Polylogarithms, regulators and Arakelov motivic complexes

2002/07/03 by A. B. Goncharov, Goncharov, A. B.
Mathematics · #Advanced Algebra and Geometry #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #Algebraic structures and combinatorial models #FOS: Mathematics #Number Theory (math.NT) #math.AG #math.NT

paper · pdf · doi:10.48550/arxiv.math/0207036

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

openalex publication_date 2002/07/03 · arxiv created 2004/06/17 · arxiv updated 2009/11/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

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 K₃ on curves.

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• The Grassmannian complex and Goncharov's motivic complex in weight 4
• Arakelov motivic cohomology II
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• Morphisms Between Classical and Infinitesimal Polylogarithmic and Grassmannian Complexes