Flips for 3-folds and 4-folds Florin Ambro Res. Inst. Math. Sci., Kyoto University Kyoto 606-8502, Japan Alessio Corti Department of Mathematics, Imperial College London Huxley Building, 180 Queen’s Gate London SW7 2AZ, UK Osamu Fujino Grad. School of Math., Nagoya University Chikusa-ku, Nagoya 464-8602, Japan Christopher D. Hacon Department of Mathematics, University of Utah 155 South 1400 East Salt Lake City UT 84112-0090, USA J´anos Koll´ar Department of Mathematics, Princeton University Fine Hall, Washington Road Princeton NJ 08544-1000, USA James McKernan Department of Mathematics, UC Santa Barbara Santa Barbara CA 93106, USA Hiromichi Takagi Grad. School of Math. Sci., The University of Tokyo 3-8-1 Komaba, Meguro-ku, Tokyo 153–8914, Japan Contents Chapter 1. Introduction Alessio Corti 1 1.1. Minimal models of surfaces 1 1.2. Higher dimensions and flips 1 1.3. The work of Shokurov 2 1.4. Minimal models of 3-folds and 4-folds 2 1.5. The aim of this book 3 1.6. Pl flips 3 1.7. b-divisors 4 1.8. Restriction and mobile b-divisors. 5 1.9. Pbd-algebras 5 1.10. Restricted systems and 3-fold pl flips 6 1.11. Shokurov’s finite generation conjecture 6 1.12. What is log terminal? 7 1.13. Special termination and reduction to pl flips. 8 1.14. The work of Hacon and McKernan: adjoint algebras 8 1.15. Mobile b-divisors on weak klt del Pezzo surfaces 9 1.16. The CCS Conjecture 9 1.17. Kodaira’s canonical bundle formula and subadjunction 10 1.18. Finite generation on non-klt surfaces 10 1.19. The glossary 10 1.20. The book as a whole 10 1.21. Prerequisites 11 1.22. Acknowledgements 11 Chapter 2. 3-fold flips after Shokurov Alessio Corti 13 2.1. Introduction 13 2.1.1. Statement and brief history of the problem 13 2.1.2. Summary of the chapter 14 2.1.3. Other surveys 14 2.2. Background 14 2.2.1. Summary 15 2.2.2. Discrepancy 15 2.2.3. Klt and plt 16 2.2.4. Inversion of adjunction 17 2.2.5. The flipping conjecture 17 2.2.6. Reduction to pl flips 18 2.2.7. Plan of the proof 19 iii iv CONTENTS 2.2.8. Log resolution 19 2.3. The language of function algebras and b-divisors 20 2.3.1. Function algebras 20 2.3.2. b-divisors 21 2.3.3. Saturated divisors and b-divisors 24 2.3.4. Mobile b-divisors 26 2.3.5. Restriction and saturation 28 2.3.6. Pbd-algebras: terminology and first properties 29 2.3.7. The limiting criterion 30 2.3.8. Function algebras and pbd-algebras 31 2.3.9. The finite generation conjecture 32 2.3.10. A Shokurov algebra on a curve is finitely generated 33 2.4. Finite generation on surfaces and existence of 3-fold flips 34 2.4.1. The finite generation conjecture implies existence of pl flips 34 2.4.2. Linear systems on surfaces 35 2.4.3. The finite generation conjecture on surfaces 37 2.5. Acknowledgements 40 Chapter 3. What is log terminal? Osamu Fujino 41 3.1. What is log terminal? 41 3.2. Preliminaries on Q-divisors 42 3.3. Singularities of pairs 43 3.4. Divisorially log terminal 44 3.5. Resolution lemma 45 3.6. Whitney umbrella 46 3.7. What is a log resolution? 48 3.8. Examples 49 3.9. Adjunction for dlt pairs 50 3.10. Miscellaneous comments 52 3.11. Acknowledgements 53 Chapter 4. Special termination and reduction to pl flips Osamu Fujino 55 4.1. Introduction 55 4.2. Special termination 56 4.3. Reduction theorem 60 4.4. A remark on the log MMP 63 4.5. Acknowledgements 65 Chapter 5. Extension theorems and the existence of flips Christopher Hacon and James McKernan 67 5.1. Introduction 67 5.1.1. The conjectures of the MMP 67 5.1.2. Previous results 69 5.1.3. Sketch of the proof 69 5.1.4. Notation and conventions 72 5.2. The real minimal model program 74 5.3. Finite generation 77 CONTENTS v 5.3.1. Generalities on Finite generation 77 5.3.2. Adjoint Algebras 79 5.4. Multiplier ideal sheaves and extension results 80 5.4.1. Definition and first properties of multiplier ideal sheaves 80 5.4.2. Extending Sections 84 5.4.3. The Restricted Algebra is an Adjoint Algebra 91 5.5. Acknowledgements 95 Chapter 6. Saturated mobile b-divisors on weak del Pezzo klt surfaces Alessio Corti, James McKernan and Hiromichi Takagi 97 6.1. Introduction 97 6.2. Example 98 6.3. Preliminaries 99 6.4. Mobile b-divisors of Iitaka dimension one 101 Chapter 7. Confined divisors James