Extensions of Motives and the Fundamental Group

TL;DR

This paper constructs new extensions of the Mixed Hodge structure on the fundamental group of algebraic curves, linking motivic cohomology cycles to iterated integrals and generalizing previous results.

Contribution

It introduces a novel method to relate motivic cohomology cycles with extensions of the fundamental group's Hodge structure, expanding on prior work by Colombo and Collino.

Findings

Provides a new iterated integral expression for the regulator.

Generalizes Colombo's construction to broader classes of curves.

Connects motivic cohomology with Hodge-theoretic extensions.

Abstract

In this paper we construct extensions of the Mixed Hodge structure on the fundamental group of a pointed algebraic curve. These extensions correspond to the regulator of certain explicit motivic cohomology cycles in the self product of the curve which were first constructed by Bloch and Beilinson. This leads to a new iterated integral expression for the regulator. Our result is a generalization of a result of Colombo's where she constructs the extension corresponding to a motivic cycle class in the Jacobian of a hyperelliptic curve constructed by Collino. This is to appear in the Mathematical Proceedings of the Indian Academy of Sciences.