Effectivity of Iitaka fibrations and pluricanonical systems of polarized pairs

Caucher Birkar; De-Qi Zhang

doi:10.1007/s10240-016-0080-x

Effectivity of Iitaka fibrations and pluricanonical systems of polarized pairs

Birkar, Caucher; Zhang, De-Qi 2016-01-18 00:00:00 EFFECTIVITY OF IITAKA FIBRATIONS AND PLURICANONICAL SYSTEMS OF POLARIZED PAIRS by CAUCHER BIRKAR and DE-QI ZHANG ABSTRACT For every smooth complex projective variety W of dimension d and nonnegative Kodaira dimension, we show the existence of a universal constant m depending only on d and two natural invariants of the very general ﬁbres of an Iitaka ﬁbration of W such that the pluricanonical system |mK | deﬁnes an Iitaka ﬁbration. This is a consequence of a more general result on polarized adjoint divisors. In order to prove these results we develop a generalized theory of pairs, singularities, log canonical thresholds, adjunction, etc. CONTENTS 1. Introduction ....................................................... 283 2. Preliminaries ...................................................... 288 3. Effective birationality of K + B + nM ......................................... 291 4. Generalized polarized pairs .............................................. 297 5. Bounds on the number of coefﬁcients of B and M .................................. 308 i i 6. ACC for generalized lc thresholds ........................................... 315 7. Global ACC ....................................................... 317 8. Proof of main results . ................................................. 326 Acknowledgements ..................................................... 330 References ......................................................... 330 1. Introduction We work over the complex number ﬁeld C. However, our results hold over any algebraically closed ﬁeld of characteristic zero. Effectivity of Iitaka ﬁbrations. — Let W be a smooth projective variety of Kodaira dimension κ(W) ≥ 0. Then by a well-known construction of Iitaka, there is a birational morphism V → W from a smooth projective variety V, and a contraction V → Xonto a projective variety X such that a (very) general ﬁbre F of V → X is smooth with Kodaira dimension zero, and dim X is equal to the Kodaira dimension κ(W).The map W X is referred to as an Iitaka ﬁbration of W, which is unique up to birational equivalence. For any sufﬁciently divisible natural number m, the pluricanonical system |mK | deﬁnes an Iitaka ﬁbration. When dim W = 2, in 1970, Iitaka [12]provedthatif m is any natural number divisible by 12 and m ≥ 86, then |mK | deﬁnes an Iitaka ﬁbration (Fabrizio Catanese informed us that Iitaka proved this result for compact complex surfaces but the algebraic case goes back to Enriques). It has since been a question whether something similar holds in higher dimension. More precisely (cf. [8]): DOI 10.1007/s10240-016-0080-x 284 CAUCHER BIRKAR, DE-QI ZHANG Conjecture 1.1 (Effective Iitaka ﬁbration). — Let W be a smooth projective variety of dimension d and Kodaira dimension κ(W) ≥ 0. Then there is a natural number m depending only on d such that the pluricanonical system |mK | deﬁnes an Iitaka ﬁbration for any natural number m divisible by m . W d In this paper, we show a version of the conjecture as formulated in [24, Ques- tion 0.1] holds, that is, by assuming that some invariants of the very general ﬁbres of the Iitaka ﬁbration are bounded. Without these extra assumptions the above conjecture seems out of reach at the moment because most likely one needs the abundance conjec- ture to deal with the very general ﬁbres. For example, when κ(W) = 0, the conjecture is equivalent to the effective nonvanishing h (W, m K ) = 0 which is obviously related d W to the abundance conjecture. Note that there is also a log version of the conjecture for pairs: see [11, Conjecture 1.2, Theorem 1.4] and the references therein, where the au- thors conﬁrmed this log version when the boundary divisor is big over the generic point of the base of the log Iitaka ﬁbration. We recall some deﬁnitions before stating our result. Using the notation above, let W be a smooth projective variety of Kodaira dimension κ(W) ≥ 0and V → Xan Iitaka ﬁbration from a resolution V of W. For a very general ﬁbre F of V → X, let b := min{u ∈ N ||uK |=∅}. F F Let F be a smooth model of the Z/(b )-cover of F ramiﬁed over the unique divisor in |b K |.Then F still has Kodaira dimension zero, but with |K |=∅.Notethat F F F dim F = dim F = dim W − dim X = dim W − κ(W) and we denote this number by d . We call the Betti number β := dim H (F, C) the middle Betti number of F. Theorem 1.2. —Let W be a smooth projective variety of dimension d and Kodaira dimension κ(W) ≥ 0. Then there is a natural number m(d, b ,β ) depending only on d , b and β such that the F F F F pluricanonical system |mK | deﬁnes an Iitaka ﬁbration whenever the natural number m is divisible by m(d, b ,β ). The theorem is an almost immediate consequence of 1.3 below. The proof is given at the end of Section 8. When X is of general type, the numbers b ,β do not play any F F role so m(d, b ,β ) depends only on d . F F Here is a brief history of partial cases of Theorem 1.2: • when dim W = 2[12], • when κ(W) = 1[7], • when W is of general type [8, 21] (see also [23]), EFFECTIVITY OF IITAKA FIBRATIONS 285 • when κ(W) = 2[24] (see also [22]), • when dim W = 3[7, 8, 14, 21, 24] (see also [6]), • when X is non-uniruled, V → X has maximal variation and its general ﬁbres have good minimal models [19] (see also [5]), • when V → X has zero variation and its general ﬁbres have good minimal models [13]. Note that the above references show that Conjecture 1.1 holds when dim W ≤ 3. Effective birationality for polarized pairs of general type. — Let W be a smooth projective variety of nonegative Kodaira dimension. After replacing W birationally we can assume the Iitaka ﬁbration W → X is a morphism. Applying the canonical bundle formula of [7] (which is based on [16]), perhaps after replacing W and X birationally, there is a Q- boundary B and a nef Q-divisor M on X such that for any natural number m divisible by 0 0 b we have a natural isomorphism between H (W, mK ) and H (X, m(K + B + M)). F W X In particular, if |m(K + B + M)| deﬁnes a birational map, then |mK | deﬁnes an Iitaka X W ﬁbration. Moreover, the coefﬁcients of B belong to a DCC set and the Cartier index of M is bounded in terms of b and β . Therefore we can derive Theorem 1.2 from the next result. Theorem 1.3. —Let be a DCC set of nonnegative real numbers, and d, r natural numbers. Then there is a natural number m(, d, r) depending only on , d, r such that if: (i) (X, B) is a projective lc pair of dimension d , (ii) the coefﬁcients of B are in , (iii) rM is a nef Cartier divisor, and (iv) K + B + M is big, then the linear system |m(K + B + M)| deﬁnes a birational map if m ∈ N is divisible by m(, d, r). We call (X, B + M) a polarized pair. When M = 0, the theorem is [10, Theorem 1.3]. 0 0 Note that for an R-divisor D, by |D| and H (X, D) we mean | D | and H (X, D ). Generalized polarized pairs. — In order to prove Theorem 1.3 we need to generalize the deﬁnitions of pairs, singularities, lc thresholds, adjunction, etc. We develop this theory, which is of independent interest, in some detail in Section 4 but for now we only give the deﬁnition of generalized polarized pairs. Deﬁnition 1.4. —A generalized polarized pair consists of a normal variety X equipped with projective morphisms X → X → Z where f is birational and X is normal, an R-boundary B , and an R-Cartier divisor M on X which is nef /Z such that K + B + M is R-Cartier, where M := f M. We call B the boundary part and M the nef part. ∗ 286 CAUCHER BIRKAR, DE-QI ZHANG Note that the deﬁnition is ﬂexible with respect to X, M. To be more precise, if g : Y → X is a projective birational morphism from a normal variety, then there is no harm in replacing X with Y and replacing M with g M. M where μ ≥ 0and M are For us the most interesting case is when M = μ j j j j nef /Z Cartier divisors. In many ways B + M behaves like a boundary, that is, it is as if the M were components of the boundary with coefﬁcient μ . Although the coefﬁcients of B j i belong to the real interval [0, 1] the coefﬁcients μ are only assumed to be nonnegative. Moreover, the M are not necessarily distinct. See Section 4 for more details. When X → X is the identity morphism, we recover the deﬁnition of polarized pairs which was formally introduced in [4] but appeared earlier in the subadjunction formula of [16]. If moreover M = 0, then (X , B ) is just a pair in the traditional sense. ACC for generalized lc thresholds. — The next result shows that the generalized lc thresholds satisfy ACC under suitable assumptions. We suggest the reader looks at Deﬁ- nitions 4.1 and 4.3 before continuing. Theorem 1.5. —Let be a DCC set of nonnegative real numbers and d a natural number. Then there is an ACC set depending only on ,dsuch that if (X , B + M ), M, N,and D are as in Deﬁnition 4.3 satisfying (i) (X , B + M ) is generalized lc of dimension d , (ii) M = μ M where M are nef /Z Cartier divisors and μ ∈ , j j j j (iii) N = ν N where N are nef /Z Cartier divisors and ν ∈ ,and k k k k (iv) the coefﬁcients of B and D belong to , then the generalized lc threshold of D + N with respect to (X , B + M ) belongs to . Note that the theorem is a local statement over X , so Z does not play any role and we could simply assume X → Z is the identity map. When X → X is the identity map, M = 0, and N = 0, the theorem is the usual ACC for lc thresholds [10, Theorem 1.1]. Global ACC. — The proof of the previous result requires the following global ACC. We will also use this to bound pseudo-effective thresholds (Theorem 8.1) which is in turn used in the proof of Theorem 1.3. Theorem 1.6. —Let be a DCC set of nonnegative real numbers and d a natural number. Then there is a ﬁnite subset ⊆ depending only on ,dsuch that if (X , B + M ), X → X → Z and M are as in Deﬁnition 1.4 satisfying (i) (X , B + M ) is generalized lc of dimension d , (ii) Z is a point, EFFECTIVITY OF IITAKA FIBRATIONS 287 (iii) M = μ M where M are nef Cartier divisors and μ ∈ , j j j j (iv) μ = 0 if M ≡ 0, j j (v) the coefﬁcients of B belong to ,and (vi) K + B + M ≡ 0, then the coefﬁcients of B and the μ belong to . When X → X is the identity map and M = 0, the theorem is [10, Theorem 1.5]. About this paper. — It is not hard to reduce Theorems 1.3 and 1.5 to Theorem 1.6. So most of thedifﬁcultieswefacehavetodowith 1.6. Since the statement of Theo- rems 1.3, 1.5,and 1.6 involve nef divisors which may not be semi-ample (or effectively semi-ample), there does not seem to be any easy way to reduce them to the traditional versions (i.e. without nef divisors) proved in [10] or to mimic the arguments in [10]. In- stead we need to develop new ideas and arguments and this occupies much of this paper. We brieﬂy explain the organization of the paper. In Section 3,weprove aspecial case of Theorem 1.3 (Proposition 3.4) by closely following [10]. In Section 4, we introduce generalized singularities and generalized lc thresholds, discuss the log minimal model pro- gram for generalized polarized pairs, and treat generalized adjunction. In Section 5,we give bounds, both in the local and global situations, on the numbers of components in the boundary and nef parts of generalized polarized pairs, under appropriate assump- tions. These bounds will be used in the proof of Proposition 7.2 which serves as the key inductive step toward the proof of Theorem 1.6.InSection 6,wereduceTheorem 1.5 to Theorem 1.6 in lower dimension by adapting a standard argument. In Section 7,we treat Theorem 1.6 inductively where we apply Proposition 3.4; a sketch of the main ideas is included below in this introduction. In Section 8, we give the proofs of our main results. Theorems 1.5 and 1.6 follow immediately from Sections 6 and 7.Toprove Theorem 1.3, we use Theorem 1.6 to bound certain pseudo-effective thresholds (Theorem 8.1)and use the concept of potential birationality [10] to reduce to the special case of Proposition 3.4. Finally, we extend Theorem 1.3 to allow more general coefﬁcients in the nef part of the pair (see Theorem 8.2), and deduce Theorem 1.2 from Theorem 1.3 as in [7, 24]. A few words about the proof of Theorem 1.6.— We try to explain, brieﬂy, some of the ideas used in the proof of 1.6.By[10, 1.5] we can assume M ≡ 0. The basic strategy is to modify (X , B + M ) so that the nef part has one less coefﬁcient μ and then repeat this to reach the case M = 0. Running appropriate LMMP’s we can reduce the problem to the case when X is a Q-factorial klt Fano variety with Picard number one. Moreover, some lengthy arguments show that the number of the μ is bounded (Section 5). If (X , B + M ) is not generalized klt, one can do induction: for example if B = 0, then we let S be the normalization of a component of B and use generalized adjunction (see Deﬁnition 4.7) to write K + B + M = K + B + M S S S X S 288 CAUCHER BIRKAR, DE-QI ZHANG and apply induction to the generalized lc polarized pair (S , B + M ). So we can assume S S (X , B + M ) is generalized klt. Although we cannot use the arguments of [10]toprove Theorem 1.6 but there is an exception: if we take n ∈ N to be sufﬁciently large, then following [10] closely one can show that there is m ∈ N depending only on , d such that |m(K + B + nM |)| X j deﬁnes a birational map (Proposition 3.4) where B is the sum of the birational transform of B and the reduced exceptional divisor of X → X . One can then show that there is an R-divisor D such that 0 ≤ D ∼ K + B + nM R X j where the coefﬁcients of D belong to some DCC set depending only on , d . Then the pushdown D of D satisﬁes D ∼ K + B + nM ≡ (n − μ )M ≡ ρM R X j j j 1 for some number ρ . Changing the indexes one can assume that ρ belongs to some ACC set depending only on , d.Let N = M − μ M .Now theideaistotake s, t,with s 1 1 maximal, so that K + B + sD + N + tM ≡ K + B + M X X and that (X , B + sD + N + tM ) is generalized lc. If it happens to have t = 0, then s would belong to some DCC set and we can replace B with B + sD and replace M with N which has one less summand, and repeat the process. But if t > 0, then (X , B + sD + N + tM ) is generalized lc but not generalized klt. We cannot simply apply induction because the s, t may not belong to a DCC set. For simplicity assume B + sD = 0and let S be one of its components and assume S is normal. The idea is to keep S but to remove the other components of D and increase t instead so that we get K + B +˜ sS + N + tM ≡ K + B + M X X ˜ ˜ for some ˜ s and t ≥ t where S is a component of B +˜ sS .Now it turnsout t belongs to some DCC set and we can apply induction by restricting to S . 2. Preliminaries Notation and terminology. — All the varieties in this paper are quasi-projective over C unless stated otherwise. For deﬁnitions and basic properties of singularities of pairs such as log canonical (lc), Kawamata log terminal (klt), divisorially log terminal (dlt), purely log terminal (plt), and the log minimal model program (LMMP) we refer to [18]. We recall some notation: EFFECTIVITY OF IITAKA FIBRATIONS 289 • The sets of natural, integer, rational, and real numbers are respectively denoted as N, Z, Q, R. • Divisors on normal varieties are always Weil R-divisors unless otherwise stated. • Let X → Z be a projective morphism from a normal variety. Linear equivalence, Q- linear equivalence, R-linear equivalence,and numerical equivalence over Z, between two R-divisors D , D on X are respectively denoted as D ∼ D /Z, D ∼ D /Z, 1 2 1 2 1 Q 2 D ∼ D /Z, and D ≡ D /Z. If Z is a point, we usually drop the Z. 1 R 2 1 2 • If φ : X X is a birational morphism whose inverse does not contract divi- sors, and D is an R-divisor on X, we usually write D for φ D. If X is replaced by X (resp. Y) we usually write D (resp. D )for φ D. Y ∗ • Let X, Y be normal varieties projective over some base Z, and φ : X Ya birational map/Z whose inverse does not contract any divisor. Let D be an R- Cartier divisor on X such that D is also R-Cartier. We say φ is D-negative if there ∗ ∗ is a common resolution g : W → Xand h : W → Ysuch thatE := g D − h D is effective and exceptional/Y, and Supp g E contains all the exceptional divisors of φ. ACC and DCC sets. — A sequence {a } of numbers is increasing (resp. strictly increasing) if a ≤ a (resp. a < a )for all i. The deﬁnition of a decreasing or strictly decreasing i i+1 i i+1 sequence is similar. A set of real numbers satisﬁes DCC (descending chain condition) if it does not contain a strictly decreasing inﬁnite sequence. A set of real numbers satisﬁes ACC (ascending chain condition) if it does not contain a strictly increasing inﬁnite sequence. Lemma 2.1. —Let and be sets of nonnegative real numbers. Deﬁne + ={a + b | a ∈ , b ∈ } and · ={ab | a ∈ , b ∈ }. Then the following hold: (1) If and are both ACC sets (resp. DCC sets), then + and · are also ACC sets (resp. DCC sets). (2) Let {a }⊆ and {b }⊆ be sequences of numbers. Assume that both sequences are i i increasing and that one of them is strictly increasing. Then the sequences {a + b } and {a b } are strictly i i i i increasing. (3) A statement similar to (2) holds if we replace ‘increasing’ by ‘decreasing’. (4) Let m, l ∈ N. Assume that is a DCC set and that a ≤ lfor every a ∈ . Then the set {ma| a ∈ } also satisﬁes DCC, where ma:= ma − ma , that is, the fractional part of ma. Proof. — The proof is left to the reader. 290 CAUCHER BIRKAR, DE-QI ZHANG Lemma 2.2. —Let d, r be natural numbers. Let X be a sequence of normal projective varieties of dimension d and Picard number one. Assume that D ,..., D are nonzero R-Cartier divisors on 1,i r,i X .Let λ be the numbers such that D ≡ λ D . Then possibly after replacing the sequence with an i j,i j,i j,i 1,i inﬁnite subsequence and rearranging the indexes, the sequence λ is a decreasing sequence for each j . j,i Proof.—Let ρ be the numbers such that D ≡ ρ D . Replacing the sequence j,k,i j,i j,k,i k,i we may assume that for each j, k the sequence ρ is increasing or decreasing. If ρ is j,k,i j,k,i decreasing we write j k. This relation is associative, that is, if j k and k l,then j l because ρ = ρ ρ . So we can order the sequences of divisors according to j,l,i j,k,i k,l,i this relation. Changing the indexes we may assume that r ··· 1 whichinparticular means that the λ = ρ form a decreasing sequence for each j . j,i j,1,i Minimal models and Mori ﬁbre spaces. — Let X → Z be a projective morphism of normal varieties and D an R-Cartier divisor on X. A normal variety Y projective over Z together with a birational map φ : X Y/Z whose inverse does not contract any divisor is called a minimal model of D over Zif: (1) Yis Q-factorial, (2) D = φ Dis nef /Z, and Y ∗ (3)φ is D-negative. If one can run an LMMP on D over Z which terminates with a Q-factorial model Y on which D is nef /Z, then Y is a minimal model of D over Z. On the other hand, we call Y a Mori ﬁbre space of D over Z if Y satisﬁes the above conditions with condition (2) replaced by: (2) there is an extremal contraction Y → T/Zsuch that −D is ample/T. In practice, we consider minimal models and Mori ﬁbre space for K + B + M where (X , B + M ) is a generalized polarized pair. Some notions and results of [10]. — For convenience we recall some technical notions and results of [10] which will be used in Section 3. Let X be a normal projective variety, and let D be a big Q-Cartier Q-divisor on X. We say that D is potentially birational [10, Deﬁnition 3.5.3] if for any pair x and y of general points of X, possibly switching x and y,wecan ﬁnd 0 ≤ ∼ (1 − )Dfor some 0 < < 1such that (X,) is not klt at y but (X,) is lc at x and {x} is a non-klt centre. Theorem 2.3 [10, Theorem 3.5.4]. — Let (X, B) be a klt pair, where X is projective of dimension d , and let H be an ample Q-divisor. Suppose there exist a constant γ ≥ 1 and a family V → C of subvarieties of X with the following property: if x and y are two general points of X then, possibly switching x and y, we can ﬁnd c ∈ C and 0 ≤ ∼ (1 − δ)H,for some δ> 0,such c Q that (X, B + ) is not klt at y and there is a unique non-klt place of (X, B + ) whose centre V c c c contains x. Further assume there is a divisor D on W, the normalization of V , such that the linear system |D| deﬁnes a birational map and γ H| − D is pseudo-effective. Then mH is potentially birational, where m = 2p γ + 1 and p = dim V . c EFFECTIVITY OF IITAKA FIBRATIONS 291 Theorem 2.4 [10, Theorem 4.2]. — Let be a subset of [0, 1] which contains 1.Let X be a projective variety of dimension d , and let V be a subvariety, with normalization W. Suppose we are given an R-boundary B and an R-Cartier divisor G ≥ 0, with the following properties: (1) the coefﬁcients of B belong to ; (2) (X, B) is klt; and (3) there is a unique non-klt place ν for (X, B + G), with centre V. Then there is an R-boundary B on W whose coefﬁcients belong to a | 1 − a ∈ LCT D() ∪{1} d−1 such that the difference (K + B + G)| − (K + B ) X W W W is pseudo-effective. Now suppose that V is the general member of a covering family of subvarieties of X.Let ψ : U → W be a log resolution of (W, B ),and let B be the sum of the birational transform of B and W U W the reduced exceptional divisor of ψ . Then K + B ≥ (K + B)| . U U X U The notation | and | mean pullback to W and U respectively. W U Remark 2.5. — Assume that the in 2.4 satisﬁes DCC. Then the hyperstandard set D() also satisﬁes DCC, hence the set of lc thresholds LCT (D()) satisﬁes ACC d−1 by the ACC for usual lc thresholds [10, Theorem 1.1]. Therefore, the set a | 1 − a ∈ LCT D() ∪{1} d−1 to which the coefﬁcients of B belong, also satisﬁes DCC. 3. Effective birationality of K + B + nM In this section, following [10] closely, we prove a special case of Theorem 1.3 (see 3.4) which will be used in Sections 7 and 8 in proving Theorems 1.6 and 1.3. This special case concerns effective birationality for big divisors of the form K + B + nM where (X, B) is projective lc, rM is nef and Cartier, and n/r is large enough. Running an LMMP on K + B + nM preserves the nef and Cartier properties of rM by boundedness of length of extremal rays [15] which allows one to apply the methods of [10]. In contrast if one runs an LMMP on K + B + M, the nef and Cartier properties of rM may be lost, hence one needs to consider generalized polarized pairs which will be discussed in later sections. First we prove a few lemmas. 292 CAUCHER BIRKAR, DE-QI ZHANG Lemma 3.1. —Let X be a normal projective variety, D abig Q-Cartier Q-divisor, and G a nef Q-Cartier Q-divisor on X.If D is potentially birational, then D + G is also potentially birational. In particular, |K +D + G| deﬁnes a birational map. Proof.—Write D ∼ A + B with B effective and A ample. By deﬁnition, for any pair x, y ∈ X of general points, possibly after switching x, y, there exist ∈ (0, 1) and a Q-divisor 0 ≤ ∼ (1 − )Dsuch that (X,) is not klt at y but it is lc at x and {x} is a non-klt centre. Now if ∈ (0, ) is rational, then we can ﬁnd 0 ≤ ∼ + − B + − A + 1 − G ∼ 1 − (D + G) Q Q so that (X, ) satisﬁes the same above properties as (X,) at x, y.So D + G is potentially birational. To get the last claim, just apply [9, Lemma 2.3.4 (1)]. Lemma 3.2. —Let be a DCC set of nonnegative real numbers, and d, r natural numbers. Then there is a real number t ∈ (0, 1) depending only on , d, r such that if: • (X, B) is projective lc of dimension d , • the coefﬁcients of B are in , • rM is a nef Cartier divisor, and • K + B + M is a big divisor, then K + tB + nM is a big divisor for any natural number n > 2rd . Proof. — Since M is nef, it is enough to treat the case n = 2rd + 1. We can assume 1 ∈ .Let (X, B) and M be as in the statement of the lemma. Let f : W → Xbe a log resolution and let B be the sum of the birational transform of B and the reduced exceptional divisor of f ,and let M be the pullback of M. Then we can replace (X, B) with (W, B ) and replace M with M hence it is enough to only consider log smooth W W pairs. We want to argue that, after extending if necessary, it is enough to only consider thecasewhen (X, B) is klt. If the lemma does not hold, then there is a sequence (X , B ), i i M of log smooth lc pairs and nef Q-divisors satisfying the assumptions of the lemma but such that the pseudo-effective thresholds b = min{a ≥ 0 | K + aB + nM is pseudo-effective} i X i i is a strictly increasing sequence of numbers approaching 1. Now by extending and decreasing the coefﬁcients in B which are equal to 1, we can assume that (X , B ) are i i i klt. To get a contradiction it is obviously enough to only consider this sequence hence we only need to consider the klt case. Now let (X, B) and M be as in the statement of the lemma where we assume (X, B) is log smooth klt. Let b be the pseudo-effective threshold as deﬁned above. We may assume b > 0. By Lemma 4.4(2) below, we can run an LMMP on K + bB + nM X EFFECTIVITY OF IITAKA FIBRATIONS 293 which ends with a minimal model X on which K + bB + nM is semi-ample deﬁning a contraction X → T . Since b > 0, a general ﬁbre of X → T is positive-dimensional and the restriction of B to it is big, by the bigness of K + B + nM and the deﬁnition of b. So relying on Lemma 4.4(2) once more, we can also run an LMMP/T on K + nM with scaling of bB which terminates with a Mori ﬁbre space. Denote the end result again by X and the Mori ﬁbre space structure by X → S . By Lemma 4.4(3), both LMMP’s are M-trivial and hence the Cartier and nefness of rM is preserved in the process. Now since K + bB + nM ≡ 0/S and since n > 2rd,M ≡ 0/S by boundedness of length of extremal rays [15]. In particular, if F is a general ﬁbre of X → S ,then K + := K + bB ≡ K + bB + nM ∼ 0. F X X R F F By construction, (K + B + M )| is big and M | ≡ 0, so B | is not zero and its coefﬁ- X F F F cients belong to . Therefore, b is bounded away from 1 otherwise we get a contradiction with the ACC property of [10, Theorem 1.5]. Thus there is t ∈ (0, 1) depending only t +1 on , d, r such that K + t B + nM is pseudo-effective. Now take t = . X 0 We should point out that although we have used (and continue to use) Lemma 4.4 but its proof does not rely on any of the results of this section. Lemma 3.3. —Let X be a normal projective variety, L abig R-divisor, and M anef Q-divisor which is not numerically trivial. Then vol(L + nM) goes to ∞ as n goes to ∞. Proof. — We may write L ∼ A + D where A is ample Q-Cartier and D ≥ 0. Thus ν d−ν ν vol(L + nM) ≥ vol(A + nM) ≥ n A · M where d = dim X and ν is the numerical dimension of M. Since A is ample and ν> 0, d−ν ν A · M > 0. Hence the above volume goes to inﬁnity as n goes to inﬁnity. Proposition 3.4. —Let ⊂[0, 1] be a DCC set of nonnegative real numbers and let d,rbe natural numbers. Then there exists a natural number m depending only on , d, r such that if: • n is a natural number satisfying n > 2rd and r|n, • (X, B) is projective lc of dimension d , • the coefﬁcients of B are in , • rM is a nef Cartier divisor, and • K + B + M is a big divisor, then |m(K + B + nM)| deﬁnes a birational map. Proof.— Step 1. We prove the proposition by induction on d . In particular, we may assume that the proposition holds in dimension < d . 