Asymptotics of the Yang-Mills Flow for Holomorphic Vector Bundles Over Kähler Manifolds: The Canonical Structure of the Limit

Benjamin Sibley

Published 2012-06-24, updated 2013-07-02Version 3

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 K\"ahler manifold (X,{\omega}). 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_{\infty}, which is isomorphic to the associated graded object of the Harder-Narasimhan-Seshadri filtration of (E,A_0). Moreover, E_{\infty} 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 1 and 2 complex dimensions and proves the general case of a conjecture of Bando and Siu.