Twisted Poincar\'{e} duality between Poisson homology and Poisson cohomology

TL;DR

This paper establishes a generalized twisted Poincaré duality between Poisson homology and cohomology for polynomial Poisson algebras, extending previous results to arbitrary Poisson modules and structures.

Contribution

It introduces a canonical twisting of Poisson modules derived from the modular derivation, broadening the scope of Poincaré duality in Poisson geometry.

Findings

Proves twisted Poincaré duality for polynomial Poisson algebras with arbitrary modules.

Reduces to classical Poincaré duality in unimodular cases.

Generalizes previous duality results for quadratic and linear Poisson structures.

Abstract

A version of the twisted Poincar\'{e} duality is proved between the Poisson homology and cohomology of a polynomial Poisson algebra with values in an arbitrary Poisson module. The duality is achieved by twisting the Poisson module structure in a canonical way, which is constructed from the modular derivation. In the case that the Poisson structure is unimodular, the twisted Poincar\'{e} duality reduces to the Poincar\'{e} duality in the usual sense. The main result generalizes the work of Launois-Richard \cite{LR} for the quadratic Poisson structures and Zhu \cite{Zhu} for the linear Poisson structures.