j. differential geometry 56 (2000) 167-188 A CANONICAL BUNDLE FORMULA OSAMU FUJINO & SHIGEFUMI MORI Abstract A higher dimensional analogue of Kodaira’s canonical bundle formula is obtained. As applications, we prove that the log-canonical ring of a klt pair with κ ≤ 3is finitely generated, and that there exists an effectively computable natural number M such that |MKX | induces the Iitaka fibering for every algebraic threefold X with Kodaira dimension κ =1. 1. Introduction If f : X → C is a minimal elliptic surface over C, then the relative canonical divisor KX/C is expressed as ∗ mP − 1 ∗ (1) KX/C = f L + f (P ), mP P where L is a nef divisor on C and P runs over the set of points such that ∗ f (P ) is a multiple fiber withmultiplicity mP > 1. It is the key in the estimates on the plurigenera Pn(X) that the coefficients (mP − 1)/mP are ‘close’ to 1 [12]. Furthermore 12L is expressed as ∗ ∗ mP − 1 ∗ ∗ (2) 12KX/C = f j OP(1)+12 f (P )+ σQf (Q), mP P 1 where σQ is an integer ∈ [0, 12) and j : C → P is the j-function [5, (2.9)]. The computation of these coefficients is based on the explicit classification of the singular fibers of f, which made the generalization difficult. 1991 Mathematics Subject Classification. 14N30; 14E30, 14J40. Received October 10, 2000. 167 168 osamu fujino & shigefumi mori We note that L in the exact analogue of the formula (1) for the case dim X/C = 2 need not be a divisor (Example 2.7) and that the formula (2) is more natural to look at if L isallowedtobeaQ-divisor. The higher dimensional analogue of the formula (2) is treated in Section 2 as a refinement of [15, §5 Part II] and the log version in Section 4. We give the full formula only in 4.5 to avoid repetition. The estimates of the coefficients are treated in 2.8, 3.1 and 4.5. (See 3.9 on the comparison of the formula (2) and our estimates.) We note that the “coefficients” in the formula (2) are of the form 1 − 1/m except for the finite number of exceptions 1/12, ··· , 11/12. In the generalized formula, the coefficients are in a more general form (cf. 4.5.(v)), which still enjoys the DCC (Descending Chain Condition) property of Shokurov. The following are some of the applications. 1. (Corollary 5.3)If(X, ∆) is a klt pair with κ(X, KX +∆) ≤ 3, then its log-canonical ring is finitely generated. 2. (Corollary 6.2) There exists an effectively computable natural number M suchthat |MKX | induces the Iitaka fibering for ev- ery algebraic threefold X withKodaira dimension κ(X)=1. To get the analogue for an (m + 1)-dimensional X (m ≥ 3) with κ(X) = 1, it remains to show that an arbitrary m-fold F with κ(F )=0 and pg(F ) = 1 is birational to a smoothprojective model witheffectively bounded m-thBetti number. Notation. Let Z>0 (resp. Z≥0) be the set of positive (resp. non- negative) integers. We work over C in this note. Let X be a normal variety and B,B Q-divisors on X. If B − B is effective, we write B B or B ≺ B. We write B ∼ B if B − B is a principal divisor on X (linear equivalence of Q-divisors). Let B+,B− be the effective Q-divisors on X without common ir- reducible components suchthat B+ − B− = B. They are called the positive and the negative parts of B. Let f : X → C be a surjective morphism. Let Bh,Bv be the Q- divisors on X with Bh + Bv = B suchthatan irreducible component of Supp B is contained in Supp Bh iff it is mapped onto C. They are called the horizontal and the vertical parts of B over C. B is said to be horizontal (resp. vertical ) over C if B = Bh (resp. B = Bv). The phrase “over C” might be suppressed if there is no danger of confusion. a canonical bundle formula 169 As for other notions, we mostly follow [14]. However we introduce a slightlydifferent terminology to distinguishthepairs withnon-effective boundaries (cf. [10]). A pair (X, D) consists of a normal variety X and a Q-divisor D.If KX + D is Q-Cartier, we can pull it back by an arbitrary resolution f : Y → X and obtain the formula ∗ KY = f (KX + D)+ aiEi, i where Ei are prime divisors and ai ∈ Q. The pair (X, D) is said to be sub klt (resp. sub lc) if ai > −1 (resp. ≥−1) for every resolution f and every