Haupt--Kapovich theorem revisited

TL;DR

This paper revisits Haupt--Kapovich theorem, exploring the realization of cohomology classes by abelian differentials, and extends the understanding to pairs and triples, revealing new insights into Teichmüller space and algebraic geometry of curves.

Contribution

It extends Haupt--Kapovich theorem to pairs and triples of cohomology classes, providing new perspectives on Teichmüller space and algebraic geometry.

Findings

Characterization of cohomology classes realizable by abelian differentials.

New local description of Teichmüller space.

Connections to algebraic geometry of curves.

Abstract

A theorem of O. Haupt, rediscovered by M. Kapovich and celebrated by his proof invoking Ratner theory, describes the set of de Rham cohomology classes on a topological orientable surface, which can be realized by an abelian differential in some respective complex structure, in purely topological terms. We make an attempt to describe similarly pairs and triples of cohomology classes, which can be realized by abelian differentials in some complex structure. This leads us to some interesting problems in algebraic geometry of curves, and gives an unexpected local description of the Teichm\"uller space.