294 CAUCHER BIRKAR, DE-QI ZHANG Fix β> 0. Pick (X, B),M, and n as in the proposition. Assume that vol(K + B + nM)>β . We ﬁrst prove the result for such (X, B),M, and n. At the end, in Steps 6 and 7, we treat the general case. As in the proof of Lemma 3.2, by extending , by taking a log resolution of (X, B), and by decreasing the coefﬁcients of B, we can assume that (X, B) is klt. Step 2. By Lemma 3.2,K + bB + nMis big for some b ∈ (0, 1) depending only on , d, r.Thus vol K + (b + 1)B + nM = vol (K + bB + nM + K + B + nM) X X 1 1 > vol (K + B + nM) > β. 2 2 Replacing b by (b + 1) we may assume that vol(K + bB + nM)>β := β. Moreover, there exists a natural number p depending only on and b (and hence only on , d, r) and there exists a boundary B such that pB is an integral divisor and bB ≤ B ≤ B: this follows from the fact that we can ﬁnd p so that λ − bλ> for every nonzero λ ∈ which in turn implies that for each λ we can ﬁnd an integer 0 ≤ i ≤ p such that bλ ≤ ≤ λ. By the calculation above, vol(K + B + nM)>β . Replacing B with B ,and β with β , we can assume ={i/p | 0 ≤ i ≤ p} and that pB is integral. Step 3. Applying Lemma 4.4(2) below, we can replace X with the lc model (= ample model) of K + B + nM so that we can assume that K + B + nM is ample keeping rM X X nef and Cartier. Since vol(K + B + nM)>β , there is a natural number k > 0 depending only on d,β,suchthat vol k(K + B + nM) >(2d) . Applying [10, Lemma 7.1] to the log pair (X, B) and the big divisor k(K + B + nM), we get a covering family V → C of subvarieties of X such that if x and y are two general points of X, then we may ﬁnd c ∈ Cand 0 ≤ ∼ k(K + B + nM) c R X such that (X, B + ) is not klt at y butitislcat x and there is a unique non-klt place of (X, B + ) whose centre is equal to V which contains x. c c EFFECTIVITY OF IITAKA FIBRATIONS 295 Step 4. Let H := 2k(K + B + nM). In this step we make the necessary preparations in order to apply [10, Theorem 3.5.4] (= Theorem 2.3 above). To do this we need to ﬁnd a natural number γ , depending only on , d, r, and ﬁnd a divisor D on the normalization WofV such that γ H| − D is pseudo-effective and |D| deﬁnes a birational map. c W If dim W = dim V = 0, then γ, D exist trivially (and H is potentially birational). So assume that dim W ≥ 1. Now applying the adjunction formula of [10, Theorem 4.2] (= Theorem 2.4 above) to the klt pair (X, B) and the divisor , and taking into account Remark 2.5, we can ﬁnd a boundary B on W whose coefﬁcients belong to a DCC set uniquely determined by , d , such that the difference (∗)(K + B + )| − (K + B ) X c W W W is a pseudo-effective divisor. Further, let ψ : U → W be a log resolution of (W, B ) and let B be the sum of the strict transform of B and the reduced exceptional divisor of ψ . U W Then K + B ≥ (K + B)| . U U X U Denote by M := M| , the pullback of M to U, by the composition U U U → W → V → X which is birational onto its image. Then K + B + M ≥ (K + B + M)| . U U U X U Hence K + B + M is big because (K + B + M)| is big being the pullback of the U U U X U big divisor K + B + M to a smooth model of the general subvariety V . X c Since the coefﬁcients of B belong to the DCC set , since rM is a nef Cartier U U divisor, and since n > 2rd , the induction hypothesis implies that |m(K + B + nM )| U U U deﬁnes a birational map for some m > 0 depending only on (and hence on )and d, r.Thus |m(K + B + nM )| also deﬁnes a birational map since it contains the direct W W W image of |m(K + B + nM )| where M denotes the pullback of M to W. U U U W Note that the difference (K + B + nM + )| − (K + B + nM ) X c W W W W ∼ (k + 1)(K + B + nM)| − (K + B + nM ) R X W W W W is a pseudo-effective divisor by (∗) above. Now let D := m(K + B + nM ) and let γ W W W be the smallest natural number satisfying γ ≥ m(k + 1)/2k.Then γ H| − D is a pseudo- effective divisor and |D| deﬁnes a birational map as required. Step 5. By Step 4 and Theorem 2.3, m H = 2m k(K + B + nM) X 296 CAUCHER BIRKAR, DE-QI ZHANG is potentially birational for some m ≤ 2(d − 1) γ + 1. Thus by Lemma 3.1, 2m kp(K + B + nM) + nM is also potentially birational and K + 2m kp(K + B + nM) + nM X X deﬁnes a birational map where p is as in Step 2 (recall that pB is an integral divisor). Since K + 2m kp(K + B + nM) + nM ≤ 2m kp + 1 (K + B + nM) X X X the linear system 2m kp + 1 (K + B + nM) also deﬁnes a birational map. Now the number m := 2m kp + 1only depends on the data , d, r,β . Step 6. Now we go back to Step 1. We will show that there exist a natural number q and a real number α> 0 depending only on , d, r,suchthatif (X, B),M, n are as in the statement of the proposition and if n ≥ q,then vol(K + B + nM)>α. If this is not true, then we can ﬁnd a sequence (X , B ),M , n satisfying the assumptions of the proposition i i i i such that the n form a strictly increasing sequence approaching ∞ and the vol(K + i X B + n M ) approach 0. By replacing X with a minimal model of K + B + n M ,wemay i i i i X i i i assume that K + B + n M is nef. We can also assume that ν , the numerical dimension X i i i of M , is independent of i. We may assume ν> 0 otherwise we can get a contradiction using [10, Theorem 1.3]. By Lemma 3.3,for each i, there is n the largest natural number divisible by r such that vol(K + B + n M )< 1. We show that the volume vol(K + B + (2n − 1)M ) is X i i X i i i i i i bounded from above. This follows from 2 > vol 2 K + B + n M X i i i i = vol K + B + 2n − 1 M + K + B + M X i i X i i i i i > vol K + B + 2n − 1 M X i i i i where we use the assumption that K + B + M is big. X i i On the other hand, since vol K + B + n + r M ≥ 1, X i i i i EFFECTIVITY OF IITAKA FIBRATIONS 297 by Steps 2–5 above, we may assume that there is an m depending only on , d, r such that m K + B + n + r M X i i i i deﬁnes a birational map for every i. In particular, there exist resolutions f : Y → X such i i i that P := f m K + B + n + r M ∼ H + G i X i i i i i i i where H is big and base point free and G is effective. So we can calculate i i 2 m > vol m K + B + 2n − 1 M X i i i i d ν ∗ d−ν ∗ ν = P + m n − r − 1 f M ≥ m n − r − 1 H · f M i i i i i i i i d−ν ∗ ν 1 which gives a contradiction as lim(n − r − 1) =∞ and H · f M ≥ . i i i Step 7. Let q,α be as in Step 6. In this step we show that there is β> 0 depending only on , d, r such that vol(K + B + nM)>β for any (X, B),M, n as in the statement n−1 of the proposition. We may assume q > n otherwise we can use Step 6. Let s = .Then q−1 vol(K + B + nM) = vol (1 − s)(K + B + M) + s(K + B + qM) X X X d d ≥ s vol(K + B + qM)> s α ≥ =: β. (q − 1) This completes the proof of the proposition. 4. Generalized polarized pairs In this section, we deﬁne generalized lc and klt singularities, discuss some of their basic properties, and then deﬁne generalized lc thresholds for generalized polarized pairs. Next we consider running the log minimal model program for these pairs, and use it to extract divisors with generalized log discrepancy <1. Then we deﬁne generalized adjunc- tion and discuss DCC and ACC properties of coefﬁcients in the boundary and nef parts of generalized polarized pairs under this adjunction. Generalized singularities. — We already deﬁned generalized polarized pairs in the in- troduction. Now we deﬁne their singularities. Deﬁnition 4.1. —Let (X , B + M ) be a generalized polarized pair as in 1.4 which comes with the data X → X → Z and M.Let E be a prime divisor on some birational model of X .We 298 CAUCHER BIRKAR, DE-QI ZHANG deﬁne the generalized log discrepancy of E with respect to the above generalized polarized pair as follows. After replacing X, we may assume E is a prime divisor on X. We can write K + B + M = f K + B + M X X for some R-divisor B. The generalized log discrepancy of E is deﬁned to be 1 − b where b is the coefﬁcient of E in B. We say that (X , B + M ) is generalized lc (resp. generalized klt) if the generalized log discrepancy of any prime divisor is ≥ 0 (resp. > 0). If f is a log resolution of (X , B ), then generalized lc (resp. generalized klt) is equivalent to the coefﬁcients of B being ≤ 1 (resp. < 1). If the generalized log discrepancy of E is ≤ 0, we call the image of E in X a generalized non-klt centre.If (X , B + M ) is generalized lc, a non-klt centre is also referred to as a generalized lc centre. Remark 4.2. —Weuse thenotationof 4.1. (1) Note that Z does not play any role in the deﬁnition of singularities. That is because singularities are local in nature over X , so one can simply assume X → Zis the identity map. The same applies to generalized lc thresholds deﬁned below (4.3) and in general to notions and statements that are local. (2) Assume that (X , B + M ) is generalized klt. Let D be an effective R-Cartier divisor. Then from the deﬁnitions we can easily see that (X , B + D + M ) is generalized klt with boundary part B + D and nef part M, for any small > 0. Now assume that D is ample/Z. Then for any a > 0 we can ﬁnd a boundary ∼ B + aD + M /Z such that (X , ) is klt. (3) Assume that K + B is R-Cartier and write K + B = f (K + B ) and X X X ˜ ˜ f M = M + E. By the negativity lemma [20, Lemma 1.1], E ≥ 0. Thus B = B + E ≥ B. Therefore, if (X , B + M ) is generalized lc (resp. generalized klt), then (X , B ) is lc (resp. klt). (4) Assume that M ∼ 0/X .Then (X , B + M ) is generalized lc (resp. generalized klt) iff (X , B ) is generalized lc (resp. generalized klt). Indeed in this case M = f M hence K + B = f (K + B ) which implies the claim. In this situation M does not contribute X X to the singularities even if its coefﬁcients are large. In contrast, the larger the coefﬁcients of B, the worse the singularities. (5) In general, M does contribute to singularities. For example, assume X = P and that f is the blowup of a point x . Let E be the exceptional divisor, L a line passing through x and L the birational transform of L . If B = 0and M = 2L, then we can calculate B = Ehence (X , B + M ) is gener- alized lc but not generalized klt. However, if B = L and M = 2L, then (X , B + M ) is not generalized lc because in this case B = L + 2E. EFFECTIVITY OF IITAKA FIBRATIONS 299 (6) Assume we are given a contraction X → Y/Z. We may assume f is a log resolu- tion of (X , B ). Let F be a general ﬁbre of X → Y, F the corresponding ﬁbre of X → Y, and g : F → F the induced morphism. Let B = B| , M = M| , B = g B , M = g M . F F F F F ∗ F F ∗ F Then (F , B + M ) is a generalized polarized pair with the data F → F → Zand M . F F F Moreover, K + B + M = K + B + M . F F F X In addition, B = B | and M = M | : note that since F is a general ﬁbre, B and M are F F F F R-Cartier along any codimension one point of F hence we can deﬁne these restrictions. (7) Let φ : X → X be a birational contraction from a normal variety. We can assume X X is a morphism. Let B , M be the pushdowns of B, M. Then K + B + M = φ K + B + M . X X Now assume that B is a boundary. Then we can naturally consider (X , B + M ) as a generalized polarized pair with boundary part B and nef part M. One may think of (X , B + M ) as a crepant model of (X , B + M ). Deﬁnition 4.3. —Let (X , B + M ) be a generalized polarized pair as in 1.4 which comes with the data X → X → Z and M. Assume that D on X is an effective R-divisor and that N on X is an R-divisor which is nef /Z and that D + N is R-Cartier. The generalized lc threshold of D + N with respect to (X , B + M ) (more precisely, with respect to the above data) is deﬁned as sup s | X , B + sD + M + sN is generalized lc where the pair in the deﬁnition has boundary part B + sD and nef part M + sN. By the negativity lemma, G := f (D + N ) − N ≥ 0. Thus we can write K + B + M = f K + B + M X X and K + B + sG + M + sN = f K + B + sD + M + sN . X X In particular, if (X , B + M ) is generalized lc, then the just deﬁned generalized lc thresh- old is nonnegative. However, the threshold might be +∞: this happens when D = 0and N ∼ 0/X . As pointed earlier, the generalized lc threshold is local over X , so we can usually assume X → Z is the identity map. When M = N = 0, we recover the usual lc threshold of D with respect to (X , B ). 300 CAUCHER BIRKAR, DE-QI ZHANG LMMP for generalized polarized pairs. — Let (X , B + M ) be a Q-factorial generalized lc polarized pair with data X → X → Z and M. One can ask whether one can run an LMMP/ZonK + B + M and whether it terminates. We cannot answer this question in such generality but we will put some extra assumptions under which the answer would be yes. Assume that K + B + M + A is nef /Zfor some R-Cartier divisor A ≥ 0 which is big/Z. Moreover, assume (∗) for any s ∈ (0, 1) there is a boundary ∼ B + sA + M /Zsuch that (X , + (1 − s)A ) is klt. Condition (∗) is automatically satisﬁed if A is general ample/Z and either (i) (X , B + M ) is generalized klt, or (ii) (X , B + M ) is generalized lc and (X , 0) is klt. We will show that we can run the LMMP/ZonK + B + M with scaling of A (However, we do not know whether it terminates). Let λ = min t ≥ 0 | K + B + M + tA is nef /Z . We may assume λ> 0. Replacing A with λA we may assume λ = 1. By assumption we can ﬁnd a number 0 < s < 1 and a boundary ∼ B + sA + M /Zsuch that (X , + (1 − s)A ) is klt. Now by [1, Lemma 3.1], there is an extremal ray R /Zsuch that (K + ) · R < 0and K + + (1 − s)A · R = 0. In particular, (K + B + M ) · R < 0and K + B + M + A · R = 0. Moreover, R can be contracted and its ﬂip exists if it is of ﬂipping type. If R deﬁnes a Mori ﬁbre space we stop. Otherwise let X X be the divisorial contraction or the ﬂip of R . Replacing X we may assume X X is a morphism. Then (X , B + M ) is naturally a generalized lc polarized pair with boundary part B and nef part M. More- over, K + B + M + A is nef /Zand (∗) is preserved. Repeating the process gives the LMMP. Now we show the LMMP terminates under suitable assumptions. Lemma 4.4. —Let (X , B + M ) be a Q-factorial generalized lc polarized pair of dimension d with data X → X → Z and M. Assume that (X , B + M ) satisﬁes (i) or (ii) above. Run an LMMP/Z on K + B + M with scaling of some general ample/Z R-Cartier divisor A ≥ 0. Then the following hold: EFFECTIVITY OF IITAKA FIBRATIONS 301 (1) Assume that K + B + M is not pseudo-effective/Z. Then the LMMP terminates with a Mori ﬁbre space. (2) Assume that • K + B + M is pseudo-effective/Z, • (X , B + M ) is generalized klt, and that • K + (1 + α)B + (1 + β)M is R-Cartier and big/Z for some α, β ≥ 0. Then the LMMP terminates with a minimal model X and K + B + M is semi- ample/Z, hence it deﬁnes a contraction φ : X → T /Z. If moreover a general ﬁbre of φ is positive-dimensional and if the restriction of B to it is nonzero, then we can run the LMMP/T on K + M with scaling of B which terminates with a Mori ﬁbre space of K + M over both T and Z. (3) Assume X → X is the identity morphism and that M = μ M where μ ≥ 0 and M j j j j are Cartier nef /Z divisors. Pick j and assume μ > 2d . Then the above LMMP’s are M - trivial. In particular, the LMMP’s preserve the Cartier and the nefness/Z of M .Moreover, under the assumptions of (2) and assuming φ is birational, M ≡ 0/T and M is the j j pullback of some Cartier divisor on T . Proof. — (1) Since K + B + M is not pseudo-effective/Z, the LMMP is also an LMMP on K + B + A + M with scaling of (1 − )A for some > 0. Now we can ﬁnd a boundary ∼ B + A + M /Z such that (X , + (1 − )A ) is klt. The claim then follows from [3] as the LMMP is an LMMP/ZonK + with scaling of (1 − )A . (2) As K + (1 + α)B + (1 + β)M is big/Z, it is R-linearly equivalent to some P + G over Z where P is ample and G ≥ 0. Now if > 0 is small, then (1 + ) K + B + M ∼ K + (1 − α)B + (1 − β)M + P + G X R X ∼ K + /Z R X for some such that (X , ) is klt and is big/Z. The LMMP is also an LMMP/Z on K + with scaling of (1 + )A which terminates on some model X by [3]. By the base point free theorem for klt pairs with big boundary divisor [3, Corollary 3.9.2], K + is semi-ample/Zhence K + B + M is semi-ample/Z and so it deﬁnes a X X contraction φ : X → T . Now assume a general ﬁbre of φ : X → T is positive-dimensional and the restric- tion of B to it is nonzero. In particular, this implies that K + M is not pseudo-effective X 302 CAUCHER BIRKAR, DE-QI ZHANG over T . Since K + ≡ K + B + M ≡ 0/T , X X 1 + running the LMMP/T on K + M with scaling of B is the same as running the LMMP/T on K + − τ B with scaling of τ B for some small τ> 0 and this termi- nates with a Mori ﬁbre space over T and also over Z, by [3]. Note that, − τ B ≥ 0 by construction. (3) Each step of those LMMP’s is M -trivial and preserves the Cartier and the nefness/ZofM by boundedness of the length of extremal rays and the cone theorem [15], [18, Theorem 3.7 (1) and (4)]. Under the assumptions of (2) and assuming φ is birational, to show that M is the pullback of some Cartier divisor on T ,itisenough to show that X → T decomposes into a sequence of extremal contractions which are negative with respect to certain klt pairs. We write this more precisely. Since in the proof of (2) is big/Z, we can assume ≥ C for some ample Q-divisor C . Since K + ≡ 0/T ,if X → T is not an isomorphism, then there is a (K + − C )-negative extremal ray which gives a contraction X → X /T . In particular M is the pullback of a Cartier divisor on X [18, Theorem 3.7 (4)]. Now j 2 repeat the process with X and so on. Since φ is birational by assumption, the process ends with T hence we can indeed decompose X → T into a sequence of extremal contractions as required. We will apply the LMMP to birationally extract certain divisors for a generalized polarized pair. Lemma 4.5. —Let (X , B + M ) be a generalized lc polarized pair with data X → X → Z and M.Let S ,..., S be prime divisors on birational models of X which are exceptional/X and whose 1 r generalized log discrepancies with respect to (X , B + M ) are at most 1. Then perhaps after replacing f with a high resolution, there exist a Q-factorial generalized lc polarized pair (X , B + M ) with data X → X → Z and M, and a projective birational morphism φ : X → X such that • S ,..., S appear as divisors on X , 1 r • each exceptional divisor of φ is one of the S or is a component of B ,and • K + B + M = φ (K + B + M ). X X In particular, the exceptional divisors of φ are exactly the S if (X , B + M ) is generalized klt. Proof. — Replacing X we may assume the S are divisors on X and that f is a log resolution of (X , B ).Let E , E ,... be the exceptional divisors of f where we can 1 2 assume E = S for i ≤ r.Write i i K + B + M = f K + B + M X X EFFECTIVITY OF IITAKA FIBRATIONS 303 and let = B +Ewhere E := a E and a is the generalized log discrepancy of E (by i i i i i>r deﬁnition a is equal to 1 − b where b is the coefﬁcient of E in B). Then is a boundary i i i i and K + + M = f K + B + M + E ≡ E/X X X with E ≥ 0exceptional/X . By construction, none of the S are components of E. Now run an LMMP/X on K + + M with scaling of some ample divisor. This is also an LMMP/X on E. In the course of the LMMP we arrive at a model X on which K + + M is a limit of movable/X divisors hence it is nef on the general curves/X of any exceptional divisor of X → X where , M are the pushdowns of , M. But since E is effective and exceptional/X ,E = 0 by the general negativity lemma (cf. [2, Lemma 3.3 and the proof of Theorem 3.4]). Note that since the LMMP contracts E, we have = B . So we can write K + B + M = φ K + B + M X X where φ is the morphism X → X . By construction, none of the S is contracted by the LMMP. Moreover, any exceptional divisor of φ is one of the S or is a component of B In particular, the exceptional divisors of φ are exactly the S if (X , B + M ) is generalized klt. Note that X is Q-factorial by construction. Lemma 4.6. — Under the notation and assumptions of Lemma 4.5, further assume that (X , C ) is klt for some C , and that the generalized log discrepancies of the S with respect to (X , B + M ) are < 1. Then we can construct φ so that in addition it satisﬁes: • its exceptional divisors are exactly S ,..., S ,and 1 r • if r = 1 and X is Q-factorial, then φ is an extremal contraction. Proof. — Since (X , C ) is klt and (X , B + M ) is generalized lc, X ,(1 − )B + C + (1 − )M is generalized klt for any small > 0 with boundary part := (1 − )B + C and nef part (1 − )M. Moreover, the generalized log discrepancies of the S with respect to (X , + (1 − )M ) are still less than 1. So by Lemma 4.5, there is φ : X → X which extracts exactly the S . Now further assume that r = 1and that X is Q-factorial. By construction, we can write K + + (1 − )M = φ K + + (1 − )M X X where is the sum of the birational transform of and sS for some s ∈ (0, 1).Now run an LMMP/X on K + + δS + (1 − )M for some small δ> 0 which is also 1 304 CAUCHER BIRKAR, DE-QI ZHANG an LMMP on S . Since X is Q-factorial, the last step of the LMMP is an extremal contraction X → X which contracts S , the pushdown of S ,and X X is an 1 1 isomorphism in codimension one. Thus replacing X with X we can assume φ is ex- tremal. Generalized adjunction. — We deﬁne an adjunction formula for generalized polarized pairs similar to the traditional one. Deﬁnition 4.7. —Let (X , B + M ) be a generalized polarized pair with data X → X → Z and M. Assume that S is the normalization of a component of B and S is its birational transform on X.Replacing X we may assume f is a log resolution of (X , B ). Write K + B + M = f K + B + M X X and let K + B + M := (K + B + M)| S S S X S where B = (B − S)| and M = M| . Let g be the induced morphism S → S and let B = g B S S S S S ∗ S and M = g M . Then we get the equality S ∗ S K + B + M = K + B + M S S S X which we refer to as generalized adjunction.Itisobviousthat B depends on both B and M. Now assume that (X , B + M ) is generalized lc. By Remark 4.8 below B is a boundary divisor on S , i.e. its coefﬁcients belong to [0, 1]. We consider (S , B + M ) as a generalized polarized S S pair which is determined by the boundary part B , the morphisms S → S → Z, and the nef part M . S S It is also clear that (S , B + M ) is generalized lc if (X , B + M ) is so because then S S K + B + M = g (K + B + M ) S S S S S S and the coefﬁcients of B are at most 1. Remark 4.8. —Wewillargue that the B deﬁned in 4.7 is indeed a boundary divisor on S ,if (X , B + M ) is generalized lc. The lc property immediately implies that the coefﬁcients of B do not exceed 1, hence we only have to show that B ≥ 0. Moreover, S S if K + B is R-Cartier, then B ≥ 0 follows from the usual divisorial adjunction: indeed X S in this case if B is the divisor given by the adjunction K + B = K + B S S X then it is well-known that B is a boundary divisor, and it is also clear from our deﬁnitions that B ≤ B . S S EFFECTIVITY OF IITAKA FIBRATIONS 305 In practice when we apply generalized adjunction, X will be Q-factorial, hence K + B will be R-Cartier. But for the sake of completeness we treat the general case, i.e. the non-R-Cartier K + B case. We will reduce the statement to the situation dim X = 2 in which case K + B turns out to be R-Cartier automatically. Assume dim X > 2. Let H be a general hypersurface section and G its pullback to S . Adding H to B we may assume H is a component of B .Both H and G are normal varieties. Let B be given by the generalized adjunction K + B + M = K + B + M . H H H X Since H is a general hypersurface section, B is simply the intersection of B − H with H , that is, each component of B is a component of the intersection of some component of B − H with H inheriting the same coefﬁcient. In particular, B is a boundary divisor and G is a component of B A further generalized adjunction and induction on dimension gives K + B + M = (K + B + M )| G G G H H H G where B is a boundary. But B is equal to the intersection of B − G with the ample G G S divisor G on S which implies that B is a boundary divisor too. Now we can assume dim X = 2. Since K + B + M = f K + B + M ≡ 0/X X X ˜ ˜ ˜ and since M is nef /X , there is a divisor B ≤ Bsuch thatK + B ≡ 0/X and f B = B . X ∗ Since each coefﬁcient of B is at most 1, each coefﬁcient of B is also at most 1. Therefore (X , B ) is numerically lc (see [18, Section 4.1]; note however that [18]onlyconsiders B with rational coefﬁcients but all the deﬁnitions and results that we need make sense and hold true for real coefﬁcients as well). Now by [18, Section 4.1], (X , B ) is lc. In particular, K + B is R-Cartier. So we are done by the above arguments. Proposition 4.9. — Let d be a natural number and a DCC set of nonnegative real numbers. Then there is a DCC set of nonnegative real numbers depending only on d and such that if (X , B + M ) is a generalized lc polarized pair of dimension d with data X → X → Z and M,and S is the normalization of a component of B satisfying • M = μ M where M are nef /Z Cartier divisors and μ ∈ , j j j j • the coefﬁcients of B belong to ,and • B is given by the following generalized adjunction (as in 4.7) K + B + M = K + B + M , S S S X then the coefﬁcients of B belong to . S 306 CAUCHER BIRKAR, DE-QI ZHANG Proof. — If the statement does not hold, then there exist a sequence of generalized lc polarized pairs (X , B + M ) and S ,withdata X → X → Z and M = μ M , i i i j,i j,i i i i i i satisfying the assumptions of the proposition but such that the set of the coefﬁcients of all the B put together does not satisfy DCC. Note that since the problem is local, we may assume X → Z is the identity map for each i. We may also assume f is a log resolution i i of (X , B ). Let S ⊂ X be the birational transform of S . We can assume that each B has i i S a component V with coefﬁcient a such that {a } is a strictly decreasing sequence. Let i i i a = lim a . We may assume that the K + B are R-Cartier otherwise as in Remark 4.8,by taking hypersurface sections, we reduce the problem to dimension 2 in which case this R-Cartier property holds automatically. Let B be the divisor given by the adjunction K + B = K + B . S S X i i i S ˜ ˜ It is clear from our deﬁnitions that B ≤ B .If c is the coefﬁcient of V in B ,thenwe S S i i S i i i may assume c ≤ a ≤ a + for some ﬁxed > 0so that a + < 1. Therefore, (X , B ) is i i i i plt near the generic point of (the image of) V (this follows from inversion of adjunction on surfaces [20, Corollary 3.12]) and there is a natural number l depending only on a + such that for each i there is l ≤ l so that for any Weil divisor D on X the divisor l D i i i i i is Cartier near the (image of the) generic point of V [20, Proposition 3.9]. Moreover, by [20, Corollary 3.10] we can write l − 1 d i k,i c = + b i k,i l l i i for some nonnegative integers d where b are the coefﬁcients of the components of B k,i k,i other than (the image of) S passing through V . On the other hand, shrinking X if necessary we can assume M is Q-Cartier for i j,i each j, i so we can write f M = M + E j,i j,i i j,i where the exceptional divisor E is effective by the negativity lemma. Since l M is j,i i j,i Cartier near the (image of the) generic point of V , the multiplicity of the birational trans- j,i form of V in E | is equal to for some nonnegative integer e . Therefore, i j,i S j,i l − 1 d i k,i j,i a = + b + μ . i k,i j,i l l l i i i This is a contradiction because the above expression and Lemma 2.1 show that the set {a } satisﬁes DCC, while the a form a strictly decreasing sequence. i i We will need the next technical lemma in the proof of Proposition 7.1 to treat Theorem 1.6 inductively. EFFECTIVITY OF IITAKA FIBRATIONS 307 Lemma 4.10. — Let d be a natural number and be a DCC set of nonnegative real numbers. Let (X , B + M ) be a sequence of generalized lc polarized pairs of dimension d with data X → i i i X → Z and M .Let S be the normalization of a component of B and consider the generalized i i i i i adjunction formula K + B + M = K + B + M . S S S X i i i i i i S Assume further that (1) X is Q-factorial and Z is a point, (2) B = b B where B are distinct prime divisors and b ∈ , k,i k,i i k,i k,i (3) M = μ M where M are nef Cartier divisors and μ ∈ , i j,i j,i j,i j,i (4) and one of the following holds: (i) {b } is not ﬁnite, and B | ≡ 0 for each i, or 1,i S 1,i (ii) {μ } is not ﬁnite, and M | ≡ 0 for each i. 1,i S 1,i Then the set of the coefﬁcients of all the B union the set {μ | M | ≡ 0} is not ﬁnite. S j,i j,i S Proof.