On Gieseker stability for Higgs sheaves
arXiv:1603.03100 · doi:10.1016/j.difgeo.2017.06.003
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-Hölder and Harder-Narasimhan filtrations for Higgs sheaves.
arXiv comment16 pages, some minor corrections, last section has been shortened
References in corpus (5)
- Yang-Mills equation for stable Higgs sheaves
- Approximate Hermitian-Yang-Mills structures and semistability for Higgs bundles. I: Generalities and the one-dimensional case
- Approximate Hermitian-Yang-Mills structures and semistability for Higgs bundles. II: Higgs sheaves and admissible structures
- On vanishing theorems for Higgs bundles
- -stability for Higgs sheaves over compact complex manifolds