The twistor geometry of parabolic structures in rank two

TL;DR

This paper constructs a twistor space for rank 2 parabolic structures on curves using moduli spaces of framed lambda-connections, revealing a mixed twistor structure in harmonic bundle deformations.

Contribution

It introduces a novel construction of Deligne-Hitchin twistor spaces for parabolic structures via moduli of lambda-connections and analyzes the associated mixed twistor structures.

Findings

Construction of a Deligne-Hitchin twistor space for rank 2 parabolic structures.

Identification of a mixed twistor structure with weights 0, 1, 2 in the tangent bundle.

Description of deformations of KMS structures including parabolic weights and residues.

Abstract

Let $X$ be a quasi-projective curve, compactified to $(Y,D)$ with $X=Y-D$. We construct a Deligne-Hitchin twistor space out of moduli spaces of framed $\lambda$-connections of rank $2$ over $Y$ with logarithmic singularities and quasi-parabolic structure along $D$. To do this, one should divide by a Hecke-gauge groupoid. Tame harmonic bundles on $X$ give preferred sections, and the relative tangent bundle along a preferred section has a mixed twistor structure with weights $0,1,2$. The weight $2$ piece corresponds to the deformations of the KMS structure including parabolic weights and the residues of the $\lambda$-connection.