  We study the "twisted" Poincar\'e duality of smooth Poisson manifolds, and
show that, if the modular symmetry is semisimple, that is, the modular vector
is diagonalizable, 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, and a gravity
algebra structure on the negative cyclic Poisson homology. This generalizes the
previous results obtained by Xu et al for unimodular Poisson algebras. We also
show that these two algebraic structures are preserved under Kontsevich's
deformation quantization, and in the case of polynomial algebras they are also
preserved by Koszul duality.
