A transfer principle: from periods to isoperiodic foliations

TL;DR

This paper classifies the closures of leaves in the isoperiodic foliation on the moduli space of abelian differentials, proving ergodicity on these sets and exploring implications for Hurwitz spaces and Torelli space topology.

Contribution

It introduces the theory of augmented Torelli space and analyzes the connectedness of fibers of the period map to classify leaf closures and establish ergodicity.

Findings

Classified possible leaf closures of the isoperiodic foliation.

Proved ergodicity of the foliation on these closures.

Derived consequences for the topology of Hurwitz spaces.

Abstract

We classify the possible closures of leaves of the isoperiodic foliation (sometimes called absolute period foliation) defined on the Hodge bundle, i.e. the moduli space of abelian differentials over genus $g\geq 2$ smooth curves, and prove that the foliation is ergodic on those sets. The results derive from the connectedness properties of the fibers of the period map defined on the Torelli cover of the moduli space. Some consequences on the topology of Hurwitz spaces of primitive branched coverings over elliptic curves are also obtained. To prove the results we develop the theory of augmented Torelli space, the branched Torelli cover of the Deligne-Mumford compactification of the moduli space of curves.