Contents Chapter 1. Introduction 1 Chapter 2. The finite dimensional Kobayashi-Hitchin correspondence 11 2.1. Analytic Stability, Symplectic stability 11 2.2. The Continuity Method in the finite dimensional case 15 2.3. Maximal weight functions for linear and projective actions 18 Chapter 3. A "universal" complex geometric classification problem 23 3.1. Oriented holomorphic pairs 23 3.2. The stability condition for universal oriented holomorphic pairs 24 Chapter 4. Hermitian-Einstein pairs 33 4.1. The Hermitian-Einstein equation 33 4.2. Pairs which allow Hermitian-Einstein reductions are polystable 34 Chapter 5. Polystable pairs allow Hermitian-Einstein reductions 39 5.1. The perturbed equation 39 5.2. A priori estimates for the solution s£ 41 5.3. Solving the equation (e£) for s G (0,1]. 42 5.4. Destabilizing the pair in the unbounded case 48 Chapter 6. Examples and Applications 57 6.1. Oriented holomorphic principal bundles and oriented connections 57 6.2. Moduli spaces of oriented pairs 64 6.3. Non-abelian monopoles on Gauduchon surfaces 70 Chapter 7. Appendix 77 7.1. Chern connections 77 7.2. Orbits of the adjoint action, sections in the adjoint bundle 80 7.3. Local maximal torus reductions of a if-bundle 84 7.4. Connection and maximal torus reductions 85 7.5. Analytic results 88 7.6. Weakly holomorphic parabolic reductions 89 Bibliography 95

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