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Foundation of the Minimal Model Program 2014/4/16 Version 0.01 Foundation of the minimal model program 2014/4/16 version 0.01 Osamu Fujino Department of Mathematics, Faculty of Science, Kyoto University, Kyoto 606-8502, Japan E-mail address: [email protected] 2010 Mathematics Subject Classification. Primary 14E30, 14F17; Secondary 14J05, 14E15 Abstract. We discuss various vanishing theorems. Then we es- tablish the fundamental theorems, that is, various Kodaira type vanishing theorems, the cone and contraction theorem, and so on, for quasi-log schemes. Preface This book is a completely revised version of the author's unpub- lished manuscript: • Osamu Fujino, Introduction to the minimal model program for log canonical pairs, preprint 2008. We note that the above unpublished manuscript is an expanded version of the composition of • Osamu Fujino, Vanishing and injectivity theorems for LMMP, preprint 2007 and • Osamu Fujino, Notes on the log minimal model program, preprint 2007. We also note that this book is not an introductory text book of the minimal model program. One of the main purposes of this book is to establish the funda- mental theorems, that is, various Kodaira type vanishing theorems, the cone and contraction theorem, and so on, for quasi-log schemes. The notion of quasi-log schemes was introduced by Florin Ambro in his epoch-making paper: • Florin Ambro, Quasi-log varieties, Tr. Mat. Inst. Steklova 240 (2003), 220{239. The theory of quasi-log schemes is extremely powerful. Unfortu- nately, it has not been popular yet because Ambro's paper has several difficulties. Moreover, the author's paper: • Osame Fujino, Fundamental theorems for the log minimal model program, Publ. Res. Inst. Math. Sci. 47 (2011), no. 3, 727{ 789 recovered the main result of Ambro's paper, that is, the cone and con- traction theorem for normal pairs, without using the theory of quasi-log schemes. Note that the author's approach in the above paper is suffi- cient for the fundamental theorems of the minimal model program for log canonical pairs. iii iv PREFACE Recently, the author proved that every quasi-projective semi log canonical pair has a natural quasi-log structure which is compatible with the original semi log canonical structure in • Osamu Fujino, Fundamental theorems for semi log canonical pairs, Algebraic Geometry 1 (2014), no. 2, 194{228. This result shows that the theory of quasi-log schemes is indispens- able for the study of semi log canonical pairs. Now the importance of the theory of quasi-log schemes is increasing. In this book, we will establish the foundation of quasi-log schemes. One of the author's main contributions in the above papers is to introduce the theory of mixed Hodge structures on cohomology groups with compact support to the minimal model program systematically. By pursuing this approach, we can naturally obtain a correct general- ization of the Fujita{Kawamata semipositivity theorem in • Osamu Fujino, Taro Fujisawa, Variations of mixed Hodge struc- ture and semipositivity theorems, to appear in Publ. Res. Inst. Math. Sci. This new powerful semipositivity theorem leads to the proof of the projectivity of the coarse moduli spaces of stable varieties in • Osamu Fujino, Semipositivity theorems for moduli problems, preprint 2012. Note that a stable variety is a projective semi log canonical variety with ample canonical divisor. Anyway, the theory of quasi-log schemes seems to be indispensable for the study of higher-dimensional algebraic varieties and its impor- tance is increasing now. On page 57 in • J´anosKoll´ar,Shigefumi Mori, Birational geometry of alge- braic varieties, Cambridge University Press, 1998, which is a standard text book on the minimal model program, the authors wrote: Log canonical: This is the largest class where discrep- ancy still makes sense. It contains many cases that are rather complicated from the cohomological point of view. Therefore it is very hard to work with. On page 209, they also wrote: The theory of these so-called semi-log canonical (slc for short) pairs is not very much different from the lc case but it needs some foundational work. PREFACE v By the author's series of papers including this book, we greatly improve the situation around log canonical pairs and semi log canonical pairs from the cohomological point of view. Acknowledgments. The author would like to thank Professors Shigefumi Mori, Yoichi Miyaoka, Noboru Nakayama, Daisuke Mat- sushita, and Hiraku Kawanoue, who were the members of the seminars when he was a graduate student at RIMS, Kyoto. In the seminars at RIMS in 1997, he learned the foundation of the minimal model program by reading a draft of [KoMo]. He thanks Professors Takao Fujita, Noboru Nakayama, Hiromichi Takagi, Florin Ambro, Hiroshi Sato, Takeshi