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H 1 Invariance of plurigenera for foliations on surfaces

Auteurs : Floris, Enrica (Auteur de la Conférence)
CIRM (Editeur )

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    Résumé : Let $X$ be a smooth algebraic surface. A foliation $F$ on $X$ is, roughly speaking, a subline bundle $T_F$ of the tangent bundle of $X$. The dual of $T_F$ is called the canonical bundle of the foliation $K_F$. In the last few years birational methods have been successfully used in order to study foliations. More precisely, geometric properties of the foliation are translated into properties of the canonical bundle of the foliation. One of the most important invariants describing the properties of a line bundle $L$ is its Kodaira dimension $\kappa(L)$, which measures the growth of the global sections of $L$ and its tensor powers. The Kodaira dimension of a foliation $F$ is defined as the Kodaira dimension of its canonical bundle $\kappa(K_F)$. In their fundamental works, Brunella and McQuillan give a classfication of foliations on surfaces on the model of Enriques-Kodaira classification of surfaces. The next step is the study of the behaviour of families of foliations. Brunella proves that, for a family of foliations $(X_t, F_t)$ of dimension one on surfaces, satisfying certain hypotheses of regularity, the Kodaira dimension of the foliation does not depend on $t$. By analogy with Siu's Invariance of Plurigenera, it is natural to ask whether for a family of foliations $(X_t, F_t)$ the dimensions of global sections of the canonical bundle and its powers depend on $t$. In this talk we will discuss to which extent an Invariance of Plurigenera for foliations is true and under which hypotheses on the family of foliations it holds.

    Codes MSC :
    14E30 - Minimal model program (Mori theory, extremal rays)
    14J10 - Families, moduli, classification: algebraic theory
    53C12 - Foliations (differential geometric aspects) [See also 57R30, 57R32]

      Informations sur la Vidéo

      Langue : Anglais
      Date de publication : 03/12/14
      Date de captation : 25/11/14
      Collection : Research talks ; Algebraic and Complex Geometry
      Format : quicktime ; audio/x-aac
      Durée : 00:45:40
      Domaine : Algebraic & Complex Geometry
      Audience : Chercheurs ; Doctorants , Post - Doctorants
      Download : https://videos.cirm-math.fr/2014-11-25_Floris.mp4

    Informations sur la rencontre

    Nom de la rencontre : Algebraic geometry and complex geometry / Géométrie algébrique et géométrie complexe
    Organisateurs de la rencontre : Broustet, Amaël ; Pasquier, Boris
    Dates : 24/11/14 - 28/11/14
    Année de la rencontre : 2014
    URL Congrès : http://math.univ-lille1.fr/~broustet/GAG...

    Citation Data

    DOI : 10.24350/CIRM.V.18641003
    Cite this video as: Floris, Enrica (2014). Invariance of plurigenera for foliations on surfaces. CIRM. Audiovisual resource. doi:10.24350/CIRM.V.18641003
    URI : http://dx.doi.org/10.24350/CIRM.V.18641003


    Bibliographie

    1. Brunella, M. (2001). Invariance par déformations de la dimension de Kodaira d'un feuilletage sur une surface. In E. Ghys, P. de la Harpe, V.F.R. Jones, V. Sergiescu & T. Tsuboi (Eds.), Essays on geometry and related topics (pp. 113-132). Genève : L'Enseignement mathématique. (Monographie de L'Enseignement mathématique, 38) -

    2. Brunella, M. (2000). Birational geometry of foliations. Rio de Janeiro: Instituto de Matemática Pura e Aplicada (IMPA). -

    3. Brunella, M. (1997). Feuilletages holomorphes sur les surfaces complexes compactes. Annales Scientifiques de l'École Normale Supérieure, 30(5), 569-594 - https://eudml.org/doc/82443

    4. Brunella, M. (1999). Courbes entières et feuilletages holomorphes. L'Enseignement Mathématique, 45(1-2), 195-216 -

    5. Kodaira, K. (1963). On stability of compact submanifolds of complex manifolds. American Journal of Mathematics, 85(1), 79-94 - http://dx.doi.org/10.2307/2373187

    6. Kodaira, K. (1964). On the structure of compact complex analytic surfaces, I. American Journal of Mathematics, 86(4), 751-798 - http://dx.doi.org/10.2307/2373157

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