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

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Cited by in corpus (40)

• From Atiyah Classes to Homotopy Leibniz Algebras
• Poisson Manifolds, Lie Algebroids, Modular Classes: a Survey
• Generalized Kahler geometry
• Lie theory of multiplicative tensors
• Poisson sigma model on the sphere
• Stable generalized complex structures
• Cohomology of skew-holomorphic Lie algebroids
• Matched pairs of Courant algebroids
• Exponential map and $L_{\infty }$ algebra associated to a Lie pair
• Foliated groupoids and their infinitesimal data
• Poisson structures on almost complex Lie algebroids
• On holomorphic maps and Generalized Complex Geometry
• Lie algebroid cohomology as a derived functor
• Local classification of generalized complex structures
• Holomorphic Koszul-Brylinski Homology
• Poisson Cohomology of holomorphic toric Poisson manifolds. I
• Invariant connections and PBW theorem for Lie groupoid pairs
• Unobstructed deformations of generalized complex structures induced by $C^{\infty }$ logarithmic symplectic structures and logarithmic Poisson structures
• Glanon groupoids
• Dirac geometry and integration of Poisson homogeneous spaces
• Dorfman connections and Courant algebroids
• Moser Lemma in Generalized Complex Geometry
• From Hypercomplex to Holomorphic Symplectic Structures
• On the local and global classification of generalized complex structures
• Connections in holomorphic Lie algebroids
• Real Forms of Holomorphic Hamiltonian Systems
• Hyperkahler metrics near Lagrangian submanifolds and symplectic groupoids
• On generalized Kähler geometry on compact Lie groups
• Foliated Lie and Courant Algebroids
• Degeneracy of Holomorphic Poisson Spectral Sequence
• The Atiyah class of generalized holomorphic vector bundles
• On almost complex Lie algebroids
• Quasi-classical generalized CRF structures
• Linear generalised complex structures
• A heterotic sigma model with novel target geometry
• Stability of fixed points in Poisson geometry and higher Lie theory
• Courant-Nijenhuis algebroids
• Frobenius Structures and Generalized Deformation of Kodaira Manifolds
• On $\mathcal{O}$-operators on modules over Lie algebras
• Holomorphic Poisson Structures and its Cohomology on Nilmanifolds