Holonomy on Poisson manifolds and the modular class

The linear holonomy of a Poisson structure, introduced in the present paper , generalizes the linearized holonomy of a regular symplectic foliation. Fo r singular Poisson structures the linear holonomy is defined for the lifts of tangential paths to the cotangent bundle. The linear holonomy is closely related to the modular class. Namely. the logarithm of the determinant of the linear holonomy is equal to the integral of the modular vector field al ong such a lift. This assertion relies on the notion of the integral of a v ector field along a cotangent path un a Poisson manifold, which is also int roduced in the paper. We then prove that for locally unimodular Poisson manifolds the modular cla ss is an invariant of Morita equivalence.