Theory of Stein spaces

Cover......Page 1
Title Page 1......Page 5
Copyright 2004......Page 6
Title Page 2......Page 7
Copyright 1979......Page 8
Dedication......Page 9
Contents......Page 11
Introduction......Page 19
2. Sums of Sheaves, Subsheaves, and Restrictions......Page 27
4. Presheaves and the Section Functor r......Page 28
6. The Sheaf Conditions "1 and Y2......Page 29
8. Image Sheaves......Page 30
1. Sheaves of Groups, Rings, and R-Modules......Page 31
2. Sheaf Homomorphisms and Subsheaves......Page 32
3. Quotient Sheaves......Page 33
5. Algebraic Reduction......Page 34
7. On the Exactness of r and r......Page 35
1. Finite Sheaves......Page 36
3. Coherent Sheaves......Page 37
5. The Functors and A'......Page 38
6. The Functor Yom and Annihilator Sheaves......Page 39
3. Complex Spaces......Page 40
2. Differentiable and Complex Manifolds......Page 41
3. Complex Spaces and Holomorphic Maps......Page 42
5. Analytic Sets......Page 44
6. Dimension Theory......Page 45
7. Reduction of Complex Spaces......Page 46
8. Normal Complex Spaces......Page 47
1. Soft Sheaves......Page 48
2. Softness of the Structure Sheaves of Differentiable Manifolds......Page 49
4. Exactness of the Functor I for Flabby and Soft Sheaves......Page 51
1. Cohomology of Complexes......Page 54
2. Flabby Cohomology Theory......Page 56
3. The Formal de Rham Lemma......Page 58
2. tech Cohomology......Page 59
1. tech Complexes......Page 60
3. Refinements and the tech Cohomology Modules fI9(X, S)......Page 61
5. The Vanishing Theorem for Compact Blocks......Page 63
6. The Long Exact Cohomology Sequence......Page 64
1. The Canonical Resolution of a Sheaf Relative to a Cover......Page 66
2. Acyclic Covers......Page 68
4. The Isomorphism Theorem fI;(X, 9') = fl9(X, 50) = H4(X, Y)......Page 69
1. Closed and Finite Maps......Page 71
3. The Exactness of the Functor f.......Page 72
4. The Isomorphisms HQ(X, 9') = H4(Y, f.(.9'))......Page 73
1. Continuity of Roots......Page 74
2. The General Weierstrass Division Theorem......Page 75
3. The Weierstrass Homomorphism OB 4 n.(0,4)......Page 76
4. The Coherence of the Direct Image Functor n.......Page 77
1. The Projection Theorem......Page 78
2. Finite Holomorphic Maps (Local Case)......Page 79
3. Finite Holomorphic Maps and Coherence......Page 80
1. Tangent Vectors......Page 82
2. Vector Fields......Page 84
3. Complex r-vectors......Page 85
5. Complex Valued Differential Forms......Page 86
7. Lifting Differential Forms......Page 88
8. The de Rham Cohomology Groups......Page 89
1. The Sheaves d' o r and fl'......Page 90
2. The Sheaves and S1.......Page 92
3. The Derivatives a and d......Page 93
4. Holomorphic Liftings of (p, q)-forms......Page 96
3. The Lemma of Grothendieck......Page 97
1. Area Integrals and the Operator T......Page 98
2. The Commutivity of T with Partial Differentiation......Page 99
3. The Cauchy Integral Formula and the Equation (a/dz)(Tf) =f......Page 100
4. A Lemma of Grothendieck......Page 101
1. The Solution of the a-problem for Compact Product Sets......Page 103
2. The Dolbeault Cohomology Groups......Page 105
3. The Analytic de Rham Theory......Page 106
Supplement to Section 4.1. A Theorem of Hartogs......Page 107
1. The Lemma of Cousin......Page 109
2. Bounded Holomorphic Matrices......Page 111
3. The Lemma of Cartan......Page 113
2. Attaching Sheaf Epimorphisms......Page 115
1. An Approximation Theorem of Runge......Page 116
2. The Attaching Lemma for Epimorphisms of Sheaves......Page 118
3. Theorems A and B......Page 121
2. The Formulations of Theorems A and B and the Reduction of Theorem B to Theorem A......Page 122
3. The Proof of Theorem A for Compact Blocks......Page 124
1. Stein Sets and Consequences of Theorem B......Page 126
2. Construction of Compact Stein Sets Using the Coherence Theorem for Finite Maps......Page 127
3. Exhaustions of Complex Spaces by Compact Stein Sets......Page 128
4. The Equations H9(H, ') = 0 for q >_ 2......Page 129
5. Stein Exhaustions and the Equation H1(X, 91) = 0......Page 130
1. The Holomorphically Convex Hull......Page 134
2. Holomorphically Convex Spaces......Page 135
3. Stones......Page 137
