aa r X i v : . [ m a t h . AG ] N ov REMARKS ON THE ABUNDANCE CONJECTURE
KENTA HASHIZUME
Abstract.
We prove the abundance theorem for log canonical n -folds such that the boundary divisor is big assuming the abundanceconjecture for log canonical ( n − Contents
1. Introduction 12. Notations and definitions 33. Proof of the main theorem 54. Minimal model program in dimension four 9References 101.
Introduction
One of the most important open problems in the minimal modeltheory for higher-dimensional algebraic varieties is the abundance con-jecture. The three-dimensional case of the above conjecture was com-pletely solved (cf. [KeMM] for log canonical threefolds and [F1] forsemi log canonical threefolds). However, Conjecture 1.1 is still open indimension ≥
4. In this paper, we deal with the abundance conjecturein relative setting.
Conjecture 1.1 (Relative abundance) . Let π : X → U be a projectivemorphism of varieties and ( X, ∆) be a (semi) log canonical pair. If K X + ∆ is π -nef, then it is π -semi-ample. Hacon and Xu [HX1] proved that Conjecture 1.1 for log canonicalpairs and Conjecture 1.1 for semi log canonical pairs are equivalent(see also [FG]). If ( X, ∆) is Kawamata log terminal and ∆ is big, then Date : 2015/10/30, version 0.21.2010
Mathematics Subject Classification.
Primary 14E30; Secondary 14J35.
Key words and phrases. abundance theorem, big boundary divisor, good minimalmodel, finite generation of adjoint ring.
Conjecture 1.1 follows from the usual Kawamata–Shokurov base pointfree theorem in any dimension. This special case of Conjecture 1.1plays a crucial role in [BCHM]. Therefore, it is natural to considerConjecture 1.1 for log canonical pairs ( X, ∆) under the assumptionthat ∆ is big.In this paper, we prove the following theorem. Theorem 1.2 (Main Theorem) . Assume Conjecture 1.1 for log canon-ical ( n − -folds. Then Conjecture 1.1 holds for any projective mor-phism π : X → U and any log canonical n -fold ( X, ∆) such that ∆ isa π -big R -Cartier R -divisor. We prove it by using the log minimal model program (log MMP, forshort) with scaling. A key gradient is termination of the log minimalmodel program with scaling for Kawamata log terminal pairs such thatthe boundary divisor is big (cf. [BCHM]). For details, see Section 3.By the above theorem, we obtain the following results in the minimalmodel theory for 4-folds.
Theorem 1.3 (Relative abundance theorem) . Let π : X → U be aprojective morphism from a normal variety to a variety, where the di-mension of X is four. Let ( X, ∆) be a log canonical pair such that ∆ is a π -big R -Cartier R -divisor. If K X + ∆ is π -nef, then it is π -semi-ample. Corollary 1.4 (Log minimal model program) . Let π : X → U bea projective morphism of normal quasi-projective varieties, where thedimension of X is four. Let ( X, ∆) be a log canonical pair such that ∆ is a π -big R -Cartier R -divisor. Then any log MMP of ( X, ∆) withscaling over U terminates with a good minimal model or a Mori fiberspace of ( X, ∆) over U . Moreover, if K X + ∆ is π -pseudo-effective,then any log MMP of ( X, ∆) over U terminates. Corollary 1.5 (Finite generation of adjoint ring) . Let π : X → U be a projective morphism from a normal variety to a variety, wherethe dimension of X is four. Let ∆ • = (∆ , · · · , ∆ n ) be an n -tuple of π -big Q -Cartier Q -divisors such that ( X, ∆ i ) is log canonical for any ≤ i ≤ n . Then the adjoint ring R ( π, ∆ • ) = M ( m , ··· , m n ) ∈ ( Z ≥ ) n π ∗ O X ( x n X i =1 m i ( K X + ∆ i ) y ) is a finitely generated O U -algebra. We note that we need to construct log flips for log canonical pairsto run the log minimal model program. Fortunately, the existence of
EMARKS ON THE ABUNDANCE CONJECTURE 3 log flips for log canonical pairs is known for all dimensions (cf. [S1] forthreefolds, [F2] for 4-folds and [B3] or [HX2] for all higher dimensions).Therefore we can run the log minimal model program for log canonicalpairs in all dimensions. By the above corollaries, we can establishalmost completely the minimal model theory for any log canonical 4-fold ( X, ∆) such that ∆ is big.The contents of this paper are as follows. In Section 2, we collectsome notations and definitions for reader’s convenience. In Section 3,we prove Theorem 1.2. In Section 4, we discuss the log minimal modelprogram for log canonical 4-folds and prove Theorem 1.3, Corollary 1.4and Corollary 1.5.Throughout this paper, we work over the complex number field. Acknowledgments.
