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

Streamlined, minor corrections added, definition of the volume form independent of the choice of a d-th root of unity