Instantons and curves on class VII surfaces

2007/04/20 by Andrei Teleman, Teleman, Andrei
Mathematics · #32G13 #53C07 #53C55 #Algebraic Geometry (math.AG) #Complex Variables (math.CV) #Differential Geometry (math.DG) #FOS: Mathematics #Geometric Topology (math.GT) #math.AG #math.CV #math.DG #math.GT #msc:32G13 #msc:53C07 #msc:53C55

paper · pdf · doi:10.48550/arxiv.0704.2634

LaTeX 48 pages; RV: minor corrections, new paragraph dedicated to the structure of the moduli space around the circles of reductions; RV: minor corrections, to appear in Annals of Mathematics

arxiv created 2009/09/15 · arxiv updated 2009/12/01

Abstract

We develop a general strategy, based on gauge theoretical methods, to prove existence of curves on class VII surfaces. We prove that, for b₂=2, every minimal class VII surface has a cycle of rational curves hence, by a result of Nakamura, is a global deformation of a one parameter family of blown up primary Hopf surfaces. The case b₂=1 has been solved in a previous article. The fundamental object intervening in our strategy is the moduli space \mathcal M^\pst(0,\mathcal K) of polystable bundles \mathcal E with c₂(\mathcal E)=0, det(\mathcal E)=\mathcal K. For large b₂ the geometry of this moduli space becomes very complicated. The case b₂=2 treated here in detail requires new ideas and difficult techniques of both complex geometric and gauge theoretical nature.

Related

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