DOCSLIB.ORG

3-Fold Log Flips

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
3-Fold Log Flips Home Search Collections Journals About Contact us My IOPscience 3-FOLD LOG FLIPS This article has been downloaded from IOPscience. Please scroll down to see the full text article. 1993 Russian Acad. Sci. Izv. Math. 40 95 (http://iopscience.iop.org/1468-4810/40/1/A04) View the table of contents for this issue, or go to the journal homepage for more Download details: IP Address: 195.37.209.182 The article was downloaded on 19/09/2010 at 20:19 Please note that terms and conditions apply. H3B Pocc. AKa«. HayK Russian Aead. Sci. Izv. Math. Cep. MaxeM. TOM 56 (1992), J\fc 1 Vol. 40 (1993), No. 1 3-FOLD LOG FLIPS* UDC 512.7 V. V. SHOKUROV ABSTRACT. We prove that 3-fold log flips exist. We deduce the existence of log canon- ical and Q-factorial log terminal models, as well as a positive answer to the inversion problem for log canonical and log terminal adjunction. CONTENTS §0. § 1. Singularities and models §2. The covering trick §3. Adjunction of log divisors §4. Two terminations §5. Complementary log divisors §6. Special flips §7. Exceptional special flips §8. Index 2 special flips §9. Applications Appendix by Y. Kawamata: The minimal discrepancy of a 3-fold terminal singularity §10. Commentary by M. Reid Bibliography §0. Let X be a normal algebraic (or analytic) 3-fold with a marked Q-divisor Β = Βχ (the boundary of X); we write Κ — Κ χ , and consider the log canonical divisor K+B as in Kawamata-Matsuda-Matsuki [8]. Suppose that /: X —* Ζ is a birational contraction of X such that Κ + Β is numerically nonpositive relative to /. A flip of / is a birational (or bimeromorphic) modification X -?Λ X+ Ζ where /+ is a small birational contraction whose modified log canonical divisor + + Kx+ + B is numerically positive relative to / ; it is known (see [25] (2.13), and [8], 5-1-11) that the flip tr/ is unique if it exists. 1991 Mathematics Subject Classification. Primary 14E05, 14E35, 14J30. *Editor's note. The present translation includes the author's substantially corrected version of §8, a section §10 of translator's comments, and a number of other minor corrections and additions. A number of the translator's footnotes have been moved into the main text with the author's permission. All the remaining footnotes are by the translator. ©1993 American Mathematical Society 1064-5632/93 $1.00+ $.25 per page 95 96 V. V. SHOKUROV Theorem. The flip of f exists if \B\ = 0 and Κ + Β is log terminal. Corollary. Let f:X—>Z be a projective morphism of algebraic (or analytic) 3-folds, and suppose that \_B\ = 0 and K + B is log terminal. Then for every extremal face R of the Kleiman-Mori cone ~NE(X/Z) (in the analytic case, of NE(A'/Z ; W), where W c Ζ is compact) contained in the halfspace Κ + Β < 0, the contraction morphism contR associated with R is either a fiber space of log Fanos over a base of dimension < 2, or has a flip VCR (respectively, the same statement over a neighborhood of W). The proof of the theorem, or more precisely of the equivalent Theorem 1.9, con- sists of a series of reductions. First of all, in §6, the construction of the flip reduces to the special case, which is classified according to its complementary index. This classification is similar, and in fact closely related to, Brieskorn's classification of log terminal surface singularities (see [2] and [4]). Index 1 special flips exist, and correspond to the flops or 0-flips of [7], (6.1), or [11], (6.6). Next, in §7, we con- struct exceptional index 2 flips, and carry out a reduction of the existence of the remaining exceptional flips of index 3, 4, and 6 to the case of special flips of index 1 or 2. The proof is completed by the reduction in §8 of special index 2 flips to exceptional index 2 flips. Here a significant role is played by a result of Kawamata on the minimal discrepancy of a terminal 3-fold singularity; this proof is given in the Appendix kindly provided by Professor Kawamata. Furthermore, as we will see in the proof, a flip can be decomposed as a composite of resolutions of singularities, birational contractions given by the eventual freedom theorem and the contraction theorem ([8], 3-1-2 and 3-2-1), and flips of types I-IV, defined in §2. We note also that the proof does not use Mori's flips ([16], (0.2.5)) in the case 5 = 0 when Κ itself is terminal, and so gives a new approach to proving the existence of Mori's flips. In addition to the definitions and related general