Séminaires : Séminaire d'Analyse et Géométrie

Equipe(s) :
Responsables :O. Biquard, A. Deruelle, I. Itenberg, X. Ma
Email des responsables : {olivier.biquard, alix.deruelle, ilia.itenberg, xiaonan.ma}@imj-prg.fr
Salle : 15–25.502
Adresse :Jussieu

Orateur(s) Songyan Xie - Université de Paris-Sud,
Titre On the ampleness of the cotangent bundles of complete intersections
Horaire14:00 à 15:00
RésumeWe present the proof of the Debarre Ampleness Conjecture: The cotangent bundle of the intersection X =$H_ 1 $ ∩⋯∩$H_c$ of 𝑐⩾N/2 generic hypersurfaces $H_i$ in ℂℙ$_N$ of high degrees $d_1$,...,$d_c$ ≫1 is ample.
First of all, we provide a geometric interpretation of symmetric differential forms in projective spaces. Thereby, we construct Brotbek’s symmetric differential forms on X, where the defining hypersurfaces $H_ 1 $,...,$H_c$ are generalized Fermat-type. Moreover, we exhibit unveiled families of lower degree symmetric differential forms on all possible intersections of X with coordinate hyperplanes. Thereafter, we introduce what we call the ‘moving coefficients method’ and the ‘product coup’ to settle the Debarre Ampleness Conjecture. In addition, we obtain an effective lower degree bound: $d_1$,...,$d_c$ ⩾$N^(N^2)$.
This talk is based on our first paper available on arXiv: http://arxiv.org/abs/1510.06323. If time allows, we will also talk about the further developments in our second paper: http://arxiv.org/abs/1601.05133.