Coherent sheaves, Chern classes, and superconnections on compact complex-analytic manifolds

Abstract

A twist-closed enhancement of the bounded derived category $\mathcal{D}^b_{\mathrm{coh}} (X)$ of complexes of $\mathcal{O}_X$-modules with coherent cohomology is constructed by means of the DG-category of $\overline\partial$-superconnections. The machinery of $\overline\partial$-superconnections is applied to define Chern classes and Bott-Chern classes of objects in the category, in particular, of coherent sheaves.