facts, the introductory § 1 contains a statement of the main results proved in §§6-8; applications of these are given in §9. The main technique is contained in §§2-5. This research has been supported in part by the Taniguchi Foundation, the Max Planck Gesellschaft, the NSF, the Deutsche Forschungsgemeinschaft SFB 170, and the Hironaka and Kajima funds. This paper was completed during a stay at the Institute for Advanced Studies, Princeton, and I would like to thank the I. A. S. for hospitality and support. § 1. SINGULARITIES AND MODELS We generally use the terminology and notation of [8], [25] and [26]. The geometric objects we work with are either normal complex analytic spaces or normal algebraic varieties over a base field k of characteristic 0. The first is the analytic case, the second the algebraic case. For example, a morphism is either a holomorphic or a regular map; and a modification is assumed to be bimeromorphic in the analytic case, and birational in the algebraic case. A contraction is a proper morphism /: X —> Υ with f»<fx = @γ , and is projective if / is. If /: X —» Υ is a contraction between varieties of the same dimension, that is, dim X = dim Υ , then / is one-to-one at the generic point, and is a modification; such an / is a birational contraction or blowdown when Υ is viewed as constructed from X , or an extraction or blowup when X is viewed as constructed from Υ (see (10.8.4) for notes on terminology). An extraction or birational contraction whose exceptional set has codimension > 2 is small. We write p(X/Y) (or p(X/Y; W) in the analytic case) for the relative Picard number of / (respectively, of / over a compact analytic subset W c Y). As a rule, in the analytic case, we work infinitesimally over suitable neighborhoods of W , 3-FOLD LOG FLIPS 97 even when we do not say so explicitly; W is often projective, the fiber of a projective morphism, having tubular neighborhoods over Stein domains. (This condition is a particular case of weakly l-complete in the sense of [18], (0.4); it means in particular that Serre vanishing can be used e.g. in the proof of Corollary 5.19 below.) An extrac- tion or birational contraction / is extremal (over W) if p(X/ Y) = 1 (respectively p(X/Y;W)=l). A divisor is usually understood as an M-Weil divisor D = Σ diD,, with Z), distinct prime Weil divisors on X and d,• € Ε, called the multiplicity of Z), in D. This terminology will be generalized later to include the multiplicity of D at prime divisors of an extraction Υ —* X; see just before Lemma 8.7, and (10.8.5). We say that D is a Q-divisor (respectively, an integral divisor) if dt e Q (or dt e Z) for all Z),. Note that a Q-divisor is Q-Cartier if and only if it is K-Cartier. More-or-less by definition, E-Cartier divisors with support in a finite union (J Z), form a vector subspace defined over Q of the space φ EZ)(- of all divisors supported in |J Z), (in the analytic case, in a neighborhood of any compact subset of X). 1.1. Negativity of a birational contraction. Let f:X-+Z be a birational contraction and D an R-Cartier divisor. Suppose that (i) / contracts all components of D with negative multiplicities; (ii) D is numerically nonpositive relative to /; and for each D,, either D, has multiplicity 0 in D, or D is not numerically 0 over the general point of f{D,). Then D is effective. Moreover, for each £>, either D = 0 in a neighborhood of the general fiber of f: Z), —> /(A) or d,• > 0. Proo/(compare [Pagoda], (0.14)). First of all, passing to a general hyperplane section (that is, a general element of a very ample linear system) and using induction on the dimension of X reduces 1.1 to the assertion over a fixed point Ρ e Ζ ; that is, after replacing Ζ by a suitable neighborhood of Ρ if necessary, we can assume that all the components D with di < 0 are contracted to Ρ. It is enough to prove the assertion on some blowup of X. Since the assertion is local over Ζ , we can also assume that X has an effective divisor Ε contracted by / that is numerically nonpositive and not numerically 0 over Ρ; for example, we could take Ε to be the difference f*H - f~lH, where Η is a general hyperplane section through Ρ. (Throughout the paper, / or /~' applied to a divisor denotes its birational image or birational transform, never the set-theoretic image, see below and (10.8.3).) Using resolution of the base locus by Hironaka, we can assume that \f~lH\ is a free linear system, which guarantees that Ε = f*H - /"' Η is numerically nonposi- tive.
