Boundedness of Complements for Log Calabi–Yau Threefolds

In this paper, we study the theory of complements, introduced by Shokurov, for Calabi–Yau type varieties with the coefficient set [0, 1]. We show that there exists a finite set of positive integers \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal {N}$$\end{document}N, such that if a threefold pair \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(X/Z\ni z,B)$$\end{document}(X/Z∋z,B) has an \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbb {R}$$\end{document}R-complement which is klt over a neighborhood of z, then it has an n-complement for some \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$n\in \mathcal {N}$$\end{document}n∈N. We also show the boundedness of complements for \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbb {R}$$\end{document}R-complementary surface pairs.


Introduction
We work over the field of complex numbers C.
The theory of complements (for Fano varieties) was introduced by Shokurov when he proved the existence of flips for threefolds [45].It originates from his earlier work on anti-canonical systems on Fano threefolds [44].The boundedness of complements [4,26,47] played an important role in various contexts in the study of Fano varieties, including the solution of the Borisov-Alexeev-Borisov conjecture (boundedness of Fano varieties) [4,5] and the Yau-Tian-Donaldson conjecture (the existence of We first sketch the proof of Theorem 1.3.If X is of Fano type, then (X , B) is N 1 -complementary for some finite set of positive integers N 1 by Theorem 2.19; here, (X , B) being N 1 -complementary means that (X , B) is n-complementary for some n ∈ N 1 (see Definition 2.12).Thus, we may assume that X is not of Fano type and κ(X , B − B 1 ) ≤ 1, where 1 := (N 1 , {0, 1}) is a hyperstandard set and B 1 is a Q-divisor with coefficients in 1 such that 0 ≤ B 1 ≤ B (see Definition 2.1).Suppose that κ(X , B − B 1 ) = κ(X , B − B 2 ) = 1, where N 2 is a finite set of positive integers given by Theorem 2.20 and 2 := (N 1 ∪ N 2 , {0, 1}).In this case, we claim that (X , B) is N 2 -complementary.Indeed, although X is not of Fano type, by Lemma 2.15 we can still run an MMP on −(K X + B 2 ) and get a good minimal model X such that −(K X + B 2 ) is semi-ample and hence defines a contraction π : X → Z , where D denotes the strict transform of D on X for any R-divisor D on X .Then, we run an MMP on −(K X + B 1 ) over Z and reach a model X on which −(K X + B 1 ) is semi-ample over Z , where D denotes the strict transform of D on X for any R-divisor D on X .As κ(X , B − B 1 ) = κ(X , B − B 2 ) = 1, the natural morphism π : X → Z is the contraction defined by −(K X + B 1 ) over Z .By the similar arguments as in [4,Proposition 6.3] and using Effective Adjunction [43,Conjecture 7.13.3 and Theorem 8.1], there exists a positive integer p which only depends on 1 such that p(K X + B 1 ) ∼ p(π Z +M π ,Z ) and pM π is base point free, where B (1) Z and M π are given by the canonical bundle formula for (X , B 1 ) over Z in Proposition 3.3.It follows that p(K X + B 2 ) ∼ p(π ) * (K Z + B (2) Z + M π ,Z ) and pM π is base point free, where B (2) Z and M π are given by the canonical bundle formula for (X , B 2 ) over Z .We may assume that p | n for any n ∈ N 2 .As pM π is base point free, one can find an effective Q-divisor M Z such that pM Z ∼ pM π ,Z , M Z has no common components with B (2) Z , and (Z , B (2) Z + M Z ) has an n-complement for some n ∈ N 2 .Then, we can lift this complement to X and get an n-complement of (X , B); see Proposition 3.5.If κ(X , B − B 2 ) = 0, then we can easily show that n 0 (K X + B) ∼ 0 for some positive integer n 0 which only depends on 2 ; see Lemma 2.18.Hence, N 1 ∪ N 2 ∪ {n 0 } has the required property.Now, we sketch the proof of Theorem 1.1.The main strategy is similar.One of the key steps is to construct a positive integer n 0 and finite sets of positive integers N i (i = 1, 2, 3) such that then (X , B) is N 3 -complementary, and (4) if κ(X , B − B 3 ) = 0, then (X , B) is n 0 -complementary, where i := ( i j=1 N j , {0, 1}) for any 1 ≤ i ≤ 3; see Sect.6 for the details.However, there are some issues when we construct these finite sets.One issue is that when we apply the canonical bundle formula, Effective Adjunction is still open when the relative dimension is ≥ 2. But in our setting we can give a positive answer to Effective Adjunction; see Proposition 3.4 for the details.On the other hand, there is also an issue when we try to lift complements from lower dimensional varieties.More precisely, it may happen that some components of Supp B have images of codimension ≥ 2 in Z .Therefore, we must lift complements more carefully; see Proposition 3.5 and Sect.6 for the details.
Structure of the paper.We outline the organization of the paper.In Sect.2, we introduce some notation and tools which will be used in this paper, and prove certain basic results.In Sect.3, we recall the canonical bundle formula, some well-known results, as well as some new results.In Sect.4, we prove the boundedness of complements for sdlt curves.In Sect.5, we prove Theorem 1.3.In Sect.6, we prove Theorem 1.1.In Sect.7, we prove Theorem 1.2.

Arithmetic of Sets
Definition 2.1 (1) We say that a set ⊆ [0, 1] satisfies the descending chain condition (DCC) if any decreasing sequence a 1 ≥ a 2 ≥ • • • in stabilizes.We say that satisfies the ascending chain condition (ACC) if any increasing sequence a 1 ≤ a 2 ≤ • • • in stabilizes.
. Indeed, for any r ∈ R , there exist r ∈ R and l ∈ Z >0 such that r = r /l.Thus, (N , (R)) ⊇ (N , (R )), and the converse inclusion follows similarly.
(2) (N , ) is the hyperstandard set associated to the following finite set: (3) If N 1 and N 2 are two finite sets of positive integers, then Then, (N 1 , ) = (R 1 ) by (2).Therefore, The following lemma was observed by the second named author.It will play an important role in the proof of the main theorems.
and n is a positive integer such that nR ⊆ Z.Then, for any b ∈ [0, 1], we have Proof Without loss of generality, we may assume that 1 .
Suppose on the contrary that there exists an integer k such that The first inequality above gives us that l −lk +m < (n +1)r , and thus l −lk +m ≤ nr as nr ∈ Z.Therefore, we have

