2016/10/07 by Samuel Trautwein, Trautwein, Samuel
Mathematics · #14L24 (Secondary) #53C07 (Primary) #53D20 #Differential Geometry (math.DG) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Geometric and Algebraic Topology #Geometry and complex manifolds
paper · pdf · doi:10.48550/arxiv.1610.02245
openalex publication_date 2016/10/07 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
The symplectic vortex equations admit a variational description as global\nminimum of the Yang-Mills-Higgs functional. We study its negative gradient flow\non holomorphic pairs (A,u) where A is a connection on a principal\nG-bundle P over a closed Riemann surface \Σ and u: P \→ X\nis an equivariant map into a K "ahler Hamiltonian G-manifold. The connection\nA induces a holomorphic structure on the K "ahler fibration P\×G X and\nwe require that u descends to a holomorphic section of this fibration.\n We prove a Lojasiewicz type gradient inequality and show uniform convergence\nof the negative gradient flow in the W1,2\× W2,2-topology when X\nis equivariantly convex at infinity with proper moment map, X is\nholomorphically aspherical and its K "ahler metric is analytic.\n As applications we establish several results inspired by finite dimensional\nGIT: First, we prove a certain uniqueness property for the critical points of\nthe Yang-Mills-Higgs functional which is the analogue of the Ness uniqueness\ntheorem. Second, we extend Mundet's Kobayashi-Hitchin correspondence to the\npolystable and semistable case. The arguments for the polystable case lead to a\nnew proof in the stable case. Third, in proving the semistable correspondence,\nwe establish the moment-weight inequality for the vortex equation and prove the\nanalogue of the Kempf existence and uniqueness theorem.\n