  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.
