Almost homogeneous Poisson manifolds

We prove that any connected holomorphic Poisson manifold has an open and dense symplectic leaf which is a pseudo-Kähler submanifold, and we define a new obstruction to study the equivariance of momentum map for tangential Poisson action. Some properties of almost homogeneous Poisson manifolds are studied and we prove that any compact symplectic Poisson homogeneous space is a torus bundle over a dressing orbit.