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

TL;DR

This paper links complex hyperbolic metrics on moduli spaces of genus zero curves to singular Kähler-Einstein metrics, providing volume formulas via boundary divisor intersections and relating these metrics to line bundle Chern classes.

Contribution

It demonstrates that hyperbolic metrics on ${ m M}_{0,n}$ are singular Kähler-Einstein metrics and connects them to line bundle Chern classes, enabling explicit volume computations.

Findings

Volumes computed via boundary divisor intersections

Metrics correspond to first Chern classes of line bundles

Formulas derived for rational weights

Abstract

We show that the complex hyperbolic metrics defined by Deligne-Mostow and Thurston on ${\mathcal{M}}_{0,n}$ are singular K\"ahler-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.