ON MINIMAL LOG DISCREPANCIES ON VARIETIES WITH FIXED GORENSTEIN INDEX YUSUKENAKAMURA Abstract. We generalize the rationality theorem of the accumulation 5 pointsoflogcanonicalthresholdswhichwasprovedbyHacon,McKernan, 1 0 and Xu. Further, we apply the rationality to the ACC problem on the 2 minimal log discrepancies. We study the set of log discrepancies on varieties with fixed Gorenstein index. As a corollary, we prove that n theminimal log discrepancies ofthree-dimensional canonical pairs with a J fixedcoefficients satisfy theACC. 6 2 ] G 1. Introduction A Theminimallogdiscrepancy(mldforshort)wasintroducedbyShokurov, . h in order to reduce the conjecture of terminations of flips to a local problem at about singularities. Recently, this has been a fundamental invariant in the m minimal model program. There are two conjectures on mld’s, the ACC [ (ascending chain condition) conjecture and the LSC (lower semi-continuity) conjecture. Shokurov showed that these two conjectures imply the conjec- 1 v ture of terminations of flips [22]. 8 In this paper, we consider the ACC conjecture. For an R-divisor D and a 4 subset I ⊂ R, we write D ∈ I when all the non-zero coefficients of D belong 2 to I. Further, for a subset I ⊂ R, we say that I satisfies the ascending chain 6 0 condition (resp. the descending chain condition) when there is no infinite 1. increasing (resp. decreasing) sequence ai ∈ I. ACC (resp. DCC) stands for 0 the ascending chain condition (resp. the descending chain condition). 5 1 Conjecture 1.1 (ACC conjecture [21, Conjecture 4.2]). Fix d ∈ Z and a >0 : v subset I ⊂ [0,1] which satisfies the DCC. Then the following set i X A(d,I) := {mld (X,∆) |(X,∆) is a log pair, dimX = d, ∆ ∈ I, x ∈X} x r a satisfies the ACC, where x is a closed point of X. WearemainlyinterestedinthecasewhenI isafiniteset. Thisisbecause, the ACC conjecture for an arbitrary finite set I and the LSC conjecture imply the termination of flips [22]. The ACC conjecture is known for d ≤ 2 by Alexeev [1] and Shokurov [20], and for toric pairs by Ambro [3]. Kawakita [11] proved the ACC con- jectureon theinterval [1,3] forthree-dimensional smoothvarieties. Further, Kawakita [10] proved that the ACC conjecture is true for fixed variety X and a finite set I. More generally, he proved the discreteness of the set of log discrepancies for log triples (see Subsection 2.1 for the definition) {a (X,∆,a) |(X,∆,a) is lc, a∈ I, E ∈ D } E X 1 2 YUSUKENAKAMURA when the pair (X,∆) is fixed and I is a finite set. Here, we denoted by D X the set of all divisor over X. Further, a = ari is an R-ideal sheaf with i coefficients r in I. The purpose of this paper is to generalize this results to i Q the family of the varieties with fixed Gorenstein index. Theorem 1.2. Fix d∈ Z , r ∈ Z and a finite subset I ⊂ [0,+∞). Then >0 >0 the following set B(d,r,I) := {a (X,a) | (X,a) ∈ P(d,r), a ∈ I, E ∈ D } ⊂ [0,+∞) E X is discrete in R. Here we denote by P(d,r) the set of all d-dimensional lc pairs (X,a) such that rK is a Cartier divisor. X Sincemld (X,a) = a (X,a) holdsfor someE ∈ D , we get thefollowing x E X Corollary. Corollary 1.3. Fix d ∈ Z , r ∈ Z and a finite subset I ⊂ [0,+∞). >0 >0 Then the following set A′(d,r,I) := {mld (X,a) |X ∈ P(d,r), a ∈ I, x ∈ X}⊂ [0,+∞) x is discrete in R. Here we denote by P(d,r) the set of all d-dimensional lc pairs (X,a) such that rK is a Cartier divisor. X Corollary 1.3 does not imply the finiteness of A′(d,r,I), because we do