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

2023/07/04 by Duc-Manh Nguyen, Nguyen, Duc-Manh
Mathematics · #14H10 #30F60 #32G13 #32G15 #Algebraic Geometry (math.AG) #Complex Variables (math.CV) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Geometric Topology (math.GT) #Geometry and complex manifolds #Meromorphic and Entire Functions

paper · pdf · doi:10.48550/arxiv.2307.01728

openalex publication_date 2023/07/04 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Let ℙΩ^dM_0,n(κ), where κ=(k₁,…,k_n), be a stratum of (projectivized) d-differentials in genus 0. We prove a recursive formula which relates the volume of ℙΩ^dM_0,n(κ) 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 π, 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 2π. This generalizes the results of [28] and [24].

Related

• Masur-Veech volumes, frequencies of simple closed geodesics and intersection numbers of moduli spaces of curves
• A remark on spaces of flat metrics with cone singularities of constant\n sign curvatures
• The Incidence Variety Compactification of strata of d-differentials in genus 0
• Finite isoresidual covers in strata of k-differentials
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