DOCSLIB.ORG

Algebraic Varieties: Minimal Models and Finite Generation

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Algebraic Varieties: Minimal Models and Finite Generation Algebraic Varieties: Minimal Models and Finite Generation Chen Jiang and Yujiro Kawamata August 28, 2020 2 Preface 3 4 Contents 1 Algebraic varieties with boundaries 11 1.1 Q-divisors and R-divisors.................... 11 1.2 Rational maps and birational maps............... 15 1.3 Canonical divisors........................ 21 1.4 Intersection numbers and numerical geometry......... 24 1.5 Cones of curves and cones of divisors.............. 28 1.5.1 Pseudo-effective cones and nef cones.......... 28 1.5.2 Kleiman's criterion and Kodaira's lemma....... 31 1.6 The Hironaka desingularization theorem............ 36 1.7 The Kodaira vanishing theorem................. 38 1.8 The covering trick........................ 40 1.9 Generalizations of the Kodaira vanishing theorem....... 43 1.10 KLT singularities......................... 47 1.11 LC, DLT, PLT singularities................... 51 1.11.1 Various singularities................... 52 1.11.2 The sub-adjunction formula............... 56 1.11.3 Terminal singularities and canonical singularities... 59 1.12 Minimality and log minimality................. 61 1.13 The curve case and the surface case.............. 64 1.13.1 The curve case...................... 64 1.13.2 Minimal model of algebraic surfaces.......... 65 1.13.3 Finite generation and the classification of algebraic surfaces.......................... 68 1.13.4 Rational singularities................... 69 1.13.5 The classification of DLT surface singularities 1.... 71 1.13.6 The classification of DLT surface singularities 2.... 75 1.13.7 The Zariski decomposition................ 78 1.14 The three-dimensional case................... 79 2 The minimal model program 85 2.1 The base point free theorem................... 85 2.1.1 Proof of the base point free theorem.......... 86 2.1.2 Paraphrasing and generalization............ 91 5 6 CONTENTS 2.2 The effective base point free theorem.............. 93 2.3 The rationality theorem..................... 96 2.4 The cone theorem........................ 100 2.4.1 The contraction theorem................. 100 2.4.2 The cone theorem.................... 103 2.4.3 Contraction morphisms in dimensions 2 and 3..... 106 2.4.4 The cone theorem for cone of divisors......... 109 2.5 Types of contraction morphisms and the minimal model pro- gram................................ 111 2.5.1 Classification of contraction morphisms........ 112 2.5.2 Flips............................ 114 2.5.3 Decrease of canonical divisors.............. 117 2.5.4 Existence and termination of flips........... 118 2.5.5 Minimal models and canonical models......... 120 2.5.6 The minimal model program.............. 123 2.6 Minimal model program with scaling.............. 124 2.7 Existence of rational curves................... 126 2.7.1 Deformation of morphisms............... 127 2.7.2 The bend-and-break method.............. 128 2.8 Length of extremal rays..................... 132 2.9 Divisorial Zariski decomposition................ 134 2.10 Polyhedral decomposition of cone of divisors.......... 140 2.10.1 Rationality of sections of nef cones........... 140 2.10.2 Polyhedral decomposition according to canonical models142 2.10.3 Polyhedral decomposition according to minimal models144 2.10.4 Applications of polyhedral decompositions....... 147 2.11 Multiplier ideal sheaves..................... 151 2.11.1 Multiplier ideal sheaves................. 152 2.11.2 Adjoint ideal sheaves................... 154 2.12 Extension theorems........................ 159 2.12.1 Extension theorems 1.................. 159 2.12.2 Extension theorems 2.................. 165 3 The finite generation theorem 169 3.1 Setting of the inductive proof.................. 169 3.2 PL flips.............................. 173 3.2.1 Restriction of canonical rings to divisors........ 174 3.2.2 The existence of PL flips................. 