not know the boundedness of A′(d,r,I). Hence Corollary 1.3 shows the finiteness of A′(d,r,I) modulo the BDD (boundedness) conjecture, which states the boundedness of minimal log discrepancies. Conjecture 1.4 (BDD conjecture). For fixed d ∈ Z , there exists a >0 real number a(d) such that mld(X) ≤ a(d) holds for any Q-Gorenstein d- dimensional normal variety X. The BDD conjecture is known only for d≤ 3 [18]. In arbitrary dimensions, the conjecture is known for the set of varieties with bounded multiplicity [9]. AsacorollaryofCorollary1.3,wecanprovetheACCforthree-dimensional canonical pairs. Corollary 1.5. If I ⊂ [0,1] is a finite subset, the following set {mld (X,∆) |(X,∆) is a canonical pair, dimX = 3, ∆ ∈ I, x ∈ X}, x denoted by Acan(3,I), satisfies the ACC. Further, 1 is the only accumulation point of Acan(3,I). Theorem1.2isprovedbyinductionondim Span (I∪{1}),thedimension Q Q oftheQ-vectorspacegeneratedbyI∪{1}. Intheinductivestep,weneedthe following theorem, which is about a perturbation of an irrational coefficient of log canonical pairs. Theorem 1.6. Fix d ∈Z>0. Let r1,...,rc′ be positive real numbers and let r0 = 1. Assume that r0,...,rc′ are Q-linearly independent. Let s1,...,sc : Rc′+1 → RbeQ-linearfunctionsfromRc′+1 toR. Assumethatsi(r0,...,rc′) ∈ R for each i. Then there exists a positive real number ǫ > 0 such that the ≥0 following holds: For any Q-Gorenstein normal variety X of dimension d and Q-CartiereffectiveWeildivisorsD1,...,Dc onX, if(X, 1≤i≤csi(r0,...,rc′)Di) P MLD’S ON VARIETIES WITH FIXED GORENSTEIN INDEX 3 is lc, then (X, 1≤i≤csi(r0,...,rc′−1,t)Di) is also lc for any t satisfying |t−rc′| ≤ ǫ. P Remark 1.7. The positive real number ǫ in Theorem 1.6 does not depend on X, but depends only on d, r1,...,rc′, and s1,...,sc. Kawakita [10] proved this theorem for a fixed variety X using a method of generic limit, and prove the discreteness of log discrepancies for fixed X. When c′ = 1 and each s satisfies s (R2 ) ⊂ R , this theorem just states i i ≥0 ≥0 the rationality of accumulation points of log canonical thresholds proved by Hacon, McKernan, and Xu [8, Theorem 1.11]. Actually, the proof of Theorem 1.6 heavily depends on their argument. We also note that the rationality of accumulation points of log canonical thresholds on smooth varietieswasprovedbyKolla´r[14,Theorem7]andbydeFernexandMusta¸t˘a [5, Corollary 1.4] using a method of generic limit. Thepaperisorganizedasfollows: InSection2,wereviewsomedefinitions and facts from the minimal model theory. Further we list some results on the ACC for log canonical thresholds by Hacon, McKernan, and Xu [8]. In Section 3, we prove the key proposition (Theorem 3.8) which is necessary to prove Theorem 1.6. The essential idea of proof is due to the paper [8]. In Section 4, we prove Theorem 1.6. In Section 5, we prove the main theorem (Theorem 1.2) and the corollaries. Notation and convention. Throughoutthispaper,wework over thefield of complex numbers C. • For an R-divisor D and a subset I ⊂ R, we write D ∈I when all the non-zero coefficients of D belong to I. • For an R-ideal sheaf A = ari and a subset I ⊂ R, we write A ∈ I i when all the non-zero coefficients r of A belong to I. i Q 2. Preliminaries 2.1. Minimal log discrepancies. We recall some notations in the theory of singularities in the minimalmodelprogram. For more details we refer the reader [15]. A log pair (X,∆) is a normal variety X and an effective R-divisor ∆ such that K +∆ is