Holomorphic Poisson Manifolds and Holomorphic Lie Algebroids

TL;DR

This paper explores the structure of holomorphic Poisson manifolds and Lie algebroids, providing new characterizations, cohomology computations, and connections to generalized complex structures within complex differential geometry.

Contribution

It introduces a characterization of holomorphic Poisson structures via Poisson Nijenhuis structures and links holomorphic Lie algebroids to elliptic Lie algebroids, advancing the understanding of their cohomology.

Findings

Holomorphic Poisson structures characterized by Poisson Nijenhuis structures.

Holomorphic Lie algebroid cohomology is isomorphic to elliptic Lie algebroid cohomology.

Elliptic Lie algebroid coincides with Dirac structure in holomorphic Poisson manifolds.

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 $A=(T^*X)_\pi$, such an elliptic Lie algebroid coincides with the Dirac structure corresponding to the associated generalized complex structure of the holomorphic Poisson manifold.