Abe, Masayuki Kawakita, Yoshinori Gongyo, Yoshinori Namikawa, and Hiromu Tanaka for discussions, comments, and ques- tions. He also would like to thank Professor J´anosKoll´arfor giving him many comments on the preliminary version of this book and showing him many examples. Finally, the author thanks Professors Shigefumi Mori, Shigeyuki Kondo, Takeshi Abe, and Yukari Ito for warm encour- agement during the preparation of this book. The author was partially supported by the Grant-in-Aid for Young Scientists (A) ]20684001 and ]24684002 from JSPS. April 16, 2014 Osamu Fujino Contents Preface iii Guide for the reader xi Chapter 1. Introduction 1 1.1. Mori's cone and contraction theorem 1 1.2. What is a quasi-log scheme? 3 1.3. Motivation 5 1.4. Background 10 1.5. Comparison with the unpublished manuscript 11 1.6. Related papers 12 1.7. Notation and convention 13 Chapter 2. Preliminaries 15 2.1. Divisors, Q-divisors, and R-divisors 15 2.2. Kleiman{Mori cone 22 2.3. Singularities of pairs 24 2.4. Iitaka dimension, movable and pseudo-effective divisors 36 Chapter 3. Classical vanishing theorems and some applications 41 3.1. Kodaira vanishing theorem 42 3.2. Kawamata{Viehweg vanishing theorem 48 3.3. Viehweg vanishing theorem 55 3.4. Nadel vanishing theorem 60 3.5. Miyaoka vanishing theorem 61 3.6. Koll´arinjectivity theorem 63 3.7. Enoki injectivity theorem 64 3.8. Fujita vanishing theorem 68 3.9. Applications of Fujita vanishing theorem 76 3.10. Tanaka vanishing theorems 79 3.11. Ambro vanishing theorem 80 3.12. Kov´acs'scharacterization of rational singularities 82 3.13. Basic properties of dlt pairs 84 3.14. Elkik{Fujita vanishing theorem 91 3.15. Method of two spectral sequences 95 vii viii CONTENTS 3.16. Toward new vanishing theorems 98 Chapter 4. Minimal model program 103 4.1. Fundamental theorems for klt pairs 103 4.2. X-method 105 4.3. MMP for Q-factorial dlt pairs 107 4.4. BCHM and some related results 112 4.5. Fundamental theorems for normal pairs 120 4.6. Lengths of extremal rays 126 4.7. Shokurov polytope 130 4.8. MMP for lc pairs 137 4.9. Non-Q-factorial MMP 145 4.10. MMP for log surfaces 148 4.11. On semi log canonical pairs 153 Chapter 5. Injectivity and vanishing theorems 157 5.1. Main results 157 5.2. Simple normal crossing pairs 160 5.3. Du Bois complexes and Du Bois pairs 165 5.4. Hodge theoretic injectivity theorems 169 5.5. Relative Hodge theoretic injectivity theorem 174 5.6. Injectivity, vanishing, and torsion-free theorems 176 5.7. Vanishing theorems of Reid{Fukuda type 182 5.8. From SNC pairs to NC pairs 188 5.9. Examples 193 Chapter 6. Fundamental theorems for quasi-log schemes 201 6.1. Overview 201 6.2. On quasi-log schemes 203 6.3. Basic properties of quasi-log schemes 206 6.4. On quasi-log structures of normal pairs 215 6.5. Basepoint-free theorem for quasi-log schemes 217 6.6. Rationality theorem for quasi-log schemes 221 6.7. Cone theorem for quasi-log schemes 226 6.8. On quasi-log Fano schemes 232 6.9. Basepoint-free theorem of Reid{Fukuda type 233 Chapter 7. Some supplementary topics 239 7.1. Alexeev's criterion for S3 condition 239 7.2. Cone singularities 247 7.3. Francia's flip revisited 251 7.4. A sample computation of a log flip 253 7.5. A non-Q-factorial flip 256 CONTENTS ix Bibliography 259 Index 271 Guide for the reader In Chapter 1, we start with Mori's cone theorem for smooth pro- jective varieties and his contraction theorem for smooth threefolds. It is one of the starting points of the minimal model program. So the minimal model program is sometimes called Mori's program. We also explain some examples of quasi-log schemes, the motivation of our van- ishing theorems, the background of this book, the author's related pa- pers, and so on, for the reader's convenience. Chapter 2 collects several definitions and preliminary results. Almost all the topics in this chap- ter are well known to the experts and are indispensable for the study of the minimal model program. We recommend the reader to be familiar with them. In Chapter 3, we discuss various Kodaira type vanishing theorems and several applications. Although this chapter contains sev- eral new results and arguments, almost all the results are standard and are known to the experts. Chapter 4 is a survey on the minimal model program. We discuss the basic results of the minimal model program, the recent results by Birkar{Cascini{Hacon{McKernan, and various results on log canonical pairs, log surfaces, semi log canonical pairs by the author, and so on, without proof. Chapter 5 is devoted to the injectivity, vanishing, and torsion-free theorems for reducible vari- eties. They are generalizations of Koll´ar'scorresponding results from the mixed Hodge theoretic viewpoint and play crucial roles in the the- ory of quasi-log schemes.
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