Coherent Sheaves, Chern Classes, and Superconnections on Compact Complex-Analytic Manifolds

We construct a twist-closed enhancement of the category ${\mathcal{D}}_{\mathrm{c}\mathrm{o}\mathrm{h}}^b(X)$, the bounded derived category of complexes of ${\mathcal{O}}_X$-modules with coherent cohomology, by means of the DG-category of $\bar{\mathrm{\partial }}$-superconnections. Then we apply the techniques of $\bar{\mathrm{\partial }}$-superconnections to define Chern classes and Bott-Chern classes of objects in the category, in particular, of coherent sheaves.