Volume 20, issue 5 (2020)

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Moduli space of meromorphic differentials with marked horizontal separatrices

Corentin Boissy

Algebraic & Geometric Topology 20 (2020) 2373–2412
Abstract

We study framed translation surfaces corresponding to meromorphic differentials on compact Riemann surfaces, for which a horizontal separatrix is marked for each pole or zero. Such geometric structures naturally appear when studying flat geometry surfaces “near” the Deligne–Mumford boundary.

We compute the number of connected components of the corresponding strata, and give a simple topological invariant that distinguishes them. In particular we see that for g > 0, there are at most two such components, except in the hyperelliptic case.

Keywords
meromorphic differentials, translation surfaces, moduli spaces
Mathematical Subject Classification 2010
Primary: 32G15
Secondary: 30F30, 57R15
References
Publication
Received: 6 April 2018
Revised: 5 April 2019
Accepted: 1 December 2019
Published: 4 November 2020
Authors
Corentin Boissy
CNRS, Institut Mathématique de Toulouse
Université de Toulouse
UMR5219
Toulouse
France