Load more

Recommended publications
  • Introduction
    SEMINAR WS 16/17 BERKOVICH SPACES, BIRATIONAL GEOMETRY AND MOTIVIC ZETA FUNCTIONS Talk II on Birational geometry and MMP Efstathia Katsigianni 10.11.2016 Introduction The goal of this talk is to present the notions that have to do with the Minimal model Program and will be used in the course of the seminar, as well as the main theorems of this area. For this we rely entirely on the references listed in the end. Given a projective variety X defined over an algebraically closed field k, one could try to find another projective variety birationally isomorphic to X which is the simplest possible. We define on the set CX of all (isomorphism classes of) projective varieties birationally isomorphic to X a relation as fol- lows: if X0, X1 are in CX , we write X0 < X1, whenever there is a birational morphism X0 ! X1. This defines an ordering on CX and we look for minimal elements in CX or for the smallest element of it. For example, each irreducible curve is birational to a unique smooth pro- jective curve, thus the investigation of smooth projective curves is equivalent to the study of all curves up to birational equivalence. When X is a smooth surface, it has a smooth minimal model obtained by contracting all exceptional curves on it. If X is not uniruled, this minimal model has nef canonical divisor and is the smallest element in cX. When X is uniruled, this minimal model is not unique. Smooth projective varieties with nef canonical bundles are minimal in the above sense: Proposition.
    [Show full text]
  • UC Berkeley UC Berkeley Electronic Theses and Dissertations
    UC Berkeley UC Berkeley Electronic Theses and Dissertations Title Cox Rings and Partial Amplitude Permalink https://escholarship.org/uc/item/7bs989g2 Author Brown, Morgan Veljko Publication Date 2012 Peer reviewed|Thesis/dissertation eScholarship.org Powered by the California Digital Library University of California Cox Rings and Partial Amplitude by Morgan Veljko Brown A dissertation submitted in partial satisfaction of the requirements for the degree of Doctor of Philosophy in Mathematics in the Graduate Division of the University of California, BERKELEY Committee in charge: Professor David Eisenbud, Chair Professor Martin Olsson Professor Alistair Sinclair Spring 2012 Cox Rings and Partial Amplitude Copyright 2012 by Morgan Veljko Brown 1 Abstract Cox Rings and Partial Amplitude by Morgan Veljko Brown Doctor of Philosophy in Mathematics University of California, BERKELEY Professor David Eisenbud, Chair In algebraic geometry, we often study algebraic varieties by looking at their codimension one subvarieties, or divisors. In this thesis we explore the relationship between the global geometry of a variety X over C and the algebraic, geometric, and cohomological properties of divisors on X. Chapter 1 provides background for the results proved later in this thesis. There we give an introduction to divisors and their role in modern birational geometry, culminating in a brief overview of the minimal model program. In chapter 2 we explore criteria for Totaro's notion of q-amplitude. A line bundle L on X is q-ample if for every coherent sheaf F on X, there exists an integer m0 such that m ≥ m0 implies Hi(X; F ⊗ O(mL)) = 0 for i > q.
