Dynamics of the absolute period foliation of a stratum of holomorphic\n 1-forms

2021/09/26 by Karl Winsor, Winsor, Karl
Mathematics · #Algebraic Geometry and Number Theory #Mathematical Dynamics and Fractals #Geometry and complex manifolds

paper · pdf · doi:10.48550/arxiv.2109.12669

Abstract

Let \C be a connected component of a stratum of the moduli space\nof holomorphic 1-forms of genus g. We show that the absolute period\nfoliation of \C is ergodic on the area-1 locus, and that the\nnon-dense leaves lie in an explicit countable union of suborbifolds, subject to\na mild assumption on \C. We show similar results for subspaces of\n\C defined by topological restrictions on the absolute periods. We\nobtain these dynamical results by showing that for a typical positive\ncohomology class in H¹(S_g;\ℂ), the associated space of isoperiodic\nforms in \C is connected. Lastly, we show that certain covering\nconstructions provide examples of spaces of isoperiodic forms with positive\ndimension and infinitely many connected components.\n

Related

• A transfer principle: from periods to isoperiodic foliations
• Dynamical properties of the absolute period foliation
• A criterion for density of the isoperiodic leaves in rank 1 affine invariant suborbifolds
• Isoperiodic meromorphic forms with at least three simple poles
• Isoperiodic dynamics in rank 1 affine invariant orbifolds