—Let V be a prime divisor on S . As in the proof of Proposition 4.9,the coefﬁcient of V in B is of the form i S l − 1 d e i k,i j,i a = + b + μ i k,i j,i l l l i i i where l is a natural number and d , e are nonnegative integers which are contributed i k,i j,i by the B and M respectively. k,i j,i Now assume (i) of (4) holds. Since {b } is not ﬁnite, we can assume b < 1for 1,i 1,i each i which in particular means B is not equal to the image of S .Thus B | is a 1,i i 1,i nonzero effective divisor for each i. Choose V to be a component of B | . Then the set i S 1,i {a } cannot be ﬁnite by Lemma 2.1 because {b } is not ﬁnite and d is positive. i 1,i 1,i Next assume (ii) of (4) holds. Although M | is not numerically trivial by assump- 1,i tion but M | may be numerically trivial for some i.If M | is not numerically trivial 1,i S 1,i S i i for inﬁnitely many i,thenobviously theset {μ | M | ≡ 0} is not ﬁnite and we are j,i j,i S done. So we may assume M | is numerically trivial for every i. Recall from the proof 1,i S of Proposition 4.9 that we can assume f M = M + E with E ≥ 0. Now we can j,i j,i j,i i j,i choose V so that e = 0for each i: indeed since M | ≡ 0but M | ≡ 0, we deduce i 1,i S 1,i S 1,i i that E | = 0 and that its pushdown to S is also not zero; thus the components of the 1,i S i i pushdown of E | are components of B , hence we can choose V to be one of these 1,i S S i components. Again this shows that {a } cannot be ﬁnite because {μ } is not ﬁnite and i 1,i e > 0. 1,i 308 CAUCHER BIRKAR, DE-QI ZHANG 5. Bounds on the number of coefﬁcients of B and M i i A well-known fact says that if (X, B) is a lc pair, then near each point x ∈ Xthe number of components of B with coefﬁcient ≥ b > 0 is bounded in terms of b and di- mension of X. There is also a global version of this fact. In this section, we prove similar local and global statements bounding the number of the coefﬁcients of B and the μ in M = μ M of a generalized lc polarized pair (X , B + M ) under certain assumptions. j j These bounds will be used in the proof of Proposition 7.2. We start with a global statement for pairs which can also be applied to generalized polarized pairs. Proposition 5.1. — Let d be a natural number and b a positive real number. Let (X, B) be a projective lc pair of dimension d such that (i) B ≥ B where B ≥ 0 are big R-Cartier divisors, k k (ii) B = b B is the irreducible decomposition and b ≥ bfor everyj, k, and k j,k j,k j,k (iii) K + B + P ≡ 0 for some pseudo-effective R-Cartier divisor P. Then the number of the B is at most (d + 1)/b, that is, r ≤ (d + 1)/b. Proof.—Let (Y,) be a Q-factorial dlt model of (X, B − B ) and f : Y → X the corresponding morphism. By deﬁnition, is the sum of the reduced exceptional B . Moreover, since (X, B) is lc, divisor of f and the birational transform of B − Supp( B ) does not contain the image of any exceptional divisor of f ,hence f B is k k equal to the birational transform of B . In particular, f B is big and it inherits the same k k coefﬁcients as B . Moreover, by letting B := + f B we get k Y k ∗ ∗ ∗ ∗ K + B + f P = K + + f B + f P = f (K + B + P) ≡ 0. Y Y Y k X Now by replacing (X, B) with (Y, B ) and replacing P with f P we can assume that r r (X, 0) is Q-factorial klt. Moreover, by adding B − B to P we can assume B = B . k k 1 1 If P ≡ 0, then K + B is not pseudo-effective so we can run an LMMP on K + B X X which terminates with a Mori ﬁbre space, by Lemma 4.4(1). But if P ≡ 0, then K is not pseudo-effective as B is big, and we can run an LMMP on K which terminates with a Mori ﬁbre space [3]. Note that in both cases the LMMP preserves the lc property of (X, B) and the Q-factorial klt property of (X, 0): in the ﬁrst case the klt property of ˜ ˜ (X, 0) is preserved since the LMMP is also an LMMP on K + B for some klt (X, B); in the second case the lc property of (X, B) is preserved as K + B ≡ 0. Also note that in either case the LMMP does not contract any B because B is big (although some of k k its components may be contracted). So in either case replacing X with the Mori ﬁbre space obtained we may assume that we already have a K -negative Mori ﬁbre structure X → T. EFFECTIVITY OF IITAKA FIBRATIONS 309 Let F be a general ﬁbre of X → T. Since B is big, B | is big too. Restricting to F k k F and applying induction on dimension we can reduce the problem to the case dim T = 0, that is, when X is a Q-factorial klt Fano variety of Picard number one. Pick a small number > 0. For each j, k take a rational number b ≤ b such that b ≥ b − .Let j,k j,k j,k B = b B . Then there is P ≥ 0such thatK + B + P ≡ 0and (X, B + P ) is j,k X k j j,k lc. Now by [17, Corollary 18.24], r(b − ) ≤ b ≤ d + 1. j,k k j Therefore taking the limit when approaches 0 we get rb ≤ d + 1hence r ≤ (d + 1)/b. Next we prove a result similar to Proposition 5.1, though not as sharp, for the nef part of generalized polarized pairs. Proposition 5.2. — Let d be a natural number and b a positive real number. Assume that the ACC for generalized lc thresholds (Theorem 1.5) holds in dimension d . Then there is a natural number p depending only on d, b such that if (X , B + M ) is a generalized lc polarized pair of dimension d with data X → X → Z and M satisfying (i) Z is a point, (ii) M = μ M where M are nef Cartier divisors and μ ≥ b, j j j j (iii) M is a big Q-Cartier divisor for every j , and (iv) K + B + M + P ≡ 0 for some pseudo-effective R-Cartier divisor P , then the number of the μ is at most p, that is, r ≤ p. Before giving the proof we prove a related local statement. Proposition 5.3. — Let d be a natural number and b a positive real number. Assume that Theorem 1.5 and Proposition 5.2 hold in dimension < d . Then there is a natural number q depending only on d,bsuch that if (X , B + M ) is a Q-factorial generalized lc polarized pair of dimension d with data X → X → Z and M,and if (i) x ∈ X is a (not necessarily closed) point, (ii) M = μ M where M are nef /Z Cartier divisors and μ ≥ b, j j j j (iii) M is not relatively numerically zero over any neighborhood of x ,for every j,and (iv) (X , 0) is klt, then the number of the μ is at most q, that is, r ≤ q. Proof.— Step 1.Let C be the closure of x .By (iii), the codimension of C in X is at least two. By adding appropriate divisors to B and shrinking X we can assume C is a generalized lc centre of (X , B + M ):tobemoreprecise,let W bethe blowup of 310 CAUCHER BIRKAR, DE-QI ZHANG X along C ; we can assume X → X factors through W; now take a general sufﬁciently ample divisor on W and let A be its pullback to X; if α is the generalized lc threshold of A near x with respect to (X , B + M ),then (X , B + αA + M ) is generalized lc near x with boundary part B + αA and nef part M, and C is a generalized lc centre of (X , B + αA + M ); the point is that after shrinking X we can assume f A = A + E where E = 0 is effective with large coefﬁcients, and that every component of E maps onto C so adding αA creates deeper singularities only along C . Now we may replace B with B + αA . Step 2. By Step 1, we can assume that there is a prime divisor S on X mapping onto C whose generalized log discrepancy with respect to (X , B + M ) is 0. Since (X , 0) is Q-factorial klt, by Lemma 4.6, there is an extremal birational contraction φ : X → X which extracts S , the birational transform of S, and X is Q-factorial. We can write K + B + M = φ K + B + M X X where B is the sum of S and the birational transform of B ,and M is the pushdown of M. Writing K + B + M = f K + B + M X X we can see that B is just the pushdown of B. We claim that M is not numerically trivial over any neighborhood of x for any j which in turn implies that M is ample/X . If this is not true for some j , then we can write ˜ ˜ ˜ f M = M + E where E ≥ 0 and S is not a component of E . But then for any general j j j j −1 closed point y ∈ C ,the ﬁbre f {y } is not inside Supp E , so the ﬁbre does not intersect ˜ ˜ Supp E ,by[18, Lemma 3.39(2)]. Therefore, E = 0 over the generic point of C ,thatis j j over x ,hence M is numerically trivial over some neighborhood of x , a contradiction. Step 3. We can assume the induced map g : X X is a morphism. To ease notation we replace S with its normalization and denote the induced morphism S → S by h. By generalized adjunction and usual adjunction, we can write K + B + M = K + B + M ≡ 0/C S S S X and K + = K + B . S S X Write g M = M + E where E ≥ 0is exceptional/X .Then j j j M = h M | + h E | ∗ j S ∗ j S and M = μ h M | + μ h E | = M + μ h E | j ∗ j S j ∗ j S S j ∗ j S S EFFECTIVITY OF IITAKA FIBRATIONS 311 and B = + μ h (E | ). S S j ∗ j S Let V be a prime divisor on S and b be its coefﬁcient in B . Then, by the proof V S of Proposition 4.9, 1 μ n j j b ≥ 1 − + l l for some natural number l and integers n ≥ 0. Moreover, n > 0 if V is a component of j j h (E | ). This in particular shows that there is a natural number s depending only on b ∗ j S such that V is a component of h (E | ) for at most s of the j because μ n ≤ 1. ∗ j S j j Step 4.Let F be a general ﬁbre of the induced map S → C and F the correspond- ing ﬁbre of S → C . Restricting to F as in Remark 4.2(6), we get K + B + M = (K + B + M )| ≡ 0. F F F S S S F Also we get K + := (K + )| . F F S S F Denote the morphism F → F by e. Since F is a general ﬁbre, restricting Weil divisors on S to F makes sense, and if P is a Weil divisor on S, then we have (h P)| = e (P| ). ∗ F ∗ F Therefore, M = e (E | ) + e (M | ), M = μ e (M | ), ∗ j F ∗ j F F j ∗ j F and B = B | = + μ h (E | ) = + μ e (E | ). F S F S j ∗ j S F j ∗ j F Since F may not be Q-factorial, we need to make some further constructions. Let (H , ) be a Q-factorial dlt model of (F , ) and ψ : H → F the corresponding H F morphism. By deﬁnition K + = ψ (K + ) H H F F and the exceptional divisors of ψ all appear with coefﬁcient 1 in .Moreover, we can write K + B + M = ψ (K + B + M ) ≡ 0 H H H F F F where B is the sum of the birational transform of B and the reduced exceptional H F divisor of ψ . 312 CAUCHER BIRKAR, DE-QI ZHANG We can assume c : F H is a morphism. By construction, ψ M | = c (E | ) + c (M | ) F ∗ j F ∗ j F which is big, and M = μ c (M | ) and B = + μ c (E | ). H j ∗ j F H H j ∗ j F Moreover, since the exceptional divisors of ψ are components of , the divisor μ c (E | ) has no exceptional component, so it is just the birational transform of j ∗ j F μ e (E | ). j ∗ j F Step 5. Run an LMMP on K . It terminates with a Mori ﬁbre space H → T and the generalized lc property of (H , B + M ) is preserved by the LMMP. Since H H c (E | ) + c (M | ) is big, its pushdown to H is also big, hence ample over T .Let G be ∗ j F ∗ j F a general ﬁbre of the above Mori ﬁbre space. Then restriction to G gives K + B + M = (K + B + M )| ≡ 0. G G G H H H G By construction, M = μ a (M | )| where we can assume a : F H is a mor- j ∗ j F G G phism. Applying Proposition 5.2 and rearranging the indexes, we can assume that there is a natural number t depending only on d, b such that a (M | )| ≡ 0for every j > t. ∗ j F But then a (E | )| is big for each j > t. ∗ j F For each j > t choose a component W of a (E | ) which is ample over T .By j ∗ j F construction, W is the birational transform of a component U of e (E | ) = (h (E | ))| j j ∗ j F ∗ j S F and U in turn is a component of V ∩ F for some component V of h (E | ).Moreover, j j j ∗ j S W = W if and only if U = U if and only if V = V .ByStep3,for each k,V = V k j k j k j k j for at most s of the j.Thusfor each k,W = W for at most s of the j . On the other k j hand, by Steps 3 and 4, the V appear as components of B with coefﬁcient ≥ min{b, }, j S r−t and there exist at least such components. Similarly the W appear as components 1 r−t of B with coefﬁcient ≥ min{b, }, and there exist at least such components. Now 2 s r−t apply Proposition 5.1 to (G , B ) to deduce that is bounded hence r is bounded by some q. Proof of Proposition 5.2. — We argue by induction on the dimension d.The case d = 1 is clear. Suppose that the proposition holds in dimension < d . Step 1. Since (X , B + M ) is generalized lc and K + B is R-Cartier, (X , B ) is lc. Let (X , B ) be a Q-factorial dlt model of (X , B ) and φ : X → X the corresponding morphism. We may assume X X is a morphism. For each j,wehave φ M = M + j j E where E ≥ 0is exceptional/X and M is the pushdown of M .So j j j K + B + μ E + M = φ K + B + M X j X j EFFECTIVITY OF IITAKA FIBRATIONS 313 where M is the pushdown of M. Since the exceptional divisors of φ are components of B and since (X , B + M ) is generalized lc, we deduce E = 0for every j,hence ∗ ∗ M = φ M for every j and M = φ M . Thus we may replace X with X , hence assume j j that (X , B ) is Q-factorial dlt. Step 2. If P ≡ 0, then K + B + M is not pseudo-effective and so we can run an LMMP on K + B + M which terminates with a Mori ﬁbre space, by Lemma 4.4(1). But if P ≡ 0, then K + B is not pseudo-effective as M is big and so we can run an LMMP on K + B which terminates with a Mori ﬁbre space. Note that in both cases the generalized lc property of (X , B + M ) is preserved: in the second case we use the fact K + B + M ≡ 0. Also note that in both cases none of the M is contracted by the LMMP since M is big. In either case we can replace X with the Mori ﬁbre space hence we may assume we already have a Mori ﬁbre structure X → T .Let F be a general ﬁbre of this ﬁbre space. Since M is big, M | is big too. Restricting to F and applying j j induction on dimension we can reduce the problem to the case dim T = 0, that is, when X is a Fano variety of Picard number one. Step 3. Perhaps after changing the indexes we may write M ≡ λ M such that j 1 λ ≥ 1for every j.Now we deﬁne μ ˜ as follows: initially let μ ˜ = μ ;nextdecrease μ ˜ and j j j j 2 instead increase μ ˜ as much as possible so that X , B + μ ˜ M j=2 is generalized lc and K + B + μ ˜ M + P ≡ 0. X j Either we hit a generalized lc threshold, i.e. (X , B + μ ˜ M ) is generalized lc but j=2 j not generalized klt, or that we reach μ ˜ = 0. If the ﬁrst case happens, we stop. But if the second case happens we repeat the process by decreasing μ ˜ and increasing μ ˜ ,and so 3 1 on. We show that the above process involves only a bounded number of the μ .Let l be the smallest number such that μ ˜ = μ for every j > l . We want to show that l is bounded j j depending only on d, b. We can assume l > 1. By construction, μ ˜ ≥ μ λ ≥ μ ≥ (l − 1)b 1 j j j j≤l−1 j≤l−1 so it is enough to show that μ ˜ is bounded depending only on d, b.If M is not numerically 1 1 trivial over X , then the generalized lc threshold of M with respect to (X , B ) is ﬁnite and bounded from above by Theorem 1.5, and this in turn implies boundedness of μ ˜ .But if M is numerically trivial over X , then again μ ˜ is bounded from above but for a different 1 1 314 CAUCHER BIRKAR, DE-QI ZHANG reason: by the cone theorem X can be covered by curves such that −(K + B ) · ≤ 2d which in turn implies that μ ˜ M · ≤ 2d hence μ ˜ M · ≤ 2d where ⊂ Xis the 1 1 1 birational transform of . This is possible only if μ ˜ is bounded from above since M is 1 1 big and Cartier and hence M · ≥ 1. If at the end of the process μ ˜ = 0for every j ≥ 2, then the above arguments show that r is indeed bounded by some number p.But if μ ˜ > 0for some j ≥ 2, then we replace Mwith μ ˜ M and replace P with P +˜ μ M where l is as above, and rearrange the j j l j=l l indexes. We can then assume that (X , B + M ) is generalized lc but not generalized klt. Step 4. The arguments of Step 3 show that, after replacing X, we can assume that there is a prime divisor S on X exceptional over X whose generalized log discrepancy with respect to (X , B + M ) is 0. Since (X , 0) is Q-factorial klt, by Lemma 4.6,there is an extremal contraction φ : X → X which extracts S , the birational transform of S. We can write K + B + M = φ K + B + M X X where B is the sum of S and the birational transform of B and M is the pushdown of M. Since ρ(X ) = 1and φ is extremal, ρ(X ) = 2. Moreover, K + B + M + P ≡ 0 where P is the pullback of P on X . Since ρ(X ) = 1, P and so P is semi-ample, hence we may assume that (X , B + P + M ) is generalized lc with boundary part B + P and nef part M. Since S is a component of B , (X , B − δS + P + M ) is generalized lc where δ> 0 is small, and −δS ≡ K + B − δS + P + M . So by Lemma 4.4(1), we can run an LMMP on −S which ends up with a Mori ﬁbre space X → T . Note that by construction X has Picard number one or two: in any case one of the extremal rays of X corresponds to the Fano contraction X → T and S is positive on this ray. We may assume that both g : X X and h : X X are morphisms. Step 5. Consider the case dim T > 0. Then the Picard number ρ(X ) = 2, hence X X is an isomorphism in codimension one. Moreover, by restricting to the general ﬁbres of X → T and applying induction we may assume M ≡ 0/T for all but a bounded number of j.For anysuch j,M is not big, hence M is not big too. Thus M j j j is ample/X otherwise M would be the pullback of M which is big, a contradiction. j j Let C := φ(S ) and let x be the generic point of C .Then M ≡ 0/T implies that M is not numerically trivial over any neighborhood of x . Now apply Proposition 5.3 to j EFFECTIVITY OF IITAKA FIBRATIONS 315 (X , B + P + M ) at x to bound the number of such j . Therefore r is indeed bounded by some number p depending only on d, b. Step 6. We can now assume dim T = 0. Let X → X be the last step of the ˜ ˜ LMMP which contracts some divisor R .Let x be the generic point of the image of R . ˜ ˜ For each j , either M is ample over X or M is ample over X where M is the push- j j j down of M via X X which we can assume to be a morphism. So either M is j j not numerically trivial over any neighborhood of x or that it is not numerically triv- ial over any neighborhood of x . Now apply Proposition 5.3 to (X , B + P + M ) and (X , B + P + M ) at x and x to bound r by some number p depending only on d, b. 6. ACC for generalized lc thresholds In this section, we reduce the ACC for generalized lc thresholds (Theorem 1.5)to the Global ACC (Theorem 1.6) in lower dimension by adapting a standard argument due to Shokurov. We create an appropriate generalized lc centre of codimension one and restrict to it to do induction. Proposition 6.1. — Assume that Theorem 1.6 holds in dimension ≤ d − 1. Then Theorem 1.5 holds in dimension d . Proof. — Applying induction we may assume that Theorem 1.5 holds in dimension ≤ d − 1. If Theorem 1.5 does not hold in dimension d , then there exist a sequence of generalized lc polarized pairs (X , B + M ) of dimension d with data X → X → Z i i i i i i and M = μ M , and divisors D and N = ν N satisfying the assumptions of i j,i j,i i k,i k,i the theorem but such that the generalized lc thresholds t of D + N with respect to i i (X , B + M ) form a strictly increasing sequence of numbers. We may assume that 0 < i i i t < ∞ for every i. Since the problem is local over X , we can assume X → Z is the i i i i identity morphism. Moreover, we can discard any μ and ν if they are zero. j,i k,i By deﬁnition, X , B + t D + M + t N i i i i i i i is generalized lc with boundary part B + t D and nef part M + t N but i i i i i i X , B + a D + M + a N i i i i i i i is not generalized lc for any a > t . i i If B = B + t D for inﬁnitely many i, then we can easily get a contradiction i i i as the t can be calculated in terms of the coefﬁcients of B and D .Thuswemay assume i i 316 CAUCHER BIRKAR, DE-QI ZHANG that B = B + t D for every i. In particular, this means that there is a generalized lc i i i centre of X , B + t D + M + t N i i i i i i i of codimension ≥ 2 which is not a generalized lc centre of (X , B + M ). i i i We may assume that the given morphism f : X → X is a log resolution of i i (X , B + t D ).Let := B + t D and let R := M + t N . We can write i i i i i i i i i i i i K + + R = f K + + R + E X i i X i i i i where is the sum of the birational transform of and the reduced exceptional divisor of f ,and E ≥ 0is exceptional/X . Then the pair (X , ) is lc but not klt; more precisely i i i i there is a component of which is not a component of E ; moreover, there is such i i a component which is exceptional/X by the last paragraph. In addition, the set of the coefﬁcients of all the union with {μ , t ν } satisﬁes the DCC by Lemma 2.1. i j,i i k,i Run an LMMP/X on K + + R with scaling of some ample divisor which is X i i i i also an LMMP/X on E . Since E is effective and exceptional/X , the LMMP ends on a i i i i model X on which E = 0 (as in the proof of Lemma 4.5). In particular, i i K + + R ≡ 0/X . i i i Let S be a component of exceptional/X but not a component of E . Since i i i the LMMP only contracts components of E , this S is not contracted/X .Deﬁne by i i S the generalized adjunction K + + R = K + + R . S S S X i i i i i i S Then the set of the coefﬁcients of all the satisﬁes DCC by Proposition 4.9.Bycon- struction K + + R ≡ 0/X . S S S i i i Let S → V be the contraction given by the Stein factorization of S → X and let F be i i i i i a general ﬁbre of S → V . We can write i i K + + R = (K + + R )| ≡ 0 F F F S S S F i i i i i i i as in Remark 4.2 (6): here = | and R = R | is the pushdown of R | = F S F F S F i F i i i i i i (M + t N )| where F is the ﬁbre of S → V corresponding to F . i i i F i i i i i Suppose that we can choose the S such that (∗) the set of the coefﬁcients of all the together with {μ | M | ≡ 0}∪ F j,i j,i F {t ν | N | ≡ 0} does not satisfy ACC. i k,i k,i F i EFFECTIVITY OF IITAKA FIBRATIONS 317 But then (∗) contradicts Theorem 1.6.Soitisenoughtoﬁnd the S so that (∗) holds. We will show that (∗) holds if for each i we can ﬁnd S satisfying: (∗∗)(D + N )| is not numerically trivial i i where D on X is the birational transform of D and D is the pushdown of D ;here i i i i i we can assume g : X X is a morphism. Indeed, let B be the sum of the birational i i i transform of B plus the reduced exceptional divisor of f ,and B its pushdown on X .By i i i generalized adjunction we can write K + B + M = K + B + M . S S S X i i i i i i S Write g (N ) = N + Q .Then N | = N + Q where N is the pushdown of N | i i F F F F i F i i i i i i i i and Q is the pushdown of Q | .If N | ≡ 0for every i,then (∗) is satisﬁed. So we can F i F i F i i assume N | ≡ 0for every i,hence by (∗∗) we have i F D + N ≡ D + Q = 0 F F i i i i for every i where D := D | .But now = B + t D and = B + t (D + Q ) F F i S S i S S i i i i i i i i i i where D := D | and Q is the pushdown of Q | . Moreover, since D + Q = 0 S S S i S S S i i i i i i i near F ,Proposition 4.9 and its proof show that the set of the coefﬁcients of all the near F does not satisfy ACC. Thus the set of the coefﬁcients of all the does not satisﬁes ACC, hence (∗) holds. Finally we show that (∗∗) holds. By the negativity lemma, we can write f D + N = D + N + P i i i i i i where P ≥ 0is exceptional/X . By the deﬁnition of t and the assumption B i i i i B + t D , there is a component of P which is a component of but not a compo- i i i i i nent of E .Infactany componentof P not contracted/X , is of this kind. Since P = 0 i i i i is exceptional/X , by the negativity lemma [3, Lemma 3.6.2], there is a component S i i of P with a covering family of curves C (contracted over X )suchthat P · C < 0. So i i i (D + N ) · C > 0 for such curves C, hence (D + N )| is not numerically trivial over i i i i general points of V which implies that we can choose the S so that (∗∗) holds. 7. Global ACC In this section, we show that Global ACC (Theorem 1.6) in dimension < d and ACC for generalized lc thresholds (Theorem 1.5 ) in dimension d together imply Global ACC in dimension d . We ﬁrst deal with the pairs which are generalized lc but not gen- eralized klt. For the general case, we will use Proposition 3.4 and do induction on the number of summands in the nef part of the pair, as illustrated in the introduction. The starting point of the induction is the important result [10, Theorem 1.5] which proves the statement when the nef part is zero. 318 CAUCHER BIRKAR, DE-QI ZHANG Proposition 7.1. — Assume Theorem 1.6 holds in dimension ≤ d − 1. Then Theorem 1.6 holds in dimension d for those (X , B + M ) which are not generalized klt. Proof.— Step 1. Extending we can assume 1 ∈ . If the proposition does not hold, then there is a sequence of generalized lc but not klt polarized pairs (X , B + M ) i i i with data X → X → Z and M = μ M satisfying the assumptions of 1.6 but such i i i j,i j,i that the set of the coefﬁcients of all the B together with the μ does not satisfy ACC. j,i We may assume that f : X → X is a log resolution of (X , B ).Let B be the sum i i i i i i of the birational transform of B and the reduced exceptional divisor of f . We can write K + B + M = f K + B + M + E X i i X i i i i i where E ≥ 0is exceptional/X . We can run an LMMP/X on K + B + M with scaling i X i i i i i of some ample divisor which contracts E and terminates with some model (as in the proof of Lemma 4.5). Moreover, by the generalized non-klt assumption, we can choose f so that there is a prime divisor S on X which is a component of B but not a component of i i i E , hence it is not contracted by the LMMP. Replacing X with the model given by the LMMP allows us to assume that (X , B ) is Q-factorial dlt and that we have a component i i S of B i i Step 2. Write B = b B where B are the distinct irreducible components of k,i i k,i k,i B . If the set of all the coefﬁcients b is not ﬁnite, then we may assume that the b form k,i 1,i a strictly increasing sequence in which case we let P := B . On the other hand, if the i 1,i set of all the coefﬁcients b is ﬁnite, then the set of all the μ is not ﬁnite hence we could k,i j,i assume that the μ form a strictly increasing sequence in which case we let P := M . 1,i i 1,i In either case we can run an LMMP on K + B + M − P ≡− P i i i i for some small > 0 which ends with a Mori ﬁbre space, by Lemma 4.4(1). The gen- eralized lc (and non-klt) property of (X , B + M ) is preserved by the LMMP because i i i K + B + M ≡ 0. i i Step 3. We ﬁrst consider the case when S is not contracted by the LMMP in Step 2, for inﬁnitely many i. Replacing the sequence we can assume this holds for every i. In this case, we replace X with the Mori ﬁbre space constructed, hence we can assume we already have a Mori ﬁbre structure X → T and that P is ample/T .Let F be a general i i i i i ﬁbre of X → T . Then we can write i i K + B + M = K + B + M F F F X i i i i i i F where B = B | and M = M | . Moreover, since P | is ample, the set of the coefﬁ- F F F F F i i i i i i i i cients of all the B together with the set {μ | M | ≡ 0} is not ﬁnite where F is the F j,i j,i F i i EFFECTIVITY OF IITAKA FIBRATIONS 319 ﬁbre of X → T corresponding to F . So applying induction we can assume dim T = 0. i i i In particular, P | is not numerically trivial. Now assume the LMMP of Step 2 contracts S at some step, for inﬁnitely many i. Replacing the sequence we can assume this holds for every i. Replacing X we can assume S is contracted by the ﬁrst step of the LMMP, say X → X .Then P is ample over X , i i i i i hence P | is not numerically trivial. From now on we assume that P | is not numerically trivial. Step 4. Apply generalized adjunction to get K + B + M = K + B + M ≡ 0. S S S X i i i i i i S By Proposition 4.9, the coefﬁcients of B belong to a DCC set depending only on d and .Moreover, M is the pushdown of M| = μ M | . Thus by induction the set S S j,i j,i S i i of the coefﬁcients of all the B together with the set {μ | M | ≡ 0} is ﬁnite. But this S j,i j,i S contradicts Lemma 4.10. Proposition 7.2. — Assume that Theorem 1.6 holds in dimension ≤ d − 1 and that Theorem 1.5 holds in dimension d . Then Theorem 1.6 holds in dimension d . Proof.