    [Show full text]
  • Effective Bounds for the Number of MMP-Series of a Smooth Threefold
    EFFECTIVE BOUNDS FOR THE NUMBER OF MMP-SERIES OF A SMOOTH THREEFOLD DILETTA MARTINELLI Abstract. We prove that the number of MMP-series of a smooth projective threefold of positive Kodaira dimension and of Picard number equal to three is at most two. 1. Introduction Establishing the existence of minimal models is one of the first steps towards the birational classification of smooth projective varieties. More- over, starting from dimension three, minimal models are known to be non-unique, leading to some natural questions such as: does a variety admit a finite number of minimal models? And if yes, can we fix some parameters to bound this number? We have to be specific in defining what we mean with minimal mod- els and how we count their number. A marked minimal model of a variety X is a pair (Y,φ), where φ: X 99K Y is a birational map and Y is a minimal model. The number of marked minimal models, up to isomorphism, for a variety of general type is finite [BCHM10, KM87]. When X is not of general type, this is no longer true, [Rei83, Example 6.8]. It is, however, conjectured that the number of minimal models up to isomorphism is always finite and the conjecture is known in the case of threefolds of positive Kodaira dimension [Kaw97]. In [MST16] it is proved that it is possible to bound the number of marked minimal models of a smooth variety of general type in terms of the canonical volume. Moreover, in dimension three Cascini and Tasin [CT18] bounded the volume using the Betti numbers.
    [Show full text]
  • AN INTRODUCTION to MORI THEORY: the CASE of SURFACES Contents Introduction 1 Part 1. Preliminaries 2 1.1. Basic Notation 3 1.2
    AN INTRODUCTION TO MORI THEORY: THE CASE OF SURFACES MARCO ANDREATTA Contents Introduction 1 Part 1. Preliminaries 2 1.1. Basic notation 3 1.2. Morphisms associated to line bundles; ample line bundles 4 1.3. Riemann - Roch and Hodge Index theorems 5 Part 2. Mori theory for surfaces 6 2.1. Mori-Kleiman cone 6 2.2. Examples 9 2.3. The rationality lemma and the cone theorem 11 2.4. Castelnuovo contraction theorem 16 2.5. Base point freeness, BPF 16 2.6. Minimal model program for surfaces 20 Part 3. Birational theory for surfaces 22 3.1. Castelnuovo rationality criterium 22 3.2. Factorization of birational morphisms 23 3.3. Singularities and log singularities. 25 3.4. Castelnuovo-Noether theorem and some definitions in the Sarkisov program 26 References 32 Introduction One of the main results of last decades algebraic geometry was the foun- dation of Mori theory or Minimal Model Program, for short MMP, and its proof in dimension three. Minimal Model Theory shed a new light on what 1991 Mathematics Subject Classification. Primary 14E30, 14J40 ; Secondary 14J30, 32H02. Key words and phrases. Fano Mori spaces, contractions, base point freeness, sub- adjunction, extremal ray. 1 2 MARCO ANDREATTA is nowadays called higher dimensional geometry. In mathematics high num- bers are really a matter of circumstances and here we mean greater than or equal to 3. The impact of MMP has been felt in almost all areas of algebraic geometry. In particular the philosophy and some of the main new objects like extremal rays, Fano-Mori contractions or spaces and log varieties started to play around and give fruitful answer to different problems.
    [Show full text]
  • Resolutions of Quotient Singularities and Their Cox Rings Extended Abstract
    RESOLUTIONS OF QUOTIENT SINGULARITIES AND THEIR COX RINGS EXTENDED ABSTRACT MAKSYMILIAN GRAB The resolution of singularities allows one to modify an algebraic variety to obtain a nonsingular variety sharing many properties with the original, singular one. The study of nonsingular varieties is an easier task, due to their good geometric and topological properties. By the theorem of Hironaka [29] it is always possible to resolve singularities of a complex algebraic variety. The dissertation is concerned with resolutions of quotient singularities over the field C of complex numbers. Quotient singularities are the singularities occuring in quotients of algebraic varieties by finite group actions. A good local model of such a singularity n is a quotient C /G of an affine space by a linear action of a finite group. The study of quotient singularities dates back to the work of Hilbert and Noether [28], [40], who proved finite generation of the ring of invariants of a linear finite group action on a ring of polynomials. Other important classical result is due to Chevalley, Shephard and Todd [10], [46], who characterized groups G that yield non-singular quotients. Nowa- days, quotient singularities form an important class of singularities, since they are in some sense typical well-behaved (log terminal) singularities in dimensions two and three (see [35, Sect. 3.2]). Methods of construction of algebraic varieties via quotients, such as the classical Kummer construction, are being developed and used in various contexts, including higher dimensions, see [1] and [20] for a recent example of application. In the well-studied case of surfaces every quotient singularity admits a unique minimal resolution.
