TY - JOUR

T1 - Holomorphic Poisson manifolds and holomorphic Lie algebroids

AU - Laurent-Gengoux, Camille

AU - Stiénon, Mathieu

AU - Xu, Ping

PY - 2008

Y1 - 2008

N2 - 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.

AB - 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.

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U2 - 10.1093/imrn/rnn088

DO - 10.1093/imrn/rnn088

M3 - Article

AN - SCOPUS:70350153960

VL - 2008

JO - International Mathematics Research Notices

JF - International Mathematics Research Notices

SN - 1073-7928

IS - 1

M1 - rnn088

ER -