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 ≤ 3 is finitely generated, and that there exists an effectively computable natural number M such that |MK_{X}| 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 K_{X/C} is expressed as

K_{X/C} = f^{∗}L + ∑_{P} \frac{m_{P} − 1}{m_{P}}f^{∗}(P), (1)

where L is a nef divisor on C and P runs over the set of points such that f^{∗}(P) is a multiple fiber with multiplicity m_{P} > 1. It is the key in the estimates on the plurigenera P_{n}(X) that the coefficients (m_{P} − 1)/m_{P} are ‘close’ to 1 [12]. Furthermore 12L is expressed as

12K_{X/C} = f^{∗}j^{∗}O_{ℙ}(1) + 12 ∑_{P} \frac{m_{P} − 1}{m_{P}}f^{∗}(P) + ∑ σ_{Q}f^{∗}(Q), (2)

where σ_{Q} is an integer ∈ [0, 12) and j : C → ℙ^{1} 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.

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 is allowed to be a ℚ-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, Δ) is a klt pair with κ(X, K_{X} +Δ) ≤ 3, then its log-canonical ring is finitely generated.

2. (Corollary 6.2) There exists an effectively computable natural number M such that |MK_{X}| induces the Iitaka fibering for every algebraic threefold X with Kodaira 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 p_{g}(F) = 1 is birational to a smooth projective model with effectively bounded m-th Betti number.

Notation. Let ℤ_{>0} (resp. ℤ_{≥0}) be the set of positive (resp. nonnegative) 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 irreducible components such that B_{+} − B_{−} = B. They are called the positive and the negative parts of B.

Let f : X → C be a surjective morphism. Let B^{h}, B^{v} be the ℚdivisors on X with B^{h} + B^{v} = B such that an irreducible component of Supp B is contained in Supp B^{h} 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 = B^{h} (resp. B = B^{v}). The phrase “over C” might be suppressed if there is no danger of confusion.

As for other notions, we mostly follow [14]. However we introduce a slightly different terminology to distinguish the pairs with non-effective boundaries (cf. [10]).

A pair (X, D) consists of a normal variety X and a ℚ-divisor D. If K_{X} + D is ℚ-Cartier, we can pull it back by an arbitrary resolution f : Y → X and obtain the formula

K_{Y} = f^{∗}(K_{X} + D) + ∑_{i} a_{i}E_{i},

where E_{i} are prime divisors and a_{i} ∈ ℚ. The pair (X, D) is said to be sub klt (resp. sub lc) if a_{i} > −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 second author’s lecture “On a canonical bundle formula” at Algebraic Geometry Workshop in Hokkaido University June 1994.

The authors were partially supported by the Grant-in-Aid for Scientific 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. Kollár for helpful conversations and Professor K. Ohno for pointing out mistakes in an earlier version.

2. Semistable part of K_{X/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 such that

(i) X has only canonical singularities, and

(ii) the generic fiber F of f is a geometrically irreducible variety with Kodaira dimension κ(F) = 0. We fix the smallest b ∈ ℤ_{>0} such that the b-th plurigenus P_{b}(F) is non-zero.

Proposition 2.2. There exists one and only one ℚ-divisor D modulo linear equivalence on C with a graded O_{C}-algebra isomorphism

⊕_{i≥0}O(⌊iD⌋) ≅ ⊕_{i≥0}(f_{∗}O(ibK_{X/C}))^{∗∗},

where M^{∗∗} denotes the double dual of M.

Furthermore, the above isomorphism induces the equality

bK_{X} = f^{∗}(bK_{C} + D) + B,

where B is a ℚ-divisor on X such that f_{∗}O_{X}(⌊iB_{+}⌋) = O_{C} (∀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 such that

(f_{∗}O(ibcK_{X/C}))^{∗∗} = {(f_{∗}O(bcK_{X/C}))^{∗∗}}^{⊗i} (∀i > 0).