178 3.3 The special termination..................... 181 3.4 The existence and the finiteness of minimal models...... 184 3.5 The non-vainishing theorem................... 190 3.6 Summary............................. 195 3.7 Algebraic fiber spaces...................... 199 3.7.1 Algebraic fiber spaces and toroidal geometry..... 199 CONTENTS 7 3.7.2 The weak semi-stable reduction theorem and the semi- positivity theorem.................... 201 3.8 The finite generation theorem.................. 203 3.9 Generalizations of the minimal model theory......... 203 3.9.1 The equivariant minimal model theory......... 204 3.9.2 The MMP over algebraically non-sclosed fields.... 204 3.10 Remaining problems....................... 204 3.10.1 The abundance conjecture................ 204 3.10.2 Case of numerical Kodaira dimension zero....... 204 3.10.3 Generalization to positive characteristics........ 204 3.11 Related topics........................... 204 3.11.1 Boundedness results................... 204 3.11.2 Minimal log discrepancies................ 204 3.11.3 The Sarkisov program.................. 204 3.11.4 Rationally connected varieties.............. 204 3.11.5 The category of smooth algebraic varieties....... 204 Bibliography 205 8 CONTENTS Introduction 9 10 CONTENTS Chapter 1 Algebraic varieties with boundaries fChapter 1g In this chapter, we introduce basic concepts of algebraic varieties with boundaries, where a boundary of an algebraic variety in this book is a di- visor with real coefficients. Using the language of numerical geometry, we define cones of curves and cones of divisors. According to the Hironaka desingularization theorem, it is possible to use birational modifications to make algebraic varieties smooth and divisors normal crossing. We focus on adjoint divisors of algebraic varieties with boundaries, and introduce defini- tions of KLT pairs and DLT pairs. We explain how to use the covering trick to generalize the Kodaira vanishing theorem for smooth projective varieties to KLT or DLT pairs. Also we discuss the classification of algebraic varieties and singularities in lower dimensions. 1.1 Q-divisors and R-divisors fsection 1.1g The linear equivalence class of a divisor defines a coherent sheaf associated to this divisor which is called its divisorial sheaf. In many situations in algebraic geometry, we deal with coherent sheaves. But in this book, we mainly focus on divisors. It is like dealing with differential forms themselves instead of cohomology classes of differential forms in differential geometry. Fix a base field k. An algebraic variety X is an irreducible reduced separated scheme of finite type over k. An algebraic variety X is said to be non-singular if for every point P on X, the local ring OX;P of the structure sheaf OX at P is a regular local ring. A point P with this property is called a non-singular point of X. In this book we mostly work over a field of characteristic zero, and we will mainly use the word smooth instead of non-singular. When dim X = n, X is smooth if and only if for every closed point P on X, the maximal ideal mP of OX;P 11 12 CHAPTER 1. ALGEBRAIC VARIETIES WITH BOUNDARIES is generated by n elements x1; : : : ; xn. Such x1; : : : ; xn are called the regular system of parameters or local coordinates. If k = C, this is equivalent to that the set of closed points of X forms a complex manifold. For an algebraic variety X, the set of all smooth points Reg(X) is a non-empty open subset of X, and its complement Sing(X) = X n Reg(X), which is a proper closed subset of X, is called the singular locus. An algebraic variety X is said to be normal if the local ring at every point is an integrally closed domain. Since normal local rings of dimension 1 are regular, the singular locus of a normal algebraic variety is a closed subset of codimension at least 2. Every algebraic variety X can be easily modified into a normal one: there is a unique finite morphism f : Xν ! X from a normal algebraic variety which is isomorphic over