2012
DOI: 10.4007/annals.2012.176.2.4
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Căldăraru's conjecture and Tsygan's formality

Abstract: In this paper we complete the proof of Cȃldȃraru'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 quasiisomorphisms introduced by Kontsevich and Shoikhet. (The existence of the latter was conjectured … Show more

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Cited by 29 publications
(40 citation statements)
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“…For us, the modified HKR isomorphism has two main advantages over its untwisted cousin. Firstly [14,13], it intertwines the action (6. From now on we restrict attention to the case of interest: S a K3 surface, X a cubic fourfold, and P ∈ D(S × X) the kernel of a full and faithful embedding D(S) → D(X).…”
Section: (Hochschild) (Co)homologymentioning
confidence: 99%
“…For us, the modified HKR isomorphism has two main advantages over its untwisted cousin. Firstly [14,13], it intertwines the action (6. From now on we restrict attention to the case of interest: S a K3 surface, X a cubic fourfold, and P ∈ D(S × X) the kernel of a full and faithful embedding D(S) → D(X).…”
Section: (Hochschild) (Co)homologymentioning
confidence: 99%
“…Besides, the multiplication morphisms of V ′ and V ′′ are given by the composition Proof. We can assume that M = A, and the lemma follows directly from (8).…”
Section: 11mentioning
confidence: 99%
“…as an exact sequence of A-modules, the A-module structure on K • being given after the choice of a retraction σ of the Atiyah sequence (8). It gives a morphism…”
Section: Admissible Complexesmentioning
confidence: 99%
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“…More concretely, the differential has the form δΓ = T 1 (m, Γ) ± T 1 (Γ, m). 8 The name stands for "Swiss Cheese Graphs", in analogy with the "Swiss Cheese" operad. 9 Although there is an obvious such action.…”
Section: Example 33mentioning
confidence: 99%