Choose an embedding ϕ : (f_{∗}O(bK_{X/C}))^{∗∗} ⊂ ℚ(C) into the function field of C, and we can define a Weil divisor cD by

ϕ^{⊗c} : (f_{∗}O(bcK_{X/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

f^{∗}O_{C}(cD) ⊂ O_{X}(bcK_{X/C}) on X \ f^{−1}(some codim 2 subset of C).

Extending it to X, we obtain a ℚ-divisor B such that B = bK_{X/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 L_{X/C}. It is obvious that L_{X/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 L_{X′/C} as our definition of L_{X/C}.

Proposition 2.4 (Viehweg). Let π : C′ → C be a finite surjective morphism from a nonsingular l-fold C′ and let \overline{f}′ : \overline{X}′ → C′ be a nonsingular model of X ×_{C} C′ → C′. Then there is a natural relation

L_{\overline{X}′/C′} ≺ π^{∗}L_{X/C}.

Furthermore if X ×_{C} C′ has a semistable resolution over a neighborhood of a codimension 1 point P′ of C′, or if X_{π(P′)} has only canonical singularities, then P′ ∉ Supp(π^{∗}L_{X/C} − L_{\overline{X}′/C′}).

Proof. Except for the last assertion, this is due to [22, §3] (cf. [15, (4.10)]). If X_{π(P′)} has only canonical singularities, then X ×_{C} C′ has only canonical singularities in a neighborhood of f′^{−1}(P′) by [20, Proposition 7] or [9] because so does the generic fiber F of f. Thus 2.4 follows.

q.e.d.

Corollary 2.5. There exists one and only one ℚ-divisor L_{X/C}^{ss} (≺ L_{X/C}) such that

(i) π^{∗}L_{X/C}^{ss} ≺ L_{\overline{X}′/C′} for arbitrary π : C′ → C as in 2.4, and

(ii) π^{∗}L_{X/C}^{ss} = L_{\overline{X}′/C′} at P′ if π in 2.4 is such that X ×_{C} C′ → C′

has a semistable resolution \overline{X}′ → C′ over a neighborhood of P′ or X_{π(P′)} has only canonical singularities.

There exists an effective divisor Σ ⊂ C such that every birational morphism 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 of X ×_{C} C′ and f′ : X′ → C′ the induced morphism. Then L_{X′/C′}^{ss} is nef.

Proof. When C′/C is Galois with group G, L_{\overline{X}′/C′} is G-invariant and therefore descends to a ℚ-subdivisor of L_{X/C}. The minimum L_{X/C}^{ss} (≺ L_{X/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 explicitly stated there. In [15, (5.13)], First our Σ is a divisor containing the discriminant locus of f and h constructed in the proof of [15, (5.15.2)] (which are f′ and h′ in our 2.6). Then [15, (5.14.1)] shows that our L_{X/C}^{ss}/b is equal to P_{f′,bc} for sufficiently divisible c ∈ ℤ_{>0}. Finally [15, (5.15.3)] shows that P_{f′,bc} is nef. q.e.d.

Remark 2.6. Under the notation of 2.1, consider the following construction. Since dim |bK_{F} | = 0, there exists a Weil divisor W on X such that

(i) W^{h} is effective and f_{∗}O(iW^{h}) = O_{C} for all i > 0, and

(ii) bK_{X} − W is a principal divisor (ψ) for some non-zero rational function ψ on X.

Let s : Z → X be the normalization of X in ℚ(X)(ψ^{1/b}). Then the above proof actually shows the following:

Fix resolutions X′ and Z′ of X and Z. We write f′ : X′ → C and h′ : Z′ → C. Then as the divisor Σ ⊂ C in 2.5, we can take an arbitrary effective divisor Σ ⊂ C such that f′ : X′ → C and h′ : Z′ → C are smooth over C \ Σ.

