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Home , Iitaka dimension, Kodaira dimension, Minimal model program

MINIMAL MODEL THEORY OF NUMERICAL ZERO

YOSHINORI GONGYO

Abstract. We prove the existence of good minimal models of numerical Kodaira dimension 0.

Contents 1. Introduction 1 2. Preliminaries 3 3. Log with scaling 7 4. Zariski decomposition in the sense of Nakayama 9 5. Existence of minimal model in the case where ν = 0 10 6. Abundance theorem in the case where ν = 0 12 References 14

1. Introduction Throughout this paper, we work over C, the complex number field. We will make use of the standard notation and definitions as in [KM] and [KMM]. The minimal model conjecture for smooth varieties is the following: Conjecture 1.1 (Minimal model conjecture). Let X be a smooth pro- jective variety. Then there exists a minimal model or a Mori fiber space of X. This conjecture is true in dimension 3 and 4 by Kawamata, Koll´ar, Mori, Shokurov and Reid (cf. [KMM], [KM] and [Sho2]). In the case where KX is numerically equivalent to some effective divisor in dimen- sion 5, this conjecture is proved by Birkar (cf. [Bi1]). When X is of general type or KX is not pseudo-effective, Birkar, Cascini, Hacon Date: 2010/9/21, version 4.03. 2000 Mathematics Subject Classification. 14E30. Key words and phrases. minimal model, the , numerical Kodaira dimension. 1 2 YOSHINORI GONGYO and McKernan prove Conjecture 1.1 for arbitrary dimension ([BCHM]). Moreover if X has maximal Albanese dimension, Conjecture 1.1 is true by [F2]. In this paper, among other things, we show Conjecture 1.1 in the case where ν(KX ) = 0 (for the definition of ν, see Definition 2.6):

Theorem 1.2. Let X be a smooth such that ν(KX ) = 0. Then there exists a good minimal model Xm of X. In particular, ∼ KXm Q 0. In fact, we construct log minimal models for divisorial log terminal pairs (Corollary 5.5). The key idea of this proof is to apply the Zariski decomposition in the sense of Nakayama for the log canonical divisor when the scale, in the log minimal model program with scaling, con- verges to zero. Moreover, from [G], we recover the abundance theorem for log canonical pairs in the case where ν = 0 proved by [CKP] and [Ka]: Theorem 1.3 (=Theorem 6.1). Let X be a normal projective variety and ∆ an effective R-divisor. Suppose that (X, ∆) is a log canonial pair such that ν(KX +∆) = 0. Then KX +∆ is abundant, i.e. ν(KX +∆) = κ(KX + ∆). In the first time, Nakayama proved the above theorem for klt pairs in [N]. Siu also gave the analytic proof of Theorem 1.3 when X is smooth and ∆ = 0 (cf. [Siu]). The results of [CKP], [Ka] and [Siu] depend on [Sim] and [Bu]. In this paper, we show Theorem 1.3 by a different way from [CKP], [Ka] and [Siu]. In our proof of Theorem 1.3, we do not need results of [Sim] and [Bu]. Our proof depends on [BCHM] and [G]. We summarize the contents of this paper. In Section 2, we define several base loci, the Kodaira dimension and the numerical Kodaira dimension, and collect some properties of these. In Section 3, we review a log minimal model program with scaling and works of Birkar–Cascini– Hacon–McKernan. In Section 4, we introduce the Zariski decomposition in the sense of Nakayama and collect some properties of it. Section 5 is devoted to the proof of the existence of minimal models in the case where ν = 0. In Section 6, we prove Theorem 1.3. Notation and Definition 1.4. Let K be the real number field R or the rational number field Q. We set K>0 = {x ∈ K|x > 0}. Let X be a normal variety and let ∆ be an effective K-divisor such that KX + ∆ is K-Cartier. Then we can define the discrepancy a(E,X, ∆) ∈ K for every prime divisor E over X. If a(E,X, ∆) ≥ −1 (resp. > −1) for every E, then (X, ∆) is called log canonical (resp. kawamata log terminal). We sometimes abbreviate log canon- ical (resp. kawamata log terminal) to lc (resp. klt). MINIMAL MODEL THEORY IN THE CASE WHERE ν = 0 3

