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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 )

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    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 complexe
    Organisateurs de la rencontre : Greb, Daniel ; Höring, Andreas ; Toma, Matei
    Dates : 09/04/2018 - 13/04/2018
    Année de la rencontre : 2018
    URL Congrès : https://conferences.cirm-math.fr/1789.html

    Citation 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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