    [Show full text]
  • Algebraic Surfaces and Hyperbolic Geometry
    Current Developments in Algebraic Geometry MSRI Publications Volume 59, 2011 Algebraic surfaces and hyperbolic geometry BURT TOTARO We describe the Kawamata–Morrison cone conjecture on the structure of Calabi–Yau varieties and more generally klt Calabi–Yau pairs. The conjecture is true in dimension 2. We also show that the automorphism group of a K3 surface need not be commensurable with an arithmetic group, which answers a question by Mazur. 1. Introduction Many properties of a projective algebraic variety can be encoded by convex cones, such as the ample cone and the cone of curves. This is especially useful when these cones have only finitely many edges, as happens for Fano varieties. For a broader class of varieties which includes Calabi–Yau varieties and many rationally connected varieties, the Kawamata–Morrison cone conjecture predicts the structure of these cones. I like to think of this conjecture as what comes after the abundance conjecture. Roughly speaking, the cone theorem of Mori, Kawamata, Shokurov, Kollár, and Reid describes the structure of the curves on a projective variety X on which the canonical bundle K X has negative degree; the abundance conjecture would give strong information about the curves on which K X has degree zero; and the cone conjecture fully describes the structure of the curves on which K X has degree zero. We give a gentle summary of the proof of the cone conjecture for algebraic surfaces, with plenty of examples [Totaro 2010]. For algebraic surfaces, these cones are naturally described using hyperbolic geometry, and the proof can also be formulated in those terms.
    [Show full text]
  • The Birational Geometry of Tropical Compactifications
    University of Pennsylvania ScholarlyCommons Publicly Accessible Penn Dissertations Spring 2010 The Birational Geometry of Tropical Compactifications Colin Diemer University of Pennsylvania, [email protected] Follow this and additional works at: https://repository.upenn.edu/edissertations Part of the Algebraic Geometry Commons Recommended Citation Diemer, Colin, "The Birational Geometry of Tropical Compactifications" (2010). Publicly Accessible Penn Dissertations. 96. https://repository.upenn.edu/edissertations/96 This paper is posted at ScholarlyCommons. https://repository.upenn.edu/edissertations/96 For more information, please contact [email protected]. The Birational Geometry of Tropical Compactifications Abstract We study compactifications of subvarieties of algebraic tori using methods from the still developing subject of tropical geometry. Associated to each ``tropical" compactification is a polyhedral object called a tropical fan. Techniques developed by Hacking, Keel, and Tevelev relate the polyhedral geometry of the tropical variety to the algebraic geometry of the compactification. eW compare these constructions to similar classical constructions. The main results of this thesis involve the application of methods from logarithmic geometry in the sense of Iitaka \cite{iitaka} to these compactifications. eW derive a precise formula for the log Kodaira dimension and log irregularity in terms of polyhedral geometry. We then develop a geometrically motivated theory of tropical morphisms and discuss the induced map on tropical fans. Tropical fans with similar structure in this sense are studied, and we show that certain natural operations on a tropical fan correspond to log flops in the sense of birational geometry. These log flops are then studied via the theory of secondary polytopes developed by Gelfand, Kapranov, and Zelevinsky to obtain polyhedral analogues of some results from logarithmic Mori theory.