i. Furthermore, (X, D) is said to be klt (resp. lc) if D is effective. Acknowledgements. This note is an expanded version of the sec- ond author’s lecture “On a canonical bundle formula” at Algebraic Ge- ometry Workshop in Hokkaido University June 1994. The authors were partially supported by the Grant-in-Aid for Scien- tific Research, the Ministry of Education, Science, Sports and Culture of Japan. The second author was also partially supported by the Inamori Foundation. We would like to thank Professors H. Clemens and J. Koll´ar for helpful conversations and Professor K. Ohno for pointing out mistakes in an earlier version. 2. Semistable part of KX/C In this section, we refine the results of [15, §5, Part II] after putting the basic results together. 2.1. Let f : X → C be a surjective morphism of a normal projective variety X of dimension n = m + l to a nonsingular projective l-fold C suchthat (i) X has only canonical singularities, and (ii) the generic fiber F of f is a geometrically irreducible variety withKodaira dimension κ(F ) = 0. We fix the smallest b ∈ Z>0 such that the b-thplurigenus Pb(F ) is non-zero. Proposition 2.2. There exists one and only one Q-divisor D modulo linear equivalence on C with a graded OC -algebra isomorphism ∼ ∗∗ ⊕i≥0O(iD) = ⊕i≥0(f∗O(ibKX/C )) , 170 osamu fujino & shigefumi mori where M ∗∗ denotes the double dual of M. Furthermore, the above isomorphism induces the equality ∗ bKX = f (bKC + D)+B, where B is a Q-divisor on X such that f∗OX (iB+)=OC (∀i>0) and codim(f(Supp B−) ⊂ C) ≥ 2. We note that for an arbitrary open set U of C, D|U and B|f −1(U) depend only on f|f −1(U). Proof. By [15, (2.6.i)], there exists c>0 suchthat ∗∗ ∗∗ ⊗i (f∗O(ibcKX/C )) = {(f∗O(bcKX/C )) } (∀i>0). ∗∗ Choose an embedding φ :(f∗O(bKX/C )) ⊂ Q(C) into the function field of C, and we can define a Weil divisor cD by ⊗c ∗∗ ∼ φ :(f∗O(bcKX/C )) = O(cD) ⊂ Q(C). D modulo linear equivalence does not depend on the choice of φ. Since taking the double dual has no effect on codimension 1 points, there is a natural inclusion ∗ −1 f OC (cD) ⊂OX (bcKX/C )onX \ f (some codim 2 subset of C). ∗ Extending it to X, we obtain a Q-divisor B suchthat B = bKX/C −f D. It is easy to see that B satisfies the required conditions. q.e.d. Definition 2.3. Under the notation of 2.2, we denote D by LX/C . It is obvious that LX/C depends only on the birational equivalence class of X over C. If X has bad singularities, then we take a nonsingular model X of X and use LX/C as our definition of LX/C . Proposition 2.4 (Viehweg). Let π : C → C be a finite surjec- tive morphism from a nonsingular l-fold C and let f : X → C be a nonsingular model of X ×C C → C . Then there is a natural relation ∗ L ≺ π L . X /C X/C Furthermore if X ×C C has a semistable resolution over a neighbor- hood of a codimension 1 point P of C ,orifXπ(P ) has only canonical ∗ singularities, then P ∈ Supp(π L − L ). X/C X /C a canonical bundle formula 171 Proof. Except for the last assertion, this is due to [22, §3] (cf. [15, (4.10)]). If Xπ(P ) has only canonical singularities, then X ×C C has only canonical singularities in a neighborhood of f −1(P )by[20, Propo- sition 7] or [9] because so does the generic fiber F of f.Thus2.4 follows. q.e.d. Corollary 2.5. There exists one and only one Q-divisor ss LX/C (≺ LX/C ) such that ∗ ss (i) π L ≺ L for arbitrary π : C → C as in 2.4, and X/C X /C ∗ ss (ii) π L = L at P if π in 2.4 is such that X × C → C X/C X /C C has a semistable resolution X → C over a neighborhood of P or Xπ(P ) has only canonical singularities. There exists an effective divisor Σ ⊂ C such that every birational mor- phism π : C → C from a nonsingular projective l-fold with π∗(Σ) an snc divisor has the following property: Let X be a projective resolution ss of X ×C C and f : X → C the induced morphism. Then LX/C is nef. Proof. When C /C is Galois withgroup G, L is G-invariant and X /C ss therefore descends to a Q-subdivisor of LX/C . The minimum LX/C (≺ LX/C ) of all the descents exists by 2.4, whence the uniqueness follows. The last assertion is proved in [15, §5, part II] though it is not explic- itly stated there.