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 | Goldman W.M. Rank one Higgs bundles and representations of fundamental groups of Riemann surfaces / Goldman W.M., Xia E.Z. - Providence, R.I.: American Mathematical Society, 2008. - 69 p. - (Memoirs of the American Mathematical Society; Vol. 193, N 904). - ISBN 082184136X
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Introduction .................................................... 1
1. Equivalences of deformation theories ......................... 4
2. The Betti and de Rham deformation theories and their moduli
spaces ....................................................... 5
2.1. The Betti groupoid ...................................... 6
2.2. The de Rham groupoid .................................... 8
2.3. Equivalence of de Rham and Betti groupoids ............. 14
3. The Dolbeault groupoid ...................................... 17
3.1. Holomorphic line bundles ............................... 17
3.2. The moduli spaces ...................................... 21
3.3. Geometric structure of the Dolbeault moduli space ...... 23
4. Equivalence of de Rham and Dolbeault groupoids .............. 25
4.1. Construction of the equivalence ........................ 25
4.2. Higgs coordinates ...................................... 33
4.3. Involutions ............................................ 36
5. Hyperkahler geometry on the moduli space .................... 37
5.1. The quaternionic structure ............................. 37
5.2. The Riemannian metric .................................. 39
5.3. Complex-symplectic structure ........................... 40
5.4. Quaternionization ...................................... 42
6. The twistor space ........................................... 43
6.1. The complex projective line ............................ 43
6.2. The twistor space as a smooth vector bundle ............ 48
6.3. A holomorphic atlas for the twistor space .............. 49
6.4. The twistor lines ...................................... 51
6.5. The real structure on the twistor space ................ 52
6.6. Symplectic geometry of the twistor space ............... 53
6.7. The lattice quotient ................................... 55
6.8. Functions and flows .................................... 56
7. The moduli space and the Riemann period matrix .............. 59
7.1. Coordinates for the Betti moduli space ................. 59
7.2. Abelian differentials and their periods ................ 60
7.3. Flat connections ....................................... 62
7.4. Higgs fields ........................................... 64
7.5. The  *-action in terms of the period matrix ............ 65
7.6. The *-action and the real points ...................... 65
Bibliography ................................................... 67
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