On jets, extensions and characteristic classes II

Abstract

In this paper we define the generalized Atiyah classes $c_{\mathcal {J}}(\mathcal {E})$ and $c_{\mathcal {O}_X}(\mathcal {E})$ of a quasi-coherent sheaf $\mathcal {E}$ with respect to a pair $(\mathcal {I},d)$, where $\mathcal {I}$ is a left and right $\mathcal {O}_X$-module and $d$ a derivation. We relate this class to the structure of left and right modules on the first order jet bundle $\mathcal {J}^1_{\mathcal {I}}(\mathcal {E})$. In the main result of the paper we show $c_{\mathcal {O}_X}(\mathcal {E})=0$ if and only if there is an isomorphism $\mathcal {J}^1_{\mathcal {I}}(\mathcal {E})^{left} \cong \mathcal {J}^1_{\mathcal {I}}(\mathcal {E})^{right}$ as $\mathcal {O}_X$-modules. We also give explicit examples where $c_{\mathcal {O}_X}(\mathcal {E})\neq 0$ using jet bundles of line bundles on the projective line. Hence the classes $c_{\mathcal {J}}(\mathcal {E})$ and $c_{\mathcal {O}_X}(\mathcal {E})$ are nontrivial. The classes we introduce generalize the classical Atiyah class.