The author would like to thank his supervisorProfessor Osamu Fujino for many useful advice and suggestions. Heis grateful to Professor Yoshinori Gongyo for giving information aboutthe latest studies of the minimal model theory. He also thanks mycolleagues for discussions.2.
Notations and definitions
In this section, we collect some notations and definitions. We willfreely use the standard notations in [BCHM]. Here we write down someimportant notations and definitions for reader’s convenience. (Divisors) . Let X be a normal variety. WDiv R ( X ) is the R -vectorspace with canonical basis given by the prime divisors of X . A variety X is called Q -factorial if every Weil divisor is Q -Cartier. Let π : X → U be a morphism from a normal variety to a variety and let D = P a i D i be an R -divisor on X . Then D is a boundary R -divisor if 0 ≤ a i ≤ i . The round down of D , denoted by x D y , is P x a i y D i where x a i y is the largest integer which is not greater than a i . D is pseudo-effectiveover U (or π -pseudo-effective ) if D is π -numerically equivalent to thelimit of effective R -divisors modulo numerically equivalence over U . D is nef over U (or π -nef ) if it is R -Cartier and ( D · C ) ≥ C on X contained in a fiber of π . D is big over U (or π -big ) if it is R -Cartier and there exists a π -ample divisor A and aneffective divisor E such that D ∼ R , U A + E . D is semi-ample over U (or π -semi-ample ) if D is an R ≥ -linear combination of semi-ample Cartierdivisors over U , or equivalently, there exists a morphism f : X → Y toa variety Y over U such that D is R -linearly equivalent to the pullbackof an ample R -divisor over U . KENTA HASHIZUME (Singularities of pairs) . Let π : X → U be a projective morphismfrom a normal variety to a variety and ∆ be an effective R -divisor suchthat K X + ∆ is R -Cartier. Let f : Y → X be a birational morphism.Then f is called a log resolution of the pair ( X, ∆) if f is projective, Y is smooth, the exceptional locus Ex( f ) is pure codimension one andSupp f − ∗ ∆ ∪ Ex( f ) is simple normal crossing. Suppose that f is a logresolution of the pair ( X, ∆). Then we may write K Y = f ∗ ( K X + ∆) + X b i E i where E i are distinct prime divisors on Y . Then the log discrepancy a ( E i , X, ∆) of E i with respect to ( X, ∆) is 1 + b i . The pair ( X, ∆)is called Kawamata log terminal ( klt , for short) if a ( E i , X, ∆) > f of ( X, ∆) and any E i on Y . ( X, ∆) is called logcanonical ( lc , for short) if a ( E i , X, ∆) ≥ f of( X, ∆) and any E i on Y . ( X, ∆) is called divisorially log terminal ( dlt ,for short) if ∆ is a boundary R -divisor and there exists a log resolution f : Y → X of ( X, ∆) such that a ( E, X, ∆) > f -exceptionaldivisor E on Y . Definition 2.3 (log minimal models) . Let π : X → U be a projectivemorphism from a normal variety to a variety and let ( X, ∆) be a logcanonical