MINIMAL MODEL THEORY OF NUMERICAL KODAIRA DIMENSION 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 minimal model program with scaling 7 4. Zariski decomposition in the sense of Nakayama 9 5. Existence of minimal model in the case where º = 0 10 6. Abundance theorem in the case where º = 0 12 References 14 1. Introduction Throughout this paper, we work over C, the complex number ¯eld. We will make use of the standard notation and de¯nitions 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 ¯ber 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 e®ective divisor in dimen- sion 5, this conjecture is proved by Birkar (cf. [Bi1]). When X is of general type or KX is not pseudo-e®ective, Birkar, Cascini, Hacon Date: 2010/9/21, version 4.03. 2000 Mathematics Subject Classi¯cation. 14E30. Key words and phrases. minimal model, the abundance conjecture, 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 º(KX ) = 0 (for the de¯nition of º, see De¯nition 2.6): Theorem 1.2. Let X be a smooth projective variety such that º(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 º = 0 proved by [CKP] and [Ka]: Theorem 1.3 (=Theorem 6.1). Let X be a normal projective variety and ¢ an e®ective R-divisor. Suppose that (X; ¢) is a log canonial pair such that º(KX +¢) = 0. Then KX +¢ is abundant, i.e. º(KX +¢) = ·(KX + ¢). In the ¯rst 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 di®erent 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 de¯ne 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 º = 0. In Section 6, we prove Theorem 1.3. Notation and De¯nition 1.4. Let K be the real number ¯eld R or the rational number ¯eld Q. We set K>0 = fx 2 Kjx > 0g. Let X be a normal variety and let ¢ be an e®ective K-divisor such that KX + ¢ is K-Cartier. Then we can de¯ne the discrepancy a(E; X; ¢) 2 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 º = 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 ¼ : 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 jD=Sj = fEjD »Z;S E ¸ 0g of D over S. The base locus of the linear system jD=Sj is denoted by BsjD=Sj. When S = Spec C, we denote simply jDj and BsjDj.P R r For an -Weil divisor D = j=1 djDj such that Dj is a prime divisor 6 6 p q Pfor every j and Di = Dj for i = j, we de¯neP 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 de¯ned by x · pxq < x + 1 (resp. x ¡ 1 < xxy · x). The fractional part fDg 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 Caucher Birkar, 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 De¯nition 2.1 (Classical Iitaka dimension, cf. [N, II, 3.2, De¯nition]). Let X be a normal projective variety and D an R-Cartier divisor on X. If jxmDyj= 6 ;, we put a dominant rational map ÁjxmDyj : X 99K Wm; with respect to the complete linear system of xmDy. We de¯ne the Classical Iitaka dimension ¶(D) of D as the following: ¶(D) = maxfdimWmg if H0(X; xmDy) =6 0 for some positive integer m or ¶(D) = ¡1 other- wise. 4 YOSHINORI GONGYO Lemma 2.2. Let Y be a normal projective variety, ' : 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) ¶('¤D) = ¶('¤D + E) for a '-exceptional e®ective R-divisor E, and (2) ¶('¤D) = ¶(D). Proof. (1) and (2) follows from [N, II, 3.11, Lemma]. ¤ The following is remarked by Shokurov: 0 0 Remark 2.3. In general, ¶(D) may not coincide with ¶(D ) if D »R D . For example, let X be the P1, P and Q closed points in X such that P =6 Q and a irrational number. Set D = a(P ¡ Q). Then ¶(D) = ¡1 in spite of the fact that D »R 0. However, fortunately, ¶(D) coincides with ¶(D0) if D and D0 are 0 e®ective divisors such that D »R D ([Ch, Corollary 2.1.4]). Hence it seems reasonable that we de¯ne the following as the Iitaka (Kodaira) dimension for R-divisors. De¯nition 2.4 (Invariant Iitaka dimension, [Ch, De¯nition 2.2.1], cf. [CS, Section 7]). Let X be a normal projective variety and D an R- Cartier divisor on X. We de¯ne the invariant Iitaka dimension ·(D) of D as the following: ·(D) = ¶(D0) 0 0 if there exists an e®ective divisor D such that D »R D or ·(D) = ¡1 otherwise. Let (X; ¢) be a log canonical. Then we call ·(KX + ¢) the log Kodaira dimension of (X; ¢). If ¢ = 0, we simply say ·(KX ) is the Kodaira dimension of X. Remark 2.5. We remark the following: (i) In [Ch, De¯nition 2.2.1], ·(¢) is denoted as I(¢). Moreover the above log Kodaira dimension coincides with the log Ko- daira dimension in [Ch, De¯nition 2.2.3] because ·(KX + ¢) = log ·(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 e®ective 0 0 Q-divisor D such that D »Q D . Then ·(D) = ¶(D). MINIMAL MODEL THEORY IN THE CASE WHERE º = 0 5 De¯nition 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) = maxfk 2 Z¸0j lim sup m dimH (X; xmDy + A) > 0g m!1 if H0(X; xmDy + A) =6 0 for in¯nitely many m 2 N or σ(D; A) = ¡1 otherwise. We de¯ne º(D) = maxfσ(D; A)jA is a divisor on Y g: Lemma 2.7. Let Y be a smooth projective variety, ' : Y ! X a projective birational morphism onto a normal projective variety, and let D be an R-Cartier divisor on X.