— Step 1. If the statement is not true, then there is a sequence of generalized lc polarized pairs (X , B + M ) with data X → X → Z and M = μ M satisfying i i i j,i j,i i i i i the assumptions of 1.6 but such that the set of the coefﬁcients of all the B and all the μ j,i put together satisﬁes DCC but not ACC. Write B = b B where B are the distinct k,i i k,i k,i irreducible components of B . As in Steps 1 and 2 of the proof of Proposition 7.1, we can reduce the problem to the situation in which X is a Q-factorial klt variety with a Mori ﬁbre space structure. Restricting to the general ﬁbres of the ﬁbration and applying induction we can in addition assume X is Fano of Picard number one. For each i,let σ(M ) be the number of the μ . Then, by Propositions 5.1 and 5.2, i j,i we can assume that the number of the components of B plus σ(M ) is bounded. Thus we can assume that the number of the components of B and σ(M ) are both independent of i. We just write σ instead of σ(M ). We will do induction on the number σ.By[10, Theorem 1.5], the proposition holds when σ = 0, i.e. when M = 0for every i. So we can assume σ> 0. We may also assume that σ is minimal with respect to all sequences as above, even if is extended to a larger set. Replacing the sequence we may assume that the numbers b and μ form a (not k,i j,i necessarily strict) increasing sequence for each k and each j , because they all belong to the DCC set . By deﬁnition, b ≤ 1. We show that the μ are also bounded from above, k,i j,i i.e. lim μ < +∞ for every j : this follows from the same arguments as in Step 3 of the i j,i proof of Proposition 5.2 by considering the generalized lc threshold of M with respect to j,i 320 CAUCHER BIRKAR, DE-QI ZHANG (X , B + M − μ M ) if M ≡ 0/X for inﬁnitely many i, or by applying boundedness j,i j,i i i i j,i i of the length of extremal rays otherwise. Step 2.ByProposition 7.1, we may assume that (X , B + M ) is generalized klt for i i i every i. In particular, (X , B + (1 + )M ) is generalized klt, and K + B + (1 + )M is i X i i i i i i ample for some small > 0, noting that the Picard number ρ(X ) = 1. We may assume that f : X → X is a log resolution of (X , B ) and can write i i i i i K + B + (1 + )M = f K + B + (1 + )M + E X i i i X i i i i i i where B is the sum of the birational transform of B and the reduced exceptional divisor of f ,and E ≥ 0is exceptional/X .So K + B + (1 + )M is big, and since the μ i i X i i i j,i i i are bounded from above, we deduce that K + B + nM is also big for some ﬁxed X i j,i natural number n 1 independent of i. Now by Proposition 3.4, there exists a natural number m, independent of i,such that |m(K + B + nM )| deﬁnes a birational map for every i. In particular, X i j,i H X , m K + B + nM = 0 i X i j,i so m K + B + nM ∼ mD X i j,i i for some integral divisor mD ≥ 0. The coefﬁcients of mD belong to N,aDCCset.Now i i let D be the R-divisor so that mD is the sum of mD and the fractional part m(K + i i i X B + nM ). Since K + nM is Cartier, mD = mD +mB . On the other hand, i j,i X j,i i i i since the coefﬁcients of B belong to the DCC set ∩[0, 1], the coefﬁcients of mB i i belong to a DCC set as well by Lemma 2.1. Therefore, the coefﬁcients of mD and hence of D belong to a DCC set, depending only on . By extending we can assume that the coefﬁcients of D belong to . By construction, 0 ≤ D ∼ K + B + nM i R X i j,i which in turn implies that 0 ≤ D ∼ K + B + nM R X i i j,i = K + B + M + (n − μ )M ≡ (n − μ )M . X j,i j,i i i j,i j,i Note that we can assume n − μ > 0for every j, i. j,i EFFECTIVITY OF IITAKA FIBRATIONS 321 Step 3. By Lemma 2.2, replacing the sequence X and reordering the indexes j , we may assume that M ≡ λ M so that for each j the numbers λ form a decreasing j,i j,i j,i 1,i sequence. By Step 2, we get D ≡ (n − μ )M ≡ (n − μ )λ M =: ρ M j,i j,i j,i i i j,i 1,i 1,i where we have deﬁned ρ := (n − μ )λ . i j,i j,i For each j,the numbers n − μ and λ form decreasing sequences hence the ρ also form j,i j,i i a decreasing sequence by Lemma 2.1. Now let N := μ M and let u be the generalized lc threshold of D with i j,i j,i i j≥2 i respect to (X , B + N ). Since D ≡ ρ M we get i i i i 1,i K + B + N + u D ≡ K + B + N + u ρ M . X i X i i i i i i i 1,i i i Assume that u ρ ≥ μ for every i.Let v ≤ u be the number so that i i 1,i i i K + B + N + v D ≡ K + B + M ≡ 0 X i X i i i i i i i 1,i that is, v = .Asthe μ form an increasing sequence and the ρ form a decreasing i 1,i i sequence, the v form an increasing sequence. Moreover, if the μ form a strictly in- i 1,i creasing sequence, then the v also form a strictly increasing sequence. Thus the set of the coefﬁcients of all the B + v D together with the {μ | j ≥ 2} is a DCC set but not i j,i i i ACC. Now (X , B + v D + N ) is generalized lc with boundary part B + v D and nef i i i i i i i i part N ,and σ(N )<σ which contradicts the minimality assumption on σ in Step 1. i i Therefore, from now on we may assume that u ρ <μ for every i. i i 1,i 1,i Step 4.Fix i.Let be the set of those elements (α, β) ∈[0, ]×[0,μ ] such i 1,i that K + B + N + αD + β M ≡ K + B + M X X i i i 1,i i i i i which is equivalent to αρ + β = μ .Notethat (0,μ ) ∈ hence = ∅.Now let i 1,i 1,i i i s = sup α | (α, β) ∈ , X , B + αD + N + β M is generalized lc i i i i i i 1,i where the pair in the deﬁnition has boundary part B + αD and nef part N + β M . i 1,i i i Letting t = μ − s ρ we get (s , t ) ∈ . i 1,i i i i i i We show that s is actually a maximum hence in particular X , B + s D + N + t M i i i i i i 1,i 322 CAUCHER BIRKAR, DE-QI ZHANG l l l is generalized lc. If not, then there is a sequence (α ,β ) ∈ such that the α form a strictly increasing sequence approaching s and the β form a strictly decreasing sequence approaching t . Since X , B + α D + N + t M i i i i 1,i is generalized lc, the generalized lc threshold of D with respect to (X , B + N + t M ) i i i i 1,i is at least lim α = s by Theorem 1.5.So X , B + s D + N + t M i i i i i i 1,i is also generalized lc. Hence s is indeed a maximum. Note that s ≤ u . i i i Step 5. Since the coefﬁcients of D belong to and since u is the generalized lc threshold of D with respect to (X , B + N ), u is bounded from above by Theorem 1.5. i i i i Thus s is also bounded from above. So we may assume the s and the t each form an i i i increasing or a decreasing sequence hence s = lim s and t = lim t exist. Since the μ i i 1,i form an increasing sequence and the ρ form a decreasing sequence, the s or the t form i i i an increasing sequence. We will show that in fact the t form an increasing sequence. Assume otherwise, that is, assume the t form a decreasing sequence. We can as- sume it is strictly decreasing. Then the s form a strictly increasing sequence. Since X , B + s D + N + tM i i i i 1,i is generalized lc, we may assume that X , B + sD + N + tM i i i i 1,i is generalized lc too, by Theorem 1.5.Now we canﬁnd ˜ s > s such that (˜ s , t) ∈ ,that i i i i is, ˜ s ρ + t = μ . Since the μ form an increasing sequence and the ρ form a decreasing i i 1,i 1,i i sequence, the ˜ s form an increasing sequence. Moreover, since t < t ≤ μ ≤ lim μ and s(lim ρ ) + t = lim(s ρ + t ) = lim μ i 1,i 1,i i i i i 1,i we deduce lim ρ > 0. Thus as lim(˜ s ρ + t) = lim μ , i i 1,i we get lim ˜ s = lim s = s. In particular this means s ≥˜ s > s ,hence i i i i X , B +˜ s D + N + tM i i i i 1,i is generalized lc which contradicts the maximality assumption of s in Step 4. So we have proved that the t form an increasing sequence. Now by deﬁnition s is i i the generalized lc threshold of D with respect to X , B + N + t M . i i i 1,i EFFECTIVITY OF IITAKA FIBRATIONS 323 So they form a decreasing sequence by Theorem 1.5. Step 6. The purpose of this step is to modify B so that we can assume s = lim s = 0. ˜ ˜ ˜ ˜ Let t be the number so that sρ + t = μ .As s ≥ s, t ≥ t ≥ 0, hence (s, t ) ∈ . Since i i i 1,i i i i i i the μ (resp. ρ ) form an increasing (resp. decreasing) sequence, the t form an increasing 1,i i i sequence. Moreover, lim t = lim(μ − sρ ) = lim(μ − s ρ ) = lim t = t i 1,i i 1,i i i i which implies t ≤ t. We claim that (∗) X , B + sD + N + t M i i i i 1,i is generalized lc. Indeed, let c be the generalized lc threshold of M with respect to 1,i (X , B + sD + N ).Then c ≥ t and by Theorem 1.5, we may assume that the c form a i i i i i i i decreasing sequence. Thus c ≥ lim c ≥ lim t = t ≥ t i i i i and the claim follows. Nowwedeﬁne theboundary C := B + sD on X where B ,asinStep2,isthe i i i i sum of the birational transform of B and the reduced exceptional divisor of X → X , i i and D is the birational transform of D .Then C = B + sD and i i i i i X , C + N + t M i i i 1,i is generalized lc by (∗),and K + C + N + t M ≡ 0. X i i i 1,i Moreover, the set of the coefﬁcients of all the C union the set {μ | j ≥ 2}∪{t } satisﬁes j,i i DCC but not ACC (note that if the μ form a strictly increasing sequence, then so do 1,i the t ). s −s On the other hand, let G := D + sD and let r := .Then i i i 1+s 0 ≤ G ∼ K + C + nM i R X i j,i and G = (1 + s)D ,and i i K + C + r G + N + t M = K + B + s D + N + t M ≡ 0. X i i X i i i i i 1,i i i i 1,i i i The equality also shows X , C + r G + N + t M i i i i i i 1,i 324 CAUCHER BIRKAR, DE-QI ZHANG is generalized lc and that r is the generalized lc threshold of G with respect to (X , C + i i i N + t M ). Therefore extending , replacing B with C , replacing μ with t , replacing i i i 1,i i i 1,i D with G , and replacing s with r allow us to assume that s = lim s = 0. i i i i i Step 7. After replacing X we may assume that there is a prime divisor S on X i i i whose generalized log discrepancy with respect to the generalized lc polarized pair X , B + s D + N + t M i i i i i i 1,i is 0: this follows from our choice of s , t . i i First assume that S is not contracted over X for every i which means that S is i i a component of B + s D .Let d be the coefﬁcient of S in D and let p be the real i i i i i i i number such that K + B + s d S + N + p M ≡ 0. X i i i i i i 1,i Obviously p ≤ μ , and equality holds if and only if s d S ≡ 0, i.e., s d = 0. Since s d S ≤ i 1,i i i i i i i s D and K + B + s D + N + t M ≡ 0 X i i i i i 1,i we have t ≤ p . Then from lim s = 0and μ := lim μ = lim t we arrive at lim p = μ . i i i 1 1,i i i 1 So we may assume that the p form an increasing sequence approaching μ . i 1 Let w be the generalized lc threshold of M with respect to 1,i X , B + s d S + N . i i i i i i Then w ≥ t . Applying Theorem 1.5, we can assume that the w form a decreasing i i i sequence. Then w ≥ lim w ≥ lim t = μ = lim p ≥ p i i i 1 i i which implies that X , B + s d S + N + p M i i i i i i i 1,i is generalized lc with boundary part := B + s d S and nef part R := N + p M . i i i i i 1,i i i i The set of the coefﬁcients of all the union the set {μ | j ≥ 2}∪{p } satisﬁes DCC. j,i i Therefore, by Proposition 7.1, we may assume that p is a constant independent of i. Now μ = lim p = p ≤ μ ≤ lim μ = μ 1 i i 1,i 1,i 1 Thus p = μ ,hence s d = 0, = B ,and R = M .Inother words, (X , B + M ) is not i 1,i i i i i i i i i i generalized klt. This contradicts Proposition 7.1. EFFECTIVITY OF IITAKA FIBRATIONS 325 So after replacing the sequence we may assume that S is exceptional over X for every i. Step 8. By Lemma 4.6, there is an extremal contraction g : X → X extracting S i i i with X being Q-factorial. We can assume X X is a morphism. We can write i i K + B + s D + N + t M = g K + B + s D + N + t M ≡ 0 X i i X i i i i i 1,i i i i i 1,i i i where B is the pushdown of B , D is the birational transform of D ,M is the pushdown i i i 1,i of M ,and N is the pushdown of N .Now S is a component of B . By Lemma 4.4(1) 1,i i i i i we can run the −S -LMMP which terminates on some Mori ﬁbre space X → T .We i i i may assume that dim T = 0for every i,or dim T > 0for every i. Replacing X we may i i assume X X is a log resolution of (X , B + s D ). i i i i i i Since (X , B + M ) is generalized lc and K + B + M ≡ 0, we deduce that K + X X i i i i i i B + M is pseudo-effective. Thus K + B + M is pseudo-effective too. Moreover, by i i X i i construction K + B + s D + N + t M ≡ 0. X i i i i i 1,i So there is the largest number q ∈[t ,μ ] such that i i 1,i K + B + N + q M ≡ 0/T . X i i i 1,i i From s = lim s = 0we get lim t = lim μ = μ from which we derive lim q = μ .Sowe i i 1,i 1 i 1 may assume that the q form an increasing sequence approaching μ .Let w be the gener- i 1 i alized lc threshold of M with respect to the generalized lc polarized pair (X , B + N ). 1,i i i i Then w ≥ t as i i X , B + s D + N + t M i i i i i i 1,i is generalized lc. Moreover, by Theorem 1.5 we can assume the w form a decreasing sequence, hence q ≤ μ ≤ μ = lim t ≤ lim w ≤ w . i 1,i 1 i i i So the pair (X , B + N + q M ) is generalized lc. But the pair is not generalized klt i i i 1,i because S is a component of B i i Step 9. Assume that dim T = 0for every i. Applying Proposition 7.1,wecan assume that the set of the coefﬁcients of all the B union the set {μ |j ≥ 2}∪{q } is ﬁnite. j,i i In particular, this means we can assume q = μ = μ for every i,and that μ = μ for i 1,i 1 j,i j every j, i where μ := lim μ . On the other hand, assume that dim T > 0for every i. j i j,i If M ≡ 0/T ,then q = μ .But if M ≡ 0/T , then by restricting to the general i 1,i 1,i i 1,i i ﬁbres of X → T and applying induction, we deduce that {q } is ﬁnite, hence q = μ i i 1 i i for i 1; so we can assume q = μ = μ . Moreover, by restricting to the general ﬁbres i 1,i 1 326 CAUCHER BIRKAR, DE-QI ZHANG of X → T and applying induction once more, we may assume that the set of the i i horizontal/T coefﬁcients of all the B together with the set {μ | M ≡ 0/T } is ﬁnite. j,i i i j,i i The last paragraph shows that in either case dim T = 0ordimT > 0, we can i i assume (∗∗) K + B + M ≡ 0/T . i i i Let B be obtained from B by replacing the coefﬁcient b with b := lim b .Let M be i i k,i k i k,i i obtained from M by replacing μ with μ = lim μ .Then K + B + M is ample be- i j,i j i j,i X i i cause ρ(X ) = 1 and because either b < b for some k or μ <μ for some j.Moreover, k,i k j,i j by Theorem 1.5, we can assume (X , B + M ) is generalized lc. Thus i i i K + B + M ≥ f K + B + M X i i X i i i i is big. This in turn implies that K + B + M is big too. On the other hand, by the i i last paragraph, we may assume that on the general ﬁbres F of X → T we have: i i i B | = B | and M | ≡ M | . This contradicts (∗∗). F F F F i i i i i i i i 8. Proof of main results In this section, we prove our main results stated in the introduction. Proof of Theorem 1.5 and Theorem 1.6. — By Proposition 6.1,Theorem 1.6 in dimen- sion < d implies Theorem 1.5 in dimension d . On the other hand, by Proposition 7.2, Theorem 1.6 in dimension < d and Theorem 1.5 in dimension d imply Theorem 1.6 in dimension d . So both theorems follow inductively the case d = 1 being trivial. Next we prove a result bounding pseudo-effective thresholds which will be needed for the proof of Theorem 1.3. Theorem 8.1. — Let d be a natural number and a DCC set of nonnegative real numbers. Then there is a real number e ∈ (0, 1) depending only on ,dsuch that if: • (X, B) is projective lc of dimension d , • M = μ M where M are nef Cartier divisors, j j j • the coefﬁcients of B and the μ are in ,and • K + B + M is a big divisor, then K + eB + eM is a big divisor. Proof. — It sufﬁces to show the assertion: there is an e ∈ (0, 1) depending only on , d such that K + eB + eM is pseudo-effective; because then 1 1 vol K + (e + 1)(B + M) = vol (K + B + M + K + eB + eM) X X X 2 2 EFFECTIVITY OF IITAKA FIBRATIONS 327 ≥ vol (K + B + M) > 0 and hence K + e B + e Mis big for e := (1 + e) ∈ (0, 1). If there is no e as in the last paragraph, then there is a sequence of pairs (X , B ) i i and divisors M = μ M satisfying the assumptions of the theorem but such that the i j,i j,i pseudo-effective thresholds e of B + M form a strictly increasing sequence approach- i i i ing 1: by deﬁnition K + e B + e M is pseudo-effective but K + c B + c M is not X i i i i X i i i i i i pseudo-effective for any c < e . i i We can extend and replace the X , B so that we may assume (X , B ) is log i i i i smooth klt. By Lemma 4.4(2), we can run an LMMP on K + e B + e M which ends X i i i i with a minimal model X on which K + e B + e M is semi-ample deﬁning a contraction X i i i i i X → T . Since K + B + M is big and K + e B + e M ≡ 0/T ,wededucethat B + M X X i i i i i i i i i i i i i is big over T . Replacing X we may assume that X X is a log resolution of (X , B ).Let F i i i i i i be a general ﬁbre of X → T and F the corresponding ﬁbre of X → T . By restricting i i i i i to F we get K + e B + e M := K + e B + e M | ≡ 0. F i F i F X i i F i i i i i i i This contradicts Theorem 1.6 because e B + e M is big hence nonzero for every i, i F i F i i so the set of the coefﬁcients of all the e B union with the set {e μ | M | ≡ 0} is not i F i j,i j,i F ﬁnite. Proof of Theorem 1.3. — As usual by taking a log resolution we may assume (X, B) is log smooth. By Theorem 8.1, there exist a rational number e ∈ (0, 1) depending only on , d, r such that K + eB + eMis big, soK + eB + Mis also big. As in Step 2 of X X the proof of Proposition 3.4, there is p ∈ N depending only on e,, r such that r|p and for any nonzero λ ∈ we can ﬁnd γ ∈[eλ, λ) such that pγ is an integer. In particular, we can ﬁnd a boundary such that eB ≤ ≤ B, p is Cartier, K + + M is big, and (X,) is klt. Replacing B with we canthenassume ={ | 0 ≤ i ≤ p − 1} and that (X, B) is klt. By Proposition 3.4, there exist l, n ∈ N depending only on , d, r such that r|n and that |l(K + B + nM)| deﬁnes a birational map. By replacing l with pl we can assume p|l . There is a resolution φ : W → Xsuch that φ l(K + B + nM) ∼ H + G where H is big and base point free and G ≥ 0. Perhaps after replacing l with (2d + 1)l , we can also assume that H is potentially birational [9, Lemma 2.3.4]. Applying Theorem 8.1 once more, there exist rational numbers s, u ∈ (0, 1) de- pending only on , d, r such that K + sB + uM is big. Perhaps after replacing s, u,we X 328 CAUCHER BIRKAR, DE-QI ZHANG can choose a sufﬁciently large natural number q so that qs is integral and divisible by p, qu is integral and divisible by r, qs + l qu + ln s := < 1, and = 1. q + l + 1 q + l + 1 Let X be a minimal model of K + sB + uM, which exists by Lemma 4.4(2). We can assume that the induced map ψ : W X is a morphism. Since X is a minimal model, ∗ ∗ φ (K + sB + uM) = ψ K + sB + uM + E X X where E is effective. Let D = ψ K + sB + uM . Since H is potentially birational, by Lemma 3.1, qD + H is potentially birational and |K +qD + H| deﬁnes a birational map. Thus ∗ ∗ K + φ q(K + sB + uM) + φ l(K + B + nM) W X X also deﬁnes a birational map which in turn implies that K + q(K + sB + uM) + l(K + B + nM) X X X deﬁnes a birational map. Hence the linear system (q + l + 1) K + s B + M deﬁnes a birational map. Therefore (q + l + 1)(K + B + M) also deﬁnes a birational map. By construction r|qu and r|ln,so r|(q + l + 1 = qu + ln). Now put a := m(, d, r) := q + l + 1. Then aMis Cartier, and for any b ∈ N, the linear system |b a(K + B + M) deﬁnes a birational map. But since aM is Cartier and B is effective, b a(K + B + M) ≤ ba(K + B + M) X X which means |m(K + B + M)| also deﬁnes a birational map where m = ba. Next we prove a result similar to 1.3 but we allow a more general nef part M. This result is not used elsewhere in this paper. Theorem 8.2. — Let d be a natural number and a DCC set of nonnegative real numbers. Then there is a natural number m depending only on , d such that if: EFFECTIVITY OF IITAKA FIBRATIONS 329 • (X, B) is projective lc of dimension d , • M = μ M where M are nef Cartier divisors, j j j • the coefﬁcients of B and the μ are in ,and • K + B + M is big, then the linear system | m(K + B) + mμ M | deﬁnes a birational map. X j j Proof. — As usual we may assume (X, B) is log smooth. By Theorem 8.1,there exists a rational number e ∈ (0, 1) depending only on , d such that K + eB + eMis pseudo-effective. As in the proof of 1.3, there is p ∈ N depending only on e, such that we can ﬁnd a boundary ≤ Band numbers ν ∈[eμ ,μ ] such that p and pNare j j j Cartier divisors and K + + Nis big whereN = ν M . X j j Applying Theorem 1.3, there is l ∈ N depending only on p, d (hence only on , d ) such that |l(K + + N)| deﬁnes a birational map and p|l . Replacing l by a multiple we caninaddition assume that l(K + + N) is potentially birational. Then by Lemma 3.1, l(K + + N) + α M X j j is potentially birational for any 0 ≤ α ∈ Z,and K + l(K + + N) + α M X X j j deﬁnes a birational map. Since ν ≤ μ ,wecan take α so that lν + α = (l + 1)μ j j j j j j Therefore (l + 1)K + l + (l + 1)μ M X j j deﬁnes a birational map which in turn implies that (l + 1)(K + B) + (l + 1)μ M X j j deﬁnes a birational map because l ≤ (l + 1)B . Now put m = l + 1. Proof of Theorem 1.2. — Replacing W we can assume the Iitaka ﬁbration I : W X is a morphism, i.e. can assume V = W using the notation before Theorem 1.2. Also we can assume κ(W) ≥ 1 otherwise there is nothing to prove. Let b := b and β := β .Let F F N = N(β) = lcm m ∈ N | ϕ(m) ≤ β where ϕ denotes Euler’s ϕ-function. Let bNu − v A(b, N) := u,v ∈ N,v ≤ bN bNu 330 CAUCHER BIRKAR, DE-QI ZHANG which is a DCC subset of the interval [0, 1). By theresultsof[7] (which is summarized in [24, Lemma 1.2]), replacing W and X by high enough resolutions, we may assume that X is smooth and that there exist a boundary B on X (the discriminant part of I : W → X) and a nef Q-divisor M (the moduli part of I : W → X) such that –NbMis Cartier, – B has simple normal crossing support with coefﬁcients in A(b, N), –K + B + Mis big, – wehaveisomorphisms 0 0 H (W, mbK ) H X, mb(K + B + M) W X for every m ∈ N,and – the rational map deﬁned by |mbK | is birational to the Iitaka ﬁbration I : W → X if and only if |mb(K + B + M)| gives rise to a birational map. By letting = A(b, N) and r = Nb, and applying Theorem 1.3, there is a con- stant m(, d, r) depending only on , d, r, (hence depending only on d, b,β)suchthat |m(K + B + M)| deﬁnes a birational map for any m ∈ N divisible by m(, d, r).Now simply let m(d, b ,β ) = bm(, d, r). Acknowledgements The ﬁrst author was partially supported by a grant of the Leverhulme Trust. Part of this work was done when the ﬁrst author visited National University of Singapore in April 2014. Part of this work was done when the ﬁrst author visited National Taiwan University in August-September 2014 with the support of the Mathematics Division (Taipei Ofﬁce) of the National Center for Theoretical Sciences. The visit was arranged by Jungkai A. Chen. He wishes to thank them all. The second author was partially supported by an ARF of National University of Singapore. The authors would like to thank the referee for the very useful corrections and suggestions which helped to simplify and clarify some of the proofs. REFERENCES 1. C. BIRKAR, On existence of log minimal models, Compos. Math., 145 (2009), 1442–1446. 2. C. BIRKAR, Existence of log canonical ﬂips and a special LMMP, Publ. Math. Inst. Hautes Études Sci., 115 (2012), 325– 3. C. BIRKAR,P.CASCINI,C.HACON and J. M KERNAN, Existence of minimal models for varieties of log general type, J. Am. Math. Soc., 23 (2010), 405–468. 4. C. BIRKAR and Z. HU, Log canonical pairs with good augmented base loci, Compos. Math., 150 (2014), 579–592. 5. G. Di CERBO, Uniform bounds for the Iitaka ﬁbration, Ann. Sc.Norm. Super. Pisa Cl.Sci.(5), 13 (2014), 1133–1143. EFFECTIVITY OF IITAKA FIBRATIONS 331 6. J. CHEN and M. CHEN, Explicit birational geometry of threefolds of general type, I, Ann. Sci. Éc. Norm. Super., 43 (2010), 365–394. 7. O. FUJINO and S. MORI, A canonical bundle formula, J. Differ. Geom., 56 (2000), 167–188. 8. C. HACON and J. M KERNAN, Boundedness of pluricanonical maps of varieties of general type, Invent. Math., 166 (2006), 1–25. 9. C. D. HACON,J.M KERNAN and C. XU, On the birational automorphisms of varieties of general type, Ann. Math. (2), 177 (2013), 1077–1111. 10. C. D. HACON,J.M KERNAN and C. XU, ACC for log canonical thresholds, Ann. Math. (2), 180 (2014), 523–571. 11. C. D. HACON and C. XU, Boundedness of log Calabi-Yau pairs of Fano type, Math. Res. Lett. (to appear), arXiv:1410.8187. 12. S. IITAKA, Deformations of compact complex surfaces, II, J. Math.Soc.Jpn., 22 (1970), 247–261. 13. X. JIANG, On the pluricanonical maps of varieties of intermediate Kodaira dimension, Math. Ann., 356 (2013), 979– 14. Y. KAWAMATA, On the plurigenera of minimal algebraic 3-folds with K ≡ 0, Math. Ann., 275 (1986), 539–546. 15. Y. KAWAMATA, On the length of an extremal rational curve, Invent. Math., 105 (1991), 609–611. 16. Y. KAWAMATA, Subadjunction of log canonical divisors. II, Am. J. Math., 120 (1998), 893–899. 17. J. KOLLÁR, et al., Flips and abundance for algebraic threefolds, Astérisque, 211 (1992). 18. J. KOLLÁR and S. MORI, Birational geometry of algebraic varieties, Cambridge Tracts in Math., vol. 134, Cambridge Univ. Press, Cambridge, 1998. 19. G. PACIENZA, On the uniformity of the Iitaka ﬁbration, Math. Res. Lett., 16 (2009), 663–681. 20. V. V. SHOKUROV, 3-fold log ﬂips, With an appendix by Yujiro Kawamata, Russ. Acad. Sci. Izv. Math., 40 (1993), 95–202. 21. S. TAKAYAMA, Pluricanonical systems on algebraic varieties of general type, Invent. Math., 165 (2006), 551–587. 22. G. TODOROV and C. XU, On Effective Log Iitaka Fibration for 3-folds and 4-folds, Algebra Number Theory, 3 (2009), 697–710. 23. H. TSUJI, Pluricanonical systems of projective varieties of general type I, Osaka J. Math., 43 (2006), 967–995. 24. E. VIEHWEG and D.-Q. ZHANG, Effective Iitaka ﬁbrations, J. Algebraic Geom., 18 (2009), 711–730. C. B. DPMMS, Centre for Mathematical Sciences, University of Cambridge, Wilberforce Road, Cambridge CB3 0WB, UK cb496@dpmms.cam.ac.uk D.-Q. Z. Department of Mathematics, National University of Singapore, 10 Lower Kent Ridge Road, Singapore 119076, Singapore matzdq@nus.edu.sg Manuscrit reçu le 16 décembre 2014 Manuscrit accepté le 14 décembre 2015 publié en ligne le 18 janvier 2016. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Publications mathématiques de l'IHÉS Springer Journals http://www.deepdyve.com/lp/springer-journals/effectivity-of-iitaka-fibrations-and-pluricanonical-systems-of-Hjr8YahUk7

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References (26)

Gianluca Pacienza (2007)
On the uniformity of the Iitaka fibration
Mathematical Research Letters, 16
C. Birkar (2011)
Existence of log canonical flips and a special LMMP
Publications mathématiques de l'IHÉS, 115
(1993)
3-fold log flips, With an appendix by Yujiro Kawamata
J. Kollár (1992)
Flips and Abundance for Algebraic Threefolds
Astérisque, 211
(1970)
Deformations of compact complex surfaces
O. Fujino, S. Mori (2000)
A CANONICAL BUNDLE FORMULA
Journal of Differential Geometry, 56
Gianluca Pacienza (2007)
PLURICANONICAL SYSTEMS ON PROJECTIVE VARIETIES OF GENERAL TYPE
(1998)
Subadjunction of log canonical divisors
G. Cerbo (2014)
Uniform bounds for the Iitaka fibration
Ann. Sc. Norm. Super. Pisa Cl. Sci. (5), 13
Y. Kawamata (1991)
On the length of an extremal rational curve
Inventiones mathematicae, 105
Jungkai Chen, Meng Chen (2007)
Explicit birational geometry of threefolds of general type
arXiv: Algebraic Geometry
Christopher Hacon, J. McKernan (2005)
Boundedness of pluricanonical maps of varieties of general type
Inventiones mathematicae, 166
Christopher Hacon, J. McKernan (2006)
Existence of minimal models for varieties of log general type
Journal of the American Mathematical Society, 23
C. Birkar (2012)
Existence of log canonical flips and a special LMMP
Publ. Math. Inst. Hautes Études Sci., 115