pair. Let π ′ : Y → U be a projective morphism from anormal variety to U and φ : X Y be a birational map over U suchthat φ − does not contract any divisors. Set ∆ Y = φ ∗ ∆. Then the pair( Y, ∆ Y ) is a log minimal model of ( X, ∆) over U if(1) K Y + ∆ Y is nef over U , and(2) for any φ -exceptional prime divisor D on X , we have a ( D, X, ∆) < a ( D, Y, ∆ Y ) . A log minimal model ( Y, ∆ Y ) of ( X, ∆) over U is called a good minimalmodel if K Y + ∆ Y is semi-ample over U .Finally, let us recall the definition of semi log canonical pairs. Definition 2.4 (semi log canonical pairs, cf. [F4, Definition 4.11.3]) . Let X be a reduced S scheme. We assume that it is pure n -dimensionaland normal crossing in codimension one. Let X = ∪ X i be the irre-ducible decomposition and let ν : X ′ = ∐ X ′ i → X = ∪ X i be thenormalization. Then the conductor ideal of X is defined by cond X = H om O X ( ν ∗ O X ′ , O X ) ⊂ O X and the conductor C X of X is the subscheme defined by cond X . Since X is S scheme and normal crossing in codimension one, C X is a reducedclosed subscheme of pure codimension one in X . EMARKS ON THE ABUNDANCE CONJECTURE 5
Let ∆ be a boundary R -divisor on X such that K X + ∆ is R -Cartierand Supp ∆ does not contain any irreducible component of C X . An R -divisor Θ on X ′ is defined by K X ′ + Θ = ν ∗ ( K X + ∆) and we setΘ i = Θ | X ′ i . Then ( X, ∆) is called semi log canonical ( slc , for short) if( X ′ i , Θ i ) is lc for any i .3. Proof of the main theorem
In this section, we prove Theorem 1.2. Before the proof, let us recallthe useful theorem called dlt blow-up by Hacon.
Theorem 3.1 (cf. [F3, Theorem 10.4], [KK, Theorem 3.1]) . Let X bea normal quasi-projective variety of dimension n and let ∆ be an R -divisor such that ( X, ∆) is log canonical. Then there exists a projectivebirational morphism f : Y → X from a normal quasi-projective variety Y such that (1) Y is Q -factorial, and (2) if we set ∆ Y = f − ∗ ∆ + X E : f -exceptional E, then ( Y, ∆ Y ) is dlt and K Y + ∆ Y = f ∗ ( K X + ∆) .Proof of Theorem 1.2. Without loss of generality, we can assume that U is affine. By Theorem 3.1, we may assume that X is Q -factorial.Let V be the finite dimensional subspace in WDiv R ( X ) spanned by allcomponents of ∆. We set N = { B ∈ V | ( X, B ) is log canonical and K X + B is π -nef } . Then N is a rational polytope in V (cf. [F4, Theorem 4.7.2 (3)], [S2,6.2. First Main Theorem]). Since ∆ is π -big, there are finitely many π -big Q -Cartier Q -divisors ∆ , · · · , ∆ l ∈ N and positive real numbers r , · · · , r l such that P li =1 r i = 1 and P li =1 r i ∆ i = ∆. Since we have K X + ∆ = P li =1 r i ( K X + ∆ i ), it is sufficient to prove that K X + ∆ i is π -semi-ample for any i . Therefore we may assume that ∆ is a Q -divisor.By using Theorem 3.1 again, we may assume that ( X, ∆) is dlt.If x ∆ y = 0, then ( X, ∆) is klt and Theorem 