Kontsevich's graph complex, GRT, and the deformation complex of the sheaf of polyvector fields

• 成果类型: Article
• 署名作者: Dolgushev, V. A.; Rogers, C. L.; Willwacher, T. H.
• 刊物名称: ANNALS OF MATHEMATICS
• ISSN/ISSBN: 0003-486X
• DOI: 10.4007/annals.2015.182.3.2
• 发表日期: 2015
• 页码: 855-943
• 关键词: mixed tate motives; complete intersections; differential forms; formality theorem; lie-algebras; COHOMOLOGY; operads; QUANTIZATION; coefficients; conjecture

摘要

We generalize Kontsevich's construction of L-infinity-derivations of polyvector fields from the affine space to an arbitrary smooth algebraic variety. More precisely, we construct a map (in the homotopy category) from Kontsevich's graph complex to the deformation complex of the sheaf of polyvector fields on a smooth algebraic variety. We show that the action of Deligne-Drinfeld elements of the Grothendieck-Teichmuller Lie algebra on the cohomology of the sheaf of polyvector fields coincides with the action of odd components of the Chern character. Using this result, we deduce that the (A) over cap -genus in the Calaque-Van den Bergh formula for the isomorphism between harmonic and Hochschild structures can be replaced by a generalized (A) over cap -genus.