Construction of Hyperkähler Metrics for Complex Adjoint Orbits

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To my father and my mother
Contents Acknowledgements 3 Declaration 4 Summary 5 1 Introduction 6 2 Preliminaries 9 2.1 Conventions . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 2.2 Nahm’s Equations and Spectral Curves . . . . . . . . . . . . 10 2.3 Hyperkähler Manifolds . . . . . . . . . . . . . . . . . . . . . . 16 2.4 Twistor Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . 17 3 Spectral Curve and the Kähler Potential for Complex Semisim- ple Orbits 19 3.1 The Spectral Curve . . . . . . . . . . . . . . . . . . . . . . . . 20 3.2 Regularity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 3.3 Regular Nahm Solutions and Line bundles on S . . . . . . . . 27 3.3.1 Reality . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 1
Page 1: Construction of Hyperkähler Metric
Page 5 and 6: Acknowledgements I would like to th
Page 7 and 8: Summary We study hyperkähler metri
Page 9 and 10: tion to certain regular lines. We c
Page 11 and 12: Chapter 2 Preliminaries The materia
Page 13 and 14: 2.2 Nahm’s Equations and Spectral
Page 19 and 20: 2.4 Twistor Spaces 17 (iv) If the c
Page 21 and 22: Chapter 3 Spectral Curve and the K
Page 23 and 24: 3.1 The Spectral Curve 21 for smoot
Page 25 and 26: 3.1 The Spectral Curve 23 the secti
Page 27 and 28: 3.2 Regularity 25 Remark It is not
Page 29 and 30: 3.3 Regular Nahm Solutions and Line
Page 47 and 48: 3.4 The Kähler Potential for Integ
Page 49 and 50: 3.4 The Kähler Potential for Integ
Page 51 and 52: 3.5 Example: Kähler Potential for
Page 53 and 54:
3.5 Example: Kähler Potential for
Page 55 and 56:
3.5 Example: Kähler Potential for
Page 57 and 58:
4.1 The Kronheimer-Biquard Moduli S
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Page 61 and 62:
Page 63 and 64:
Page 65 and 66:
4.2 The Twistor Space 63 Here A(0,
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4.2 The Twistor Space 65 Remark. As
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4.3 Regular Sections of g c ⊗ O(2
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4.3 Regular Sections of g c ⊗ O(2
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4.4 Regular Twistor Lines 71 projec
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4.4 Regular Twistor Lines 73 It fol
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4.4 Regular Twistor Lines 75 Lemma
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4.4 Regular Twistor Lines 77 as req
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4.5 Construction of the Metric 79 d
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4.5 Construction of the Metric 81
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4.5 Construction of the Metric 83 f
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4.5 Construction of the Metric 85 S
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4.5 Construction of the Metric 87 I
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4.5 Construction of the Metric 89 [
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Construction of Hyperkähler Metrics for Complex Adjoint Orbits

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