Reg(X), which is called the normalization of X. Normality can be determined by Serre's criterion ([102]): Theorem 1.1.1. An algebraic variety X is normal if and only if the fol- lowing 2 conditions are satisfied: (1)( R1) Its singular locus is a closed subset of codimension at least 2. (2)( S2) For any open subset U and any closed subset Z of codimension at least 2, the restriction map Γ(U; OX ) ! Γ(U n Z; OX ) is bijective. From now on we always assume that X is a normal algebraic variety. A prime divisor on X is a closed subvariety of codimension 1. A divisor is P a formal finite sum of prime divisors D = diDi. Unless otherwise stated, the coefficients di are integers, and Di are distinct prime divisors. In other words, divisors are elements in the free abelian group Z1(X) generated by all prime divisors on X. D is said to be effective if all coefficients di are non- negative. D is said to be reduced if all coefficients di = 1. For two divisors D; D0, we write the inequality D ≥ D0 if D − D0 is an effective divisor. Let D be a prime divisor on X and P be the generic point of D, then the local ring OX;P is a discrete valuation ring with quotient field k(X). For a rational function h 2 k(X)∗1.1.0.1, the divisor div(h) is defined as X div(h) = vP (h)D; which is known to be a finite sum. Here the sum runs over all prime divisors D, P is the generic point of the prime divisor D, and vP is the valuation of the discrete valuation ring OX;P . Any divisor of the form div(h) for some h 2 k(X) is called a principal divisor. For a divisor D, the corresponding divisorial sheaf OX (D) is defined as the following: for any open subset U of X, ∗ 1.1.0.2 Γ(U; OX (D)) = fh 2 k(X) j div(h)jU + DjU ≥ 0g [ f0g: 1.1.0.1corrction 1.1.0.2correction 1.1. Q-DIVISORS AND R-DIVISORS 13 Also we define 0 0 H (X; D) = H (X; OX (D)): If a non-zero global section s of OX (D) corresponds to a rational function h, we define the divisor of s by div(s) = div(h) + D; which is an effective divisor.
Load more

Recommended publications
  • Higher Dimensional Algebraic Geometry (In Honour of Professor Yujiro Kawamata's Sixtieth Birthday)
    Higher Dimensional Algebraic Geometry (in honour of Professor Yujiro Kawamata's sixtieth birthday) Date : January 7-11, 2013 Venue : Auditorium, Graduate School of Mathematical Sciences, the University of Tokyo Title and Abstract : 1) Arnaud Beauville Title: The L¥"uroth problem Abstract: The L¥"uroth problem asks whether every field $K$ with $¥mathbb{C}¥subset K ¥subset ¥mathbb{C}(x_1,¥ldots ,x_n)$ is of the form $¥mathbb{C}(y_1,¥ldots ,y_p)$. In geometric terms, if an algebraic variety can be parametrized by rational functions, can one find a one-to-one such parametrization? After a brief historical survey, I will recall the counter-examples found in the 70's; then I will describe a quite simple (and new) counter-example. If time permits I will explain the relation with the study of finite simple groups of birational automorphisms of $¥mathbb{P}^3$. 2) Caucher Birkar Title: Singularities in Fano type fibrations. Abstract: I will discuss singularities that appear on the base of a Fano type fibration, and mention the conjectures relevant to this problem. I will try to explain my results in this direction assuming boundednessof the log general fibres of the fibration. 1 / 8 3) Fabrizio Catanese Title: (Some) Topological and transcendental methods in classification and moduli theory Abstract: The structure of the moduli spaces of surfaces and higher dimensional varieties is usually rather difficult to get hold of. Sometimes there are lucky situations, like the description of varieties whose universal cover is a polydisk or a symmetric bounded domain (joint work with Di Scala). Some other times the homotopy type of an algebraic variety may determine the structure of the moduli spaces, as for curves, Abelian varieties, Kodaira surfaces, varieties isogenous to a product, Beauville surfaces.
    [Show full text]
  • 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.
