On Gieseker stability for Higgs sheaves

S. A. H. Cardona; O. Mata-Gutiérrez

doi:10.1016/j.difgeo.2017.06.003

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arXiv · 1603.03100

On Gieseker stability for Higgs sheaves

S. A. H. Cardona·O. Mata-Gutiérrez

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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.

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S. A. H. Cardona, O. Mata-Gutiérrez. 2016-03-09. On Gieseker stability for Higgs sheaves. https://doi.org/10.1016/j.difgeo.2017.06.003

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