bracket) satisfying Leibniz's rule; then the function can be described afor a unique smooth bivector field . Conversely, given any smooth bivector field on , the same formula defines an almost Lie bracket that automatically obeys Leibniz's rule. Then the following integrability conditions are equivalent: satisfies the Jacobi identity (hence it is a Poisson bracket); satisfies (hence it a Poisson bivector); the map is a Lie algebra homomorphism, i.e. the Hamiltonian vector fields satisfy ; the graph defines a Dirac structure, i.e. a Lagrangian subbundle which is closed under the standard Courant bracket. A Poisson structure without any of the four requirements above is also called an almost Poisson structure. Holomorphic Poisson structures The definition of Poisson structure for real smooth manifolds can be also adapted to the complex case. A holomorphic Poisson manifold is a complex manifold whose sheaf of holomorphic functions is a sheaf of Poisson algebras. Equivalently, recall that a holomorphic bivector field on a complex manifold is a section such that . Then a holomorphic Poisson structure on is a holomorphic bivector field satisfying the equation . Holomorphic Poisson manifolds can be characterised also in terms of Poisson-Nijenhuis structures. Many results for real Poisson structures, e.g. regarding their integrability, extend also to holomorphic ones. Holomorphic Poisson structures appear naturally in the context of generalised complex structures: locally, any generalised complex manifold is the product of a symplectic manifold and a holomorphic Poisson manifold. Symplectic leaves A Poisson manifold is naturally partitioned into regularly immersed symplectic manifolds of possibly different dimensions, called its symplectic leaves. These arise as the maximal integral submanifolds of the completely integrable singular foliation spanned by the Hamiltonian vector fields. Rank of a Poisson structure Recall that any bivector field can be regarded as a skew homomorphism . The image consists therefore of the values of all Hamiltonian vector fields evaluated at every . The rank of at a point is the rank of the induced linear mapping . A point is called regular for a Poisson structure on if and only if the rank of is constant on an open neighborhood of ; otherwise, it is called a singular point. Regular points form an open dense subspace ; when , i.e. the map is of constant rank, the Poisson structure is called regular. Examples of regular Poisson structures include trivial and nondegenerate structures (see below). The regular case For a regular Poisson