    [Show full text]
  • A SNAPSHOT of the MINIMAL MODEL PROGRAM 1. Introduction
    A SNAPSHOT OF THE MINIMAL MODEL PROGRAM BRIAN LEHMANN Abstract. We briefly review the main goals of the minimal model pro- gram. We then discuss which results are known unconditionally, which are known conditionally, and which are still open. 1. Introduction This survey paper reviews the main goals and results of the Minimal Model Program (henceforth MMP). The paper has three parts. In Section 2, we give a very gentle introduction to the main ideas and conjectures of the MMP. We emphasize why the results are useful for many different areas of algebraic geometry. In Sections 3-6, we take a \snapshot" of the MMP: we describe which results are currently known unconditionally, which are known conditionally, and which are wide open. We aim to state these results precisely, but in a manner which is as useful as possible to as wide a range of mathematicians as possible. Accordingly, this paper becomes more and more technical as we go. The hope is this paper will be helpful for mathematicians looking to apply the results of the MMP in their research. In Section 7, we briefly overview current directions of research which use the MMP. Since the foundations of the MMP are discussed previously, this section focuses on applications. The choice of topics is not comprehensive and is idiosyncratically based on my own knowledge. These notes are not intended to be a technical introduction to the MMP. There are many good introductions to the techniques of the MMP already: [KM98], [Mat02], [HK10], [Kol13b], and many others. These notes are also not intended to be a historical introduction.
    [Show full text]
  • Full Text (PDF Format)
    Current Developments in Mathematics, 2007 Finite Generation of a Canonical Ring Yujiro Kawamata Abstract. The purpose of this note is to review an algebraic proof of the finite generation theorem due to Birkar-Cascini-Hacon-McKernan [5] whose method is based on the Minimal Model Program (MMP). An analytic proof by Siu [57] will be reviewed by Mihai Paun. 1. Introduction The finite generation of canonical rings was a problem considered by Zariski [64], and the proof in the case of dimension 2 due to Mumford [41] in the appendix of [64] is one of the motivations towards the minimal model theory of higher dimensional algebraic varieties. Let X be a smooth projective variety defined over a field k, D a divisor on X,andOX (D) the associated invertible sheaf. Many problems in algebraic geometry are translated into questions on the vector space of holomorphic 0 0 sections H (X, D)=H (X, OX (D)). The Riemann-Roch problem is to determine this vector space. For example, the Riemann-Roch theorem tells n − p p O us that the alternating sum p=0( 1) dim H (X, X (D)) is expressed in terms of topological invariants. Instead of considering a single vector space, we look at the graded ring ∞ R(X, D)= H0(X, mD) m=0 called the section ring for the pair (X, D), where we use the additive notation for divisors instead of multiplicative one for sheaves; we have OX (mD)= ⊗m OX (D) . There are obvious multiplication homomorphisms 0 0 0 H (X, m1D) ⊗ H (X, m2D) → H (X, (m1 + m2)D) and the section ring becomes a graded algebra over the base field k = 0 H (X, OX ).
    [Show full text]
  • A First Glimpse at the Minimal Model Program
    Contemporary Mathematics A FIRST GLIMPSE AT THE MINIMAL MODEL PROGRAM CHARLES CADMAN, IZZET COSKUN, KELLY JABBUSCH, MICHAEL JOYCE, SÁNDOR J. KOVÁCS, MAX LIEBLICH, FUMITOSHI SATO, MATT SZCZESNY, AND JING ZHANG CONTENTS 1. Minimal models of surfaces 2 2. Bend and Break 5 3. The cone of curves 8 4. Introduction to the Minimal Model Program 10 5. Running the Minimal Model Program 13 6. Where next? 16 7. Minimal models of surfaces, revisited 16 8. Examples 18 References 24 This article is an account of the afternoon sessions at Snowbird that were devoted to an introduction to the Minimal Model Program and led by Sándor Kovács. The sections were written by the participants based on the discussion there and then kneaded together into one article by the fifth named author. Despite trying to make the presentation cohesive, the reader may detect a different style of writing in each section. We hope that this will only entertain rather than annoy the reader. Most participants had no prior knowledge of the Minimal Model Program other than a general (strong) background in algebraic geometry. Readers with similar preparation form our targeted audience. As the title suggests this article is only a glimpse at this beautiful and important area. If we succeed in raising the reader’s interest, we suggest further investigation. A natural next step could be Kollár’s introductory, but much more detailed article [Kol87]. We also recommend consulting other survey articles, such as [Rei87] and [Mor87], or the technically more demanding [KMM87] and [Kol91]. For 1991 Mathematics Subject Classification. 14E30. Key words and phrases.
