On the Deformation Quantization of Affine Algebraic Varieties

Since the fundamental work of Bayen et al. [1] in the seventies, a lot of effort has been dedicated to show the existence of deformations of a Poisson manifold. Some landmarks in this way were the proof of the existence of differential star products for symplectic manifolds which was done independently by De Wilde and Lecomte [2] and Fedosov [5], using different constructions. It turned out that the star products on a symplectic manifold are classified, up to equivalence, by the de Rham cohomology H(M). Etingof and Kazhdan showed the existence of star products for another class of Poisson manifolds, the Poisson–Lie groups. Kontsevich gave the proof of existence and classification of star products on an arbitrary Poisson manifolds as a consequence of his formality theorem [7]. Tamarkin [9] gave another