Polylogarithms, regulators, and Arakelov motivic complexes

Abstract

We construct an explicit regulator map from the weight n n Bloch higher Chow group complex to the weight n n Deligne complex of a regular projective complex algebraic variety X X . We define the weight n n Arakelov 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. Soulé. We relate the Grassmannian n n -logarithms to the geometry of the symmetric space S L n ( C ) / S U ( n ) SL_n(\mathcal {C})/SU(n) . For n = 2 n=2 we recover Lobachevsky’s formula expressing the volume of an ideal geodesic simplex in the hyperbolic space via the dilogarithm. Using the relationship with symmetric spaces we construct the Borel regulator on K 2 n − 1 ( C ) K_{2n-1}(\mathcal {C}) via the Grassmannian n n -logarithms. We study the Chow dilogarithm and prove a reciprocity law which strengthens Suslin’s reciprocity law for Milnor’s group K 3 M K^M_3 on curves. Our note,“Chow polylogarithms and regulators”, can serve as an introduction to this paper.