    [Show full text]
  • Name: Yujiro Kawamata Research Field: Algebraic Geometry
    Name: Yujiro Kawamata Research Field: Algebraic Geometry (Birational Geometry of Higher Dimensional Algebraic Varieties) Key Words: algebraic variety, Kodaira dimension, vanishing theorem, base point free theorem, minimal model program, log terminal singularity, derived category Current research Invesigating the birational geometry of higher dimensional algebraic varieties by making the best use of the combinations of algebraic, geometric and analytic methods. The topics include the Kodaira dimension of algebraic fiber spaces, applications of the Hodge theory, a birational characterization of abelian varieties, classification of logarithmic algebraic varieties, singularities of pairs, extensions of vanishing theorems, minimal model program, base point free theorem, logarithmic deformation theory, applications of multiplier ideal sheaves, derived category of coherent sheaves on algebraic varieties. Prerequisites It takes long time to understand basic concepts in algebraic geometry because this field of mathematics has many interactions with other fields. But thanks to this fact, algebraic geometry is useful and has diverse applications. The language of algebraic geometry is abstract and it is difficult to grasp the meaning of technical terms at first. It is better to start reading books with concrete examples. For example, a textbook by Kirwan on algebraic curves is a nice book showing how mathematics learned up to the second year at the university is applied to algebraic geometry. A text book by Reid is also easily accesible. Shafarevich's textbook is longer but very good. One can proceed to the scheme theory after getting acquainted to the algebraic varieties. Hartshorne's is the standard textbook in scheme theory..
    [Show full text]
  • On Minimal Models and Canonical Models of Elliptic Fourfolds with Section
    University of California Santa Barbara On Minimal Models and Canonical Models of Elliptic Fourfolds with Section A dissertation submitted in partial satisfaction of the requirements for the degree Doctor of Philosophy in Mathematics by David Wen Committee in charge: Professor David R. Morrison, Chair Professor Mihai Putinar Professor Ken Goodearl June 2018 The Dissertation of David Wen is approved. Professor Mihai Putinar Professor Ken Goodearl Professor David R. Morrison, Chair June 2018 On Minimal Models and Canonical Models of Elliptic Fourfolds with Section Copyright c 2018 by David Wen iii To those who believed in me and to those who didn't for motivating me to prove them wrong. iv Acknowledgements I would like to thank my advisor, Professor David R. Morrison, for his advice, guidance and encouragement throughout my graduate studies and research. Additionally, I would like to thank the UCSB math department and my colleagues in the department for their support, advice and conversations. v Curriculum Vitae David Wen [email protected] Department of Mathematics South Hall, Room 6607 University of California Santa Barbara, CA 93106-3080 Citizenship: U.S.A. Education Ph.D Candidate, Mathematics University of California, Santa Barbara; Expected June 2018 Thesis Advisor: David R. Morrison Thesis Title: On Minimal Models and Canonical Models of Elliptic Fourfolds with Section M.A., Mathematics University of California, Santa Barbara; June 2014 B.S., Mathematics, Computer Science minor, Summa Cum Laude New York University, Tandon School of
    [Show full text]
  • Log Minimal Models for Arithmetic Threefolds
    LOG MINIMAL MODELS FOR ARITHMETIC THREEFOLDS by Paul Andrew Egbert A dissertation submitted to the faculty of The University of Utah in partial fulfillment of the requirements for the degree of Doctor of Philosophy Department of Mathematics The University of Utah May 2016 Copyright c Paul Andrew Egbert 2016 All Rights Reserved The University of Utah Graduate School STATEMENT OF DISSERTATION APPROVAL The dissertation of Paul Andrew Egbert has been approved by the following supervisory committee members: Christopher Hacon , Chair(s) 8 March 2016 Date Approved Mladen Bestvina , Member 8 March 2016 Date Approved Tommaso Defernex , Member 8 March 2016 Date Approved Yuan-Pin Lee , Member 15 March 2016 Date Approved Chenyang Xu , Member 16 March 2016 Date Approved by Peter Trapa , Chair/Dean of the Department/College/School of Mathematics and by David B. Kieda , Dean of The Graduate School. ABSTRACT I study the existence of log minimal models for a Kawamata log-terminal pair of relative dimension two over a Dedekind domain. This generalizes the semistable result of Kawamata. Also I prove a result on the invariance of log plurigenera for such pairs, generalizing the result of Suh. To extend the result from discrete valuation rings to Dedekind domains, some computability results are given for basepoint-freeness, vanishing of cohomology, and finite generation of log-canonical and adjoint rings on a mixed characteristic family of surfaces. To my wife Lifang and my dog Pigu. CONTENTS ABSTRACT :::::::::::::::::::::::::::::::::::::::::::::::::::::::: iii ACKNOWLEDGEMENTS ::::::::::::::::::::::::::::::::::::::::::: vi CHAPTERS :::::::::::::::::::::::::::::::::::::::::::::::::::::::: 1 1. INTRODUCTION ::::::::::::::::::::::::::::::::::::::::::::::: 1 2. THE MINIMAL MODEL PROGRAM ::::::::::::::::::::::::::::: 5 2.1 Arithmetic Schemes . .5 2.2 Intersections and Positivity .
