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

Abstract

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 $i\Lambda F(A_t)$ approaches in $L^2$ 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 $E_{\infty }$ is isomorphic to $Gr(E{)}^{\star \star }$, verifying a conjecture of Bando and Siu.

arXiv commentCorrection of a few typos, and some small changes in phrasing and exposition to increase readability, as requested by the referee. The main result is unchanged. Final version to appear in the American Journal of Mathematics. 34 pages

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• Remarks on the Yang-Mills flow on a compact Kahler manifold
• A note on curvature estimate of the Hermitian-Yang-Mills flow