Example 2.7. Let F be a K3 surface with a free involution ι : F → F so that E = F/{1, ι} is an Enriques surface. Let j : ℙ^{1} → ℙ^{1} be the involution x ↦ −x, so that 0 and ∞ are the only fixed points. Let f : X = ℙ^{1} × F/{1, j × ι} → C = ℙ^{1}/{1, j} be the map induced by the first projection. Then f is smooth over C \ {0, ∞}, and f^{∗}(0) = 2E_{0} and f^{∗}(∞) = 2E_{∞}, where E_{0} ≃ E_{∞} ≃ E. Using K_{E_{t}} ∼ K_{X} + E_{t}|_{E_{t}} ≁ 0 for t = 0 and ∞, one easily sees that K_{X/C} ∼ f^{∗}O(1) and that L_{X/C}^{ss} = \frac{1}{2}(0) − \frac{1}{2}(∞). Thus L_{X/C}^{ss} is only a ℚ-divisor fitting in the analogue of the formula (1):

K_{X/C} = f^{∗}L_{X/C}^{ss} + ∑_{t=0,∞} \frac{1}{2}f^{∗}(t).

The main part (i) of the following is included in 4.5.(v) (see also 4.7). We still leave it here for reference.

Proposition 2.8. Under the notation and the assumptions of 2.5, let N ∈ ℤ_{>0} be such that Nf^{∗}(L_{X/C}^{ss}) is a Weil divisor. Then we have

L_{X/C} = L_{X/C}^{ss} + ∑_{P} s_{P} P,

where s_{P} ∈ ℚ for every codimension 1 point P of C such that:

(i) For each P, there exist u_{P} , v_{P} ∈ ℤ_{>0} such that 0 < v_{P} ≤ bN and s_{P} = (bNu_{P} − v_{P})/(Nu_{P}).

(ii) s_{P} = 0 if f^{∗}(P) has only canonical singularities or if X → C has a semistable resolution in a neighborhood of P.

In particular, s_{P} depends only on f|_{f^{−1}(U)} where U is an open set of C containing P.

The assertion (ii) is proved in 2.5.(ii).

3. Bounding the denominator

In applications, it is important to bound the denominator of L_{X/C}^{ss}.

Theorem 3.1. Let E → F be the cover associated to the b-th root of the unique element of |bK_{F} |. Let \overline{E} be a nonsingular projective model of E and let B_{m} be its m-th Betti number. Then there is a natural number N = N(B_{m}) depending only on B_{m} such that NL_{X/C}^{ss} is a divisor.

Remark 3.2. We have N(x) = lcm{y ∈ ℤ_{>0} | φ(y) ≤ x}, where φ(y) is Euler’s function.

3.3. To prove 3.1, we can replace the base space C by a general hyperplane-section H and X by f^{∗}H. Repeating this procedure, we may assume that C is a curve in the rest of this section.

We use the construction given in 2.6, i.e., let W, s : Z → X and h = f ∘ s : X → C be as in 2.6. We note that the branch locus of s is contained in the singular locus of X and Supp W.

Lemma 3.4 ([15, (5.15.8)]). bL_{Z/C}^{ss} = L_{X/C}^{ss}.

Proof. Because of the definition 2.3, we may replace C with its base change and change the model X birationally. Hence we may assume that X is smooth and W_{red}^{h} + f^{∗}Σ is a reduced snc divisor, where Σ is a reduced effective divisor of C such that f^{−1}(Σ) contains W^{v} and f is smooth over C \ Σ. We may also assume that h : Z → C has a semistable resolution h′ : Z′ → C and that h is smooth over C \ Σ. Hence K_{X} + W_{red}^{h} + f^{∗}Σ is log-canonical and

K_{Z} + (s^{−1}W^{h})_{red} + h^{∗}Σ = s^{∗}(K_{X} + W_{red}^{h} + f^{∗}Σ)

is also log-canonical (cf. [13, (20.3)] for the explicit statement, which goes back to [19, §1]). Hence for all n > 0 we have

h′_{∗}O(n(K_{Z′} + (s′^{−1}W^{h})_{red} + h′^{∗}Σ)) = h_{∗}O(n(K_{Z} + (s^{−1}W^{h})_{red} + h^{∗}Σ)),

where s′ : Z′ → X is the induced morphism. By 2.6.(i), we see

h_{∗}O(bn(K_{Z} + (s^{−1}W^{h})_{red} + h^{∗}Σ)) = h′_{∗}O(bnK_{Z′}) ⊗ O(bnΣ)
= O(bnL_{Z/C}^{ss} + bnK_{C} + bnΣ).

