  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.
