English
Holomorphic Poisson structures - lecture 4
Virtualconference
Auteurs : Pym, Brent (Auteur de la Conférence)
CIRM (Editeur )
14J10 - Families, moduli, classification: algebraic theory
37F75 - Holomorphic foliations and vector fields
53D17 - Poisson manifolds; Poisson groupoids and algebroids
Ressources complémentaires :
https://www.cirm-math.com/uploads/2/6/6/0/26605521/2020-cirm_poisson_4.pdf
https://www.cirm-math.com/uploads/2/6/6/0/26605521/2020-cirm_poisson_discussion.pdf
CIRM (Editeur )
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Résumé :
The notion of a Poisson manifold originated in mathematical physics, where it is used to describe the equations of motion of classical mechanical systems, but it is nowadays connected with many different parts of mathematics. A key feature of any Poisson manifold is that it carries a canonical foliation by even-dimensional submanifolds, called its symplectic leaves. They correspond physically to regions in phase space where the motion of a particle is trapped.
I will give an introduction to Poisson manifolds in the context of complex analytic/algebraic geometry, with a particular focus on the geometry of the associated foliation. Starting from basic definitions and constructions, we will see many examples, leading to some discussion of recent progress towards the classification of Poisson brackets on Fano manifolds of small dimension, such as projective space.
Keywords : Poisson bracket; holomorphic foliation; algebraic variety
Codes MSC : I will give an introduction to Poisson manifolds in the context of complex analytic/algebraic geometry, with a particular focus on the geometry of the associated foliation. Starting from basic definitions and constructions, we will see many examples, leading to some discussion of recent progress towards the classification of Poisson brackets on Fano manifolds of small dimension, such as projective space.
Keywords : Poisson bracket; holomorphic foliation; algebraic variety
14J10 - Families, moduli, classification: algebraic theory
37F75 - Holomorphic foliations and vector fields
53D17 - Poisson manifolds; Poisson groupoids and algebroids
Ressources complémentaires :
https://www.cirm-math.com/uploads/2/6/6/0/26605521/2020-cirm_poisson_4.pdf
https://www.cirm-math.com/uploads/2/6/6/0/26605521/2020-cirm_poisson_discussion.pdf
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Informations sur la Vidéo
Réalisateur : Hennenfent, GuillaumeLangue : Anglais Date de publication : 06/05/2020 Date de captation : 04/05/2020 Sous collection : Research School arXiv category : Algebraic Geometry ; Spectral Theory ; Mathematical Physics Domaine : Algebraic & Complex Geometry Format : MP4 (.mp4) - HD Durée : 00:42:26 Audience : Researchers Download : https://videos.cirm-math.fr/2020-04-28_Pym_Part4.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 Congrès : https://www.chairejeanmorlet.com/2251.html
Données de citation
DOI : 10.24350/CIRM.V.19630803Citer cette vidéo: Pym, Brent (2020). Holomorphic Poisson structures - lecture 4. CIRM. Audiovisual resource. doi:10.24350/CIRM.V.19630803 URI : http://dx.doi.org/10.24350/CIRM.V.19630803 |
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
- POLISHCHUK, A. Algebraic geometry of Poisson brackets. Journal of Mathematical Sciences, 1997, vol. 84, no 5, p. 1413-1444 - https://doi.org/10.1007/BF02399197
- PYM, Brent. Constructions and classifications of projective Poisson varieties. Letters in mathematical physics, 2018, vol. 108, no 3, p. 573-632 - https://doi.org/10.1007/s11005-017-0984-5
- DUFOUR, Jean-Paul et ZUNG, Nguyen Tien. Poisson structures and their normal forms. Springer Science & Business Media, 2006. - https://doi.org/10.1007/b137493
- LAURENT-GENGOUX, Camille, PICHEREAU, Anne, et VANHAECKE, Pol. Poisson structures. Springer Science & Business Media, 2012 - https://doi.org/10.1007/978-3-642-31090-4
- BONDAL, Alexey I. Non-commutative deformations and Poisson brackets on projective spaces. Max-Planck-Institut für Mathematik, 1993. - http://www.mi-ras.ru/~akuznet/math/Bondal%20Non-commutative%20deformations%20and%20Poisson%20brackets%20on%20projective%20spaces.pdf
- LORAY, Frank, PEREIRA, Jorge Vitorio, et TOUZET, Frédéric. Singular foliations with trivial canonical class. arXiv preprint arXiv:1107.1538, 2011. - https://arxiv.org/abs/1107.1538
- 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
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