By a similar computation, we get

f_{∗}O(bn(K_{X} + W_{red}^{h} + f^{∗}Σ)) = O(nL_{X/C}^{ss} + bnK_{C} + bnΣ).

Whence the natural inclusion bL_{Z/C}^{ss} ≻ L_{X/C}^{ss} becomes an equality. q.e.d.

3.5. Let \overline{h} : \overline{Z} → C be a nonsingular projective model of Z → C whose generic fiber is isomorphic to \overline{E}. Let π : C′ → C be a finite Galois morphism with group G such that \overline{Z} ×_{C} C′ has a semistable resolution \overline{h}′ : \overline{Z}′ → C′ whose generic fiber is isomorphic to \overline{E}. We give names as in the next diagram.

\overline{Z} ⟵^{σ} \overline{Z}′
\overline{h}↓  ↓\overline{h}′
C ⟵_{π} C′

By 2.5, we have

\overline{h}′_{∗}O(K_{\overline{Z}′/C′}) = O(π^{∗}L_{Z/C}^{ss}).

We note that π^{∗}L_{Z/C}^{ss} is a Weil divisor [15, (2.6.ii)].

3.6. Due to its birational invariance, \overline{h}′_{∗}O(K_{\overline{Z}′/C′}) is G-linearized. Let P′ ∈ C′ and let G_{P′} be the stabilizer. Then G_{P′} acts on \overline{h}′_{∗}O(K_{\overline{Z}′/C′}) ⊗ ℂ(P′) through a character χ_{P′} : G_{P′} → ℂ^{∗}. Then NL_{Z/C}^{ss} is a divisor iff χ_{P′}^{N} = 1 for all P′ because O(NL_{Z/C}^{ss}) should be (π_{∗}O(Nπ^{∗}L_{Z/C}^{ss}))^{G}.

We can now localize everything on neighborhoods of P = π(P′) and P′. Let e be the ramification index of π at P′. Let z be a local coordinate for the germ (C, P) and z′ = z^{1/e} for (C′, P′). We have a natural homomorphism G_{P′} → μ_{e} = {x ∈ ℂ | x^{e} = 1}. Let H′ be the canonical extension of H′_{∘} = O_{C′_{∘}} ⊗ (R^{m}\overline{h}′_{∘}ℂ\overline{Z}′_{∘})_{prim} to C′, where

C′_{∘} = C′ \ \{P′\}, \overline{Z}′_{∘} = \overline{Z}′ \ \overline{h}′^{−1}(P′) and \overline{h}′_{∘} : \overline{Z}′_{∘} → C′_{∘} is the restriction of \overline{h}′. By its construction, H′ admits a μ_{e}-action. Let ζ = e^{2πi/e} ∈ μ_{e}.

We quote the following to compare χ_{P′} and H′.

Proposition 3.7 [7, Proposition 1]). There exists a μ_{e}-equivariant injection

\overline{h}′_{∗}O(K_{\overline{Z}′/C′}) ⊗ ℂ(P′) ⊂ H′ ⊗ ℂ(P′).

3.8. Thus if ℓ is the order of χ_{P′}, then φ(ℓ) ≤ B_{m}, where φ(ℓ) is Euler’s function. Let N(x) = lcm{ℓ | φ(ℓ) ≤ x}. Then we see that N(B_{m})L_{Z/C}^{ss} is a divisor. Hence 3.1 is proved.

Remark 3.9. When f : X → C is a minimal elliptic surface, we mentioned the formula (2). By 2.5 and [5, (2.9)], we have b = 1, 12L_{X/C}^{ss} ≃ j^{∗}O_{ℙ}(1) and

s_{P} = \{ \begin{matrix}(m_{P} − 1)/m_{P} & if & m_{P} > 1 \\ σ_{P} /12 & if & m_{P} = 1. \end{matrix}

Our estimates are the following. Since B_{1}(F) = 2 and N(2) = 12, our 3.1 shows that 12L_{X/C}^{ss} is a Weil divisor and the estimates 2.8.(i) is compatible with the above.

