The limit of the Yang-Mills flow on semi-stable bundles

TL;DR

This paper investigates the behavior of the Yang-Mills flow on semi-stable bundles over compact Kahler manifolds, establishing a precise isomorphism between the limiting reflexive sheaf and the graded Seshadri filtration's stable quotients.

Contribution

It constructs an explicit isomorphism between the limiting reflexive sheaf and the double dual of the stable quotients of the Seshadri filtration, clarifying the structure of the limit.

Findings

The Yang-Mills flow converges to a Hermitian-Einstein metric on semi-stable bundles.

The limiting reflexive sheaf is isomorphic to the double dual of the stable quotients of the Seshadri filtration.

The structure of the limit is explicitly characterized in terms of the bundle's filtration.

Abstract

By the work of Hong and Tian it is known that given a holomorphic vector bundle E over a compact Kahler manifold X, the Yang-Mills flow converges away from an analytic singular set. If E is semi-stable, then the limiting metric is Hermitian-Einstein and will decompose the limiting bundle into a direct sum of stable bundles. Bando and Siu prove this limiting bundle can be extended to a reflexive sheaf E' on all of X. In this paper, we construct an isomorphism between E' and the double dual of the stable quotients of the graded Seshadri filtration of E.