Volume forms on moduli spaces of d–differentials

Nguyen, Duc-Manh¹

¹ Institut de Mathématiques de Bordeaux, Université de Bordeaux, CNRS UMR 5251, Talence, France

Geometry & topology, Tome 26 (2022) no. 7, pp. 3173-3220.

Résumé

Given $d\in \mathbb{N}$, $g\in \mathbb{N}\cup \{0\}$ and an integral vector $\kappa =(k_1,\dots ,k_n)$ such that $k_i>-d$ and $k_1+\cdots +k_n=d(2g-2)$, let ${\mathrm{\Omega }}^d{\mathcal{ℳ}}_{g,n}(\kappa )$ 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 ${\mathrm{\Omega }}^d{\mathcal{ℳ}}_{g,n}(\kappa )$ carries a canonical volume form that is parallel with respect to its affine complex manifold (orbifold) structure, and that the total volume of $\mathbb{P}{\mathrm{\Omega }}^d{\mathcal{ℳ}}_{g,n}(\kappa )={\mathrm{\Omega }}^d{\mathcal{ℳ}}_{g,n}(\kappa )∕{ℂ}^{\ast }$ with respect to the measure induced by this volume form is finite.

Détail

DOI : 10.2140/gt.2022.26.3173

Keywords: differentials on Riemann surfaces, moduli space, flat surfaces

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