Hochschild cohomology and Atiyah classes

TL;DR

This paper establishes a deep isomorphism between Hochschild cohomology and poly-vector fields on smooth algebraic varieties, utilizing the HKR-morphism twisted by the Todd genus, with broad applicability via Lie algebroid frameworks.

Contribution

It proves that the HKR-morphism twisted by the Todd genus induces an isomorphism between Hochschild cohomology and poly-vector fields, compatible with algebraic structures, in a general Lie algebroid setting.

Findings

Isomorphism between Hochschild cohomology and poly-vector fields.

Compatibility of the isomorphism with Lie bracket and cup product.

Applicability to a broad class of geometric settings.

Abstract

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 The latter compatibility is an unpublished result by Kontsevich.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.