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

Abstract

We show that the complex hyperbolic metrics defined by Deligne-Mostow and Thurston on M0,n are singular K\ahler-Einstein metrics when $M_{0,n}$ is embedded in the Deligne-Mumford-Knudsen compactification$\bar{M}_{0,n}$. As a consequence, we obtain a formula computing the volumes of $M_{0,n}$ with respect to these metrics using intersection of boundary divisors of $\bar{M}_{0,n}$. 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 $\bar{M}_{0,n}$, from which other formulas computing the same volumes are derived.