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H 1 Fano foliations 0 - Algebraicity of smooth formal schemes and applications to foliations - lecture 1

Auteurs : Druel, Stéphane (Auteur de la Conférence)
CIRM (Editeur )

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    Résumé : In the last few decades, much progress has been made in birational algebraic geometry. The general viewpoint is that complex projective manifolds should be classified according to the behavior of their canonical class. As a result of the minimal model program (MMP), every complex projective manifold can be built up from 3 classes of (possibly singular) projective varieties, namely, varieties $X$ for which $K_X$ satisfies $K_X<0$, $K_X\equiv 0$ or $K_X>0$. Projective manifolds $X$ whose anti-canonical class $-K_X$ is ample are called Fano manifolds.

    Techniques from the MMP have been successfully applied to the study of global properties of holomorphic foliations. This led, for instance, to Brunella's birational classification of foliations on surfaces, in which the canonical class of the foliation plays a key role. In recent years, much progress has been made in higher dimensions. In particular, there is a well developed theory of Fano foliations, i.e., holomorphic foliations $F$ on complex projective varieties with ample anti-canonical class $-K_F$.

    This mini-course is devoted to reviewing some aspects of the theory of Fano Foliations, with a special emphasis on Fano foliations of large index. We start by proving a fundamental algebraicity property of Fano foliations, as an application of Bost's criterion of algebraicity for formal schemes. We then introduce and explore the concept of log leaves. These tools are then put together to address the problem of classifying Fano foliations of large index.

    Keywords : algebraic geometry; foliations

    Codes MSC :
    37F75 - Holomorphic foliations and vector fields


    Ressources complémentaires :
    https://www.cirm-math.com/uploads/2/6/6/0/26605521/algebraization.pdf
    https://www.cirm-math.com/uploads/2/6/6/0/26605521/problems.pdf

    FANO FOLIATIONS - Druel, Stéphane; Aroujo, Carolina

     

     

    CODIMENSION ONE FOLIATIONS WITH PSEUDO-EFFECTIVE CONORMAL BUNDLE - Touzet Frédéric

     

     

    HOLOMORPHIC POISSON STRUCTURES - Pym, Brent

     

     

    COMPLETE HOLOMORPHIC VECTOR FIELDS AND THEIR SINGULAR POINTS - Guillot, Adolfo

     

     

    MINIMAL MODEL PROGRAM - Cascini, Paolo; Spicer, Calum

     

     


      Informations sur la Vidéo

      Réalisateur : Hennenfent, Guillaume
      Langue : Anglais
      Date de publication : 06/05/2020
      Date de captation : 30/04/2020
      Collection : Research School ; Algebraic and Complex Geometry
      Durée : 00:42:55
      Domaine : Algebraic & Complex Geometry
      Audience : Chercheurs ; Doctorants , Post - Doctorants
      Download : https://videos.cirm-math.fr/2020-04-30_Druel_Part1.mp4

    Informations sur la Rencontre Virtuelle

    Nom de la rencontre : Jean-Morlet Chair 2020 - Research School: Geometry and Dynamics of Foliations / Chaire Jean-Morlet 2020 - Ecole : Géométrie et dynamiques des feuilletages
    Organisateurs de la rencontre : Druel, Stéphane ; Pereira, Jorge Vitório ; Rousseau, Erwan
    Dates : 18/05/2020 - 22/05/2020
    Année de la rencontre : 2020
    URL Congrès : https://www.cirm-math.com/virtual-event-2251.html

    Citation Data

    DOI : 10.24350/CIRM.V.19630603
    Cite this video as: Druel, Stéphane (2020). Fano foliations 0 - Algebraicity of smooth formal schemes and applications to foliations - lecture 1.CIRM . Audiovisual resource. doi:10.24350/CIRM.V.19630603
    URI : http://dx.doi.org/10.24350/CIRM.V.19630603


    Voir aussi

    Bibliographie

    1. Araujo Carolina, Druel Stéphane: Characterization of generic projective space bundles and algebraicity of foliations. Comment. Math. Helv. 94 (2019), 833-853 - http://dx.doi.org/10.4171/CMH/475

    2. ARAUJO, Carolina et DRUEL, Stéphane. On Fano foliations 2. In : Foliation Theory in Algebraic Geometry. Springer, Cham, 2016. p. 1-20. - https://doi.org/10.1007/978-3-319-24460-0_1

    3. ARAUJO, Carolina et DRUEL, Stéphane. On fano foliations. Advances in Mathematics, 2013, vol. 238, p. 70-118. - https://doi.org/10.1016/j.aim.2013.02.003

    4. CAMPANA, Frédéric et PĂUN, Mihai. Foliations with positive slopes and birational stability of orbifold cotangent bundles. Publications mathématiques de l'IHÉS, 2019, vol. 129, no 1, p. 1-49. - https://doi.org/10.1007/s10240-019-00105-w

    5. BOGOMOLOV, Fedor et MCQUILLAN, Michael. Rational curves on foliated varieties. In : Foliation theory in algebraic geometry. Springer, Cham, 2016. p. 21-51. - http://dx.doi.org/10.1007/978-3-319-24460-0_2

    6. Bost, Jean-Benoît. Algebraic leaves of algebraic foliations over number fields. Publications Mathématiques de l'IHÉS, Tome 93 (2001) , pp. 161-221 - http://www.numdam.org/item/PMIHES_2001__93__161_0/

    7. Bost, J.-B.; Germs of analytic varieties in algebraic varieties: canonical metrics and arithmetic algebraization theorems ii. In: Adolphson, A., Baldassarri, F., Berthelot, P., Katz, N., Loeser, F. (eds.) Geometric Aspects of Dwork Theory, vol. I. Walter de
      Gruyter II, Berlin (2004). -

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