4. Exhaustions by Stones and Weakly Holomorphically Convex Spaces......Page 138
5. Holomorphic Convexity and Unbounded Holomorphic Functions......Page 139
1. Analytic Blocks......Page 142
3. Holomorphically Convex Spaces......Page 143
1. Good Semi-norms......Page 144
2. The Compatibility Theorem......Page 145
3. The Convergence Theorem......Page 146
4. The Approximation Theorem......Page 147
5. Exhaustions by Analytic Blocks are Stein Exhaustions......Page 149
1. Standard Constructions......Page 151
2. Stein Coverings......Page 153
3. Differences of Complex Spaces......Page 154
4. The Spaces C2{0} and t:'(0)......Page 156
5. Classical Examples......Page 160
1. The Cousin I Problem......Page 162
2. The Cousin II Problem......Page 164
3. Poincarb Problem......Page 165
4. The Exact Exponential Sequence 0 - 1- O O -. 1......Page 168
5. Oka's Principle......Page 170
1. Divisors and Locally-Free Sheaves of Rank 1......Page 172
2. The Isomorphism HI(X, O') -+ LF(X)......Page 173
3. The Group of Divisor Classes on a Stein Space......Page 174
1. Cycles and Global Holomorphic Functions......Page 176
3. The Reduction Theorem......Page 178
4. Differential Forms on Stein Manifolds......Page 180
5. Topological Properties of Stein Spaces......Page 182
1. An Induction Principle......Page 183
2. The Equations H'(B, OB) = = H'-'(B, OB) = 0......Page 185
3. Representation of 1......Page 187
4. The Character Theorem......Page 188
Goto 188 /FitH 555. Frechet Spaces......Page 189
1. The Topology of Compact Convergence......Page 190
2. The Uniqueness Theorem......Page 191
3. The Existence Theorem......Page 192
4. Properties of the Canonical Topology......Page 194
6. Reduced Complex Spaces and Compact Convergence......Page 196
7. Convergent Series......Page 197
1. Characters and Character Ideals......Page 202
2. Finiteness Lemma for Character Ideals......Page 203
3. The Homeomorphism EE: X -T(T)......Page 206
4. Complex Analytic Structure on T(T)......Page 207
1. The Space 0,,(B)......Page 213
2. The Bergman Inequality......Page 214
3. The Hilbert Space 0,',(B)......Page 215
5. The Schwarz Lemma......Page 216
1. Monotonicity......Page 217
2. The Subdegree......Page 218
3. Construction of Monotone Orthogonal Bases by Means of Minimal Functions......Page 219
1. Existence......Page 220
2. The Hilbert Space CR(U, ,9')......Page 222
3. The Hilbert Space Zj(U, 9')......Page 223
4. Refinements......Page 224
1. The Smoothing Lemma......Page 226
2. Finiteness Lemma......Page 227
3. Proof of the Finiteness Theorem......Page 228
1. Divisors and Locally Free Sheaves......Page 230
1. Divisors of Meromorphic Sections......Page 231
2. The Sheaves F(D)......Page 232
3. The Sheaves O(D)......Page 233
1. The Sequence 0 -. F(D) -. F(D') - F -a 0......Page 234
2. The Characteristic Theorem and the Existence Theorem......Page 235
4. The Degree Equation......Page 236
1. The Genus of Riemann-Roch......Page 237
2. Applications......Page 238
1. Locally Free Subsheaves......Page 239
3. The Canonical Divisors......Page 240
1. The Chern Function......Page 241
3. The Riemann-Roch Theorem......Page 242
1. The C-homomorphism O(np)(X) -. Hom(H'(X, O(D)), H'(X, O(D + np)))......Page 243
3. The Equation H'(X, K) = 0......Page 244
1. The Principal Part Distributions with Respect to a Divisor......Page 245
2. The Equation H'(X, O(D)) = I(D)......Page 246
4. The Inequality Dim. (X) J < 1......Page 247
5. The Residue Calculus......Page 248
6. The Duality Theorem......Page 249
1. The Equation i(D) = I(K - D)......Page 251
2. The Formula of Riemann-Roch......Page 252
4. Theorem A for Sheaves O(D)......Page 253
5. The Existence of Meromorphic Differential Forms......Page 254
7. Theorems A and B for Locally Free Sheaves......Page 255
8. The Hodge Decomposition of H'(X, C)......Page 257
1. The Number .r)......Page 258
2. Maximal Subsheaves......Page 259
3. The Inequality u(g) = u(F) + 2g......Page 260
4. The Splitting Criterion......Page 261
6. Existence of the Splitting......Page 263
7. Uniqueness of the Splitting......Page 264
Bibliography......Page 266
Subject Index......Page 269
Table of Symbols......Page 274
Addendum......Page 277
Errors and Misprints......Page 281