    [Show full text]
  • Endomorphisms of Smooth Projective 3-Folds with Non-Negative Kodaira Dimension
    i i Publ. RIMS, Kyoto Univ. 38 (2002), 33–92 Endomorphisms of Smooth Projective 3-Folds with Non-Negative Kodaira Dimension By Yoshio Fujimoto∗ Abstract Let X be a nonsingular projective 3-fold with non-negative Kodaira dimension κ(X) ≥ 0 which admits a nonisomorphic surjective morphism f : X → X onto itself. If κ(X) = 0 or 2, a suitable finite ´etale covering X˜ of X is isomorphic to an abelian 3-fold or the direct product E × S of an elliptic curve E and a nonsingular algebraic surface S with κ(S)=κ(X). §1. Introduction The main purpose of this paper is to study the structure of a nonsingular projective 3-fold X with a surjective morphism f : X → X onto itself which is not an isomorphism. We call it a nontrivial surjective endomorphism of X. Let f : X → X be a surjective morphism from a nonsingular projective variety X onto itself. Then f is a finite morphism and if the Kodaira dimension κ(X)ofX is non-negative, f is a finite ´etale covering. Moreover, if X is of general type or a compact hyperbolic manifold in the sense of Kobayashi [Kob], then X admits no nontrivial surjective endomorphisms (cf. [I], [Kob]). Surjec- tive endomorphisms of a projective manifold with negative Kodaira dimension have been studied by many algebraic geometers (cf. [CS], [D], [HM], [PS]) from the viewpoint of Lazarsfeld’s conjecture (cf. [PS], Problem 1). Notably, in [HM, Main Theorem] and [PS, Proposition 2], it has been proved that a ra- tional homogeneous manifold of Picard number 1 has no nontrivial surjective endomorphisms unless it is isomorphic to the complex projective space Pn.
    [Show full text]
  • Resolving 3-Dimensional Toric Singularities
    Resolving 3-dimensional toric singularities ∗Dimitrios I. Dais Mathematics Department, Section of Algebra and Geometry, University of Ioannina GR-45110 Ioannina, Greece. E-mail: [email protected] Abstract This paper surveys, in the first place, some basic facts from the classification theory of normal complex singularities, including details for the low dimensions 2 and 3. Next, it describes how the toric singularities are located within the class of rational singularities, and recalls their main properties. Finally, it focuses, in particular, on a toric version of Reid’s desingularization strategy in dimension three. 1 Introduction There are certain general qualitative criteria available for the rough classification of singularities of complex varieties. The main ones arise: • from the study of the punctual algebraic behaviour of these varieties (w.r.t. local rings associated to singular points) [algebraic classification] • from an intrinsic characterization for the nature of the possible exceptional loci w.r.t. any desingularization [rational, elliptic, non-elliptic etc.] arXiv:math/0110278v1 [math.AG] 25 Oct 2001 • from the behaviour of “discrepancies” (for Q-Gorenstein normal complex varieties) [adjunction-theoretic classification] ∗GSRT-fellow supported by the European Union and the Greek Ministry of Research and Technology. 2000 Mathematics Subject Classification. Primary 14M25; Secondary 14B05, 32S05. Keywords: canonical singularities, toric singularities. 1 2 dimitrios i. dais • Algebraic Classification. At first we recall some fundamental definitions from commutative algebra (cf. [52], [54]). Let R be a commutative ring with 1. The height ht(p) of a prime ideal p of R is the supremum of the lengths of all prime ideal chains which are contained in p, and the dimension of R is defined to be dim (R) := sup {ht (p) |p prime ideal of R } .
