2007/07/31 by Camille Laurent-Gengoux, Mathieu Stiénon, Mathieu Stienon +1 · 4 citations
Mathematics · Physics and Astronomy · #Advanced Topics in Algebra #Algebraic structures and combinatorial models #Homotopy and Cohomology in Algebraic Topology #hep-th #math-ph #math.DG #math.MP
paper · pdf · doi:10.1093/imrn/rnn088
published as International Mathematics Research Notices (2008) Vol. 2008 : article ID rnn088, 46 pages · 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
arxiv created 2008/07/14 · openalex publication_date 2008/08/21 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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 (T0,1X, A1,0), in the sense of Lu. The holomorphic Lie algebroid cohomology of A is isomorphic to the cohomology of the elliptic Lie algebroid T0,1X ⋈ 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.