Assume that (X, ∆) is log canonical. If E is a prime divisor over X such that a(E,X, ∆) = −1, then cX (E) is called a log canonical center (lc center, for short) of (X, ∆), where cX (E) is the closure of the image of E on X. For the basic properties of log canonical centers, see [F1, Section 9]. Let π : X → S be a projective morphism of normal quasi-projective varieties and D a Z-Cartier divisor on X. We set the complete linear system |D/S| = {E|D ∼Z,S E ≥ 0} of D over S. The base locus of the linear system |D/S| is denoted by Bs|D/S|. When S = Spec C, we denote simply |D| and Bs|D|.∑ R r For an -Weil divisor D = j=1 djDj such that Dj is a prime divisor ̸ ̸ p q for∑ every j and Di = Dj for i = j, we define∑ the round-up D = r p q x y r x y j=1 dj Dj (resp. the round-down D = j=1 dj Dj), where for every real number x, pxq (resp. xxy) is the integer defined by x ≤ pxq < x + 1 (resp. x − 1 < xxy ≤ x). The fractional part {D} of D denotes D − xDy. Acknowledgments. The author wishes to express his deep gratitude to Professor Osamu Fujino for various comments and discussions. He would like to thank his supervisor Professor Hiromichi Takagi for care- fully reading a preliminary version of this paper and valuable com- ments. He also wishes to thank Professor , Professor Daisuke Matsushita, Professor Noboru Nakayama and Professor Mihai P˘aunfor valuable comments. Moreover he would like to thank Pro- fessor Vyacheslav V. Shokurov for serious remarks about the Kodaira dimensions of R-divisors. He is partially supported by the Research Fel- lowships of the Japan Society for the Promotion of Science for Young Scientists.

2. Preliminaries Definition 2.1 (Classical , cf. [N, II, 3.2, Definition]). Let X be a normal projective variety and D an R-Cartier divisor on X. If |xmDy| ̸= ∅, we put a dominant rational map

φ|xmDy| : X 99K Wm, with respect to the complete linear system of xmDy. We define the Classical Iitaka dimension ι(D) of D as the following:

ι(D) = max{dimWm} if H0(X, xmDy) ≠ 0 for some positive integer m or ι(D) = −∞ other- wise. 4 YOSHINORI GONGYO

Lemma 2.2. Let Y be a normal projective variety, ϕ : Y → X a projective birational morphism onto a normal projective variety, and let D be an R-Cartier divisor on X. Then it holds the following: (1) ι(ϕ∗D) = ι(ϕ∗D + E) for a ϕ-exceptional effective R-divisor E, and (2) ι(ϕ∗D) = ι(D). Proof. (1) and (2) follows from [N, II, 3.11, Lemma]. ¤

The following is remarked by Shokurov:

′ ′ Remark 2.3. In general, ι(D) may not coincide with ι(D ) if D ∼R D . For example, let X be the P1, P and Q closed points in X such that P ≠ Q and a irrational number. Set D = a(P − Q). Then ι(D) = −∞ in spite of the fact that D ∼R 0. However, fortunately, ι(D) coincides with ι(D′) if D and D′ are ′ effective divisors such that D ∼R D ([Ch, Corollary 2.1.4]). Hence it seems reasonable that we define the following as the Iitaka (Kodaira) dimension for R-divisors. Definition 2.4 (Invariant Iitaka dimension, [Ch, Definition 2.2.1], cf. [CS, Section 7]). Let X be a normal projective variety and D an R- Cartier divisor on X. We define the invariant Iitaka dimension κ(D) of D as the following:

