Geometry of Maurer-Cartan Elements on Complex Manifolds

Zhuo Chen Tsinghua University Mathieu Stienon Penn State University Ping Xu Penn State University

Differential Geometry mathscidoc:1702.10003

Communications in Mathematical Physics, 297, (1), 169-187, 2010
The semi-classical data attached to stacks of algebroids in the sense of Kashiwara and Kontsevich are Maurer-Cartan elements on complex manifolds, which we call extended Poisson structures as they generalize holomorphic Poisson structures. A canonical Lie algebroid is associated to each Maurer-Cartan element. We study the geometry underlying these Maurer-Cartan elements in the light of Lie algebroid theory. In particular, we extend Lichnerowicz-Poisson cohomology and Koszul-Brylinski homology to the realm of extended Poisson manifolds; we establish a sufficient criterion for these to be finite dimensional; we describe how homology and cohomology are related through the Evens-Lu-Weinstein duality module; and we describe a duality on Koszul-Brylinski homology, which generalizes the Serre duality of Dolbeault cohomology.
Complex Manifolds, Extended Poisson structures, Maurer-Cartan Elements, Evens-Lu-Weinstein duality module
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  title={Geometry of Maurer-Cartan Elements on Complex Manifolds},
  author={Zhuo Chen, Mathieu Stienon, and Ping Xu},
  booktitle={Communications in Mathematical Physics},
Zhuo Chen, Mathieu Stienon, and Ping Xu. Geometry of Maurer-Cartan Elements on Complex Manifolds. 2010. Vol. 297. In Communications in Mathematical Physics. pp.169-187.
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