Batalin-Vilkovisky algebra structure on Poisson manifolds with diagonalizable modular symmetry

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.