Asymptotics of the Yang–Mills flow for holomorphic vector bundles over Kähler manifolds: The canonical structure of the limit

Abstract

Abstract In the following article we study the limiting properties of the Yang–Mills flow associated to a holomorphic vector bundle E over an arbitrary compact Kähler manifold (X,ω). In particular we show that the flow is determined at infinity by the holomorphic structure of E. Namely, if we fix an integrable unitary reference connection A 0 defining the holomorphic structure, then the Yang–Mills flow with initial condition A 0 converges (away from an appropriately defined singular set) in the sense of the Uhlenbeck compactness theorem to a holomorphic vector bundle E ∞, which is isomorphic to the associated graded object of the Harder–Narasimhan–Seshadri filtration of (E,A 0). Moreover, E ∞ extends as a reflexive sheaf over the singular set as the double dual of the associated graded object. This is an extension of previous work in the cases of one and two complex dimensions and proves the general case of a conjecture of Bando and Siu.