Hermitian structures on the derived category of coherent sheaves

TL;DR

This paper develops a theoretical framework for hermitian structures on the derived category of coherent sheaves, extending Bott-Chern classes and defining morphisms with hermitian structures on tangent complexes.

Contribution

It introduces hermitian structures on derived category objects, extends Bott-Chern class theory, and defines a new category with morphisms having hermitian structures on tangent complexes.

Findings

Hermitian structures are defined on objects of the derived category.

Bott-Chern classes are extended to these hermitian structures.

A new category with hermitian morphisms on tangent complexes is introduced.

Abstract

The main objective of the present paper is to set up the theoretical basis and the language needed to deal with the problem of direct images of hermitian vector bundles for projective non-necessarily smooth morphisms. To this end, we first define hermitian structures on the objects of the bounded derived category of coherent sheaves on a smooth complex variety. Secondly we extend the theory of Bott-Chern classes to these hermitian structures. Finally we introduce the category $\oSm_{\ast/\CC}$ whose morphisms are projective morphisms with a hermitian structure on the relative tangent complex.