Convergence properties of the Yang-Mills flow on Kaehler surfaces

Details Convergence properties of the Yang-Mills flow on Kaehler surfaces Convergence properties of the Yang-Mills flow on Kaehler surfaces

2004-10-04

arxiv.org/abs/math/0410055v1

Mathematics

Differential Geometry

30 pages. To appear in Crelle's Journal

Scientific paper

Let $E$ be a hermitian complex vector bundle over a compact K\"ahler surface $X$ with K\"ahler form $\omega$, 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 $(X,\omega)$ 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 $\omega$-Harder-Narasimhan-Seshadri filtration of the holomorphic bundle $(E,D^{\prime\prime})$. This generalizes to K\"ahler surfaces the known result on Riemann surfaces and proves, in this case, a conjecture of Bando and Siu.

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