Geometry of Maurer-Cartan Elements on Complex Manifolds

TL;DR

This paper explores the geometry of Maurer-Cartan elements on complex manifolds, introducing extended Poisson structures and extending classical cohomology theories to this new setting.

Contribution

It generalizes holomorphic Poisson structures to extended Poisson structures and develops associated Lie algebroid theory and duality results.

Findings

Extended Lichnerowicz-Poisson cohomology and Koszul-Brylinski homology are established.

Finite dimensionality criteria for these (co)homologies are provided.

A duality between homology and cohomology is described, generalizing Serre duality.

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.