---
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.'
address:
- 'Tsinghua University, Department of Mathematics'
- 'Université Paris Diderot, Institut de Mathématiques de Jussieu'
- 'Pennsylvania State University, Department of Mathematics'
author:
- Zhuo Chen
- Mathieu Stiénon
- Ping Xu
title: |
    Geometry of Maurer-Cartan Elements\
    on Complex Manifolds
---

[^1]

[^2]

Introduction
============

Due to their close connection to mirror symmetry, noncommutative deformations of complex manifolds have recently generated increasing interest [@MR1855264; @Bondal]. Kashiwara-Kontsevich’s stacks of algebroids are one way of substantiating the abstract concept of quantum complex manifolds (or 