Poisson cohomology, Koszul duality, and Batalin-Vilkovisky algebras

We study the noncommutative Poincar\'e duality between the Poisson homology and cohomology of unimodular Poisson algebras, and show that Kontsevich's deformation quantization as well as Koszul duality preserve the corresponding Poincar\'e duality. As a corollary, the Batalin-Vilkovisky algebra structures that naturally arise in these cases are all isomorphic.