E. Viehweg, De-Qi Zhang (2007)
Effective Iitaka fibrations
Journal of Algebraic Geometry, 18
Y. Kawamata (1986)
On the plurigenera of minimal algebraic 3-folds withK≋0
Mathematische Annalen, 275
Gabriele Cerbo (2011)
Uniform bounds for the Iitaka fibration
arXiv: Algebraic Geometry
Christopher Hacon, J. McKernan, Chenyang Xu (2012)
ACC for log canonical thresholds
Annals of Mathematics, 180
Meng Chen (2008)
On pluricanonical systems of algebraic varieties of general type
arXiv: Algebraic Geometry
Shigeetj Iitaka (2010)
Birational Geometry of Algebraic Varieties
C. Birkar, Zhengyu Hu (2013)
Log canonical pairs with good augmented base loci
Compositio Mathematica, 150
Christopher Hacon, J. McKernan, Chenyang Xu (2010)
On the birational automorphisms of varieties of general type
Annals of Mathematics, 177
Christopher Hacon, Chenyang Xu (2014)
Boundedness of log Calabi–Yau pairs of Fano type
Mathematical Research Letters, 22
Xiao-Tian Jiang (2010)
On the pluricanonical maps of varieties of intermediate Kodaira dimension
Mathematische Annalen, 356
Gueorgui Todorov, Chenyang Xu (2008)
On Effective log Iitaka Fibration for 3-folds and 4-folds
arXiv: Algebraic Geometry
C. Birkar (2007)
On existence of log minimal models
Compositio Mathematica, 146

Publisher: Springer Journals
Copyright: Copyright © 2016 by IHES and Springer-Verlag Berlin Heidelberg
Subject: Mathematics; Mathematics, general; Algebra; Analysis; Geometry; Number Theory
ISSN: 0073-8301
eISSN: 1618-1913
DOI: 10.1007/s10240-016-0080-x
Publisher site: See Article on Publisher Site

Abstract

EFFECTIVITY OF IITAKA FIBRATIONS AND PLURICANONICAL SYSTEMS OF POLARIZED PAIRS by CAUCHER BIRKAR and DE-QI ZHANG ABSTRACT For every smooth complex projective variety W of dimension d and nonnegative Kodaira dimension, we show the existence of a universal constant m depending only on d and two natural invariants of the very general ﬁbres of an Iitaka ﬁbration of W such that the pluricanonical system |mK | deﬁnes an Iitaka ﬁbration. This is a consequence of a more general result on polarized adjoint divisors. In order to prove these results we develop a generalized theory of pairs, singularities, log canonical thresholds, adjunction, etc. CONTENTS 1. Introduction ....................................................... 283 2. Preliminaries ...................................................... 288 3. Effective birationality of K + B + nM ......................................... 291 4. Generalized polarized pairs .............................................. 297 5. Bounds on the number of coefﬁcients of B and M .................................. 308 i i 6. ACC for generalized lc thresholds ........................................... 315 7. Global ACC ....................................................... 317 8. Proof of main results . ................................................. 326 Acknowledgements ..................................................... 330 References ......................................................... 330 1. Introduction We work over the complex number ﬁeld C. However, our results hold over any algebraically closed ﬁeld of characteristic zero. Effectivity of Iitaka ﬁbrations. — Let W be a smooth projective variety of Kodaira dimension κ(W) ≥ 0. Then by a well-known construction of Iitaka, there is a birational morphism V → W from a smooth projective variety V, and a contraction V → Xonto a projective variety X such that a (very) general ﬁbre F of V → X is smooth with Kodaira dimension zero, and dim X is equal to the Kodaira dimension κ(W).The map W X is referred to as an Iitaka ﬁbration of W, which is unique up to birational equivalence. For any sufﬁciently divisible natural number m, the pluricanonical system |mK | deﬁnes an Iitaka ﬁbration. When dim W = 2, in 1970, Iitaka [12]provedthatif m is any natural number divisible by 12 and m ≥ 86, then |mK | deﬁnes an Iitaka ﬁbration (Fabrizio Catanese informed us that Iitaka proved this result for compact complex surfaces but the algebraic case goes back to Enriques). It has since been a question whether something similar holds in higher dimension. More precisely (cf. [8]): DOI 10.1007/s10240-016-0080-x 284 CAUCHER BIRKAR, DE-QI ZHANG Conjecture 1.1 (Effective Iitaka ﬁbration). — Let W be a smooth projective variety of dimension d and Kodaira dimension κ(W) ≥ 0. Then there is a natural number m depending only on d such that the pluricanonical system |mK | deﬁnes an Iitaka ﬁbration for any natural number m divisible by m . W d In this paper, we show a version of the conjecture as formulated in [24, Ques- tion 0.1] holds, that is, by assuming that some invariants of the very general ﬁbres of the Iitaka ﬁbration are bounded. Without these extra assumptions the above conjecture seems out of reach at the moment because most likely one needs the abundance conjec- ture to deal with the very general ﬁbres. For example, when κ(W) = 0, the conjecture is equivalent to the effective nonvanishing h (W, m K ) = 0 which is obviously related d W to the abundance conjecture. Note that there is also a log version of the conjecture for pairs: see [11, Conjecture 1.2, Theorem 1.4] and the references therein, where the au- thors conﬁrmed this log version when the boundary divisor is big over the generic point of the base of the log Iitaka ﬁbration. We recall some deﬁnitions before stating our result. Using the notation above, let W be a smooth projective variety of Kodaira dimension κ(W) ≥ 0and V → Xan Iitaka ﬁbration from a resolution V of W. For a very general ﬁbre F of V → X, let b := min{u ∈ N ||uK |=∅}. F F Let F be a smooth model of the Z/(b )-cover of F ramiﬁed over the unique divisor in |b K |.Then F still has Kodaira dimension zero, but with |K |=∅.Notethat F F F dim F = dim F = dim W − dim X = dim W − κ(W) and we denote this number by d . We call the Betti number β := dim H (F, C) the middle Betti number of F. Theorem 1.2. —Let W be a smooth projective variety of dimension d and Kodaira dimension κ(W) ≥ 0. Then there is a natural number m(d, b ,β ) depending only on d , b and β such that the F F F F pluricanonical system |mK | deﬁnes an Iitaka ﬁbration whenever the natural number m is divisible by m(d, b ,β ). The theorem is an almost immediate consequence of 1.3 below. The proof is given at the end of Section 8. When X is of general type, the numbers b ,β do not play any F F role so m(d, b ,β ) depends only on d . F F Here is a brief history of partial cases of Theorem 1.2: • when dim W = 2[12], • when κ(W) = 1[7], • when W is of general type [8, 21] (see also [23]), EFFECTIVITY OF IITAKA FIBRATIONS 285 • when κ(W) = 2[24] (see also [22]), • when dim W = 3[7, 8, 14, 21, 24] (see also [6]), • when X is non-uniruled, V → X has maximal variation and its general ﬁbres have good minimal models [19] (see also [5]), • when V → X has zero variation and its general ﬁbres have good minimal models [13]. Note that the above references show that Conjecture 1.1 holds when dim W ≤ 3. Effective birationality for polarized pairs of general type. — Let W be a smooth projective variety of nonegative Kodaira dimension. After replacing W birationally we can assume the Iitaka ﬁbration W → X is a morphism. Applying the canonical bundle formula of [7] (which is based on [16]), perhaps after replacing W and X birationally, there is a Q- boundary B and a nef Q-divisor M on X such that for any natural number m divisible by 0 0 b we have a natural isomorphism between H (W, mK ) and H (X, m(K + B + M)). F W X In particular, if |m(K + B + M)| deﬁnes a birational map, then |mK | deﬁnes an Iitaka X W ﬁbration. Moreover, the coefﬁcients of B belong to a DCC set and the Cartier index of M is bounded in terms of b and β . Therefore we can derive Theorem 1.2 from the next result. Theorem 1.3. —Let be a DCC set of nonnegative real numbers, and d, r natural numbers. Then there is a natural number m(, d, r) depending only on , d, r such that if: (i) (X, B) is a projective lc pair of dimension d , (ii) the coefﬁcients of B are in , (iii) rM is a nef Cartier divisor, and (iv) K + B + M is big, then the linear system |m(K + B + M)| deﬁnes a birational map if m ∈ N is divisible by m(, d, r). We call (X, B + M) a polarized pair. When M = 0, the theorem is [10, Theorem 1.3]. 0 0 Note that for an R-divisor D, by |D| and H (X, D) we mean | D | and H (X, D ). Generalized polarized pairs. — In order to prove Theorem 1.3 we need to generalize the deﬁnitions of pairs, singularities, lc thresholds, adjunction, etc. We develop this theory, which is of independent interest, in some detail in Section 4 but for now we only give the deﬁnition of generalized polarized pairs. Deﬁnition 1.4. —A generalized polarized pair consists of a normal variety X equipped with projective morphisms X → X → Z where f is birational and X is normal, an R-boundary B , and an R-Cartier divisor M on X which is nef /Z such that K + B + M is R-Cartier, where M := f M. We call B the boundary part and M the nef part. ∗ 286 CAUCHER BIRKAR, DE-QI ZHANG Note that the deﬁnition is ﬂexible with respect to X, M. To be more precise, if g : Y → X is a projective birational morphism from a normal variety, then there is no harm in replacing X with Y and replacing M with g M. M where μ ≥ 0and M are For us the most interesting case is when M = μ j j j j nef /Z Cartier divisors. In many ways B + M behaves like a boundary, that is, it is as if the M were components of the boundary with coefﬁcient μ . Although the coefﬁcients of B j i belong to the real interval [0, 1] the coefﬁcients μ are only assumed to be nonnegative. Moreover, the M are not necessarily distinct. See Section 4 for more details. When X → X is the identity morphism, we recover the deﬁnition of polarized pairs which was formally introduced in [4] but appeared earlier in the subadjunction formula of [16]. If moreover M = 0, then (X , B ) is just a pair in the traditional sense. ACC for generalized lc thresholds. — The next result shows that the generalized lc thresholds satisfy ACC under suitable assumptions. We suggest the reader looks at Deﬁ- nitions 4.1 and 4.3 before continuing. Theorem 1.5. —Let be a DCC set of nonnegative real numbers and d a natural number. Then there is an ACC set depending only on ,dsuch that if (X , B + M ), M, N,and D are as in Deﬁnition 4.3 satisfying (i) (X , B + M ) is generalized lc of dimension d , (ii) M = μ M where M are nef /Z Cartier divisors and μ ∈ , j j j j (iii) N = ν N where N are nef /Z Cartier divisors and ν ∈ ,and k k k k (iv) the coefﬁcients of B and D belong to , then the generalized lc threshold of D + N with respect to (X , B + M ) belongs to . Note that the theorem is a local statement over X , so Z does not play any role and we could simply assume X → Z is the identity map. When X → X is the identity map, M = 0, and N = 0, the theorem is the usual ACC for lc thresholds [10, Theorem 1.1]. Global ACC. — The proof of the previous result requires the following global ACC. We will also use this to bound pseudo-effective thresholds (Theorem 8.1) which is in turn used in the proof of Theorem 1.3. Theorem 1.6. —Let be a DCC set of nonnegative real numbers and d a natural number. Then there is a ﬁnite subset ⊆ depending only on ,dsuch that if (X , B + M ), X → X → Z and M are as in Deﬁnition 1.4 satisfying (i) (X , B + M ) is generalized lc of dimension d , (ii) Z is a point, EFFECTIVITY OF IITAKA FIBRATIONS 287 (iii) M = μ M where M are nef Cartier divisors and μ ∈ , j j j j (iv) μ = 0 if M ≡ 0, j j (v) the coefﬁcients of B belong to ,and (vi) K + B + M ≡ 0, then the coefﬁcients of B and the μ belong to . When X → X is the identity map and M = 0, the theorem is [10, Theorem 1.5]. About this paper. — It is not hard to reduce Theorems 1.3 and 1.5 to Theorem 1.6. So most of thedifﬁcultieswefacehavetodowith 1.6. Since the statement of Theo- rems 1.3, 1.5,and 1.6 involve nef divisors which may not be semi-ample (or effectively semi-ample), there does not seem to be any easy way to reduce them to the traditional versions (i.e. without nef divisors) proved in [10] or to mimic the arguments in [10]. In- stead we need to develop new ideas and arguments and this occupies much of this paper. We brieﬂy explain the organization of the paper. In Section 3,weprove aspecial case of Theorem 1.3 (Proposition 3.4) by closely following [10]. In Section 4, we introduce generalized singularities and generalized lc thresholds, discuss the log minimal model pro- gram for generalized polarized pairs, and treat generalized adjunction. In Section 5,we give bounds, both in the local and global situations, on the numbers of components in the boundary and nef parts of generalized polarized pairs, under appropriate assump- tions. These bounds will be used in the proof of Proposition 7.2 which serves as the key inductive step toward the proof of Theorem 1.6.InSection 6,wereduceTheorem 1.5 to Theorem 1.6 in lower dimension by adapting a standard argument. In Section 7,we treat Theorem 1.6 inductively where we apply Proposition 3.4; a sketch of the main ideas is included below in this introduction. In Section 8, we give the proofs of our main results. Theorems 1.5 and 1.6 follow immediately from Sections 6 and 7.Toprove Theorem 1.3, we use Theorem 1.6 to bound certain pseudo-effective thresholds (Theorem 8.1)and use the concept of potential birationality [10] to reduce to the special case of Proposition 3.4. Finally, we extend Theorem 1.3 to allow more general coefﬁcients in the nef part of the pair (see Theorem 8.2), and deduce Theorem 1.2 from Theorem 1.3 as in [7, 24]. A few words about the proof of Theorem 1.6.— We try to explain, brieﬂy, some of the ideas used in the proof of 1.6.By[10, 1.5] we can assume M ≡ 0. The basic strategy is to modify (X , B + M ) so that the nef part has one less coefﬁcient μ and then repeat this to reach the case M = 0. Running appropriate LMMP’s we can reduce the problem to the case when X is a Q-factorial klt Fano variety with Picard number one. Moreover, some lengthy arguments show that the number of the μ is bounded (Section 5). If (X , B + M ) is not generalized klt, one can do induction: for example if B = 0, then we let S be the normalization of a component of B and use generalized adjunction (see Deﬁnition 4.7) to write K + B + M = K + B + M S S S X S 288 CAUCHER BIRKAR, DE-QI ZHANG and apply induction to the generalized lc polarized pair (S , B + M ). So we can assume S S (X , B + M ) is generalized klt. Although we cannot use the arguments of [10]toprove Theorem 1.6 but there is an exception: if we take n ∈ N to be sufﬁciently large, then following [10] closely one can show that there is m ∈ N depending only on , d such that |m(K + B + nM |)| X j deﬁnes a birational map (Proposition 3.4) where B is the sum of the birational transform of B and the reduced exceptional divisor of X → X . One can then show that there is an R-divisor D such that 0 ≤ D ∼ K + B + nM R X j where the coefﬁcients of D belong to some DCC set depending only on , d . Then the pushdown D of D satisﬁes D ∼ K + B + nM ≡ (n − μ )M ≡ ρM R X j j j 1 for some number ρ . Changing the indexes one can assume that ρ belongs to some ACC set depending only on , d.Let N = M − μ M .Now theideaistotake s, t,with s 1 1 maximal, so that K + B + sD + N + tM ≡ K + B + M X X and that (X , B + sD + N + tM ) is generalized lc. If it happens to have t = 0, then s would belong to some DCC set and we can replace B with B + sD and replace M with N which has one less summand, and repeat the process. But if t > 0, then (X , B + sD + N + tM ) is generalized lc but not generalized klt. We cannot simply apply induction because the s, t may not belong to a DCC set. For simplicity assume B + sD = 0and let S be one of its components and assume S is normal. The idea is to keep S but to remove the other components of D and increase t instead so that we get K + B +˜ sS + N + tM ≡ K + B + M X X ˜ ˜ for some ˜ s and t ≥ t where S is a component of B +˜ sS .Now it turnsout t belongs to some DCC set and we can apply induction by restricting to S . 2. Preliminaries Notation and terminology. — All the varieties in this paper are quasi-projective over C unless stated otherwise. For deﬁnitions and basic properties of singularities of pairs such as log canonical (lc), Kawamata log terminal (klt), divisorially log terminal (dlt), purely log terminal (plt), and the log minimal model program (LMMP) we refer to [18]. We recall some notation: EFFECTIVITY OF IITAKA FIBRATIONS 289 • The sets of natural, integer, rational, and real numbers are respectively denoted as N, Z, Q, R. • Divisors on normal varieties are always Weil R-divisors unless otherwise stated. • Let X → Z be a projective morphism from a normal variety. Linear equivalence, Q- linear equivalence, R-linear equivalence,and numerical equivalence over Z, between two R-divisors D , D on X are respectively denoted as D ∼ D /Z, D ∼ D /Z, 1 2 1 2 1 Q 2 D ∼ D /Z, and D ≡ D /Z. If Z is a point, we usually drop the Z. 1 R 2 1 2 • If φ : X X is a birational morphism whose inverse does not contract divi- sors, and D is an R-divisor on X, we usually write D for φ D. If X is replaced by X (resp. Y) we usually write D (resp. D )for φ D. Y ∗ • Let X, Y be normal varieties projective over some base Z, and φ : X Ya birational map/Z whose inverse does not contract any divisor. Let D be an R- Cartier divisor on X such that D is also R-Cartier. We say φ is D-negative if there ∗ ∗ is a common resolution g : W → Xand h : W → Ysuch thatE := g D − h D is effective and exceptional/Y, and Supp g E contains all the exceptional divisors of φ. ACC and DCC sets. — A sequence {a } of numbers is increasing (resp. strictly increasing) if a ≤ a (resp. a < a )for all i. The deﬁnition of a decreasing or strictly decreasing i i+1 i i+1 sequence is similar. A set of real numbers satisﬁes DCC (descending chain condition) if it does not contain a strictly decreasing inﬁnite sequence. A set of real numbers satisﬁes ACC (ascending chain condition) if it does not contain a strictly increasing inﬁnite sequence. Lemma 2.1. —Let and be sets of nonnegative real numbers. Deﬁne + ={a + b | a ∈ , b ∈ } and · ={ab | a ∈ , b ∈ }. Then the following hold: (1) If and are both ACC sets (resp. DCC sets), then + and · are also ACC sets (resp. DCC sets). (2) Let {a }⊆ and {b }⊆ be sequences of numbers. Assume that both sequences are i i increasing and that one of them is strictly increasing. Then the sequences {a + b } and {a b } are strictly i i i i increasing. (3) A statement similar to (2) holds if we replace ‘increasing’ by ‘decreasing’. (4) Let m, l ∈ N. Assume that is a DCC set and that a ≤ lfor every a ∈ . Then the set {ma| a ∈ } also satisﬁes DCC, where ma:= ma − ma , that is, the fractional part of ma. Proof. — The proof is left to the reader. 290 CAUCHER BIRKAR, DE-QI ZHANG Lemma 2.2. —Let d, r be natural numbers. Let X be a sequence of normal projective varieties of dimension d and Picard number one. Assume that D ,..., D are nonzero R-Cartier divisors on 1,i r,i X .Let λ be the numbers such that D ≡ λ D . Then possibly after replacing the sequence with an i j,i j,i j,i 1,i inﬁnite subsequence and rearranging the indexes, the sequence λ is a decreasing sequence for each j . j,i Proof.—Let ρ be the numbers such that D ≡ ρ D . Replacing the sequence j,k,i j,i j,k,i k,i we may assume that for each j, k the sequence ρ is increasing or decreasing. If ρ is j,k,i j,k,i decreasing we write j k. This relation is associative, that is, if j k and k l,then j l because ρ = ρ ρ . So we can order the sequences of divisors according to j,l,i j,k,i k,l,i this relation. Changing the indexes we may assume that r ··· 1 whichinparticular means that the λ = ρ form a decreasing sequence for each j . j,i j,1,i Minimal models and Mori ﬁbre spaces. — Let X → Z be a projective morphism of normal varieties and D an R-Cartier divisor on X. A normal variety Y projective over Z together with a birational map φ : X Y/Z whose inverse does not contract any divisor is called a minimal model of D over Zif: (1) Yis Q-factorial, (2) D = φ Dis nef /Z, and Y ∗ (3)φ is D-negative. If one can run an LMMP on D over Z which terminates with a Q-factorial model Y on which D is nef /Z, then Y is a minimal model of D over Z. On the other hand, we call Y a Mori ﬁbre space of D over Z if Y satisﬁes the above conditions with condition (2) replaced by: (2) there is an extremal contraction Y → T/Zsuch that −D is ample/T. In practice, we consider minimal models and Mori ﬁbre space for K + B + M where (X , B + M ) is a generalized polarized pair. Some notions and results of [10]. — For convenience we recall some technical notions and results of [10] which will be used in Section 3. Let X be a normal projective variety, and let D be a big Q-Cartier Q-divisor on X. We say that D is potentially birational [10, Deﬁnition 3.5.3] if for any pair x and y of general points of X, possibly switching x and y,wecan ﬁnd 0 ≤ ∼ (1 − )Dfor some 0 < < 1such that (X,) is not klt at y but (X,) is lc at x and {x} is a non-klt centre. Theorem 2.3 [10, Theorem 3.5.4]. — Let (X, B) be a klt pair, where X is projective of dimension d , and let H be an ample Q-divisor. Suppose there exist a constant γ ≥ 1 and a family V → C of subvarieties of X with the following property: if x and y are two general points of X then, possibly switching x and y, we can ﬁnd c ∈ C and 0 ≤ ∼ (1 − δ)H,for some δ> 0,such c Q that (X, B + ) is not klt at y and there is a unique non-klt place of (X, B + ) whose centre V c c c contains x. Further assume there is a divisor D on W, the normalization of V , such that the linear system |D| deﬁnes a birational map and γ H| − D is pseudo-effective. Then mH is potentially birational, where m = 2p γ + 1 and p = dim V . c EFFECTIVITY OF IITAKA FIBRATIONS 291 Theorem 2.4 [10, Theorem 4.2]. — Let be a subset of [0, 1] which contains 1.Let X be a projective variety of dimension d , and let V be a subvariety, with normalization W. Suppose we are given an R-boundary B and an R-Cartier divisor G ≥ 0, with the following properties: (1) the coefﬁcients of B belong to ; (2) (X, B) is klt; and (3) there is a unique non-klt place ν for (X, B + G), with centre V. Then there is an R-boundary B on W whose coefﬁcients belong to a | 1 − a ∈ LCT D() ∪{1} d−1 such that the difference (K + B + G)| − (K + B ) X W W W is pseudo-effective. Now suppose that V is the general member of a covering family of subvarieties of X.Let ψ : U → W be a log resolution of (W, B ),and let B be the sum of the birational transform of B and W U W the reduced exceptional divisor of ψ . Then K + B ≥ (K + B)| . U U X U The notation | and | mean pullback to W and U respectively. W U Remark 2.5. — Assume that the in 2.4 satisﬁes DCC. Then the hyperstandard set D() also satisﬁes DCC, hence the set of lc thresholds LCT (D()) satisﬁes ACC d−1 by the ACC for usual lc thresholds [10, Theorem 1.1]. Therefore, the set a | 1 − a ∈ LCT D() ∪{1} d−1 to which the coefﬁcients of B belong, also satisﬁes DCC. 3. Effective birationality of K + B + nM In this section, following [10] closely, we prove a special case of Theorem 1.3 (see 3.4) which will be used in Sections 7 and 8 in proving Theorems 1.6 and 1.3. This special case concerns effective birationality for big divisors of the form K + B + nM where (X, B) is projective lc, rM is nef and Cartier, and n/r is large enough. Running an LMMP on K + B + nM preserves the nef and Cartier properties of rM by boundedness of length of extremal rays [15] which allows one to apply the methods of [10]. In contrast if one runs an LMMP on K + B + M, the nef and Cartier properties of rM may be lost, hence one needs to consider generalized polarized pairs which will be discussed in later sections. First we prove a few lemmas. 292 CAUCHER BIRKAR, DE-QI ZHANG Lemma 3.1. —Let X be a normal projective variety, D abig Q-Cartier Q-divisor, and G a nef Q-Cartier Q-divisor on X.If D is potentially birational, then D + G is also potentially birational. In particular, |K +D + G| deﬁnes a birational map. Proof.—Write D ∼ A + B with B effective and A ample. By deﬁnition, for any pair x, y ∈ X of general points, possibly after switching x, y, there exist ∈ (0, 1) and a Q-divisor 0 ≤ ∼ (1 − )Dsuch that (X,) is not klt at y but it is lc at x and {x} is a non-klt centre. Now if ∈ (0, ) is rational, then we can ﬁnd 0 ≤ ∼ + − B + − A + 1 − G ∼ 1 − (D + G) Q Q so that (X, ) satisﬁes the same above properties as (X,) at x, y.So D + G is potentially birational. To get the last claim, just apply [9, Lemma 2.3.4 (1)]. Lemma 3.2. —Let be a DCC set of nonnegative real numbers, and d, r natural numbers. Then there is a real number t ∈ (0, 1) depending only on , d, r such that if: • (X, B) is projective lc of dimension d , • the coefﬁcients of B are in , • rM is a nef Cartier divisor, and • K + B + M is a big divisor, then K + tB + nM is a big divisor for any natural number n > 2rd . Proof. — Since M is nef, it is enough to treat the case n = 2rd + 1. We can assume 1 ∈ .Let (X, B) and M be as in the statement of the lemma. Let f : W → Xbe a log resolution and let B be the sum of the birational transform of B and the reduced exceptional divisor of f ,and let M be the pullback of M. Then we can replace (X, B) with (W, B ) and replace M with M hence it is enough to only consider log smooth W W pairs. We want to argue that, after extending if necessary, it is enough to only consider thecasewhen (X, B) is klt. If the lemma does not hold, then there is a sequence (X , B ), i i M of log smooth lc pairs and nef Q-divisors satisfying the assumptions of the lemma but such that the pseudo-effective thresholds b = min{a ≥ 0 | K + aB + nM is pseudo-effective} i X i i is a strictly increasing sequence of numbers approaching 1. Now by extending and decreasing the coefﬁcients in B which are equal to 1, we can assume that (X , B ) are i i i klt. To get a contradiction it is obviously enough to only consider this sequence hence we only need to consider the klt case. Now let (X, B) and M be as in the statement of the lemma where we assume (X, B) is log smooth klt. Let b be the pseudo-effective threshold as deﬁned above. We may assume b > 0. By Lemma 4.4(2) below, we can run an LMMP on K + bB + nM X EFFECTIVITY OF IITAKA FIBRATIONS 293 which ends with a minimal model X on which K + bB + nM is semi-ample deﬁning a contraction X → T . Since b > 0, a general ﬁbre of X → T is positive-dimensional and the restriction of B to it is big, by the bigness of K + B + nM and the deﬁnition of b. So relying on Lemma 4.4(2) once more, we can also run an LMMP/T on K + nM with scaling of bB which terminates with a Mori ﬁbre space. Denote the end result again by X and the Mori ﬁbre space structure by X → S . By Lemma 4.4(3), both LMMP’s are M-trivial and hence the Cartier and nefness of rM is preserved in the process. Now since K + bB + nM ≡ 0/S and since n > 2rd,M ≡ 0/S by boundedness of length of extremal rays [15]. In particular, if F is a general ﬁbre of X → S ,then K + := K + bB ≡ K + bB + nM ∼ 0. F X X R F F By construction, (K + B + M )| is big and M | ≡ 0, so B | is not zero and its coefﬁ- X F F F cients belong to . Therefore, b is bounded away from 1 otherwise we get a contradiction with the ACC property of [10, Theorem 1.5]. Thus there is t ∈ (0, 1) depending only t +1 on , d, r such that K + t B + nM is pseudo-effective. Now take t = . X 0 We should point out that although we have used (and continue to use) Lemma 4.4 but its proof does not rely on any of the results of this section. Lemma 3.3. —Let X be a normal projective variety, L abig R-divisor, and M anef Q-divisor which is not numerically trivial. Then vol(L + nM) goes to ∞ as n goes to ∞. Proof. — We may write L ∼ A + D where A is ample Q-Cartier and D ≥ 0. Thus ν d−ν ν vol(L + nM) ≥ vol(A + nM) ≥ n A · M where d = dim X and ν is the numerical dimension of M. Since A is ample and ν> 0, d−ν ν A · M > 0. Hence the above volume goes to inﬁnity as n goes to inﬁnity. Proposition 3.4. —Let ⊂[0, 1] be a DCC set of nonnegative real numbers and let d,rbe natural numbers. Then there exists a natural number m depending only on , d, r such that if: • n is a natural number satisfying n > 2rd and r|n, • (X, B) is projective lc of dimension d , • the coefﬁcients of B are in , • rM is a nef Cartier divisor, and • K + B + M is a big divisor, then |m(K + B + nM)| deﬁnes a birational map. Proof.