Geometric Models and Variation of Weights on Moduli of Parabolic Higgs Bundles over the Riemann Sphere: a Case Study

TL;DR

This paper constructs explicit geometric models for moduli spaces of semi-stable rank two parabolic Higgs bundles over the Riemann sphere, analyzing their behavior under weight variation and wall-crossing.

Contribution

It introduces a general combinatorial approach to construct and analyze moduli spaces of parabolic Higgs bundles in genus zero, applicable beyond the specific case studied.

Findings

Explicit geometric models for moduli spaces are constructed.

The behavior under weight variation and wall-crossing is thoroughly analyzed.

The techniques are applicable to a broad class of moduli problems in genus zero.

Abstract

We construct explicit geometric models for moduli spaces of semi-stable strongly parabolic Higgs bundles over the Riemann sphere, in the case of rank two, four marked points, arbitrary degree, and arbitrary weights. The mechanism of construction relies on elementary geometric and combinatorial techniques, based on a detailed study of orbit stability of (in general non-reductive) bundle automorphism groups on certain carefully crafted spaces. The aforementioned techniques are not exclusive to the case we examine, and this work elucidates a general approach to construct arbitrary moduli spaces of semi-stable parabolic Higgs bundles in genus 0, which is encoded into the combinatorics of weight polytopes. We also present a comprehensive analysis of the geometric models' behavior under variation of parabolic weights and wall-crossing, which is concentrated on their nilpotent cones.