The Van den Bergh duality and the modular symmetry of a Poisson variety

Abstract.

We consider a smooth Poisson affine variety with the trivial canon-ical bundle over ${\mathbb{C}}$. For such a variety the deformation quantization algebra $A_\hbar$ obeys the conditions of the Van den Bergh duality theorem and the corresponding dualizing module is determined by an outer automorphism of $A_\hbar$ intrinsic to $A_\hbar$. We show how this automorphism can be expressed in terms of the modular class of the corresponding Poisson variety. We also prove that the Van den Bergh dualizing module of the deformation quantization algebra $A_\hbar$ is free if and only if the corresponding Poisson structure is unimodular.