Let be a normal projective manifold, equipped with an effective ‘orbifold’ divisor , such that the pair is log-canonical. We first define the notion of ‘orbifold cotangent bundle’ , living on any suitable ramified cover of . 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: is generically semi-positive if is pseudo-effective. Using the deep results of the LMMP, we immediately get a statement conjectured by E. Viehweg: if is smooth, and if is a reduced divisor with simple normal crossings on such that some tensor power of contains the injective image of a big line bundle, then 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 has ‘maximal variation’, then is of log-general type.
Nous définissons la notion de ‘fibré cotangent orbifolde’ pour une paire 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 : est génériquement semi-positif si est pseudo-effectif. Nous en déduisons, à l’aide des résultats récents du PMML, un énoncé conjecturé par E. Viehweg : si est lisse, et si est un diviseur réduit à croisements normaux simples sur tel qu’une puissance tensorielle de contienne un fibré en droites ‘big’, alors 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 a une ‘variation’ maximale, égale à , alors est de type log-général.
Keywords: Orbifold cotangent bundle, generic semi-positivity, canonically polarised manifolds
Mot clés : Fibré cotangent orbifolde, semi-positivité générique, variétés canoniquement polarisées
Campana, Frédéric 1; Păun, Mihai 2
@article{AIF_2015__65_2_835_0, author = {Campana, Fr\'ed\'eric and P\u{a}un, Mihai}, title = {Orbifold generic semi-positivity: an application to families of canonically polarized manifolds}, journal = {Annales de l'Institut Fourier}, pages = {835--861}, publisher = {Association des Annales de l{\textquoteright}institut Fourier}, volume = {65}, number = {2}, year = {2015}, doi = {10.5802/aif.2945}, language = {en}, url = {https://aif.centre-mersenne.org/articles/10.5802/aif.2945/} }
TY - JOUR AU - Campana, Frédéric AU - Păun, Mihai TI - Orbifold generic semi-positivity: an application to families of canonically polarized manifolds JO - Annales de l'Institut Fourier PY - 2015 SP - 835 EP - 861 VL - 65 IS - 2 PB - Association des Annales de l’institut Fourier UR - https://aif.centre-mersenne.org/articles/10.5802/aif.2945/ DO - 10.5802/aif.2945 LA - en ID - AIF_2015__65_2_835_0 ER -
%0 Journal Article %A Campana, Frédéric %A Păun, Mihai %T Orbifold generic semi-positivity: an application to families of canonically polarized manifolds %J Annales de l'Institut Fourier %D 2015 %P 835-861 %V 65 %N 2 %I Association des Annales de l’institut Fourier %U https://aif.centre-mersenne.org/articles/10.5802/aif.2945/ %R 10.5802/aif.2945 %G en %F 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. https://aif.centre-mersenne.org/articles/10.5802/aif.2945/
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