Lemma 2.4
Let N be a finite set of positive integers, a hyperstandard set, and n ∈ N .Suppose that b, b + ∈ [0, 1] such that nb + ∈ Z and By the construction, (n + 1)b = (n + 1)b , which implies that

Divisors
Let F be either the rational number field Q or the real number field R. Let X be a normal variety and WDiv(X ) the free abelian group of Weil divisors on X .Then, an F-divisor is defined to be an element of WDiv(X ) F := WDiv(X ) ⊗ F.
A b-divisor on X is an element of the projective limit where the limit is taken over all the pushforward homomorphisms f * : WDiv(Y ) → WDiv(X ) induced by proper birational morphisms f : Y → X .In other words, a b-divisor D on X is a collection of Weil divisors D Y on higher models Y of X that are compatible with respect to pushforward.The divisor D Y is called the trace of D on the birational model Y .A b-F-divisor is defined to be an element of WDiv(X ) ⊗ F, and the trace of a b-F-divisor is defined similarly.
The Cartier closure of an where D Y is an F-Cartier F-divisor on a birational model Y of X ; in this situation, we say D descends to Y .Moreover, if D Y is a Cartier divisor, then we say D is b-Cartier.Let X → Z be a projective morphism.Then, a b-F-divisor is nef (respectively, base point free) over Z if it descends to a nef (respectively, base point free) over Z F-divisor on some birational model of X .
Assume that ⊆ [0, 1] is a set, and Assume that N is a finite set of positive integers and is a hyperstandard set.We define If N = {n} (respectively, N = ∅), we may write B n_ (respectively, B ) instead of B N _ .
Definition 2.5 (1) We say π : X → Z is a contraction if X and Z are normal quasiprojective varieties, π is a projective morphism, and We say that a birational map φ : X Y is a birational contraction if φ is projective and φ −1 does not contract any divisors.Lemma 2.6 Suppose that τ : Z → Z and Z → Z are contractions.Suppose that H (respectively, H ) is an R-Cartier R-divisor on Z (respectively, Z ) which is ample over Z (respectively, Z ).Then, H + τ * H is ample over Z for any 0 < 1.

Generalized Pairs and Singularities
In this paper, we usually discuss the (sub-)pair in the relative setting (X /Z z, B); we refer the reader to [11, §2] (cf.[6,36]).Moreover, if the (sub-)pair (X /Z z, B) is (sub-)lc over z for any z ∈ Z , then we say (X /Z , B) is (sub-)lc.
Here, we briefly discuss the analogous concepts for generalized pairs, and refer the reader to [7,22,24] for further details.Definition 2.7 A generalized pair (g-pair for short) (X /Z , B + M) consists of a contraction X → Z , an effective R-divisor B on X , and a nef/Z b-R-divisor M on X , such that Let (X /Z , B + M) be a g-pair and f : W → X a log resolution of (X , Supp B) to which M descends.We may write for some R-divisor B W on W .Let E be a prime divisor on W .The log discrepancy of E with respect to (X , B + M) is defined as We say (X /Z , B + M) is generalized lc or glc (respectively, generalized klt or gklt) if a(E, X , B + M) ≥ 0 (respectively, > 0) for any prime divisor E over X .
We say that two g-pairs (X /Z , B + M) and (X /Z , B + M ) are crepant if X is birational to X , M = M , and f * (K X + B +M X ) = ( f ) * (K X + B +M X ) for some common resolution f : W → X and f : W → X .We also call (X /Z , B + M) a crepant model of (X /Z , B + M).Lemma 2.8 Let d be a positive integer and ⊆ [0, 1] a DCC set.Then, there is a positive real number depending only on d and satisfying the following.Assume that (X , B) is a projective klt pair of dimension d such that K X + B ∼ R 0 and B ∈ .Then, (X , B) is -lc.Lemma 2.11 Let X → Z be a contraction, and D 2 ≥ D 1 two effective R-Cartier R-divisors on X .Suppose that X X is a sequence of steps of the D 1 -MMP over Z .Let D 2 be the strict transform of D 2 on X .Then, κ(X /Z , D 2 ) = κ(X /Z , D 2 ).

Proof
Proof Pick a positive real number such that X X is also a sequence of steps of the where D 1 is the strict transform of D 1 on X .

Complements
Definition 2. 12 We say that a pair Let n be a positive integer.An n-complement of (X /Z z, B) is a pair (X /Z z, B + ), such that over some neighborhood of z, we have Let N be a non-empty set of positive integers.We say that (X /Z z, B) is Ncomplementary (respectively, n-complementary, R-complementary) if (X /Z z, B) has an m-complement (respectively, n-complement, R-complement) for some m ∈ N .If (X /Z z, B) has an R-complement (respectively, n-complement) for any z ∈ Z , then we say that (X /Z , B) is R-complementary (respectively, n-complementary).
Note that if z ∈ z is a closed point and (X /Z z , B + ) is an R-complement (respectively, an n-complement) of (X /Z z , B), then (X /Z z, B + ) is an Rcomplement (respectively, an n-complement) of (X /Z z, B).Hence, when proving the existence of complements we may assume that z ∈ Z is a closed point.
The following lemma is well-known to experts.We will use the lemma frequently without citing it in this paper.
The following lemma is an easy consequence of Lemmas 2.3 and 2.4.

Lemma 2.14 Let
We will use the following lemma frequently in this paper.
Proof According to [11,Lemma 4.2], possibly shrinking Z near z, there exist a positive real number u and a surface (respectively, threefold) pair (X , ), such that the coefficients of are at most 1 (respectively, (X , ) is dlt) and −u( In both cases, we can run an MMP on K X + over Z and reach a good minimal model X over Z by [18,32,48] This finishes the proof.

Boundedness of Complements
We propose a conjecture on the boundedness of complements and collect some useful results.
For a positive integer l and a non-empty set N ⊆ Z >0 , we say N is divisible by l, denoted by l | N , if l | n for any n ∈ N .

Conjecture 2.16 Let d, l be two positive integers and
Then, there exists a finite set of positive integers N divisible by l depending only on d, l and satisfying the following.
Assume that Remark 2.17 (1) In Conjecture 2.16, we do not assume (X /Z z, B) is lc.
(2) One can not remove the klt assumption in Conjecture 2.16 when d ≥ 3; see [47,Example 11].However, we will show Conjecture 2.16 for R-complementary surface pairs without the klt assumption; see Theorem 1.3.