κ(D) = ι(D′) ′ ′ if there exists an effective divisor D such that D ∼R D or κ(D) = −∞ otherwise. Let (X, ∆) be a log canonical. Then we call κ(KX + ∆) the log Kodaira dimension of (X, ∆). If ∆ = 0, we simply say κ(KX ) is the Kodaira dimension of X. Remark 2.5. We remark the following: (i) In [Ch, Definition 2.2.1], κ(·) is denoted as I(·). Moreover the above log Kodaira dimension coincides with the log Ko- daira dimension in [Ch, Definition 2.2.3] because κ(KX + ∆) = log κ(KY +∆Y ) for a log canonical pair (X, ∆) and a log resolution Y → X ([Ch, Corollary 2.2.6 (2)]), and

(ii) Let D be a Q-Cartier divisor such that there exists an effective ′ ′ Q-divisor D such that D ∼Q D . Then κ(D) = ι(D). MINIMAL MODEL THEORY IN THE CASE WHERE ν = 0 5

Definition 2.6 (Numerical Iitaka dimension for smooth varieties). Let Y be a smooth projective variety, D an R-Cartier divisor and A a Cartier divisor on Y . We set −k 0 σ(D,A) = max{k ∈ Z≥0| lim sup m dimH (X, xmDy + A) > 0} m→∞ if H0(X, xmDy + A) ≠ 0 for infinitely many m ∈ N or σ(D,A) = −∞ otherwise. We define ν(D) = max{σ(D,A)|A is a divisor on Y }. Lemma 2.7. Let Y be a smooth projective variety, ϕ : Y → X a projective birational morphism onto a normal projective variety, and let D be an R-Cartier divisor on X. Then it holds the following: (1) ν(ϕ∗D) = ν(ϕ∗D + E) for a ϕ-exceptional effective R-divisor E, (2) ν(ϕ∗D) = ν(D) when X is smooth, and k (3) ν(D) = max{k ∈ Z≥0|D ̸≡ 0} when D is nef. Proof. Let A be a divisor on Y . We take a sufficiently ample divisor H on X such that H0(Y, ϕ∗H − A) ≠ 0. Since H0(Y, xm(ϕ∗D + E)y + A) ⊆ H0(Y, xm(ϕ∗D + E)y + ϕ∗H), = H0(Y, xmϕ∗Dy + ϕ∗H) for any m > 0, it holds that ν(ϕ∗D) ≥ ν(ϕ∗D + E). Therefore (1) follows. We see (2) and (3) from [N, V, 2.7, Proposition, (4) and (6)]. ¤ Definition 2.8 (Numerical Iitaka dimension for singular varieties). Let X be a projective variety which may have singularities and D an R-Cartier divisor on X. Take ϕ : Y → X as a resolution of singularities of X. We define ν(D) = ν(ϕ∗D). Remark that ν(D) is independent of the choice of ϕ by Lemma 2.7. Let (X, ∆) be a log pair. Then we call ν(KX + ∆) the numerical log Kodaira dimension of (X, ∆). If ∆ = 0, we simply say ν(KX ) is the numerical Kodaira dimension of X. Lemma 2.9 ([N, V, 2.7, Proposition, (1)]). Let X be a projective va- riety and D and D′ R-Cartier divisors on X such that D ≡ D′. Then ν(D) = ν(D′).

Remark 2.10. ν(D) is denoted as κσ(D) in [N, V, § 2]. Moreover − + Nakayama also defined κσ (D), κσ (D) and κν(D) as some numerical Iitaka dimensions. In this paper we mainly treat in the case where 6 YOSHINORI GONGYO

− + ν(D) = 0, i.e. κσ(D) = 0. Then it holds that κσ (D) = κσ (D) = κν(D) = 0 (cf. [N, V, 2.7, Proposition (8)]). Moreover, if a log canonical pair (X, ∆) has a weakly log canonical model in the sense of Shokurov, then ν(KX + ∆) coincides with the numerical log Ko- daira dimension in the sense of Shokurov by Lemma 2.7 (cf. [Sho1, 2.4, Proposition]). Definition 2.11. Let π : X → S be a projective morphism of normal quasi-projective varieties and D an R-Cartier divisor on X. We set ∩ ∩ B(D/S) = Supp E, B≡(D/S) = Supp E,

