Orbifold generic semi-positivity: an application to families of canonically polarized manifolds

[Semi-positivité orbifolde : une application aux familles de variétés canoniquement polarisées]

Campana, Frédéric ; Păun, Mihai

Annales de l'Institut Fourier, Tome 65 (2015), p. 835-861 / Harvested from Numdam

Résumé

Nous définissons la notion de ‘fibré cotangent orbifolde’ $\Omega ^1(X, \Delta )$ pour une paire $(X,\Delta )$ log-canonique : ce fibré est défini sur des revêtement cycliques adéquats. Nous formulons et démontrons ensuite une version orbifolde du théorème de semi-positivité générique de Y. Miyaoka : $\Omega ^1(X, \Delta )$ est génériquement semi-positif si $K_X+\Delta $ est pseudo-effectif. Nous en déduisons, à l’aide des résultats récents du PMML, un énoncé conjecturé par E. Viehweg : si $X$ est lisse, et si $\Delta $ est un diviseur réduit à croisements normaux simples sur $X$ tel qu’une puissance tensorielle de $\Omega ^1_X(Log(\Delta ))$ contienne un fibré en droites ‘big’, alors $K_X+\Delta $ est lui-même ‘big’. Les travaux de Viehweg-Zuo impliquent alors la conjecture d’hyperbolicité de V.I. Shafarevich : si une famille algébrique de variétés projectives canoniquement polarisées et paramétrée par une variété quasi-projective irréductible lisse $B$ a une ‘variation’ maximale, égale à $dim(B)$, alors $B$ est de type log-général.

Abstract

Let $X$ be a normal projective manifold, equipped with an effective ‘orbifold’ divisor $\Delta $, such that the pair $(X, \Delta )$ is log-canonical. We first define the notion of ‘orbifold cotangent bundle’ $\Omega ^1(X, \Delta )$, living on any suitable ramified cover of $X$. We are then in position to formulate and prove (in a completely different way) an orbifold version of Y. Miyaoka’s generic semi-positivity theorem: $\Omega ^1(X, \Delta )$ is generically semi-positive if $K_X+ \Delta $ is pseudo-effective. Using the deep results of the LMMP, we immediately get a statement conjectured by E. Viehweg: if $X$ is smooth, and if $\Delta $ is a reduced divisor with simple normal crossings on $X$ such that some tensor power of $\Omega ^1(X, \Delta )=\Omega ^1_X(Log (\Delta ))$ contains the injective image of a big line bundle, then $K_X+ \Delta $ is big.

This implies, by fundamental results of Viehweg-Zuo, the ‘Shafarevich-Viehweg hyperbolicity conjecture’: if an algebraic family of canonically polarized manifolds parametrised by a quasi-projective manifold $B$ has ‘maximal variation’, then $B$ is of log-general type.

Détail

Publié le : 2015-01-01
DOI : https://doi.org/10.5802/aif.2945
Classification:  14D05,  14D22,  14E22,  14E30,  14J40,  32J25
Mots clés: Fibré cotangent orbifolde, semi-positivité générique, variétés canoniquement polarisées

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