Connected components of the strata of the moduli space of meromorphic differentials

TL;DR

This paper investigates the topology of moduli spaces of meromorphic differentials on Riemann surfaces, determining the number of connected components in various genera, revealing complex structures especially in genus one.

Contribution

It extends the understanding of moduli space stratification by computing connected components for meromorphic differentials, generalizing known results from Abelian differentials.

Findings

Up to three components in genus ≥ 2 similar to Abelian case

Arbitrarily many components in genus one distinguished by topological invariants

Provides explicit counts and descriptions of connected components

Abstract

In this paper, we study the translation surfaces corresponding to meromorphic differentials on compact Riemann surfaces. We compute the number of connected components of the corresponding strata of the moduli space. We show that in genus greater than or equal to two, one has up to three components with a similar description as the one of Kontsevich and Zorich for the moduli space of Abelian differentials. In genus one, one can obtain an arbitrarily large number of connected components that are easily distinghished by a simple topological invariant.