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Volume forms on moduli spaces of $d$–differentials

Duc-Manh Nguyen

Geometry & Topology 26 (2022) 3173–3220
Abstract

Given d , g {0} and an integral vector κ = (k1,,kn) such that ki > d and k1 + + kn = d(2g 2), let Ωdg,n(κ) denote the moduli space of meromorphic d–differentials on Riemann surfaces of genus g whose zeros and poles have orders prescribed by κ. We show that Ωdg,n(κ) carries a canonical volume form that is parallel with respect to its affine complex manifold (orbifold) structure, and that the total volume of Ωdg,n(κ) = Ωdg,n(κ) with respect to the measure induced by this volume form is finite.

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Keywords
differentials on Riemann surfaces, moduli space, flat surfaces
Mathematical Subject Classification
Primary: 30F60, 32G15, 51H25
References
Publication
Received: 16 February 2021
Revised: 23 July 2021
Accepted: 24 August 2021
Published: 23 January 2023
Proposed: Benson Farb
Seconded: Paul Seidel, Anna Wienhard
Authors
Duc-Manh Nguyen
Institut de Mathématiques de Bordeaux
Université de Bordeaux
CNRS UMR 5251
Talence
France