On Gieseker stability for Higgs sheaves

In this article 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. Here we prove some properties of Higgs sheaves 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; 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; then we prove that a classical property of morphisms between Gieseker semistable sheaves also holds in the Higgs case; as a consequence of this and because of an existing relation between Mumford-Takemoto stability and Gieseker stability for Higgs sheaves, we obtain certain properties concerning the existence of Hermitian-Yang-Mills metrics, simplesness and extensions. Finally, we make some comments about Jordan-Hölder and Harder-Narasimhan filtrations for Higgs sheaves.