Lemma 2.18
Let ⊆ [0, 1] ∩ Q be a hyperstandard set.Then, there exists a positive integer n depending only on satisfying the following.
Proof By Lemma 2.15, we may run an MMP on −(K X + B ) ∼ R B − B which terminates with a good minimal model X .Let B be the strict transform of B on X .Since κ(X , B − B ) = 0 and B − B is semi-ample, we see that B = B and, therefore, K X + B ∼ R 0. By [11, Proposition 6.4, Theorem 1.1] and [12, Theorem 2.14], there is a positive integer n depending only on such that It follows that n(K X + B) ∼ 0 as (X , B) and (X , B ) are crepant.
We will use the following results on the boundedness of complements.Assume that

Canonical Bundle Formulas
For the definition and basic properties of the canonical bundle formula, we refer the reader to [4,15,27,30].Briefly speaking, suppose that (X /Z , B) is a sub-pair and φ : X → T is a contraction over Z , such that (X , B) is lc over the generic point of T and K X + B ∼ R,T 0. Then, there exist a uniquely determined R-divisor B T and a nef over Z b-R-divisor M φ which is determined only up to R-linear equivalence, such that (T /Z , B T + M φ ) is a g-sub-pair and Here, B (respectively, M φ ) is called the discriminant part (respectively, a moduli part) of the canonical bundle formula for (X /Z , B) over T .Moreover, if (X /Z , B) is an lc (respectively, klt) pair, then (T /Z , B T + M φ ) is a glc (respectively, gklt) g-pair.
It is worthwhile to point out that M φ only depends on (X , B) over the generic point of T (cf.[4, 3.4 (2)]), and there are many choices of M φ , some of which could behave badly.But we can always choose one with the required properties, e.g., Propositions 3.3 and 3.4.

Lemma 3.1 Notation as above.
(1) Assume that (X , B) is a klt pair.Then, there exists a crepant model ( T , B T + M φ ) → (T , B T + M φ ) such that for any prime divisor P ⊆ Supp B which is vertical over T , the image of P on T is a prime divisor.
(2) Suppose that there is an R-divisor G T on T such that (T , Proof (1) According to [ (2) Suppose on the contrary that (X , B + G) is not sub-lc.Let P be a non-sub-lc place of (X , B + G), i.e., a(P , X , B + G) < 0. It is clear that Center X (P ) ⊆ Supp G which is vertical over T .We can find birational morphisms f : X → X and g : T → T such that X → T is a contraction, P is a prime divisor on X , and the image Q of P on T is a prime divisor (cf.[34, VI, Theorem 1.3]).We may write and T .Then, T is the discriminant part of the canonical bundle formula for (X /Z , ) over T ; see [43,Lemma 7.4 (ii)].Since (T , By the definition of the canonical bundle formula, (X , ) is sub-lc over the generic point of Q .In particular, mult Lemma 3.2 Let p be a positive integer, (X , B) and (X , B ) two lc pairs, and φ : Let B T (respectively, B T ) and M φ be the discriminant part and a moduli part of the canonical bundle formula for (X , B) (respectively, (X , B )) ∩ Q be a finite set, and := (R).Then, there exist a positive integer p and a hyperstandard set ⊆ [0, 1] ∩ Q depending only on satisfying the following.
Assume that (X /Z , B) is an lc pair of dimension ≤ 3 and φ : X → T is a contraction over Z such that dim T > 0, B ∈ , and K X + B ∼ Q,T 0. Then, we can choose a moduli part M φ of the canonical bundle formula for (X , B) over T , such that B T ∈ , pM φ is b-Cartier, and where B T is the discriminant part of the canonical bundle formula for (X , B) over T .Moreover, if dim T = dim X − 1, then pM φ is base point free over Z .Assume that (X /Z , B) is a klt threefold pair and φ : X → T is a contraction over Z , such that dim T = 1, B ∈ , and K X + B ∼ Q,Z 0. Then, we can choose a moduli part M φ of the canonical bundle formula for (X , B) over T , such that

Proof
and pM φ is base point free over Z , where B T is the discriminant part of the canonical bundle formula for (X , B) over T .
where p 1 is given by Proposition 3.3 depending only on , and M φ is a moduli part chosen as in Proposition 3.3.Therefore, in what follows, we may assume that dim Z = 0, i.e., Z is a point.
If K T ≡ 0, then T = P 1 .Let M φ be a moduli part chosen as in Proposition 3.3.Then, p 1 M φ,T is base point free.Now, assume that K T ≡ 0 and in particular, B T = 0. Let F be a general fiber of X → T and According to [11, Proposition 6.4 and Theorem 1.1], r (K F + B F ) ∼ 0 for some positive integer r depending only on .Then, there exist a rational function α ∈ K (X ) and an R-Cartier R-divisor We conclude that p := p 1 p 2 r has the required property.

Lifting Complements
Now, we turn to the following technical statement on lifting complements via the canonical bundle formula.(1) for any prime divisor P ⊆ Supp B which is vertical over T , the image of P on T is a prime divisor, (2) pM T ∼ Z pM φ,T and M T ∧ B T = 0, and (3) (T /Z z, B T + M T ) is n-complementary for some z ∈ Z.
Then, (X /Z z, B) is also n-complementary.
Proof Let X be the normalization of the main component of X × T T .Denote by f : X → X and φ : X → T the induced morphisms.We may write K X + B = f * (K X + B) for some R-divisor B .Note that by our assumption, we have

Let (T /Z
z, B + T + M T ) be an n-complement of (T /Z z, B T + M T ).We remark that as p | n, B + T ≥ 0. Possibly shrinking Z near z, we may assume that Hence, n(K X + B + ) ∼ Z 0. According to Lemma 3.1 (2), the sub-pair (X /Z z, B + ) is sub-lc, and thus (X /Z z, B + ) is also sub-lc.It suffices to prove that Let P ⊆ Supp B + be a prime divisor.If P is horizontal over T , then mult P B + = mult P B and there is nothing to prove.Therefore, we may assume that Q, the image of P on T , is a prime divisor.Let b P := mult P B, b Hence, Moreover, as 1 − b Q is the lc threshold of φ * Q with respect to (X , B ) over the generic point of Q, we know where the last equality holds as We finish the proof.

