Electronic ISBN:  9781470404673 
Product Code:  MEMO/183/863.E 
97 pp 
List Price:  $63.00 
MAA Member Price:  $56.70 
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Book DetailsMemoirs of the American Mathematical SocietyVolume: 183; 2006MSC: Primary 53; 32; 58;
We prove a very general KobayashiHitchin 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 KobayashiHitchin correspondence relates the complex geometric concept “polystable oriented holomorphic pair” to the existence of a reduction solving a generalized HermitianEinstein equation. The proof is based on the UhlenbeckYau 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 nonKählerian framework: Oriented holomorphic structures, Quotspaces, oriented holomorphic pairs and oriented vortices, nonabelian SeibergWitten monopoles.

Table of Contents

Chapters

1. Introduction

2. The finite dimensional KobayashiHitchin correspondence

3. A “universal” complex geometric classification problem

4. HermitianEinstein pairs

5. Polystable pairs allow HermitianEinstein reductions

6. Examples and Applications

7. Appendix


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We prove a very general KobayashiHitchin 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 KobayashiHitchin correspondence relates the complex geometric concept “polystable oriented holomorphic pair” to the existence of a reduction solving a generalized HermitianEinstein equation. The proof is based on the UhlenbeckYau 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 nonKählerian framework: Oriented holomorphic structures, Quotspaces, oriented holomorphic pairs and oriented vortices, nonabelian SeibergWitten monopoles.

Chapters

1. Introduction

2. The finite dimensional KobayashiHitchin correspondence

3. A “universal” complex geometric classification problem

4. HermitianEinstein pairs

5. Polystable pairs allow HermitianEinstein reductions

6. Examples and Applications

7. Appendix