Hochschild cohomology and Atiyah classes

By Damien Calaque, Michel van den Bergh. 2007

103 citations recorded by the source corpus.

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 The latter compatibility is an unpublished result by Kontsevich.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.

Subfields: Algebra and Number Theory, Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/a21d1e76926bce3bdafd9ac25205336618461512

Citations in the corpus

• Hodge theory and derived categories of cubic fourfolds (2012, unknown): “Firstly [14, 13], it intertwines the action (6.8) of HH ∗ ( Y ) on HH ∗ ( Y ) with the interior multiplication of H ∗ (Λ ∗ T Y ) on H ∗ (Ω ∗ Y ).”