CHAPTER 1 Introduction In order to understand the aim and the motivation of this article let us consider the following classical complex geometric moduli problems: Holomorphic structures with fixed determinant. Let X be a compact complex n-dimensional manifold and E a differentiate rank r vector bundle on X. We fix a holomorphic structure C on the determinant line bundle detE of X. The problem is to classify all holomorphic structures £ on E which induce the fixed holomorphic structure C on det E modulo the group T(X, SL(E)) of automorphisms of determinant 1. In order to get a Hausdorff moduli spaces with good properties one considers only stable (or more general semistable) holomorphic structures. The stability con- dition depends on the choice of a Gauduchon metric on X, or more generally a Hermitian metric g (see [LT]) which plays the same role as the choice of a polar- ization of the base manifold in algebraic geometry. £ is called ^-stable if (1.1) fMgiF) Hg(£) for any nontrivial subsheaf T C £ with torsion free quotient. Such a subsheaf must be reflexive. Note that the slope map fig is not a topological invariant in the general non-Kahlerian framework, but it is always a holomorphic invariant. The classical Kobayashi-Hitchin correspondence ([Kol], [Dol], [Do2], [UY1], [UY2], [Bu], [LY], [LT]) states that a holomorphic structure is stable if and only if it is simple (i. e. it admits no non-trivial trace free infinitesimal automorphisms) and admits a Hermitian-Einstein metric, i. e. a metric h solving the Hermit ian- Einstein equation: 2-7T lA °Fh = (n-l)\Vol g (xf9{mE ' General solutions of the Hermitian-Einstein equation correspond to polystable holo- morphic structures, i. e. to bundles which are direct sums of stable bundles of the same slope. From a historical point of view, around 1980 the correspondence was indepen- dently suggested by Hitchin and Kobayashi (according to Donaldson [Dol] who first linked the two names together1). First indications that a connection between these two concepts might exist had been e.g. Kobayashi's proof that a Hermitian- Einstein bundle is stable in the sense of Bogomolov [Ko4], and the first author's proof of the Chern class inequality for Hermitian-Einstein bundles [Liil]. The "sim- ple" implication in this case, i.e. the polystability of Hermitian-Einstein bundles, was first written down in Japanese by Kobayashi in [Ko3] and announced in [Kol] the first author then found a simpler proof [Lii2]. The "difficult" implication, i.e. We thank the referee for pointing out this fact and reference [KO]. 1
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