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

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].