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

Abstract

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 B₂ ⊕ S²B₁, where S² 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).

à Alexandre Grothendieck pour son 60^e anniversaire