Relative Deligne cohomology and Cheeger-Simons characters

TL;DR

This paper explores the relationship between relative Deligne cohomology and Cheeger-Simons characters, establishing connections between different definitions and extending formulas for holonomy, transgression, and integration to the relative setting.

Contribution

It relates two frameworks for relative Cheeger-Simons characters, analyzes three meaningful relative groups, and extends key formulas to the relative case.

Findings

Identified three meaningful relative groups in Deligne cohomology.

Connected different definitions of relative Cheeger-Simons characters.

Extended formulas for holonomy, transgression, and integration to the relative case.

Abstract

In the paper [1] (arXiv:math/0408333) the authors discuss two possible definitions of the relative Cheeger-Simons characters, the second one fitting into a long exact sequence. Here we relate that picture to the one of the relative Deligne cohomology groups, defined via the mapping cone: we show that there are three meaningful relative groups, and we analyze the corresponding definition of relative Cheeger-Simons characters in each case. We then extract another definition, corresponding to the one fitting into a long exact sequence in [1]. Finally we show how the explicit formulas for the holonomy, the transgression maps and the integration can be extended to the relative case.