Construction of Hyperkähler Metrics for Complex Adjoint Orbits

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3.3 Regular Nahm Solutions and Line bundles on S 42 The conditions we must impose are f1(0) = f2(0) = f3(0), ⎛ f ⎜ det ⎜ ⎝ ′ 1(0) f 1 λ1 ′ 2(0) 1 λ2 f ′ ⎞ ⎟ = 0, ⎠ ⎛ ⎜ det ⎜ ⎝ 3(0) 1 λ3 ˜ f1(0) = ˜ f2(0) = ˜ f3(0), from which we easily get that α1 and β1 must satisfy ⎧ ⎪⎨ ⎪⎩ d exp(λ1(s + a) + λ 2 1c) α1 + exp(λ1(s + a) + λ 2 1c) 3 i=1 3 i=1 ˜f ′ 1(0) 1 λ1 ˜f ′ 2(0) 1 λ2 ˜f ′ 3(0) 1 λ3 exp(λi(s + a) + λ 2 i c) j=i (λi − λj) ⎞ ⎟ = 0, ⎠ β1 = 0, exp(−λi(s + a) − λ2 i c) j=i (λi − λj) α1 − b β1 = 0. The condition (3.35) fails at a point s exactly when this system has a non-zero solution, that is to say if and only if ⎛ ⎜ det ⎜ ⎝ 3 i=1 in other words if where τ(t) = bd + d exp(−λi(s + a) − λ 2 i c) j=i (λi − λj) 3 i=1 3 i=1 τ(s + a) = 0, exp(λit + λ2 i c) j=i (λi 3 − λj) i=1 Now since the λ ′ is are real, we clearly have lim |τ(t)| = ∞; t→−∞ exp(λi(s + a) + λ 2 i c) j=i (λi − λj) −b ⎞ exp(−λit − λ2 i c) j=i (λi . − λj) ⎟ = 0, ⎟ ⎠
3.3 Regular Nahm Solutions and Line bundles on S 43 in particular, τ(t) is non-vanishing for large values of |t|. If a0 = a0(b, c, d) is such that τ(t) = 0 for t < a0, we see that if a < a0, then τ(s + a) = 0 for s ∈ (−∞, 0]. This means that given b, c, d, then for each a < a0(b, c, d) the line bundle M with transition function as in (3.36) is such that ML s has no sections = 0, for all s ∈ (−∞, 0]. One may state this in a slightly different way as follows: given any line bundle N on S of degree 0, there exists a0 ∈ R such that M = NL a satisfies the condition above for any a < a0. In the general case (i.e. for any k), writing the transition function for M in the form k−1 exp(aη/ζ) exp( (η/ζ) i qi(ζ)), we can show similarly that the condition H 0 (S; ML s (k − 2)) = 0 is given by τ(s + a) = 0, where τ(t) is a polynomial in exp(±λit), i = 1, 2, . . . k. Reasoning as before we see, since the λ ′ is are real, that τ is non-vanishing for i=2 large values of |t|, thus proving Propositon 3.5. Example In order to illustrate the discussion above, consider the follow- ing example, going back to the case k = 3. Let N = O be the trivial bundle (that is, a = b = c = d = 0.) We obtain 3 exp(λit) τ(t) = j=i (λi 3 exp(−λit) − λj) j=i (λi ; − λj) i=1 elementary calculus shows that t = 0 is the only zero of τ, so that and so, if a < 0, i=1 H 0 (L s (k − 2)) = 0 for s ∈ (−∞, 0), H 0 (L a L s (k − 2)) = 0 for s ∈ (−∞, 0].
Page 1 and 2: Construction of Hyperkähler Metric
Page 3 and 4: Contents Acknowledgements 3 Declara
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 43: 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 57 and 58: 4.1 The Kronheimer-Biquard Moduli S
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
Page 71 and 72: 4.3 Regular Sections of g c ⊗ O(2
Page 73 and 74: 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
Page 79 and 80: 4.4 Regular Twistor Lines 77 as req
Page 81 and 82: 4.5 Construction of the Metric 79 d
Page 83 and 84: 4.5 Construction of the Metric 81
Page 85 and 86: 4.5 Construction of the Metric 83 f
Page 87 and 88: 4.5 Construction of the Metric 85 S
Page 89 and 90: 4.5 Construction of the Metric 87 I
Page 91 and 92: 4.5 Construction of the Metric 89 [

Construction of Hyperkähler Metrics for Complex Adjoint Orbits

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