Semilocal properties of modular classes of Poisson manifolds

Velasco-Barreras Eduardo

We study the behavior of the modular class of an orientable Poisson manifold and formulate some unimodularity criteria in the semilocal context, around a possibly singular symplectic leaf. Our results give a generalization to the singular case of the well-known results of A. Abouqateb and M. Boucetta for regular Poisson manifolds related to the notion of the Reeb class. We show that the unimodularity of the transverse Poisson structure of the leaf is a necessary condition for the semilocal unimodularity property. Furthermore, we introduce a generalized Reeb class, which takes values in the first cohomology of the De Rham - Casimr complex associated to the (singular) symplectic leaf and gives an obstruction to the unimodularity.