A Formality Theorem for Poisson Manifolds

Let M be a differential manifold. Using different methods, Kontsevich and Tamarkin have proved a formality theorem, which states the existence of a Lie homomorphism 'up to homotopy' between the Lie algebra of Hochschild cochains on C∞(M) and its cohomology (Γ(M, ΛTM), [−, −]S). Suppose M is a Poisson manifold equipped with a Poisson tensor π; then one can deduce from this theorem the existence of a star product \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$\star$$ \end{document} on C∞(M). In this Letter we prove that the formality theorem can be extended to a Lie (and even Gerstenhaber) homomorphism 'up to homotopy' between the Lie (resp. Gerstenhaber 'up to homtoptopy') algebra of Hochschild cochains on the deformed algebra (C∞(M), *) and the Poisson complex (Γ(M, ΛTM), [π, −]S). We will first recall Tamarkin's proof and see how the formality maps can be deduced from Etingof and Kazhdan's theorem using only homotopies formulas. The formality theorem for Poisson manifolds will then follow.