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Chapter 1 Covering Spaces 1

1．The Definition of Riemann Surfaces 1

2．Elementary Properties of Holomorphie Mappings 10

3．Homotopy of Curves.The Fundamental Group 13

4．Branched and Unbranched Coverings 20

5．The Universal Covering and Covering Transformations 31

6．Sheaves 40

7．Analytic Continuation 44

8．Algebraic Functions 48

9．Differential Forms 59

10．The Integration of Differential Forms 68

11．Linear Differential Equations 81

Chapter 2 Compact Riemann Surfaces 96

12．Cohomology Groups 96

13．Dolbeault's Lemma 104

14．A Finiteness Theorem 109

15．The Exact Cohomology Sequence 118

16．The Riemann-Roch Theorem 126

17．The Serre Duality Theorem 132

18．Functions and Differential Forms with Prescribed Principal Parts 146

19．Harmonic Differential Forms 153

20．Abel's Theorem 159

21．The Jacobi Inversion Problem 166

Chapter 3 Non-compact Riemann Surfaces 175

22．The Dirichlet Boundary Value Problem 175

23．Countable Topology 185

24．Weyl's Lemma 190

25．The Runge Approximation Theorem 196

26．The Theorems of Mittag-Leffler and Weierstrass 201

27．The Riemann Mapping Theorem 206

28．Functions with Prescribed Summands of Automorphy 214

29．Line and Vector Bundles 219

30．The Triviality of Vector Bundles 228

31．The Riemann-Hilbert Problem 231

Appendix 237

A．Partitions of Unity 237

B．Topological Vector Spaces 238

References 243

Symbol Index 247

Author and Subject Index 249