Boundedness of Complements for sdlt Curves
Definition 4. 1 We say X is a semismooth curve if X is a reduced scheme of dimension 1, every irreducible component of X is normal, and all of its singularities are simple normal crossing points.
Let X be a semismooth curve, and let B ≥ 0 be an R-divisor on X .We say (X , B) is sdlt if B is supported in the smooth locus of X and B ≤ 1. Definition 4.2 Let X be a semismooth curve, and B ≥ 0 an R-divisor on X , such that (X , B) is sdlt.We say that (X , The following theorem is a generalization of [45, 5.2.2] and [33, 19.4 Theorem] where the case l = 1 is proved.Theorem 4.3 Let l be a positive integer.Then, there exists a finite set of positive integers N sdlt divisible by l depending only on l satisfying the following. Assume that X is a semismooth curve, connected but not necessarily complete, and B ≥ 0 is an R-divisor on X , such that (1) (X , B) is sdlt, (2) X has at least one complete component, (3) each incomplete component of X does not meet any other incomplete component of X , (4) the union of the complete components of X is connected, and (5) −(K X + B) is nef on each complete component of X .
Then, there exists an n-semi-complement (X , B + ) of (X , B) in a neighborhood of the union of the complete components of X for some n ∈ N sdlt .
Proof Let X 0 be a complete component of X , and let {P 1 , . . ., P k } := X 0 ∩ Sing X .Then, deg(K X | X 0 ) = 2g − 2 + k, where g is the genus of X 0 .Since deg(K X | X 0 ) ≤ 0, there are four possibilities: We remark that B could not meet the components of type (i) or (ii) as deg( If X 0 is of type (i), then X = X 0 and B = 0.In this case, (X , B) is l-complementary.If X 0 is of type (iv), then X = X 0 and X ∼ = P 1 .By Theorem 2.20, there exists a finite set of positive integers N divisible by l depending only on l, such that (X , B) is N -complementary.Now, suppose that any complete component of X is either of type (ii) or of type (iii).Note that each component of type (ii) (respectively, type (iii)) can only meet other components at two points (respectively, one point).By assumptions (3) and ( 4), the entire curve X must form a chain or a cycle.If X is a cycle, then B = 0 and K X ∼ 0 by Lemma 4.4.Otherwise, by Lemma 4.5, it suffices to construct B + such that (X , B + ) is an n-complement of (X , B) on each component of X .Note that possibly shrinking X near the union of the complete components, for any positive integer n, (X , B) is an n-complement of itself on each incomplete component and each complete component of type (ii).Since X has at most two complete components of type (iii), by Lemma 4.6 there exists a finite set of positive integers N divisible by l depending only on l, such that (X , B) has an n-complement on each complete component of type (iii) for some n ∈ N .Therefore, by Lemma 4.5, (X , B) has an n-semi-complement for some n ∈ N .
Let N sdlt := N ∪ N and we are done.
Lemma 4.4 Let X = m i=1 X i be a semismooth curve which is a cycle of irreducible curves X i .Suppose that X i ∼ = P 1 for any 1 ≤ i ≤ m.Then, K X ∼ 0.
Proof For each integer m ≥ j ≥ 2, we construct a semismooth curve Y j in a smooth projective surface S j such that Y j is a cycle of j complete rational curves and the union of a line and a conic which is semismooth.Then, K S 2 + Y 2 ∼ 0 and thus Suppose that we have constructed a semismooth curve Y j−1 contained in a smooth projective surface S j−1 , such that Y j−1 is a cycle of j − 1 complete rational curves and K S j−1 + Y j−1 ∼ 0. Let π j : S j → S j−1 be the blow-up of S j−1 at one snc point of Y j−1 , and E j the exceptional divisor of π j .Let Y j = (π j ) −1 * Y j−1 ∪ E j .Then, we get a semismooth curve Y j ⊆ S j , which is a cycle of j complete rational curves, such that K S j + Y j ∼ 0. Since X is analytically isomorphic to Y m , by [28, Appendix B, Theorem 2.1], K Y m ∼ 0 implies K X ∼ 0. Lemma 4.5 Let X = m i=1 X i be a semismooth curve which is a chain of irreducible curves X i .Suppose that D is an R-divisor on X , supported in the smooth locus of X , such that D| X i ∼ 0 for any 1 ≤ i ≤ m.Then, D ∼ 0.
Proof Let X (i) := i j=1 X j for 1 ≤ i ≤ m, and P i := X i ∩ X i+1 = X (i) ∩ X i+1 for 1 ≤ i ≤ m − 1.We will prove by induction that D| X (i) ∼ 0 for any 1 ≤ i ≤ m.Suppose that D| X (i−1) ∼ 0 for some integer i ≥ 2.Then, there exist a rational function α i−1 on X (i−1) and a rational function β i on X i , such that D| X (i−1) = (α i−1 ) and D| X i = (β i ).Since P i−1 is not contained in the support of D, α i−1 and β i are non-zero regular functions near P i−1 .Replacing β i by α i−1 (P i−1 ) β i (P i−1 ) β i , we may assume that α i−1 (P i−1 ) = β i (P i−1 ).Then, there exists a rational function α i on Proof Without loss of generality, we may assume that a i , b i < 1 for any i.Then, it suffices to prove For any positive integer n and non-negative real numbers c, d, we have Thus, possibly replacing (a i , a j ) by (a i +a j , 0) (respectively, (b i , b j ) by (b i +b j , 0)), we may assume that a i +a j ≥ 1 (respectively, b i +b j ≥ 1) for any i = j.In particular, we may assume that k = 2 and By Dirichlet prime number theorem, there exist three distinct prime numbers q j such that l | q j −1 for any j ∈ {1, 2, 3}.Let n j := q j −1, and N := {n 1 , n 2 , n 3 }.We claim that there exists n ∈ N satisfying (4.1).It suffices to show that both (n j + 1)a 1 and (n j + 1)b 1 are not integers for some j ∈ {1, 2, 3}.Otherwise, by the pigeonhole principle, we may assume that a 1 ∈ 1 n j +1 Z ∩ [0, 1) = 1 q j Z ∩ [0, 1) for two indices j ∈ {1, 2, 3}, which is absurd.

