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Holomorphic Poisson structures - lecture 4
Virtualconference
Authors : Pym, Brent (Author of the conference)
CIRM (Publisher )
14J10 - Families, moduli, classification: algebraic theory
37F75 - Holomorphic foliations and vector fields
53D17 - Poisson manifolds; Poisson groupoids and algebroids
Additional resources :
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 (Publisher )
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Abstract :
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
MSC Codes : 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
Additional resources :
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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Information on the Video
Film maker : Hennenfent, GuillaumeLanguage : English Available date : 06/05/2020 Conference Date : 04/05/2020 Subseries : Research School arXiv category : Algebraic Geometry ; Spectral Theory ; Mathematical Physics Mathematical Area(s) : Algebraic & Complex Geometry Format : MP4 (.mp4) - HD Video Time : 00:42:26 Targeted Audience : Researchers Download : https://videos.cirm-math.fr/2020-04-28_Pym_Part4.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.19630803Cite this video as: 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 |
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
- 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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