On the Gieseker Harder-Narasimhan filtration for principal bundles

TL;DR

This paper presents examples of orthogonal and symplectic bundles where the Gieseker Harder-Narasimhan filtration of the underlying vector bundle does not match any parabolic reduction of the original principal bundle, highlighting a discrepancy in stability filtrations.

Contribution

The paper provides explicit examples demonstrating that the Gieseker Harder-Narasimhan filtration can differ from parabolic reductions in orthogonal and symplectic bundles, revealing new insights into stability theory.

Findings

Counterexamples for orthogonal bundles

Counterexamples for symplectic bundles

Discrepancy between filtrations and reductions

Abstract

We give an example of an orthogonal bundle where the Harder-Narasimhan filtration, with respect to Gieseker semistability, of its underlying vector bundle does not correspond to any parabolic reduction of the orthogonal bundle. A similar example is given for the symplectic case.