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El. knyga: Flips for 3-folds and 4-folds

Edited by (Department of Mathematics, Imperial College, London)
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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.
List of Contributors
ix
Introduction
1(17)
Minimal models of surfaces
1(1)
Higher dimensions and flips
1(1)
The work of Shokurov
2(1)
Minimal models of 3-folds and 4-folds
3(1)
The aim of this book
3(1)
PI flips
4(1)
b-divisors
5(1)
Restriction and mobile b-divisors
6(1)
Pbd-algebras
6(1)
Restricted systems and 3-fold pi flips
7(1)
Shokurov's finite generation conjecture
8(1)
What is log terminal?
9(1)
Special termination and reduction to pi flips
10(1)
The work of Hacon and McKernan: adjoint algebras
10(4)
Saturated mobile b-divisors on weak del Pezzo kit surfaces
14(1)
The CCS Conjecture
14(1)
Kodaira's canonical bundle formula and adjunction
14(1)
Finite generation on non-kit surfaces
15(1)
The glossary
15(1)
The book as a whole
15(1)
Termination?
16(1)
The work of Siu
16(1)
Pre-requisites
16(2)
3-fold flips after Shokurov
18(31)
Introduction
18(2)
Statement and brief history of the problem
18(1)
Summary of the chapter
19(1)
Other surveys
20(1)
Background
20(6)
Summary
20(1)
Discrepancy
20(2)
Kit and pit
22(1)
Inversion of adjunction
22(1)
The flip conjecture
23(1)
Reduction of kit flips to pi flips
23(1)
Plan of the proof
24(1)
Log resolution
25(1)
The language of function algebras and b-divisors
26(14)
Function algebras
26(1)
b-divisors
27(3)
Saturated divisors and b-divisors
30(2)
Mobile b-divisors
32(2)
Restriction and saturation
34(2)
Pbd-algebras: terminology and first properties
36(1)
The limiting criterion
37(1)
Function algebras and pbd-algebras
37(1)
The finite generation conjecture
38(1)
A Shokurov algebra on a curve is finitely generated
39(1)
Finite generation on surfaces and existence of 3-fold flips
40(9)
The finite generation conjecture implies existence of pi flips
40(2)
Linear systems on surfaces
42(3)
The finite generation conjecture on surfaces
45(4)
What is log terminal?
49(14)
What is log terminal?
49(1)
Preliminaries on Q-divisors
50(2)
Singularities of pairs
52(1)
Divisorially log terminal
52(1)
Resolution lemma
53(1)
Whitney umbrella
54(2)
What is a log resolution?
56(2)
Examples
58(1)
Adjunction for dlt pairs
59(1)
Miscellaneous comments
60(3)
Special termination and reduction to pi flips
63(13)
Introduction
63(1)
Special termination
64(4)
Reduction theorem
68(4)
The non-Q-factorial minimal model program
72(4)
Extension theorems and the existence of flips
76(35)
Introduction
76(7)
The conjectures of the MMP
76(2)
Previous results
78(1)
Sketch of the proof
78(4)
Notation and conventions
82(1)
The real minimal model program
83(4)
Finite generation
87(8)
Generalities on finite generation
87(2)
Adjoint algebras
89(2)
Basic properties of the mobile PART
91(4)
Multiplier ideal sheaves and extension results
95(16)
Definition and first properties of multiplier ideal sheaves
95(5)
Extending sections
100(7)
The restricted algebra is an adjoint algebra
107(4)
Saturated mobile b-divisors on weak del Pezzo kit Surfaces
111(10)
Introduction
111(1)
Example
112(1)
Preliminaries
113(2)
Mobile b-divisors of Iitaka dimension one
115(6)
Confined divisors
121(13)
Introduction
121(1)
Diophantine approximation and descent
122(2)
Confined b-divisors
124(2)
The CCS conjecture
126(2)
The surface case
128(1)
A strategy for the general case
129(5)
Kodaira's Canonical bundle formula and adjunction
134(29)
Introduction
134(2)
Fujita's canonical bundle formula
136(2)
The general canonical bundle formula
138(4)
Fibre spaces of log Kodaira dimension 0
142(6)
Kawamata's canonical bundle formula
148(2)
Adjunction
150(2)
Tie breaking
152(2)
Log canonical purity
154(2)
Ambro's seminormality theorem
156(2)
Covering tricks and semipositivity of f*WX/Y
158(5)
Non-kit techniques
163(8)
Introduction
163(1)
Preliminary
163(3)
Codimension one adjunction
163(1)
The discriminant of a log pair
164(2)
Non-kit finite generation
166(5)
Glossary
171(10)
Bibliography 181(6)
Index 187


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