Polylogarithms, regulators and Arakelov motivic complexes

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.