Holomorphic maps between configuration spaces of Riemann surfaces

TL;DR

This paper classifies holomorphic maps between configuration spaces of Riemann surfaces across different genera, extending known theorems and linking to the de Franchis problem, with implications for complex geometry and group theory.

Contribution

It provides a complete classification of holomorphic maps between configuration spaces for various genera, extending the Tameness Theorem and addressing open questions in the field.

Findings

Classified holomorphic maps between configuration spaces of complex plane for large n

Extended Tameness Theorem to cases where m ≤ 2n

Linked higher genus cases to the de Franchis problem

Abstract

We prove a suite of results classifying holomorphic maps between configuration spaces of Riemann surfaces; we consider both the ordered and unordered setting as well as the cases of genus zero, one, and at least two. We give a complete classification of all holomorphic maps $\operatorname{Conf}_n(\mathbb{C})\to \operatorname{Conf}_m(\mathbb{C})$ provided that $n\ge 5$ and $m\le 2n$ extending the Tameness Theorem of Lin, which is the case $m = n$. We also give a complete classification of holomorphic maps between ordered configuration spaces of Riemann surfaces of genus at most one (answering a question of Farb), and show that the higher genus setting is closely linked to the still-mysterious ``effective de Franchis problem''. The main technical theme of the paper is that holomorphicity allows one to promote group-theoretic rigidity results to the space level.