The cycle complex over P1P1 minus 3 points: Toward multiple zeta value cycles

In this paper, we construct a family of algebraic cycles in Bloch's cycle complex over P1P1 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 (P1∖{0,1,∞})×(P1∖{1})2p−1(P1∖{0,1,∞})×(P1∖{1})2p−1 and in weight greater than or equal to 2, they naturally extend as equidimensional cycles over A1A1.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 to 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.