Instantons and curves on class VII surfaces | Annals of Mathematics

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 was solved in a previous article. The fundamental object intervening in our strategy is the moduli space pst(0,) of polystable bundles  with c2()=0, det()=. 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. We explain the substantial obstacles which must be overcome in order to extend our methods to the case b2≥3.

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数学年刊(Annals of Mathematics)

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