Orbifold generic semi-positivity: an application to families of canonically polarized manifolds
Annales de l'Institut Fourier, Volume 65 (2015) no. 2, p. 835-861

Let X be a normal projective manifold, equipped with an effective ‘orbifold’ divisor Δ, such that the pair (X,Δ) is log-canonical. We first define the notion of ‘orbifold cotangent bundle’ Ω 1 (X,Δ), 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: Ω 1 (X,Δ) is generically semi-positive if K X +Δ 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 Δ is a reduced divisor with simple normal crossings on X such that some tensor power of Ω 1 (X,Δ)=Ω X 1 (Log(Δ)) contains the injective image of a big line bundle, then K X +Δ 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.

Nous définissons la notion de ‘fibré cotangent orbifolde’ Ω 1 (X,Δ) pour une paire (X,Δ) 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 : Ω 1 (X,Δ) est génériquement semi-positif si K X +Δ 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 Δ est un diviseur réduit à croisements normaux simples sur X tel qu’une puissance tensorielle de Ω X 1 (Log(Δ)) contienne un fibré en droites ‘big’, alors K X +Δ 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.

DOI : https://doi.org/10.5802/aif.2945
Classification:  14D05,  14D22,  14E22,  14E30,  14J40,  32J25
Keywords: Orbifold cotangent bundle, generic semi-positivity, canonically polarised manifolds
@article{AIF_2015__65_2_835_0,
     author = {Campana, Fr\'ed\'eric and P\u aun, Mihai},
     title = {Orbifold generic semi-positivity: an application to families of canonically polarized manifolds},
     journal = {Annales de l'Institut Fourier},
     publisher = {Association des Annales de l'institut Fourier},
     volume = {65},
     number = {2},
     year = {2015},
     pages = {835-861},
     doi = {10.5802/aif.2945},
     language = {en},
     url = {http://www.numdam.org/item/AIF_2015__65_2_835_0}
}
Campana, Frédéric; Păun, Mihai. Orbifold generic semi-positivity: an application to families of canonically polarized manifolds. Annales de l'Institut Fourier, Volume 65 (2015) no. 2, pp. 835-861. doi : 10.5802/aif.2945. http://www.numdam.org/item/AIF_2015__65_2_835_0/

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