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H 1 Improved bounds for the Kobayashi conjecture on generic hyperbolicity

Auteurs : Demailly, Jean-Pierre (Auteur de la Conférence)
CIRM (Editeur )

Résumé : A famous conjecture of Kobayashi from the 1970s asserts that a generic algebraic hypersurface of sufficiently large degree $d\geq d_n$ in the complex projective space of dimension $n+1$ is hyperbolic. Yum-Tong Siu introduced several fundamental ideas that led recently to a proof of the conjecture. In 2016, Damian Brotbek gave a new geometric argument based on the use of Wronskian operators and on an analysis of the geometry of Semple jet bundles. Shortly afterwards, Ya Deng obtained effective degree bounds by means of a refined technique. Our goal here will be to explain a drastically simpler proof that yields an improved (though still non optimal) degree bound, e.g. $d_n=[(en)^{2n+2}/5]$. We will also present a more general approach that could possibly lead to optimal bounds.

Keywords : Kobayashi hyperbolic variety; directed manifold; genus of a curve; jet bundle; jet differential; jet metric; Chern connection and curvature; negativity of jet curvature; variety of general type; Kobayashi conjecture; Green-Griffiths conjecture; Lang conjecture

Codes MSC :
14J40 - Algebraic $n$-folds ($n>4$)
32L10 - Sheaves and cohomology of sections of holomorphic vector bundles, general results
32Q45 - Hyperbolic and Kobayashi hyperbolic manifolds
53C55 - Hermitian and Kählerian manifolds (global differential geometry)

 Informations sur la Vidéo Langue : Anglais Date de publication : 12/04/2018 Date de captation : 11/04/2018 Collection : Research talks ; Algebraic and Complex Geometry Format : MP4 Durée : 00:53:55 Domaine : Algebraic & Complex Geometry Audience : Chercheurs ; Doctorants , Post - Doctorants Download : https://videos.cirm-math.fr/2018-04-11_Demailly.mp4 Informations sur la rencontre Nom de la rencontre : French-German meeting on complex algebraic geometry / Rencontre franco-allemande en géométrie algébrique complexeOrganisateurs de la rencontre : Greb, Daniel ; Höring, Andreas ; Toma, MateiDates : 09/04/2018 - 13/04/2018 Année de la rencontre : 2018 URL Congrès : https://conferences.cirm-math.fr/1789.htmlCitation Data DOI : 10.24350/CIRM.V.19388003 Cite this video as: Demailly, Jean-Pierre (2018). Improved bounds for the Kobayashi conjecture on generic hyperbolicity. CIRM. Audiovisual resource. doi:10.24350/CIRM.V.19388003 URI : http://dx.doi.org/10.24350/CIRM.V.19388003

### Voir aussi

Bibliographie

1. Demailly, J.-P. (2018). Recent results on the Kobayashi and Green-Griffiths-Lang conjectures. - https://arxiv.org/abs/1801.04765

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