Suppose that $X$ is a smooth manifold, whose $C^{\infty}$-functions we denote by $A$. Let $D_{poly}^*(A):=\bigoplus_{n\geq -1}Hom(A^{\otimes n+1},A)$ be the Lie algebra of polydifferential operators with the Hochschild differential and the Gerstenhaber bracket. A version of the Hochschild-Kostant-Rosenberg theorem shows the cohomology of $D_{poly}^*(A)$ is isomorphic to $\bigoplus_{n\geq -1}\wedge^{n+1}T_X$, polyvector fields. The graded vector space of polyvector fields is also a Lie algebra with the Schouten-Nijenhius bracket. However, the HKR-isomorphism is not a morphism of Lie algebras. The formality theorem of Kontsevich shows the HKR-isomorphism can be corrected to an $L_{\infty}$-morphism whose first term is the HKR-isomorphism. The upshot is this proves star-products on $C^{\infty}(X)$ correspond to formal Poisson structures.

In the article: M. de Wilde and P. B. A Lecomte: *An homotopy formula for the Hochschild cohomology*, Compositio Mathematica, tome 96, no. 1 (1995), the authors construct an explicit homotopy for the HKR-isomorphism. From this explicit homotopy the authors construct a star-product on $\frak g^*$, the dual of the Lie algebra $\frak g$. My question is the following: can one do the same thing for the general case of a Poisson manifold. In particular, is the explicit homotopy which induces Kontsevich's $L_{\infty}$-quasiisomorphism known? I believe that one has to exist since two $L_{\infty}$-algebras are quasiisomorphic if and only if they are homotopy equivalent. But, can you write down the explicit homotopy from the quasiisomorphism.