The Atiyah class of a dg-vector bundle

Abstract

We introduce the notions of Atiyah class and Todd class of a differential graded vector bundle with respect to a differential graded Lie algebroid. We prove that the space of vector fields on a dg-manifold with homological vector field $Q$ admits a structure of L-infinity algebra with the Lie derivative $L_Q$ as unary bracket, and the Atiyah cocycle corresponding to a torsion-free affine connection as binary bracket.Comment: 7 pages. To appear in Compte Rendus Mat

Atiyah and Todd classes arising from integrable distributions

“…In fact, the category of DG vector bundles and the category of locally free DG-modules are equivalent(cf. [13]).…”

“…. For any positive integer k, one can form α k M , the image of α ⊗k M under the natural map The scalar Atiyah classes [13] of the DG manifold (M, Q) are defined by…”

Kapranov’s construction of $\operatorname{sh}$ Leibniz algebras

“…Atiyah classes of dg Lie algebroids and Lie pairs. In this section, we briefly recall Atiyah classes of dg vector bundles with respect to a dg Lie algebroid defined in [23] and Atiyah classes of Lie pairs defined in [8] (see [9] for the equivalence between the two types of Atiyah classes arising from integrable distributions), and show that both of them can be viewed as twisted Atiyah classes.…”

“…Mehta-Sténon-Xu[23]). Let (M, Q M ) be a smooth dg manifold, where M = (M, O M ) is a smooth Z-graded manifold, and Q M is a homological vector field on M.Then A = (C ∞ (M), Q M ) is a cdga.…”

Introduction to graded geometry

“…In recent developments (see [1,15]) formal polynomial functions on graded manifolds have been considered with respect to a Z-graded vector space that has a non-trivial part of degree zero. This means that, in Definition 3.1, we could choose W such that dim W 0 1.…”