    [Show full text]
  • Boundedness Results for Singular Fano Varieties, and Applications to Cremona Groups 3
    BOUNDEDNESS RESULTS FOR SINGULAR FANO VARIETIES, AND APPLICATIONS TO CREMONA GROUPS STEFAN KEBEKUS Abstract. is survey paper reports on work of Birkar, who confirmed a long-standing conjecture of Alexeev and Borisov-Borisov: Fano varieties with mild singularities form a bounded family once their dimension is fixed. Following Prokhorov-Shramov, we explain how this boundedness result implies that birational automorphism groups of projective spaces satisfy the Jordan property, answering a question of Serre in the positive. Contents 1. Main results 1 2. Notation, standard facts and known results 5 3. b-Divisors and generalised pairs 7 4. Boundedness of complements 10 5. Effective birationality 13 6. Bounds for volumes 17 7. Bounds for lc thresholds 18 8. Application to the Jordan property 22 References 24 1. Main results roughout this paper, we work over the field of complex numbers. 1.1. Boundedness of singular Fano varieties. A normal, projective variety X is called Fano if a negative multiple of its canonical divisor class is Cartier and if the associated line bundle is ample. Fano varieties appear throughout geometry and have been studied intensely, in many contexts. For the purposes of this talk, we remark that Fanos with sufficiently mild singularities constitute one of the fundamental variety classes in bira- tional geometry. In fact, given any projective manifold X , the Minimal Model Programme arXiv:1812.04506v3 [math.AG] 9 Apr 2019 (MMP) predicts the existence of a sequence of rather special birational transformations, known as “divisorial contractions” and “flips”, as follows, (1) (2) (n) X = X (0) α ❴❴❴❴❴❴ / X (1) α ❴❴❴❴❴❴ / ··· α ❴❴❴❴❴❴ / X (n).
    [Show full text]
  • The Work of Hacon and Mckernan
    THE WORK OF HACON AND MCKERNAN ALESSIO CORTI Contents 1. Introduction 1 2. Birational Geometry 4 3. Minimal Models of Surfaces 4 4. Line Bundles and the Kleiman Criterion 6 5. The Canonical Line Bundle 7 6. The Cone Theorem 9 7. The Contraction Theorem 9 8. Minimal Models in Higher Dimensions 11 9. The Minimal Model Program with Scaling 15 10. Finite Generation of Adjoint Rings 16 11. What Remains to Do: Dichotomy and Trichotomy 18 References 19 1. Introduction This note is a sketch of the theory of minimal models of algebraic varieties and the recent advances of Hacon, McKernan and their col- laborators [HM06, BCHM09, HM09]. The classification theory of algebraic varieties—nonsingular, say, or mildly singular—aims, in the first instance, to establish thefollowing basic dichotomy, which I state before defining some key words.Thegoal is to divide, up to surgery (called ‘birational equivalence’inalgebraic geometry), all proper algebraic varieties into two basic classes: (1) Varieties X with nef canonical line bundle. (2) Fibre spaces X → Z such that the anti-canonical line bundle ∗ KX is ample on all general fibres Xz. Recall that a line bundle L on an algebraic variety X is nef if: c1(L)[C]= c1(L) ≥ 0 !C Date:24th November 2009. 1 2 ALESSIO CORTI for all proper (i.e., compact) algebraic curves C ⊂ X.ThefirstChern class c1(L)ofalinebundleisrepresentedbythecurvatureform i ω = ∂∂ log h 2π of a Hermitian metric h on L:thus,nefisacohomologicalversionof ‘non-negatively curved.’ On a proper algebraic manifold X,themost basic line bundle is the canonical line bundle top ∗ KX = TX , " and to say that KX is nef is another way to say that X is, in some weak sense, nonpositively curved.
