Complex Poisson Manifolds Bronis\l aw Przybylski Abstract. We discuss some elementary properties of complex Poisson manifolds which are introduced by analogy to (real) Poisson ones. The properties are formulated in the language of sheaf theory and concern holomorphic geometrical objects. In particular, we speak of Hamiltonian morphisms which should be characteristic of a future formulation of complex Hamiltonian mechanics. Moreover, we consider the canonical complex symplectic structure on the holomorphic cotangent manifold of a complex manifold which can be regarded as a complex analogue of the real symplectic structure of a phase space. Keywords. Complex Poisson manifolds, Hamiltonian morphisms, conjugate pairs. MS classification. 32C10, 58F05.