On the geometric quantization of the symplectic leaves of poisson manifolds

In the paper, we establish some conditions which ensure one of the following: (i) the existence of the pullback of the quantization bundle of a Poisson manifold to a quantization bundle of a symplectic leaf, (ii) the existence of the projection of a quantization bundle from a presymplectic realization of a Poisson manifold to the manifold or to its symplectic leaves. The main case is that of an isotropic realization. The paper ends by a discussion of the notion of a polarization of a Poisson manifold.