Algebraic Geometry

I: Algebraic Curves. Algebraic Manifolds and Schemes......Page 0001
Contents......Page 0004
Introduction......Page 0008
1.1. Complex Chart; Complex Coordinates......Page 0019
1.3. Complex Analytic Manifolds......Page 0020
1.4. Mappings of Complex Manifolds......Page 0022
1.6. Riemann Surfaces......Page 0023
1.7. Differentiable Manifolds......Page 0025
2.2. Meromorphic Functions on a Riemann Surface......Page 0026
2.3. Meromorphic Functions with Prescribed Behaviour at Poles......Page 0028
2.4. Multiplicity of a Mapping; Order of a Function......Page 0029
2.6. Divisors on Riemann Surfaces......Page 0030
2.7. Finite Mappings of Riemann Surfaces......Page 0032
2.9. The Universal Covering......Page 0033
2.10. Continuation of Mappings......Page 0034
2.11. The Riemann Surface of an Algebraic Function......Page 0035
3.1. Orientability......Page 0038
3.2. Triangulability......Page 0039
3.3. Development; Topological Genus......Page 0040
3.4. Structure of the Fundamental Group......Page 0041
3.6. The Hurwitz Formulae......Page 0042
3.7. Homology and Cohomology; Betti Numbers......Page 0044
3.8. Intersection Product; Poincaré Duality......Page 0045
4.1. Tangent Vectors; Differentiations......Page 0047
4.2. Differential Forms......Page 0048
4.3. Exterior Differentiations; de Rham Cohomology......Page 0049
4.4. Kähler and Riemann Metrics......Page 0050
4.5. Integration of Exterior Differentials; Green's Formula......Page 0051
4.6. Periods; de Rham Isomorphism......Page 0054
4.7. Holomorphic Differentials; Geometric Genus; Riemann's Bilinear Relations......Page 0055
4.8. Meromorphic Differentials; Canonical Divisors......Page 0057
4.9. Meromorphic Differentials with Prescribed Behaviour at Poles; Residues......Page 0059
4.10. Periods of Meromorphic Differentials......Page 0060
4.11. Harmonic Differentials......Page 0061
4.12. Hilbert Space of Differentials; Harmonic Projection......Page 0062
4.13. Hodge Decomposition......Page 0064
4.14. Existence of Meromorphic Differentials and Functions......Page 0065
§5. Classification of Riemann Surfaces......Page 0068
5.2. Uniformization......Page 0069
5.3. Types of Riemann Surfaces......Page 0070
5.4. Automorphisms of Canonical Regions......Page 0071
5.6. Riemann Surfaces of Parabolic Type......Page 0072
5.7. Riemann Surfaces of Hyperbolic Type......Page 0074
5.8. Automorphic Forms; Poincaré Series......Page 0077
5.9. Quotient Riemann Surfaces; the Absolute Invariant......Page 0078
5.10. Moduli of Riemann Surfaces......Page 0079
6.1. Function Spaces and Mappings Associated with Divisors......Page 0082
6.2. Riemann-Roch Formula; Reciprocity Law for Differentials of the First and Second Kind......Page 0085
6.3. Applications of the Riemann-Roch Formula to Problems of Existence of Meromorphic Functions and Differentials......Page 0087
6.4. Compact Riemann Surfaces are Projective......Page 0088
6.5. Algebraic Nature of Projective Models; Arithmetic Riemann Surfaces......Page 0089
6.6. Models of Riemann Surfaces of Genus 1......Page 0090
1.1. Algebraic Varieties; Zariski Topology......Page 0092
1.2. Regular Functions and Mappings......Page 0093
1.4. Irreducibility; Dimension......Page 0096
1.6. Singular and Nonsingular Points on Varieties......Page 0097
1.7. Rational Functions, Mappings and Varieties......Page 0099
1.8. Differentials......Page 0105
1.9. Comparison Theorems......Page 0107
1.10. Lefschetz Principle......Page 0108
2.1. Multiplicity of a Mapping; Ramification......Page 0109
2.2. Divisors......Page 0110
2.3. Intersection of Plane Curves......Page 0112
2.4. The Hurwitz Formulae......Page 0114
2.6. Comparison Theorems (Continued)......Page 0115
2.9. First Applications......Page 0116
2.10. Riemann Count......Page 0120
3.1. Linear Systems......Page 0121
