Introduction to Several Complex Variables: Lectures by Lipman Bers 1962-1963

Preface......Page 1
Contents......Page 2
1. Preliminaries......Page 5
2. An inequality......Page 9
3. Proof of Hartogs' Theorem1......Page 11
4. Holomorphic mappings......Page 14
1. Examples and definitions......Page 16
2. Convexity with respect to a family of functions......Page 18
3. Domains of convergence of power series......Page 22
4. Bergman domains......Page 26
5. Analytic polyhedra......Page 28
1. Plurisubharmonic and pseudoconvex functions......Page 30
2. Pseudoconvex domains......Page 34
3. Solution of the Levi Problem for tube domains......Page 39
1. Weierstrass Preparation Theorem......Page 42
2. Rings of power series......Page 48
3. Meromorphic functions......Page 52
4. Removable singularities......Page 54
5. Complex manifolds......Page 57
1. The additive Problem formulated......Page 63
2. Reformulation of the Cousin Problem......Page 65
3. Reduction of the Cousin Problem to non-homogeneous Cauchy-Riemann equations......Page 67
1. Cohomology of a complex manifold with holomorphic functions as coefficients......Page 73
2. Applications......Page 78
3. Other cohomologies......Page 80
1. Ring of differential forms inadomain......Page 84
2. Differential forms on manifolds......Page 88
3. Poincar6 Lemmas......Page 89
2 De Rham's......Page 93
2. Dolbeault's Theorem......Page 97
3 Complex de Rham Theorem......Page 100
1. The Multiplicative Problem, formulated......Page 102
2. The Multiplicative Cousin Problem is not always solvable......Page 104
3. The solution of the Multiplicative Cousin Problem for polydiscs......Page 107
4. Characteristic classes (From C.II to C.I)......Page 110
1. Preliminaries......Page 114
2. Polynomial polyhedra......Page 116
3. Runge domains......Page 117
2. Applications of the Fundamental Lemma......Page 119
3. Preparation for the proof of the Fundamental Lemma......Page 121
4. Proof of the Fundamental Lemma......Page 124
1. Relative convexity......Page 132
2. Unbounded regions ofholomorphy......Page 133
3. The Behnke-Stein Theorem......Page 134
4. Applications to the Levi Problem......Page 136
1. Reduction to a finiteness statement......Page 138
2. Reduction to an extension property......Page 141
3. Proof of Proposition 2......Page 144
1. Exact sequences......Page 146
2. Differential operators......Page 148
3. Graded groups......Page 151
4. Sheaves and pre-sheaves......Page 153
5. Exact sequences of sheaves and cohomology......Page 154
6. Applications of the exact cohomology sequence theorem......Page 159
7. Proof of the exact cohomology sequence theorem......Page 162
1. Definitions......Page 166
2. Oka's coherence theorem......Page 167
3. Ueierstrass Preparation Theorem, revisited......Page 169
4. The third step......Page 172
5. Consequences of Oka's theorem......Page 175
6. The sheaf of ideals of a variety......Page 177
2. First step of the proof......Page 179
3. Reduction of (3) to on holomorphic matrices......Page 181
4. Proof of Cartants theorem on holomorphic matrices.......Page 184
5. New proof of the Oka-Weil Approximation Theorem......Page 188
6. Fundamental theorems for regions of holomorphy (semi-local form)......Page 189
2. Preparations for the proof......Page 191
3. Proof of Theorem A......Page 195
4. Proof of Theorem B......Page 198
5. Applications of the fundamental theorems......Page 200
1. Definition and examples......Page 205
2. An approximation theorem......Page 206
4. Characterization of Stein manifolds......Page 207
Appendix......Page 209