A description of a result of Deligne by log higher Albanese map

TL;DR

This paper explores Deligne's work on nilpotent quotients of fundamental groups using log higher Albanese maps, providing explicit q-expansions at boundary points within the framework of log mixed Hodge theory.

Contribution

It introduces a novel application of log mixed Hodge theory to describe Deligne's results on fundamental group quotients with explicit boundary behavior.

Findings

Derived q-expansions of higher Albanese maps at boundary points

Connected polylogarithms to nilpotent quotients of fundamental groups

Extended the understanding of boundary behavior in log higher Albanese maps

Abstract

In a joint work [9] with Kazuya Kato and Chikara Nakayama, log higher Albanese manifolds was constructed as an application of log mixed Hodge theory with group action. In this framework, we describe a work of Deligne in [3] on some nilpotent quotients of the fundamental group of the projective line minus three points, where polylogarithms appear. As a result, we have $q$-expansions of higher Albanese maps at boundary points, i.e., log higher Albanese maps over the boundary.