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Poisson cohomology and canonical homology of Poisson manifolds. (English) Zbl 0870.53026
The paper gives an account of the recent development of the cohomology theory of Poisson manifolds.
In the first part the authors discuss the relationship between the Lichnerowicz-Poisson cohomology and the deRham cohomology of a Poisson manifold, resp. between their coeffective version.
The second part of the paper is devoted to the canonical cohomology of a Poisson manifold. The authors recall the problem of finding suitable conditions for the existence of symplectically harmonic representatives of deRham cohomology classes (Problem A). Then they introduce two spectral sequences related to the canonical cohomology and discuss the problem of the degeneracy of one of them (Problem B). In the last paragraph the authors show the independence of Problems A and B.

##### MSC:
 53C15 General geometric structures on manifolds (almost complex, almost product structures, etc.) 58A14 Hodge theory in global analysis 37J99 Dynamical aspects of finite-dimensional Hamiltonian and Lagrangian systems