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Fano foliations 3 - Classification of Fano foliations of large index - lecture 4
Virtualconference
Authors : Araujo, Carolina (Author of the conference)
CIRM (Publisher )
14E30 - Minimal model program (Mori theory, extremal rays)
37F75 - Holomorphic foliations and vector fields
14M22 - Rationally connected varieties
Additional resources :
https://www.cirm-math.com/uploads/2/6/6/0/26605521/lecture3_public.pdf
https://www.cirm-math.com/uploads/2/6/6/0/26605521/problems.pdf
CIRM (Publisher )
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Abstract :
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
MSC Codes : 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
14E30 - Minimal model program (Mori theory, extremal rays)
37F75 - Holomorphic foliations and vector fields
14M22 - Rationally connected varieties
Additional resources :
https://www.cirm-math.com/uploads/2/6/6/0/26605521/lecture3_public.pdf
https://www.cirm-math.com/uploads/2/6/6/0/26605521/problems.pdf
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Information on the Video
Film maker : Hennenfent, GuillaumeLanguage : English Available date : 14/05/2020 Conference Date : 12/05/2020 Subseries : Research School arXiv category : Algebraic Geometry ; Commutative Algebra Mathematical Area(s) : Algebraic & Complex Geometry Format : MP4 (.mp4) - HD Video Time : 00:52:30 Targeted Audience : Researchers Download : https://videos.cirm-math.fr/2020-05-13_Arujo_Part2.mp4 
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Information on the Event
Event Title : Jean-Morlet Chair 2020 - Research School: Geometry and Dynamics of Foliations / Chaire Jean-Morlet 2020 - Ecole : Géométrie et dynamiques des feuilletagesEvent Organizers : Druel, Stéphane ; Pereira, Jorge Vitório ; Rousseau, Erwan Dates : 18/05/2020 - 22/05/2020 Event Year : 2020 Event URL : https://www.chairejeanmorlet.com/2251.html
Citation Data
DOI : 10.24350/CIRM.V.19632103Cite this video as: Araujo, Carolina (2020). Fano foliations 3 - Classification of Fano foliations of large index - lecture 4. CIRM. Audiovisual resource. doi:10.24350/CIRM.V.19632103 URI : http://dx.doi.org/10.24350/CIRM.V.19632103 |
See Also
- [Virtualconference] Hyperkähler structures on symplectic realizations of holomorphic Poisson surfaces / Author of the conference Mayrand, Maxence.
- [Virtualconference] MMP & foliations on 3-folds : applications / Author of the conference Svaldi, Roberto.
- [Virtualconference] Regular foliations on rationally connected manifolds / Author of the conference Figueredo, João Paulo.
- [Virtualconference] Boundedness of minimal partial du Val resolutions of canonical surface foliations / Author of the conference Chen, Yen-An.
- [Virtualconference] Topics on the Poisson varieties of dimension at least four / Author of the conference Okumura, Katsuhiko.
- [Virtualconference] A model-theoretic analysis of geodesic equations in negative curvature / Author of the conference Jaoui, Rémi.
- [Virtualconference] On symmetries of transversely projective foliations / Author of the conference Lo Bianco, Federico.
- [Virtualconference] Higher Ramanujan foliations / Author of the conference Jardim Da Fonseca, Tiago.
- [Virtualconference] Dynamics of Jouanolou foliation / Author of the conference Deroin, Bertrand.
Bibliography
- 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
- KOLLÁR, János, MIYAOKA, Yoichi, MORI, Shigefumi, et al. Rational connectedness and boundedness of Fano manifolds. Journal of Differential Geometry, 1992, vol. 36, no 3, p. 765-779. - http://dx.doi.org/10.4310/jdg/1214453188
- Kobayashi, Shoshichi; Ochiai, Takushiro. Characterizations of complex projective spaces and hyperquadrics. J. Math. Kyoto Univ. 13 (1973), no. 1, 31--47 - http://dx.doi.org/10.1215/kjm/1250523432
- MUKAI, Shigeru. Fano 3—f0lds. Complex Projective Geometry: Selected Papers, 1992, vol. 179, p. 255. - https://doi.org/10.1017/CBO9780511662652.018
- Araujo, C., Druel, S. & Kovács, S. Cohomological characterizations of projective spaces and hyperquadrics. Invent. math. 174, 233 (2008) - http://dx.doi.org/10.1007/s00222-008-0130-1
- Wahl, J.M. A cohomological characterization of ℙn . Invent Math 72, 315–322 (1983). - http://dx.doi.org/10.1007/BF01389326
- CHO, Koji, MIYAOKA, Yoichi, SHEPHERD-BARRON, Nicholas I., et al. Characterizations of projective space and applications to complex symplectic manifolds. In : Higher dimensional birational geometry. Mathematical Society of Japan, 2002. p. 1-88. - http://dx.doi.org/10.2969/aspm/03510001
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