On Goncharov's regulator and higher arithmetic Chow groups

TL;DR

This paper proves that Goncharov's regulator aligns with Beilinson's and Burgos-Feliu's constructions, confirming the consistency of higher arithmetic Chow groups across different frameworks for projective arithmetic varieties.

Contribution

It establishes the equivalence of Goncharov's regulator with Beilinson's and compares Goncharov's higher arithmetic Chow groups with those of Burgos and Feliu.

Findings

Goncharov's regulator agrees with Beilinson's regulator.

Goncharov's higher arithmetic Chow groups coincide with Burgos and Feliu's for projective arithmetic varieties.

The paper provides a direct comparison between different regulator constructions.

Abstract

In this paper we show that the regulator defined by Goncharov from higher algebraic Chow groups to Deligne-Beilinson cohomology agrees with Beilinson's regulator. We give a direct comparison of Goncharov's regulator to the construction given by Burgos and Feliu. As a consequence, we show that the higher arithmetic Chow groups defined by Goncharov agree, for all projective arithmetic varieties over an arithmetic field, with the ones defined by Burgos and Feliu.