This book is printed on acid-free paper. Preface.V When in 1993, encouraged by several colleagues, wc decided to write this book, we had two main reasons to do so.On the one hand, we were able to give a complete proof of the Kobayashi-Hitchin correspondence, i.e. the natural isomorphy of the moduli spaces of stable holomorphic structures respectively irreducible Hermitian-Einstein connections in a differentiate complex vector bundle over a compact complex manifold. In particular, we could give this proof in the most general Hermitian context, whereas in most of the existing literature on this subject only algebraic or Kahler manifolds were considered; even for these cases, there was no single reference containing a complete proof of the correspondence in detail.On the other hand, the Kobayashi-Hitchin correspondence had found important applications. In Donaldson theory, it had been used in the algebraic context to compute moduli spaces of iastantons by algebraic-geometric methods, and thus had been an important tool in proving spectacular results in 4-dimcnsional differential topology. (In fact, this was our main motivation.) Furthermore, it had been used in the non-Kahler case to give a new and comparatively simple proof of Bogomolov's theorem on surfaces of type VIIoTherefore, we thought it might be useful to present a complete, as far as possible self-contained, and hopefully readable proof of the correspondence and some of its applications.Although at the end of 1994 it became apparent that many results in Donaldson theory can be proved in a much simpler way, by means of the newly discovered Scibcrg-Witten invariants, and using only a very simple variant of the KobayashiHitchin correspondence, wc still think that this fundamental result is important and interesting enough to justify the publication of this book.Acknowledgements.

We prove that any class V II surface with b2 = 1 has curves. This implies the "Global Spherical Shell conjecture" in the case b2 = 1:Any minimal class V II surface with b2 = 1 admits a global spherical shell, hence it is isomorphic to one of the surfaces in the known list.By the results in [LYZ], [Te1], which treat the case b2 = 0 and give complete proofs of Bogomolov's theorem, one has a complete classification of all class V II-surfaces with b2 ∈ {0, 1}.Therefore the intersection form H 2 (X, Z)/Tors × H 2 (X, Z)/Tors → Z is negative definite so, by Donaldson's first theorem, it is trivial over Z.Class V II surfaces are not classified yet. This is probably the most important gap in the Enriques-Kodaira classification table. The case b 2 = 0 is completely understood:Theorem 1.1 Every class V II-surface with b 2 = 0 is isomorphic to either a Hopf surface or an Inoue surface. This result was stated by Bogomolov a long time ago [Bo1], [Bo2], but his proof is long and difficult to follow (see [BHPV] p. 230); complete proofs appeared in [Te1] and [LYZ]. Both proofs are based on the Kobayashi-Hitchin correspondence on non-Kählerian surfaces (see [Bu1], [LY], [LT1]) applied to a single holomorphic bundle: the tangent bundle of the surface.The main result of this paper is: Theorem 1.2 Let X be a class V II surface with b 2 (X) = 1. Then X has an effective divisor C > 0 withwhere c Q 1 stands for the first Chern class in rational cohomology. Using Theorem 11.2 in [Na], one concludes that Corollary 1.3 Any minimal class V II surface X with b 2 (X) = 1 possesses a spherical shell, hence it belongs to the known class of surfaces.

We develop a general strategy, based on gauge theoretical methods, to prove existence of curves on class VII surfaces. We prove that, for b 2 D 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 b 2 D 1 was solved in a previous article. The fundamental object intervening in our strategy is the moduli space ᏹ pst .0; / of polystable bundles Ᏹ with c 2 .Ᏹ/ D 0, det.Ᏹ/ D . For large b 2 the geometry of this moduli space becomes very complicated. The case b 2 D 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 b 2 3. Contents

IntroductionRecently, Seiberg and Witten [W] introduced new invariants of 4-manifolds, which are defined by counting solutions of a certain non-linear differential equation.The new invariants are expected to be equivalent to Donaldson's polynomial-invariants-at least for manifolds of simple type [KM 1]-and they have already found important applications, like e.g. in the proof of the Thom conjecture by Kronheimer and Mrowka [KM 2].Nevertheless, the equations themselves remain somewhat mysterious, especially from a mathematical point of view.The present paper contains our attempt to understand and to generalize the Seiberg-Witten equations by coupling them to connections in unitary vector bundles, and to relate their solutions to more familiar objects, namely stable pairs. Fix a Spin c -structure on a Riemannian 4-manifold X, and denote by Σ ± the associated spinor bundles. The equations which we will study are: * Partially supported by: AGE-Algebraic Geometry in Europe, contract No ER-BCHRXCT940557 (BBW 93.0187), and by SNF, Proposition 1.4 Let J be a g-orthogonal almost complex structure on X, compatible with the orientation. i) The spinor bundles Σ ± J ofP J are:is given by Λ 20 J ⊕Λ 02 J ⊕Λ 00 ω g ∋ (λ 20 , λ 02 , ω g ) Γ −→ 2 −i − * (λ 20 ∧ ·) λ 02 ∧ · i ∈ End 0 (Λ 00 ⊕Λ 02 ).

We prove a very general Kobayashi-Hitchin correspondence on arbitrary compact Hermitian manifolds, and we discuss differential geometric properties of the corresponding moduli spaces. This correspondence refers to moduli spaces of "universal holomorphic oriented pairs". Most of the classical moduli problems in complex geometry (e. g. holomorphic bundles with reductive structure groups, holomorphic pairs, holomorphic Higgs pairs, Witten triples, arbitrary quiver moduli problems) are special cases of this universal classification problem. Our Kobayashi-Hitchin correspondence relates the complex geometric concept "polystable oriented holomorphic pair" to the existence of a reduction solving a generalized Hermitian-Einstein equation. The proof is based on the Uhlenbeck-Yau continuity method. Using idea from Donaldson theory, we further introduce and investigate canonical Hermitian metrics on such moduli spaces. We discuss in detail remarkable classes of moduli spaces in the non-Kählerian framework: Oriented holomorphic structures, Quot-spaces, oriented holomorphic pairs and oriented vortices, non-abelian Seiberg-Witten monopoles.