Item

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.