PART I PRELIMINARIES
1 Holomorphic functions ........................................ 3
1.1 Simple examples: algebraic functions .................... 3
1.2 Analytic continuation: differential equations ........... 5
Exercises ............................................... 8
2 Surface topology ............................................ 10
2.1 Classification of surfaces ............................. 10
2.2 Discussion: the mapping class group .................... 21
Exercises ................................................... 24
PART II BASIC THEORY
3 Basic definitions ........................................... 29
3.1 Riemann surfaces and holomorphic maps .................. 29
3.2 Examples ............................................... 31
3.2.1 First examples .................................. 31
3.2.2 Algebraic curves ................................ 32
3.2.3 Quotients ....................................... 36
Exercises ................................................... 40
4 Maps between Riemann surfaces ............................... 42
4.1 General properties ..................................... 42
4.2 Monpdromy and the Riemann Existence Theorem ............ 45
4.2.1 Digression into algebraic topology .............. 45
4.2.2 Monodromy of covering maps ...................... 48
4.2.3 Compactifying algebraic curves .................. 50
4.2.4 The Riemann surface of a holomorphic function ... 54
4.2.5 Quotients ....................................... 55
Exercises ................................................... 56
PART III DEEPER THEORY
5 Calculus on surfaces ........................................ 57
5.1 Smooth surfaces ........................................ 57
5.1.1 Cotangent spaces and 1-forms .................... 57
5.1.2 2-forms and integration ......................... 61
5.2 de Rham cohomology ..................................... 67
5.2.1 Definition and examples ......................... 67
5.2.2 Cohomology with compact support, and Poincare
duality ......................................... 70
5.3 Calculus on Riemann surfaces ........................... 73
5.3.1 Decomposition of the 1-forms .................... 73
5.3.2 The Laplace operator and harmonic functions ..... 77
5.3.3 The Dirichlet norm .............................. 78
Exercises ................................................... 81
6 Elliptic functions and integrals ............................ 82
6.1 Elliptic integrals ..................................... 82
6.2 The Weierstrass p function ............................. 86
6.3 Further topics ......................................... 88
6.3.1 Theta functions ................................. 88
6.3.2 Classification .................................. 91
Exercises ................................................... 96
7 Applications of the Euler characteristic .................... 97
7.1 The Euler characteristic and meromorphic forms ......... 97
7.1.1 Topology ........................................ 97
7.1.2 Meromorphic forms ............................... 99
7.2 Applications .......................................... 100
7.2.1 The Riemann-Hurwitz formula .................... 100
7.2.2 The degree-genus formula ....................... 102
7.2.3 Real structures and Harnack's bound ............ 103
7.2.4 Modular curves ................................. 105
Exercises .................................................. 107
8 Meromorphic functions and the Main Theorem for compact
Riemann surfaces ........................................... 111
8.1 Consequences of the Main Theorem ...................... 113
8.2 The Riemann-Roch formula .............................. 115
Exercises .................................................. 117
9 Proof of the Main Theorem .................................. 118
9.1 Discussion and motivation ............................. 118
9.2 The Riesz Representation Theorem ...................... 120
9.3 The heart of the proof ................................ 122
9.4 Weyl's Lemma .......................................... 126
Exercises .................................................. 129
10 The Uniformisation Theorem ................................. 131
10.1 Statement ............................................. 131
10.2 Proof of the analogue of the Main Theorem ............. 133
10.2.1 Set-up ......................................... 133
10.2.2 Classification of behaviour at infinity ........ 135
10.2.3 The main argument .............................. 138
10.2.4 Proof of Proposition 30 ........................ 141
Exercises .................................................. 142
PART IV FURTHER DEVELOPMENTS
11 Contrasts in Riemann surface theory ........................ 147
11.1 Algebraic aspects ..................................... 147
11.1.1 Fields of meromorphic functions ................ 147
11.1.2 Valuations ..................................... 151
11.1.3 Connections with algebraic number theory ....... 156
11.2 Hyperbolic surfaces ................................... 159
11.2.1 Definitions .................................... 159
11.2.2 Models of the hyperbolic plane ................. 161
11.2.3 Self-isometries ................................ 163
11.2.4 Hyperbolic surfaces ............................ 164
11.2.5 Geodesies ...................................... 164
11.2.6 Discussion ..................................... 167
11.2.7 The Gauss-Bonnet Theorem ....................... 169
11.2.8 Right-angled hexagons .......................... 171
11.2.9 Closed geodesies ............................... 173
Exercises .................................................. 177
12 Divisors, line bundles and Jacobians ....................... 178
12.1 Cohomology and line bundles ........................... 178
12.1.1 Sheaves and cohomology ......................... 178
12.1.2 Line bundles ................................... 182
12.1.3 Line bundles and projective embeddings ......... 188
12.1.4 Divisors and unique factorisation .............. 194
12.2 Jacobians of Riemann surfaces ......................... 196
12.2.1 The Abel-Jacobi Theorem ........................ 196
12.2.2 Abstract theory ................................ 198
12.2.3 Geometry of symmetric products ................. 199
12.2.4 Remarks in the direction of algebraic
topology ....................................... 201
12.2.5 Digression into projective geometry ............ 203
Exercises .................................................. 208
13 Moduli and deformations .................................... 210
13.1 Almost-complex structures, Beltrami differentials
and the integrability theorem ......................... 210
13.2 Deformations and cohomology ........................... 214
13.3 Appendix .............................................. 220
Exercises .................................................. 226
14 Mappings and moduli ........................................ 227
14.1 Diffeomorphisms of the plane .......................... 227
14.2 Braids, Dehn twists and quadratic singularities ....... 229
14.2.1 Classification of branched covers .............. 229
14.2.2 Monodromy and Dehn twists ...................... 236
14.2.3 Plane curves ................................... 240
14.3 Hyperbolic geometry ................................... 243
14.4 Compactification of the moduli space .................. 249
14.4.1 Collars and cusps .............................. 252
Exercises .................................................. 259
15 Ordinary differential equations ............................ 260
15.1 Conformal mapping ..................................... 260
15.2 Periods of holomorphic forms and ordinary
differential equations ................................ 267
15.2.1 The hypergeometric equation .................... 267
15.2.2 The Gauss-Manin connection ..................... 269
15.2.3 Singular points ................................ 272
Exercises .................................................. 281
References .................................................... 282
Index ......................................................... 285
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