Intersection theory and volumes of moduli spaces of flat metrics on the sphere

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 Koziarz and Nguyen (Ann Sci l’Éc Normale Supér 51(6):1549–1597, 2018) and McMullen (Am J Math 139(1):261–291, 2017).