— Step 1. We prove the proposition by induction on d . In particular, we may assume that the proposition holds in dimension < d . 294 CAUCHER BIRKAR, DE-QI ZHANG Fix β> 0. Pick (X, B),M, and n as in the proposition. Assume that vol(K + B + nM)>β . We ﬁrst prove the result for such (X, B),M, and n. At the end, in Steps 6 and 7, we treat the general case. As in the proof of Lemma 3.2, by extending , by taking a log resolution of (X, B), and by decreasing the coefﬁcients of B, we can assume that (X, B) is klt. Step 2. By Lemma 3.2,K + bB + nMis big for some b ∈ (0, 1) depending only on , d, r.Thus vol K + (b + 1)B + nM = vol (K + bB + nM + K + B + nM) X X 1 1 > vol (K + B + nM) > β. 2 2 Replacing b by (b + 1) we may assume that vol(K + bB + nM)>β := β. Moreover, there exists a natural number p depending only on and b (and hence only on , d, r) and there exists a boundary B such that pB is an integral divisor and bB ≤ B ≤ B: this follows from the fact that we can ﬁnd p so that λ − bλ> for every nonzero λ ∈ which in turn implies that for each λ we can ﬁnd an integer 0 ≤ i ≤ p such that bλ ≤ ≤ λ. By the calculation above, vol(K + B + nM)>β . Replacing B with B ,and β with β , we can assume ={i/p | 0 ≤ i ≤ p} and that pB is integral. Step 3. Applying Lemma 4.4(2) below, we can replace X with the lc model (= ample model) of K + B + nM so that we can assume that K + B + nM is ample keeping rM X X nef and Cartier. Since vol(K + B + nM)>β , there is a natural number k > 0 depending only on d,β,suchthat vol k(K + B + nM) >(2d) . Applying [10, Lemma 7.1] to the log pair (X, B) and the big divisor k(K + B + nM), we get a covering family V → C of subvarieties of X such that if x and y are two general points of X, then we may ﬁnd c ∈ Cand 0 ≤ ∼ k(K + B + nM) c R X such that (X, B + ) is not klt at y butitislcat x and there is a unique non-klt place of (X, B + ) whose centre is equal to V which contains x. c c EFFECTIVITY OF IITAKA FIBRATIONS 295 Step 4. Let H := 2k(K + B + nM). In this step we make the necessary preparations in order to apply [10, Theorem 3.5.4] (= Theorem 2.3 above). To do this we need to ﬁnd a natural number γ , depending only on , d, r, and ﬁnd a divisor D on the normalization WofV such that γ H| − D is pseudo-effective and |D| deﬁnes a birational map. c W If dim W = dim V = 0, then γ, D exist trivially (and H is potentially birational). So assume that dim W ≥ 1. Now applying the adjunction formula of [10, Theorem 4.2] (= Theorem 2.4 above) to the klt pair (X, B) and the divisor , and taking into account Remark 2.5, we can ﬁnd a boundary B on W whose coefﬁcients belong to a DCC set uniquely determined by , d , such that the difference (∗)(K + B + )| − (K + B ) X c W W W is a pseudo-effective divisor. Further, let ψ : U → W be a log resolution of (W, B ) and let B be the sum of the strict transform of B and the reduced exceptional divisor of ψ . U W Then K + B ≥ (K + B)| . U U X U Denote by M := M| , the pullback of M to U, by the composition U U U → W → V → X which is birational onto its image. Then K + B + M ≥ (K + B + M)| . U U U X U Hence K + B + M is big because (K + B + M)| is big being the pullback of the U U U X U big divisor K + B + M to a smooth model of the general subvariety V . X c Since the coefﬁcients of B belong to the DCC set , since rM is a nef Cartier U U divisor, and since n > 2rd , the induction hypothesis implies that |m(K + B + nM )| U U U deﬁnes a birational map for some m > 0 depending only on (and hence on )and d, r.Thus |m(K + B + nM )| also deﬁnes a birational map since it contains the direct W W W image of |m(K + B + nM )| where M denotes the pullback of M to W. U U U W Note that the difference (K + B + nM + )| − (K + B + nM ) X c W W W W ∼ (k + 1)(K + B + nM)| − (K + B + nM ) R X W W W W is a pseudo-effective divisor by (∗) above. Now let D := m(K + B + nM ) and let γ W W W be the smallest natural number satisfying γ ≥ m(k + 1)/2k.Then γ H| − D is a pseudo- effective divisor and |D| deﬁnes a birational map as required. Step 5. By Step 4 and Theorem 2.3, m H = 2m k(K + B + nM) X 296 CAUCHER BIRKAR, DE-QI ZHANG is potentially birational for some m ≤ 2(d − 1) γ + 1. Thus by Lemma 3.1, 2m kp(K + B + nM) + nM is also potentially birational and K + 2m kp(K + B + nM) + nM X X deﬁnes a birational map where p is as in Step 2 (recall that pB is an integral divisor). Since K + 2m kp(K + B + nM) + nM ≤ 2m kp + 1 (K + B + nM) X X X the linear system 2m kp + 1 (K + B + nM) also deﬁnes a birational map. Now the number m := 2m kp + 1only depends on the data , d, r,β . Step 6. Now we go back to Step 1. We will show that there exist a natural number q and a real number α> 0 depending only on , d, r,suchthatif (X, B),M, n are as in the statement of the proposition and if n ≥ q,then vol(K + B + nM)>α. If this is not true, then we can ﬁnd a sequence (X , B ),M , n satisfying the assumptions of the proposition i i i i such that the n form a strictly increasing sequence approaching ∞ and the vol(K + i X B + n M ) approach 0. By replacing X with a minimal model of K + B + n M ,wemay i i i i X i i i assume that K + B + n M is nef. We can also assume that ν , the numerical dimension X i i i of M , is independent of i. We may assume ν> 0 otherwise we can get a contradiction using [10, Theorem 1.3]. By Lemma 3.3,for each i, there is n the largest natural number divisible by r such that vol(K + B + n M )< 1. We show that the volume vol(K + B + (2n − 1)M ) is X i i X i i i i i i bounded from above. This follows from 2 > vol 2 K + B + n M X i i i i = vol K + B + 2n − 1 M + K + B + M X i i X i i i i i > vol K + B + 2n − 1 M X i i i i where we use the assumption that K + B + M is big. X i i On the other hand, since vol K + B + n + r M ≥ 1, X i i i i EFFECTIVITY OF IITAKA FIBRATIONS 297 by Steps 2–5 above, we may assume that there is an m depending only on , d, r such that m K + B + n + r M X i i i i deﬁnes a birational map for every i. In particular, there exist resolutions f : Y → X such i i i that P := f m K + B + n + r M ∼ H + G i X i i i i i i i where H is big and base point free and G is effective. So we can calculate i i 2 m > vol m K + B + 2n − 1 M X i i i i d ν ∗ d−ν ∗ ν = P + m n − r − 1 f M ≥ m n − r − 1 H · f M i i i i i i i i d−ν ∗ ν 1 which gives a contradiction as lim(n − r − 1) =∞ and H · f M ≥ . i i i Step 7. Let q,α be as in Step 6. In this step we show that there is β> 0 depending only on , d, r such that vol(K + B + nM)>β for any (X, B),M, n as in the statement n−1 of the proposition. We may assume q > n otherwise we can use Step 6. Let s = .Then q−1 vol(K + B + nM) = vol (1 − s)(K + B + M) + s(K + B + qM) X X X d d ≥ s vol(K + B + qM)> s α ≥ =: β. (q − 1) This completes the proof of the proposition. 4. Generalized polarized pairs In this section, we deﬁne generalized lc and klt singularities, discuss some of their basic properties, and then deﬁne generalized lc thresholds for generalized polarized pairs. Next we consider running the log minimal model program for these pairs, and use it to extract divisors with generalized log discrepancy <1. Then we deﬁne generalized adjunc- tion and discuss DCC and ACC properties of coefﬁcients in the boundary and nef parts of generalized polarized pairs under this adjunction. Generalized singularities. — We already deﬁned generalized polarized pairs in the in- troduction. Now we deﬁne their singularities. Deﬁnition 4.1. —Let (X , B + M ) be a generalized polarized pair as in 1.4 which comes with the data X → X → Z and M.Let E be a prime divisor on some birational model of X .We 298 CAUCHER BIRKAR, DE-QI ZHANG deﬁne the generalized log discrepancy of E with respect to the above generalized polarized pair as follows. After replacing X, we may assume E is a prime divisor on X. We can write K + B + M = f K + B + M X X for some R-divisor B. The generalized log discrepancy of E is deﬁned to be 1 − b where b is the coefﬁcient of E in B. We say that (X , B + M ) is generalized lc (resp. generalized klt) if the generalized log discrepancy of any prime divisor is ≥ 0 (resp. > 0). If f is a log resolution of (X , B ), then generalized lc (resp. generalized klt) is equivalent to the coefﬁcients of B being ≤ 1 (resp. < 1). If the generalized log discrepancy of E is ≤ 0, we call the image of E in X a generalized non-klt centre.If (X , B + M ) is generalized lc, a non-klt centre is also referred to as a generalized lc centre. Remark 4.2. —Weuse thenotationof 4.1. (1) Note that Z does not play any role in the deﬁnition of singularities. That is because singularities are local in nature over X , so one can simply assume X → Zis the identity map. The same applies to generalized lc thresholds deﬁned below (4.3) and in general to notions and statements that are local. (2) Assume that (X , B + M ) is generalized klt. Let D be an effective R-Cartier divisor. Then from the deﬁnitions we can easily see that (X , B + D + M ) is generalized klt with boundary part B + D and nef part M, for any small > 0. Now assume that D is ample/Z. Then for any a > 0 we can ﬁnd a boundary ∼ B + aD + M /Z such that (X , ) is klt. (3) Assume that K + B is R-Cartier and write K + B = f (K + B ) and X X X ˜ ˜ f M = M + E. By the negativity lemma [20, Lemma 1.1], E ≥ 0. Thus B = B + E ≥ B. Therefore, if (X , B + M ) is generalized lc (resp. generalized klt), then (X , B ) is lc (resp. klt). (4) Assume that M ∼ 0/X .Then (X , B + M ) is generalized lc (resp. generalized klt) iff (X , B ) is generalized lc (resp. generalized klt). Indeed in this case M = f M hence K + B = f (K + B ) which implies the claim. In this situation M does not contribute X X to the singularities even if its coefﬁcients are large. In contrast, the larger the coefﬁcients of B, the worse the singularities. (5) In general, M does contribute to singularities. For example, assume X = P and that f is the blowup of a point x . Let E be the exceptional divisor, L a line passing through x and L the birational transform of L . If B = 0and M = 2L, then we can calculate B = Ehence (X , B + M ) is gener- alized lc but not generalized klt. However, if B = L and M = 2L, then (X , B + M ) is not generalized lc because in this case B = L + 2E. EFFECTIVITY OF IITAKA FIBRATIONS 299 (6) Assume we are given a contraction X → Y/Z. We may assume f is a log resolu- tion of (X , B ). Let F be a general ﬁbre of X → Y, F the corresponding ﬁbre of X → Y, and g : F → F the induced morphism. Let B = B| , M = M| , B = g B , M = g M . F F F F F ∗ F F ∗ F Then (F , B + M ) is a generalized polarized pair with the data F → F → Zand M . F F F Moreover, K + B + M = K + B + M . F F F X In addition, B = B | and M = M | : note that since F is a general ﬁbre, B and M are F F F F R-Cartier along any codimension one point of F hence we can deﬁne these restrictions. (7) Let φ : X → X be a birational contraction from a normal variety. We can assume X X is a morphism. Let B , M be the pushdowns of B, M. Then K + B + M = φ K + B + M . X X Now assume that B is a boundary. Then we can naturally consider (X , B + M ) as a generalized polarized pair with boundary part B and nef part M. One may think of (X , B + M ) as a crepant model of (X , B + M ). Deﬁnition 4.3. —Let (X , B + M ) be a generalized polarized pair as in 1.4 which comes with the data X → X → Z and M. Assume that D on X is an effective R-divisor and that N on X is an R-divisor which is nef /Z and that D + N is R-Cartier. The generalized lc threshold of D + N with respect to (X , B + M ) (more precisely, with respect to the above data) is deﬁned as sup s | X , B + sD + M + sN is generalized lc where the pair in the deﬁnition has boundary part B + sD and nef part M + sN. By the negativity lemma, G := f (D + N ) − N ≥ 0. Thus we can write K + B + M = f K + B + M X X and K + B + sG + M + sN = f K + B + sD + M + sN . X X In particular, if (X , B + M ) is generalized lc, then the just deﬁned generalized lc thresh- old is nonnegative. However, the threshold might be +∞: this happens when D = 0and N ∼ 0/X . As pointed earlier, the generalized lc threshold is local over X , so we can usually assume X → Z is the identity map. When M = N = 0, we recover the usual lc threshold of D with respect to (X , B ). 300 CAUCHER BIRKAR, DE-QI ZHANG LMMP for generalized polarized pairs. — Let (X , B + M ) be a Q-factorial generalized lc polarized pair with data X → X → Z and M. One can ask whether one can run an LMMP/ZonK + B + M and whether it terminates. We cannot answer this question in such generality but we will put some extra assumptions under which the answer would be yes. Assume that K + B + M + A is nef /Zfor some R-Cartier divisor A ≥ 0 which is big/Z. Moreover, assume (∗) for any s ∈ (0, 1) there is a boundary ∼ B + sA + M /Zsuch that (X , + (1 − s)A ) is klt. Condition (∗) is automatically satisﬁed if A is general ample/Z and either (i) (X , B + M ) is generalized klt, or (ii) (X , B + M ) is generalized lc and (X , 0) is klt. We will show that we can run the LMMP/ZonK + B + M with scaling of A (However, we do not know whether it terminates). Let λ = min t ≥ 0 | K + B + M + tA is nef /Z . We may assume λ> 0. Replacing A with λA we may assume λ = 1. By assumption we can ﬁnd a number 0 < s < 1 and a boundary ∼ B + sA + M /Zsuch that (X , + (1 − s)A ) is klt. Now by [1, Lemma 3.1], there is an extremal ray R /Zsuch that (K + ) · R < 0and K + + (1 − s)A · R = 0. In particular, (K + B + M ) · R < 0and K + B + M + A · R = 0. Moreover, R can be contracted and its ﬂip exists if it is of ﬂipping type. If R deﬁnes a Mori ﬁbre space we stop. Otherwise let X X be the divisorial contraction or the ﬂip of R . Replacing X we may assume X X is a morphism. Then (X , B + M ) is naturally a generalized lc polarized pair with boundary part B and nef part M. More- over, K + B + M + A is nef /Zand (∗) is preserved. Repeating the process gives the LMMP. Now we show the LMMP terminates under suitable assumptions. Lemma 4.4. —Let (X , B + M ) be a Q-factorial generalized lc polarized pair of dimension d with data X → X → Z and M. Assume that (X , B + M ) satisﬁes (i) or (ii) above. Run an LMMP/Z on K + B + M with scaling of some general ample/Z R-Cartier divisor A ≥ 0. Then the following hold: EFFECTIVITY OF IITAKA FIBRATIONS 301 (1) Assume that K + B + M is not pseudo-effective/Z. Then the LMMP terminates with a Mori ﬁbre space. (2) Assume that • K + B + M is pseudo-effective/Z, • (X , B + M ) is generalized klt, and that • K + (1 + α)B + (1 + β)M is R-Cartier and big/Z for some α, β ≥ 0. Then the LMMP terminates with a minimal model X and K + B + M is semi- ample/Z, hence it deﬁnes a contraction φ : X → T /Z. If moreover a general ﬁbre of φ is positive-dimensional and if the restriction of B to it is nonzero, then we can run the LMMP/T on K + M with scaling of B which terminates with a Mori ﬁbre space of K + M over both T and Z. (3) Assume X → X is the identity morphism and that M = μ M where μ ≥ 0 and M j j j j are Cartier nef /Z divisors. Pick j and assume μ > 2d . Then the above LMMP’s are M - trivial. In particular, the LMMP’s preserve the Cartier and the nefness/Z of M .Moreover, under the assumptions of (2) and assuming φ is birational, M ≡ 0/T and M is the j j pullback of some Cartier divisor on T . Proof. — (1) Since K + B + M is not pseudo-effective/Z, the LMMP is also an LMMP on K + B + A + M with scaling of (1 − )A for some > 0. Now we can ﬁnd a boundary ∼ B + A + M /Z such that (X , + (1 − )A ) is klt. The claim then follows from [3] as the LMMP is an LMMP/ZonK + with scaling of (1 − )A . (2) As K + (1 + α)B + (1 + β)M is big/Z, it is R-linearly equivalent to some P + G over Z where P is ample and G ≥ 0. Now if > 0 is small, then (1 + ) K + B + M ∼ K + (1 − α)B + (1 − β)M + P + G X R X ∼ K + /Z R X for some such that (X , ) is klt and is big/Z. The LMMP is also an LMMP/Z on K + with scaling of (1 + )A which terminates on some model X by [3]. By the base point free theorem for klt pairs with big boundary divisor [3, Corollary 3.9.2], K + is semi-ample/Zhence K + B + M is semi-ample/Z and so it deﬁnes a X X contraction φ : X → T . Now assume a general ﬁbre of φ : X → T is positive-dimensional and the restric- tion of B to it is nonzero. In particular, this implies that K + M is not pseudo-effective X 302 CAUCHER BIRKAR, DE-QI ZHANG over T . Since K + ≡ K + B + M ≡ 0/T , X X 1 + running the LMMP/T on K + M with scaling of B is the same as running the LMMP/T on K + − τ B with scaling of τ B for some small τ> 0 and this termi- nates with a Mori ﬁbre space over T and also over Z, by [3]. Note that, − τ B ≥ 0 by construction. (3) Each step of those LMMP’s is M -trivial and preserves the Cartier and the nefness/ZofM by boundedness of the length of extremal rays and the cone theorem [15], [18, Theorem 3.7 (1) and (4)]. Under the assumptions of (2) and assuming φ is birational, to show that M is the pullback of some Cartier divisor on T ,itisenough to show that X → T decomposes into a sequence of extremal contractions which are negative with respect to certain klt pairs. We write this more precisely. Since in the proof of (2) is big/Z, we can assume ≥ C for some ample Q-divisor C . Since K + ≡ 0/T ,if X → T is not an isomorphism, then there is a (K + − C )-negative extremal ray which gives a contraction X → X /T . In particular M is the pullback of a Cartier divisor on X [18, Theorem 3.7 (4)]. Now j 2 repeat the process with X and so on. Since φ is birational by assumption, the process ends with T hence we can indeed decompose X → T into a sequence of extremal contractions as required. We will apply the LMMP to birationally extract certain divisors for a generalized polarized pair. Lemma 4.5. —Let (X , B + M ) be a generalized lc polarized pair with data X → X → Z and M.Let S ,..., S be prime divisors on birational models of X which are exceptional/X and whose 1 r generalized log discrepancies with respect to (X , B + M ) are at most 1. Then perhaps after replacing f with a high resolution, there exist a Q-factorial generalized lc polarized pair (X , B + M ) with data X → X → Z and M, and a projective birational morphism φ : X → X such that • S ,..., S appear as divisors on X , 1 r • each exceptional divisor of φ is one of the S or is a component of B ,and • K + B + M = φ (K + B + M ). X X In particular, the exceptional divisors of φ are exactly the S if (X , B + M ) is generalized klt. Proof. — Replacing X we may assume the S are divisors on X and that f is a log resolution of (X , B ).Let E , E ,... be the exceptional divisors of f where we can 1 2 assume E = S for i ≤ r.Write i i K + B + M = f K + B + M X X EFFECTIVITY OF IITAKA FIBRATIONS 303 and let = B +Ewhere E := a E and a is the generalized log discrepancy of E (by i i i i i>r deﬁnition a is equal to 1 − b where b is the coefﬁcient of E in B). Then is a boundary i i i i and K + + M = f K + B + M + E ≡ E/X X X with E ≥ 0exceptional/X . By construction, none of the S are components of E. Now run an LMMP/X on K + + M with scaling of some ample divisor. This is also an LMMP/X on E. In the course of the LMMP we arrive at a model X on which K + + M is a limit of movable/X divisors hence it is nef on the general curves/X of any exceptional divisor of X → X where , M are the pushdowns of , M. But since E is effective and exceptional/X ,E = 0 by the general negativity lemma (cf. [2, Lemma 3.3 and the proof of Theorem 3.4]). Note that since the LMMP contracts E, we have = B . So we can write K + B + M = φ K + B + M X X where φ is the morphism X → X . By construction, none of the S is contracted by the LMMP. Moreover, any exceptional divisor of φ is one of the S or is a component of B In particular, the exceptional divisors of φ are exactly the S if (X , B + M ) is generalized klt. Note that X is Q-factorial by construction. Lemma 4.6. — Under the notation and assumptions of Lemma 4.5, further assume that (X , C ) is klt for some C , and that the generalized log discrepancies of the S with respect to (X , B + M ) are < 1. Then we can construct φ so that in addition it satisﬁes: • its exceptional divisors are exactly S ,..., S ,and 1 r • if r = 1 and X is Q-factorial, then φ is an extremal contraction. Proof. — Since (X , C ) is klt and (X , B + M ) is generalized lc, X ,(1 − )B + C + (1 − )M is generalized klt for any small > 0 with boundary part := (1 − )B + C and nef part (1 − )M. Moreover, the generalized log discrepancies of the S with respect to (X , + (1 − )M ) are still less than 1. So by Lemma 4.5, there is φ : X → X which extracts exactly the S . Now further assume that r = 1and that X is Q-factorial. By construction, we can write K + + (1 − )M = φ K + + (1 − )M X X where is the sum of the birational transform of and sS for some s ∈ (0, 1).Now run an LMMP/X on K + + δS + (1 − )M for some small δ> 0 which is also 1 304 CAUCHER BIRKAR, DE-QI ZHANG an LMMP on S . Since X is Q-factorial, the last step of the LMMP is an extremal contraction X → X which contracts S , the pushdown of S ,and X X is an 1 1 isomorphism in codimension one. Thus replacing X with X we can assume φ is ex- tremal. Generalized adjunction. — We deﬁne an adjunction formula for generalized polarized pairs similar to the traditional one. Deﬁnition 4.7. —Let (X , B + M ) be a generalized polarized pair with data X → X → Z and M. Assume that S is the normalization of a component of B and S is its birational transform on X.Replacing X we may assume f is a log resolution of (X , B ). Write K + B + M = f K + B + M X X and let K + B + M := (K + B + M)| S S S X S where B = (B − S)| and M = M| . Let g be the induced morphism S → S and let B = g B S S S S S ∗ S and M = g M . Then we get the equality S ∗ S K + B + M = K + B + M S S S X which we refer to as generalized adjunction.Itisobviousthat B depends on both B and M. Now assume that (X , B + M ) is generalized lc. By Remark 4.8 below B is a boundary divisor on S , i.e. its coefﬁcients belong to [0, 1]. We consider (S , B + M ) as a generalized polarized S S pair which is determined by the boundary part B , the morphisms S → S → Z, and the nef part M . S S It is also clear that (S , B + M ) is generalized lc if (X , B + M ) is so because then S S K + B + M = g (K + B + M ) S S S S S S and the coefﬁcients of B are at most 1. Remark 4.8. —Wewillargue that the B deﬁned in 4.7 is indeed a boundary divisor on S ,if (X , B + M ) is generalized lc. The lc property immediately implies that the coefﬁcients of B do not exceed 1, hence we only have to show that B ≥ 0. Moreover, S S if K + B is R-Cartier, then B ≥ 0 follows from the usual divisorial adjunction: indeed X S in this case if B is the divisor given by the adjunction K + B = K + B S S X then it is well-known that B is a boundary divisor, and it is also clear from our deﬁnitions that B ≤ B . S S EFFECTIVITY OF IITAKA FIBRATIONS 305 In practice when we apply generalized adjunction, X will be Q-factorial, hence K + B will be R-Cartier. But for the sake of completeness we treat the general case, i.e. the non-R-Cartier K + B case. We will reduce the statement to the situation dim X = 2 in which case K + B turns out to be R-Cartier automatically. Assume dim X > 2. Let H be a general hypersurface section and G its pullback to S . Adding H to B we may assume H is a component of B .Both H and G are normal varieties. Let B be given by the generalized adjunction K + B + M = K + B + M . H H H X Since H is a general hypersurface section, B is simply the intersection of B − H with H , that is, each component of B is a component of the intersection of some component of B − H with H inheriting the same coefﬁcient. In particular, B is a boundary divisor and G is a component of B A further generalized adjunction and induction on dimension gives K + B + M = (K + B + M )| G G G H H H G where B is a boundary. But B is equal to the intersection of B − G with the ample G G S divisor G on S which implies that B is a boundary divisor too. Now we can assume dim X = 2. Since K + B + M = f K + B + M ≡ 0/X X X ˜ ˜ ˜ and since M is nef /X , there is a divisor B ≤ Bsuch thatK + B ≡ 0/X and f B = B . X ∗ Since each coefﬁcient of B is at most 1, each coefﬁcient of B is also at most 1. Therefore (X , B ) is numerically lc (see [18, Section 4.1]; note however that [18]onlyconsiders B with rational coefﬁcients but all the deﬁnitions and results that we need make sense and hold true for real coefﬁcients as well). Now by [18, Section 4.1], (X , B ) is lc. In particular, K + B is R-Cartier. So we are done by the above arguments. Proposition 4.9. — Let d be a natural number and a DCC set of nonnegative real numbers. Then there is a DCC set of nonnegative real numbers depending only on d and such that if (X , B + M ) is a generalized lc polarized pair of dimension d with data X → X → Z and M,and S is the normalization of a component of B satisfying • M = μ M where M are nef /Z Cartier divisors and μ ∈ , j j j j • the coefﬁcients of B belong to ,and • B is given by the following generalized adjunction (as in 4.7) K + B + M = K + B + M , S S S X then the coefﬁcients of B belong to . S 306 CAUCHER BIRKAR, DE-QI ZHANG Proof. — If the statement does not hold, then there exist a sequence of generalized lc polarized pairs (X , B + M ) and S ,withdata X → X → Z and M = μ M , i i i j,i j,i i i i i i satisfying the assumptions of the proposition but such that the set of the coefﬁcients of all the B put together does not satisfy DCC. Note that since the problem is local, we may assume X → Z is the identity map for each i. We may also assume f is a log resolution i i of (X , B ). Let S ⊂ X be the birational transform of S . We can assume that each B has i i S a component V with coefﬁcient a such that {a } is a strictly decreasing sequence. Let i i i a = lim a . We may assume that the K + B are R-Cartier otherwise as in Remark 4.8,by taking hypersurface sections, we reduce the problem to dimension 2 in which case this R-Cartier property holds automatically. Let B be the divisor given by the adjunction K + B = K + B . S S X i i i S ˜ ˜ It is clear from our deﬁnitions that B ≤ B .If c is the coefﬁcient of V in B ,thenwe S S i i S i i i may assume c ≤ a ≤ a + for some ﬁxed > 0so that a + < 1. Therefore, (X , B ) is i i i i plt near the generic point of (the image of) V (this follows from inversion of adjunction on surfaces [20, Corollary 3.12]) and there is a natural number l depending only on a + such that for each i there is l ≤ l so that for any Weil divisor D on X the divisor l D i i i i i is Cartier near the (image of the) generic point of V [20, Proposition 3.9]. Moreover, by [20, Corollary 3.10] we can write l − 1 d i k,i c = + b i k,i l l i i for some nonnegative integers d where b are the coefﬁcients of the components of B k,i k,i other than (the image of) S passing through V . On the other hand, shrinking X if necessary we can assume M is Q-Cartier for i j,i each j, i so we can write f M = M + E j,i j,i i j,i where the exceptional divisor E is effective by the negativity lemma. Since l M is j,i i j,i Cartier near the (image of the) generic point of V , the multiplicity of the birational trans- j,i form of V in E | is equal to for some nonnegative integer e . Therefore, i j,i S j,i l − 1 d i k,i j,i a = + b + μ . i k,i j,i l l l i i i This is a contradiction because the above expression and Lemma 2.1 show that the set {a } satisﬁes DCC, while the a form a strictly decreasing sequence. i i We will need the next technical lemma in the proof of Proposition 7.1 to treat Theorem 1.6 inductively. EFFECTIVITY OF IITAKA FIBRATIONS 307 Lemma 4.10. — Let d be a natural number and be a DCC set of nonnegative real numbers. Let (X , B + M ) be a sequence of generalized lc polarized pairs of dimension d with data X → i i i X → Z and M .Let S be the normalization of a component of B and consider the generalized i i i i i adjunction formula K + B + M = K + B + M . S S S X i i i i i i S Assume further that (1) X is Q-factorial and Z is a point, (2) B = b B where B are distinct prime divisors and b ∈ , k,i k,i i k,i k,i (3) M = μ M where M are nef Cartier divisors and μ ∈ , i j,i j,i j,i j,i (4) and one of the following holds: (i) {b } is not ﬁnite, and B | ≡ 0 for each i, or 1,i S 1,i (ii) {μ } is not ﬁnite, and M | ≡ 0 for each i. 1,i S 1,i Then the set of the coefﬁcients of all the B union the set {μ | M | ≡ 0} is not ﬁnite. S j,i j,i S Proof.