Limiting Properties of the Yang-Mills flow on Kahler Manifolds

Limiting Properties of the Yang-Mills flow on Kahler Manifolds

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Benjamin Sibley, University of Maryland
Fine Hall 314

In this talk, I will give a result about the limit of the Yang-Mills fl‡ow associated to a holomorphic vector bundle E over an arbitrary Kähler manifold (X;ω). In particular, this theorem says that the ‡flow is determined at infi…nity by the holomorphic structure of E. Namely, if we …fix an integrable unitary reference connection A0 de…fining the holomorphic structure, then the Yang-Mills ‡flow with initial condition A0, converges (away from an appropriately defi…ned 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 …ltration of (E;A0). Moreover, E extends as a re‡exive 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.