[Propriétés de courbure des modules des variétés canoniquement polarisées—une analogie avec les modules des variétés de Calabi–Yau]
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 de variétés canoniquement polarisées, les faisceaux images directes 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 of canonically polarized varieties, the direct image sheaves 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.
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@article{CRMATH_2014__352_10_835_0,
author = {Schumacher, Georg},
title = {Curvature properties for moduli of canonically polarized {manifolds{\textemdash}An} analogy to moduli of {Calabi{\textendash}Yau} manifolds},
journal = {Comptes Rendus. Math\'ematique},
pages = {835--840},
publisher = {Elsevier},
volume = {352},
number = {10},
year = {2014},
doi = {10.1016/j.crma.2014.08.008},
language = {en},
url = {http://www.numdam.org/articles/10.1016/j.crma.2014.08.008/}
}
TY - JOUR AU - Schumacher, Georg TI - Curvature properties for moduli of canonically polarized manifolds—An analogy to moduli of Calabi–Yau manifolds JO - Comptes Rendus. Mathématique PY - 2014 SP - 835 EP - 840 VL - 352 IS - 10 PB - Elsevier UR - http://www.numdam.org/articles/10.1016/j.crma.2014.08.008/ DO - 10.1016/j.crma.2014.08.008 LA - en ID - CRMATH_2014__352_10_835_0 ER -
%0 Journal Article %A Schumacher, Georg %T Curvature properties for moduli of canonically polarized manifolds—An analogy to moduli of Calabi–Yau manifolds %J Comptes Rendus. Mathématique %D 2014 %P 835-840 %V 352 %N 10 %I Elsevier %U http://www.numdam.org/articles/10.1016/j.crma.2014.08.008/ %R 10.1016/j.crma.2014.08.008 %G en %F CRMATH_2014__352_10_835_0
Schumacher, Georg. Curvature properties for moduli of canonically polarized manifolds—An analogy to moduli of Calabi–Yau manifolds. Comptes Rendus. Mathématique, Tome 352 (2014) no. 10, pp. 835-840. doi : 10.1016/j.crma.2014.08.008. http://www.numdam.org/articles/10.1016/j.crma.2014.08.008/
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