1.2 follows from [BCHM,Corollary 3.9.2]. Thus we may assume that x ∆ y = 0. Let k be apositive integer such that k ( K X + ∆) is Cartier. Pick a sufficientlysmall positive rational number ǫ such that ∆ − ǫ x ∆ y is big over U and (2 kǫ · dim X ) / (1 − ǫ ) < . By [BCHM, Lemma 3.7.5], there is aboundary Q -divisor ∆ ′ , which is the sum of a general π -ample Q -divisorand an effective divisor, such that K X + (∆ − ǫ x ∆ y ) ∼ Q , U K X + ∆ ′ KENTA HASHIZUME and ( X, ∆ ′ ) is klt. By [BCHM, Theorem E], the ( K X + ∆ ′ )-log MMPwith scaling a π -ample divisor X = X X · · · X i · · · over U terminates. Since ∆ − ǫ x ∆ y ∼ Q , U ∆ ′ , it is also the log MMP of( X, ∆ − ǫ x ∆ y ) over U . Let ∆ i be the birational transform of ∆ on X i .Then ( X m , ∆ m − ǫ x ∆ m y ) is a log minimal model or a Mori fiber space h : X m → Z of ( X, ∆ − ǫ x ∆ y ) over U for some m ∈ Z > .Let f i : X i → V i be the contraction morphism of the i -th step ofthe log MMP over U , that is, X i +1 = V i or X i +1 → V i is the flipof f i over U . Then K X i + ∆ i is nef over U and f i -trivial for any i ≥
1. Indeed, by the induction on i , it is sufficient to prove that K X + ∆ is f -trivial and K X + ∆ is nef over U . Recall that k is apositive integer such that k ( K X + ∆) is Cartier. We show that K X + ∆is f -trivial and k ( K X + ∆ ) is a nef Cartier divisor over U . Since K X + ∆ is nef over U , for any ( K X + ∆ − ǫ x ∆ y )-negative extremalray over U , it is also a ( K X + ∆ − x ∆ y )-negative extremal ray over U .Then we can find a rational curve C on X contracted by f such that0 < − ( K X + ∆ − x ∆ y ) · C ≤ X by [F3, Theorem 18.2]. By thechoice of ǫ , we have0 ≤ k ( K X + ∆) · C = k − ǫ (cid:0) ( K X + ∆ − ǫ x ∆ y ) · C − ǫ ( K X + ∆ − x ∆ y ) · C (cid:1) < kǫ − ǫ · X < . Since k ( K X + ∆) is Cartier, k ( K X + ∆) · C is an integer. Then wehave k ( K X + ∆) · C = 0 and thus K X + ∆ is f -trivial. By the conetheorem (cf. [F4, Theorem 4.5.2]), there is a Cartier divisor D on V such that k ( K X + ∆) ∼ f ∗ D . Since k ( K X + ∆) is nef over U , D is alsonef over U . Let g : W → X and g ′ : W → X be a common resolutionof X X . Then g ∗ ( K X + ∆) = g ′∗ ( K X + ∆ ) by the negativitylemma because K X + ∆ is f -trivial. Then k ( K X + ∆ ) is the pullbackof D and therefore it is a nef Cartier divisor over U . Thus, K X i + ∆ i is nef over U and f i -trivial for any i ≥ X i X i +1 and thenegativity lemma, we see that K Xi + ∆ i is semi-ample over U if andonly if K X i +1 + ∆ i +1 is semi-ample over U for any 1 ≤ i ≤ m − X, ∆) with ( X m , ∆ m ), we may assume that X is a logminimal model or a Mori fiber space h : X → Z of ( X, ∆ − ǫ x ∆ y ) over U . We note that after replacing ( X, ∆) with ( X m , ∆ m ), ( X, ∆) is lcbut not necessarily dlt. EMARKS ON THE ABUNDANCE CONJECTURE 7
Case 1. X is a Mori fiber space h : X → Z of ( X, ∆ − ǫ x ∆ y ) over U . Proof of Case 1.
