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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.
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