  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.
