Polarized Poisson structures

We study various properties of polarized Poisson structure subordinate to the polarized symplectic structure.We study also the notion of a polarized Poisson manifold, i.e., a Poisson manifold foliated by coisotropic submanifolds. We show that the characteristic distribution of a polarized Poisson structure is completely integrable and its leaves are symplectic. In the particular case when the foliation is Lagrangian, we show that, the polarized Poisson manifold is also foliated by polarized symplectic leaves, and we prove Darboux’s theorem corresponding to a Lagrangian foliation with respect to a polarized Poisson manifold.