Volume forms on moduli spaces of d–differentials

Abstract

Given $d\in \mathbb{N}$, $g\in \mathbb{N}\cup \{0\}$ and an integral vector $\kappa =(k_1,\dots ,k_n)$ such that 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.