Geometry of Maurer-Cartan Elements on Complex Manifolds

Details Geometry of Maurer-Cartan Elements on Complex Manifolds Geometry of Maurer-Cartan Elements on Complex Manifolds

2009-04-26

arxiv.org/abs/0904.4062v2

Comm. Math. Phys. 297 (2010), no. 1, 169-187

Mathematics

Differential Geometry

final version to appear in Comm. Math. Phys

Scientific paper

10.1007/s00220-010-1029-4

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.

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