no. 4
Remarks on the Lichnerowicz-Poisson cohomology
Annales de l'Institut Fourier, Tome 40 (1990) no. 4, pp. 951-963.

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.

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     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},
     zbl = {0708.58010},
     mrnumber = {92c:58155},
     language = {en},
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}
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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. https://aif.centre-mersenne.org/articles/10.5802/aif.1243/

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