THE CANONICAL SPECTRAL SEQUENCES FOR POISSON MANIFOLDS-ASTERISK

For a compact symplectic manifold M of dimension 2n, Brylinski proved that the canonical homology group H-k(can)(M) is isomorphic to the de Rham cohomology group H2n-k(M), and the first spectral sequence {E-r(M )} degenerates at E-1(M). In this paper, we show that these isomorphis ms do not exist for an arbitrary Poisson manifold. More precisely, we exhibit an example of a five-dimensional compact Poisson manifold M-5 for which H-1(can)(M-5) is not isomorphic to H-4(M-5), and E-1(M-5) is not isomorphic to E-2(M-5).