The moduli stack of parabolic bundles over the projective line, quiver representations, and the Deligne-Simpson problem

TL;DR

This paper investigates the conditions under which the moduli stack of parabolic bundles over the projective line is 'very good' and applies this to analyze solutions to the Deligne-Simpson problem, connecting geometric structures with representation theory.

Contribution

It establishes a sufficient condition for the moduli stack of parabolic bundles on the projective line to be 'very good' and explores its implications for the Deligne-Simpson problem.

Findings

Identified conditions for the 'very good' property of the moduli stack

Connected the geometric property to solutions of the Deligne-Simpson problem

Extended the concept of 'very good' stacks to genus zero cases

Abstract

In "Quantization of Hitchin's Integrable System and Hecke Eigensheaves", Beilinson and Drinfeld introduced the "very good" property for a smooth complex equidimensional stack. They prove that for a semisimple complex group G, the moduli stack of G-bundles over a smooth complex projective curve X is "very good", as long as X has genus g > 1. In the case of the projective line, when g = 0, this is not the case. However, the result can sometimes be extended to the projective line by introducing additional parabolic structure at a collection of marked points and slightly modifying the definition of a "very good" stack. We provide a sufficient condition for the moduli stack of parabolic vector bundles over the complex projective line to be very good. We then use this property to study the space of solutions to the Deligne-Simpson problem.