Asymptotics of the Yang-Mills Flow for Holomorphic Vector Bundles over Kahler Manifolds
Asymptotics of the Yang-Mills Flow for Holomorphic Vector Bundles over Kahler Manifolds
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Date
2013
Authors
Sibley, Benjamin Caleb
Advisor
Wentworth, Richard A
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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.