A weight two phenomenon for the moduli of rank one local systems on open varieties

TL;DR

This paper explores the structure of the moduli space of rank one local systems on open varieties, revealing a weight two phenomenon related to monodromy and Higgs fields, with implications for understanding harmonic bundles.

Contribution

It introduces a new perspective on the twistor space of local systems, highlighting a weight two phenomenon linked to monodromy and Higgs residues in open varieties.

Findings

The twistor space maps to a weight two space of monodromy transformations.

Invariant sections form a real 3-dimensional space parameterized by residues and weights.

The structure connects harmonic bundles with monodromy and Higgs field data.

Abstract

The twistor space of representations on an open variety maps to a weight two space of local monodromy transformations around a divisor component at infinty. The space of $\sigma$-invariant sections of this slope-two bundle over the twistor line is a real 3 dimensional space whose parameters correspond to the complex residue of the Higgs field, and the real parabolic weight of a harmonic bundle.