Complex analytic sets

Cover......Page 1
Title Page......Page 2
Copyright Page......Page 3
Series Editor's Preface......Page 4
Table of Contents......Page 6
Preface......Page 12
List of notations......Page 16
1.1. Weierstrass' preparation theorem......Page 22
1.2. Dependence of roots on parameters......Page 24
1.3. Discriminant set......Page 26
1.4. Factorization into irreducible factors......Page 28
1.5. Multiplicity of zeros. Divisor of a holomorphic function......Page 29
2.1......Page 31
2.2. Simplest topological properties......Page 34
2.3. Regular and singular points......Page 35
2.4. Dimension......Page 38
2.5. Regularity in P_n and C^{n+1}......Page 39
2.6. Principal analytic sets......Page 40
2.7. Critical points......Page 43
2.8. Local representation of sets of codimension 1......Page 44
2.9. Minimal defining functions......Page 46
3.1. Proper maps......Page 47
3.2. Exception of variables......Page 48
3.3. Corollaries......Page 50
3.4. Existence of proper projections......Page 52
3.5. On the dimension......Page 53
3.6. Almost single-sheeted projections......Page 55
3.7. Local representation of analytic sets......Page 57
3.8. Images of analytic sets......Page 59
4.1. Definitions......Page 61
4.2. Canonical defining functions......Page 63
4.3. Analytic covers as analytic sets......Page 66
4.4. The theorem of Remmert-Stein-Shiffman......Page 67
4.5. Analyticity of sng A......Page 68
5.1. Connected components of reg A......Page 71
5.2. Decomposition by dimension. Analyticity of sng A and S(A)......Page 72
5.3. Irreducibility......Page 73
5.4. Irreducible components......Page 75
5.5. Stratifications......Page 78
5.6. Intersections of analytic sets......Page 80
5.7. The number of defining functions......Page 82
5.8. A theorem on proper maps......Page 84
6.1. Local parametrization......Page 86
6.2. Normalization and uniformization......Page 88
6.3. Maximum principle......Page 90
7.1. Chow's theorem......Page 92
7.2. Closure of affine algebraic sets......Page 93
7.3. Algebraic sets as analytic covers......Page 94
7.4. Some criteria for being algebraic......Page 96
8.1. Definitions and simplest properties......Page 98
8.2. The tangent cone and maps......Page 100
8.3. The tangent cone and the a-process......Page 101
8.4. Analytic description......Page 102
8.5. Tangent vectors and one-dimensional sections......Page 105
8.6. Deviation......Page 107
9.1. Definitions and simplest properties......Page 110
9.2. Hierarchy and analyticity......Page 112
9.3. Tangent space......Page 114
9.4. Whitney cones and projections......Page 115
9.5. Singularities of codimension 1. Puiseux normalization......Page 117
10.1. Multiplicity of projections......Page 121
10.2. Multiplicity of maps......Page 124
10.3. Multiplicities and initial polynomials......Page 129
10.4. Bezout's theorem......Page 134
10.5. Milnor numbers......Page 136
11.1. Multiplicity of an analytic set at a point......Page 139
11.2. Multiplicities and the tangent cone......Page 140
11.3. Degree of an algebraic set......Page 144
11.4. Multiplicity sets......Page 146
11.5. Hooomorphic chains......Page 149
11.6. The tangent cone as chain......Page 151
11.7. Dependence of the tangent cone on parameters......Page 152
12.1. The case of complementary codimensions......Page 155
12.2. Some properties of indices......Page 159
12.3. Intersections of holomorphic chains......Page 162
12.4. Properties of intersection chains......Page 165
12.5. Multiplicities and transversality......Page 169
12.6. Multiplicities of fibers of holomorphic maps......Page 171
13.1. Hermitian manifolds......Page 174
13.2. Volume forms......Page 177
13.3. Wirtinger's inequality......Page 180
13.4. Integration in P......Page 182
13.5. Integration over incidence manifolds. Crofton's formula......Page 183
13.6. Relation between projective and affine volumes......Page 188
14.1. Lelong's theorem......Page 191
14.2. Properties of integrals over analytic sets......Page 193
14.3. Stokes' theorem......Page 194
14.4. Analytic sets as minimal surfaces......Page 199
14.5. Tangential and normal components of volume......Page 201
14.6. Volumes of analytic subsets of a ball......Page 203
14.7. Volumes of algebraic sets......Page 206
15.1. Lelong numbers......Page 208
15.2. Integral representations......Page 211
15.3. Lower bounds for volumes......Page 214
15.4. Areas of projections......Page 217
15.5. Sequences of analytic sets......Page 222
16.1. Sequences of holomorphic chains......Page 225
16.2. Intersection chains as currents......Page 227
16.3. Formulas of Poincare-Lelong......Page 231
16.4. Jensen formulas......Page 236
17.1. Blaschke's condition......Page 241
17.2. Metrical conditions of algebraicity......Page 243
17.3. Growth estimates of hyperplane sections......Page 246
17.4. Converse estimates......Page 249
17.5. Corollaries and generalizations......Page 252
18.1. Singularities of small codimensions......Page 256
18.2. Infectiousness of continuation......Page 258
18.3. Removing pluripolar singularities. Bishop's theorems......Page 262
18.4. Continuation across R^n......Page 264
18.5. Obstructions of small CR-dimensions......Page 268
18.6. "Hartogs' lemma" for analytic sets......Page 271
19.1. Regularity near the boundary......Page 274
19.2. Boundary uniqueness theorems......Page 277
19.3. Plateau's problem for analytic sets......Page 278
19.4. Preparation lemmas......Page 280
19.5. Boundaries of analytic covers......Page 284
19.6. The Harvey-Lawson theorem......Page 287
19.7. On singularities of analytic films......Page 291
20. Analytic continuation......Page 292
20.1. On continuation of analytic sets......Page 293
20.2. Compact singularities......Page 294
20.3. Continuation across pseudoconcave surfaces......Page 295
20.4. Continuation across an edge......Page 299
20.5. The symmetry principle......Page 301
A1. 1. Holomorphic functions in C^n......Page 304
A1.2. Plurisubharmonic functions......Page 311
A1.3. Holomorphic continuation along sections......Page 314
A1.4. Removable singularities of bounded functions......Page 318
A1.5. Removable singularities of continuous functions......Page 320
A2.1. Holomorphic maps......Page 321
A2.2. The implicit function theorem and the rank theorem......Page 324
A2.3. Complex manifolds in C^n......Page 328
A2.4. Real manifolds in C^n......Page 330
A3.1. Abstract complex manifolds......Page 333
A3.2. Complex projective space P_n......Page 335
A3.3. Complex planes in P_n......Page 338
A3.4. The Grassmannians G(k,n)......Page 339
A3.5. Incidence manifolds and the a-process......Page 340
A3.6. Imbedding of Grassmannians into P_N......Page 342
A4.1. Exterior algebra......Page 344
A4.2. Differential forms......Page 346
A4.3. Integration of forms. Stokes' theorem......Page 350
A4.4. Fubini's theorem......Page 351
A4.5. Positive forms......Page 354
A5.1. Definitions. Positive currents......Page 356
A5.2. The operators d,dbar, and integral representations......Page 359
A5.3. Regularization......Page 361
A5.4. The dbar-problem and the jump theorem......Page 362
A6. I. Definition and simplest properties......Page 365
A6.2. H_m on an m-dimensional manifold......Page 367
A6.3. The Lemma concerning fibers......Page 369
A6.4. Sections and projections......Page 370
References......Page 372
References added in proof......Page 384
Index......Page 386