Screen LibraryThe Kobayashi-Hitchin correspondence /
By the Kobayashi-Hitchin correspondence, the authors of this book mean the isomorphy of the moduli spaces Mst of stable holomorphic - resp. MHE of irreducible Hermitian-Einstein - structures in a differentiable complex vector bundle on a compact complex manifold. They give a complete proof of this r...
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| Published: |
World Scientific Pub. Co.,
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| Publisher Address: | Singapore : |
| Publication Dates: | 1995. |
| Literature type: | eBook |
| Language: | English |
| Subjects: | |
| Online Access: |
http://www.worldscientific.com/worldscibooks/10.1142/2660#t=toc |
| Summary: |
By the Kobayashi-Hitchin correspondence, the authors of this book mean the isomorphy of the moduli spaces Mst of stable holomorphic - resp. MHE of irreducible Hermitian-Einstein - structures in a differentiable complex vector bundle on a compact complex manifold. They give a complete proof of this result in the most general setting, and treat several applications and some new examples. After discussing the stability concept on arbitrary compact complex manifolds in chapter 1, the authors consider, in chapter 2, Hermitian-Einstein structures and prove the stability of irreducible Hermitian-Einstein bundles. This implies the existence of a natural map I from MHE to Mst which is bijective by the result of (the rather technical) chapter 3. In chapter 4 the moduli spaces involved are studied in detail, in particular it is shown that their natural analytic structures are isomorphic via I. Also a comparison theorem for moduli spaces of instantons resp. stable bundles is proved; this is the form in which the Kobayashi-Hitchin has been used in Donaldson theory to study differentiable structures of complex surfaces. The fact that I is an isomorphism of real analytic spaces is applied in chapter 5 to show the openness of the stability condition and the existence of a natural Hermitian metric in the moduli space, and to study, at least in some cases, the dependence of Mst on the base metric used to define stability. Another application is a rather simple proof of Bogomolov's theorem on surfaces of type VII0. In chapter 6, some moduli spaces of stable bundles are calculated to illustrate what can happen in the general (i.e. not necessarily Kahler) case compared to the algebraic or Kahler one. Finally, appendices containing results, especially from Hermitian geometry and analysis, in the form they are used in the main part of the book are included. |
| Carrier Form: | 1 online resource (viii,254pages) |
| Bibliography: | Includes bibliographical references (pages 242-250) and index. |
| ISBN: | 9789812815439 |
| Index Number: | QA601 |
| CLC: | O187 |
| Contents: | ch. 0. Introduction -- ch. 1. Preparations and basic material. 1.1. Holomorphic structures and integrable connections. 1.2. Gauduchon metrics. 1.3. Degree maps. 1.4. Stability of vector bundles -- ch. 2. Hermitian-Einstein connections and metrics. 2.1. Definitions and first results. 2.2. Vanishing theorem and Chern class inequality. 2.3. Stability of Hermitian-Einstein bundles -- ch. 3. Existence of Hermitian-Einstein metrics in stable bundles. 3.1. The strategy of the proof. 3.2. The continuity method: first step. 3.3. The continuity method: second step. 3.4. The construction of a destabilising subsheaf -- ch. 4. The Kobayashi-Hitchin correspondence. 4.1. Summary. 4.2. Moduli spaces of connections. 4.3. Moduli spaces of holomorphic structures. 4.4. Isomorphy of moduli spaces. 4.5. Local models. 4.6. Instantons and Hermitian-Einstein connections -- ch. 5. Applications. 5.1 Openness of the stability property. 5.2 Dependence on the base metric. 5.3 The natural Hermitian metric in the moduli space. 5.4 A proof of Bogomolov's theorem on surfaces of type VII[symbol] -- ch. 6. Examples of moduli spaces. 6.1. The algebraic case. 6.2. Non-K hler principal elliptic fibre bundles over curves. 6.4. SL(2, C)-bundlcs on principal elliptic bundles over curves of genus 1. 6.5. SL(2, C)-bundles on primary elliptic Hopf surfaces -- ch. 7. Appendices. 7.1. Hermitian geometry. 7.2. Elliptic operators. 7.3. Sobolev spaces. 7.4. Local diagonalisation. 7.5. Analytic subspaces of a Banach manifold. |