  In this paper, we show that the twisted Poincar\'e duality between Poisson
homology and cohomology can be derived from the Serre invertible bimodule. This
gives another definition of a unimodular Poisson algebra in terms of its
Poisson Picard group. We also achieve twisted Poincar\'e duality for Hochschild
(co)homology of Poisson bimodules using rigid dualizing complex. For a smooth
Poisson affine variety with the trivial canonical bundle, we prove that its
enveloping algebra is a Calabi-Yau algebra if the Poisson structure is
unimodular.
