A transfer principle: from periods to isoperiodic foliations

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 $g\ge 2$ 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.

(96 pages and 14 figures) Revised and improved version to appear in GAFA-Geometric and Functional Analysis

References in corpus (6)

• Compactification of strata of abelian differentials
• Geometry of the Weil-Petersson completion of Teichmüller space
• Rel leaves of the Arnoux-Yoccoz surfaces
• Relative homological representations of framed mapping class groups
• A criterion for density of the isoperiodic leaves in rank 1 affine invariant suborbifolds
• Dynamical properties of the absolute period foliation

Cited by in corpus (7)

• Translation surfaces and periods of meromorphic differentials
• Reconstructing orbit closures from their boundaries
• Isoperiodic meromorphic forms: two simple poles
• Period realization of meromorphic differentials with prescribed invariants
• Dynamics of the absolute period foliation of a stratum of holomorphic 1-forms
• Haupt--Kapovich theorem revisited
• Dynamical properties of the absolute period foliation