The book provides an introduction to the theory of functions of several complex variables and their singularities, with special emphasis on topological aspects. The topics include Riemann surfaces, holomorphic functions of several variables, classification and deformation of singularities, fundamentals of differential topology, and the topology of singularities. The aim of the book is to guide the reader from the fundamentals to more advanced topics of recent research. All the necessary prerequisites are specified and carefully explained. The general theory is illustrated by various examples and applications.
Readership: Graduate students and research mathematicians interested in several complex variables and complex algebraic geometry.
Author(s): Wolfgang Ebeling
Series: Graduate Studies in Mathematics 83
Publisher: American Mathematical Society
Year: 2007
Language: English
Pages: xviii+312
Foreword to the English translation ix
Introduction xi
List of figures xiii
List of tables xvii
Chapter 1. Riemann surfaces 1
§1.1. Riemann surfaces 1
§1.2. Homotopy of paths, fundamental groups 9
§1.3. Coverings 13
§1.4. Analytic continuation 24
§1.5. Branched meromorphic continuation 29
§1.6. The Riemann surface of an algebraic function 33
§1.7. Puiseux expansion 40
§1.8. The Riemann sphere 41
Chapter 2. Holomorphic functions of several variables 43
§2.1. Holomorphic functions of several variables 43
§2.2. Holomorphic maps and the implicit function theorem 57
§2.3. Local rings of holomorphic functions 60
§2.4. The Weierstrass preparation theorem 63
§2.5. Analytic sets 74
§2.6. Analytic set germs 76
§2.7. Regular and singular points of analytic sets 84
§2.8. Map germs and homomorphisms of analytic algebras 89
§2.9. The generalized Weierstrass preparation theorem 96
§2.10. The dimension of an analytic set germ 101
§2.11. Elimination theory for analytic sets 109
Chapter 3. Isolated singularities of holomorphic functions 113
§3.1. Differentiable manifolds 113
§3.2. Tangent bundles and vector fields 119
§3.3. Transversality 125
§3.4. Lie groups 127
§3.5. Complex manifolds 134
§3.6. Isolated critical points 140
§3.7. The universal unfolding 144
§3.8. Morsifications 149
§3.9. Finitely determined function germs 158
§3.10. Classification of simple singularities 165
§3.11. Real morsifications of the simple curve singularities 171
Chapter 4. Fundamentals of differential topology 181
§4.1. Differentiable manifolds with boundary 181
§4.2. Riemannian metric and orientation 183
§4.3. The Ehresmann fibration theorem 186
§4.4. The holonomy group of a differentiable fiber bundle 189
§4.5. Singular homology groups 194
§4.6. Intersection numbers 200
§4.7. Linking numbers 209
§4.8. The braid group 211
§4.9. The homotopy sequence of a differentiable fiber bundle 214
Chapter 5. Topology of singularities 223
§5.1. Monodromy and variation 223
§5.2. Monodromy group and vanishing cycles 226
§5.3. The Picard-Lefschetz theorem 229
§5.4. The Milnor fibration 238
§5.5. Intersection matrix and Coxeter-Dynkin diagram 249
§5.6. Classical monodromy, variation, and the Seifert form 252
§5.7. The action of the braid group 259Contents vii
§5.8. Monodromy group and vanishing lattice 269
§5.9. Deformation 277
§5.10. Polar curves and Coxeter-Dynkin diagrams 283
§5.11. Unimodal singularities 292
§5.12. The monodromy groups of the isolated hypersurface singularities 298
Bibliography 303
Index 307