First, note that in this case K X + ∆ is h -trivial by theabove discussion. Moreover x ∆ y is ample over Z . By the cone theorem(cf. [F4, Theorem 4.5.2]), there exists a Q -Cartier Q -divisor Ξ on Z such that K X + ∆ ∼ Q , U h ∗ Ξ. Since x ∆ y is ample over Z , Supp x ∆ y dominates Z . In particular, there exists a component of x ∆ y , whichwe put T , such that T dominates Z . Let f : ( Y, ∆ Y ) → ( X, ∆) be adlt blow-up (see Lemma 3.1) and e T be the strict transform of T on Y .Then K X +∆ is semi-ample over U if and only if K Y +∆ Y is semi-ampleover U . Furthermore, we have K e T + Diff(∆ Y − e T ) = (( h ◦ f ) | e T ) ∗ Ξ since e T dominates Z . Thus it is sufficient to prove that K e T + Diff(∆ Y − e T ) issemi-ample over U . Since ( Y, ∆ Y ) is dlt, e T is normal by [KM, Corollary5.52]. By [K, 17.2. Theorem], we see that the pair ( e T ,
Diff(∆ Y − e T ))is lc. Then K e T + Diff(∆ Y − e T ) is semi-ample over U by the relativeabundance theorem for log canonical ( n − (cid:3) Case 2. X is a log minimal model of ( X, ∆ − ǫ x ∆ y ) over U . Proof of Case 2.
In this case, both K X + ∆ and K X + ∆ − ǫ x ∆ y arenef over U . By [BCHM, Corollary 3.9.2], K X + ∆ − ǫ x ∆ y is semi-ample over U . Therefore we may assume that x ∆ y = 0, and thereexists a sufficiently large and divisible positive integer l such that both l ( K X + ∆) and lǫ x ∆ y are Cartier and π ∗ π ∗ O X ( l ( K X + ∆ − ǫ x ∆ y )) → O X ( l ( K X + ∆ − ǫ x ∆ y ))is surjective. Then, in the following diagram, π ∗ π ∗ O X ( l ( K X + ∆ − ǫ x ∆ y )) | X \ x ∆ y (cid:15) (cid:15) / / π ∗ π ∗ O X ( l ( K X + ∆)) | X \ x ∆ y (cid:15) (cid:15) O X ( l ( K X + ∆ − ǫ x ∆ y )) | X \ x ∆ y ∼ = / / O X ( l ( K X + ∆)) | X \ x ∆ y the left vertical morphism is surjective. Moreover, the lower horizontalmorphism is an isomorphism. Therefore the right vertical morphism issurjective. Thus π ∗ π ∗ O X ( l ( K X + ∆)) → O X ( l ( K X + ∆)) is surjectiveoutside of x ∆ y .Next, set D = Diff(∆ − x ∆ y ) and consider the following exact se-quence0 → O X ( l ′ ( K X + ∆) − x ∆ y ) → O X ( l ′ ( K X + ∆)) → O x ∆ y ( l ′ ( K x ∆ y + D )) → , KENTA HASHIZUME where l ′ is a sufficiently large and divisible positive integer such that1 /l ′ ≤ ǫ . Then we have l ′ ( K X + ∆) − x ∆ y = l ′ ( K X + ∆ − l ′ x ∆ y ) . Moreover, ∆ − (1 /l ′ ) x ∆ y is big over U and K X + ∆ − (1 /l ′ ) x ∆ y is nefover U . Since ( X, ∆ − (1 /l ′ ) x ∆ y ) is klt, by [BCHM, Lemma 3.7.5], wemay find a π -big Q -Cartier Q -divisor A + B , where A ≥ Q -divisor over U and B ≥
