Uniqueness of equivariant singular Bott-Chern classes

TL;DR

This paper establishes the uniqueness of equivariant singular Bott-Chern classes by axiomatic characterization, generalizing previous work and providing tools for applications in Arakelov geometry.

Contribution

It introduces a natural axiomatic framework that ensures the uniqueness of equivariant singular Bott-Chern classes, extending prior non-equivariant theories.

Findings

Construction of Bismut's equivariant Bott-Chern currents is unique under the axioms.

Proves a concentration formula applicable in Arakelov geometry.

Generalizes non-equivariant theories to the equivariant setting.

Abstract

In this paper, we shall discuss possible theories of defining equivariant singular Bott-Chern classes and corresponding uniqueness property. By adding a natural axiomatic characterization to the usual ones of equivariant Bott-Chern secondary characteristic classes, we will see that the construction of Bismut's equivariant Bott-Chern singular currents provides a unique way to define a theory of equivariant singular Bott-Chern classes. This generalizes J. I. Burgos Gil and R. Li\c{t}canu's discussion to the equivariant case. As a byproduct of this study, we shall prove a concentration formula which can be used to prove an arithmetic concentration theorem in Arakelov geometry.