Polylogarithms, regulators and Arakelov motivic complexes

TL;DR

This paper constructs explicit regulator maps linking higher Chow groups to Deligne complexes, relates Grassmannian n-logarithms to symmetric space geometry, and develops Borel regulators with applications to algebraic K-theory and reciprocity laws.

Contribution

It introduces an explicit regulator map from Bloch Higher Chow groups to Deligne complexes and connects Grassmannian n-logarithms to symmetric space geometry, advancing Arakelov motivic theory.

Findings

Constructed explicit regulator maps for higher Chow groups.

Linked Grassmannian n-logarithms to symmetric space geometry.

Proved a reciprocity law strengthening Suslin's law.

Abstract

We construct an explicit regulator map from the weigh n Bloch Higher Chow group complexto the weight n Deligne complex of a regular complex projective algebraic variety X. We define the Arakelovweight n motivic complex as the cone of this map shifted by one. Its last cohomology group is (a version of) the Arakelov Chow group defined by H. Gillet. and C.Soule. We relate the Grassmannian n-logarithms (defined as in [G5]) to geometry of the symmetric space for GL_n(C). For n=2 we recover Lobachevsky's formula for the volume of an ideal geodesic tetrahedron via the dilogarithm. Using the relationship with symmetric spaces we construct the Borel regulator on K_{2n-1}(C) via the Grassmannian n-logarithms. We study the Chow dilogarithm and prove a reciprocity law which strengthens Suslin's reciprocity law for Milnor's K_3 on curves.