We study holomorphic Poisson manifolds and holomorphic Lie algebroids from the viewpoint of real Poisson geometry. We give a characterization of holomorphic Poisson structures in terms of the Poisson Nijenhuis structures of Magri-Morosi and describe a double complex that computes the holomorphic Poisson cohomology. A holomorphic Lie algebroid structure on a vector bundle A → X is shown to be equivalent to a matched pair of complex Lie algebroids (T 0.1 X, A1'0), in the sense of Lu. The holomorphic Lie algebroid cohomology of Ais isomorphic to the cohomology of the elliptic Lie algebroid T0.1 X A1'0. In the case when (X, π) is a holomorphic Poisson manifold and A = (T* X)π, such an elliptic Lie algebroid coincides with the Dirac structure corresponding to the associated generalized complex structure of the holomorphic Poisson manifold.
All Science Journal Classification (ASJC) codes