0, such that (
X, A + B ) is klt and∆ − (1 /l ′ ) x ∆ y ∼ Q , U A + B . In particular, ( X, B ) is klt. Furthermore, A + ( l ′ − K X + ∆ − (1 /l ′ ) x ∆ y ) is ample over U . Thus we have l ′ ( K X + ∆) − x ∆ y ∼ Q , U K X + A + ( l ′ − K X + ∆ − l ′ x ∆ y ) + B and R π ∗ O X ( l ′ ( K X + ∆) − x ∆ y ) = 0 (cf. [KaMM, Theorem 1-2-5]).Then π ∗ O X ( l ′ ( K X + ∆)) → π ∗ O x ∆ y ( l ′ ( K x ∆ y + D )) is surjective and thus π ∗ π ∗ O X ( l ′ ( K X + ∆)) ⊗ O x ∆ y → π ∗ π ∗ O x ∆ y ( l ′ ( K x ∆ y + D )) is surjective.We can check that the pair ( x ∆ y , D ) is semi log canonical. Indeed,since ( X, ∆ − ǫ x ∆ y ) is klt and since X is Q -factorial, by [KM, Corollary5.25], x ∆ y is Cohen–Macaulay. In particular, x ∆ y satisfies the S condition. Moreover, since ( X, ∆) is lc, x ∆ y is normal crossing incodimension one. We also see that D does not contain any irreduciblecomponent of C x ∆ y by [C, 16.6 Proposition]. Therefore ( x ∆ y , D ) is semilog canonical by [K, 17.2 Theorem]. Since K x ∆ y + D = ( K X + ∆) | x ∆ y is nef over U , K x ∆ y + D is semi-ample over U by [HX1, Theorem 1.4]and the relative abundance theorem for log canonical ( n − π ∗ π ∗ O X ( l ′ ( K X + ∆)) ⊗ O x ∆ y (cid:15) (cid:15) / / π ∗ π ∗ O x ∆ y ( l ′ ( K x ∆ y + D )) (cid:15) (cid:15) O X ( l ′ ( K X + ∆)) ⊗ O x ∆ y ∼ = / / O x ∆ y ( l ′ ( K x ∆ y + D ))the right vertical morphism and the upper horizontal morphism areboth surjective. Furthermore, the lower horizontal morphism is anisomorphism. Therefore the left vertical morphism is surjective. Then π ∗ π ∗ O X ( l ′ ( K X +∆)) → O X ( l ′ ( K X +∆)) is surjective in a neighborhoodof x ∆ y .Therefore, π ∗ π ∗ O X ( l ( K X + ∆)) → O X ( l ( K X + ∆)) is surjective forsome sufficiently large and divisible positive integer l . So we are done. (cid:3) Thus, in both case, K X + ∆ is semi-ample over U . Therefore wecomplete the proof. (cid:3) EMARKS ON THE ABUNDANCE CONJECTURE 9 Minimal model program in dimension four
In this section, we discuss the log minimal model for log canonical4-folds and prove Theorem 1.3, Corollary 1.4 and Corollary 1.5.
Proof of Theorem 1.3.
It immediately follows from Theorem 1.2 sincethe relative abundance theorem for log canonical 3-folds holds (cf. [F1,Theorem A.2]). (cid:3)
Proposition 4.1.
