Asymptotics of the Yang-Mills Flow for Holomorphic Vector Bundles over Kahler Manifolds

Abstract

In this thesis we study the limiting properties of the Yang-Mills flow

associated to a holomorphic vector bundle $E$ over an arbitrary K"{a}hler

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.

Chapter 1 is an introduction and a review of the background material. Chapter 2 gives the proof of several critical intermediate results, including the existence of an approximate critical hermitian structure. Chapter 3 concludes the proof of the main theorem.