R-Cartier. If X is Q-Gorenstein, we sometimes identify X X with the log pair (X,0). An R-ideal sheaf on X is a formal productar1···ars, wherea ,...,a are 1 s 1 s ideal sheaves on X and r ,...,r are positive real numbers. For a log pair 1 s (X,∆) and an R-ideal sheaf a, we call (X,∆,a) a log triple. When ∆ = 0 (resp. A = O ), we sometimes drop ∆ (resp. A) and write (X,a) (resp. X (X,∆)). For a proper birational morphism f : X′ → X from a normal variety X′ and a primedivisor E on X′, the log discrepancy of (X,∆,a) at E is defined as aE(X,∆,a) := 1+coeffE(KX′ −f∗(KX +∆))−ordEa, where ord a := s r ord a . The image f(E) is called the center of E E i=1 i E i on X, and we denote it by c (E). For a closed subset Z of X, the minimal X P 4 YUSUKENAKAMURA log discrepancy (mld for short) over Z is defined as mld (X,∆,a) := inf a (X,∆,a). Z E cX(E)⊂Z In the above definition, the infimum is taken over all prime divisors E on X′ with the center c (E) ⊂ Z, where X′ is a higher birational model of X, X that is, X′ is the source of some proper birational morphism X′ → X. Remark 2.1. It is known that mld (X,∆,a) is in R ∪{−∞} and that if Z ≥0 mld (X,∆,a) ≥0, then theinfimumon therighthandsidein thedefinition Z is actually the minimum. Remark 2.2. Let D beeffective Weil divisors on X, and a := O (−D )the i i X i corresponding ideal sheaves. When X is Q-Gorenstein and D are Cartier i divisors, we can identify (X, r D ) and (X, ari). Indeed, for any divisor i i i E over X, we have a (X, r D ) = a (X, ari). E Pi i E Qi Forsimplicityofnotation,wewritemld (X,∆,a)insteadofmld (X,∆,a) P x Q {x} for a closed point x of X, and write mld(X,∆,a) instead of mld (X,∆,a). X Wesaythatthepair(X,∆,a)islogcanonical (lc forshort)ifmld(X,∆,a) ≥ 0. Further, we say that the pair (X,∆,a) is Kawamata log terminal (klt for short) if mld(X,∆,a) > 0. When E is a divisor over X such that a (X,∆,a) ≤ 0, the center c (E) is called a non-klt center. E X Wesaythatthepair(X,∆,a)iscanonical (resp.terminal)ifa (X,∆,a) ≥ E 1 (resp. > 1) for any exceptional divisor E over X. 2.2. Extraction of divisors. In this subsection, we recall some known results on extractions of divisors. We can extract a divisor whose log discrepancy is at most one. Theorem 2.3. Let (X,∆) be a klt pair, and let E be a divisor over X such that a (X,∆) ≤ 1. Then there exists a projective birational morphism E π : Y → X such that Y is Q-factorial and the only exceptional divisor is E. Proof. This is the special case of [4, Corollary 1.4.3]. (cid:3) When (X,∆) is lc, we can find a modification which is dlt. We call a log pair (X,∆) divisorial log terminal (dlt for short) when there exists a log resolution f :Y → X such that a (X,∆) > 0 for any f-exceptional divisor E E on Y. Theorem 2.4 (dlt modification). Let (X,∆) be a lc pair. Then there exists a projective birational morphism f :Y → X with the following properties: • Y is Q-factorial. • (Y,∆ ) is dlt, where we define ∆ as K +∆ = f∗(K +∆). Y Y Y Y X • a (X,∆) = 0 for every f-exceptional divisor E. E Proof. See [6, Theorem 10.4] for instance. (cid:3) 2.3. ACC for log canonical thresholds. In Section 3, we need the fol- lowing ACC properties proved by Hacon, McKernan, and Xu [8]. Theorem 2.5 (Hacon, McKernan, Xu [8, Theorem 1.4]). Fix d ∈ Z and >0 a subset I ⊂ [0,1] satisfying the DCC. Then there is a finite subset I ⊂ I with the following property: If (X,∆) 0 is a log pair such that MLD’S ON VARIETIES WITH FIXED GORENSTEIN INDEX 5 • (X,∆) is lc, dimX = d, ∆ ∈I, and • there exists a non-klt center Z ⊂ X which is contained in every component of ∆, then ∆ ∈ I . 