Atiyah and Todd classes arising from integrable distributions

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 $(T_X^{0,1}[1],\bar{\partial })$. 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 $(T_X^{0,1}[1],\bar{\partial })$.

19 pages, the contexts are rearranged, typos removed

References in corpus (2)

• The Atiyah class of a dg-vector bundle
• Exponential map and $L_{\infty }$ algebra associated to a Lie pair

Cited by in corpus (6)

• Dg manifolds, formal exponential maps and homotopy Lie algebras
• Hochschild cohomology of dg manifolds associated to integrable distributions
• Atiyah classes and Todd classes of pullback dg Lie algebroids associated with Lie pairs
• Keller admissible triples and Duflo theorem
• Internal symmetry of the $L_{⩽3}$ algebra arising from a Lie pair
• Atiyah and Todd classes of regular Lie algebroids