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Volume forms on moduli spaces of $d$–differentials

### Duc-Manh Nguyen

Geometry & Topology 26 (2022) 3173–3220
##### Abstract

Given $d\in ℕ$, $g\in ℕ\cup \left\{0\right\}$ and an integral vector $\kappa =\left({k}_{1},\dots ,{k}_{n}\right)$ such that ${k}_{i}>-d$ and ${k}_{1}+\cdots +{k}_{n}=d\left(2g-2\right)$, let ${\mathrm{\Omega }}^{d}{\mathsc{ℳ}}_{g,n}\left(\kappa \right)$ denote the moduli space of meromorphic $d$–differentials on Riemann surfaces of genus $g$ whose zeros and poles have orders prescribed by $\kappa$. We show that ${\mathrm{\Omega }}^{d}{\mathsc{ℳ}}_{g,n}\left(\kappa \right)$ carries a canonical volume form that is parallel with respect to its affine complex manifold (orbifold) structure, and that the total volume of $ℙ{\mathrm{\Omega }}^{d}{\mathsc{ℳ}}_{g,n}\left(\kappa \right)={\mathrm{\Omega }}^{d}{\mathsc{ℳ}}_{g,n}\left(\kappa \right)∕{ℂ}^{\ast }$ with respect to the measure induced by this volume form is finite.

##### Keywords
differentials on Riemann surfaces, moduli space, flat surfaces
##### Mathematical Subject Classification
Primary: 30F60, 32G15, 51H25
##### 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