0 Theorem 2.6 (Hacon, McKernan, Xu [8, Theorem 1.5]). Fix d ∈ Z and >0 a subset I ⊂ [0,1] satisfying the DCC. Then there is a finite subset I ⊂ I with the following property: If (X,∆) 0 is a projective log pair such that • (X,∆) is lc, dimX = d, ∆ ∈I, and • K +∆ ≡ 0, X then ∆ ∈ I . 0 3. Accumulation points of log canonical thresholds The goal of this section is to prove Corollary 3.9. It is a generalization of [8, Theorem 1.11] and necessary for the proof of Theorem 1.6. Usually, the log canonical threshold is defined as follows: for a lc pair (X,∆) and a Q-Cartier Z-Weil effective divisor M, LCT(∆;M) := sup{c ∈ R | (X,∆+cM) is lc}. ≥0 However, for the proof of Theorem 1.6, we need to treat the case when M is not effective. According to this reason, we introduce the new threshold set L (I). It no longer satisfies the ACC, but we can prove the rationality d of the accumulation points (Corollary 3.9). Corollary 3.9 easily follows from Theorem 3.6 and Theorem 3.8. They are proved in essentially the same way of the proof of Proposition 11.5 and Proposition 11.7 in [8]. For the reader’s convenience, we follow the proof of Proposition 11.5 and Proposition 11.7 in [8], and use as same notations as possible. First, we introduce some notations. For a subset I ⊂ [0,+∞), we define I as follows: + I := {0}∪ r | l ∈ Z ,r ,...,r ∈ I . + i >0 1 l 1≤i≤l (cid:8) X (cid:9) This becomes a discrete set if I is discrete. When D are finitely many i distinct prime divisors and d (t) : R → R are R-linear functions, then we i call the formal finite sum d (t)D a linear functional divisor. i i i Definition 3.1 (Dc(I)). PFix c ∈ R≥0 and a subset I ⊂ [0,+∞). For a linear functional divisor ∆(t) = d (t)D , we write ∆(t)∈ D (I) whenthe i i i c following conditions are satisfied: P • Each d (t) is equal to 1 or the form of m−1+f+kt, where m ∈ Z , i m >0 f ∈ I , and k ∈ Z. + • Further, f+kt above can bewritten as f+kt = (f +k t), where j j j f ∈ I ∪{0}, k ∈ Z, and f +k c≥ 0 hold for each j. j j j j P Further, by abuse of notation, we also write d (t) ∈ D (I) if d (t) satisfies i c i the above conditions. The form of the coefficient d (t) is preserved by adjunction. i 6 YUSUKENAKAMURA Lemma 3.2. Fix c ∈ R and a subset I ⊂ [0,1]. Let X be a Q-factorial ≥0 normal variety and ∆(t)= d (t)D be a linear functional divisor on 0≤i≤c i i X. Assume the following conditions: P • ∆(t)∈ D (I), and (X,∆(c)) is lc. c • d (t) = 1, and d (c) > 0 for each i. 0 i n Let S be the normalization of S := D . Define a linear functional divisor 0 n ∆Sn(t) on S by adjunction: (KX +∆(t))|Sn = KSn +∆Sn(t). Then, ∆Sn(t) ∈ Dc(I) holds. Proof. The statement follows from [16, Proposition 16.6]. We give a sketch of proof. Let p ∈ S be a codimension one point of S. Suppose that (X,D ) is not plt at p. Then p 6∈ SuppD for any i ≥ 1 0 i and coeffpDiffSn(0) = 0 or 1 [16, Proposition 16.6.1-2]. Hence, we have coeffp∆Sn(t) = 0 or 1 for any t. Suppose that (X,D0) is plt at p. Then coeffpDiffSn(0) = mm−1 holds for some m ∈ Z , and mD becomes Cartier at p for any Weil divisor D >0 [16, Proposition 16.6.3]. Hence, coeffp∆Sn(t) is the form of m−1 1 n −1+f +k t j j j + , m m n j j X where nj−1+fj+kjt is the form as in the definition of D (I). We can prove nj c that such form also satisfies the condition in the definition of D (I) by easy c calculation (cf. [19, Lemma 4.4]). (cid:3) We define L (I), the set of all log canonical thresholds derived from co- d efficients I. Definition 3.3 (L (I)). Let d ∈ Z and let I ⊂ [0,+∞) be a subset. d >0 We define L (I) ⊂ R as follows: c ∈ L (I) if and only if there exist a d ≥0 d Q-Gorenstein normal varieties X, and a linear functional divisor ∆(t) with the following conditions: • dimX ≤ d, ∆(t)∈ D (I), c • ∆(a) is R-Cartier for any a ∈ R, • (X,∆(c)) is lc, and • (X,∆(c+ǫ)) is not lc for any ǫ > 0, or (X,∆(c−ǫ)) is not lc for any ǫ > 0. Remark 3.4. When we say that (X,∆) is a lc pair, we assume that ∆ is effective. Therefore, we say that (X,∆) is not lc when ∆ is not effective. Further, we define G (I), the set of all numerically trivial thresholds d derived from coefficients I. Definition 3.5 (G (I)). Let d ∈ Z and let I ⊂ [0,+∞) be a subset. d >0 We define G (I) ⊂ R as follows: c ∈ G (I) if and only if there exist a d ≥0 d Q-factorial normalprojective variety X, and a linear functional divisor ∆(t) with the following conditions: • dimX ≤ d, ∆(t)∈ D (I), c MLD’S ON VARIETIES WITH FIXED GORENSTEIN INDEX 7 • (X,∆(c)) is lc, and K +∆(c) ≡ 0. X • K +∆(c′) 6≡ 0 for some c′ 6= c (equivalently for all c′ 6=c). X By the following theorem, we can reduce a local problem to a global problem. Theorem 3.6. Let d ≥ 2 and I ⊂ [0,+∞) be a subset. Then, L (I) ⊂ d G (I) holds. d−1 Lemma 3.7. Let c ∈ R and I ⊂ [0,+∞) be a subset. Suppose that there ≥0 exists an R-linear function d(t) :R → R with the following conditions: • d(t) ∈ D (I), and d(t) is not a constant function. c • d(c) = 0 or 1. Then c ∈ G (I) for any d ≥ 1. Especially, f ∈ G (I) holds for any d ≥ 1, d k d f ∈ I ∪{0}, and k ∈ Z . >0 Proof. We can easily construct on a curve. (cid:3) Proof of Theorem 3.6. Let c ∈ L (I), and let (X,∆(t)) be as in Definition d 3.3. Assume that (X,∆(c+ǫ)) is not lc for any ǫ > 0 (the same proof works in the other case). We may write ∆(t) = d (t)D with distinct prime i i i divisors D . By Lemma 3.7, we may assume that d (c) > 0 for any i. Then i i P ∆(c+ǫ) ≥ 0 holds for sufficiently small ǫ > 0. Let f : Y → X be a dlt modification (Theorem 2.4) of (X,∆(c)). Then Y is Q-factorial and we can write K +T +∆′(c) = f∗(K +∆(c)), Y X where ∆′(t) is the strict transform of ∆(t), and T is the sum of the excep- tional divisors. Since the pair (Y,T +∆′(c)) is dlt, there exists a divisor E on Y such that a (X,∆(c)) = 0, a (X,∆(c+ǫ)) < 0 E E for any ǫ > 0. If E is not f-exceptional, then d (c) = 1 holds for some d (t) i i which is not identically one. In this case c∈ G (I) by Lemma 3.7. d−1 In what follows, we assume that E is f-exceptional and so a component of SuppT. By adjunction, we can define a linear functional divisor ∆ (t) E on E such that (K +T +∆′(t))| = K +∆ (t). Y E E E Here, ∆ (t) ∈ D (I) holds by Lemma 3.2. E c Let F be a general fiber of E → f(E). Define ∆ (t) as F (K +∆ (t))| = K +∆ (t). E E F F F Then (F,∆ (t)) satisfies F • dimF ≤ d−1, F is projective, • ∆ (t) ∈ D (I), F c • K +∆ (c) = f∗(K +∆(c))| ≡ 0, and F F X F • (F,∆ (c)) is lc. F Hence (F,∆ (t)) satisfies all conditions in Definition 3.5 except for K + F F ∆ (c′) 6≡ 0 for some c′. F We may write ∆(t) = ∆+tM with an R-divisor ∆ and a Q-divisor M. Write M = M − M , where M ≥ 0 and M ≥ 0 have no common + − + − 8 YUSUKENAKAMURA components. Since a (X,∆+(c+ǫ)M) < a (X,∆+cM) = 0, it follows E E that ord M > ord M ≥ 0. Possibly replacing E by other component of E + E − T, we may assume that ord M ·ord M ≤ ord M ·ord M E − Ej + Ej − E + for any component E ⊂ SuppT. We may take ǫ ≥ ǫ > 0 such that j 1 2 a (X,∆+(c+ǫ )M −ǫ M )= 0. Note that E 1 2 + ǫ (ord M −ord M ) = ǫ ord M 1 E + E − 2 E + holds. Then we have 0 ≡ f∗(K +∆+(c+ǫ )M −ǫ M )| X 1 2 + F = (K +T +U +∆′+(c+ǫ )M′−ǫ M′ )| Y 1 2 + F = K +∆ (c+ǫ )+U| −ǫ M′ | , F F 1 F 2 + F where we set U = (ǫ ord M −ǫ ord M )E . 