Exploring isoperiodic foliations

Abstract

Strata of meromorphic $1$-forms are endowed with the atlas of period coordinates given by the periods of the differential on arcs joining the $n$ zeroes of the differential. Fixing the periods on absolute homology classes defines the isoperiodic foliation.

Isoperiodic leaves are complex manifolds of dimension $n-1$ endowed with a translation structure inherited from the period atlas of the stratum. In this talk, we give a description of some elementary examples.

In genus zero, isoperiodic leaves are compact and their translation structures have finite order singularities. On the other hand, for nonzero genus, In nonzero genus, even the simplest example displays wild behavior.

For strata H(1,1,-2) of meromorphic $1$-forms with one double pole and two simple zeroes on an elliptic curve, isoperiodic leaves are complex curves of infinite genus and their translation structures are given by differential forms with essential singularities.