3.2. Mappings of Curves into ¥n......Page 0123
3.3. Generic Hyperplane Sections......Page 0124
3.4. Geometrical Interpretation of the Riemann-Roch Formula......Page 0126
3.5. Clifford's Inequality......Page 0127
3.6. Castelnuovo's Inequality......Page 0129
3.7. Space Curves......Page 0130
3.8. Projective Normality......Page 0131
3.9. The Ideal of a Curve; Intersections of Quadrics......Page 0132
3.10. Complete Intersections......Page 0135
3.11. The Simplest Singularities of Curves......Page 0137
3.13. Dual Curves......Page 0138
3.15. Correspondence of Branches; Dual Formulae......Page 0140
1.1. Algebraic Groups......Page 0142
1.3. Algebraic Complex Tori; Polarized Tori......Page 0143
1.4. Theta Function and Riemann Theta Divisor......Page 0148
1.5. Principally Polarized Abelian Varieties......Page 0150
1.6. Points of Finite Order on Abelian Varieties......Page 0151
1.7. Elliptic Curves......Page 0153
2.1. Principal Divisors on Riemann Surfaces......Page 0157
2.2. Inversion Problem......Page 0158
2.4. Picard Varieties and their Universal Property......Page 0159
2.5. Polarization Divisor of the Jacobian of a Curve; Poincaré Formulae......Page 0161
2.6. Jacobian of a Curve of Genus 1......Page 0164
References......Page 0166
Contents......Page 0170
Introduction......Page 0175
1.1. Base Field......Page 0177
1.3. Algebraic Subsets......Page 0178
1.4. Systems of Algebraic Equations; Ideals......Page 0179
1.5. Hilbert's Nullstellensatz......Page 0180
2.1. Affine Varieties......Page 0181
2.2. Abstract Affine Varieties......Page 0182
2.3. Affine Schemes......Page 0183
2.5. Intersection of Subvarieties......Page 0184
2.6. Fibres of a Morphism......Page 0185
2.7. The Zariski Topology......Page 0186
2.8. Localization......Page 0187
2.9. Quasi-affine Varieties......Page 0188
2.10. Affine Algebraic Geometry......Page 0189
3.1. Projective Space......Page 0190
3.2. Atlases and Varieties......Page 0191
3.3. Gluing......Page 0192
3.5. Projective Varieties......Page 0193
4.1. Definitions......Page 0194
4.2. Products of Varieties......Page 0195
4.3. Equivalence Relations......Page 0196
4.4. Projection......Page 0197
4.6. The Segre Embedding......Page 0198
5.1. Algebraic Groups......Page 0199
5.2. Vector Bundles......Page 0200
5.4. Constructions with Bundles......Page 0201
6.2. Sheaves......Page 0202
6.3. Sheaves of Modules......Page 0203
6.4. Coherent Sheaves of Modules......Page 0204
6.5. Ideal Sheaves......Page 0205
6.6. Constructions of Varieties......Page 0206
7.1. Differential of a Regular Function......Page 0207
7.2. Tangent Space......Page 0208
7.3. Tangent Cone......Page 0209
7.5. Normal Bundle......Page 0210
7.7. Sheaves of Differentials......Page 0211
1.1. Irreducible Varieties......Page 0213
1.2. Noetherian Spaces......Page 0214
1.4. Rational Maps......Page 0215
1.5. Graph of a Rational Map......Page 0216
1.6. Blowing up a Point......Page 0217
2.1. Quasi-finite Morphisms......Page 0219
2.2. Finite Morphisms......Page 0220
2.5. Normalization Theorems......Page 0221
2.7. Normal Varieties......Page 0222
2.8. Finite Morphisms Are Open......Page 0223
3.2. Properties of Complete Varieties......Page 0224
3.4. Example of a Complete Nonprojective Variety......Page 0225
3.6. The Connectedness Theorem......Page 0227
3.7. The Stein Factorization......Page 0228
4.2. Dimension and Finite Morphisms......Page 0229
4.3. Dimension of a Hypersurface......Page 0230
4.6. Dimension of Intersections in Affine Space......Page 0231
4.7. The Generic Smoothness Theorem......Page 0232
5.2. Unramified Morphisms......Page 0233
5.3. Embedding of Projective Varieties......Page 0234
5.4. Étale Morphisms......Page 0235
5.6. The Degree of a Finite Morphism......Page 0236
5.7. The Principle of Conservation of Number......Page 0237
6.2. Local Irreducibility......Page 0238
