Aomoto Dilogarithms, Mixed Hodge Structures and Motivic Cohomology of Pairs of Triangles on the Plane

  • A. A. Beilinson
  • A. B. Goncharov
  • V. V. Schechtman
  • A. N. Varchenko
Part of the Progress in Mathematics book series (MBC)


It is known that a group of linear combinations of polytopes in R3 considered up to movements with respect to cutting of polytopes may be embedded into ℝ ⊗ ℝ/2πℤ ⊕ ℝ; this embedding assigns to a polytope its Dehn invariant and volume [C]. The study of motivic cohomology of a projective plane with two distinguished families of projective lines leads to an analogous problem: to describe a group of linear combinations of pairs of triangles on a plane considered up to the action of PGL(3), with respect to a cutting of any triangle of a pair. It turns out that this group is isomorphic up to 12—torsion to B2S2B1, where S 2 B1 is the symmetric square of the multiplicative group of a ground field, and B2 — the Bloch group of this field. This is the first main result of the paper (see Theorems 2.12, 3.8 and 3.6.2).


Hopf Algebra Canonical Isomorphism Hodge Structure Free Abelian Group Admissible Pair 
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Copyright information

© Springer Science+Business Media New York 2007

Authors and Affiliations

  • A. A. Beilinson
    • 1
  • A. B. Goncharov
    • 2
  • V. V. Schechtman
    • 3
  • A. N. Varchenko
    • 4
  1. 1.Department of MathematicsM.I.T.CambridgeUSA
  2. 2.Scientific Council on CyberneticsAcademy of Sciences of the USSRMoscowUSSR
  3. 3.Institute of problems of microelectronics technology and superpure materialsAcademy of Sciences of the USSRMoscowUSSR
  4. 4.Department of MathematicsGubkin MINGMoscowUSSR

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