D∼S,RE≥0 D≡S E≥0 and ∩ B+(D/S) = B(D − ϵA/S),

ϵ∈R>0 where A be a π-ample divisor. We remark that these definitions are independent of the choice of A. When S = Spec C, we denote simply B(D), B≡(D) and B+(D). Lemma 2.12. Let π : X → S be a projective morphism of nor- mal quasi-projective varieties and D a π-big R-Cartier divisor on X. Let Z1,Z2,...,Zk be Zariski closed subsets of X. Suppose that Zi ̸⊆ B+(D/S) for all i. Then there exist a π-ample Q-divisor A and an effective R-divisor B such that Zi ̸⊆ Supp B and

A + B ∼R,S D. Proof. By [ELMNP, Proposition 1.5], we can take a π-ample R-divisor A′ and a sufficiently small positive number ϵ such that D − ϵA′ is a Q-Cartier divisor and ′ B+(D/S) = B(D − ϵA /S). Moreover, from [BCHM, Lemma 3.5.3], it holds that ∩ B(D − ϵA′/S) = Bs|md(D − ϵA′)/S|, m≥1 where d is a positive integer such that d(D − ϵA′) is an integral divisor. ′ Therefore we can take m ≫ 0 such that Zi ̸⊆ Bs|md(D − ϵA )/S| for all i. Thus a general member B′ ∈ |md(D − ϵA′)/S| satisfies that ′ Zi ̸⊆ Supp B for any i. We decompose ′′ ′ A + A ∼R,S ϵA MINIMAL MODEL THEORY IN THE CASE WHERE ν = 0 7 such that A is a π-ample Q-divisor and A′′ is an effective R-divisor ′′ which satisfies Zi ̸⊆ Supp A for any i. When we put 1 B = B′ + A′′, md Lemma 2.12 follows. ¤ We introduce a dlt blow-up. The following theorem is originally proved by Hacon: Theorem 2.13 (Dlt blow-up, [F1, Theorem 10.4], [KK, Theorem 3.1]). Let X be a normal quasi-projective variety and ∆ an effective K-divisor on X such that KX + ∆ is K-Cartier. Suppose that (X, ∆) is log canonical. Then there exists a projective birational morphism ϕ : Y → X from a normal quasi-projective variety with the following properties: (i) Y is Q-factorial, (ii) a(E,X, ∆) = −1 for every ϕ-exceptional divisor E on Y , and (iii) for ∑ −1 Γ = ϕ∗ ∆ + E, E:ϕ-exceptional ∗ it holds that (Y, Γ) is dlt and KY + Γ = ϕ (KX + ∆). The above theorem is very useful for studying log canonical singu- larities (cf. [F1], [G], [KK]).

3. Log minimal model program with scaling In this section, we review a log minimal model program with scaling and introduce works by Birkar–Cascini–Hacon–McKernan. Lemma 3.1 (cf. [Bi1, Lemma 2.1] and [F1, Theorem 18.9]). Let π : X → S be a projective morphism of normal quasi-projective varieties and (X, ∆) a Q-factorial projective log canonical pair such that ∆ is a K-divisor. Let H be an effective K-divisor such that KX + ∆ + H is π-nef and (X, ∆ + H) is log canonical. Suppose that KX + ∆ is not π-nef. We put

λ = inf{α ∈ R|KX + ∆ + αH is π-nef}.