6.3. Factorial Varieties......Page 0239
6.4. Subvarieties of Higher Codimension......Page 0240
6.6. The Cohen-Macaulay Property......Page 0241
7.1. Fundamental Points......Page 0242
7.2. Zariski's Main Theorem......Page 0243
7.4. The Exceptional Variety of a Birational Morphism......Page 0244
7.6. A Criterion for Normality......Page 0245
1.1. External Geometry of a Variety......Page 0247
1.2. The Universal Linear Section......Page 0248
1.3. Hyperplane Sections......Page 0249
1.4. The Connectedness Theorem......Page 0250
1.5. The Ruled Join......Page 0251
1.6. Applications of the Connectedness Theorem......Page 0252
2.1. Definition of the Degree......Page 0253
2.2. Theorem of Bézout......Page 0254
2.3. Degree and Codimension......Page 0255
2.4. Degree of a Linear Projection......Page 0256
2.5. The Hilbert Polynomial......Page 0257
3.1. Cartier Divisors......Page 0258
3.2. Weil Divisors......Page 0259
3.4. Functoriality......Page 0260
3.5. Excision Theorem......Page 0261
3.6. Divisors on Curves......Page 0262
4.1. Families of Divisors......Page 0263
4.3. Linear Systems without Base Points......Page 0264
4.4. Ample Systems......Page 0265
4.5. Linear Systems and Rational Maps......Page 0266
4.7. Linear and Projective Normality......Page 0268
5.2. Direct Image of a Cycle......Page 0269
5.3. Rational Equivalence of Cycles......Page 0270
5.4. Excision Theorem......Page 0271
5.6. Segre Classes of Vector Bundles......Page 0272
5.7. The Splitting Principle......Page 0273
6.2. Deformation to the Normal Cone......Page 0274
6.4. The Chow Ring......Page 0275
6.5. The Chow Ring of Projective Space......Page 0276
6.6. The Chow Ring of a Grassmannian......Page 0277
6.7. Intersections on Surfaces......Page 0278
7.1. Cycles in P^n......Page 0279
7.3. Prom Divisors to Cycles......Page 0280
7.6. Lines on a Cubic......Page 0281
7.7. The Five Conics Problem......Page 0282
Chapter 4. Schemes......Page 0283
1.2. Equations over a Field......Page 0284
1.4. The Prime Spectrum......Page 0285
1.5. Comparison with Varieties......Page 0286
2.2. Topology on the Spectrum......Page 0287
2.5. Example: the Affine Line......Page 0288
2.6. Example: the Abstract Vector......Page 0289
3.2. Examples......Page 0290
3.4. Properties of Schemes......Page 0291
3.5. Properties of Morphisms......Page 0292
3.7. Flat Morphisms......Page 0293
4.2. Geometrization......Page 0294
4.4. Families of Algebraic Schemes......Page 0295
4.5. Smooth Families......Page 0296
References......Page 0297
II: Cohomology of Algebraic Varieties. Algebraic Surfaces......Page 0311
Contents......Page 0317
Introduction......Page 0322
1.1 The Idea of Homology......Page 0323
1.4 Cohomology......Page 0324
1.5 Sheaves......Page 0325
1.6 Cohomology of Sheaves......Page 0326
2.1 Exact Sequences......Page 0327
2.2 Complexes......Page 0328
2.4 Filtered Complexes......Page 0329
2.5 Spectral Sequences......Page 0330
2.6 Bicomplexes......Page 0331
2.8 Products......Page 0332
3.1 Presheaves......Page 0333
3.2 Sheaves......Page 0334
3.4 Abelian Sheaves......Page 0335
3.5 Flabby Sheaves......Page 0336
4.1 Construction of Cohomology......Page 0337
4.2 Hypercohomology......Page 0338
4.3 Higher Direct Images......Page 0339
4.4 Cohomology of a Covering......Page 0340
Chapter 2. Cohomology of Coherent Sheaves......Page 0342
1.1 Quasi-Coherent Sheaves......Page 0343
1.2 Serre's Theorem......Page 0344
1.3 The Koszul Complex......Page 0345
1.4 A Theorem on Affine Coverings......Page 0346
1.6 Higher Direct Images......Page 0347
1.8 Cohomology of Open Inclusions......Page 0348
2.1 Sheaves on P^n and Graded Modules......Page 0349
2.2 Applications to Invertible Sheaves......Page 0350
2.3 Applications to Coherent Sheaves......Page 0351
2.4 Regular Sheaves......Page 0352
2.5 The Euler Characteristic......Page 0353