Proposition 4.7 Let l be a positive integer. Then, there exists a finite set of positive integers N divisible by l depending only on l satisfying the following.
Assume that (X /Z z, B) is a surface pair such that z is a closed point, (X , B) is dlt, S := B = 0, B − S is big over Z and K X + B ∼ R,Z 0. Then, over a neighborhood of z, and (S, B S ) has an n-semi-complement for some n ∈ N , where Proof Let N sdlt and N 1 be finite sets of positive integers divisible by l given by Theorem 4.3 and Theorem 2.20, respectively, which only depend on l.We will show that N := N sdlt ∪ N 1 has the required property.
It is clear that S is a semismooth curve, and (S, B S ) is sdlt.We first show that S is connected over a neighborhood of z.Otherwise, there exists a contraction φ : X → T to a curve T such that the general fiber F of φ is P 1 and each connected component of S is horizontal over T ; see Shokurov's connectedness lemma [33, 17.4 Theorem] and [42, Propositions 3.3.1 and 3.3.2](see also [45, 5.7 Connectedness lemma], [21,Corollary 1.3]).Note that if dim Z = 1, then we take R 0, a contradiction.Thus, S is connected over a neighborhood of z.Possibly shrinking Z near z, we may assume that K X + B ∼ R 0 and thus K S + B S is trivial on each complete component of S. If S has two irreducible incomplete components S 1 and S 2 that S 1 ∩ S 2 = ∅ over any neighborhood of z, then by assumption, we have B = S = S 1 + S 2 over a neighborhood of z.In this case, B S = 0 and K S ∼ 0 over a neighborhood of z.Now, we assume that each irreducible incomplete component of S does not meet any other irreducible incomplete component of S. By the classification of dlt surface pairs (cf.[36,Corollary 5.55]), over a neighborhood of z, either the support of B S lies in the union of the complete components of S or S is irreducible and its image on Z is also a curve.In the former case, (S, B S ) has an n-semi-complement in a neighborhood of the union of complete component of S for some n ∈ N sdlt by Theorem 4.3.Therefore, over a neighborhood of z, (S, B S ) has an n-semi-complement.In the latter case, the morphism from S to its image on Z is a contraction, then (S, B S ) has an n-complement over a neighborhood of z for some n ∈ N 1 .This finishes the proof.

Conjecture 2.16 for Surfaces
In this subsection, we confirm Conjecture 2.16 for surfaces.For convenience, by (Theorem * ) d we mean Theorem * in dimension d.

Notation ( ).
Let 1 ⊆ [0, 1]∩Q be a hyperstandard set.Let p = p( 1 ) be a positive integer given by (Proposition 3.3) 2 which only depends on 1 .Let N 2 = N 2 ( p) be a finite set of positive integers divisible by p given by Theorem 2.20 which only depends on p, and let 2 := (N 2 , 1 ).

Proposition 5.1 Under Notation
Proof By Lemma 2.15 we can run an MMP on −(K X + B 2 ) ∼ R,Z B − B 2 over Z and reach a good minimal model ψ : X → X over Z , such that −(K X + B 2 ) is semi-ample over Z , where D denotes the strict transform of D on X for any R-divisor D on X .Let π : X → Z be the contraction defined by −(K X + B 2 ) over Z .By assumption, dim Z = 1.Let B (2) Z and M π be the discriminant and moduli parts of the canonical bundle formula for (X , B 2 ) over Z in Proposition 3.3.Claim 5.2 pM π is base point free over Z and Assume Claim 5.2.Then, Z , and Z + M Z ) for any z ∈ Z .Now, by our choice of N 2 , (Z /Z z, B (2) According to Proposition 3.5, (X /Z z, B (2) ) is N 2 -complementary, and hence (X /Z z, B 2 ) is also N 2 -complementary as B (2) ≥ B 2 .Thus, (X /Z z, B) is N 2 -complementary by Lemma 2.14.Therefore, it suffices to prove Claim 5.2.

Proof of Claim 5.2 According to Lemma 2.15 again, we may run an MMP on −(K
, where D denotes the strict transform of D on X for any R-divisor D on X .One can pick a positive real number , such that g : X → X is also an MMP on . Hence, the natural morphism π : X → Z is the contraction defined by B 2 − B 1 over Z .In particular, we have By Lemma 3.2 and Proposition 3.3, we see that pM π is base point free, and and (X , B 2 ) are crepant.Therefore, We complete the proof.Theorem 5.3 Let l be a positive integer and ⊆ [0, 1]∩Q a hyperstandard set.Then, there exists a finite set of positive integers N divisible by l depending only on l and satisfying the following.
Assume that Proof Let N 1 = N 1 (l, ) be a finite set of positive integers divisible by l given by (Theorem 2.19) 2 which only depends on l and , and let 1 := (N 1 , ).Let p = p(l, 1 ) be a positive integer divisible by l given by (Proposition 3.3) 2 which only depends on l and 1 .Let N 2 = N 2 ( p) be a finite set of positive integers divisible by p given by Theorem 2.20 which only depends on p, and let 2 := (N 1 ∪ N 2 , ).
Let n CY = n CY (l, 2 ) be a positive integer divisible by l given by Lemma 2.18 which only depends on l and 2 .We will show that the finite set N := N 1 ∪ N 2 ∪ {n CY } has the required property.
Possibly replacing z by a closed point of z, we may assume that z is a closed point.Suppose that Possibly replacing (X , B) by a small Q-factorialization of (X , B + ) and shrinking Z near z, we may assume that (X , B) is Q-factorial klt and Therefore, we only need to consider the following three cases: by the choice of n CY (see Lemma 2.18).We finish the proof.

