Haupt's theorem for strata of abelian differentials

TL;DR

This paper refines Haupt's theorem by incorporating divisor data, providing a more precise criterion for when a homology character can be realized by a holomorphic 1-form on a topological surface.

Contribution

It introduces a refined version of Haupt's theorem that accounts for divisor data, enhancing the understanding of abelian differentials on surfaces.

Findings

Refined Haupt's theorem incorporating divisor data

Necessary and sufficient conditions for realization of homology characters

Enhanced criteria for holomorphic 1-forms on surfaces

Abstract

Let S be a closed topological surface. Haupt's theorem provides necessary and sufficient conditions for a complex-valued character of the first integer homology group of S to be realized by integration against a complex-valued 1-form that is holomorphic with respect to some complex structure on S. We prove a refinement of this theorem that takes into account the divisor data of the 1-form.