2007/08/31 by Damien Calaque, Michel Van den Bergh · 1 citation
Mathematics · #math.KT #math.QA
paper · pdf · doi:10.1016/j.aim.2010.01.012
published as Advances in Mathematics 224 (2010), no. 5, 1839--1889 · Reference to work of Cattaneo, Felder and Willwacher added
arxiv created 2007/11/02 · arxiv updated 2010/10/06
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 The latter compatibility is an unpublished result by Kontsevich.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.