Let π : X → U be a projective morphism of normalquasi-projective varieties, where the dimension of X is four. Let ( X, ∆) be a log canonical pair and let A be an effective R -divisor such that ( X, ∆ + A ) is log canonical and K X + ∆ + A is π -nef. Then we canrun the log MMP of ( X, ∆) with scaling A over U and this log MMPwith scaling terminates.Proof. We can run the log MMP of ( X, ∆) with scaling A over U by[F4, Remark 4.9.2]. Therefore we only have to prove the terminationof the log MMP with scaling.Suppose by contradiction that we get an infinite sequence of bira-tional maps by running the log MMP with scaling A ( X = X , ∆ = ∆ , λ A ) · · · ( X i , ∆ i , λ i A i ) · · · over U , where A i is the birational transform of A on X i and λ i = inf { µ ∈ R ≥ | K X i + ∆ i + µA i is nef over U } for every i ≥
1. Let X i → V i be the contraction morphism of the i -thstep of the ( K X +∆)-log MMP with scaling A over U . Note that by [B2,Lemma 3.8], the log MMP with scaling terminates for all Q -factorialdlt 4-folds. By the same argument as in the proof of [F4, Lemma 4.9.3],we obtain the following diagram( Y , Ψ ) α (cid:15) (cid:15) / / ❴❴ · · · / / ❴❴ ( Y k = Y , Ψ k = Ψ ) α (cid:15) (cid:15) / / ❴❴ · · · / / ❴❴ ( Y i , Ψ i ) α i (cid:15) (cid:15) / / ❴❴ · · · ( X , ∆ ) / / ❴❴❴❴❴❴❴❴❴ ( X , ∆ ) / / ❴❴❴❴❴ · · · / / ❴❴ ( X i , ∆ i ) / / ❴❴ · · · such that(1) ( Y i , Ψ i ) is Q -factorial dlt and K Y i + Ψ i = α ∗ i ( K X i + ∆ i ),(2) the sequence of birational maps( Y i , Ψ i ) · · · ( Y k i i , Ψ k i i ) = ( Y i +1 , Ψ i +1 )is a finite number of steps of the ( K Y i + Ψ i )-log MMP over V i for any i ≥
1, and (3) the sequence of the upper horizontal birational maps is an in-finite sequence of divisorial contractions and log flips of the( K Y + Ψ )-log MMP over U .For every i ≥ ≤ j < k i , let A ji be the birational transform of α ∗ A on Y ji and let λ ji = inf { µ ∈ R ≥ | K Y ji + Ψ ji + µA ji is nef over U } . Then we have λ ji = λ i for any i ≥ ≤ j < k i . Indeed, since K X i + ∆ i + λ i A i is nef over U and it is also trivial over V i , there is an R -Cartier divisor D , which is nef over U , on V i such that K X i + ∆ i + λ i A i is R -linearly equivalent to the pullback of D . Since A i = α ∗ i A i , bythe condition (1), K Y i + Ψ i + λ i A i is also R -linearly equivalent to thepullback of D . Thus K Y i + Ψ i + λ i A i is nef over U . Moreover, bythe condition (2), K Y ji + Ψ ji + λ i A ji is also R -linearly equivalent to thepullback of D . Therefore K Y ji + Ψ ji + λ i A ji is nef over U and trivialover V i for any 0 ≤ j < k i . We also see that K Y ji + Ψ ji + µA ji is not nefover V i for any µ ∈ [0 , λ i ) by the condition (2). In particular it is notnef over U . Therefore we have λ ji = λ i for any i ≥ ≤ j < k i .By these facts, we can identify the sequence of birational maps( Y , Ψ ) · · · ( Y ji , Ψ ji ) ( Y j +1 i , Ψ j +1 i ) · · · with an infinite sequence of birational maps of the ( K Y +Ψ )-log MMPwith scaling A = α ∗ A over U . But then it must terminate by [B2,Lemma 3.8]. It contradicts to our assumption. So we are done. (cid:3) Proof of Corollary 1.4.
The first half of the assertions immediately fol-lows from Proposition 4.1 and Theorem 1.3. For the latter half, if K X + ∆ is π -pseudo-effective then it is π -effective by the first halfof this corollary. By [B1, Main Theorem 1.3], termination of any logMMP follows. So we are done. (cid:3) Proof of Corollary 1.5.
Without loss of generality, we can assume that U is affine. Then the assertion follows from Proposition 4.1 and Theo-rem 1.3 with the same argument as in the proof of [H, Lemma 3.2] andthe discussion of [H, Section 4]. (cid:3) References [B1] C. Birkar, Ascending chain condition for log canonical thresholds andtermination of log flips, Duke Math. J. (2007), no. 1, 173–180.[B2] C. Birkar, On existence of log minimal models, Compos. Math. (2010), no. 4, 919–928.
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