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

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