Translation surfaces and periods of meromorphic differentials

Abstract

Let S $S$ be an oriented surface of genus g $g$ and n $n$ punctures. The periods of any meromorphic differential on S $S$ , with respect to a choice of complex structure, determine a representation χ : Γ g , n → C $\chi :\Gamma _{g,n} \rightarrow \mathbb {C}$ where Γ g , n $\Gamma _{g,n}$ is the first homology group of S $S$ . We characterise the representations that thus arise, that is, lie in the image of the period map Per : Ω M g , n → Hom ( Γ g , n , C ) $\textsf {Per}:\Omega \mathcal {M}_{g,n}\rightarrow \textsf {Hom}(\Gamma _{g,n}, {\mathbb {C}})$ . This generalises 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 $S$ with the prescribed holonomy χ $\chi$ . Along the way, we describe a connection with the Hurwitz problem concerning the existence of branched covers with prescribed branching data.