McKernan 105 7.1. Introduction 105 7.2. Diophantine Approximation and Descent 106 7.3. Confined b-divisors 107 7.4. The CCS conjecture 109 7.5. The surface case 111 7.6. A strategy for the general case 112 7.7. Acknowledgements 115 Chapter 8. Kodaira’s canonical bundle formula and subadjunction Janos´ Kollar´ 117 8.1. Introduction 117 8.2. Fujita’s canonical bundle formula 118 8.3. The general canonical bundle formula 120 8.4. Fiber spaces of log Kodaira dimension 0 124 8.5. Kawamata’s canonical bundle formula 129 8.6. Subadjunction 131 8.7. Log canonical purity 133 8.8. Ambro’s seminormality theorem 134 8.9. Covering tricks and Semipositivity of f∗ωX/Y 136 8.10. Log Fano varieties 140 8.11. Acknowledgements 141 Chapter 9. Non-klt techniques Florin Ambro 143 9.1. Introduction 143 9.2. Preliminary 143 9.2.1. Codimension one adjunction 143 9.2.2. The discriminant of a log pair 144 9.3. Non-klt FGA 146 Chapter 10. Glossary Alessio Corti 151 vi CONTENTS Bibliography 161 Index 165 CHAPTER 1 Introduction Alessio Corti 1.1. Minimal models of surfaces In this book, we generalise the following theorem to 3-folds and 4-folds: Theorem 1.1.1 (Minimal model theorem for surfaces). Let X be a nonsingular projective surface. There is a birational morphism f : X → X0 to a nonsingular projective surface X0 satisfying one of the following conditions: 0 X is a minimal model: KX0 is nef, that is KX0 · C ≥ 0 for every curve C ⊂ X0; or X0 is a Mori fibre space: X0 =∼ P2 or X0 is a P1-bundle over a nonsingu- lar curve T . This result is well known; the classical proof runs more or less as follows. If X does not satisfy the conclusion, then X contains a −1-curve, that is, a nonsingular 2 rational curve E ⊂ X such that KX · E = E = −1. By the Castelnuovo con- tractibility theorem, a −1-curve can always be contracted: there exists a morphism f : E ⊂ X → P ∈ X1 which maps E ⊂ X to a nonsingular point P ∈ X1 and restricts to an isomorphism X r E → X1 r {P }. Viewed from P ∈ X1, this is just the blow up of P ∈ X1. Either X1 satisfies the conclusion, or we can continue contracting a −1-curve in X1. Every time we contract a −1-curve, we decrease the rank of the N´eron-Severi group; hence the process must terminate (stop). 1.2. Higher dimensions and flips The conceptual framework generalising this result to higher dimensions is the well known Mori program or Minimal Model Program. The higher dimensional analog f : X → X1 of the contraction of a −1-curve is an extremal divisorial con- traction. Even if we start with X nonsingular, X1 can be singular. This is not a problem: we now know how to handle the relevant classes of singularities, for exam- ple the class of terminal singularities. The problem is that, in higher dimensions, we meet a new type of contraction: Definition 1.2.1. A small contraction is a birational morphism f : X → Z with connected fibres such the exceptional set C ⊂ X is of codimension codimX C ≥ 2. When we meet a small contraction f : X → Z, the singularities on Z are so bad that the canonical class of Z does not even have a Chern class in H2(Z, Q): it does not make sense to form the intersection number KZ · C with algebraic curves C ⊂ Z. We need a new type of operation, called a flip: 1 2 1. INTRODUCTION Definition 1.2.2. A small contraction f : X → Z is a flipping contraction if KX is anti-ample along the fibres of f. The flip of f is a new small contraction 0 0 0 f : X → Z such that KX0 is ample along the fibres of f It is conjectured that flips exist and that every sequence of flips terminates (that is, there exists no infinite sequence of flips). 1.3. The work of Shokurov Mori [Mor88] first proved that flips exist if X is 3-dimensional. It was known (and, in any case, it is easy to prove) that 3-fold flips terminate, thus Mori’s theorem implied the minimal model theorem in dimension 3. Recently, Shokurov [Sho03] completed in dimension ≤ 4 a program to con- struct flips started in [Sho92, FA92]. In fact, Shokurov shows that a more general type of flip exists. We consider perturbations of the canonical class KX of the P form KX + biBi, where Bi ⊂ X are prime divisors and 0 ≤ bi < 1. For the P program to work, the pair X, biBi needs to have klt singularities.