Approximate Hermitian-Yang-Mills structures and semistability for Higgs bundles. II: Higgs sheaves and admissible structures

Abstract

We study the basic properties of Higgs sheaves over compact Kähler manifolds and we establish some results concerning the notion of semistability; in particular, we show that any extension of semistable Higgs sheaves with equal slopes is semistable. Then, we use the flattening theorem to construct a regularization of any torsion-free Higgs sheaf and we show that it is in fact a Higgs bundle. Using this, we prove that any Hermitian metric on a regularization of a torsion-free Higgs sheaf induces an admissible structure on the Higgs sheaf. Finally, using admissible structures we proved some properties of semistable Higgs sheaves.

18 pages; some typos corrected

References in corpus (2)

• Yang-Mills equation for stable Higgs sheaves
• Existence of approximate Hermitian-Einstein structures on semi-stable Higgs bundles

Cited by in corpus (8)

• The Hitchin--Kobayashi Correspondence for Quiver Bundles over Generalized Kähler Manifolds
• On vanishing theorems for Higgs bundles
• Approximate Hermitian-Yang-Mills structures on semistable principal Higgs bundles
• On Gieseker stability for Higgs sheaves
• $T$-stability for Higgs sheaves over compact complex manifolds
• On a functional of Kobayashi for Higgs bundles
• Non-existence of Higgs fields on Calabi-Yau Manifolds
• On $2k$-Hitchin's equations and Higgs bundles: a survey