Almost Homogeneous Poisson Spaces

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