On Gieseker stability for Higgs sheaves

TL;DR

This paper explores Gieseker stability for Higgs sheaves, extending classical stability notions to this context and establishing fundamental properties and relations with other stability concepts.

Contribution

It generalizes Gieseker stability to Higgs sheaves, proving key properties, relations with Mumford-Takemoto stability, and implications for Hermitian-Yang-Mills metrics and filtrations.

Findings

Gieseker stability for Higgs sheaves can be characterized using Higgs subsheaves.

Classical relations between Gieseker and Mumford-Takemoto stability extend to Higgs sheaves.

Direct sums of Gieseker semistable Higgs sheaves are Gieseker semistable if normalized Hilbert polynomials match.

Abstract

We review the notion of Gieseker stability for torsion-free Higgs sheaves. This notion is a natural generalization of the classical notion of Gieseker stability for torsion-free coherent sheaves. We prove some basic properties that are similar to the classical ones for torsion-free coherent sheaves over projective algebraic manifolds. In particular, we show that Gieseker stability for torsion-free Higgs sheaves can be defined using only Higgs subsheaves with torsion-free quotients; and we show that a classical relation between Gieseker stability and Mumford-Takemoto stability extends naturally to Higgs sheaves. We also prove that a direct sum of two Higgs sheaves is Gieseker semistable if and only if the Higgs sheaves are both Gieseker semistable with equal normalized Hilbert polynomial and we prove that a classical property of morphisms between Gieseker semistable sheaves also holds in the Higgs case; as a consequence of this and the existing relation between Mumford-Takemoto stability and Gieseker stability, we obtain certain properties concerning the existence of Hermitian-Yang-Mills metrics, simplesness and extensions in the Higgs context. Finally, we make some comments about Jordan-H\"older and Harder-Narasimhan filtrations for Higgs sheaves.