Then λ ∈ K>0 and there exists an extremal ray R ⊆ NE(X/S) such that (KX + ∆).R < 0 and (KX + ∆ + λH).R = 0. Definition 3.2 (Log minimal model program with scaling). Let π : X → S be a projective morphism of normal quasi-projective varieties and (X, ∆) a Q-factorial projective divisorial log terminal pair such 8 YOSHINORI GONGYO that ∆ is a K-divisor. Let H be an effective K-divisor such that KX + ∆ + H is π-nef and (X, ∆ + H) is divisorial log terminal. We put

λ1 = inf{α ∈ R|KX + ∆ + αH is π-nef}.

If KX + ∆ is not π-nef, then λ1 > 0. By Lemma 3.1, there exists an extremal ray R1 ⊆ NE(X/S) such that (KX + ∆).R1 < 0 and (KX + ∆ + λ1H).R1 = 0. We consider an extremal contraction with respect to this R1. If it is a divisorial contraction or a flipping contraction, let

(X, ∆) 99K (X1, ∆1) be the divisorial contraction or its flip. Since KX1 +∆1 +λ1H1 is π-nef, we put { ∈ R| } λ2 = inf α KX1 + ∆1 + αH1 is π-nef , where H1 is the strict transform of H on X1. Then we find an extremal ray R2 by the same way as the above. We may repeat the process. We call this program a log minimal model program with scaling of H over S. When this program runs as the following:

(X0, ∆0) = (X, ∆) 99K (X1, ∆1) 99K ··· 99K (Xi, ∆i) ··· , then λ1 ≥ λ2 ≥ λ3 ..., { ∈ R| } where λi = inf α KXi−1 + ∆i−1 + αHi−1 is π-nef and Hi−1 is the strict transform of H on Xi−1. The following theorems are slight generalizations of [BCHM, Corol- lary 1.4.1] and [BCHM, Corollary 1.4.2]. These seem to be well-known for the experts. Theorem 3.3 (cf. [BCHM, Corollary 1.4.1]). Let π : X → S be a projective morphism of normal quasi-projective varieties and (X, ∆) be a Q-factorial projective divisorial log terminal pair such that ∆ is an R-divisor. Suppose that ϕ : X → Y is a flipping contraction of (X, ∆). Then there exists the log flip of ϕ.

Proof. Since −(KX + ∆) is ϕ-ample, so is −(KX + ∆ − ϵx∆y) for a sufficiently small ϵ > 0. Because ρ(X/Y ) = 1, it holds that KX + ∆ ∼R,Y c(KX + ∆ − ϵx∆y) for some positive number c. By [BCHM, Corollary 1.4.1], there exists the log flip of (X, ∆−ϵx∆y). This log flip is also the log flip of (X, ∆) since KX + ∆ ∼R,Y c(KX + ∆ − ϵx∆y). ¤ Theorem 3.4 (cf. [BCHM, Corollary 1.4.2]). Let π : X → S be a projective morphism of normal quasi-projective varieties and (X, ∆) be a Q-factorial projective divisorial log terminal pair such that ∆ is an MINIMAL MODEL THEORY IN THE CASE WHERE ν = 0 9

R-divisor. Suppose that there exists a π-ample R-divisor A on X such that ∆ ≥ A. Then any sequences of log flips starting from (X, ∆) with scaling of H over S terminate, where H satisfies that (X, ∆ + H) is divisorial log terminal and KX + ∆ + H is π-nef. The above theorem is proved by the same argument as the proof of [BCHM, Corollary1.4.2] because [BCHM, Theorem E] holds on the above setting. 4. Zariski decomposition in the sense of Nakayama In this section, we introduce the Zariski decomposition in the sense of Nakayama for a pseudo-effective divisor. Definition 4.1 (cf. [N, III, 1.13, Definition]). Let π : X → S be a projective morphism of normal quasi-projective varieties and D an R- Cartier divisor. We call that D is a limit of movable R-divisors over S if [D] ∈ Mov(X/S) ⊆ N1(X/S) where Mov(X/S) is the closure of the convex cone spanned by classes of fixed part free Z-Cartier divisors over S. When S = Spec C, we denote simply Mov(X). Lemma 4.2. Let π : X → S be a projective morphism of normal quasi- projective varieties and D an R-Cartier divisor. Suppose that B(D/S) dose not coincide with X and has no codimension 1 components. Then there exist fixed part free Z-Cartier divisors {Zi} over S and positive numbers {α } such that i ∑ 1 [D] = αi[Zi] ∈ N (X/S). i Proof. This follows from [BCHM, Proposition 3.5.4]. ¤ Definition 4.3 (cf. [N, III, 1.6, Definition and 1.12, Definition]). Let X be a smooth projective variety and B an R-big divisor. We define ′ ′ σΓ(B) = {multΓB |B ≡ B ≥ 0} for a prime divisor Γ. Let D be a pseudo-effective divisor. Then we define the following:

σΓ(D) = lim σΓ(D + ϵA) ϵ→0+ for some ample divisor A. We remark that σΓ(D) is independent of the choice of A. Moreover the above two definitions coincide for a big divisor because a function σΓ(·) on Big(X) is continuous where Big(X) := {[B] ∈ N1(X)|B is big} (cf. [N, III, 1.7, Lemma]). We set ∑ N(D) = σΓ(D)Γ and P(D) = D − N(D). Γ:prime divisor 10 YOSHINORI GONGYO

We remark that N(D) is a finite sum. We call the decomposition D ≡ P(D) + N(D) the σ-decomposition of D. We say that P(D) (resp. N(D)) is the positive part (resp. negative part) of D. Moreover we say that the decomposition D ≡ P(D) + N(D) is the Zariski decomposition in the sense of Nakayama of D if P(D) is nef. Proposition 4.4. Let X be a smooth projective variety and D a pseudo- effective R-divisor on X. Then it holds the following:

(1) σΓ(D) = limϵ→0+ σΓ(D + ϵE) for a pseudo-effective divisor E, and (2) ν(D) = 0 if and only if D ≡ N(D). Proof. (1) follows from [N, III, 1.4, Lemma]. (2) follows from [N, V, 2.7, Proposition (8)]. ¤

5. Existence of minimal model in the case where ν = 0 Lemma 5.1 (cf. [BCHM, Lemma 3.7.3]). Let π : X → S be a pro- jective morphism of normal quasi-projective varieties and (X, ∆) a Q- factorial projective divisorial log terminal pair such that ∆ is an R- divisor, and H an effective π-big R-divisor such that (X, ∆ + H) is divisorial log terminal. Suppose that no lc centers of (X, ∆) are con- tained in B+(H/S). Then there exist an effective π-ample Q-divisor A and an effective R-divisor B such that H ∼R,S B+A and (X, ∆+B+A) is divisorial log terminal.

Proof. Since no lc centers of (X, ∆) are contained in B+(H/S), there exist a π-ample Q-divisor A′ and an effective R-divisor B′ such that Supp B′ contains no lc centers of (X, ∆) and ′ ′ A + B ∼R,S H ′′ ′ from Lemma 2.12. We take a general effective Q-divisor A ∼Q,S A and a sufficiently small number δ ∈ Q>0 such that (X, ∆ + (1 − δ)H + δ(A′′ + B′)) is dlt. When we put A = δA′′ and B = (1 − δ)H + δB′, Lemma 5.1 holds. ¤ Theorem 5.2. Let X be a Q-factorial projective variety and ∆ an effective R-divisor such that (X, ∆) is divisorial log terminal. Suppose that ν(KX + ∆) = 0. Then any log minimal model programs starting from (X, ∆) with scaling of H terminate, where H satisfies that H ≥ A for some effective Q-ample divisor A and (X, ∆ + H) is divisorial log terminal and KX + ∆ + H is nef. MINIMAL MODEL THEORY IN THE CASE WHERE ν = 0 11

Proof. Let (X, ∆) 99K (X1, ∆1) be a divisorial contraction or a log flip. Remark that it holds that

