Atiyah and Todd classes arising from integrable distributions

Abstract

In this paper, we study the Atiyah class and the Todd class of the DG manifold (F[1],dF) corresponding to an integrable distribution F⊂TKM=TM⊗RK, where K=R or ℂ. We show that these two classes are canonically identical to those of the Lie pair (TKM,F). As a consequence, the Atiyah class of a complex manifold X is isomorphic to the Atiyah class of the corresponding DG manifold (TX0,1[1],∂̄). Moreover, if X is a compact Kähler manifold, then the Todd class of X is also isomorphic to the Todd class of the corresponding DG manifold (TX0,1[1],∂̄).