Caldararu's conjecture and Tsygan's formality

• 成果类型: Article
• 署名作者: Calaque, Damien; Rossi, Carlo A.; Van den Bergh, Michel
• 刊物名称: ANNALS OF MATHEMATICS
• ISSN/ISSBN: 0003-486X
• DOI: 10.4007/annals.2012.176.2.4
• 发表日期: 2012
• 页码: 865-923
• 关键词: deformation quantization; hochschild cohomology

摘要

In this paper we complete the proof of Caldararu's conjecture on the compatibility between the module structures on differential forms over polyvector fields and on Hochschild homology over Hochschild cohomology. In fact we show that twisting with the square root of the Todd class gives an isomorphism of precalculi between these pairs of objects. Our methods use formal geometry to globalize the local formality quasi-isomorphisms introduced by Kontsevich and Shoikhet. (The existence of the latter was conjectured by Tsygan.) We also rely on the fact - recently proved by the first two authors - that Shoikhet's quasi-isomorphism is compatible with cap products after twisting with a Maurer-Cartan element.