Proof of
and Let ˜ := ( R) and ˜ 1 := ( R1 ).It is clear that (R 1 ) = ({n}, ) and ˜ 1 = ({n}, ˜ ).By [33, 16.7 Corollary], if B ∈ , then B S ∈ ˜ , and if B ∈ ({n}, ), then B S ∈ ({n}, ˜ ).Therefore, ˜ has the required property.Proposition 5.5 Let (X /Z z, B) be a surface pair such that (X , B) is Q-factorial dlt and −(K X + B) is nef and big over a neighborhood of z.Let S := B and K S + B S := (K X + B)| S .Suppose that S intersects X z , the fiber of X → Z over z, and (S, B S ) has a monotonic n-semi-complement (S, B + S ) over a neighborhood of z.Then, (X /Z z, B) is n-complementary.
Proof Possibly replacing z by a closed point of z and shrinking Z near z, we may assume that z is a closed point, (X , B) is Q-factorial dlt, −(K X + B) is nef and big over Z , and n(K S + B + S ) ∼ 0. Let g : W → X be a log resolution of (X , B) such that g is an isomorphism over the snc locus of (X , B) (cf.[35,Theorem 10.45]), and let S W be the strict transform It remains to show that (X , B + ) is lc over a neighborhood of z.Let V be the non-lc locus of (X , B + ).There exists a real number a ∈ (0, 1), such that the non-klt locus of (X , a B is nef and big over Z , by Shokurov-Kollár connectedness principle (cf.[33, 17.4 Theorem]), (S ∪ V ) ∩ X z is connected.Recall that by assumption, S ∩ X z = ∅.Hence, V ∩ X z = ∅ and (X , B + ) is lc over a neighborhood of z.Theorem 5.6 Let l be a positive integer and ⊆ [0, 1]∩Q a hyperstandard set.Then, there exists a finite set of positive integers N divisible by l depending only on l and satisfying the following.
Assume that (X /Z z, B) is a surface pair such that Proof Let ˜ := ( R) be the hyperstandard set associated to the finite set R ⊆ [0, 1] ∩ Q given by Proposition 5.4 which only depends on .Possibly replacing l by a multiple, we may assume that l R ⊆ Z ≥0 .Let N 0 = N 0 (l, ) be a finite set of positive integers divisible by l given by Theorem 5.3 which only depends on l and .Let N 1 = N 1 (l) be a finite set of positive integers divisible by l given by Proposition 4.7 which only depends on l, and let 1 := (N 1 , ).Let p = p(l, 1 ) be a positive integer divisible by l given by (Proposition 3.3) 2 which only depends on l and 1 .Let N 2 = N 2 ( p) be a finite set of positive integers divisible by p given by Theorem 2.20 which only depends on p, and let 2 := (N 1 ∪ N 2 , ).Let n CY = n CY (l, 2 ) be a positive integer divisible by l given by Lemma 2.18 which only depends on l and 2 .We will show that the finite set N := N 0 ∪ N 1 ∪ N 2 ∪ {n CY } has the required property.
Possibly replacing z by a closed point of z, we may assume that z is a closed point.If (X /Z z, B N _ ) has a klt R-complement, then so does (X /Z z, B N 0 _ ), and hence (X /Z z, B) is N 0 -complementary by the choice of N 0 .Therefore, we may assume that (X /Z z, B N _ ) has an R-complement (X /Z z, B N _ + G) which is not klt.Possibly replacing (X /Z z, B) by a Q-factorial dlt modification of (X /Z z, B N _ + G) and shrinking Z near z, we may assume that (X , B) is Q-factorial dlt, K X + B ∼ R,Z 0, and S ∩ X z = ∅, where S := B = 0 and X z is the fiber of X → Z over z.
Therefore, we only need to consider the following three possibilities: ( Since −(K X + B) − ψ * (−(K X + B n_ )) is nef over X , and −B + B n_ ≤ 0, by the negativity lemma, Note that B n_ ,S ∈ ({n}, ˜ ) by Proposition 5.4, and the support of −(K X + B) − ψ * (−(K X + B n_ )) does not contain any component of S.Then, Let B S be the strict transform of B S on S .Since (B S ) n_ ˜ , B n_ ,S ∈ ({n}, ˜ ), and B S ≥ B n_ ,S , we deduce that (B S ) n_ ˜ ≥ B n_ ,S .Hence, (S , B n_ ,S ) has a monotonic n-semi-complement over a neighborhood of z.By Proposition 5.5 and Lemma 2.14, (X /Z z, B n_ ) has a monotonic n-complement.Since ψ is −(K X + B n_ )-negative, (X /Z z, B n_ ) has a monotonic n-complement (X /Z z, B + ).By Lemma 2.14, (X /Z z, B + ) is an n-complement of (X /Z z, B).
Proof of Theorem 1. 3 The theorem follows by Theorem 5.6.

