**Graduate Studies in Mathematics**

Volume: 83;
2007;
312 pp;
Hardcover

MSC: Primary 32;
Secondary 58

**Print ISBN: 978-0-8218-3319-3
Product Code: GSM/83**

List Price: $69.00

AMS Member Price: $55.20

MAA Member Price: $62.10

**Electronic ISBN: 978-1-4704-2113-7
Product Code: GSM/83.E**

List Price: $65.00

AMS Member Price: $52.00

MAA Member Price: $58.50

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#### Supplemental Materials

# Functions of Several Complex Variables and Their Singularities

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*Wolfgang Ebeling*

Translated by Philip G. Spain

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.

#### Reviews & Endorsements

The book contains a lot of illustrative pictures and diagrams substantially helping the geometric intuition of the reader.

-- European Mathematical Society Newsletter

#### Table of Contents

# Table of Contents

## Functions of Several Complex Variables and Their Singularities

- Cover Cover11 free
- Title iii4 free
- Copyright iv5 free
- Contents v6 free
- Foreword to the English translation ix10 free
- Introduction xi12 free
- List of figures xiii14 free
- List of tables xvii18 free
- Chapter 1. Riemann surfaces 120 free
- Chapter 2. Holomorphic functions of several variables 4362
- § 2.1. Holomorphic functions of several variables 4362
- §2.2. Holomorphic maps and the implicit function theorem 5776
- § 2.3. Local rings of holomorphic functions 6079
- §2.4. The Weierstrass preparation theorem 6382
- §2.5. Analytic sets 7493
- §2.6. Analytic set germs 7695
- §2.7. Regular and singular points of analytic sets 84103
- §2.8. Map germs and homomorphisms of analytic algebras 89108
- §2.9. The generalized Weierstrass preparation theorem 96115
- §2.10. The dimension of an analytic set germ 101120
- § 2.11. Elimination theory for analytic sets 109128

- Chapter 3. Isolated singularities of holomorphic functions 113132
- § 3.1. Differentiable manifolds 113132
- §3.2. Tangent bundles and vector fields 119138
- § 3.3. Transversality 125144
- §3.4. Lie groups 127146
- §3.5. Complex manifolds 134153
- §3.6. Isolated critical points 140159
- §3.7. The universal unfolding 144163
- §3.8. Modifications 149168
- §3.9. Finitely determined function germs 158177
- §3.10. Classification of simple singularities 165184
- § 3.11. Real morsifications of the simple curve singularities 171190

- Chapter 4. Fundamentals of differential topology 181200
- § 4.1. Differentiable manifolds with boundary 181200
- §4.2. Riemannian metric and orientation 183202
- § 4.3. The Ehresmann fibration theorem 186205
- §4.4. The holonomy group of a differentiable fiber bundle 189208
- §4.5. Singular homology groups 194213
- §4.6. Intersection numbers 200219
- §4.7. Linking numbers 209228
- §4.8. The braid group 211230
- §4.9. The homotopy sequence of a differentiable fiber bundle 214233

- Chapter 5. Topology of singularities 223242
- § 5.1. Monodromy and variation 223242
- §5.2. Monodromy group and vanishing cycles 226245
- § 5.3. The Picard- Lefschetz theorem 229248
- §5.4. The Milnor fibration 238257
- §5.5. Intersection matrix and Coxeter- Dynkin diagram 249268
- §5.6. Classical monodromy, variation, and the Seifert form 252271
- §5.7. The action of the braid group 259278
- §5.8. Monodromy group and vanishing lattice 269288
- §5.9. Deformation 277296
- §5.10. Polar curves and Coxeter- Dynkin diagrams 283302
- § 5.11. Unimodal singularities 292311
- §5.12. The monodromy groups of the isolated hypersurface singularities 298317

- Bibliography 303322
- Index 307326
- Back Cover Back Cover1334