Harder-Narasimhan stratification for the moduli stack of parabolic vector bundles

TL;DR

This paper develops a Harder-Narasimhan stratification for the moduli stack of parabolic vector bundles on a curve, proving schematicity, completeness, and an analogue of Behrend's conjecture, with comparisons to $ heta$-stratifications.

Contribution

It introduces a new stratification for parabolic vector bundles, establishing schematic and completeness properties, and extends Behrend's conjecture to this setting.

Findings

Stratification is schematic and each stratum is complete.

Established an analogue of Behrend's conjecture for parabolic bundles.

Compared the stratification approach with $ heta$-stratification methods.

Abstract

For every set of parabolic weights, we construct a Harder-Narasimhan stratification for the moduli stack of parabolic vector bundles on a curve. It is based on the notion of parabolic slope, introduced by Mehta and Seshadri. We also prove that the stratification is schematic, that each stratum is complete, and establish an analogue of Behrend's conjecture for parabolic vector bundles. A comparison with recent $\Theta$-stratification approaches is discussed.