Construction of Hyperkähler Metrics for Complex Adjoint Orbits

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3.3.2 Description of H 0 (S; ML s (k − 1)) . . . . . . . . . . . 32 3.3.3 The Endomorphisms Ãj . . . . . . . . . . . . . . . . . 39 3.3.4 The Theta Divisor Condition . . . . . . . . . . . . . . 40 3.4 The Kähler Potential for Integral Orbits . . . . . . . . . . . . 45 3.5 Example: Kähler Potential for the Eguchi-Hanson metric . . . 49 4 Twistor Geometry of Complex Adjoint Orbits 54 4.1 The Kronheimer-Biquard Moduli Space . . . . . . . . . . . . . 55 4.2 The Twistor Space . . . . . . . . . . . . . . . . . . . . . . . . 62 4.3 Regular Sections of g c ⊗ O(2) . . . . . . . . . . . . . . . . . . 67 4.4 Regular Twistor Lines . . . . . . . . . . . . . . . . . . . . . . 70 4.5 Construction of the Metric . . . . . . . . . . . . . . . . . . . . 78 Bibliography 90 2
Acknowledgements I would like to thank my supervisor, Professor Nigel Hitchin, for many things: for having kindly accepted me as his student in unusual circumstances; for introducing me to research in Mathematics through his enthusiastic style; and for his generous advice, encouragement and patience. This work would not have been possible without his constant help. Throughout my period of study in Warwick I enjoyed the benefit of stim- ulating mathematical discussions; I especially thank Claudio Arezzo, Gavin Brown and Peter Gothen for those. I am also grateful to the latter for helping me with TeX. I would also like to thank Dietmar Salamon for being my official supervisor for the last year and having patiently helped in bureaucratic matters. I have been fortunate in having the company of many friends at the Mathematics Institute, and I take this opportunity to thank each of them. I especially thank my parents, sisters, sister-in-law and brother for their unfailing help and support. Finally, I acknowledge financial support from CNPq, the National Re- search Council from Brazil. 3
Page 1 and 2: Construction of Hyperkähler Metric
Page 3: Contents Acknowledgements 3 Declara
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
Page 59 and 60:
Page 61 and 62:
Page 63 and 64:
Page 65 and 66:
4.2 The Twistor Space 63 Here A(0,
Page 67 and 68:
4.2 The Twistor Space 65 Remark. As
Page 69 and 70:
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
Page 75 and 76:
4.4 Regular Twistor Lines 73 It fol
Page 77 and 78:
4.4 Regular Twistor Lines 75 Lemma
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4.4 Regular Twistor Lines 77 as req
Page 81 and 82:
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
Page 87 and 88:
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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