Comptes Rendus
no. 10
Algebraic geometry/Analytic geometry
Curvature properties for moduli of canonically polarized manifolds—An analogy to moduli of Calabi–Yau manifolds
[Propriétés de courbure des modules des variétés canoniquement polarisées—une analogie avec les modules des variétés de Calabi–Yau]

Présenté par : 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.

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.

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.

Reçu le :
Accepté le :
Publié le :
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/

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