Since the 1960s, there has been a flowering in higher-dimensional complex analysis. Both classical and new results in this area have found numerous applications in analysis, differential and algebraic geometry, and, in particular, contemporary mathematical physics. In many areas of modern mathematics, the mastery of the foundations of higher-dimensional complex analysis has become necessary for any specialist. Intended as a first study of higher-dimensional complex analysis, this book covers the theory of holomorphic functions of several complex variables, holomorphic mappings, and submanifolds of complex Euclidean space.
Author(s): B. V. Shabat
Series: Translations of Mathematical Monographs Pt. 2
Publisher: American Mathematical Society
Year: 1992
Language: English
Pages: 380
Contents......Page 4
Foreword to the Third Edition......Page 8
1. The space C^n......Page 10
2. The simplest domains......Page 15
3. The concept of holomorphy......Page 21
4. Pluriharmonic functions......Page 24
5. Simplest properties of holomorphic functions......Page 27
6. The fundamental theorem of Hartogs......Page 33
7. Power series......Page 38
8. Other series......Page 42
9. Properties of holomorphic mappings......Page 48
10. Biholomorphic mappings......Page 53
11. Fatou's example......Page 63
Problems......Page 67
12. The concept of a manifold......Page 70
13. Complexification of Minkowski space......Page 76
14. Stokes's formula......Page 86
15. The Cauchy-Poincare theorem......Page 92
16. Maxwell's equations......Page 95
17.Submanifoldsof C^n......Page 105
18. Wirtinger's theorem......Page 110
19. The Fubini-Study form and related topics......Page 117
20. The concept of a covering......Page 121
21. Fundamental groups and coverings......Page 124
22. Riemann domains......Page 130
23. The Weierstrass preparation theorem......Page 132
24. Properties of analytic sets......Page 138
25. Local structure......Page 145
26. The concept of a bundle......Page 149
27. The tangent and cotangent bundles......Page 152
28. The concept of a sheaf......Page 157
Problems......Page 161
29. The formulas of Martinelli-Bochner and Leray......Page 164
30. Weil's formula......Page 170
31. Extension from the boundary......Page 175
32. Hartogs's theorem and the removal of singularities......Page 182
33. The concept of a domain of holomorphy......Page 186
34. Holomorphic convexity......Page 190
35. Properties of domains of holomorphy......Page 194
36. The continuity principle......Page 197
37. Local pseudoconvexity......Page 201
38. Plurisubharmonic functions......Page 208
39. Pseudoconvex domains......Page 215
40. One-sheeted envelopes......Page 221
41. Multiple-sheeted envelopes......Page 226
42. Analyticity of the set of singularities......Page 232
Problems......Page 237
43. The concept of a meromorphic function......Page 240
44. The first Cousin problem......Page 243
45. Solution of the first problem......Page 247
46. Cohomology groups......Page 251
47. Exact sequences of sheaves......Page 256
48. Localized first Cousin problem......Page 258
49. Second Cousin problem......Page 262
50. Applications of the Cousin problems......Page 268
51. Solution of the Levi problem......Page 271
52. Other applications......Page 273
53. The theory of Martinelli......Page 281
54. The theory of Leray......Page 287
55. Logarithmic residue......Page 295
Problems......Page 301
56. The Bergman metric......Page 304
57. The Caratheodory metric......Page 311
58. The Kobayashi metric......Page 314
59. Criteria for hyperbolicity......Page 317
60. Generalizations of Picard's theorem......Page 326
61. Mappings of strictly pseudoconvex domains......Page 336
62. Correspondence of boundaries......Page 341
63. A symmetry principle......Page 345
64. Vector fields......Page 350
65. Boundary properties of functions......Page 356
66. Uniqueness theorems and propositions......Page 361
Problems......Page 368
Appendix. Complex Potential Theory......Page 370
Subject Index......Page 378
