The mixed Hodge structure on the fundamental groups of the Collino surfaces

TL;DR

This paper computes the mixed Hodge structure on the fundamental group of a specific open set of a symmetric square of a hyperelliptic curve, linking it to Abel-Jacobi invariants and extension classes.

Contribution

It explicitly describes the mixed Hodge structure on the fundamental group of Collino surfaces, connecting it to classical invariants and linear maps.

Findings

The fundamental group is isomorphic to the integral Heisenberg group.

The second extension class relates to the Abel-Jacobi invariant of the canonical class.

The structure involves a specific F_2-linear map.

Abstract

Collino proved that the fundamental group of a certain Zariski open set of the symmetric square of a hyperelliptic curve is isomorphic to the integral Heisenberg group. We compute the mixed Hodge structure on this fundamental group, and show that the second extension class is expressed by the Abel-Jacobi invariant of the canonical class and the marked points of the hyperelliptic curve, together with a certain F_2-linear map.