Atiyah and Todd classes arising from integrable distributions

TL;DR

This paper establishes a canonical equivalence between the Atiyah and Todd classes of certain DG manifolds derived from integrable distributions and those of associated Lie pairs, with implications for complex and Kähler manifolds.

Contribution

It demonstrates that Atiyah and Todd classes of DG manifolds from integrable distributions are identical to those of related Lie pairs, linking classical and derived geometric invariants.

Findings

Atiyah and Todd classes of DG manifolds match those of Lie pairs.

Atiyah class of a complex manifold is isomorphic to that of a related DG manifold.

Todd class of a compact Kähler manifold is isomorphic to that of a related DG manifold.

Abstract

In this paper, we study the Atiyah class and Todd class of the DG manifold $(F[1],d_F)$ corresponding to an integrable distribution $F \subset T_{\mathbb{K}} M = TM \otimes_{\mathbb{R}} \mathbb{K}$, where $\mathbb{K} = \mathbb{R}$ or $\mathbb{C}$. We show that these two classes are canonically identical to those of the Lie pair $(T_{\mathbb{K}} M, F)$. As a consequence, the Atiyah class of a complex manifold $X$ is isomorphic to the Atiyah class of the corresponding DG manifold $(T^{0,1}_X[1],\bar{\partial})$. Moreover, if $X$ is a compact K\"ahler manifold, then the Todd class of $X$ is also isomorphic to the Todd class of the corresponding DG manifold $(T^{0,1}_X[1],\bar{\partial})$.