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H 1 Fano foliations 1 - Definition, examples and first properties - lecture 2

Auteurs : Araujo, Carolina (Auteur de la Conférence)
CIRM (Editeur )

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 : birational geometry; Fano varieties; holomorphic foliations

Ressources complémentaires :

#### MINIMAL MODEL PROGRAM

 Informations sur la Vidéo Réalisateur : Hennenfent, Guillaume Langue : Anglais Date de publication : 14/05/2020 Date de captation : 11/05/2020 Collection : Research School ; Algebraic and Complex Geometry Durée : 00:41:24 Domaine : Algebraic & Complex Geometry Audience : Chercheurs ; Doctorants , Post - Doctorants Download : https://videos.cirm-math.fr/2020-05-11_Arujo_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 feuilletagesOrganisateurs de la rencontre : Druel, Stéphane ; Pereira, Jorge Vitório ; Rousseau, ErwanDates : 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.19632003 Cite this video as: Araujo, Carolina (2020). Fano foliations 1 - Definition, examples and first properties - lecture 2.CIRM . Audiovisual resource. doi:10.24350/CIRM.V.19632003 URI : http://dx.doi.org/10.24350/CIRM.V.19632003

### 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. LORAY, Frank, PEREIRA, Jorge Vitório, et TOUZET, Frédéric. Singular foliations with trivial canonical class. Inventiones mathematicae, 2018, vol. 213, no 3, p. 1327-1380. - https://doi.org/10.1007/s00222-018-0806-0

5. CERVEAU, Dominique et NETO, A. Lins. Irreducible components of the space of holomorphic foliations of degree two in CP (n), n≥ 3. Annals of mathematics, 1996, p. 577-612. - http://dx.doi.org/10.2307/2118537

6. 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

7. Araujo, C., Druel, S. & Kovács, S. Cohomological characterizations of projective spaces and hyperquadrics. Invent. math. 174, 233 (2008) - https://doi.org/10.1007/s00222-008-0130-1

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