Hopf algebras of multiple polylogarithms, and holomorphic 1-forms

TL;DR

This paper explores the relationship between multiple polylogarithms, their associated holomorphic 1-forms, and the underlying algebraic structures, providing new insights into their Hodge-theoretic and Hopf algebraic properties.

Contribution

It introduces a novel association of holomorphic 1-forms to multiple polylogarithms and connects this to Goncharov's Hopf algebra, enriching the understanding of their algebraic and geometric structures.

Findings

Holomorphic 1-forms relate to the symbol and variation matrix of polylogarithms.

The 1-forms define a lift of the variation of mixed Hodge structures.

A new map from the Chevalley-Eilenberg complex to the de Rham complex is established.

Abstract

We associate to a multiple polylogarithm a holomorphic 1-form on the universal abelian cover of its domain. We relate the 1-forms to the symbol and variation matrix and show that the 1-forms naturally define a lift of the variation of mixed Hodge structure associated to a polylogarithm. The results are conveniently described in terms of a variant H of Goncharov's Hopf algebra of multiple polylogarithms. In particular, we show that the association of a 1-form to a multiple polylogarithm induces a map from the Chevalley-Eilenberg complex of the Lie coalgebra of indecomposables of H to the de Rham complex.