3.1 The Finiteness Theorem......Page 0354
3.3 Sketch of the Proof......Page 0355
3.5 Continuous Families of Sheaves......Page 0357
3.7 The Lemma on Equivalent Complex......Page 0358
3.8 The Constancy of Euler Characteristic......Page 0359
4.2 The General Riemann Problem......Page 0360
4.3 Chern Classes......Page 0361
4.4 The Riemann-Roth-Hirzebruch Theorem......Page 0363
4.6 Principle of the Proof......Page 0364
5.2 Duality for Curves......Page 0365
5.3 The Serre Duality......Page 0366
5.4 The Hodge Index Theorem......Page 0367
5.5 General Duality......Page 0368
5.6 Duality on Cohen-Macaulay Schemes......Page 0369
6.2 A Degeneration Theorem......Page 0370
6.3 Reduction to Finite Fields......Page 0371
6.4 The Finite Field Case......Page 0372
6.5 The Cartier Operators......Page 0373
6.6 Vanishing Theorems......Page 0374
6.8 Crystalline Cohomology......Page 0375
1.1 Classical Topology......Page 0377
1.2 Properties of the Classical Topology......Page 0378
1.4 The Borel-Moore Homology......Page 0379
1.5 The Intersection Theory......Page 0380
1.6 The Lefschetz Formula......Page 0381
2.2 The Comparison Theorem......Page 0383
2.4 The Weak Lefschetz Theorem......Page 0384
2.6 The Connectedness Theorem......Page 0385
2.8 The Exponential Sequence......Page 0386
3.1 Weight Filtration......Page 0387
3.3 Assembling and Sorting out......Page 0388
3.5 Continuity of Weights......Page 0389
§4. Algebraic Approach to Classical Topology......Page 0390
4.2 Grothendieck's Idea......Page 0391
4.3 Nice Neighborhoods......Page 0392
4.5 Algebraic Coverings......Page 0393
4.6 Instructive Example......Page 0394
1.1 Finite Fields......Page 0395
1.2 Equations over Finite Fields......Page 0397
1.3 Zeta Functions......Page 0398
1.4 Weil's Theorem......Page 0399
1.5 Proof of Weil's Theorem......Page 0400
1.6 The Weil Conjectures......Page 0401
1.7 Weil's Cohomology......Page 0402
2.2 Étale Coverings......Page 0403
2.3 Algebraic Fundamental Group......Page 0404
2.5 Construction of Coverings......Page 0406
3.1 Étale Presheaves......Page 0407
3.2 Étale Sheaves......Page 0408
3.3 Category of Sheaves......Page 0409
3.5 Étale Localization......Page 0410
4.1 Abelian Sheaves......Page 0411
4.3 Galois Cohomology......Page 0412
4.5 Torsors......Page 0413
4.7 Acyclicity of Finite Morphisms......Page 0414
5.1 Outline of Strategy......Page 0415
5.3 Cohomology of O^*......Page 0416
5.4 Cohomology of Complete Curves......Page 0417
5.6 Open Curves......Page 0418
6.3 Cohomology with Compact Support......Page 0419
6.6 Specialization and Vanishing Cycles......Page 0422
6.7 Acyclicity of Smooth Morphisms......Page 0423
6.8 Étale Monodromy......Page 0424
7.2 Finiteness......Page 0425
7.4 Poincare Duality: Orientation......Page 0426
7.5 Poincare Duality: Pairing......Page 0427
7.7 The Weak Lefschetz Theorem......Page 0428
7.10 L-Functions......Page 0429
8.2 Main Theorem......Page 0431
8.3 Outline of Proof......Page 0433
8.5 The Hard Lefschetz Theorem......Page 0434
8.6 Theorem on Invariant Subspace......Page 0435
8.7 Tate's Conjecture......Page 0436
Bibliography......Page 0437
References......Page 0438
Contents......Page 0443
Introduction......Page 0446
§1. Main Invariants......Page 0447
§2. Examples......Page 0450
3.1 Divisors......Page 0459
3.2 Algebraic Equivalence......Page 0460
3.3 Linear Equivalence......Page 0462
3.4 Picard and Albanese Varieties......Page 0465
3.5 Divisors on Fibrations......Page 0466
4.1 Main Properties......Page 0467
4.2 Adjunction Formula......Page 0470
5.1 Riemann-Roch Theorem......Page 0472
5.2 The Cone of Effective Classes of Curves......Page 0473
6.1 \sigma-Process......Page 0479
6.2 Birational Transformations......Page 0482
6.3 Contraction......Page 0487
7.1 The Main Theorem......Page 0490
7.2 Proof of the Main Theorem......Page 0493