ν(KX + ∆) = ν(KX1 + ∆1) from Lemma 2.7 (1) and the negativity lemma. If some lc centers of (X, ∆) are contained in Supp (H−A), then (X, ∆+H) is not log canon- ical. Hence, by the assumption, no lc centers of (X, ∆) are contained in B+(H). By the negativity lemma, no lc centers of (X1, ∆1) are contained in B+(H1), where H1 is the strict transform of H on X1 (cf. [BCHM, Lemma 3.10.11]). From Lemma 5.1, there exist an effective Q- ample divisor A1 and an effective R-divisor B1 such that H1 ∼R B1+A1,

(X1, ∆1 + B1 + A1) is divisorial log terminal and KX1 + ∆1 + A1 + B1 is nef. We may replace (X, ∆,H) with (X1, ∆1,A1 + B1). Therefore it suffices to show termination of such log flips. Assume that there exists a sequence of log flips

(Xi, ∆i) 99K (Xi+1, ∆i+1) with scaling of H does not terminate where (X0, ∆0) = (X, ∆). Let {λi} be as in Definition 3.2. We set

λ = lim λi. i→∞ ̸ 1 If λ = 0, the sequence is composed by (KX + ∆ + 2 λH)-log flips. Thus the sequence terminates by Theorem 3.4. We may assume that λ = 0. Then we see the following:

Claim 5.3. KX + ∆ is a limit of movable R-divisors.

Proof of Claim 5.3. Let (X, ∆) = (X0, ∆0) 99K (Xi, ∆i) be a composi- tion of the log flips. We take a log resolution pi : Wi → X of (X, ∆+H) and a log resolution qi : Wi → X of (Xi, ∆i + Hi) such that the fol- lowing diagram commutes, where Hi is the strict transform of H on Xi:

Wi C }} CC pi }} CCqi }} CC ~}} C! / X Xi.

Let Gi be an ample divisor on Xi and ϵi a sufficiently small rational number. Since KXi +∆i +λiHi is nef, we see that KXi +∆i +λiHi +ϵiGi is ample. By the negativity lemma, it holds that

∗ ∗ pi (KX + ∆ + λiH + ϵiGi,X ) = qi (KXi + ∆i + λiHi + ϵiGi) + Ei 12 YOSHINORI GONGYO where Gi,X is the strict transform of Gi on X and Ei is an effective pi-exceptional and qi-exceptional divisor. Thus it holds that ∗ B(pi (KX + ∆ + λiH + ϵiGi,X )) = Supp Ei.

Since B(KX + ∆ + λiH + ϵiGi,X ) ⊆ pi(Supp Ei), we see that KX + ∆ + λiH + ϵiGi,X is movable from Lemma 4.2. Hence, if we take {ϵi} such that ϵi → 0, Claim 5.3 follows. ¤ Let ϕ : Y → X be a log resolution of (X, ∆). We consider the σ-decomposition ∗ ∗ ∗ ϕ (KX + ∆) ≡ P(ϕ (KX + ∆)) + N(ϕ (KX + ∆)) (Definition 4.3). Since ∗ ν(ϕ (KX + ∆)) = ν(KX + ∆) = 0, ∗ we see P(ϕ (KX + ∆)) = 0 by Proposition 4.4 (2). In particular, this is the Zariski decomposition in the sense of Nakayama. Moreover we see the following claim: ∗ Claim 5.4. N(ϕ (KX + ∆)) is a ϕ-exceptional divisor. Proof of Claim 5.4. Let G be an ample divisor on X and ϵ a sufficiently small positive number. By Proposition 4.4 (1), it holds that ∗ ∗ Supp N(ϕ (KX + ∆)) ⊆ Supp N(ϕ (KX + ∆ + ϵG)). ∗ If it holds that ϕ∗(N(ϕ (KX + ∆))) ≠ 0, we see that B≡(KX + ∆ + ϵG) has codimension 1 components. This is a contradiction to Claim 5.3. ∗ Thus N(ϕ (KX + ∆)) is a ϕ-exceptional divisor. ¤

Hence KX + ∆ ≡ 0, in particular, KX + ∆ is nef. This is a contra- diction to the assumption. ¤ Corollary 5.5. Let X be a Q-factorial projective variety and ∆ an effective R-divisor such that (X, ∆) is divisorial log terminal. Suppose that ν(KX + ∆) = 0. Then there exists a log minimal model of (X, ∆).

