Hochschild cohomology and Atiyah classes

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• 作者：Damien Calaque ; Michel Van den Bergh
• 关键词：Deformation quantization ; Atiyah classes
• 刊名：Advances in Mathematics
• 出版年：2010
• 出版时间：1 August 2010
• 年：2010
• 卷：224
• 期：5
• 页码：1839-1889
• 全文大小：469 K

文摘

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 The latter compatibility is an unpublished result by Kontsevich.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.