Polylogarithms, regulators, and Arakelov motivic complexes

By A. Goncharov. 2002

48 citations recorded by the source corpus.

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. Soule. We relate the Grassmannian n–logarithms (defined as in [G5]) to the geometry of symmetric space SLn(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 K2n−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 3 on curves. Our note [G5] can serve as an introduction to this paper.

Subfields: Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/5d91b9e564acda98ed60b401eeec6c25f5b21711

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Citations in the corpus

Moduli spaces of convex projective structures on surfaces (2004, background): “Here is a yet another, the third definition of this invariant borrowed from ([9], Section 4.”
Moduli spaces of local systems and higher Teichmüller theory (2003, background): “The skew symmetry property of the component C U guarantees that this definition does not depend on the ordering of the edges at v0. The condition δ ◦ w4(Γ ,e Γ) = w3 ◦ δ(Γ ,e Γ) is equivalent to the cocycle condition ii) for C U . Let (Γ ,e Γ) ∈ G5 (S).”