6. Abundance theorem in the case where ν = 0 In this section, we prove the abundance theorem in the case where ν = 0 for an R-divisor: Theorem 6.1. Let X be a normal projective variety and ∆ an effective R-divisor. Suppose that (X, ∆) is a log canonical pair such that ν(KX + ∆) = 0. Then ν(KX +∆) = κ(KX +∆). Moreover, if ∆ is a Q-divisor, then ν(KX + ∆) = κ(KX + ∆) = ι(KX + ∆). First we extends [G, Theorem 1.2] to an R-divisor. MINIMAL MODEL THEORY IN THE CASE WHERE ν = 0 13

Lemma 6.2 (cf. [FG, Theorem 3.1]). Let X be a normal projective variety and ∆ an effective K-divisor. Suppose that (X, ∆) is a log canonical pair such that KX + ∆ ≡ 0. Then KX + ∆ ∼K 0 Proof. If K = Q, then the statement is nothing∑ but [G, Theorem 1.2]. K R From now on, we assume that = .⊕ Let i Bi be the irreducible decomposition of Supp ∆. We put V = RBi. Then it is well known i that L = {∆ ∈ V | (X, ∆) is log canonical} is a rational polytope in V . We can also check that

N = {∆ ∈ L | KX + ∆ is nef} is a rational polytope and ∆ ∈ N (cf. [Bi2, Proposition 3.2] and [Sho1, 6.2 First Main theorem]). We note that N is known as Shokurov’s polytope. Therefore, we can write ∑k KX + ∆ = ri(KX + ∆i) i=1 such that

(i) ∆i is an effective Q-divisor such that ∆i ∈ N for every i, (ii) (X, ∆i) is log canonical for every i,∑ and ∈ R k (iii) 0 < ri < 1, ri for every i, and i=1 ri = 1. Since KX + ∆ is numerically trivial and KX + ∆i is nef for every i, KX + ∆i is numerically trivial for every i. By [G, Theorem 1.2], we see that KX + ∆ ∼R 0. ¤ Proof of Theorem 6.1. By taking a dlt blow-up (Theorem 2.13), we may assume that (X, ∆) is a Q-factorial dlt pair. By Corollary 5.5, there exists a log minimal model (Xm, ∆m) of (X, ∆). From Lemma ≡ 2.7 (3), it holds that KXm + ∆m 0. By Lemma 6.2, it holds that

κ(KXm + ∆m) = 0. Lemma 2.2 implies that κ(KX + ∆) = 0. If ∆ is a Q-divisor, then there exists an effective Q-divisor E such that KX + ∆ ∼Q E by Corollary 5.5 and Lemma 6.2. Thus we see that κ(KX + ∆) = ι(KX + ∆). We finish the proof of Theorem 6.1. ¤ Corollary 6.3. Let π : X → S be a projective surjective morphism of normal quasi-projective varieties, and let (X, ∆) be a projective log canonical pair such that ∆ is an effective K-divisor. Suppose that ν(KF + ∆F ) = 0 for a general fiber F , where KF + ∆F = (KX + ∆)|F . Then there exists an effective K-divisor D such that KX + ∆ ∼K,π D. 14 YOSHINORI GONGYO

Proof. This follows from Theorem 6.1 and [BCHM, Lemma 3.2.1]. ¤ References

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Graduate School of Mathematical Sciences, the University of Tokyo, 3-8-1 Komaba, Meguro-ku, Tokyo 153-8914, Japan. E-mail address: [email protected]