Cycle complex over P1 minus 3 points : toward multiple zeta values cycles

Abstract. In this paper, the author constructs a family of algebraic cycles in Bloch’s cycle complex over P1 minus three points which are expected to correspond to multiple polylogarithms in one variable. Elements in this family of weight p are in the cubical cycle group of codimension p in (P1 \{0, 1,∞})× (P1 \ {1})2p−1 and are, in weight greater or equal to 2, naturaly extended as equidimensional cycles over over A1. This allows to consider their fibers at the point 1 and this is one of the main differences with the work of Gangl, Goncharov and Levin. Considering the fiber at 1 makes it possible to think of these cycles as corresponding to weight n multiple zeta values. After the introduction, the author recalls some properties of Bloch’s cy-cle complex and enlightens the difficulties on a few examples. Then a large section is devoted to the combinatorial situation involving the combinatoric of trivalent trees. In the last section, two families of cycles are constructed as solutions to a “differential system ” in Bloch’s cycle complex. One of thi