A GIT interpretration of the Harder-Narasimhan filtration

TL;DR

This paper demonstrates that the GIT instability analysis via Kempf's maximally destabilizing 1-parameter subgroup aligns with the classical Harder-Narasimhan filtration for unstable torsion free sheaves on smooth projective varieties.

Contribution

It establishes a precise correspondence between GIT destabilization and the Harder-Narasimhan filtration, connecting geometric invariant theory with sheaf stability theory.

Findings

GIT unstable points correspond to sheaves with non-trivial Harder-Narasimhan filtrations.

Kempf's destabilizing subgroup induces the Harder-Narasimhan filtration.

The result bridges GIT stability analysis and classical sheaf theory.

Abstract

An unstable torsion free sheaf on a smooth projective variety gives a GIT unstable point in certain Quot scheme. To a GIT unstable point, Kempf associates a "maximally destabilizing" 1-parameter subgroup, and this induces a filtration of the torsion free sheaf. We show that this filtration coincides with the Harder-Narasimhan filtration.