Translation surfaces and periods of meromorphic differentials

TL;DR

This paper characterizes the representations arising from periods of meromorphic differentials on surfaces, generalizing Haupt's classical result and describing the image of the period map for various strata, with geometric proofs and connections to branched covers.

Contribution

It extends Haupt's classical result to meromorphic differentials and characterizes the period map's image for different strata, using geometric constructions.

Findings

Characterization of period representations for meromorphic differentials

Description of the period map's image for prescribed zeros and poles

Connection established between translation structures and the Hurwitz problem

Abstract

Let $S$ be an oriented surface of genus $g$ and $n$ punctures. The periods of any meromorphic differential on $S$, with respect to a choice of complex structure, determine a representation $\chi:\Gamma_{g,n} \to\mathbb C$ where $\Gamma_{g,n}$ is the first homology group of $S$. We characterize the representations that thus arise, that is, lie in the image of the period map $\textsf{Per}:\Omega\mathcal{M}_{g,n}\to \textsf{Hom}(\Gamma_{g,n},\mathbb{C})$. This generalizes a classical result of Haupt in the holomorphic case. Moreover, we determine the image of this period map when restricted to any stratum of meromorphic differentials, having prescribed orders of zeros and poles. Our proofs are geometric, as they aim to construct a translation structure on $S$ with the prescribed holonomy $\chi$. Along the way, we describe a connection with the Hurwitz problem concerning the existence of branched covers with prescribed branching data.