Hochschild cohomology and Atiyah classes

Understand

In this paper we prove that on a smooth algebraic variety the HKR-morphism twisted by the square root of the Todd genus gives an isomorphism between the sheaf of poly-vector fields and the sheaf of poly-differential operators, both considered as derived Gerstenhaber algebras.

• In particular we obtain an isomorphism between Hochschild cohomology and the cohomology of poly-vector fields which is compatible with the Lie bracket and the cupproduct.
• The latter compatibility is an unpublished result by Kontsevich.
• Our proof is set in the framework of Lie algebroids and so applies without modification in much more general settings as well.