Holomorphic Poisson Structures and Groupoids

We study holomorphic Poisson manifolds, holomorphic Lie algebroids and holomorphic Lie groupoids 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 which 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 0,1X,A1,0), in the sense of Lu. The holomorphic Lie algebroid cohomology of A is isomorphic to the cohomology of the elliptic Lie algebroid T 0,1X ⊲⊳ 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. We also show that a holomorphic Lie algebroid is integrable if, and only if, its underlying real Lie algebroid is integrable. Thus the integrability criteria of Crainic-Fernandes do also apply in the holomorphic context without any modification. Research supported by the European Union through the FP6 Marie Curie R.T.N. ENIGMA (Contract number MRTN-CT-2004-5652). Research partially supported by NSF grants DMS-0306665 and DMS-0605725 & NSA grant H98230-06-1-0047 1