Some structures on polarized Poisson manifolds

The present paper shows that every polarized Poisson manifold admits a torsion-free connection that preserves the foliation. Moreover, each leaf of this foliation carries a flat, torsion-free connection, which equips it with a locally affine structure. Besides, we prove Darboux's theorem for a coisotropic foliation with respect to a polarized Poisson manifold.