7.3 Uniqueness of a Minimal Model......Page 0495
8.1 Main Results......Page 0498
8.2 Discussion of Theorem 1......Page 0500
8.3 The Castelnuovo - de Franchis Inequality......Page 0502
8.4 Discussion of Theorem 2......Page 0503
9.1 Moduli......Page 0506
9.2 Geography of Surfaces......Page 0508
9.3 Almost Rational Surfaces......Page 0511
10.1 Families of Groups......Page 0512
10.2 Singular Fibers......Page 0516
10.3 Jacobian Fibration......Page 0522
10.4 Classification......Page 0524
10.5 Applications......Page 0526
11.1 Enriques Surfaces......Page 0528
11.2 Abelian Surfaces......Page 0530
11.3 Bi-elliptic Surfaces......Page 0533
12.1 Main Invariants......Page 0535
12.2 Projective Geometry......Page 0536
12.3 Topology......Page 0537
12.4 Analytic Geometry......Page 0538
12.5 Applications......Page 0540
13.1 Ruled Surfaces......Page 0542
13.2 Rational Surfaces......Page 0546
13.3 Del Pezzo Surfaces......Page 0548
14.1 Meromorphic Functions......Page 0553
14.2 Cohomology......Page 0555
14.3 Surfaces with a(X) = 0 or a(X) = 1......Page 0557
14.4 Uniformization......Page 0559
15.1 Counterexamples to Bertini's Theorem......Page 0560
15.2 Quotients by a Nonreduced Group Scheme......Page 0561
15.3 Nonreducibility of the Picard Scheme......Page 0562
15.5 Absence of Analogs of the Theorems of Lefschetz and Lüroth......Page 0563
15.7 Changes in Classification......Page 0564
Bibliography......Page 0566
References......Page 0567
Name Index......Page 0571
Subject Index......Page 0573
III: Complex Algebraic Varieties. Algebraic Curves and Their Jacobians......Page 0579
Contents......Page 584
Introduction......Page 586
§1. Algebraic Varieties......Page 594
§2. Complex Manifolds......Page 599
§3. A Comparison Between Algebraic Varieties and Analytic Spaces......Page 602
§4. Complex Manifolds as C^\infty Manifolds......Page 607
§5. Connections on Holomorphic Vector Bundles......Page 611
§6. Hermitian Manifolds......Page 616
§7. Kähler Manifolds......Page 621
§8. Line Bundles and Divisors......Page 632
§9. The Kodaira Vanishing Theorem......Page 637
§10. Monodromy......Page 643
§1. Classifying Space......Page 649
§2. Complex Tori......Page 660
§3. The Period Mapping......Page 667
§4. Variation of Hodge Structures......Page 671
§5. Torelli Theorems......Page 672
§6. Infinitesimal Variation of Hodge Structures......Page 680
§1. Algebraic Curves......Page 683
§2. The Cubic Threefold......Page 691
§3. K3 Surfaces and Elliptic Pencils......Page 698
§4. Hypersurfaces......Page 712
§5. Counterexamples to Torelli Theorems......Page 723
§1. Definition of mixed Hodge structures......Page 726
§2. Mixed Hodge structure on the Cohomology of a Complete Variety with Normal Crossings......Page 732
§3. Cohomology of Smooth Varieties......Page 739
§4. The Invariant Subspace Theorem......Page 748
§5. Hodge Structure on the Cohomology of Smooth Hypersurfaces......Page 751
§6. Further Development of the Theory of Mixed Hodge Structures......Page 759
§1. Degenerations of Manifolds......Page 766
§2. The Limit Hodge Structure......Page 771
§3. The Clemens-Schmid Exact Sequence......Page 773
§4. An Application of the Clemens-Schmid Exact Sequence to the Degeneration of Curves......Page 779
§5. An Application of the Clemens-Schmid Exact Sequence to Surface Degenerations. The Relationship Between the Numerical Invariants of the Fibers X_t and X_0......Page 782
§6. The Epimorphicity of the Period Mapping for K3 Surfaces......Page 788
Comments on the bibliography......Page 794
References......Page 796
Contents......Page 802
1.1. Theory of Burnchall-Chaundy-Krichever......Page 804
1.2. Deformation of Commuting Differential Operators......Page 807
1.3. Kadomtsev-Petviashvili Equations......Page 809
1.4. Finite Dimensional Solutions of the KP Hierarchy......Page 810
