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

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 A₀ defining the holomorphic structure, then the Yang-Mills flow with initial condition A₀, 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,A₀). 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.

Department dc:contributor.department

Mathematics

Year dc:date.issued

2013

Author dc:creator

• Sibley, Benjamin Caleb

Advisor dc:contributor.advisor

• Wentworth, Richard A

Handle dc:identifier.uri

http://hdl.handle.net/1903/14056

OAI identifier oai:identifier

oai:drum.lib.umd.edu:1903/14056