Caldararu's conjecture and Tsygan's formality

Abstract

In this paper we complete the proof of Caldararu's conjecture on the compatibility between the module structures on differential forms over poly-vector 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 product after twisting with a Maurer-Cartan element.

48 pages, 2 figures; a number of additional explanations added (suggested by the referees)

References in corpus (7)

• Hochschild cohomology and Atiyah classes
• The Atiyah class, Hochschild cohomology and the Riemann-Roch theorem
• Formality of the homotopy calculus algebra of Hochschild (co)chains
• Compatibility with cap-products in Tsygan's formality and homological Duflo isomorphism
• Formality for Lie algebroids
• Hochschild cohomology for Lie algebroids
• Shoikhet's Conjecture and Duflo Isomorphism on (Co)Invariants

Cited by in corpus (10)

• Hodge theory and derived categories of cubic fourfolds
• The Homotopy Braces Formality Morphism
• Integral Transforms and Deformations of K3 Surfaces
• Compatibility with cap-products in Tsygan's formality and homological Duflo isomorphism
• Deformations of algebras in noncommutative geometry
• Derived equivalences of hyperkähler varieties
• Noncommutative calculus and operads
• Categorical Saito theory, I: A comparison result
• A note on a question of Markman
• Sheaves of Twisted Cherednik Algebras as Universal Filtered Formal Deformations