2006/11/05 by Graeme Wilkin, Wilkin, Graeme · 1 citation
Mathematics · #58E15 #Differential Geometry (math.DG) #FOS: Mathematics #Symplectic Geometry (math.SG) #math.DG #math.SG #msc:58E15
paper · pdf · doi:10.48550/arxiv.math/0611113
45 pages, to appear Communications in Analysis and Geometry
arxiv created 2008/06/06 · arxiv updated 2009/12/01
Here we prove the necessary analytic results to construct a Morse theory for the Yang-Mills-Higgs functional on the space of Higgs bundles over a compact Riemann surface. The main result is that the gradient flow with initial conditions (A'', ϕ) converges to a critical point of this functional, the isomorphism class of which is given by the graded object associated to the Harder-Narasimhan-Seshadri filtration of (A'', ϕ). In particular, the results of this paper show that the failure of hyperkähler Kirwan surjectivity for rank 2 fixed determinant Higgs bundles does not occur because of a failure of the existence of a Morse theory.