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.

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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.

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O. IntroductionThe purpose of this paper is to prove that certain moduli spaces of connections in complex vector bundles over compact complex manifolds carry natural complex structures.Let E be any differentiable complex vector bundle over a compact complex manifold M. In the first part of our paper we give a differential geometric construction for the set J/'(E) of isomorphism classes of simple holomorphic bundles differentiably equivalent to E. This moduli space is a locally Hausdorff complex analytic space of finite dimension. In general it is non-reduced. Our approach is inspired by the fundamental papers of Atiyah-Hitchin-Singer [2], Singer [15] and Itoh's construction [5]. We obtain d/t'~(E) as a closed subspace in the moduli space ~(E) of all simple semi-connections of type (0,1) in E; this space ~r is a (non separated) complex analytic Hilbert manifold (after completion with respect to suitable L2-Sobolev norms). We use Kuranishi's method to describe local models for ~//~(E); locally around a point lEa] ~ ~r this space is the zero set of a (finite dimensional) complex analytic map H a :HI(End(EA))~H2(End(EA)) between cohomotogy spaces of the bundle End(Ea) of holomorphic endomorphisms..//S(E) is smooth at [EA] if H2(sl(Ea)) vanishes, where sI(EA) is the bundle of tracefree holomorphic endomorphisms of E A.We have to mention several papers which are related to this result. In [14] Norton constructs moduli spaces of simple holomorphic bundles on compact K~ihler manifolds; he also discusses the non-Hausdorff phenomena, but his construction is completely different from ours. Also for K~ihler manifolds our result has been obtained independently by Kobayashi [9].The second part of our paper treats moduli spaces of Yang-Mills connections over K~ihler manifolds. Let M be a compact Kiihler manifold with Kiihler form 09 and E a complex vector bundle over M with a fixed Hermitian metric h. An * Supported by the Heisenberg program of the DFG