## Positivity in Algebraic Geometry I: Classical Setting: Line Bundles and Linear SeriesThis two volume work on Positivity in Algebraic Geometry contains a contemporary account of a body of work in complex algebraic geometry loosely centered around the theme of positivity. Topics in Volume I include ample line bundles and linear series on a projective variety, the classical theorems of Lefschetz and Bertini and their modern outgrowths, vanishing theorems, and local positivity. Volume II begins with a survey of positivity for vector bundles, and moves on to a systematic development of the theory of multiplier ideals and their applications. A good deal of this material has not previously appeared in book form, and substantial parts are worked out here in detail for the first time. At least a third of the book is devoted to concrete examples, applications, and pointers to further developments. Volume I is more elementary than Volume II, and, for the most part, it can be read without access to Volume II. |

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### Contents

Ample and Nef Line Bundles | 7 |

11A Divisors and Line Bundles | 8 |

11B Linear Series | 12 |

11C Intersection Numbers and Numerical Equivalence | 15 |

11D RiemannRoch | 20 |

12 The Classical Theory | 24 |

12A Cohomological Properties | 25 |

12B Numerical Properties | 33 |

24B Graded Families of Ideals | 176 |

Notes | 183 |

Geometric Manifestations of Positivity | 185 |

31A Topology of Affine Varieties | 186 |

31B The Theorem on Hyperplane Sections | 192 |

31C Hard Lefschetz Theorem | 199 |

32 Projective Subvarieties of Small Codimension | 201 |

32B Hartshornes Conjectures | 204 |

12C Metric Characterizations of Amplitude | 39 |

13 QDivisors and RDivisors | 44 |

13B RDivisors and Their Amplitude | 48 |

14 Nef Line Bundles and Divisors | 50 |

14A Definitions and Formal Properties | 51 |

14B Kleimans Theorem | 53 |

14C Cones | 59 |

14D Fujitas Vanishing Theorem | 65 |

15 Examples and Complements | 70 |

15B Products of Curves | 73 |

15C Abelian Varieties | 79 |

15D Other Varieties | 80 |

15E Local Structure of the Nef Cone | 82 |

15F The Cone Theorem | 86 |

16 Inequalities | 88 |

16B Mixed Multiplicities | 91 |

17 Amplitude for a Mapping | 94 |

18 CastelnuovoMumford Regularity | 98 |

18A Definitions Formal Properties and Variants | 99 |

18B Regularity and Complexity | 107 |

18C Regularity Bounds | 110 |

18D Syzygies of Algebraic Varieties | 115 |

Notes | 119 |

Linear Series | 121 |

21A Basic Definitions | 122 |

21B Semiample Line Bundles | 128 |

21C litaka Fibration | 133 |

22 Big Line Bundles and Divisors | 139 |

22B Pseudoeffective and Big Cones | 145 |

22C Volume of a Big Divisor | 148 |

23 Examples and Complements | 157 |

23A Zariskis Construction | 158 |

23B Cutkoskys Construction | 159 |

23C Base Loci of Nef and Big Linear Series | 164 |

23D The Theorem of Campana and Peternell | 166 |

23E Zariski Decompositions | 167 |

24 Graded Linear Series and Families of Ideals | 172 |

33 Connectedness Theorems | 207 |

33B Theorem of Fulton and Hansen | 210 |

33C Grothendiecks Connectedness Theorem | 212 |

34 Applications of the FultonHansen Theorem | 213 |

34A Singularities of Mappings | 214 |

34B Zaks Theorems | 219 |

34C Zariskis Problem | 227 |

35 Variants and Extensions | 231 |

35B Higher Connectivity for Mappings to Projective Space | 233 |

Notes | 237 |

Vanishing Theorems | 239 |

41 Preliminaries | 240 |

41B Covering Lemmas | 242 |

42 Kodaira and Nakano Vanishing Theorems | 248 |

43 Vanishing for Big and Nef Line Bundles | 252 |

43B Some Applications | 257 |

44 Generic Vanishing Theorem | 261 |

Notes | 267 |

Local Positivity | 269 |

52 Lower Bounds | 278 |

52B Multiplicities of Divisors in Families | 282 |

52C Proof of Theorem 525 | 286 |

53 Abelian Varieties | 290 |

53B Proof of Theorem 536 | 297 |

53C Complements | 301 |

54 Local Positivity Along an Ideal Sheaf | 303 |

54B Complexity Bounds | 308 |

Notes | 312 |

Appendices | 313 |

Projective Bundles | 315 |

Cohomology and Complexes | 317 |

B2 Complexes | 320 |

325 | |

Glossary of Notation | 359 |

365 | |

### Other editions - View all

Positivity in Algebraic Geometry II: Positivity for Vector Bundles, and ... R.K. Lazarsfeld No preview available - 2004 |

Positivity in Algebraic Geometry II: Positivity for Vector Bundles, and ... R.K. Lazarsfeld No preview available - 2004 |