Aomoto Dilogarithms, Mixed Hodge Structures and Motivic Cohomology of Pairs of Triangles on the Plane
Aomoto Dilogarithms, Mixed Hodge Structures and Motivic Cohomology of Pairs of Triangles on the Plane
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青本二对数、混合Hodge结构和平面上三角形对的动机上同调
DOI：
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发表时间：
2007
期刊：
影响因子：
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通讯作者：
A. Varchenko
中科院分区：
文献类型：
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作者：
A. Beilinson;A. Goncharov;V. Schechtman;A. Varchenko
已知R3中的多面体的一组线性组合可以嵌入到多面体的切割运动中;这种嵌入赋予多面体其Dehn不变量和体积[C]。研究一个具有两个特殊射影直线族的射影平面的运动上同调，会引出一个类似的问题：描述一个平面上的三角形对的线性组合群，它们在PGL（3）作用下，关于一对三角形中任意三角形的切割。证明了这个群同构到B2 <$S2 B1，其中S2 B1是基域的乘法群的对称平方，B2是这个域的Bloch群。这是本文的第一个主要结果（见定理2.12，3.8和3.6.2）。
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 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).