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This is a revised version of 4. We construct an explicit regulator map from Bloch's Higher Chow group complex to the Deligne complex of a complex algebraic variety X. We define the 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-logarithms (defined as in [G5]) to the geometry of symmetric space SL_n(C)/SU(n). For 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_{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 group K^M_3 on curves. |