Simplicial Abel-Jacobi maps and reciprocity laws

TL;DR

This paper constructs explicit morphisms linking higher Chow groups to Deligne cohomology and explores reciprocity laws that shed light on the functional equations of complex polylogarithms.

Contribution

It introduces explicit complexes for cycle-class maps and demonstrates how reciprocity laws clarify functional equations of polylogarithms.

Findings

Explicit morphism from higher Chow groups to Deligne cohomology

Reciprocity laws for currents related to polylogarithm equations

Enhanced understanding of functional equations of polylogarithms

Abstract

We describe an explicit morphism of complexes that induces the cycle-class maps from (simplicially described) higher Chow groups to rational Deligne cohomology. The reciprocity laws satisfied by the currents we introduce for this purpose are shown to provide a clarifying perspective on functional equations satisfied by complex-valued di- and trilogarithms.