Boundedness of Complements for Threefolds
We will prove the following theorem which is stronger than Theorem 1.1.Theorem 6.1 Let l be a positive integer and ⊆ [0, 1]∩Q a hyperstandard set.Then, there exists a finite set of positive integers N divisible by l depending only on l and satisfying the following.
Assume that (X /Z z, B) is a threefold pair such that (X /Z z, B N _ ) has a klt R-complement.Then, (X /Z z, B) is N -complementary.
Proof Let N 1 = N 1 (l, ) be a finite set of positive integers divisible by l given by (Theorem 2.19) 3 which only depends on l and , and set 1 := (N 1 , ).Let p 1 = p 1 (l, 1 ) be a positive integer divisible by l given by (Proposition 3.3) 3 and Proposition 3.4 which only depends on l and 1 .Let N 2 = N 2 ( p 1 ) be a finite set of positive integers divisible by p 1 given by Theorems 2.20 and 1.3 which only depends on p 1 , and set 2 := (N 1 ∪ N 2 , ).Let p 2 = p 2 ( p 1 , 2 ) be a positive integer divisible by p 1 given by (Proposition 3.3) 3 and Proposition 3.4 which only depends on p 1 and 2 .Let N 3 = N 3 ( p 2 ) be a finite set of positive integers divisible by p 2 given by Theorems 2.20 and 1.3 which only depends on p 2 , and set 3 := (N 1 ∪N 2 ∪N 3 , ).
Let n CY = n CY (l, 3 ) be a positive integer divisible by l given by Lemma 2.18 which only depends on l and 3 .We will show that N := N 1 ∪ N 2 ∪ N 3 ∪ {n CY } has the required property.
Replacing z by a closed point of z, we may assume that z is a closed point.Possibly replacing (X /Z z, B) by a small Q-factorialization of a klt R-complement of (X /Z z, B N _ ) and shrinking Z near z, we may assume that (X , B) is Q-factorial klt and Lemma 2.18).Therefore, in the following, we may assume that In particular, there exist integers i, k ∈ {1, 2} such that We will show that (X /Z z, B i+1 ) is N i+1 -complementary and thus finish the proof by Lemma 2.14.By Lemma 2.15, we can run an MMP on −(K X + B i+1 ) ∼ R,Z B − B i+1 over Z and reach a good minimal model X X over Z , such that B − B i+1 is semiample over Z , where D denotes the strict transform of D on X for any R-divisor D on X .Let π : X → Z be the contraction defined by and M π be the discriminant and moduli parts of the canonical bundle formula for (X , B i+1 ) over Z in Proposition 3.3 (respectively, Proposition 3.4) if k = 2 (respectively, k = 1).Claim 6.2 p i M π is base point free over Z , and Assume Claim 6.2.As X X is an MMP on −(K X + B i+1 ) over Z , for any prime divisor P on X which is exceptional over X , we have Therefore, ).We finish the proof.
7 Proof of Theorem 1.2 Proof Since X is of Calabi-Yau type over Z , there exists a klt pair (X , C) such that ).We have and where B + Y and C Y are the strict transforms of B + and C on Y , respectively, a + i := a(E i , X , B + ) < 1, and a i := a(E i , X , C) for any i.Then, for any R-complement (X /Z z, B + ) of (X /Z z, B), B + = B over some neighborhood of z.Remark 7. 4 When dim = 0, (X , B) is strictly lc Calabi-Yau if and only if (1) holds.Lemma 7.5 Assume that (X /Z z, B) is an R-complement of itself.Then, (X /Z z, B) is strictly lc Calabi-Yau if and only if either dim z = dim Z or z is the image of an lc center of (X , B) on Z .
Proof First assume that (X /Z z, B) is strictly lc Calabi-Yau.Suppose that dim z < dim Z and z is not the image of any lc center of (X , B). Possibly shrinking Z near z, we may find an ample divisor H ≥ 0 such that z ∈ Supp H and H does not contain the image of any lc center of (X , B). Pick a positive real number , such that (X /Z z, B + π * H ) is lc and thus an R-complement of (X /Z z, B).However, B + π * H = B over any neighborhood of z, a contradiction.Now, we prove the converse direction.Assume that If z is the image of some lc center of (X , B), then z / ∈ Supp L Z as (X , B + G) is lc over a neighborhood of z.Therefore, in both cases, (X /Z z, B) is strictly lc Calabi-Yau.
Example 7. 6 Let π : X := P 1 × P 1 → Z := P 1 , z ∈ Z a closed point, and L 1 , L 2 two sections.Then, over a neighborhood of z, we have (X , L 1 + L 2 ) is lc and K X + L 1 + L 2 ∼ R,Z 0. Since K X + L 1 + L 2 + π * z ∼ R,Z 0, (X /Z z, L 1 + L 2 ) is not strictly lc Calabi-Yau.Lemma 7.7 Suppose that (X /Z z, B) is strictly lc Calabi-Yau and X X is a birational contraction over Z .Let B be the strict transform of B on X .Then, (X /Z z, B ) is strictly lc Calabi-Yau.
Proof It follows from the definition of strictly lc Calabi-Yau and the fact that (X , B) (X , B ) is crepant over some neighborhood of z.
Proposition 7.8 Let ⊆ [0, 1] ∩ Q be a DCC set.Then, there exists a positive integer I depending only on satisfying the following.If (X /Z z, B) is a strictly lc Calabi-Yau threefold pair such that B ∈ , then I (K X + B) ∼ 0 over some neighborhood of z.
Proof Possibly replacing (X /Z z, B) by a Q-factorial dlt modification, we may assume that X is Q-factorial.Since (X /Z z, B) is a strictly lc Calabi-Yau pair, by [26,Theorem 5.20], B ∈ over a neighborhood of z for some finite subset ⊆ which only depends on .According to [12,Theorem 2.14], we may find a positive integer I which only depends on such that (X /Z z, B) has a monotonic Icomplement (X /Z z, B + G) for some G ≥ 0. By assumption, G = 0 over some neighborhood of z.Thus, I (K X + B) = I (K X + B + G) ∼ 0 over some neighborhood of z.

Proof of Theorem
We first show a special case of Theorem 1.2.
For convenience, we say a pair (X , B) is klt over a closed subset Z 0 ⊆ Z , if a(E, X , B) > 0 for any prime divisor E over X such that π(Center X (E)) ⊆ Z 0 , where π : X → Z is a contraction.For two R-divisors D for any n ∈ N .By Theorem 1.1 and the construction of N , (X /Z z, B ) is ncomplementary for some n ∈ N .Thus, (X /Z z, B) is n-complementary by (7.1).
Therefore, it suffices to prove the claim.By assumption, we may find an effective R-Cartier R-divisor H 1 ∼ R,Z H such that Z \ U ⊆ Supp H 1 and (X , B + H 1 ) is lc, where H 1 := (π ) * H 1 .In particular, we have Since X is of Calabi-Yau type over Z , there exists a boundary C such that (X , C) is klt and K X + C ∼ R,Z 0. Let δ ∈ (0, 1) be a positive real number such that It is clear that (X , B ) is klt and K X + B ∼ R,Z 0. This completes the proof.
123 Proof of 7. 10 We may pick a positive real number such that (X , C + F ) is klt and that X X is a sequence of steps of the −(K X + B N _ + (1 − )F)-MMP over Z .Since one can run an MMP on −(K X + B N _ + (1 − )F ) over Z .This MMP terminates with a model X on which −(K X + B N _ + (1 − )F ) is semi-ample over Z , where D denotes the strict transform of D on X for any R-divisor D on X .

X X X Z Z ψ
For any positive real number , we infer that ψ : X X is also an MMP on −(K X + B N _ + (1 − )F) over Z , and If dim X = dim Z and z = η ∈ Z scy , then X is smooth and B + = 0 over a neighborhood of η .Otherwise, by Lemma 7.7, we know that (X /Z z , B N _ ) is strictly lc Calabi-Yau for any z ∈ Z scy , which implies that B N _ = B + = B ∈ ∩ Q over a neighborhood of z .We, therefore, see that I (K X + B N _ ) ∼ 0 over some neighborhood of z by our choice of I .Since (X , B N _ + F ) is crepant to (X , B N _ ), our claim holds.

