---
- CiteKey: "BGSV90"
- Type: bookSection
- Title: "Aomoto Dilogarithms, Mixed Hodge Structures and Motivic Cohomology of Pairs of Triangles on the Plane," 
- Author: "Beilinson, A. A.; Goncharov, A. B.; Schechtman, V. V.; Varchenko, A. N.;"  
- Publisher: "Birkhäuser,"
- Year: 2007 
- Collections: "Syllabus; Talbot 2022,"
- Keywords: "Admissible Pair"; "Canonical Isomorphism"; "Free Abelian Group"; "Hodge Structure"; "Hopf Algebra"
---

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

## Meta

- **URL**: <https://doi.org/10.1007/978-0-8176-4574-8_6>
- **URI**: <http://zotero.org/users/1049732/items/2AIFCMH9>
- **Local File**: [Beilinson et al. - 2007 - Aomoto Dilogarithms, Mixed Hodge Structures and Mo.pdf](file:///home/zack/Downloads/Zotero_Source/Birkhäuser/2007/Beilinson%20et%20al.%20-%202007%20-%20Aomoto%20Dilogarithms,%20Mixed%20Hodge%20Structures%20and%20Mo.pdf)
- **Open in Zotero**: [Zotero](zotero://select/library/items/2AIFCMH9)

## 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 S2B1 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).

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