Volume forms on moduli spaces of d–differentials

Nguyen, Duc-Manh

doi:10.2140/gt.2022.26.3173

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Volume forms on moduli spaces of $d$–differentials

Duc-Manh Nguyen

Geometry & Topology 26 (2022) 3173–3220

DOI: 10.2140/gt.2022.26.3173

Abstract

Given $d \in ℕ$ , $g \in ℕ \cup {0}$ and an integral vector $κ = (k_{1}, \dots, k_{n})$ such that $k_{i} > - d$ and $k_{1} + \dots + k_{n} = d (2 g - 2)$ , let $Ω^{d} ℳ_{g, n} (κ)$ 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 $Ω^{d} ℳ_{g, n} (κ)$ carries a canonical volume form that is parallel with respect to its affine complex manifold (orbifold) structure, and that the total volume of $ℙ Ω^{d} ℳ_{g, n} (κ) = Ω^{d} ℳ_{g, n} (κ) ∕ ℂ^{*}$ 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

Volume 26, issue 7 (2022)