Hochschild cohomology and Atiyah classes

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 The latter compatibility is an unpublished result by Kontsevich.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.

Reference to work of Cattaneo, Felder and Willwacher added

References in corpus (7)

• The relative Riemann-Roch theorem from Hochschild homology
• A Counterexample to the Quantizability of Modules
• Twisted Deformation Quantization of Algebraic Varieties
• Weak quantization of Poisson structures
• Formality for Lie algebroids
• Koszul duality in deformation quantization and Tamarkin's approach to Kontsevich formality
• The Kontsevich weight of a wheel with spokes pointing outward

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