Geometry of Maurer-Cartan Elements on Complex Manifolds
Abstract
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.
Cite
@article{arxiv.0904.4062,
title = {Geometry of Maurer-Cartan Elements on Complex Manifolds},
author = {Zhuo Chen and Mathieu Stienon and Ping Xu},
journal= {arXiv preprint arXiv:0904.4062},
year = {2017}
}
Comments
final version to appear in Comm. Math. Phys