# Volume forms on moduli spaces of d-differentials

## TL;DR

This paper establishes a canonical volume form on moduli spaces of meromorphic d-differentials on Riemann surfaces, proving the finiteness of the total volume of the projectivized space under this measure.

## Contribution

It introduces a natural volume form on these moduli spaces and proves the finiteness of their total volume, advancing understanding of their geometric structure.

## Key findings

- Existence of a canonical volume form on the moduli space.
- Finiteness of the total volume of the projectivized space.
- The volume form is parallel with respect to the affine complex structure.

## 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 $k_1+\dots+k_n=d(2g-2)$, 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 $\mathbb{P}\Omega^d\mathcal{M}_{g,n}(\kappa)=\Omega^d\mathcal{M}_{g,n}/\mathbb{C}^*$ with respect to the measure induced by this volume form is finite.