Differential complexes and Hodge theory on log-symplectic manifolds

We study certain complexes of differential forms, including reverse de Rham complexes, on (real or complex) Poisson manifolds, especially holomorphic log-symplectic ones. We relate these to the degeneracy divisor and rank loci of the Poisson bivector. In some good holomorphic cases we compute the local cohomology of these complexes. In the Kahlerian case, we deduce a relation between the multiplicity loci of the degeneracy divisor and the Hodge numbers of the manifold. We also show that vanishing of one of these Hodge numbers is related tounobstructed deformations of the normalized degeneracy divisor with its induced Poisson structure.