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Polylogarithms, regulators, and Arakelov motivic complexes


Author: A. B. Goncharov
Journal: J. Amer. Math. Soc. 18 (2005), 1-60
MSC (2000): Primary 11G55, 19F27, 14G40, 19E15
DOI: https://doi.org/10.1090/S0894-0347-04-00472-2
Published electronically: November 3, 2004
MathSciNet review: 2114816
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Abstract: We construct an explicit regulator map from the weight $n$ Bloch higher Chow group complex to the weight $n$ Deligne complex of a regular projective complex algebraic variety $X$. We define the weight $n$ 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 to the geometry of the symmetric space $SL_n(\mathcal{C})/SU(n)$. 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 $K_{2n-1}(\mathcal{C})$ 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.

Our note,``Chow polylogarithms and regulators'', can serve as an introduction to this paper.


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Additional Information

A. B. Goncharov
Affiliation: Department of Mathematics, Brown University, Providence, Rhode Island 02912
Email: sasha@math.brown.edu

DOI: https://doi.org/10.1090/S0894-0347-04-00472-2
Keywords: Polylogarithms, motivic complexes, regulators, Arakelov theory
Received by editor(s): July 31, 2002
Published electronically: November 3, 2004
Article copyright: © Copyright 2004 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.