Complex hyperbolic volume and intersection of boundary divisors in moduli spaces of genus zero curves

Abstract

We show that the complex hyperbolic metrics defined by Deligne-Mostow and Thurston on ${\mathcal{M}}_{0,n}$ are singular Kähler-Einstein metrics when ${\mathcal{M}}_{0,n}$ is embedded in the Deligne-Mumford-Knudsen compactification ${\overline{\mathcal{M}}}_{0,n}$. As a consequence, we obtain a formula computing the volumes of ${\mathcal{M}}_{0,n}$ with respect to these metrics using intersection of boundary divisors of ${\overline{\mathcal{M}}}_{0,n}$. In the case of rational weights, following an idea of Y. Kawamata, we show that these metrics actually represent the first Chern class of some line bundles on ${\overline{\mathcal{M}}}_{0,n}$, from which other formulas computing the same volumes are derived.

Added a new expression of the divisor whose self-intersection computes the volume in Theorem 1.1. Exposition improved