1.5. Solutions of the Toda Lattice......Page 811
1.6. Solution of Algebraic Equations Using Theta-Constants......Page 813
2.1. Varieties of Special Divisors and Linear Systems......Page 815
2.2. The Brill-Noether Matrix. The Brill-Noether Numbers......Page 816
2.3. Existence of Special Divisors......Page 817
2.5. Special Curves. The General Case......Page 818
2.7. Infinitesimal Theory of Special Linear Systems......Page 820
2.8. Gauss Mappings......Page 822
2.9. Sharper Bounds on Dimensions......Page 824
2.10. Tangent Cones......Page 825
3.1. Unbranched Double Covers......Page 826
3.2. Prymians and Prym Varieties......Page 827
3.3. Polarization Divisor......Page 829
3.4. Singularities of the Polarization Divisor......Page 832
3.5. Differences Between Prymians and Jacobians......Page 834
3.6. The Prym Map......Page 835
4.1. The Variety of Jacobians......Page 836
4.2. The Andreotti-Meyer Subvariety......Page 837
4.3. Kummer Varieties......Page 838
4.4. Reducedness of \Theta \cap (\Theta + p) and Trisecants......Page 839
4.5. The Characterization of Novikov-Krichever......Page 841
4.6. Schottky Relations......Page 842
References......Page 843
Index......Page 846
IV: Linear Algebraic Groups. Invariant Theory......Page 0854
Contents......Page 0858
Historical Comments......Page 0861
1.2. Morphisms......Page 0864
1.6. Non-Affine Varieties......Page 0865
2.1. The Definition of a Linear Algebraic Group......Page 0866
2.2. Some Basic Facts......Page 0870
2.3. G-Spaces......Page 0871
2.4. The Lie Algebra of an Algebraic Group......Page 0873
2.5. Quotients......Page 0875
§3. Structural Properties of Linear Algebraic Groups......Page 0876
3.1. Jordan Decomposition and Related Results......Page 0877
3.2. Diagonalizable Groups and Tori......Page 0878
3.4. Connected Solvable Groups......Page 0880
3.5. Parabolic Subgroups and Borel Subgroups......Page 0882
4.1. Groups of Rank One......Page 0885
4.2. The Root Datum and the Root System......Page 0887
4.3. Basic Properties of Reductive Groups......Page 0890
4.4. Existence and Uniqueness Theorems for Reductive Groups......Page 0894
4.5. Classification of Quasi-simple Linear Algebraic Groups......Page 0896
4.6. Representation Theory......Page 0899
Chapter 2. Linear Algebraic Groups over Arbitrary Ground Fields......Page 905
1.1. F-Structures on Affine Varieties......Page 0905
1.2. F-Structures on Arbitrary Varieties......Page 0906
1.3. Forms......Page 0907
1.4. Restriction of the Ground Field......Page 0908
2.1. Generalities About F-Groups......Page 0909
2.2. Quotients......Page 0911
2.3. Forms......Page 0912
2.4. Restriction of the Ground Field......Page 0913
3.1. F-Tori......Page 0914
3.2. F-Tori in F-Groups......Page 0916
3.3. Split Tori in F-Groups......Page 0917
4.1. Solvable Groups......Page 0918
4.2. Sections......Page 0919
4.3. Elementary Unipotent Groups......Page 0920
4.5. Basic Results About Solvable F-Groups......Page 0921
5.1. Split Reductive Groups......Page 0922
5.2. Parabolic Subgroups......Page 0923
5.3. The Small Root System......Page 0925
5.4. The Groups G(F)......Page 0929
5.5. The Spherical Tits Building of a Reductive F-Group......Page 0931
6.1. Isomorphism Theorem......Page 0932
6.2. Existence......Page 0934
6.3. Representation Theory of F-Groups......Page 0941
1.1. Algebraic Subalgebras......Page 0943
2.1. Locally Compact Fields......Page 0945
2.2. Real Lie Groups......Page 0948
3.1. Lang's Theorem and its Consequences......Page 0951
3.2. Finite Groups of Lie Type......Page 0954
3.3. Representations of Finite Groups of Lie Type......Page 0956
4.1. The Apartment and Affine Dynkin Diagram......Page 0958
4.2. The Affine Building......Page 0961
4.3. Tits System, Decompositions......Page 0964
4.4. Local Fields......Page 0965
5.1. Adele Groups......Page 0966
