Holomorphic Poisson Manifolds and Holomorphic Lie Algebroids

Abstract

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 which computes the holomorphic Poisson cohomology. A holomorphic Lie algebroid structure on a vector bundle $A\to X$ is shown to be equivalent to a matched pair of complex Lie algebroids $(T^{0,1}X,A^{1,0})$, in the sense of Lu. The holomorphic Lie algebroid cohomology of $A$ is isomorphic to the cohomology of the elliptic Lie algebroid $T^{0,1}X\bowtie A^{1,0}$. In the case when $(X,\pi )$ is a holomorphic Poisson manifold and $A=(T^{\ast }X{)}_{\pi }$, such an elliptic Lie algebroid coincides with the Dirac structure corresponding to the associated generalized complex structure of the holomorphic Poisson manifold.

29 pages, v2: paper split into two, part 1 of 2, v3: two references added, v4: final version to appear in International Mathematics Research Notices

References in corpus (3)

• Generalized complex structures and Lie brackets
• On the modular classes of Poisson-Nijenhuis manifolds
• Ehresmann doubles and Drinfel'd doubles for Lie algebroids and Lie bialgebroids

Cited by in corpus (1)

• Poisson sigma model on the sphere