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We study the behavior of the modular class of an orientable Poisson manifoldand formulate some unimodularity criteria in the semilocal context, around a(singular) symplectic leaf. Our results generalize some known unimodularitycriteria for regular Poisson manifolds related to the notion of the Reeb class.In particular, we show that the unimodularity of the transverse Poissonstructure of the leaf is a necessary condition for the semilocal unimodularproperty. Our main tool is an explicit formula for a bigraded decomposition ofmodular vector fields of a coupling Poisson structure on a foliated manifold.Moreover, we also exploit the notion of the modular class of a Poissonfoliation and its relationship with the Reeb class.

Unimodularity Criteria for Poisson Structures on Foliated Manifolds