ТРУДЫ МАТЕМАТИЧЕСКОГО ИНСТИТУТА ИМ. В.А. СТЕКЛОВА, 2003, т. 240, с. 82–219 УДК 512.7 Prelimiting Flips1 c 2003 г. V.V.Shokurov2 Поступило в мае 2002 г. You can’t always get exactly what you want. The paper discusses an inductive approach to constructing log flips. In addition to special termination and thresholds, we introduce two new ingredients: the saturation of linear systems, and families of divisors with confined singularities. We state conjectures concerning these notions in any dimension and prove them in general in dimension ≤ 2. This allows us to construct prelimiting flips (pl flips) and all log flips in dimension 4 and to prove the stabilization of an asymptotically saturated system of birationally free (b-free) divisors under certain conditions in dimension 3. In dimension 3, this stabilization upgrades pl flips to directed quasiflips. It also gives for the first time a proof of the existence of log flips that is algebraic in nature, that is, via f.g. algebras, as opposed to geometric flips. It accounts for all the currently known flips and flops, except possibly for flips arising from geometric invariant theory. 1. INTRODUCTION (83): Prelimiting contractions (83). Induction Theorem (84). Main The- orem 1.7 (85). Structure of the paper (86). History (86). Conventions (87). Acknowledge- ments (88) 2. SPECIAL TERMINATION (88): Theorem on Special Termination (88). Proof of Reduction Theorem 1.2 (90) 3. DIVISORIAL OZ -ALGEBRAS AND FLIPS (91): Definition of bss ample (BSS) (91). The algebra of flips (93). Theorem: f.g. algebras and rational divisor (96). Remark: generalized Zariski decomposition (101). Theorem: LMMP gives Zariski decomposition (102). Main The- orem: (RFA)n,d(bir) ⇒ (PLF)n,d (103). Main Lemma (105). Definition of restricted algebras (RFA) (106). The covering trick (107) 4. FINITELY GENERATED pbd ALGEBRAS (109): Definition of functional algebra (109). Trun- cation Principle (110). Definition of pbd algebra (111). Proposition: the mobile system M• of a functional algebra (113). Limiting Criterion (118). Saturation of linear systems (119). Definition of lca saturation (123). Definition of algebra of type (FGA)n (124). Conjecture FGA (124). A Pythagorean dream: (FGA) in the 1-dimensional case (125). Proposition on log canonical saturation (126). Proposition: restriction is lca saturated (129). Main corollary: (FGA)d ⇒ (RFA)n,d, and small pl flips (134) 5. SATURATION AND DESCENT (135): Descent of divisors (135). Descent data (135). Asymp- totic descent problem (136). Choice of Y in the asymptotic descent (138). Definition of predic- tion model (138). Main theorem on asymptotic descent (139). Lemma on approximation (140). Corollary on Diophantine approximation (141). Proof of Theorem 5.12 (144) 6. CANONICAL CONFINEMENT AND SATURATION (152): Definition of a canonically con- fined set of divisors (153). Definition of desirable triple (155). Conjecture on canonical con- finement (157). Remarks: generalizations of the conjecture (159). Main Theorem on canoni- cal confinement (162). Examples in dimension ≤ 2 for Conjecture 6.14 (165). Main Proposi- tion for Conjecture 6.14 in dimension 2 (166). Example: log canonical confinement and the nef condition (173). Corollary: Conjecture 6.14 in dimension 2 (173). Theorem: LMMPd,(BP)d, ∗ (CCS)d−1, and (SSB)d−1 imply (RFA)n,m(bir) (174) 1Partially supported by NSF (grants DMS-9800807 and DMS-0100991). 2Department of Mathematics, Johns Hopkins University, Baltimore, MD 21218, USA. E-mail: [email protected] 82 PRELIMITING FLIPS 83 7. RESTRICTIONS OF b-DIVISORS (175): Mobile restriction (175). Fixed restriction (175). Mixed restriction (175) 8. APPROXIMATIONS (178): Definition of freedom with tolerance τ (179). Main theorem on freedom with tolerance (179). Proposition: properties of freedom with tolerance (179). Theorem on freedom with tolerance in a family of triples (184). Theorem: the case of log Fano contrac- tions (186) 9. STABILIZATION AT A CENTRAL DIVISOR (187): Conventions (187). Theorem: existence of inductive model with a stabilized and rational central limit (189). Example: projective space Pn et al. (189). Proposition: rationality and stabilization (190). Proposition: restriction to cen- tral divisor and confined singularities (194). Proof of Theorem 9.9 (196). Proof of Adden- dum 9.21.1 (201) 10. DESTABILIZATION (204): Key: destabilizing model (204). Proof of Proposition 10.6. Con- struction of a destab (destabilizing) model (205). Corollary: conditional stabilization of the char- acteristic system (211). Corollary: conditional existence of pl flips (211) 11. THE MAIN RESULT (211): Theorem on (S+−) flips (211). Preamble (213). Trick (215). Proof of (RFA) in Main Theorem 1.7 (215) LIST OF ACRONYMS AND ABBREVIATIONS (216) 1. INTRODUCTION 1.1. Prelimiting contractions. We write f : X → X∨ to denote a prelimiting contraction, or pl contraction, with respect to a divisor S. By this we mean that f is a birational contraction such that (1) S = Si is a sum of s ≥ 1 prime Weil divisors S1,...,Ss on X that are Q-Cartier and proportional,thatis,Si ∼Q ri,jSj for some rational ri,j > 0; (2) there is a boundary B such that B =0andK + S + B is divisorially log terminal, and purely log terminal if s =1;and (3) K + S + B is numerically negative/X∨ (that is, it is minus an ample R-Cartier divisor/X∨). By conditions (1), (2) and Shokurov [41, следствие 3.8] (see also Koll´ar and others [27, Ch. 17]), the divisors Si are normal varieties, with normal varieties in their intersections and normal crossing at the generic points of these intersections. In particular, Y = Si is a normal variety, the core of f. Conditions (2) and (3) together mean that (X/X∨,S+B) is a log Fano contraction with only log terminal singularities. Thus, by the Connectedness of LCS [27, Theorem 17.4], Y is irreducible of dimension d = n − s near each fibre of f,wheres is the number of Si that intersect that fibre and n =dimX.Moreover,f induces a contraction Y → f(Y ) ⊂ X∨,and(Y,BY ) is Klt (Kawamata log terminal) [27, Definition 2.13; 41, с. 110] for the adjoint log divisor KY + BY =(K + S + B) Y . These properties establish an induction on Y in our construction of pl flips below (see Section 3 for inductive sequences and compare the proof in Example 3.40). We always consider the local situation, where f is a germ over a neighborhood of a given point −1 P ∈ X∨. In particular, all the Si intersect the central fibre X/P = f P .Thus,dimY = d = n − s ≤ n − 1 by condition (1). The core dimension d measures the difficulty of constructing pl flips in the sense indicated in Induction Theorem 1.4 (see Corollary 1.10 and also Examples 3.53 and 3.54). Sometimes, it is also reasonable to consider a contraction f that is not birational. Then S is not numerically negative/X∨, in contrast to the case of elementary pl flips. This situation occurs naturally in Section 3 in certain applications of Main Lemma 3.43 (see Example 3.38). ТРУДЫ МАТЕМАТИЧЕСКОГО ИНСТИТУТА ИМ. В.А. СТЕКЛОВА, 2003, т. 240 6* 84 SHOKUROV However, in Reduction Theorem 1.2 below, we can assume that the pl contractions f we are most interested in are elementary, that is, satisfy in addition the following conditions: (4) S is numerically negative/X∨; (5) K + S + B is strictly log terminal/X∨,thatis,X is Q-factorial and projective/X∨; (6) f is extremal in the sense that it is the contraction of an extremal ray of the Mori cone; that is, ρ(X/X∨)=1,whereρ denotes the relative Picard number; and (7) f is small, or equivalently, under the current assumptions, S is not f-exceptional. We can omit the final condition (7), but then we meet an additional well-known situation when the contraction is divisorial and is its own S-flip (cf. Example 3.45). Note also that, by (6), f is nontrivial, i.e., is not an isomorphism. (Compare a remark after Example 3.53 when f is an isomorphism.) A pl flip is the S-flip of a pl contraction f (see [45, Section 5] and Example 3.15 below). By a log flip we mean the D-flip of a birational log contraction (X/T, B)overanybaseT (see Conventions 1.14), with D = K + B, where we assume that • K + B is Klt, and •−(K + B)isnef/T (cf. Shokurov [41, с. 105, теорема]). Thus, each pl flip for which −S/X∨ is nef is also the log flip of (X/T, B)=(X/X∨, (a + r)S + B + H) for some real numbers 0 <r a<1 and for a general effective divisor H ∼R −(K + aS + B) (a complement) that is numerically ample/X∨. This holds because K +(a + r)S + B + H is Klt and ∼R rS (compare the proof of Theorem 3.33). Note also that a D-flip is uniquely determined up to isomorphism by the class of D up to ∼R and positive real scalar multiples (see Corollary 3.6). By condition (4), each elementary pl flip is also a log flip. Interest in pl flips rests on the following result: Reduction Theorem 1.2. Log flips exist in dimension n provided that el (PLF)n elementary pl flips exist in dimension n; and (ST)n special termination holds in dimension n. We discuss Termination (ST) presently in Section 2, where we sketch a proof of Reduction Theorem 1.2. In Special Termination 2.3, we also prove that (ST)n follows from the log minimal model program (LMMP; see Conventions 1.14) in dimension n − 1. By our reduction, this is sufficient for the existence of 3-fold and 4-fold log flips.