Atiyah class of DG manifolds of positive amplitude

TL;DR

This paper proves the invariance of Atiyah and Todd classes, as well as Hochschild cohomology, for DG manifolds of positive amplitude under weak equivalences, connecting to formality and Duflo--Kontsevich theorems.

Contribution

It establishes the invariance of key geometric and algebraic invariants of DG manifolds of positive amplitude under weak equivalences, extending their applicability.

Findings

Atiyah and Todd classes are invariant under weak equivalences.

Hochschild cohomology defined via poly-differential operators is invariant under weak equivalences.

Connections to Kontsevich formality and Duflo--Kontsevich theorems are confirmed.

Abstract

Behrend, Liao, and Xu showed that differential graded (DG) manifolds of positive amplitude forms a category of fibrant objects. In particular, this ensures that notion of derived intersection -- more generally, homotopy fibre product -- is well-defined up to weak equivalences. We prove that the Atiyah and Todd classes of DG manifolds of positive amplitude are invariant under the weak equivalences. As an application, we study Hochschild cohomology of DG manifolds of positive amplitude defined using poly-differential operators, which is compatible with Kontsevich formality theorem and Duflo--Kontsevich-type theorem established by Liao, Sti\'enon and Xu. We prove that this Hochschild cohomology is invariant under weak equivalences.