This book grew out of lectures on Riemann surfaces which the author gave at the universities of Munich, Regensburg and Munster. Its aim is to give an introduction to this rich and beautiful subject, while presenting methods from the theory of complex manifolds which, in the special case of one complex variable, turn out to be particularly elementary and transparent. The book is divided into three chapters. In the first chapter we consider Riemann surfaces as covering spaces and develop a few basics from topology which are needed for this. Then we construct the Riemann surfaces which arise via analytic continuation of function germs. In particular this includes the Riemann surfaces of algebraic functions. As well we look more closely at analytic functions which display a special multi-valued behavior. Examples of this are the primitives of holomorphic i-forms and the solutions of linear differential equations. The second chapter is devoted to compact Riemann surfaces. The main classical results, like the Riemann-Roch Theorem, Abel's Theorem and the Jacobi inversion problem, are presented. Sheaf cohomology is an important technical tool. But only the first cohomology groups are used and these are comparatively easy to handle. The main theorems are all derived, following Serre, from the finite dimensionality of the first cohomology group with coefficients in the sheaf of holomorphic functions. And the proof of this is based on the fact that one can locally solve inhomogeneous Cauchyยญ Riemann equations and on Schwarz' Lemma
CONTENT
1 Covering Spaces -- ยง1. The Definition of Riemann Surfaces -- ยง2. Elementary Properties of Holomorphic Mappings -- ยง3. Homotopy of Curves. The Fundamental Group -- ยง4. Branched and Unbranched Coverings -- ยง5. The Universal Covering and Covering Transformations -- ยง6. Sheaves -- ยง7. Analytic Continuation -- ยง8. Algebraic Functions -- ยง9. Differential Forms -- ยง10. The Integration of Differential Forms -- ยง11. Linear Differential Equations -- 2 Compact Riemann Surfaces -- ยง12. Cohomology Groups -- ยง13. Dolbeaultโ{128}{153}s Lemma -- ยง14. A Finiteness Theorem -- ยง15. The Exact Cohomology Sequence -- ยง16. The Riemann-Roch Theorem -- ยง17. The Serre Duality Theorem -- ยง18. Functions and Differential Forms with Prescribed Principal Parts -- ยง19. Harmonic Differential Forms -- ยง20. Abelโ{128}{153}s Theorem -- ยง21. The Jacobi Inversion Problem -- 3 Non-compact Riemann Surfaces -- ยง22. The Dirichlet Boundary Value Problem -- ยง23. Countable Topology -- ยง24. Weylโ{128}{153}s Lemma -- ยง25. The Runge Approximation Theorem -- ยง26. The Theorems of Mittag-Leffler and Weierstrass -- ยง27. The Riemann Mapping Theorem -- ยง28. Functions with Prescribed Summands of Automorphy -- ยง29. Line and Vector Bundles -- ยง30. The Triviality of Vector Bundles -- ยง31. The Riemann-Hilbert Problem -- A. Partitions of Unity -- B. Topological Vector Spaces -- References -- Symbol Index -- Author and Subject Index
