  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.
