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. relate these to the degeneracy divisor and rank loci bivector. In some good cases we compute local cohomology complexes. Kahlerian case, deduce a relation between multiplicity Hodge numbers manifold. also show that vanishing one is related tounobstructed deformations normalized with its induced structure.