1 Ej 2 Ej + j j X Note that ǫ ord M −ǫ ord M 1 Ej 2 Ej + ǫ 1 = (ord M ·ord M −ord M ·ord M ) ord M E − Ej + Ej − E + E + ≤ 0. ThereforeK +∆ (c+ǫ ) ≡ ǫ M′ | −U| ≥ ǫ M′ | . Sinceord M > 0, F F 1 2 + F F 2 + F E + itfollowsthatf(E) ⊂ SuppM andsoM′ | > 0. ThereforeK +∆ (c+ǫ ) + + F F F 1 is not numerically trivial. (cid:3) Theorem 3.8. Let d ≥ 2 and let I ⊂ [0,+∞) be a finite subset. The accumulation points of G (I) are contained in G (I). d d−1 As a corollary, we can prove the rationality of the accumulation points of L (I). d Corollary 3.9. Let d ∈ Z and let I ⊂ [0,+∞) be a finite subset. The >0 accumulation points of L (I) are contained in Span (I ∪ {1}), where we d Q denote by Span (I ∪{1}) ⊂ R the Q-vector space spanned by the elements Q of I and 1. We prove a stronger statement (cf. [8, Proposition 11.7]). Proposition 3.10. Let d ≥ 2 and let I ⊂ [0,+∞) be a finite subset. Fur- ther, let c ∈R . ≥0 Suppose that for each i∈ Z , there exist c ∈ R , a Q-factorial normal >0 i ≥0 projective variety X , and a linear functional divisor ∆ (t) on X with the i i i following conditions: • The sequence c is increasing or decreasing. Further, c is accumu- i i lating to c. • dimX ≤ d for each i. i • ∆ (t) can be written as ∆ (t) = A +B (t), where the coefficients of i i i i A are approaching one, and B (t)∈ D (I). i i ci • (X ,∆ (c )) is lc, and K +∆ (c ) ≡ 0. i i i Xi i i MLD’S ON VARIETIES WITH FIXED GORENSTEIN INDEX 9 • K +∆ (c′) 6≡ 0 for some c′ 6= c . Xi i i i i Then, c ∈ G (I) holds. d−1 Remark 3.11. If c ∈ G (I), then c satisfies the above conditions (In this i d i case, A = 0). Hence, Theorem 3.8 follows from Proposition 3.10. i IntheproofofProposition3.10,wereducetothecasewhenX hasPicard i number one, and apply the following lemma from [8]. Lemma 3.12 ([8, Lemma 11.6]). Let (X,∆) be a projective Q-factorial lc pair of dimension d and of Picard number one. Assume that K +∆≡ 0. If X the coefficients of ∆ are at least δ > 0, then ∆ has at most d+1 components. δ Proof of Proposition 3.10. Possibly replacing A and B (t), we may assume i i that the coefficient of B (t) is not identically one. We may write B (t) = i i d (t)D as in Definition 3.1. l il il By Lemma 3.7, we may assume that (I ∪{0})∩cZ = ∅. Then we may >0 P assume the following conditions on B (t). i Lemma 3.13. We may assume the following conditions: (1) When we write d (t) = m−1+f+kt as in Definition 3.1, f and k have il m only finitely many possibilities. (2) d (c ) are bounded from zero, and d (c )< 1 for any i,l. il i il i (3) d (c) > 0. il (4) The set {d (c) | i,l} satisfies the DCC. il Proof. Since (I∪{0})∩cZ =∅, possibly passing to a tail of the sequence, >0 we may assume that there exist k′ ∈ Z and ǫ ∈ R such that for any >0 >0 f ∈ I ∪{0}, k ∈ Z, and i, j j • f +k c ≥ 0 implies f +k c ≥ ǫ and k ≥−k′ unless f = k = 0. j j i j j i j j j Here, we note that I is a finite set. Let d (t) = m−1+f+kt be a coefficient of B (t). By assumption, f +kt il m i above can be written as f +kt = (f +k t), where f ∈ I, k ∈ Z, and j j j j j f +k c ≥ 0 hold for each j. j j i P Note that