4. Log-canonical bundle formula

In this section, we give the log analogue of the semistable part defined in Section 2 and give a log-canonical bundle formula.

4.1. 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 such that:

(i) (X, Δ) is a sub klt pair (assumed klt from 4.4 and on).

(ii) The generic fiber F of f is a geometrically irreducible variety with κ(F, (K_{X} + Δ)|_{F}) = 0. We fix the smallest b ∈ ℤ_{>0} such that the f_{∗}O_{X}(b(K_{X} + Δ)) ≠ 0.

The following is proved similarly to 2.2.

Proposition 4.2. There exists one and only one ℚ-divisor D modulo linear equivalence on C with a graded O_{C}-algebra isomorphism

⊕_{i≥0}O_{C}(⌊iD⌋) ≅ ⊕_{i≥0}(f_{∗}O_{X}(⌊ib(K_{X} + Δ)⌋ − ibf^{∗}K_{C}))^{∗∗}.

Furthermore, the above isomorphism induces the equality

b(K_{X} + Δ) = f^{∗}(bK_{C} + D) + B^{Δ},

where B^{Δ} is a ℚ-divisor on X such that f_{∗}O_{X}(⌊iB_{+}^{Δ}⌋) = O_{C} (∀i > 0) and codim(f(Supp B_{−}^{Δ}) ⊂ C) ≥ 2. For an arbitrary open set U ⊂ C, D|_{U} and B^{Δ}|_{f^{−1}(U)} depend only on f|_{f^{−1}(U)} and Δ|_{f^{−1}(U)}.

Definition 4.3. We denote D given in 4.2 by L_{(X,Δ)/C} or simply by L_{X/C}^{log} if there is no danger of confusion.

We set s_{P}^{Δ} := b(1 − t_{P}^{Δ}), where t_{P}^{Δ} is the log-canonical threshold of f^{∗}P with respect to (X, Δ − B^{Δ}/b) over the generic point η_{P} of P:

t_{P}^{Δ} := max{t ∈ ℝ | (X, Δ − B^{Δ}/b + tf^{∗}P) is sub lc over η_{P} }.

Note that s_{P}^{Δ} ≠ 0 only for a finite number of codimension 1 points P because there exists a nonempty Zariski open set U ⊂ C such that s_{P}^{Δ} = 0 for every prime divisor P with P ∩ U ≠ ∅. We may simply write s_{P} rather than s_{P}^{Δ} if there is no danger of confusion. We note that s_{P}^{Δ} depends only on f|_{f^{−1}(U)} and Δ|_{f^{−1}(U)} where U is an open set containing P.

We will see in 4.7 that this coincides with s_{P} introduced in 2.8. We set L_{(X,Δ)/C}^{ss} := L_{(X,Δ)/C} −∑_{P} s_{P}^{Δ}P and call it the log-semistable

part of K_{X/C}(Δ). We may simply denote it by L_{X/C}^{log,ss} if there is no danger of confusion.

We note that D, L_{(X,Δ)/C}, s_{P}^{Δ}, t_{P}^{Δ} and L_{(X,Δ)/C}^{ss} are birational invariants of (X, Δ) over C in the following sense. Let (X′, Δ′) be a projective sub klt pair and σ : X′ → X a birational morphism such that K_{X′} + Δ′ − σ^{∗}(K_{X} + Δ) is an effective σ-exceptional ℚ-divisor. Then the above invariants for f ∘ σ and (X′, Δ′) are equal to those for f and (X, Δ).

Putting the above symbols together, we have the log-canonical bundle formula for (X, Δ) over C:

b(K_{X} + Δ) = f^{∗}(bK_{C} + L_{X/C}^{log,ss}) + ∑_{P} s_{P}^{Δ}f^{∗}P + B^{Δ}, (3)

where B^{Δ} is a ℚ-divisor on X such that f_{∗}O_{X}(⌊iB_{+}^{Δ}⌋) = O_{C} (∀i > 0) and codim(f(SuppB_{−}^{Δ}) ⊂ C) ≥ 2.

