### 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 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, A^{1'0}), in the sense of Lu. The holomorphic Lie algebroid cohomology of Ais isomorphic to the cohomology of the elliptic Lie algebroid T^{0.1} X A^{1'0}. 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.

Original language | English (US) |
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Article number | rnn088 |

Journal | International Mathematics Research Notices |

Volume | 2008 |

Issue number | 1 |

DOIs | |

State | Published - Dec 1 2008 |

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### All Science Journal Classification (ASJC) codes

- Mathematics(all)

### Cite this

*International Mathematics Research Notices*,

*2008*(1), [rnn088]. https://doi.org/10.1093/imrn/rnn088