Atiyah and Todd classes arising from integrable distributions

Zhuo Chen Tsinghua University MAOSONG XIANG Peking University Ping Xu Pennsylvania State University

Differential Geometry mathscidoc:1809.10001

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\"{a}hler manifold, then the Todd class of $X$ is also isomorphic to the Todd class of the corresponding DG manifold $(T^{0,1}_X[1],\bar{\partial})$.
Atiyah class, Todd class, DG manifold, integrable distribution.
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@inproceedings{zhuoatiyah,
  title={Atiyah and Todd classes arising from integrable distributions},
  author={Zhuo Chen, MAOSONG XIANG, and Ping Xu},
  url={http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20180907140610148146153},
}
Zhuo Chen, MAOSONG XIANG, and Ping Xu. Atiyah and Todd classes arising from integrable distributions. http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20180907140610148146153.
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