Cycle complex over the projective line minus three points : toward multiple zeta values cycles

TL;DR

This paper constructs algebraic cycles over the projective line minus three points, linking them to multiple zeta values and polylogarithms, advancing the understanding of their algebraic and combinatorial structures.

Contribution

It introduces new families of algebraic cycles in Bloch's complex that relate to multiple polylogarithms and zeta values, with novel fiber analysis at specific points.

Findings

Construction of algebraic cycles corresponding to multiple polylogarithms.

Identification of cycles with fibers at 1 as related to multiple zeta values.

Development of a differential system approach for cycle families.

Abstract

In this paper, the author constructs a family of algebraic cycles in Bloch's cubical cycle complex over the projective line minus three points which are expected to correspond to multiple polylogarithms in one variable. Elements in this family are in particular equidimensional over the projective line minus three points. In weight greater or equal to $2$, they are naturaly extended as equidimensional cycle over the affine line. This allows to consider their fibers at the point 1 and this is one of the main differences with Gangl, Goncharov and Levin work where generic arguments are imposed for cycles corresponding to multiple polylogarithms in many variables. Considering the fiber at 1 make it possible to think of these cycles as corresponding multiple zeta values. After the introduction, the author recalls some properties of Bloch's cycle complex, presents the strategy and enlightens the difficulties on a few examples. Then a large section is devoted to the combinatorial situation which is related to the combinatoric of trivalent trees and to a differential on trees already introduced by Gangl Goncharov and Levin. In the last section, two families of cycles are constructed as solution to a "differential system" in Bloch cycle complex. One of this families contains only cycles with empty fiber at 0 and should correspond to multiple polylogarithms while the other contains only cycles empty at 1. The use of two such families is required in order to work with equidimimensional cycles and to insure the admissibility condition.