Intersection theory and volumes of moduli spaces of flat metrics on the sphere (with an appendix by Vincent Koziarz and Duc-Manh Nguyen)

TL;DR

This paper develops a recursive formula for volumes of moduli spaces of flat metrics on the sphere, providing new proofs of known formulas and revealing that these volumes are piecewise polynomial functions of cone angles.

Contribution

It introduces a recursive relation for volumes of strata of differentials and proves their polynomial nature, extending previous results and offering new proofs of established formulas.

Findings

Recursive volume formula for strata of differentials.

Volumes are continuous piecewise polynomial functions.

Generalization of previous results on moduli space volumes.

Abstract

Let $\mathbb{P}\Omega^d\mathcal{M}_{0,n}(\kappa)$, where $\kappa=(k_1,\dots,k_n)$, be a stratum of (projectivized) $d$-differentials in genus $0$. We prove a recursive formula which relates the volume of $\mathbb{P}\Omega^d\mathcal{M}_{0,n}(\kappa)$ to the volumes of other strata of lower dimensions in the case where none of the $k_i$ is divisible by $d$. As an application, we give a new proof of the Kontsevich's formula for the volumes of strata of quadratic differentials with simple poles and zeros of odd order, which was originally proved by Athreya-Eskin-Zorich. In another application, we show that up to some power of $\pi$, the volume of the moduli spaces of flat metrics on the sphere with prescribed cone angles is a continuous piecewise polynomial with rational coefficients function of the angles, provided none of the angles is an integral multiple of $2\pi$. This generalizes the results of [28] and [24].