5.2. Reduction Theory......Page 0969
5.3. Finiteness Results......Page 0972
References......Page 0975
Contents......Page 0980
Conventions and Notation......Page 0984
0.1. The Subject of Invariant Theory......Page 0986
0.2. Sources of Invariant Theory......Page 0988
0.3. Geometric Methods......Page 0989
0.4. Invariants of the Symmetric Group......Page 0990
0.6. Invariants of a Linear Operator......Page 0991
0.7. Unimodular Invariants of a Quadratic Form......Page 0992
0.9. Invariants of a System of Vectors......Page 0993
0.10. Applications to Projective Geometry......Page 0995
0.12. Invariants of Binary Forms......Page 0997
0.13. Invariants of Binary Polyhedral Groups......Page 0999
0.14. Invariants of a Ternary Cubic Form......Page 1002
1.1. Regular and Rational Actions......Page 1003
1.2. Embedding Theorems......Page 1005
1.3. Orbits......Page 1006
1.4. Stabilizers......Page 1008
1.5. Inheritance of Orbits......Page 1009
2.1. Introduction......Page 1010
2.2. The Graph of an Action......Page 1011
2.3. Separation of Orbits in General Position......Page 1012
2.4. Rational Quotient......Page 1013
2.5. Sections......Page 1014
2.7. Birational Classification of Actions......Page 1016
2.8. Relative Sections......Page 1017
2.9. The Rationality Problem......Page 1019
3.1. Introduction......Page 1021
3.2. Connection Between Integral and Rational Invariants......Page 1022
3.3. Basic Invariants......Page 1023
3.4. Hilbert's Theorem on Invariants......Page 1025
3.5. Constructive Invariant Theory......Page 1026
3.6. Hilbert's Fourteenth Problem......Page 1027
3.7. Grosshans Subgroups......Page 1028
3.8. Chevalley Sections......Page 1030
3.9. Properties of the Algebra of Invariants......Page 1033
3.10. Facts about Poincare Series......Page 1034
3.11. The Poincare Series of the Algebra of Invariants......Page 1036
3.12. Covariants......Page 1038
3.13. The Global Module of Covariants......Page 1040
3.14. The Algebra of Covariants......Page 1041
4.2. The Geometric Quotient......Page 1042
4.4. Construction of the Quotient for an Action of a Reductive Group on an Affine Variety......Page 1044
4.5. Igusa's Criterion......Page 1047
4.6. Construction of the Quotient for an Action of a Reductive Group on an Arbitrary Variety......Page 1048
4.7. Homogeneous Spaces......Page 1050
4.8. Homogeneous Fiber Spaces......Page 1051
5.1. Introduction......Page 1053
5.2. Asymptotic Cones......Page 1054
5.3. The Hilbert-Mumford Criterion......Page 1055
5.4. The Support Method......Page 1056
5.5. The Characteristic of a Nilpotent Element......Page 1058
5.6. Stratification and Resolution of Singularities of the Null-Cone......Page 1062
6.1. Slices: Statement of the Problem......Page 1064
6.2. Excellent Morphisms......Page 1065
6.3. Étale Slices......Page 1066
6.4. Stabilizers of Points in a Neighborhood of a Closed Orbit......Page 1068
6.6. Étale Slices and Analytic Slices......Page 1069
6.7. Structure of Fibers of the Quotient Morphism......Page 1070
6.8. The Theorem on Reaching the Boundary of an Orbit by Means of a One-Parameter Subgroup......Page 1072
6.9. Luna's Stratification......Page 1073
6.10. Sheets......Page 1076
6.11. Closedness of Orbits: Luna's Criterion......Page 1078
6.12. Closedness of Orbits: the Kempf-Ness Criterion......Page 1079
6.13. The Closed Orbit Contained in the Closure of a Given Orbit......Page 1081
6.14. The Moment Mapping......Page 1083
7.1. Introduction......Page 1085
7.2. Existence Theorems for S.G.P......Page 1086
7.3. S.G.P. for Linear Actions......Page 1089
7.4. Closed Orbits in General Position......Page 1092
7.5. S.G.P., Chevalley Sections, and Stability......Page 1093
8.1. Good Properties in Invariant Theory......Page 1095
8.2. Inheritance of Good Properties......Page 1097
