English
Fano foliations 1 - Definition, examples and first properties - lecture 2
Virtualconference
Auteurs : Araujo, Carolina (Auteur de la conférence)
CIRM (Editeur )
14E30 - Minimal model program (Mori theory, extremal rays)
37F75 - Holomorphic foliations and vector fields
14M22 - Rationally connected varieties
Ressources complémentaires :
https://www.cirm-math.com/uploads/2/6/6/0/26605521/lecture1_public.pdf
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.
Mots-Clés : birational geometry; Fano varieties; holomorphic foliations
Codes MSC : 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.
Mots-Clés : birational geometry; Fano varieties; holomorphic foliations
14E30 - Minimal model program (Mori theory, extremal rays)
37F75 - Holomorphic foliations and vector fields
14M22 - Rationally connected varieties
Ressources complémentaires :
https://www.cirm-math.com/uploads/2/6/6/0/26605521/lecture1_public.pdf
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Informations sur la Vidéo
Réalisateur : Hennenfent, GuillaumeLangue : Anglais Date de Publication : 14/05/2020 Date de Captation : 11/05/2020 Sous Collection : Research School Catégorie arXiv : Algebraic Geometry Domaine(s) : Géométrie Complexe & géométrie Algébrique Format : MP4 (.mp4) - HD Durée : 00:41:24 Audience : Chercheurs Download : https://videos.cirm-math.fr/2020-05-11_Arujo_Part1.mp4 
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Informations sur la Rencontre
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, Erwan Dates : 18/05/2020 - 22/05/2020 Année de la rencontre : 2020 URL de la Rencontre : https://www.chairejeanmorlet.com/2251.html
Données de citation
DOI : 10.24350/CIRM.V.19632003Citer cette vidéo: 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
- [Virtualconference] Hyperkähler structures on symplectic realizations of holomorphic Poisson surfaces / Auteur de la conférence Mayrand, Maxence.
- [Virtualconference] MMP & foliations on 3-folds : applications / Auteur de la conférence Svaldi, Roberto.
- [Virtualconference] Regular foliations on rationally connected manifolds / Auteur de la conférence Figueredo, João Paulo.
- [Virtualconference] Boundedness of minimal partial du Val resolutions of canonical surface foliations / Auteur de la conférence Chen, Yen-An.
- [Virtualconference] Topics on the Poisson varieties of dimension at least four / Auteur de la conférence Okumura, Katsuhiko.
- [Virtualconference] A model-theoretic analysis of geodesic equations in negative curvature / Auteur de la conférence Jaoui, Rémi.
- [Virtualconference] On symmetries of transversely projective foliations / Auteur de la conférence Lo Bianco, Federico.
- [Virtualconference] Higher Ramanujan foliations / Auteur de la conférence Jardim Da Fonseca, Tiago.
- [Virtualconference] Dynamics of Jouanolou foliation / Auteur de la conférence Deroin, Bertrand.
Bibliographie
- 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
- 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
- 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
- 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
- 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
- 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
- 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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