The cycle complex over P1 minus 3 points: Toward multiple zeta value cycles

Abstract

Abstract. In this paper, we construct a family of algebraic cycles in Bloch's cycle complex over P 1 minus three points, which are expected to correspond to multiple polylogarithms in one variable. Elements in this family of weight p belong to the cubical cycle group of codimension p in (P 1 \ {0, 1, ∞}) × (P 1 \ {1}) 2p−1 and in weight greater than or equal to 2, they naturally extend as equidimensional cycles over A 1 .Thus, we can consider their fibers at the point 1. This is one of the main differences with the work of Gangl, Goncharov and Levin. Considering the fiber of our cycles at 1 makes it possible to view these cycles as those corresponding to weight n multiple zeta values which are viewed here as the values at 1 of multiple polylogarithms.After the introduction, we recall some properties of Bloch's cycle complex, and explain the difficulties on a few examples. Then a large section is devoted to the combinatorial situation, essentially involving the combinatorics of trivalent trees in relation with the structure of the free Lie algebra on two generators. In the last section, two families of cycles are constructed as solutions to a "differential system" in Bloch's cycle complex. One of these families contains only cycles with empty fiber at 0; these correspond to multiple polylogarithms. Lie coalgebra generating its 1-minimal model is isomorphic to the above Tannakian Lie coalgebra. ContentsIn order to obtain a motive, that is a comodule, it is enough to have an element in the relevant Tannakian Lie coalgebra. Then the motive is the comodule cogenerated by this element. Since the Tannakian Lie coalgebra is a "H 0 modulo product", we write one of its elements as a class [L B ] in this H 0 . This class can be represented in the bar construction over N • by an element L B . The element L B is essentially determined by its projection onto its tensor degree 1 part, which gives an algebraic cycle L in N • . The cycle L has a decomposable boundary because the element L B leads to a class in the H 0 .Hence a first step toward explicit motives via algebraic cycles is to build algebraic cycles in N • having a decomposable boundary. The main result of this paper (Theorem 5.8) provides such algebraic cycles denoted by L 0 W ; here the indexing set W consists in Lyndon words. It is shown in [Sou14] that the motives attached to these cycles generate the Lie coalgebra associated to the Deligne-Goncharov fundamental group.However lifting algebraic cycles to objects in the bar construction requires, in general, a strong combinatorial or algebraic control of the boundaries involved. In our context, this control is insured (Theorem 5.8) by the algebraic and combinatorial structure of Ihara's special derivations studied at Proposition 4.27.The next steps in order to obtain motivic multiple zeta values are:(1) Our main theorem makes it possible to build a commutative differential graded algebra morphism between the cobar construction over the Lie coalgebra associated to Ihara's special derivations (combinatorial structure...

A relative basis for mixed Tate motives over the projective line minus three points

“…• In Section 3, we present the action of Lie(X 0 , X 1 ) on itself by Ihara's special derivations and the corresponding Lie coalgebra. From there we recall the result from [Sou12] constructing the cycles L ε W . We conclude this section by lifting the cycle to elements in the bar construction.…”

“…In [Sou12] the author produces such a family. Together with two explicit degree 1 weight 1 algebraic cycles generating the H 1 (N qf, • P 1 \{0,1,∞} ), the author obtains: Theorem.…”

Rational mixed Tate motivic graphs

“…Therefore, conjecturally, it contains all the cycles necessary to define the full category of mixed Tate motives. There has been some effort to understand subalgebras of A 1L in terms of polylogarithms and multiple logarithms [12,22]. In this paper, we study a subalgebra A × 1L ⊂ A 1L that specifically excludes the Totaro cycles.…”

“…There is little technology developed to identify minimally decomposable cycles, which define classes in H 0 (B(G 1L )), let alone in understanding relations between such. What little progress there has been [12,11,22] has been on a case by case basis. We hope to revisit this question in the future in a more systematic manner.…”

“…Soudéres [22,21] extends the family of algebraic cycles studied by Gangl, Goncharov and Levin to include those over a more general base scheme, in particular giving a rigorous construction of unital values of the multiple polylogarithms ie. multiple zeta values, as periods (and not just non-unital values of the multiple logarithms).…”