8.3. Comparison of the Algebras of Invariants of Finite and Connected Reductive Linear Groups......Page 1098
8.4. The Case of a Two-Dimensional Quotient......Page 1100
8.5. Adjoint Groups of Graded Lie Algebras (0-Groups)......Page 1101
8.6. Polar Groups......Page 1103
8.7. Enumeration of Semisimple Linear Groups with Good Properties......Page 1104
8.8. Weierstrass Sections......Page 1105
9.1. Polarization......Page 1107
9.2. Reduction of the First Fundamental Theorem......Page 1108
9.3. Invariants of Systems of Vectors and Linear Forms......Page 1110
9.4. Relations Between Invariants of Systems of Vectors and Linear Forms......Page 1111
9.5. Invariants of Tensors......Page 1113
Summary Table......Page 1116
References......Page 1120
V: Fano Varieties......Page 1142
Contents......Page 1145
Introduction......Page 1148
§1.1. Singularities......Page 1151
§1.2. On Numerical Geometry of Cycles......Page 1155
§1.3. On the Mori Minimal Model Program......Page 1157
§1.4. Results on Minimal Models in Dimension Three......Page 1161
§2.1. Definitions, Examples and the Simplest Properties......Page 1167
§2.2. Some General Results......Page 1178
§2.3. Existence of Good Divisors in the Fundamental Linear System......Page 1183
§2.4. Base Points in the Fundamental Linear System......Page 1191
§3.1. On Some Preliminary Results of Fujita......Page 1194
§3.2. Del Pezzo Varieties. Definition and Preliminary Results......Page 1197
§3.3. Nonsingular del Pezzo Varieties. Statement of the Main Theorem and Beginning of the Proof......Page 1198
§3.4. Del Pezzo Varieties with Picard Number \rho = 1. Continuation of the Proof of the Main Theorem......Page 1201
§3.5. Del Pezzo Varieties with Picard Number \rho \geq 2. Conclusion of the Proof of the Main Theorem......Page 1206
§4.1. Elementary Rational Maps: Preliminary Results......Page 1209
§4.2. Families of Lines and Conics on Fano Threefolds......Page 1215
§4.3. Elementary Rational Maps with Center along a Line......Page 1220
§4.4. Elementary Rational Maps with Center along a Conic......Page 1230
§4.5. Elementary Rational Maps with Center at a Point......Page 1239
§4.6. Some Other Rational Maps......Page 1245
§5.1. Fano Threefolds of Genus 6 and 8: Gushel's Approach......Page 1248
§5.2. A Review of Mukai's Results on the Classification of Fano Manifolds of Comdex 3......Page 1252
§6.1. Uniruledness......Page 1260
§6.2. Rational Connectedness of Fano Varieties......Page 1264
§7.1. Fano Threefolds with Picard Number \rho \geq 2 (Survey of Results of Mori and Mukai)......Page 1272
§7.2. A Survey of Results about Higher-dimensional Fano Varieties with Picard Number \rho \geq 2......Page 1285
§8.1. Intermediate Jacobian and Prym Varieties......Page 1297
§8.2. Intermediate Jacobian: the Abel-Jacobi Map......Page 1306
§8.3. The Brauer Group as a Birational Invariant......Page 1310
§9.1. Birational Automorphisms of Fano Varieties......Page 1314
§9.2. Decomposition of Birational Maps in the Context of Mori Theory......Page 1322
§10.1. Some Constructions of Unirationality......Page 1327
§10.2. Unirationality of Complete Intersections......Page 1332
§10.3. Some General Constructions of Rationality......Page 1335
§11.1. On the Classification of Three-dimensional Q-Fano Varieties......Page 1340
§11.2. Generalizations......Page 1347
§11.3. Some Particular Results......Page 1352
§11.4. Some Open Problems......Page 1356
§12.2. Fano Threefolds with \rho = 1......Page 1358
§12.3. Fano Threefolds with \rho = 2......Page 1361
§12.4. Fano Threefolds with \rho = 3......Page 1364
§12.5. Fano Threefolds with \rho = 4......Page 1367
§12.6. Fano Threefolds with \rho \geq 5......Page 1368
§12.7. Fano Fourfolds of Index 2 with \rho \geq 2......Page 1369
§12.8. Toric Fano Threefolds......Page 1370
References......Page 1371
Index......Page 1390