    [Show full text]
  • On the Three Dimensional Minimal Model Program in Positive Characteristic
    JOURNAL OF THE AMERICAN MATHEMATICAL SOCIETY Volume 28, Number 3, July 2015, Pages 711–744 S 0894-0347(2014)00809-2 Article electronically published on June 4, 2014 ON THE THREE DIMENSIONAL MINIMAL MODEL PROGRAM IN POSITIVE CHARACTERISTIC CHRISTOPHER D. HACON AND CHENYANG XU 1. Introduction The minimal model program (MMP) is one of the main tools in the classification of higher dimensional algebraic varieties. It aims to generalize to dimension ≥ 3 the results obtained by the Italian school of algebraic geometry at the beginning of the 20-th century. In characteristic 0, much progress has been made towards establishing the mini- mal model program. In particular the minimal model program is true in dimension ≤3, and in higher dimensions, it is known that the canonical ring is finitely gen- erated, flips and divisorial contractions exist, minimal models exist for varieties of general type, and we have termination of flips for the minimal model program with scaling on varieties of general type (see [BCHM10] and the references con- tained therein). The fundamental tool used in establishing these results is Nadel- Kawamata-Viehweg vanishing (a powerful generalization of Kodaira vanishing). Unluckily, vanishing theorems are known to fail for varieties in characteristic p>0 and so very little is known about the minimal model program in charac- teristic p>0. Another serious difficulty is that a resolution of singularities is not yet known in characteristic p>0 and dimension >3. The situation is as fol- lows: in dimension 2, the full minimal model program holds (see [KK, Tanaka12a] and references therein). In dimension 3, the resolution of singularities is known (see [Abhyankar98, Cutkosky04, CP08, CP09]).
    [Show full text]
  • Birational Geometry of Algebraic Varieties, by János Kollár And
    BULLETIN (New Series) OF THE AMERICAN MATHEMATICAL SOCIETY Volume 38, Number 2, Pages 267{272 S 0273-0979(01)00910-7 Article electronically published on February 7, 2001 Birational geometry of algebraic varieties,byJ´anos Koll´ar and Shigefumi Mori, with the collaboration of C. H. Clemens and A. Corti, Cambridge University Press, 1998, viii + 254 pp., ISBN 0-521-63277-3 There has been a revolutionary change in the field of birational geometry in the last twenty plus years. This is based on the theory of extremal rays initiated by Mori and is central to the investigation of minimal models called the Minimal Model Program (MMP) or Mori program. But there has been no reasonably acces- sible textbook for this theory besides more professionally oriented surveys such as [KMM87] and [K+92] (we use the reference list of the book). In this sense, this book, written by two of the main players in this development, answers a demand for a long awaited introductory textbook for the beginners in this field. The ex- position is sufficiently elementary, self-contained and comprehensive, and requires fewer prerequisites, so this book will become a standard reference. A caution is that the proof of the existence theorem of 3-dimensional flips is treated only in the easy semistable case. One has to refer to the original articles in order to study the general case. Let us briefly recall the main results of the minimal model program. We consider all the algebraic varieties which are birationally equivalent to a given algebraic variety, called birational models.
    [Show full text]