Journées de Géométrie Algébrique Poitiers-Bordeaux

Volume of complex hyperbolic structures on moduli spaces of genus zero curves

24 nov. 2016, 11:45

50m

0-6 (Poitiers)

Panneau: 3

Exposé de 50 min

Orateur

Prof. Vincent Koziarz

Description

(joint work with D. M. Nguyen): I will show that the complex hyperbolic metrics defined by Deligne-Mostow and Thurston on the moduli space of genus $0$ curves with $n$ marked points $M_{0,n}$ are singular Kaehler-Einstein metrics when $M_{0,n}$ is embedded in its Deligne-Mumford-Knudsen compactification. As a consequence, I will obtain a formula computing the volume of $M_{0,n}$ with respect to these metrics using intersection of boundary divisors of its compactification. In the case when the weights parametrizing the complex hyperbolic structures are rational, following an idea of Y. Kawamata, I will show that the associated metrics actually represent the first Chern class of some line bundles on the compactification of $M_{0,n}$, from which other formulas computing the same volumes will be derived.