  We study the ``twisted" Poincar\'e duality of smooth Poisson manifolds, and
show that, if the modular vector field is diagonalizable, then there is a mixed
complex associated to the Poisson complex, which, combining with the twisted
Poincar\'e duality, gives a Batalin-Vilkovisky algebra structure on the Poisson
cohomology. This generalizes the previous results obtained by Xu for unimodular
Poisson manifolds. We also show that the Batalin-Vilkovisky algebra structure
is preserved under Kontsevich's deformation quantization, and in the case of
polynomial algebras it is also preserved by Koszul duality.
