Last multipliers for multivectors with applications to Poisson geometry

The theory of the last multipliers as solutions of the Liouville’s transport equation, previously developed for vector fields, is extended here to general multivectors. Characterizations in terms of Witten and Marsden differentials are reobtained as well as the algebraic structure of the set of multivectors with a common last multiplier, namely Gerstenhaber algebra. Applications to Poisson bivectors are presented by obtaining that last multipliers count for ‘‘how far away’’ is a Poisson structure from being exact with respect to a given volume form. The notion of exact Poisson cohomology for an unimodular Poisson structure on $I{R}^n$ is introduced.