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

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