    [Show full text]
  • Arxiv:2108.13052V1 [Math.AG]
    EFFECTIVE RESULTS IN THE THREE-DIMENSIONAL MINIMAL MODEL PROGRAM YURI PROKHOROV Abstract. We give a brief review on recent developments in the three-dimensional minimal model program. In this note we give a brief review on recent developments in the three-dimensional minimal model program (MMP for short). Certainly, this is not a complete survey of all advances in this area. For example, we do not discuss the minimal models of varieties of non-negative Kodaira dimension, as well as, applications to birational geometry and moduli spaces. The aim of the MMP is to find a good representative in a fixed birational equivalence class of algebraic varieties. Starting with an arbitrary smooth projective variety one can perform a finite number of certain elementary transformations, called divisorial contractions and flips, and at the end obtain a variety which is simpler in some sense. Most parts of the MMP are completed in arbitrary dimension. One of the basic remaining problems is the following: Describe all the intermediate steps and the outcome of the MMP. The MMP makes sense only in dimensions 2 and for surfaces it is classical and well-known. So the first non-trivial case is the three-dimensional≥ one. It turns out that to proceed with the MMP in dimension 3 one has to work with varieties admitting certain types of very mild, so- called terminal, singularities.≥ On the other hand, dimension 3 is the last dimension where one can expect effective results: in higher dimensions classification results become very complicated and unreasonably long. We will work over the field C of complex numbers throughout.
    [Show full text]
  • Arxiv:1503.08320V2 [Math.AG] 10 Dec 2015 O Hsnto;W Eegie Ytecnet Flrecmlxstr Complex Large of Symmetry
    December 11, 2015 THE DUAL COMPLEX OF CALABI–YAU PAIRS JANOS´ KOLLAR´ AND CHENYANG XU Abstract. A log Calabi–Yau pair consists of a proper variety X and a divisor D on it such that KX +D is numerically trivial. A folklore conjecture predicts that the dual complex of D is homeomorphic to the quotient of a sphere by a finite group. The main result of the paper shows that the fundamental group of the dual complex of D is a quotient of the fundamental group of the smooth locus of X, hence its pro-finite completion is finite. This leads to a positive answer in dimension ≤ 4. We also study the dual complex of degenerations of Calabi–Yau varieties. The key technical result we prove is that, after a volume preserving birational equivalence, the transform of D supports an ample divisor. A log Calabi–Yau pair, abbreviated as logCY, is a pair (X, ∆) consisting of a proper variety X and an effective Q-divisor ∆ such that (X, ∆) is log canonical and KX + ∆ is Q-linearly equivalent to 0. Any Calabi–Yau variety X can be naturally identified with the log Calabi–Yau pair (X, 0). At the other extreme, if X is a Fano variety and ∆ ∼Q −KX is an effective divisor then (X, ∆) is also logCY (provided that it is log canonical). Here we are interested in these Fano–type logCYs, especially when ∆ is “large.” Being Fano is not preserved under birational equivalence, thus it is better to define “large” without reference to Fano varieties. There are several natural candidates for this notion; we were guided by the concepts of large complex structure limit and maximal unipotent degeneration used in Mirror Symmetry.
    [Show full text]
  • ON the EXISTENCE of FLIPS 1. Introduction the Main Result of This Paper Is: Theorem 1.1. Assume the Real MMP in Dimension
    ON THE EXISTENCE OF FLIPS CHRISTOPHER D. HACON AND JAMES MCKERNAN Abstract. Using the techniques of [17], [7], [13] and [15], we prove that flips exist in dimension n, if one assumes the termination of real flips in dimension n − 1. 1. Introduction The main result of this paper is: Theorem 1.1. Assume the real MMP in dimension n − 1. Then flips exist in dimension n. Here are two consequences of this result: Corollary 1.2. Assume termination of real flips in dimension n − 1 and termination of flips in dimension n. Then the MMP exists in dimension n. As Shokurov has proved, [14], the termination of real flips in di- mension three, we get a new proof of the following result of Shokurov [15]: Corollary 1.3. Flips exist in dimension four. Given a proper variety, it is natural to search for a good birational model. An obvious, albeit hard, first step is to pick a smooth projective model. Unfortunately there are far too many such models; indeed given any such, we can construct infinitely many more, simply by virtue of successively blowing up smooth subvarieties. To construct a unique model, or at least cut down the examples to a manageable number, we have to impose some sort of minimality on the birational model. The choice of such a model depends on the global geometry of X. One possibility is that we can find a model on which the canonical di- visor KX is nef, so that its intersection with any curve is non-negative. Date: December 31, 2005.
    [Show full text]