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

TL;DR

This paper explores the properties of Higgs sheaves on compact Kähler manifolds, establishing semistability criteria, regularization techniques, and the relationship between Hermitian metrics and admissible structures, advancing the understanding of Higgs sheaves and bundles.

Contribution

It introduces a regularization method for torsion-free Higgs sheaves and links Hermitian metrics to admissible structures, enhancing the theory of Higgs sheaves and their semistability.

Findings

Extension of semistable Higgs sheaves with equal slopes remains semistable

Regularization of torsion-free Higgs sheaves yields Higgs bundles

Hermitian metrics induce admissible structures on Higgs sheaves

Abstract

We study the basic properties of Higgs sheaves over compact K\"ahler 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.