On the Bott-Chern characteristic classes for coherent sheaves

TL;DR

This paper extends Bott-Chern characteristic classes to the derived category of coherent sheaves using superconnections, showing their independence from metrics and defining secondary forms.

Contribution

It introduces a new construction of Bott-Chern classes for coherent sheaves via superconnections, extending their applicability to derived categories.

Findings

Characteristic classes are metric-independent

Extension of Bott-Chern cohomology to derived categories

Construction of well-defined secondary Bott-Chern forms

Abstract

Using the concept of a cohesive module defined by Block, we use the theory of superconnections in the sense of Quillen to construct natural superconnections on Hermitian cohesive modules. By the Chern-Weil construction, we obtain characteristic classes with values in Bott-Chern cohomology which refines the usual deRham cohomology. The main result of this paper shows that such characteristic classes are independent of the Hermitian metric and only dependent on the homotopy class of the cohesive module in the homotopy category. This extends the Bott-Chern cohomology to the bounded derived category of complexes of analytic sheaves with coherent cohomology. We also constructed secondary Bott-Chern characteristic forms and the main technical theorem proves that they are well defined.