Convergence properties of the Yang-Mills flow on Kaehler surfaces

2004/10/04 by Georgios Daskalopoulos, Georgios D. Daskalopoulos, Daskalopoulos, Georgios D. +3
Mathematics · #Advanced Algebra and Geometry #Algebraic Geometry and Number Theory #Geometry and complex manifolds #math.AP #math.DG #msc:58E15 #msc:81T13

paper · pdf · doi:10.48550/arxiv.math/0410055

30 pages. To appear in Crelle's Journal

arxiv created 2004/10/04 · arxiv updated 2009/12/01

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^′′ 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^′′). This generalizes to Kähler surfaces the known result on Riemann surfaces and proves, in this case, a conjecture of Bando and Siu.

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