Comptes Rendus
no. 10
Algebraic geometry/Analytic geometry
Curvature properties for moduli of canonically polarized manifolds—An analogy to moduli of Calabi–Yau manifolds

Presented by: Jean-Pierre Demailly

Georg Schumacher1
1 Fachbereich Mathematik und Informatik, Philipps-Universität Marburg, Lahnberge, Hans-Meerwein-Straße, 35032 Marburg, Germany
Comptes Rendus. Mathématique, Volume 352 (2014) no. 10, pp. 835-840.

In this note we explain an analogy of moduli of canonically polarized varieties and of Calabi–Yau manifolds, when these are equipped with Kähler–Einstein forms. Given a holomorphic family f:X→S of canonically polarized varieties, the direct image sheaves Rn−qf⁎ΩX/Sp(KX/S) carry induced Hermitian metrics, whose curvatures enjoy similar properties. Due to the absence of a Torelli theorem, we construct a Finsler metric in the orbifold sense in order to conclude about the hyperbolicity of the moduli stack.

Dans cette note, nous expliquons une analogie entre les espaces de modules des variétés canoniquement polarisées et ceux des variétés de Calabi–Yau, lorsque celles-ci sont équipées de métriques de Kähler–Einstein. Étant donné une famille f:X→S de variétés canoniquement polarisées, les faisceaux images directes Rn−qf⁎ΩX/Sp(KX/S) possèdent des métriques hermitiennes induites, dont les tenseurs de courbure jouissent de propriétés analogues. En raison de l'absence de théorème de type Torelli, nous construisons une métrique de Finsler au sens orbifold afin de pouvoir conclure à l'hyperbolicité du champ de modules.

Received:
Accepted:
Published online:
DOI: 10.1016/j.crma.2014.08.008

Georg Schumacher 1

1 Fachbereich Mathematik und Informatik, Philipps-Universität Marburg, Lahnberge, Hans-Meerwein-Straße, 35032 Marburg, Germany 
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Georg Schumacher. Curvature properties for moduli of canonically polarized manifolds—An analogy to moduli of Calabi–Yau manifolds. Comptes Rendus. Mathématique, Volume 352 (2014) no. 10, pp. 835-840. doi : 10.1016/j.crma.2014.08.008. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2014.08.008/

[1] L. Ahlfors Some remarks on Teichmüller's space of Riemann surfaces, Ann. Math. (2), Volume 74 (1961), pp. 171-191

[2] L. Ahlfors Curvature properties of Teichmüller's space, J. Anal. Math., Volume 9 (1961), pp. 161-176

[3] J.-P. Demailly Algebraic criteria for Kobayashi hyperbolic projective varieties and jet differentials, 1995 (Lecture notes, Santa Cruz published online)

[4] P.A. Griffiths Periods of integrals on algebraic manifolds. III: Some global differential-geometric properties of the period mapping, Publ. Math. Inst. Hautes Études Sci., Volume 38 (1970), pp. 125-180

[5] K. Liu; X. Sun; S.T. Yau Good geometry on the curve moduli, Publ. Res. Inst. Math. Sci., Kyoto Univ., Volume 42 (2008), pp. 600-724

[6] G. Schumacher Harmonic maps of the moduli space of compact Riemann surfaces, Math. Ann., Volume 275 (1986), pp. 455-466

[7] G. Schumacher The curvature of the Petersson–Weil metric on the moduli space of Kähler–Einstein manifolds (V. Ancona et al., eds.), Complex Analysis and Geometry, University Series in Mathematics, Plenum Press, New York, 1993, pp. 339-354

[8] G. Schumacher Asymptotics of Kähler–Einstein metrics on quasi-projective manifolds and an extension theorem on holomorphic maps, Math. Ann., Volume 311 (1998), pp. 631-645

[9] G. Schumacher Positivity of relative canonical bundles and applications, Invent. Math., Volume 190 (2012), pp. 1-56

[10] G. Schumacher Positivity of relative canonical bundles for families of canonically polarized manifolds | arXiv

[11] G. Schumacher Curvature of higher direct images and applications | arXiv

[12] Y.-T. Siu Curvature of the Weil–Petersson metric in the moduli space of compact Kähler–Einstein manifolds of negative first Chern class, Aspects Math. E, Volume 9 (1986), pp. 261-298

[13] W.-K. To; S.K. Yeung Finsler Metrics and Kobayashi hyperbolicity of the moduli spaces of canonically polarized manifolds, Ann. Math. (2014) (in press)

[14] E. Viehweg; K. Zuo On the Brody hyperbolicity of moduli spaces for canonically polarized manifolds, Duke Math. J., Volume 118 (2003), pp. 103-150

[15] C. Voisin Théorie de Hodge et géométrie algébrique complexe, Société mathématique de France, Paris, 2002 (Cours spécialisés)

[16] A. Weil Œuvres scientifiques: Collected Papers, vol. II (1951–1964), Springer-Verlag, New York–Heidelberg–Berlin, 1980 (XII, 1958c, final report)

[17] S. Wolpert Chern forms and the Riemann tensor for the moduli space of curves, Invent. Math., Volume 85 (1986), pp. 119-145

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