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://www.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://www.numdam.org/articles/10.24033/asens.2381/

Bibliographie
Cité par

Arbarello, E.; Cornalba, M.; Griffiths, P. A., Grundl. math. Wiss., 268, Springer, 2011, 963 pages (ISBN: 978-3-540-42688-2) | DOI | MR | Zbl

Boucksom, S.; Eyssidieux, P.; Guedj, V.; Zeriahi, A. Monge-Ampère equations in big cohomology classes, Acta Math., Volume 205 (2010), pp. 199-262 (ISSN: 0001-5962) | DOI | MR | Zbl

Bedford, E.; Taylor, B. A. A new capacity for plurisubharmonic functions, Acta Math., Volume 149 (1982), pp. 1-40 (ISSN: 0001-5962) | DOI | MR | Zbl

Cerveau, D.; Ghys, É.; Sibony, N.; Yoccoz, J.-C., SMF/AMS Texts and Monographs, 10, Amer. Math. Soc.; Société Mathématique de France, 2003, 197 pages (ISBN: 0-8218-3228-X) | MR | Zbl

Demailly, J.-P., Surveys of Modern Mathematics, 1, International Press; Higher Education Press, 2012, 231 pages (ISBN: 978-1-57146-234-3) | MR | Zbl

Demailly, J.-P. Complex analytic and algebraic geometry (2012) ( https://www-fourier.ujf-grenoble.fr/~demailly/manuscripts/agbook.pdf )

Demailly, J.-P., Complex analysis and geometry (Univ. Ser. Math.), Plenum, 1993, pp. 115-193 | DOI | MR | Zbl

Deligne, P.; Mumford, D. The irreducibility of the space of curves of given genus, Inst. Hautes Études Sci. Publ. Math., Volume 36 (1969), pp. 75-109 (ISSN: 0073-8301) | DOI | Numdam | MR | Zbl

Deligne, P.; Mostow, G. D. Monodromy of hypergeometric functions and nonlattice integral monodromy, Inst. Hautes Études Sci. Publ. Math., Volume 63 (1986), pp. 5-89 (ISSN: 0073-8301) | DOI | Numdam | MR | Zbl

Deligne, P.; Mostow, G. D., Annals of Math. Studies, 132, Princeton Univ. Press, 1993, 183 pages (ISBN: 0-691-00096-4) | DOI | MR | Zbl

Guenancia, H. Kähler-Einstein metrics with cone singularities on klt pairs, Internat. J. Math., Volume 24 (2013) (ISSN: 0129-167X) | DOI | MR | Zbl

Hassett, B. Moduli spaces of weighted pointed stable curves, Adv. Math., Volume 173 (2003), pp. 316-352 (ISSN: 0001-8708) | DOI | MR | Zbl

Kawamata, Y., Birational algebraic geometry (Baltimore, MD, 1996) (Contemp. Math.), Volume 207, Amer. Math. Soc., 1997, pp. 79-88 | DOI | MR | Zbl

Keel, S. Intersection theory of moduli space of stable $n$ -pointed curves of genus zero, Trans. Amer. Math. Soc., Volume 330 (1992), pp. 545-574 (ISSN: 0002-9947) | DOI | MR | Zbl

Kappes, A.; Möller, M. Lyapunov spectrum of ball quotients with applications to commensurability questions, Duke Math. J., Volume 165 (2016), pp. 1-66 (ISSN: 0012-7094) | DOI | MR | Zbl

Kontsevich, M.; Manin, Y. Gromov-Witten classes, quantum cohomology, and enumerative geometry, Comm. Math. Phys., Volume 164 (1994), pp. 525-562 http://projecteuclid.org/euclid.cmp/1104270948 (ISSN: 0010-3616) | DOI | MR | Zbl

Knudsen, F. F. The projectivity of the moduli space of stable curves. II & III. The stacks $M_{g, n}$ , Math. Scand., Volume 52 (1983) (ISSN: 0025-5521) | DOI | MR | Zbl

McMullen, C. T. The Gauss-Bonnet theorem for cone manifolds and volumes of moduli spaces, Amer. J. Math., Volume 139 (2017), pp. 261-291 (ISSN: 0002-9327) | DOI | MR | Zbl

Parker, J. R., Discrete groups and geometric structures (Contemp. Math.), Volume 501, Amer. Math. Soc., 2009, pp. 1-42 | DOI | MR | Zbl

Sauter, J. K. J. Isomorphisms among monodromy groups and applications to lattices in $PU (1, 2)$ , Pacific J. Math., Volume 146 (1990), pp. 331-384 http://projecteuclid.org/euclid.pjm/1102645161 (ISSN: 0030-8730) | DOI | MR | Zbl

Smyth, D. I. Towards a classification of modular compactifications of $ℳ_{g, n}$ , Invent. math., Volume 192 (2013), pp. 459-503 (ISSN: 0020-9910) | DOI | MR | Zbl

Thurston, W. P. Shapes of polyhedra and triangulations of the sphere, The Epstein birthday Schrift (Geom. Topol. Monogr.), Volume 1, Geom. Topol. Publ. (1998), pp. 511-549 | DOI | MR

Wolpert, S. A. The hyperbolic metric and the geometry of the universal curve, J. Differential Geom., Volume 31 (1990), pp. 417-472 http://projecteuclid.org/euclid.jdg/1214444322 (ISSN: 0022-040X) | MR | Zbl

Yoshida, M. Volume formula for certain discrete reflection groups in $PU (2, 1)$ , Mem. Fac. Sci. Kyushu Univ. Ser. A, Volume 36 (1982), pp. 1-11 (ISSN: 0373-6385) | DOI | MR | Zbl

Zvonkine, D., Handbook of Teichmüller theory. Volume III (IRMA Lect. Math. Theor. Phys.), Volume 17, Eur. Math. Soc., 2012, pp. 667-716 | DOI | MR | Zbl

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