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The strong Suslin reciprocity law

Part of: Arithmetic algebraic geometry

Published online by Cambridge University Press:  01 April 2021

Daniil Rudenko
Daniil Rudenko*
Affiliation:
Department of Mathematics, The University of Chicago, Eckhart Hall, 5734 S University Ave, Chicago, IL60637, USArudenkodaniil@gmail.com
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Abstract

We prove the strong Suslin reciprocity law conjectured by A. Goncharov. The Suslin reciprocity law is a generalization of the Weil reciprocity law to higher Milnor $K$-theory. The Milnor $K$-groups can be identified with the top cohomology groups of the polylogarithmic motivic complexes; Goncharov's conjecture predicts the existence of a contracting homotopy underlying Suslin reciprocity. The main ingredient of the proof is a homotopy invariance theorem for the cohomology of the polylogarithmic motivic complexes in the ‘next to Milnor’ degree. We apply these results to the theory of scissors congruences of hyperbolic polytopes. For every triple of rational functions on a compact projective curve over $\mathbb {C}$ we construct a hyperbolic polytope (defined up to scissors congruence). The hyperbolic volume and the Dehn invariant of this polytope can be computed directly from the triple of rational functions on the curve.

Keywords

polylogarithmsK-theoryscissors congruence

MSC classification

Primary: 11G55: Polylogarithms and relations with $K$-theory
Type
Research Article
Information
Compositio Mathematica , Volume 157 , Issue 4 , April 2021 , pp. 649 - 676
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Copyright
© The Author(s) 2021

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