Preface
Chapter I. Riemann Surfaces: Basic Definitions
1.Complex Charts and Complex Structures
Complex Charts
Complex Atlases
The Definition of a Piemann Surface
Real 2-Manifolds
The Genus of a Compact Riemann Surface
Complex Manifolds
Problems 1.1
2.First Examples of Riemann Surfaces
A Remark on Defining Riemann Surfaces
The Projective Line
Complex Tori
Graphs of Holomorphic Functions
Smooth Affine Plane Curves
Problems 1.2
3.Projective Curves
The Projective Plane P2
Smooth Projective Plane Curves
Higher-Dimensional Projective Spaces
Complete Intersections
Local Complete Intersections
Problems 1.3
Further Reading
Chapter II. Functions and Maps
1.Functions on Riemann Surfaces
Holomorphic Functions
Singularities of Functions; Meromorphic Functions
Lanrent Series
The Order of a Meromorphic Function at a Point
C∞o Functions
Harmonic Functions
Theorems Inherited from One Complex Variable
Problems II.1
2.Examples of Meromorphic Functions
Meromorphic Functions on the Riemann Sphere
Meromorphic Functions on the Projective Line
Meromorphic Functions on a Complex Torus
Meromorphic Functions on Smooth Plane Curves
Smooth Projective Curves
Problems II.2
3.Holomorphic Maps Between Riemann Surfaces
The Definition of a Holomorphic Map
Isomorphisms and Automorphisms
Easy Theorems about Holomorphic Maps
Meromorphic Functions and Holomorphic Maps to the Riemann Sphere
Meromorphic Functions on a Complex Torus, Again
Problems II.3
4.Global Properties of Holomorphic Maps
