Notation and Conventions ........................................ 1
Part One: Ample Line Bundles and Linear Series
Introduction to Part One ........................................ 5
1. Ample and Nef Line Bundles ................................... 7
1.1. Preliminaries: Divisors, Line Bundles, and Linear
Series .................................................. 7
1.1.A. Divisors and Line Bundles ........................ 8
1.1.В. Linear Series ................................... 12
1.1.С. Intersection Numbers and Numerical
Equivalence ..................................... 15
l.l.D. Riemann-Roch .................................... 20
1.2. The Classical Theory ................................... 24
1.2.A. Cohomological Properties ........................ 25
1.2.В. Numerical Properties ............................ 33
1.2.С. Metric Characterizations of Amplitude ........... 39
1.3. Q-Divisors and R-Divisors .............................. 44
1.3.A. Definitions for Q-Divisors ...................... 44
1.3.B. R-Divisors and Their Amplitude .................. 48
1.4. Nef Line Bundles and Divisors .......................... 50
1.4.A. Definitions and Formal Properties ............... 51
1.4.В. Kleiman's Theorem ............................... 53
1.4.C. Cones ........................................... 59
1.4.D. Fujita's Vanishing Theorem ...................... 65
1.5. Examples and Complements ............................... 70
1.5.A. Ruled Surfaces .................................. 70
1.5.В. Products of Curves .............................. 73
1.5.С. Abelian Varieties ............................... 79
1.5.D. Other Varieties ................................. 80
1.5.E. Local Structure of the Nef Cone ................. 82
1.5.F. The Cone Theorem ................................ 86
1.6. Inequalities ........................................... 88
1.6.A. Global Results .................................. 88
1.6.B. Mixed Multiplicities ............................ 91
1.7. Amplitude for a Mapping ................................ 94
1.8. Castelnuovo-Mumford Regularity ......................... 98
1.8.A. Definitions, Formal Properties, and Variants .... 99
1.8.В. Regularity and Complexity ...................... 107
1.8.С. Regularity Bounds .............................. 110
1.8.D. Syzygies of Algebraic Varieties ................ 115
Notes ...................................................... 119
2. Linear Series .............................................. 121
2.1. Asymptotic Theory ..................................... 121
2.1.A. Basic Definitions .............................. 122
2.1.В. Semiample Line Bundles ......................... 128
2.1.C. Iitaka Fibration ............................... 133
2.2. Big Line Bundles and Divisors ......................... 139
2.2.A. Basic Properties of Big Divisors ............... 139
2.2.В. Pseudoeffective and Big Cones .................. 145
2.2.C. Volume of a Big Divisor ........................ 148
2.3. Examples and Complements .............................. 157
2.3.A. Zariski's Construction ......................... 158
2.3.В. Cutkosky's Construction ........................ 159
2.3.С. Base Loci of Nef and Big Linear Series ......... 164
2.3.D. The Theorem of Campana and Peternell ........... 166
2.3.E. Zariski Decompositions ......................... 167
2.4. Graded Linear Series and Families of Ideals ........... 172
2.4.A. Graded Linear Series ........................... 172
2.4.B. Graded Families of Ideals ...................... 176
Notes ...................................................... 183
3. Geometric Manifestations of Positivity ..................... 185
3.1. The Lefschetz Theorems ................................ 185
3.1.A. Topology of Affine Varieties ................... 186
3.1.В. The Theorem on Hyperplane Sections ............. 192
3.1.C. Hard Lefschetz Theorem ......................... 199
3.2. Projective Subvarieties of Small Codimension. 201
3.2.A. Barth's Theorem ................................ 201
3.2.В. Hartshorne's Conjectures ....................... 204
3.3. Connectedness Theorems ................................ 207
3.3.A. Bertini Theorems ............................... 207
3.3.B. Theorem of Fulton and Hansen ................... 210
3.3.С. Grothendieck's Connectedness Theorem ........... 212
3.4. Applications of the Fulton-Hansen Theorem ............. 213
3.4.A. Singularities of Mappings ...................... 214
3.4.B. Zak's Theorems ................................. 219
3.4.C. Zariski's Problem .............................. 227
3.5. Variants and Extensions ............................... 231
3.5.A. Homogeneous Varieties .......................... 231
3.5.В. Higher Connectivity ............................ 233
Notes ...................................................... 237
4. Vanishing Theorems ......................................... 239
4.1. Preliminaries ......................................... 240
4.1.A. Normal Crossings and Resolutions of
Singularities .................................. 240
4.1.В. Covering Lemmas ................................ 242
4.2. Kodaira and Nakano Vanishing Theorems ................. 248
4.3. Vanishing for Big and Nef Line Bundles ................ 252
4.3.A. Statement and Proof of the Theorem ............. 252
4.3.B. Some Applications .............................. 257
4.4. Generic Vanishing Theorem ............................. 261
Notes ...................................................... 267
5. Local Positivity ........................................... 269
5.1. Seshadri Constants .................................... 269
5.2. Lower Bounds .......................................... 278
5.2.A. Background and Statements ...................... 278
5.2.В. Multiplicities of Divisors in Families ......... 282
5.2.C. Proof of Theorem 5.2.5 ......................... 286
5.3. Abelian Varieties ..................................... 290
5.3.A. Period Lengths and Seshadri Constants .......... 290
5.3.B. Proof of Theorem 5.3.6 ......................... 297
5.3.С. Complements .................................... 301
5.4. Local Positivity Along an Ideal Sheaf ................. 303
5.4.A. Definition and Formal Properties of the
s-Invariant .................................... 303
5.4.B. Complexity Bounds .............................. 308
Notes ...................................................... 312
Appendices
A. Projective Bundles ...................................... 315
В. Cohomology and Complexes ................................ 317
B.l. Cohomology ......................................... 317
B.2. Complexes .......................................... 320
References .................................................... 325
Glossary of Notation .......................................... 359
Index ......................................................... 365
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