Ce papier commence par des remarques générales concernant la suite exacte de Mayer-Vietoris, une version covariante de la cohomologie de Lichnerowicz-Poisson et la définition d’une suite spectrale de Serre-Hochschild associée. Ensuite, on considère le cas régulier et on étudie la cohomologie de Poisson à l’aide d’une bigraduation naturelle du complexe de cochaînes de Lichnerowicz. Si le feuilletage symplectique de la variété de Poisson est transversalement riemannien ou transversalement symplectique, la suite spectrale mentionnée ci-dessus est déterminée par les cohomologies de ce feuilletage le long des feuilles. Si de plus, la structure de Poisson est transversalement constante, la suite spectrale définit alors directement la cohomologie de Lichnerowicz-Poisson.
The paper begins with some general remarks which include the Mayer-Vietoris exact sequence, a covariant version of the Lichnerowicz-Poisson cohomology, and the definition of an associated Serre-Hochshild spectral sequence. Then we consider the regular case, and we discuss the Poisson cohomology by using a natural bigrading of the Lichnerowicz cochain complex. Furthermore, if the symplectic foliation of the Poisson manifold is either transversally Riemannian or transversally symplectic, the spectral sequence mentioned above is determined by the leafwise cohomologies of the foliation, and if, moreover, the Poisson structure is transversally constant, the spectral sequence defines the Lichnerowicz-Poisson cohomology in a straightforward manner.
@article{AIF_1990__40_4_951_0,
author = {Vaisman, Izu},
title = {Remarks on the {Lichnerowicz-Poisson} cohomology},
journal = {Annales de l'Institut Fourier},
pages = {951--963},
publisher = {Institut Fourier},
address = {Grenoble},
volume = {40},
number = {4},
year = {1990},
doi = {10.5802/aif.1243},
mrnumber = {92c:58155},
zbl = {0708.58010},
language = {en},
url = {http://www.numdam.org/articles/10.5802/aif.1243/}
}
TY - JOUR AU - Vaisman, Izu TI - Remarks on the Lichnerowicz-Poisson cohomology JO - Annales de l'Institut Fourier PY - 1990 SP - 951 EP - 963 VL - 40 IS - 4 PB - Institut Fourier PP - Grenoble UR - http://www.numdam.org/articles/10.5802/aif.1243/ DO - 10.5802/aif.1243 LA - en ID - AIF_1990__40_4_951_0 ER -
Vaisman, Izu. Remarks on the Lichnerowicz-Poisson cohomology. Annales de l'Institut Fourier, Tome 40 (1990) no. 4, pp. 951-963. doi : 10.5802/aif.1243. http://www.numdam.org/articles/10.5802/aif.1243/
[BV] and , Poisson algebras and Poisson manifolds, Pitman Research Notes in Math., 174, Longman Sci., Harlow and New York, 1988. | MR | Zbl
[BT] and , Differential forms in algebraic topology, Graduate Texts in Math., 82, Springer-Verlag, New York, Heidelberg, Berlin, 1982. | MR | Zbl
[E] , Sur la cohomologie feuilletée, Composition Math., 49 (1983), 195-215. | Numdam | MR | Zbl
[F] , Cohomology of infinite dimensional Lie algebras, Consultants Bureau, New York and London, 1986. | MR | Zbl
[G] , Variétés bistructurées et opérateurs de récursion, Ann. Inst. H. Poincaré, 43 (1985), 349-357. | Numdam | MR | Zbl
[H] , Poisson cohomology and quantization, J. für Reine und Angew. Math., 408 (1990), 57-113. | MR | Zbl
[K] , Crochet de Schouten — Nijenhuis et cohomologie, In : E. Cartan et les mathématiques d'aujourd'hui, Soc. Math. de France, Astérisque, hors série, (1985), 257-271. | Numdam | MR | Zbl
[KT] and , Foliations and metrics, Progress in Math., 32, Birkhäuser, Boston, 1983, 103-152. | MR | Zbl
[L] , Les variétés de Poisson et leurs algèbres de Lie associées, J. Diff. Geometry, 12 (1977), 253-300. | MR | Zbl
[LT] and , Hodge decomposition along the leaves of a Riemannian foliation, Preprint, Urbana-Illinois, 1989.
[V1] , Cohomology and differential forms, M. Dekker Inc., New York, 1973. | MR | Zbl
[V2] , On the geometric quantization of Poisson manifolds, Preprint, Haifa, 1990.
[VK] and , Poisson manifolds and their Schouten bracket, Funct. Analysis and its Applications, 22(1) (1988), 1-9. | Zbl
[X] , Poisson cohomology of regular Poisson manifolds, Preprint, Berkeley, 1990.
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