f +kc ≤ 1 by the log canonicity. Since f +k c ≥ 0 implies i j j i f +k c ≥ ǫ and k ≥ −k′, it follows that k is bounded from below. Since j j i j c ≥ ǫ,itfollows thatk isalsoboundedfromabove. AsthesetI isdiscrete, i + f has also only finitely many possibilities. Therefore (1) follows. By (1), it follows that d (c ) ≥ min{1,ǫ}. Hence d (c ) are bounded il i 2 il i from zero. Since c are distinct, by (1), possibly passing to a subsequence, i we may assume that d (c ) 6= 1 (hence d (c ) < 1) holds for any i,l. Thus, il i il i (2) follows. (3) follows from (2) and (4) follows from (1). (cid:3) By Lemma 3.13 (2), possibly passing to a tail of the sequence, we may assume that A and B (t) have no common components, and that ⌊A ⌋ = i i i ⌊A +B (c )⌋. i i i In our setting, the following claim is important and allow the same argu- ment in [8] to work. Claim 3.14. We may assume that (X ,⌈A ⌉+B (c)) is lc for any i. i i i 10 YUSUKENAKAMURA Proof. We may write ∆ (t) = A +M +t(N+ −N−), where N+ ≥ 0 and i i i i i i N− ≥ 0 have no common components. (X ,A +M +c (N+ −N−)) is lc i i i i i i i by the assumption. First suppose c < c. Note that A +M +c N+−cN− ≥ 0 (Lemma 3.13 i i i i i i (3)). Hence (X ,A +M +c N+−cN−) is also lc. Here, the coefficients of i i i i i i M −cN− satisfy the DCC (Lemma 3.13 (4)), and the coefficient of A and i i i the sequence c are increasing. Hence by Theorem 2.5, possibly passing to i a tail of the sequence, we may assume that (X ,⌈A ⌉+M +cN+−cN−) is i i i i i lc. Suppose c > c. Then (X ,A +M +cN+ −c N−) is lc. Here, the coef- i i i i i i i ficients of M +cN+ satisfy the DCC (Lemma 3.13 (4)), and the coefficient i i of A and the sequence −c are increasing. Hence by Theorem 2.5, possibly i i passing to a tail of the sequence, we may assume that (X ,⌈A ⌉ + M + i i i cN+−cN−) is lc. (cid:3) i i Set a := mld(X ,∆ (c )) ≥ 0. Possibly passing to a subsequence, it is i i i i sufficient to treat the following two cases: (A) a is bounded away from zero. i (B) a approaches zero. i Case B We treat the case when a approaches zero from above. i STEP B-1 We reduce to the case when A 6= 0 and (X ,∆ (c )) is dlt. i i i i We may assume a ≤ 1 for any i. Take an extraction π : X′ → X of a i i i i divisor E computing mld(X ,∆ (c )) = a (Theorem 2.3 and 2.4). Then we i i i i i may write KXi′ +(1−ai)Ei +Ti+∆′i(ci)= πi∗(KXi +∆i(ci)), where T is the sum of exceptional divisors (Note that T = 0 when a > 0) i i i and ∆′(t) is the strict transform of ∆ (t). Then X′,(1−a )E +T +∆′(t) i i i i i i i satisfies the following conditions: (cid:0) (cid:1) • We may write (1 − a )E + T + ∆′(t) = A′ + B′(t) with all the i i i i i i conditions in Proposition 3.10. • ⌊A′⌋ = ⌊A′ +B′(c )⌋ and A′ 6= 0. i i i i i • X′,(1−a )E +T +∆′(c ) is dlt. i i i i i i Hence, we may replace (X ,∆ (t)) by X′,(1−a )E +T +∆′(t) . (cid:0) i i (cid:1) i i i i i STEP B-2 We are done if there exis(cid:0)ts a component Si ⊂ Supp⌊(cid:1)Ai⌋ such that (K +∆ (c′))| 6≡0. Xi i i Si Suppose that there exists a component S ⊂ Supp⌊A ⌋ such that (K + i i Xi ∆ (c′))| 6≡ 0. By adjunction, we can define ∆ (t) as follows: i i Si Si (K +∆ (t))| = K +∆ (t). Xi i Si Si Si Then (S ,∆ (t)) satisfies the following conditions: i Si • dimS ≤d−1. i • K +∆ (c′) 6≡ 0 by the assumption. Si Si i • (S ,∆ (t)) satisfies the other conditions in Proposition 3.10. i Si Hence, we may replace (X ,∆ (t)) by (S ,∆ (t)). By induction on d, it i i i Si follows that c ∈ G (I) ⊂ G (I). d−2 d−1