Limiting Properties of the Yang-Mills flow on Kahler Manifolds
Limiting Properties of the Yang-Mills flow on Kahler Manifolds
In this talk, I will give a result about the limit of the Yang-Mills fl聡ow associated to a holomorphic vector bundle E over an arbitrary K盲hler manifold (X;蠅). In particular, this theorem says that the 聡flow is determined at infi聟nity by the holomorphic structure of E. Namely, if we 聟fix an integrable unitary reference connection A0 de聟fining the holomorphic structure, then the Yang-Mills 聡flow with initial condition A0, converges (away from an appropriately defi聟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 聟ltration of (E;A0). Moreover,听E extends as a re聡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.