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 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.

Hodge theory and derived categories of cubic fourfolds

“…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).…”

Derived geometry of the first formal neighbourhood of a smooth analytic cycle

“…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).…”

“…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…”

“…This result turns out to be extremely useful in algebraic and complex geometry as well as in deformation quantization, we refer the interested reader to the non exhaustive list of papers [10], [25], [27], [12], [5], [8], [11], [22], [19], [3] as well as references therein.…”

The homotopy braces formality morphism

“…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.…”