Definition 2 . 9
The lemma follows from [4, Lemma 2.48].Let X → Z be a contraction and D an R-Cartier R-divisor on X .We denote by κ(X , D) and κ(X /Z , D) the Iitaka dimension and relative Iitaka dimension of D respectively; see [41, II, §3.b and §3.c].Definition 2.10 Let X → Z be a contraction, D an R-Cartier R-divisor on X , and φ : X Y a birational contraction of normal quasi-projective varieties over Z .We say that Y is a good minimal model of D over Z , if φ is D-negative, D Y is semi-ample over Z , and Y is Q-factorial, where D Y is the strict transform of D on Y .

Theorem 2 .
19 (cf.[47,Theorem 16]) Let d, l be two positive integers and ⊆ [0, 1] ∩ Q a hyperstandard set.Then, there exists a finite set of positive integers N divisible by l depending only on d, l and satisfying the following.

Theorem 2 .
20 ([11, Theorem 1.3]) Let l be a positive integer.Then, there exists a finite set of positive integers N divisible by l depending on l satisfying the follow- The result follows from [12, Proposition 3.1] and [16, Theorem 5.5].Proposition 3.4 Let ⊆ [0, 1]∩Q be a hyperstandard set.Then, there exists a positive integer p depending only on satisfying the following.
T ).Let b F be the second Betti number of a smooth model of the index one cover of F. By Lemma 2.8, there exists a positive real number which only depends on such that (F, B F ) is -lc.If B F = 0, then F belongs to a bounded family by [2, Theorem 6.9], and hence b F has an upper bound.If B F = 0, then K F ∼ Q 0, and hence b F ≤ 22 by the classification of surfaces.Therefore, by[17, Theorem 1.2], there exists a positive integer p 2 depending only on b F such that p 2 M φ,T ∼ 0.

Proposition 3 . 5
Let p and n be two positive integers such that p | n.Let (X /Z , B) be an lc pair and φ : X → T a contraction over Z such that dim T > 0 and K X + B ∼ R,T 0. Let B T and M φ be the discriminant part and a moduli part of the canonical bundle formula for (X , B) over T , such that p(K X + B) ∼ pφ * (K T + B T + M φ,T ) and pM φ is b-Cartier.Let (T , B T + M φ ) → (T , B T + M φ ) be a crepant model and M T an effective Q-divisor on T , such that and thus b P = b + P .If b P = 1, then r P Q = b Q = 1 and thus b + P = 1.Hence, we may assume that b Q < 1 and b P < 1.Since b P = b
complementary by Proposition 5.1.Hence, in what follows we assume that κ(X /Z , B − B 1 )+ dim Z = κ(X /Z , B − B ) + dim Z = 2.We will show that (X /Z z, B) is N 1 -complementary.In this case, both B − B and B − B 1 are big over Z .Let K S + B S := (K X + B)| S ∼ R,Z 0. By Lemma 2.14 and the choice of N 1 , (S, (B S ) n_ ˜ ) has a monotonic n-semi-complement over a neighborhood of z for some n ∈ N 1 .Note that B n_ ∈ ({n}, ) ⊆ (N 1 , ), B n_ ≤ B 1 , and B − B n_ is big over Z .According to Lemma 2.15, we may run an MMP on −(K X + B n_ ) ∼ R,Z B − B n_ over Z and reach a minimal model ψ : X → X over Z , such that B − B n_ is nef and big over Z , where D denotes the strict transform of D on X for any R-divisor D on X .No component of S is contracted by ψ and ψ S := ψ| S : S → S is an isomorphism as S ≤ B n_ ≤ B and ψ is (K X + B)-trivial.

Proposition 7 . 9
1 and D 2 on X , by D 1 ≥ D 2 (respectively, D 1 > D 2 ) over Z 0 , we mean that multE D 1 ≥ mult E D 2 (respectively, mult E D 1 > mult E D 2 ) for any prime divisor E on X with π(E) ⊆ Z 0 .By D 1 ≥ D 2 over an open subset U ⊆ Z , we mean D 1 | π −1 (U ) ≥ D 2 | π −1 (U ) .Let I be a positive integer.Assume that N is a finite set of positive integers divisible by I given by Theorem 1.1 which only depends on I .Assume that (X /Z z, B) is an R-complementary threefold pair such that X is of Calabi-Yau type over Z .Assume that is a contraction π : X → Z over Z , and an open subset U ⊆ Z , such that(1) I B ∈ Z ≥0 over U , (2) (X , B) is klt over Z \ U , and (3) −(K X + B) ∼ R (π ) * H for some R-divisor H which is ample over Z .Then, (X /Z z, B) is N -complementary.Proof Possibly shrinking Z near z, we may assume that (X , B) is lc.Set N := max n∈N n.We claim that there exists a boundary B on X such that• (X , B ) is klt, • K X + B ∼ R,Z0, and • B ≥ N N +1 B over U and B ≥ B over Z \ U .Assume the claim holds.Then, (n + 1)B ≥ n B + (n + 1){B} (7.1) is semi-ample over Z (see Lemma 2.6).In particular, X is a good minimal model of−(K X + B N _ +(1− )F) over Z .Since N σ (−(K X + B N _ +(1− )F)/Z ) = F (cf.[41, III,4.2Lemma]), by[23, Lemma 2.4], F is contracted by ψ.Hence,K X + B N _ = ψ * K X + B N _ + F .
and H z := H | Z z .By assumption H z is ample over Z z and H z is ample.According to [36, Proposition 1.45], z H z + (τ | Z z ) * H z is ample for any 0 < z 1.In particular, z H + τ * H is ample over z.It follows that z H + τ * H is ample over some neighborhood of z by [37, Theorem 1.2.17].Since Z is quasi-compact, the lemma follows.
and thus D| X (i) ∼ 0. Therefore, by induction we see that D ∼ 0.
i ≤ 1.Then, there exist positive integers n ∈ N and k ≥ k, and two sequences of non-negative real numbers {a + i } k i=1 and {b

7.1 Strictly Lc Calabi-Yau Pairs Definition 7.1
We say that X is of Calabi-Yau type over Z , if X → Z is a contraction, and there is a boundary C such that (X , C) is klt and K X + C ∼ R,Z 0.
Lemma 7.2 Suppose that X is of Calabi-Yau type over Z .Assume that (X , B + ) is lc, K X + B + ∼ R,Z 0 for some boundary B + , and f : Y → X is a projective birational morphism from a normal quasi-projective variety Y , such that a(E i , X , B + ) < 1 for any prime exceptional divisor E i of f .Then, Y is of Calabi-Yau type over Z .