The Atiyah class of DG manifolds of amplitude $+1$

TL;DR

This paper computes the Atiyah class of DG manifolds of amplitude +1, linking its vanishing to the geometric property of a clean intersection, with applications to derived intersections of smooth manifolds.

Contribution

It provides a explicit computation of the Atiyah class for DG manifolds of amplitude +1 and characterizes its vanishing in terms of intersection properties.

Findings

Atiyah class vanishes iff the intersection is clean

Computed Atiyah class for DG manifolds of amplitude +1

Applied results to derived intersections of smooth manifolds

Abstract

A DG manifold of amplitude $+1$ encodes the derived intersection of a section $s$ and the zero section of a vector bundle $E$. In this paper, we compute the Atiyah class of DG manifolds of amplitude $+1$. In particular, we show that the Atiyah class vanishes if and only if the intersection of $s$ with the zero section is a clean intersection. As an application, we study the Atiyah class of DG manifolds that encodes the derived intersection of two smooth manifolds.