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

Vincent Koziarz (1) , Duc-Manh Nguyen (2)

1 IMB - Institut de Mathématiques de Bordeaux
2 Équipe Géométrie

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.

Domaines: Topologie géométrique [math.GT]; Géométrie algébrique [math.AG]

Vincent Koziarz, Duc-Manh Nguyen. Complex hyperbolic volume and intersection of boundary divisors in moduli spaces of pointed genus zero curves.. Annales Scientifiques de l'École Normale Supérieure, 2018, ⟨10.24033/asens.2381⟩. ⟨hal-01449671⟩