$$T$$ T -stability for Higgs sheaves over compact complex manifolds

Abstract

We introduce the notion of T -stability for torsion-free Higgs sheaves as a natural generalization of the notion of T -stability for torsion-free coherent sheaves over compact complex manifolds. We prove similar properties to the classical ones for Higgs sheaves. In particular, we show that only saturated flags of torsion-free Higgs sheaves are important in the definition of T -stability. Using this, we show that this notion is preserved under dualization and tensor product with an arbitrary Higgs line bundle. Then, we prove that for a torsion-free Higgs sheaf over a compact Kähler manifold, ω-stability implies T -stability. As a consequence of this, we obtain the T -semistability of any reflexive Higgs sheaf with an admissible Hermitian-Yang-Mills metric. Finally, we prove that T -stability implies ω-stability if, as in the classical case, some additional requirements on the base manifold are assumed. In that case, we obtain the existence of admissible Hermitian-Yang-Mills metrics on any T -stable reflexive sheaf.

On Gieseker stability for Higgs sheaves

“…From these definitions it is clear that any Higgs sheaf of rank one is Gieseker stable. Now, as a consequence of (7) we get a relation between the notions of stability and Gieseker stability, which is indeed an extension of a classical proposition in [13]. To be precise we have the following result.…”

“…To be precise we have the following result. Proof: Assume first that E is stable and let F be a Higgs subsheaf of E with 0 < rk F < rk E. Then from (7) it follows that (for any integer k)…”

“…Let E be a Gieseker semistable Higgs sheaf over X. Following the classical definition for coherent sheaves [8,12,13] and in analogy to the definition of flags for Higgs sheaves [7], a Jordan-Hölder filtration of E is a family {E i } s+1 i=0 of Higgs subsheaves of E with…”

“…Let E be a torsion-free Higgs sheaf, as it was shown in [7] we can use a canonical isomorphism of coherent sheaves to construct a Higgs morphism for det E, and hence, this determinant can be considered as a Higgs line bundle det E. Now, from a classical result of coherent sheaves [13], we know that any injective morphism between torsion-free sheaves of the same rank induces an injective morphism between its determinant bundles. From all this we get the following result.…”

“…In particular, there are Bochner's vanishing theorems for Higgs bundles [6]; and there are extensions of the Hitchin-Kobayashi correspondence to polystable reflexive Higgs sheaves [2] (a reflexive Higgs sheaf is polystable if and only if it has an admissible HYM-metric) and to semistable Higgs bundles [3,14]; in this case, the differential geometric counterpart is the notion of approximate Hermitian-Yang-Mills metric (from now on abbreviated apHYM-metric). More recently [7], the notion of T -stability (also called Bogomolov stability) has been studied in the context of Higgs sheaves over compact complex manifolds. In this article, we study the notion of Gieseker stability for Higgs sheaves and we prove some basic properties, as we said before, some of these results can be seen as natural extensions of classical results for coherent sheaves.…”