Poisson cohomology of regular Poisson manifolds
Annales de l'Institut Fourier, Volume 42 (1992) no. 4, p. 967-988

The main purpose of this paper is to suggest a method of computing Poisson cohomology of a Poisson manifold by means of symplectic groupoids. The key idea is to convert the problem of computing Poisson cohomology to that of computing de Rham cohomology of certain manifolds. In particular, we shall derive an explicit formula for the Poisson cohomology of a regular Poisson manifold where the symplectic foliation is a trivial fibration.

Le but principal de cet article est de proposer une méthode pour calculer la cohomologie de Poisson d’une variété de Poisson par l’utilisation de groupoïdes symplectiques.

L’idée clé est de se ramener à calculer la cohomologie de Rham de certaines variétés. En particulier nous en déduisons une formule pour la cohomologie de Poisson d’une variété de Poisson régulière dont la feuilletage symplectique est trivial.

@article{AIF_1992__42_4_967_0,
     author = {Xu, Ping},
     title = {Poisson cohomology of regular Poisson manifolds},
     journal = {Annales de l'Institut Fourier},
     publisher = {Imprimerie Louis-Jean},
     address = {Gap},
     volume = {42},
     number = {4},
     year = {1992},
     pages = {967-988},
     doi = {10.5802/aif.1317},
     zbl = {0759.58020},
     mrnumber = {94d:58167},
     language = {en},
     url = {http://www.numdam.org/item/AIF_1992__42_4_967_0}
}
Xu, Ping. Poisson cohomology of regular Poisson manifolds. Annales de l'Institut Fourier, Volume 42 (1992) no. 4, pp. 967-988. doi : 10.5802/aif.1317. http://www.numdam.org/item/AIF_1992__42_4_967_0/

[BT] R. Bott, L.W. Tu, Differential forms in algebraic topology, Springer-Verlag, (1981).

[Br] J.-L. Brylinski, A differential complex for Poisson Manifolds, J. Diff. Geom., 28 (1988), 93-114. | MR 89m:58006 | Zbl 0634.58029

[CDW] A. Coste, P. Dazord, A. Weinstein, Groupoïdes symplectiques, Publications du Départment de Mathématiques, Université Claude Bernard Lyon I, (1987). | Numdam | MR 90g:58033 | Zbl 0668.58017

[D1] P. Dazord, Groupoïdes symplectiques et troisième théorème de Lie “non linéaire”, Lecture Notes in Mathematics, vol. 1416 (1990), 39-74. | MR 91i:58169 | Zbl 0702.58023

[D2] P. Dazord, Réalisations isotropes de Libermann, Publ. Dept. Math. Lyon, (1989).

[DD] P. Dazord, and T. Delzant, le problème general des variables actions angles, J. Diff. Geom., 26 (1987), 223-251. | MR 88j:58032 | Zbl 0634.58003

[F] D.B. Fuks, Cohomology of infinite-dimensional Lie algebras, Contemporary Soviet Mathematics, Consultants Bureau (1986). | MR 88b:17001 | Zbl 0667.17005

[Ka] M.V. Karasev, Analogues of objects of the theory of Lie groups for nonlinear Poisson brackets, Math. USSR Izvestiya, 28 (1987), 497-527. | Zbl 0624.58007

[H] J. Huebschmann, Poisson cohomology and quantization, J. reine angew. Math., 408 (1990), 57-113. | MR 92e:17027 | Zbl 0699.53037

[L] A. Lichnerowicz, Les variétés de Poisson et leurs algebres de Lie associées, J. Diff. Geom., 12 (1977), 253-300. | MR 58 #18565 | Zbl 0405.53024

[M] K. Mackenzie, Lie groupoids and Lie algebroids in differential geometry ; LMS lecture Notes Series, 124 Cambridge Univ. Press, (1987). | MR 89g:58225 | Zbl 0683.53029

[V] I. Vaisman, Remarks on the Licherowicz-Poisson cohomology, Ann. Inst. Fourier, Grenoble, 40, 4 (1990), 951-963. | Numdam | MR 92c:58155 | Zbl 0708.58010

[VK1] Yu. M. Vorob'Ev, M.V. Karasev, Corrections to classical dynamics and quantization conditions which arise in the deformation of Poisson brackets, Dokl. Akad. Nauk USSR, 247, No. 6 (1987), 1294-1298. | Zbl 0676.58026

[VK2] Yu. M. Vorob'Ev, M.V. Karasev, Poisson manifolds and the Schouten Bracket, Functional Analysis and its Applications, Vol. 22, No. 1 (1988), 1-9. | MR 89k:58011 | Zbl 0667.58018

[W1] A. Weinstein, The local structure of Poisson manifolds, J. Diff. Geom., 18 (1983), 523-557. | MR 86i:58059 | Zbl 0524.58011

[W2] A. Weinstein, Symplectic groupoids and Poisson manifolds, Bull. Amer. Math. Soc., 16 (1987), 101-104. | MR 88c:58019 | Zbl 0618.58020

[WX] A. Weinstein, P. Xu, Extensions of symplectic groupoids and quantization, J. reine angew. Math., 417 (1991), 159-189. | MR 92k:58094 | Zbl 0722.58021