—Let V be a prime divisor on S . As in the proof of Proposition 4.9,the coefﬁcient of V in B is of the form i S l − 1 d e i k,i j,i a = + b + μ i k,i j,i l l l i i i where l is a natural number and d , e are nonnegative integers which are contributed i k,i j,i by the B and M respectively. k,i j,i Now assume (i) of (4) holds. Since {b } is not ﬁnite, we can assume b < 1for 1,i 1,i each i which in particular means B is not equal to the image of S .Thus B | is a 1,i i 1,i nonzero effective divisor for each i. Choose V to be a component of B | . Then the set i S 1,i {a } cannot be ﬁnite by Lemma 2.1 because {b } is not ﬁnite and d is positive. i 1,i 1,i Next assume (ii) of (4) holds. Although M | is not numerically trivial by assump- 1,i tion but M | may be numerically trivial for some i.If M | is not numerically trivial 1,i S 1,i S i i for inﬁnitely many i,thenobviously theset {μ | M | ≡ 0} is not ﬁnite and we are j,i j,i S done. So we may assume M | is numerically trivial for every i. Recall from the proof 1,i S of Proposition 4.9 that we can assume f M = M + E with E ≥ 0. Now we can j,i j,i j,i i j,i choose V so that e = 0for each i: indeed since M | ≡ 0but M | ≡ 0, we deduce i 1,i S 1,i S 1,i i that E | = 0 and that its pushdown to S is also not zero; thus the components of the 1,i S i i pushdown of E | are components of B , hence we can choose V to be one of these 1,i S S i components. Again this shows that {a } cannot be ﬁnite because {μ } is not ﬁnite and i 1,i e > 0. 1,i 308 CAUCHER BIRKAR, DE-QI ZHANG 5. Bounds on the number of coefﬁcients of B and M i i A well-known fact says that if (X, B) is a lc pair, then near each point x ∈ Xthe number of components of B with coefﬁcient ≥ b > 0 is bounded in terms of b and di- mension of X. There is also a global version of this fact. In this section, we prove similar local and global statements bounding the number of the coefﬁcients of B and the μ in M = μ M of a generalized lc polarized pair (X , B + M ) under certain assumptions. j j These bounds will be used in the proof of Proposition 7.2. We start with a global statement for pairs which can also be applied to generalized polarized pairs. Proposition 5.1. — Let d be a natural number and b a positive real number. Let (X, B) be a projective lc pair of dimension d such that (i) B ≥ B where B ≥ 0 are big R-Cartier divisors, k k (ii) B = b B is the irreducible decomposition and b ≥ bfor everyj, k, and k j,k j,k j,k (iii) K + B + P ≡ 0 for some pseudo-effective R-Cartier divisor P. Then the number of the B is at most (d + 1)/b, that is, r ≤ (d + 1)/b. Proof.—Let (Y,) be a Q-factorial dlt model of (X, B − B ) and f : Y → X the corresponding morphism. By deﬁnition, is the sum of the reduced exceptional B . Moreover, since (X, B) is lc, divisor of f and the birational transform of B − Supp( B ) does not contain the image of any exceptional divisor of f ,hence f B is k k equal to the birational transform of B . In particular, f B is big and it inherits the same k k coefﬁcients as B . Moreover, by letting B := + f B we get k Y k ∗ ∗ ∗ ∗ K + B + f P = K + + f B + f P = f (K + B + P) ≡ 0. Y Y Y k X Now by replacing (X, B) with (Y, B ) and replacing P with f P we can assume that r r (X, 0) is Q-factorial klt. Moreover, by adding B − B to P we can assume B = B . k k 1 1 If P ≡ 0, then K + B is not pseudo-effective so we can run an LMMP on K + B X X which terminates with a Mori ﬁbre space, by Lemma 4.4(1). But if P ≡ 0, then K is not pseudo-effective as B is big, and we can run an LMMP on K which terminates with a Mori ﬁbre space [3]. Note that in both cases the LMMP preserves the lc property of (X, B) and the Q-factorial klt property of (X, 0): in the ﬁrst case the klt property of ˜ ˜ (X, 0) is preserved since the LMMP is also an LMMP on K + B for some klt (X, B); in the second case the lc property of (X, B) is preserved as K + B ≡ 0. Also note that in either case the LMMP does not contract any B because B is big (although some of k k its components may be contracted). So in either case replacing X with the Mori ﬁbre space obtained we may assume that we already have a K -negative Mori ﬁbre structure X → T. EFFECTIVITY OF IITAKA FIBRATIONS 309 Let F be a general ﬁbre of X → T. Since B is big, B | is big too. Restricting to F k k F and applying induction on dimension we can reduce the problem to the case dim T = 0, that is, when X is a Q-factorial klt Fano variety of Picard number one. Pick a small number > 0. For each j, k take a rational number b ≤ b such that b ≥ b − .Let j,k j,k j,k B = b B . Then there is P ≥ 0such thatK + B + P ≡ 0and (X, B + P ) is j,k X k j j,k lc. Now by [17, Corollary 18.24], r(b − ) ≤ b ≤ d + 1. j,k k j Therefore taking the limit when approaches 0 we get rb ≤ d + 1hence r ≤ (d + 1)/b. Next we prove a result similar to Proposition 5.1, though not as sharp, for the nef part of generalized polarized pairs. Proposition 5.2. — Let d be a natural number and b a positive real number. Assume that the ACC for generalized lc thresholds (Theorem 1.5) holds in dimension d . Then there is a natural number p depending only on d, b such that if (X , B + M ) is a generalized lc polarized pair of dimension d with data X → X → Z and M satisfying (i) Z is a point, (ii) M = μ M where M are nef Cartier divisors and μ ≥ b, j j j j (iii) M is a big Q-Cartier divisor for every j , and (iv) K + B + M + P ≡ 0 for some pseudo-effective R-Cartier divisor P , then the number of the μ is at most p, that is, r ≤ p. Before giving the proof we prove a related local statement. Proposition 5.3. — Let d be a natural number and b a positive real number. Assume that Theorem 1.5 and Proposition 5.2 hold in dimension < d . Then there is a natural number q depending only on d,bsuch that if (X , B + M ) is a Q-factorial generalized lc polarized pair of dimension d with data X → X → Z and M,and if (i) x ∈ X is a (not necessarily closed) point, (ii) M = μ M where M are nef /Z Cartier divisors and μ ≥ b, j j j j (iii) M is not relatively numerically zero over any neighborhood of x ,for every j,and (iv) (X , 0) is klt, then the number of the μ is at most q, that is, r ≤ q. Proof.— Step 1.Let C be the closure of x .By (iii), the codimension of C in X is at least two. By adding appropriate divisors to B and shrinking X we can assume C is a generalized lc centre of (X , B + M ):tobemoreprecise,let W bethe blowup of 310 CAUCHER BIRKAR, DE-QI ZHANG X along C ; we can assume X → X factors through W; now take a general sufﬁciently ample divisor on W and let A be its pullback to X; if α is the generalized lc threshold of A near x with respect to (X , B + M ),then (X , B + αA + M ) is generalized lc near x with boundary part B + αA and nef part M, and C is a generalized lc centre of (X , B + αA + M ); the point is that after shrinking X we can assume f A = A + E where E = 0 is effective with large coefﬁcients, and that every component of E maps onto C so adding αA creates deeper singularities only along C . Now we may replace B with B + αA . Step 2. By Step 1, we can assume that there is a prime divisor S on X mapping onto C whose generalized log discrepancy with respect to (X , B + M ) is 0. Since (X , 0) is Q-factorial klt, by Lemma 4.6, there is an extremal birational contraction φ : X → X which extracts S , the birational transform of S, and X is Q-factorial. We can write K + B + M = φ K + B + M X X where B is the sum of S and the birational transform of B ,and M is the pushdown of M. Writing K + B + M = f K + B + M X X we can see that B is just the pushdown of B. We claim that M is not numerically trivial over any neighborhood of x for any j which in turn implies that M is ample/X . If this is not true for some j , then we can write ˜ ˜ ˜ f M = M + E where E ≥ 0 and S is not a component of E . But then for any general j j j j −1 closed point y ∈ C ,the ﬁbre f {y } is not inside Supp E , so the ﬁbre does not intersect ˜ ˜ Supp E ,by[18, Lemma 3.39(2)]. Therefore, E = 0 over the generic point of C ,thatis j j over x ,hence M is numerically trivial over some neighborhood of x , a contradiction. Step 3. We can assume the induced map g : X X is a morphism. To ease notation we replace S with its normalization and denote the induced morphism S → S by h. By generalized adjunction and usual adjunction, we can write K + B + M = K + B + M ≡ 0/C S S S X and K + = K + B . S S X Write g M = M + E where E ≥ 0is exceptional/X .Then j j j M = h M | + h E | ∗ j S ∗ j S and M = μ h M | + μ h E | = M + μ h E | j ∗ j S j ∗ j S S j ∗ j S S EFFECTIVITY OF IITAKA FIBRATIONS 311 and B = + μ h (E | ). S S j ∗ j S Let V be a prime divisor on S and b be its coefﬁcient in B . Then, by the proof V S of Proposition 4.9, 1 μ n j j b ≥ 1 − + l l for some natural number l and integers n ≥ 0. Moreover, n > 0 if V is a component of j j h (E | ). This in particular shows that there is a natural number s depending only on b ∗ j S such that V is a component of h (E | ) for at most s of the j because μ n ≤ 1. ∗ j S j j Step 4.Let F be a general ﬁbre of the induced map S → C and F the correspond- ing ﬁbre of S → C . Restricting to F as in Remark 4.2(6), we get K + B + M = (K + B + M )| ≡ 0. F F F S S S F Also we get K + := (K + )| . F F S S F Denote the morphism F → F by e. Since F is a general ﬁbre, restricting Weil divisors on S to F makes sense, and if P is a Weil divisor on S, then we have (h P)| = e (P| ). ∗ F ∗ F Therefore, M = e (E | ) + e (M | ), M = μ e (M | ), ∗ j F ∗ j F F j ∗ j F and B = B | = + μ h (E | ) = + μ e (E | ). F S F S j ∗ j S F j ∗ j F Since F may not be Q-factorial, we need to make some further constructions. Let (H , ) be a Q-factorial dlt model of (F , ) and ψ : H → F the corresponding H F morphism. By deﬁnition K + = ψ (K + ) H H F F and the exceptional divisors of ψ all appear with coefﬁcient 1 in .Moreover, we can write K + B + M = ψ (K + B + M ) ≡ 0 H H H F F F where B is the sum of the birational transform of B and the reduced exceptional H F divisor of ψ . 312 CAUCHER BIRKAR, DE-QI ZHANG We can assume c : F H is a morphism. By construction, ψ M | = c (E | ) + c (M | ) F ∗ j F ∗ j F which is big, and M = μ c (M | ) and B = + μ c (E | ). H j ∗ j F H H j ∗ j F Moreover, since the exceptional divisors of ψ are components of , the divisor μ c (E | ) has no exceptional component, so it is just the birational transform of j ∗ j F μ e (E | ). j ∗ j F Step 5. Run an LMMP on K . It terminates with a Mori ﬁbre space H → T and the generalized lc property of (H , B + M ) is preserved by the LMMP. Since H H c (E | ) + c (M | ) is big, its pushdown to H is also big, hence ample over T .Let G be ∗ j F ∗ j F a general ﬁbre of the above Mori ﬁbre space. Then restriction to G gives K + B + M = (K + B + M )| ≡ 0. G G G H H H G By construction, M = μ a (M | )| where we can assume a : F H is a mor- j ∗ j F G G phism. Applying Proposition 5.2 and rearranging the indexes, we can assume that there is a natural number t depending only on d, b such that a (M | )| ≡ 0for every j > t. ∗ j F But then a (E | )| is big for each j > t. ∗ j F For each j > t choose a component W of a (E | ) which is ample over T .By j ∗ j F construction, W is the birational transform of a component U of e (E | ) = (h (E | ))| j j ∗ j F ∗ j S F and U in turn is a component of V ∩ F for some component V of h (E | ).Moreover, j j j ∗ j S W = W if and only if U = U if and only if V = V .ByStep3,for each k,V = V k j k j k j k j for at most s of the j.Thusfor each k,W = W for at most s of the j . On the other k j hand, by Steps 3 and 4, the V appear as components of B with coefﬁcient ≥ min{b, }, j S r−t and there exist at least such components. Similarly the W appear as components 1 r−t of B with coefﬁcient ≥ min{b, }, and there exist at least such components. Now 2 s r−t apply Proposition 5.1 to (G , B ) to deduce that is bounded hence r is bounded by some q. Proof of Proposition 5.2. — We argue by induction on the dimension d.The case d = 1 is clear. Suppose that the proposition holds in dimension < d . Step 1. Since (X , B + M ) is generalized lc and K + B is R-Cartier, (X , B ) is lc. Let (X , B ) be a Q-factorial dlt model of (X , B ) and φ : X → X the corresponding morphism. We may assume X X is a morphism. For each j,wehave φ M = M + j j E where E ≥ 0is exceptional/X and M is the pushdown of M .So j j j K + B + μ E + M = φ K + B + M X j X j EFFECTIVITY OF IITAKA FIBRATIONS 313 where M is the pushdown of M. Since the exceptional divisors of φ are components of B and since (X , B + M ) is generalized lc, we deduce E = 0for every j,hence ∗ ∗ M = φ M for every j and M = φ M . Thus we may replace X with X , hence assume j j that (X , B ) is Q-factorial dlt. Step 2. If P ≡ 0, then K + B + M is not pseudo-effective and so we can run an LMMP on K + B + M which terminates with a Mori ﬁbre space, by Lemma 4.4(1). But if P ≡ 0, then K + B is not pseudo-effective as M is big and so we can run an LMMP on K + B which terminates with a Mori ﬁbre space. Note that in both cases the generalized lc property of (X , B + M ) is preserved: in the second case we use the fact K + B + M ≡ 0. Also note that in both cases none of the M is contracted by the LMMP since M is big. In either case we can replace X with the Mori ﬁbre space hence we may assume we already have a Mori ﬁbre structure X → T .Let F be a general ﬁbre of this ﬁbre space. Since M is big, M | is big too. Restricting to F and applying j j induction on dimension we can reduce the problem to the case dim T = 0, that is, when X is a Fano variety of Picard number one. Step 3. Perhaps after changing the indexes we may write M ≡ λ M such that j 1 λ ≥ 1for every j.Now we deﬁne μ ˜ as follows: initially let μ ˜ = μ ;nextdecrease μ ˜ and j j j j 2 instead increase μ ˜ as much as possible so that X , B + μ ˜ M j=2 is generalized lc and K + B + μ ˜ M + P ≡ 0. X j Either we hit a generalized lc threshold, i.e. (X , B + μ ˜ M ) is generalized lc but j=2 j not generalized klt, or that we reach μ ˜ = 0. If the ﬁrst case happens, we stop. But if the second case happens we repeat the process by decreasing μ ˜ and increasing μ ˜ ,and so 3 1 on. We show that the above process involves only a bounded number of the μ .Let l be the smallest number such that μ ˜ = μ for every j > l . We want to show that l is bounded j j depending only on d, b. We can assume l > 1. By construction, μ ˜ ≥ μ λ ≥ μ ≥ (l − 1)b 1 j j j j≤l−1 j≤l−1 so it is enough to show that μ ˜ is bounded depending only on d, b.If M is not numerically 1 1 trivial over X , then the generalized lc threshold of M with respect to (X , B ) is ﬁnite and bounded from above by Theorem 1.5, and this in turn implies boundedness of μ ˜ .But if M is numerically trivial over X , then again μ ˜ is bounded from above but for a different 1 1 314 CAUCHER BIRKAR, DE-QI ZHANG reason: by the cone theorem X can be covered by curves such that −(K + B ) · ≤ 2d which in turn implies that μ ˜ M · ≤ 2d hence μ ˜ M · ≤ 2d where ⊂ Xis the 1 1 1 birational transform of . This is possible only if μ ˜ is bounded from above since M is 1 1 big and Cartier and hence M · ≥ 1. If at the end of the process μ ˜ = 0for every j ≥ 2, then the above arguments show that r is indeed bounded by some number p.But if μ ˜ > 0for some j ≥ 2, then we replace Mwith μ ˜ M and replace P with P +˜ μ M where l is as above, and rearrange the j j l j=l l indexes. We can then assume that (X , B + M ) is generalized lc but not generalized klt. Step 4. The arguments of Step 3 show that, after replacing X, we can assume that there is a prime divisor S on X exceptional over X whose generalized log discrepancy with respect to (X , B + M ) is 0. Since (X , 0) is Q-factorial klt, by Lemma 4.6,there is an extremal contraction φ : X → X which extracts S , the birational transform of S. We can write K + B + M = φ K + B + M X X where B is the sum of S and the birational transform of B and M is the pushdown of M. Since ρ(X ) = 1and φ is extremal, ρ(X ) = 2. Moreover, K + B + M + P ≡ 0 where P is the pullback of P on X . Since ρ(X ) = 1, P and so P is semi-ample, hence we may assume that (X , B + P + M ) is generalized lc with boundary part B + P and nef part M. Since S is a component of B , (X , B − δS + P + M ) is generalized lc where δ> 0 is small, and −δS ≡ K + B − δS + P + M . So by Lemma 4.4(1), we can run an LMMP on −S which ends up with a Mori ﬁbre space X → T . Note that by construction X has Picard number one or two: in any case one of the extremal rays of X corresponds to the Fano contraction X → T and S is positive on this ray. We may assume that both g : X X and h : X X are morphisms. Step 5. Consider the case dim T > 0. Then the Picard number ρ(X ) = 2, hence X X is an isomorphism in codimension one. Moreover, by restricting to the general ﬁbres of X → T and applying induction we may assume M ≡ 0/T for all but a bounded number of j.For anysuch j,M is not big, hence M is not big too. Thus M j j j is ample/X otherwise M would be the pullback of M which is big, a contradiction. j j Let C := φ(S ) and let x be the generic point of C .Then M ≡ 0/T implies that M is not numerically trivial over any neighborhood of x . Now apply Proposition 5.3 to j EFFECTIVITY OF IITAKA FIBRATIONS 315 (X , B + P + M ) at x to bound the number of such j . Therefore r is indeed bounded by some number p depending only on d, b. Step 6. We can now assume dim T = 0. Let X → X be the last step of the ˜ ˜ LMMP which contracts some divisor R .Let x be the generic point of the image of R . ˜ ˜ For each j , either M is ample over X or M is ample over X where M is the push- j j j down of M via X X which we can assume to be a morphism. So either M is j j not numerically trivial over any neighborhood of x or that it is not numerically triv- ial over any neighborhood of x . Now apply Proposition 5.3 to (X , B + P + M ) and (X , B + P + M ) at x and x to bound r by some number p depending only on d, b. 6. ACC for generalized lc thresholds In this section, we reduce the ACC for generalized lc thresholds (Theorem 1.5)to the Global ACC (Theorem 1.6) in lower dimension by adapting a standard argument due to Shokurov. We create an appropriate generalized lc centre of codimension one and restrict to it to do induction. Proposition 6.1. — Assume that Theorem 1.6 holds in dimension ≤ d − 1. Then Theorem 1.5 holds in dimension d . Proof. — Applying induction we may assume that Theorem 1.5 holds in dimension ≤ d − 1. If Theorem 1.5 does not hold in dimension d , then there exist a sequence of generalized lc polarized pairs (X , B + M ) of dimension d with data X → X → Z i i i i i i and M = μ M , and divisors D and N = ν N satisfying the assumptions of i j,i j,i i k,i k,i the theorem but such that the generalized lc thresholds t of D + N with respect to i i (X , B + M ) form a strictly increasing sequence of numbers. We may assume that 0 < i i i t < ∞ for every i. Since the problem is local over X , we can assume X → Z is the i i i i identity morphism. Moreover, we can discard any μ and ν if they are zero. j,i k,i By deﬁnition, X , B + t D + M + t N i i i i i i i is generalized lc with boundary part B + t D and nef part M + t N but i i i i i i X , B + a D + M + a N i i i i i i i is not generalized lc for any a > t . i i If B = B + t D for inﬁnitely many i, then we can easily get a contradiction i i i as the t can be calculated in terms of the coefﬁcients of B and D .Thuswemay assume i i 316 CAUCHER BIRKAR, DE-QI ZHANG that B = B + t D for every i. In particular, this means that there is a generalized lc i i i centre of X , B + t D + M + t N i i i i i i i of codimension ≥ 2 which is not a generalized lc centre of (X , B + M ). i i i We may assume that the given morphism f : X → X is a log resolution of i i (X , B + t D ).Let := B + t D and let R := M + t N . We can write i i i i i i i i i i i i K + + R = f K + + R + E X i i X i i i i where is the sum of the birational transform of and the reduced exceptional divisor of f ,and E ≥ 0is exceptional/X . Then the pair (X , ) is lc but not klt; more precisely i i i i there is a component of which is not a component of E ; moreover, there is such i i a component which is exceptional/X by the last paragraph. In addition, the set of the coefﬁcients of all the union with {μ , t ν } satisﬁes the DCC by Lemma 2.1. i j,i i k,i Run an LMMP/X on K + + R with scaling of some ample divisor which is X i i i i also an LMMP/X on E . Since E is effective and exceptional/X , the LMMP ends on a i i i i model X on which E = 0 (as in the proof of Lemma 4.5). In particular, i i K + + R ≡ 0/X . i i i Let S be a component of exceptional/X but not a component of E . Since i i i the LMMP only contracts components of E , this S is not contracted/X .Deﬁne by i i S the generalized adjunction K + + R = K + + R . S S S X i i i i i i S Then the set of the coefﬁcients of all the satisﬁes DCC by Proposition 4.9.Bycon- struction K + + R ≡ 0/X . S S S i i i Let S → V be the contraction given by the Stein factorization of S → X and let F be i i i i i a general ﬁbre of S → V . We can write i i K + + R = (K + + R )| ≡ 0 F F F S S S F i i i i i i i as in Remark 4.2 (6): here = | and R = R | is the pushdown of R | = F S F F S F i F i i i i i i (M + t N )| where F is the ﬁbre of S → V corresponding to F . i i i F i i i i i Suppose that we can choose the S such that (∗) the set of the coefﬁcients of all the together with {μ | M | ≡ 0}∪ F j,i j,i F {t ν | N | ≡ 0} does not satisfy ACC. i k,i k,i F i EFFECTIVITY OF IITAKA FIBRATIONS 317 But then (∗) contradicts Theorem 1.6.Soitisenoughtoﬁnd the S so that (∗) holds. We will show that (∗) holds if for each i we can ﬁnd S satisfying: (∗∗)(D + N )| is not numerically trivial i i where D on X is the birational transform of D and D is the pushdown of D ;here i i i i i we can assume g : X X is a morphism. Indeed, let B be the sum of the birational i i i transform of B plus the reduced exceptional divisor of f ,and B its pushdown on X .By i i i generalized adjunction we can write K + B + M = K + B + M . S S S X i i i i i i S Write g (N ) = N + Q .Then N | = N + Q where N is the pushdown of N | i i F F F F i F i i i i i i i i and Q is the pushdown of Q | .If N | ≡ 0for every i,then (∗) is satisﬁed. So we can F i F i F i i assume N | ≡ 0for every i,hence by (∗∗) we have i F D + N ≡ D + Q = 0 F F i i i i for every i where D := D | .But now = B + t D and = B + t (D + Q ) F F i S S i S S i i i i i i i i i i where D := D | and Q is the pushdown of Q | . Moreover, since D + Q = 0 S S S i S S S i i i i i i i near F ,Proposition 4.9 and its proof show that the set of the coefﬁcients of all the near F does not satisfy ACC. Thus the set of the coefﬁcients of all the does not satisﬁes ACC, hence (∗) holds. Finally we show that (∗∗) holds. By the negativity lemma, we can write f D + N = D + N + P i i i i i i where P ≥ 0is exceptional/X . By the deﬁnition of t and the assumption B i i i i B + t D , there is a component of P which is a component of but not a compo- i i i i i nent of E .Infactany componentof P not contracted/X , is of this kind. Since P = 0 i i i i is exceptional/X , by the negativity lemma [3, Lemma 3.6.2], there is a component S i i of P with a covering family of curves C (contracted over X )suchthat P · C < 0. So i i i (D + N ) · C > 0 for such curves C, hence (D + N )| is not numerically trivial over i i i i general points of V which implies that we can choose the S so that (∗∗) holds. 7. Global ACC In this section, we show that Global ACC (Theorem 1.6) in dimension < d and ACC for generalized lc thresholds (Theorem 1.5 ) in dimension d together imply Global ACC in dimension d . We ﬁrst deal with the pairs which are generalized lc but not gen- eralized klt. For the general case, we will use Proposition 3.4 and do induction on the number of summands in the nef part of the pair, as illustrated in the introduction. The starting point of the induction is the important result [10, Theorem 1.5] which proves the statement when the nef part is zero. 318 CAUCHER BIRKAR, DE-QI ZHANG Proposition 7.1. — Assume Theorem 1.6 holds in dimension ≤ d − 1. Then Theorem 1.6 holds in dimension d for those (X , B + M ) which are not generalized klt. Proof.— Step 1. Extending we can assume 1 ∈ . If the proposition does not hold, then there is a sequence of generalized lc but not klt polarized pairs (X , B + M ) i i i with data X → X → Z and M = μ M satisfying the assumptions of 1.6 but such i i i j,i j,i that the set of the coefﬁcients of all the B together with the μ does not satisfy ACC. j,i We may assume that f : X → X is a log resolution of (X , B ).Let B be the sum i i i i i i of the birational transform of B and the reduced exceptional divisor of f . We can write K + B + M = f K + B + M + E X i i X i i i i i where E ≥ 0is exceptional/X . We can run an LMMP/X on K + B + M with scaling i X i i i i i of some ample divisor which contracts E and terminates with some model (as in the proof of Lemma 4.5). Moreover, by the generalized non-klt assumption, we can choose f so that there is a prime divisor S on X which is a component of B but not a component of i i i E , hence it is not contracted by the LMMP. Replacing X with the model given by the LMMP allows us to assume that (X , B ) is Q-factorial dlt and that we have a component i i S of B i i Step 2. Write B = b B where B are the distinct irreducible components of k,i i k,i k,i B . If the set of all the coefﬁcients b is not ﬁnite, then we may assume that the b form k,i 1,i a strictly increasing sequence in which case we let P := B . On the other hand, if the i 1,i set of all the coefﬁcients b is ﬁnite, then the set of all the μ is not ﬁnite hence we could k,i j,i assume that the μ form a strictly increasing sequence in which case we let P := M . 