Homological properties of certain Generalized Jacobian Poisson Structures in dimension 3

The unimodularity condition for a Poisson structure (i.e., a Poisson structure with a trivial modular class) induces a Poincaré duality between its Poisson homology and its Poisson cohomology. Therefore, information about the Poisson homology of this kind Poisson structures induces by duality information about its Poisson cohomology and vise versa. However, it is not longer true in the case of a non-trivial modular class, which is the case of Generalized Jacobian Poisson Structures (GJPS). In this paper, certain GJPS are considered in dimension 3 and the properties of their Poisson homological groups and their Poisson cohomological groups are obtained. More precisely, under some assumptions, the Poincaré series of these Poisson homological groups are obtained, and these Poisson cohomological groups are computed explicitly, except the second group, which seems to be more complicated to obtain.

► The unimodularity condition for a Poisson structure (i.e., a Poisson structure with a trivial modular class) induces a Poincaré duality between its Poisson homology and its Poisson cohomology. ► Therefore, information about the Poisson homology of this kind Poisson structures induces by duality information about its Poisson cohomology and vise versa. ► However, it is not longer true in the case of a non-trivial modular class, which is the case of Generalized Jacobian Poisson Structures (GJPS). ► In this paper, certain GJPS are considered in dimension 3 and the properties of their Poisson homological groups and their Poisson cohomological groups are obtained. ► More precisely, under some assumptions, the Poincaré series of these Poisson homological groups are obtained, and these Poisson cohomological groups are computed explicitly, except the second group, which seems to be more complicated to obtain.