Almost complex Poisson manifolds
After some generalities about almost complex structures and complex manifolds, the authors define an almost complex Poisson structure, essentially by carrying over the definition of a complex Poisson structure by Lichnerowicz from a complex manifold to an almost complex one. Such a structure gives rise to two real Poisson structures. The most interesting of the examples which are given in the same section concerns the situation of a linear connection on a manifold \(M\), which gives rise to an almost complex structure \(J\) on \(TM\), from which almost complex Poisson structures can be constructed, which are not complex Poisson. It is further shown that almost complex Poisson manifolds carry a generalized foliation whose leaves are complex symplectic manifolds. Section 4 is about almost complex Poisson morphisms. The concepts of \(J\)-coisotropic and \(J\)-Lagrangian submanifolds are introduced, leading to appropriate generalizations of the classical results which relate Lagrangian submanifolds to canonical transformations and locally Hamiltonian vector fields. The last three sections are rather related to complex Poisson manifolds. Section 5 deals with holomorphic Lie algebroid structures and aspects of cohomology. Regular complex Poisson structures and foliated complex symplectic forms are discussed in Section 6. The final section is about complex Lie group actions on complex Poisson manifolds.
scientific article; zbMATH DE number 1530052