Atiyah class of DG manifolds of positive amplitude
Seokbong Seol
TL;DR
This work shows that the Atiyah and Todd classes, as well as Hochschild cohomology for DG manifolds of positive amplitude, descend to the homotopy category under weak equivalences, making them robust invariants of derived intersections. The method reduces to establishing quasi-isomorphisms of (p, q)-tensor complexes under weak equivalences, achieved via acyclic linear fibrations and a careful kernel/cokernel analysis. The Hochschild cohomology HH_⊕ is shown invariant and compatible with Gerstenhaber structures, in line with Kontsevich formality and Duflo–Kontsevich-type results adapted to the DG setting. Overall, the results provide a differential-geometric framework for characteristic classes and Hochschild theory that is stable under homotopy equivalences and suitable for derived intersection theory. The combination of Brown-type fibrant object methods with explicit DG bundle and tensor calculus yields a coherent invariance theory for characteristic classes and deformation theory in positive-amplitude DG geometry.
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, Stiénon and Xu. We prove that this Hochschild cohomology is invariant under weak equivalences.