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The Universal Kobayashi-Hitchin Correspondence on Hermitian Manifolds

M. Lübke Leiden University, Leiden, The Netherlands
A. Teleman CMI, Marseille, France
Available Formats:
Electronic ISBN: 978-1-4704-0467-3
Product Code: MEMO/183/863.E
List Price: $63.00 MAA Member Price:$56.70
AMS Member Price: $37.80 Click above image for expanded view The Universal Kobayashi-Hitchin Correspondence on Hermitian Manifolds M. Lübke Leiden University, Leiden, The Netherlands A. Teleman CMI, Marseille, France Available Formats:  Electronic ISBN: 978-1-4704-0467-3 Product Code: MEMO/183/863.E  List Price:$63.00 MAA Member Price: $56.70 AMS Member Price:$37.80
• Book Details

Memoirs of the American Mathematical Society
Volume: 1832006; 97 pp
MSC: Primary 53; 32; 58;

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 ideas 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.

• Chapters
• 1. Introduction
• 2. The finite dimensional Kobayashi-Hitchin correspondence
• 3. A “universal” complex geometric classification problem
• 4. Hermitian-Einstein pairs
• 5. Polystable pairs allow Hermitian-Einstein reductions
• 6. Examples and Applications
• 7. Appendix
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Volume: 1832006; 97 pp
MSC: Primary 53; 32; 58;

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 ideas 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.

• Chapters
• 1. Introduction
• 2. The finite dimensional Kobayashi-Hitchin correspondence
• 3. A “universal” complex geometric classification problem
• 4. Hermitian-Einstein pairs
• 5. Polystable pairs allow Hermitian-Einstein reductions
• 6. Examples and Applications
• 7. Appendix
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