Optimum weight chamber examples of moduli spaces of stable parabolic bundles in genus 0

TL;DR

This paper explicitly constructs moduli spaces of rank 2 stable parabolic bundles over the Riemann sphere for specific weight chambers, revealing their structure and applications to Hitchin's integrable system.

Contribution

It provides explicit models of moduli spaces in genus 0 for optimal weight chambers, including their geometric structure and automorphism group actions.

Findings

Moduli spaces are smooth, compact, biholomorphic to projective spaces.

Explicit description of the nilpotent cone locus in Hitchin's system.

Construction varies with even or odd degree, affecting the space's dimension.

Abstract

We present an explicit construction of the moduli spaces of rank 2 stable parabolic bundles of parabolic degree 0 over the Riemann sphere, corresponding to "optimum" open weight chambers of parabolic weights in the weight polytope. The complexity of the different moduli space' weight chambers is understood in terms of the complexity of the actions of the corresponding groups of bundle automorphisms on stable parabolic structures. For the given choices of parabolic weights, $\mathscr{N}$ consists entirely of isomorphism classes of strictly stable parabolic bundles whose underlying Birkhoff-Grothendieck splitting coefficients are constant and minimal, is constructed as a quotient of a set of stable parabolic structures by a group of bundle automorphisms, and is a smooth, compact complex manifold biholomorphic to $\left(\mathbb{C}\mathbb{P}^{1}\right)^{n-3}$ for even degree, and $\mathbb{C}\mathbb{P}^{n-3}$ for odd degree. As an application of the construction of such explicit models, we provide an explicit characterization of the nilpotent cone locus on $T^{*}\mathscr{N}$ for Hitchin's integrable system.