Convergence properties of the Yang-Mills flow on Kaehler surfaces

Abstract

Let $E$ be a hermitian complex vector bundle over a compact Kähler surface $X$ with Kähler form $ω$, and let $D$ be an integrable unitary connection on $E$ defining a holomorphic structure $D^{\prime\prime}$ on $E$. We prove that the Yang-Mills flow on $(X,ω)$ with initial condition $D$ converges, in an appropriate sense which takes into account bubbling phenomena, to the double dual of the graded sheaf associated to the $ω$-Harder-Narasimhan-Seshadri filtration of the holomorphic bundle $(E,D^{\prime\prime})$. This generalizes to Kähler surfaces the known result on Riemann surfaces and proves, in this case, a conjecture of Bando and Siu.