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

Koziarz, Vincent; Nguyen, Duc-Manh

doi:10.24033/asens.2381

Complex hyperbolic volume and intersection of boundary divisors in moduli spaces of pointed genus zero curves
[Volume des structures hyperboliques complexes et intersection des diviseurs de bord des espaces de modules de surfaces de Riemann pointées de genre zéro.]

Koziarz, Vincent ; Nguyen, Duc-Manh

Annales scientifiques de l'École Normale Supérieure, Série 4, Tome 51 (2018) no. 6, pp. 1549-1597.

Résumé
Abstract

Nous démontrons que les métriques hyperboliques complexes introduites par Deligne-Mostow et Thurston sur l'espace de modules de surfaces de Riemann de genre zéro avec $n$ points marqués $ℳ_{0, n}$ sont des métriques Kähler-Einstein singulières sur la compactification de Deligne-Mumford-Knudsen ${\bar{ℳ}}_{0, n}$ . Nous en déduisons des formules calculant le volume de $ℳ_{0, n}$ muni de ces métriques en fonction des nombres d'intersection des diviseurs de bord de ${\bar{ℳ}}_{0, n}$ . De plus, lorsque les poids sont tous rationnels, en développant une idée de Y. Kawamata, nous montrons que ces métriques sont aussi des représentants de la première classe de Chern de certains fibrés en droites sur ${\bar{ℳ}}_{0, n}$ , ce qui nous permet d'obtenir d'autres formules calculant les mêmes volumes.

We show that the complex hyperbolic metrics defined by Deligne-Mostow and Thurston on $ℳ_{0, n}$ are singular Kähler-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 volume 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.

MR Zbl

DOI : 10.24033/asens.2381

Classification : 32G15, 53C55, 14C17, 14D22.
Keywords: Moduli spaces of genus zero curves with marked points, flat surfaces, complex hyperbolic cone manifolds, singular Kähler-Einstein metrics.
Mot clés : Espaces de modules de courbes à points marqués en genre zéro, surfaces plates, variétés coniques hyperboliques complexes, métriques Kähler-Einstein singulières.

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     author = {Koziarz, Vincent and Nguyen, Duc-Manh},
     title = {Complex hyperbolic volume  and intersection of boundary divisors  in moduli spaces of pointed genus zero curves},
     journal = {Annales scientifiques de l'\'Ecole Normale Sup\'erieure},
     pages = {1549--1597},
     publisher = {Soci\'et\'e Math\'ematique de France. Tous droits r\'eserv\'es},
     volume = {Ser. 4, 51},
     number = {6},
     year = {2018},
     doi = {10.24033/asens.2381},
     mrnumber = {3940904},
     zbl = {1422.32020},
     language = {en},
     url = {http://archive.numdam.org/articles/10.24033/asens.2381/}
}

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Koziarz, Vincent; Nguyen, Duc-Manh. Complex hyperbolic volume  and intersection of boundary divisors  in moduli spaces of pointed genus zero curves. Annales scientifiques de l'École Normale Supérieure, Série 4, Tome 51 (2018) no. 6, pp. 1549-1597. doi : 10.24033/asens.2381. http://archive.numdam.org/articles/10.24033/asens.2381/

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