1,i i 1,i In either case we can run an LMMP on K + B + M − P ≡− P i i i i for some small > 0 which ends with a Mori ﬁbre space, by Lemma 4.4(1). The gen- eralized lc (and non-klt) property of (X , B + M ) is preserved by the LMMP because i i i K + B + M ≡ 0. i i Step 3. We ﬁrst consider the case when S is not contracted by the LMMP in Step 2, for inﬁnitely many i. Replacing the sequence we can assume this holds for every i. In this case, we replace X with the Mori ﬁbre space constructed, hence we can assume we already have a Mori ﬁbre structure X → T and that P is ample/T .Let F be a general i i i i i ﬁbre of X → T . Then we can write i i K + B + M = K + B + M F F F X i i i i i i F where B = B | and M = M | . Moreover, since P | is ample, the set of the coefﬁ- F F F F F i i i i i i i i cients of all the B together with the set {μ | M | ≡ 0} is not ﬁnite where F is the F j,i j,i F i i EFFECTIVITY OF IITAKA FIBRATIONS 319 ﬁbre of X → T corresponding to F . So applying induction we can assume dim T = 0. i i i In particular, P | is not numerically trivial. Now assume the LMMP of Step 2 contracts S at some step, for inﬁnitely many i. Replacing the sequence we can assume this holds for every i. Replacing X we can assume S is contracted by the ﬁrst step of the LMMP, say X → X .Then P is ample over X , i i i i i hence P | is not numerically trivial. From now on we assume that P | is not numerically trivial. Step 4. Apply generalized adjunction to get K + B + M = K + B + M ≡ 0. S S S X i i i i i i S By Proposition 4.9, the coefﬁcients of B belong to a DCC set depending only on d and .Moreover, M is the pushdown of M| = μ M | . Thus by induction the set S S j,i j,i S i i of the coefﬁcients of all the B together with the set {μ | M | ≡ 0} is ﬁnite. But this S j,i j,i S contradicts Lemma 4.10. Proposition 7.2. — Assume that Theorem 1.6 holds in dimension ≤ d − 1 and that Theorem 1.5 holds in dimension d . Then Theorem 1.6 holds in dimension d . Proof.— Step 1. If the statement is not true, then there is a sequence of generalized lc polarized pairs (X , B + M ) with data X → X → Z and M = μ M satisfying i i i j,i j,i i i i i the assumptions of 1.6 but such that the set of the coefﬁcients of all the B and all the μ j,i put together satisﬁes DCC but not ACC. Write B = b B where B are the distinct k,i i k,i k,i irreducible components of B . As in Steps 1 and 2 of the proof of Proposition 7.1, we can reduce the problem to the situation in which X is a Q-factorial klt variety with a Mori ﬁbre space structure. Restricting to the general ﬁbres of the ﬁbration and applying induction we can in addition assume X is Fano of Picard number one. For each i,let σ(M ) be the number of the μ . Then, by Propositions 5.1 and 5.2, i j,i we can assume that the number of the components of B plus σ(M ) is bounded. Thus we can assume that the number of the components of B and σ(M ) are both independent of i. We just write σ instead of σ(M ). We will do induction on the number σ.By[10, Theorem 1.5], the proposition holds when σ = 0, i.e. when M = 0for every i. So we can assume σ> 0. We may also assume that σ is minimal with respect to all sequences as above, even if is extended to a larger set. Replacing the sequence we may assume that the numbers b and μ form a (not k,i j,i necessarily strict) increasing sequence for each k and each j , because they all belong to the DCC set . By deﬁnition, b ≤ 1. We show that the μ are also bounded from above, k,i j,i i.e. lim μ < +∞ for every j : this follows from the same arguments as in Step 3 of the i j,i proof of Proposition 5.2 by considering the generalized lc threshold of M with respect to j,i 320 CAUCHER BIRKAR, DE-QI ZHANG (X , B + M − μ M ) if M ≡ 0/X for inﬁnitely many i, or by applying boundedness j,i j,i i i i j,i i of the length of extremal rays otherwise. Step 2.ByProposition 7.1, we may assume that (X , B + M ) is generalized klt for i i i every i. In particular, (X , B + (1 + )M ) is generalized klt, and K + B + (1 + )M is i X i i i i i i ample for some small > 0, noting that the Picard number ρ(X ) = 1. We may assume that f : X → X is a log resolution of (X , B ) and can write i i i i i K + B + (1 + )M = f K + B + (1 + )M + E X i i i X i i i i i i where B is the sum of the birational transform of B and the reduced exceptional divisor of f ,and E ≥ 0is exceptional/X .So K + B + (1 + )M is big, and since the μ i i X i i i j,i i i are bounded from above, we deduce that K + B + nM is also big for some ﬁxed X i j,i natural number n 1 independent of i. Now by Proposition 3.4, there exists a natural number m, independent of i,such that |m(K + B + nM )| deﬁnes a birational map for every i. In particular, X i j,i H X , m K + B + nM = 0 i X i j,i so m K + B + nM ∼ mD X i j,i i for some integral divisor mD ≥ 0. The coefﬁcients of mD belong to N,aDCCset.Now i i let D be the R-divisor so that mD is the sum of mD and the fractional part m(K + i i i X B + nM ). Since K + nM is Cartier, mD = mD +mB . On the other hand, i j,i X j,i i i i since the coefﬁcients of B belong to the DCC set ∩[0, 1], the coefﬁcients of mB i i belong to a DCC set as well by Lemma 2.1. Therefore, the coefﬁcients of mD and hence of D belong to a DCC set, depending only on . By extending we can assume that the coefﬁcients of D belong to . By construction, 0 ≤ D ∼ K + B + nM i R X i j,i which in turn implies that 0 ≤ D ∼ K + B + nM R X i i j,i = K + B + M + (n − μ )M ≡ (n − μ )M . X j,i j,i i i j,i j,i Note that we can assume n − μ > 0for every j, i. j,i EFFECTIVITY OF IITAKA FIBRATIONS 321 Step 3. By Lemma 2.2, replacing the sequence X and reordering the indexes j , we may assume that M ≡ λ M so that for each j the numbers λ form a decreasing j,i j,i j,i 1,i sequence. By Step 2, we get D ≡ (n − μ )M ≡ (n − μ )λ M =: ρ M j,i j,i j,i i i j,i 1,i 1,i where we have deﬁned ρ := (n − μ )λ . i j,i j,i For each j,the numbers n − μ and λ form decreasing sequences hence the ρ also form j,i j,i i a decreasing sequence by Lemma 2.1. Now let N := μ M and let u be the generalized lc threshold of D with i j,i j,i i j≥2 i respect to (X , B + N ). Since D ≡ ρ M we get i i i i 1,i K + B + N + u D ≡ K + B + N + u ρ M . X i X i i i i i i i 1,i i i Assume that u ρ ≥ μ for every i.Let v ≤ u be the number so that i i 1,i i i K + B + N + v D ≡ K + B + M ≡ 0 X i X i i i i i i i 1,i that is, v = .Asthe μ form an increasing sequence and the ρ form a decreasing i 1,i i sequence, the v form an increasing sequence. Moreover, if the μ form a strictly in- i 1,i creasing sequence, then the v also form a strictly increasing sequence. Thus the set of the coefﬁcients of all the B + v D together with the {μ | j ≥ 2} is a DCC set but not i j,i i i ACC. Now (X , B + v D + N ) is generalized lc with boundary part B + v D and nef i i i i i i i i part N ,and σ(N )<σ which contradicts the minimality assumption on σ in Step 1. i i Therefore, from now on we may assume that u ρ <μ for every i. i i 1,i 1,i Step 4.Fix i.Let be the set of those elements (α, β) ∈[0, ]×[0,μ ] such i 1,i that K + B + N + αD + β M ≡ K + B + M X X i i i 1,i i i i i which is equivalent to αρ + β = μ .Notethat (0,μ ) ∈ hence = ∅.Now let i 1,i 1,i i i s = sup α | (α, β) ∈ , X , B + αD + N + β M is generalized lc i i i i i i 1,i where the pair in the deﬁnition has boundary part B + αD and nef part N + β M . i 1,i i i Letting t = μ − s ρ we get (s , t ) ∈ . i 1,i i i i i i We show that s is actually a maximum hence in particular X , B + s D + N + t M i i i i i i 1,i 322 CAUCHER BIRKAR, DE-QI ZHANG l l l is generalized lc. If not, then there is a sequence (α ,β ) ∈ such that the α form a strictly increasing sequence approaching s and the β form a strictly decreasing sequence approaching t . Since X , B + α D + N + t M i i i i 1,i is generalized lc, the generalized lc threshold of D with respect to (X , B + N + t M ) i i i i 1,i is at least lim α = s by Theorem 1.5.So X , B + s D + N + t M i i i i i i 1,i is also generalized lc. Hence s is indeed a maximum. Note that s ≤ u . i i i Step 5. Since the coefﬁcients of D belong to and since u is the generalized lc threshold of D with respect to (X , B + N ), u is bounded from above by Theorem 1.5. i i i i Thus s is also bounded from above. So we may assume the s and the t each form an i i i increasing or a decreasing sequence hence s = lim s and t = lim t exist. Since the μ i i 1,i form an increasing sequence and the ρ form a decreasing sequence, the s or the t form i i i an increasing sequence. We will show that in fact the t form an increasing sequence. Assume otherwise, that is, assume the t form a decreasing sequence. We can as- sume it is strictly decreasing. Then the s form a strictly increasing sequence. Since X , B + s D + N + tM i i i i 1,i is generalized lc, we may assume that X , B + sD + N + tM i i i i 1,i is generalized lc too, by Theorem 1.5.Now we canﬁnd ˜ s > s such that (˜ s , t) ∈ ,that i i i i is, ˜ s ρ + t = μ . Since the μ form an increasing sequence and the ρ form a decreasing i i 1,i 1,i i sequence, the ˜ s form an increasing sequence. Moreover, since t < t ≤ μ ≤ lim μ and s(lim ρ ) + t = lim(s ρ + t ) = lim μ i 1,i 1,i i i i i 1,i we deduce lim ρ > 0. Thus as lim(˜ s ρ + t) = lim μ , i i 1,i we get lim ˜ s = lim s = s. In particular this means s ≥˜ s > s ,hence i i i i X , B +˜ s D + N + tM i i i i 1,i is generalized lc which contradicts the maximality assumption of s in Step 4. So we have proved that the t form an increasing sequence. Now by deﬁnition s is i i the generalized lc threshold of D with respect to X , B + N + t M . i i i 1,i EFFECTIVITY OF IITAKA FIBRATIONS 323 So they form a decreasing sequence by Theorem 1.5. Step 6. The purpose of this step is to modify B so that we can assume s = lim s = 0. ˜ ˜ ˜ ˜ Let t be the number so that sρ + t = μ .As s ≥ s, t ≥ t ≥ 0, hence (s, t ) ∈ . Since i i i 1,i i i i i i the μ (resp. ρ ) form an increasing (resp. decreasing) sequence, the t form an increasing 1,i i i sequence. Moreover, lim t = lim(μ − sρ ) = lim(μ − s ρ ) = lim t = t i 1,i i 1,i i i i which implies t ≤ t. We claim that (∗) X , B + sD + N + t M i i i i 1,i is generalized lc. Indeed, let c be the generalized lc threshold of M with respect to 1,i (X , B + sD + N ).Then c ≥ t and by Theorem 1.5, we may assume that the c form a i i i i i i i decreasing sequence. Thus c ≥ lim c ≥ lim t = t ≥ t i i i i and the claim follows. Nowwedeﬁne theboundary C := B + sD on X where B ,asinStep2,isthe i i i i sum of the birational transform of B and the reduced exceptional divisor of X → X , i i and D is the birational transform of D .Then C = B + sD and i i i i i X , C + N + t M i i i 1,i is generalized lc by (∗),and K + C + N + t M ≡ 0. X i i i 1,i Moreover, the set of the coefﬁcients of all the C union the set {μ | j ≥ 2}∪{t } satisﬁes j,i i DCC but not ACC (note that if the μ form a strictly increasing sequence, then so do 1,i the t ). s −s On the other hand, let G := D + sD and let r := .Then i i i 1+s 0 ≤ G ∼ K + C + nM i R X i j,i and G = (1 + s)D ,and i i K + C + r G + N + t M = K + B + s D + N + t M ≡ 0. X i i X i i i i i 1,i i i i 1,i i i The equality also shows X , C + r G + N + t M i i i i i i 1,i 324 CAUCHER BIRKAR, DE-QI ZHANG is generalized lc and that r is the generalized lc threshold of G with respect to (X , C + i i i N + t M ). Therefore extending , replacing B with C , replacing μ with t , replacing i i i 1,i i i 1,i D with G , and replacing s with r allow us to assume that s = lim s = 0. i i i i i Step 7. After replacing X we may assume that there is a prime divisor S on X i i i whose generalized log discrepancy with respect to the generalized lc polarized pair X , B + s D + N + t M i i i i i i 1,i is 0: this follows from our choice of s , t . i i First assume that S is not contracted over X for every i which means that S is i i a component of B + s D .Let d be the coefﬁcient of S in D and let p be the real i i i i i i i number such that K + B + s d S + N + p M ≡ 0. X i i i i i i 1,i Obviously p ≤ μ , and equality holds if and only if s d S ≡ 0, i.e., s d = 0. Since s d S ≤ i 1,i i i i i i i s D and K + B + s D + N + t M ≡ 0 X i i i i i 1,i we have t ≤ p . Then from lim s = 0and μ := lim μ = lim t we arrive at lim p = μ . i i i 1 1,i i i 1 So we may assume that the p form an increasing sequence approaching μ . i 1 Let w be the generalized lc threshold of M with respect to 1,i X , B + s d S + N . i i i i i i Then w ≥ t . Applying Theorem 1.5, we can assume that the w form a decreasing i i i sequence. Then w ≥ lim w ≥ lim t = μ = lim p ≥ p i i i 1 i i which implies that X , B + s d S + N + p M i i i i i i i 1,i is generalized lc with boundary part := B + s d S and nef part R := N + p M . i i i i i 1,i i i i The set of the coefﬁcients of all the union the set {μ | j ≥ 2}∪{p } satisﬁes DCC. j,i i Therefore, by Proposition 7.1, we may assume that p is a constant independent of i. Now μ = lim p = p ≤ μ ≤ lim μ = μ 1 i i 1,i 1,i 1 Thus p = μ ,hence s d = 0, = B ,and R = M .Inother words, (X , B + M ) is not i 1,i i i i i i i i i i generalized klt. This contradicts Proposition 7.1. EFFECTIVITY OF IITAKA FIBRATIONS 325 So after replacing the sequence we may assume that S is exceptional over X for every i. Step 8. By Lemma 4.6, there is an extremal contraction g : X → X extracting S i i i with X being Q-factorial. We can assume X X is a morphism. We can write i i K + B + s D + N + t M = g K + B + s D + N + t M ≡ 0 X i i X i i i i i 1,i i i i i 1,i i i where B is the pushdown of B , D is the birational transform of D ,M is the pushdown i i i 1,i of M ,and N is the pushdown of N .Now S is a component of B . By Lemma 4.4(1) 1,i i i i i we can run the −S -LMMP which terminates on some Mori ﬁbre space X → T .We i i i may assume that dim T = 0for every i,or dim T > 0for every i. Replacing X we may i i assume X X is a log resolution of (X , B + s D ). i i i i i i Since (X , B + M ) is generalized lc and K + B + M ≡ 0, we deduce that K + X X i i i i i i B + M is pseudo-effective. Thus K + B + M is pseudo-effective too. Moreover, by i i X i i construction K + B + s D + N + t M ≡ 0. X i i i i i 1,i So there is the largest number q ∈[t ,μ ] such that i i 1,i K + B + N + q M ≡ 0/T . X i i i 1,i i From s = lim s = 0we get lim t = lim μ = μ from which we derive lim q = μ .Sowe i i 1,i 1 i 1 may assume that the q form an increasing sequence approaching μ .Let w be the gener- i 1 i alized lc threshold of M with respect to the generalized lc polarized pair (X , B + N ). 1,i i i i Then w ≥ t as i i X , B + s D + N + t M i i i i i i 1,i is generalized lc. Moreover, by Theorem 1.5 we can assume the w form a decreasing sequence, hence q ≤ μ ≤ μ = lim t ≤ lim w ≤ w . i 1,i 1 i i i So the pair (X , B + N + q M ) is generalized lc. But the pair is not generalized klt i i i 1,i because S is a component of B i i Step 9. Assume that dim T = 0for every i. Applying Proposition 7.1,wecan assume that the set of the coefﬁcients of all the B union the set {μ |j ≥ 2}∪{q } is ﬁnite. j,i i In particular, this means we can assume q = μ = μ for every i,and that μ = μ for i 1,i 1 j,i j every j, i where μ := lim μ . On the other hand, assume that dim T > 0for every i. j i j,i If M ≡ 0/T ,then q = μ .But if M ≡ 0/T , then by restricting to the general i 1,i 1,i i 1,i i ﬁbres of X → T and applying induction, we deduce that {q } is ﬁnite, hence q = μ i i 1 i i for i 1; so we can assume q = μ = μ . Moreover, by restricting to the general ﬁbres i 1,i 1 326 CAUCHER BIRKAR, DE-QI ZHANG of X → T and applying induction once more, we may assume that the set of the i i horizontal/T coefﬁcients of all the B together with the set {μ | M ≡ 0/T } is ﬁnite. j,i i i j,i i The last paragraph shows that in either case dim T = 0ordimT > 0, we can i i assume (∗∗) K + B + M ≡ 0/T . i i i Let B be obtained from B by replacing the coefﬁcient b with b := lim b .Let M be i i k,i k i k,i i obtained from M by replacing μ with μ = lim μ .Then K + B + M is ample be- i j,i j i j,i X i i cause ρ(X ) = 1 and because either b < b for some k or μ <μ for some j.Moreover, k,i k j,i j by Theorem 1.5, we can assume (X , B + M ) is generalized lc. Thus i i i K + B + M ≥ f K + B + M X i i X i i i i is big. This in turn implies that K + B + M is big too. On the other hand, by the i i last paragraph, we may assume that on the general ﬁbres F of X → T we have: i i i B | = B | and M | ≡ M | . This contradicts (∗∗). F F F F i i i i i i i i 8. Proof of main results In this section, we prove our main results stated in the introduction. Proof of Theorem 1.5 and Theorem 1.6. — By Proposition 6.1,Theorem 1.6 in dimen- sion < d implies Theorem 1.5 in dimension d . On the other hand, by Proposition 7.2, Theorem 1.6 in dimension < d and Theorem 1.5 in dimension d imply Theorem 1.6 in dimension d . So both theorems follow inductively the case d = 1 being trivial. Next we prove a result bounding pseudo-effective thresholds which will be needed for the proof of Theorem 1.3. Theorem 8.1. — Let d be a natural number and a DCC set of nonnegative real numbers. Then there is a real number e ∈ (0, 1) depending only on ,dsuch that if: • (X, B) is projective lc of dimension d , • M = μ M where M are nef Cartier divisors, j j j • the coefﬁcients of B and the μ are in ,and • K + B + M is a big divisor, then K + eB + eM is a big divisor. Proof. — It sufﬁces to show the assertion: there is an e ∈ (0, 1) depending only on , d such that K + eB + eM is pseudo-effective; because then 1 1 vol K + (e + 1)(B + M) = vol (K + B + M + K + eB + eM) X X X 2 2 EFFECTIVITY OF IITAKA FIBRATIONS 327 ≥ vol (K + B + M) > 0 and hence K + e B + e Mis big for e := (1 + e) ∈ (0, 1). If there is no e as in the last paragraph, then there is a sequence of pairs (X , B ) i i and divisors M = μ M satisfying the assumptions of the theorem but such that the i j,i j,i pseudo-effective thresholds e of B + M form a strictly increasing sequence approach- i i i ing 1: by deﬁnition K + e B + e M is pseudo-effective but K + c B + c M is not X i i i i X i i i i i i pseudo-effective for any c < e . i i We can extend and replace the X , B so that we may assume (X , B ) is log i i i i smooth klt. By Lemma 4.4(2), we can run an LMMP on K + e B + e M which ends X i i i i with a minimal model X on which K + e B + e M is semi-ample deﬁning a contraction X i i i i i X → T . Since K + B + M is big and K + e B + e M ≡ 0/T ,wededucethat B + M X X i i i i i i i i i i i i i is big over T . Replacing X we may assume that X X is a log resolution of (X , B ).Let F i i i i i i be a general ﬁbre of X → T and F the corresponding ﬁbre of X → T . By restricting i i i i i to F we get K + e B + e M := K + e B + e M | ≡ 0. F i F i F X i i F i i i i i i i This contradicts Theorem 1.6 because e B + e M is big hence nonzero for every i, i F i F i i so the set of the coefﬁcients of all the e B union with the set {e μ | M | ≡ 0} is not i F i j,i j,i F ﬁnite. Proof of Theorem 1.3. — As usual by taking a log resolution we may assume (X, B) is log smooth. By Theorem 8.1, there exist a rational number e ∈ (0, 1) depending only on , d, r such that K + eB + eMis big, soK + eB + Mis also big. As in Step 2 of X X the proof of Proposition 3.4, there is p ∈ N depending only on e,, r such that r|p and for any nonzero λ ∈ we can ﬁnd γ ∈[eλ, λ) such that pγ is an integer. In particular, we can ﬁnd a boundary such that eB ≤ ≤ B, p is Cartier, K + + M is big, and (X,) is klt. Replacing B with we canthenassume ={ | 0 ≤ i ≤ p − 1} and that (X, B) is klt. By Proposition 3.4, there exist l, n ∈ N depending only on , d, r such that r|n and that |l(K + B + nM)| deﬁnes a birational map. By replacing l with pl we can assume p|l . There is a resolution φ : W → Xsuch that φ l(K + B + nM) ∼ H + G where H is big and base point free and G ≥ 0. Perhaps after replacing l with (2d + 1)l , we can also assume that H is potentially birational [9, Lemma 2.3.4]. Applying Theorem 8.1 once more, there exist rational numbers s, u ∈ (0, 1) de- pending only on , d, r such that K + sB + uM is big. Perhaps after replacing s, u,we X 328 CAUCHER BIRKAR, DE-QI ZHANG can choose a sufﬁciently large natural number q so that qs is integral and divisible by p, qu is integral and divisible by r, qs + l qu + ln s := < 1, and = 1. q + l + 1 q + l + 1 Let X be a minimal model of K + sB + uM, which exists by Lemma 4.4(2). We can assume that the induced map ψ : W X is a morphism. Since X is a minimal model, ∗ ∗ φ (K + sB + uM) = ψ K + sB + uM + E X X where E is effective. Let D = ψ K + sB + uM . Since H is potentially birational, by Lemma 3.1, qD + H is potentially birational and |K +qD + H| deﬁnes a birational map. Thus ∗ ∗ K + φ q(K + sB + uM) + φ l(K + B + nM) W X X also deﬁnes a birational map which in turn implies that K + q(K + sB + uM) + l(K + B + nM) X X X deﬁnes a birational map. Hence the linear system (q + l + 1) K + s B + M deﬁnes a birational map. Therefore (q + l + 1)(K + B + M) also deﬁnes a birational map. By construction r|qu and r|ln,so r|(q + l + 1 = qu + ln). Now put a := m(, d, r) := q + l + 1. Then aMis Cartier, and for any b ∈ N, the linear system |b a(K + B + M) deﬁnes a birational map. But since aM is Cartier and B is effective, b a(K + B + M) ≤ ba(K + B + M) X X which means |m(K + B + M)| also deﬁnes a birational map where m = ba. Next we prove a result similar to 1.3 but we allow a more general nef part M. This result is not used elsewhere in this paper. Theorem 8.2. — Let d be a natural number and a DCC set of nonnegative real numbers. Then there is a natural number m depending only on , d such that if: EFFECTIVITY OF IITAKA FIBRATIONS 329 • (X, B) is projective lc of dimension d , • M = μ M where M are nef Cartier divisors, j j j • the coefﬁcients of B and the μ are in ,and • K + B + M is big, then the linear system | m(K + B) + mμ M | deﬁnes a birational map. X j j Proof. — As usual we may assume (X, B) is log smooth. By Theorem 8.1,there exists a rational number e ∈ (0, 1) depending only on , d such that K + eB + eMis pseudo-effective. As in the proof of 1.3, there is p ∈ N depending only on e, such that we can ﬁnd a boundary ≤ Band numbers ν ∈[eμ ,μ ] such that p and pNare j j j Cartier divisors and K + + Nis big whereN = ν M . X j j Applying Theorem 1.3, there is l ∈ N depending only on p, d (hence only on , d ) such that |l(K + + N)| deﬁnes a birational map and p|l . Replacing l by a multiple we caninaddition assume that l(K + + N) is potentially birational. Then by Lemma 3.1, l(K + + N) + α M X j j is potentially birational for any 0 ≤ α ∈ Z,and K + l(K + + N) + α M X X j j deﬁnes a birational map. Since ν ≤ μ ,wecan take α so that lν + α = (l + 1)μ j j j j j j Therefore (l + 1)K + l + (l + 1)μ M X j j deﬁnes a birational map which in turn implies that (l + 1)(K + B) + (l + 1)μ M X j j deﬁnes a birational map because l ≤ (l + 1)B . Now put m = l + 1. Proof of Theorem 1.2. — Replacing W we can assume the Iitaka ﬁbration I : W X is a morphism, i.e. can assume V = W using the notation before Theorem 1.2. Also we can assume κ(W) ≥ 1 otherwise there is nothing to prove. Let b := b and β := β .Let F F N = N(β) = lcm m ∈ N | ϕ(m) ≤ β where ϕ denotes Euler’s ϕ-function. Let bNu − v A(b, N) := u,v ∈ N,v ≤ bN bNu 330 CAUCHER BIRKAR, DE-QI ZHANG which is a DCC subset of the interval [0, 1). By theresultsof[7] (which is summarized in [24, Lemma 1.2]), replacing W and X by high enough resolutions, we may assume that X is smooth and that there exist a boundary B on X (the discriminant part of I : W → X) and a nef Q-divisor M (the moduli part of I : W → X) such that –NbMis Cartier, – B has simple normal crossing support with coefﬁcients in A(b, N), –K + B + Mis big, – wehaveisomorphisms 0 0 H (W, mbK ) H X, mb(K + B + M) W X for every m ∈ N,and – the rational map deﬁned by |mbK | is birational to the Iitaka ﬁbration I : W → X if and only if |mb(K + B + M)| gives rise to a birational map. By letting = A(b, N) and r = Nb, and applying Theorem 1.3, there is a con- stant m(, d, r) depending only on , d, r, (hence depending only on d, b,β)suchthat |m(K + B + M)| deﬁnes a birational map for any m ∈ N divisible by m(, d, r).Now simply let m(d, b ,β ) = bm(, d, r). Acknowledgements The ﬁrst author was partially supported by a grant of the Leverhulme Trust. Part of this work was done when the ﬁrst author visited National University of Singapore in April 2014. Part of this work was done when the ﬁrst author visited National Taiwan University in August-September 2014 with the support of the Mathematics Division (Taipei Ofﬁce) of the National Center for Theoretical Sciences. The visit was arranged by Jungkai A. Chen. He wishes to thank them all. The second author was partially supported by an ARF of National University of Singapore. The authors would like to thank the referee for the very useful corrections and suggestions which helped to simplify and clarify some of the proofs. REFERENCES 1. C. BIRKAR, On existence of log minimal models, Compos. Math., 145 (2009), 1442–1446. 2. C. BIRKAR, Existence of log canonical ﬂips and a special LMMP, Publ. Math. Inst. Hautes Études Sci., 115 (2012), 325– 3. C. BIRKAR,P.CASCINI,C.HACON and J. M KERNAN, Existence of minimal models for varieties of log general type, J. Am. Math. Soc., 23 (2010), 405–468. 4. C. BIRKAR and Z. HU, Log canonical pairs with good augmented base loci, Compos. Math., 150 (2014), 579–592. 5. G. Di CERBO, Uniform bounds for the Iitaka ﬁbration, Ann. Sc.Norm. Super. Pisa Cl.Sci.(5), 13 (2014), 1133–1143. EFFECTIVITY OF IITAKA FIBRATIONS 331 6. J. CHEN and M. CHEN, Explicit birational geometry of threefolds of general type, I, Ann. Sci. Éc. Norm. Super., 43 (2010), 365–394. 7. O. FUJINO and S. MORI, A canonical bundle formula, J. Differ. Geom., 56 (2000), 167–188. 8. C. HACON and J. M KERNAN, Boundedness of pluricanonical maps of varieties of general type, Invent. Math., 166 (2006), 1–25. 9. C. D. HACON,J.M KERNAN and C. XU, On the birational automorphisms of varieties of general type, Ann. Math. (2), 177 (2013), 1077–1111. 10. C. D. HACON,J.M KERNAN and C. XU, ACC for log canonical thresholds, Ann. Math. (2), 180 (2014), 523–571. 11. C. D. HACON and C. XU, Boundedness of log Calabi-Yau pairs of Fano type, Math. Res. Lett. (to appear), arXiv:1410.8187. 12. S. IITAKA, Deformations of compact complex surfaces, II, J. Math.Soc.Jpn., 22 (1970), 247–261. 13. X. JIANG, On the pluricanonical maps of varieties of intermediate Kodaira dimension, Math. Ann., 356 (2013), 979– 14. Y. KAWAMATA, On the plurigenera of minimal algebraic 3-folds with K ≡ 0, Math. Ann., 275 (1986), 539–546. 15. Y. KAWAMATA, On the length of an extremal rational curve, Invent. Math., 105 (1991), 609–611. 16. Y. KAWAMATA, Subadjunction of log canonical divisors. II, Am. J. Math., 120 (1998), 893–899. 17. J. KOLLÁR, et al., Flips and abundance for algebraic threefolds, Astérisque, 211 (1992). 18. J. KOLLÁR and S. MORI, Birational geometry of algebraic varieties, Cambridge Tracts in Math., vol. 134, Cambridge Univ. Press, Cambridge, 1998. 19. G. PACIENZA, On the uniformity of the Iitaka ﬁbration, Math. Res. Lett., 16 (2009), 663–681. 20. V. V. SHOKUROV, 3-fold log ﬂips, With an appendix by Yujiro Kawamata, Russ. Acad. Sci. Izv. Math., 40 (1993), 95–202. 21. S. TAKAYAMA, Pluricanonical systems on algebraic varieties of general type, Invent. Math., 165 (2006), 551–587. 22. G. TODOROV and C. XU, On Effective Log Iitaka Fibration for 3-folds and 4-folds, Algebra Number Theory, 3 (2009), 697–710. 23. H. TSUJI, Pluricanonical systems of projective varieties of general type I, Osaka J. Math., 43 (2006), 967–995. 24. E. VIEHWEG and D.-Q. ZHANG, Effective Iitaka ﬁbrations, J. Algebraic Geom., 18 (2009), 711–730. C. B. DPMMS, Centre for Mathematical Sciences, University of Cambridge, Wilberforce Road, Cambridge CB3 0WB, UK cb496@dpmms.cam.ac.uk D.-Q. Z. Department of Mathematics, National University of Singapore, 10 Lower Kent Ridge Road, Singapore 119076, Singapore matzdq@nus.edu.sg Manuscrit reçu le 16 décembre 2014 Manuscrit accepté le 14 décembre 2015 publié en ligne le 18 janvier 2016.

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Effectivity of Iitaka fibrations and pluricanonical systems of polarized pairs

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Effectivity of Iitaka fibrations and pluricanonical systems of polarized pairs

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