non-compact Riemann surfaces, revealing a rigidity phenomenon similar to that in compact cases.

Findings

Proper holomorphic maps are determined by maps of the underlying surfaces.

Provides a condition for hyperbolicity of symmetric products.

Extends results from bounded planar domains to non-compact surfaces.

Abstract

We show that the structure of proper holomorphic maps between the $n$-fold symmetric products, $n\geq 2$, of a pair of non-compact Riemann surfaces $X$ and $Y$, provided these are reasonably nice, is very rigid. Specifically, any such map is determined by a proper holomorphic map of $X$ onto $Y$. This extends existing results concerning bounded planar domains, and is a non-compact analogue of a phenomenon observed in symmetric products of compact Riemann surfaces. Along the way, we also provide a condition for the complete hyperbolicity of all $n$-fold symmetric products of a non-compact Riemann surface.