This edited collection of chapters, authored by leading experts, provides a complete and essentially self-contained construction of 3-fold and 4-fold klt flips. A large part of the text is a digest of Shokurov's work in the field and a concise, complete and pedagogical proof of the existence of 3-fold flips is presented. The text includes a ten page glossary and is accessible to students and researchers in algebraic geometry.
Author(s): Alessio Corti
Series: Oxford Lecture Series in Mathematics and Its Applications
Publisher: Oxford University Press, USA
Year: 2007
Language: English
Pages: 184
Contents......Page 6
List of Contributors......Page 10
1.2 Higher dimensions and flips......Page 12
1.3 The work of Shokurov......Page 13
1.5 The aim of this book......Page 14
1.6 Pl flips......Page 15
1.7 b-divisors......Page 16
1.9 Pbd-algebras......Page 17
1.10 Restricted systems and 3-fold pl flips......Page 18
1.11 Shokurov’s finite generation conjecture......Page 19
1.12 What is log terminal?......Page 20
1.14 The work of Hacon and M[sup(c)]Kernan: adjoint algebras......Page 21
1.17 Kodaira’s canonical bundle formula and adjunction......Page 25
1.20 The book as a whole......Page 26
1.23 Pre-requisites......Page 27
2.1.1 Statement and brief history of the problem......Page 29
2.1.2 Summary of the chapter......Page 30
2.2.2 Discrepancy......Page 31
2.2.4 Inversion of adjunction......Page 33
2.2.6 Reduction of klt flips to pl flips......Page 34
2.2.7 Plan of the proof......Page 35
2.2.8 Log resolution......Page 36
2.3.1 Function algebras......Page 37
2.3.2 b-divisors......Page 38
2.3.3 Saturated divisors and b-divisors......Page 41
2.3.4 Mobile b-divisors......Page 43
2.3.5 Restriction and saturation......Page 45
2.3.6 Pbd-algebras: terminology and first properties......Page 47
2.3.8 Function algebras and pbd-algebras......Page 48
2.3.9 The finite generation conjecture......Page 49
2.3.10 A Shokurov algebra on a curve is finitely generated......Page 50
2.4.1 The finite generation conjecture implies existence of pl flips......Page 51
2.4.2 Linear systems on surfaces......Page 53
2.4.3 The finite generation conjecture on surfaces......Page 56
3.1 What is log terminal?......Page 60
3.2 Preliminaries on Q-divisors......Page 61
3.4 Divisorially log terminal......Page 63
3.5 Resolution lemma......Page 64
3.6 Whitney umbrella......Page 65
3.7 What is a log resolution?......Page 67
3.8 Examples......Page 69
3.9 Adjunction for dlt pairs......Page 70
3.10 Miscellaneous comments......Page 71
4.1 Introduction......Page 74
4.2 Special termination......Page 75
4.3 Reduction theorem......Page 79
4.4 The non-Q-factorial minimal model program......Page 83
5.1.1 The conjectures of the MMP......Page 87
5.1.3 Sketch of the proof......Page 89
5.1.4 Notation and conventions......Page 93
5.2 The real minimal model program......Page 94
5.3.1 Generalities on finite generation......Page 98
5.3.2 Adjoint algebras......Page 100
5.3.3 Basic properties of the mobile part......Page 102
5.4.1 Definition and first properties of multiplier ideal sheaves......Page 106
5.4.2 Extending sections......Page 111
5.4.3 The restricted algebra is an adjoint algebra......Page 118
6.1 Introduction......Page 122
6.2 Example......Page 123
6.3 Preliminaries......Page 124
6.4 Mobile b-divisors of Iitaka dimension one......Page 126
7.1 Introduction......Page 132
7.2 Diophantine approximation and descent......Page 133
7.3 Confined b-divisors......Page 135
7.4 The CCS conjecture......Page 137
7.5 The surface case......Page 139
7.6 A strategy for the general case......Page 140
8.1 Introduction......Page 145
8.2 Fujita’s canonical bundle formula......Page 147
8.3 The general canonical bundle formula......Page 149
8.4 Fibre spaces of log Kodaira dimension 0......Page 153
8.5 Kawamata’s canonical bundle formula......Page 159
8.6 Adjunction......Page 161
8.7 Tie breaking......Page 163
8.8 Log canonical purity......Page 165
8.9 Ambro’s seminormality theorem......Page 167
8.10 Covering tricks and semipositivity of f[sub(*)]ω[sub(X /Y)]......Page 169
9.2.1 Codimension one adjunction......Page 174
9.2.2 The discriminant of a log pair......Page 175
9.3 Non-klt finite generation......Page 177
B......Page 182
C......Page 184
E......Page 185
L......Page 186
M......Page 187
N......Page 188
Q......Page 189
T......Page 190
Z......Page 191
Bibliography......Page 192
C......Page 198
L......Page 199
W......Page 200
