Aomoto dilogarithms, mixed Hodge structures and motivic cohomology of pairs of triangles on the plane. (English) Zbl 0737.14003

Beilinson, A. A.; Goncharov, A. B.; Schechtman, V. V.; Varchenko, A. N.

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.

Reviewer: J.Steenbrink (Nijmegen)

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)

Keywords:

motivic cohomology; mixed Hodge structure; Hopf algebra; Hodge structure of Tate type; \(n\)-logarithm; \(K\)-theory; Bloch group; Aomoto dilogarithm; Euler dilogarithms

Citations:

Zbl 0717.00008