The Yang-Mills flow and the Atiyah-Bott formula on compact Kahler manifolds

Original · EN

We study the Yang-Mills flow on a holomorphic vector bundle E over a compact Kahler manifold X. Along a solution of the flow, we show the curvature iΛF(Aₜ) approaches in L² an endomorphism with constant eigenvalues given by the slopes of the quotients from the Harder-Narasimhan filtration of E. This proves a sharp lower bound for the Hermitian-Yang-Mills functional and thus the Yang-Mills functional, generalizing to arbitrary dimension a formula of Atiyah and Bott first proven on Riemann surfaces. Furthermore, we show any reflexive extension to all of X of the limiting bundle E∞ is isomorphic to Gr(E)⋆⋆, verifying a conjecture of Bando and Siu.