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

TL;DR

This paper develops a new DG-category framework using $ar ext{∂}$-superconnections to enhance the derived category of coherent sheaves on complex-analytic manifolds, enabling novel definitions of Chern and Bott-Chern classes.

Contribution

It introduces a twist-closed DG-category enhancement of the derived category of coherent sheaves using $ar ext{∂}$-superconnections, facilitating the definition of characteristic classes.

Findings

Constructed a twist-closed DG-category enhancement of ${ m D}^b_{ m coh}(X)$.

Defined Chern and Bott-Chern classes for coherent sheaves using $ar ext{∂}$-superconnections.

Provided a new framework for characteristic classes in complex geometry.

Abstract

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.