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.Comment: Version 3: It is the final version, as it will appear in JAMS. 71 pages, 12 figure

Motivic Zeta Functions of the Quartic and its Mirror Dual

We use a formula of Bultot to compute the motivic zeta function for the toric degeneration of the quartic K3 and its Gross-Siebert mirror dual degeneration. We check for this explicit example that the identification of the logarithm of the monodromy and the mirror dual Lefschetz operator works at an integral level