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Aomoto dilogarithms, mixed Hodge structures and motivic cohomology of pairs of triangles on the plane. (English) Zbl 0737.14003
The Grothendieck Festschrift, Collect. Artic. in Honor of the 60th Birthday of A. Grothendieck. Vol. I, Prog. Math. 86, 135-172 (1990).
[For the entire collection see Zbl 0717.00008.]
Let $$L=(L_ 0,\ldots,L_ r)$$, $$M=(M_ 0,\ldots,M_ s)$$ be configurations of hyperplanes in $$\mathbb{P}^ n(\mathbb{C})$$. Then the group $$H^ n(\mathbb{P}^ n\backslash L,M\backslash L;\mathbb{Z})$$ carries a mixed Hodge structure such that $$Gr_ i^ W=0$$ for $$i$$ odd and $$Gr^ W_{2i}=G_ i(L,M)$$ is purely of type $$(i,i)$$ (such mixed Hodge structures are called “of Tate type”). Moreover there are complexes $$C_ i(L,M)$$, only depending on the combinatorics of $$(L,M)$$, with $$G_ i(L,M)\cong H^ 0(C_ i(L,M))$$; these complexes are the $$E_ 1$$-terms of the weight spectral sequence.
Let $${\mathcal H}_ n$$ be the abelian group with elements $$(H,x,y)$$ where $$H$$ is a mixed Hodge structure, $$x\in Gr^ W_{2n}H$$ and $$y\in(W_ 0H)^*$$, up to suitable equivalence and with a suitable addition; let $${\mathcal H}=\oplus_{n\in\mathbb{N}}{\mathcal H}_ n$$, $${\mathcal H}_ \mathbb{Q}={\mathcal H}\otimes_ \mathbb{Z}\mathbb{Q}$$. There is a natural Hopf algebra structure on $${\mathcal H}_ \mathbb{Q}$$ and an equivalence of categories between mixed $$\mathbb{Q}$$-Hodge structures of Tate type and graded comodules $$M$$ over $${\mathcal H}_ \mathbb{Q}$$ with $$\dim_ \mathbb{Q} M<\infty$$.
For $$L=(L_ 0,\ldots,L_ n)$$, $$M=(M_ 0,\ldots,M_ n)$$ configurations in $$\mathbb{P}^ n(k)$$, $$k$$ an arbitrary field, one disposes of a natural generator $$\omega_ L$$ of $$G_ n(L,M)$$ (the one lifting to a generator of $$F^ nH^ n(\mathbb{P}^ n\backslash L,M\backslash L))$$ and a natural cycle $$\Delta_ M\in G_ 0(L,M)^*$$. A lifting of $$\Delta_ M$$ to $$H_ n(\mathbb{P}^ n\backslash L,M\backslash L)$$ defines $$a_ n(L,M):=\int_{\Delta_ M}\omega_ L$$, an Aomoto $$n$$-logarithm.
For $$L,M$$ configurations of hyperplanes in $$\mathbb{P}^ n(k)$$ the construction of $$G_ i(L,M)$$ still makes sense and one defines $${\mathcal H}^ n_{\mathcal M}(\mathbb{P}^ n\backslash L,M\backslash L):=\oplus^ n_{i=0}G_ i(L,M)$$ as a graded abelian group. The analogue of a mixed Hodge structure of Tate type is defined as follows. A graded Hopf algebra $${\mathcal A}(k)$$ is constructed; its generators in degree $$n$$ are symbols $$(L;M)$$ with $$L=(L_ 0,\ldots,L_ n)$$, $$M=(M_ 0,\ldots,M_ n)$$ configurations in $$\mathbb{P}^ n(k)$$ without common faces, up to suitable equivalence. In case $$n=2$$ and configurations $$L,M$$ in $$\mathbb{P}^ 2(k)$$, mappings $$G^*_ i(L,M)\otimes G_ j(L,M)\to{\mathcal A}_{j-i}$$ are constructed, endowing $${\mathcal H}^ 2_{\mathcal M}(\mathbb{P}^ 2\backslash L,M\backslash L)$$ with the structure of a graded comodule over $${\mathcal A}(k)$$. For $$k=\mathbb{C}$$, one disposes of a Hopf algebra morphism $${\mathcal A}_ \mathbb{Q}\to{\mathcal H}_ \mathbb{Q}$$ such that the mixed Hodge structure is obtained by the forgetful functor from $${\mathcal A}_ \mathbb{Q}$$-comodules to $${\mathcal H}_ \mathbb{Q}$$-comodules.
The groups $${\mathcal A}_ n$$ are closely related to the $$K$$-theory of the field $$k$$. It is shown that the Bloch group $$B_ 2(k)$$ is isomorphic to the quotient of $${\mathcal A}_ 2(k)$$ by an explicitly defined subgroup $$\Sigma$$. By a further analysis of $${\mathcal A}_ 2$$ it is shown that an Aomoto dilogarithm can be expressed in terms of Euler dilogarithms, products of logarithms and multiples of $$\pi^ 2/6$$. The proofs of the main results of the paper will be published later.

##### MSC:
 14C35 Applications of methods of algebraic $$K$$-theory in algebraic geometry 14D07 Variation of Hodge structures (algebro-geometric aspects) 14A20 Generalizations (algebraic spaces, stacks)