@KS1; @KS2].

It is well known that the semi-classical data attached to quantum real manifolds (i.e. star-algebras) are Poisson structures [@MR0496157; @MR0496158]. The cotangent bundle of a real Poisson manifold $(M,\pi)$ is endowed with a canonical Lie algebroid structure denoted by $(T^*M)_{\pi}$. This Lie algebroid structure plays a central role in Poisson geometry. For instance, the Lichnerowicz-Poisson cohomology is simply the Lie algebroid cohomology of $(T^*M)_{\pi}$ with trivial coefficients. Evens-Lu-Weinstein discovered a procedure for constructing a canonical module over a given Lie algebroid. With the canonical module of $(T^*M)_{\pi}$ at hand, they interpreted Koszul-Brylinski homology as a Lie algebroid cohomology. According to Kontsevich’s formality theorem and Tsygan’s chain formality theorem, the Hochschild cohomology and Hochschild homology of a star algebra are isomorphic to the Lichnerowicz-Poisson cohomology and Koszul-Brylinski homology of the underlying Poisson manifold.

In the context of complex geometry, the semiclassical data associated to quantum complex manifolds are solutions of the Maurer-Cartan equation in the derived global sections $R{\Gamma( X,\wedge^\bullet {T{\boldsymbol{X}}}[1])}$ of the sheaf of graded Lie algebras $\wedge^\bullet {T{\boldsymbol{X}}}[1]$ of polyvector fields on ${\boldsymbol{X}}$, which, according to Kontsevich’s formality theorem, classify the deformations of stacks of algebroids up to gauge transformations [@MR1855264; @Y1; @MR2348030]. More precisely, a Maurer-Cartan 