We need to pass to a certain birational model f′ : X′ → C′ to understand the log-semistable part more clearly and to make the logcanonical bundle formula more useful.

4.4. From now on we assume that (X, Δ) is klt.

Let g : Y → X be a log resolution of (X, Δ) with Θ a ℚ-divisor on Y such that K_{Y} + Θ = g^{∗}(K_{X} + Δ). Let Σ ⊂ C be an effective divisor satisfying the following conditions:

(1) h := f ∘ g is smooth and Supp Θ^{h} is relatively normal crossing over C \ Σ.

(2) h(Supp Θ^{v}) ⊂ Σ.

(3) f is flat over C \ Σ.

Let π : C′ → C be a birational morphism from a nonsingular projective variety such that:

(i) Σ′ := π^{−1}(Σ) is an snc divisor.

(ii) π induces an isomorphism C′ \ Σ′ ≅ C \ Σ.

(iii) The irreducible component X_{1} of X ×_{C} C′ dominating C′ is flat over C′.

Let X′ be the normalization of X_{1}, and f′ : X′ → C′ the induced morphism. Let g′ : Y′ → X′ be a log resolution such that Y′\h′^{−1}(Σ′) ≅ Y ×_{X} X_{1} \ α^{−1}(Σ′), where h′ := f′ ∘ g′ and α : Y ×_{X} X_{1} → C′ (see [21, Resolution Lemma] or [1, Theorem 12.4]). Let Θ′ be the ℚ-divisor on Y′ such that K_{Y′} +Θ′ = (τ ∘g′)^{∗}(K_{X} +Δ), where τ : X′ → X is the induced morphism. Furthermore, we can assume that Supp(h′^{−1}(Σ′) ∪ Θ′) is an snc divisor, and h′(Supp(Θ′)^{v}) ⊂ Σ′.

Later we treat horizontal or vertical divisors on X, X′ or Y′ over C without referring to C. Note that a ℚ-divisor on X′ or Y′ is horizontal (resp. vertical) over C iff it is horizontal (resp. vertical) over C′.

We note that the horizontal part (Θ′)_{−}^{h} of the negative part Θ′_{−} of Θ′ is g′-exceptional.

Y ⟵ Y ×_{X} X_{1} ⟵ Y′
g↓  ↓  ↓^{g′}
X ⟵ X_{1} ⟵ X′
f↓  ↓  ↓^{f′}
C ⟵_{π} C′  C′

The following formula is the main theorem of this section.

Theorem 4.5 (Log-canonical bundle formula). Under the above notation and assumptions, let Ξ be a ℚ-divisor on Y′ such that (Y′, Ξ) is sub klt and Ξ − Θ′ is effective and exceptional over X. (Note that Ξ exists since (X, Δ) is klt.) Then the log-canonical bundle formula

b(K_{Y′} + Ξ) = (h′)^{∗}(bK_{C′} + L_{(Y′,Ξ)/C′}^{ss}) + ∑_{P} s_{P}^{Ξ}(h′)^{∗}(P) + B^{Ξ}

for (Y′, Ξ) over C′ has the following properties:

(i) h′_{∗}O_{Y′}(⌊iB_{+}^{Ξ}⌋) = O_{C′} for all i > 0.

(ii) B_{−}^{Ξ} is g′-exceptional and codim(h′(Supp B_{−}^{Ξ}) ⊂ C′) ≥ 2.

(iii) The following holds for every i > 0:

H^{0}(X, ⌊ib(K_{X} + Δ)⌋) = H^{0}(Y′, ⌊ib(K_{Y′} + Ξ)⌋)
= H^{0}(C′, ⌊ibK_{C′} + iL_{Y′/C′}^{log,ss} + ∑ i s_{P}^{Ξ}P ⌋).

(iv) L_{Y′/C′}^{log,ss} is nef.

If furthermore (Y′, Ξ) is klt, then:

(v) Let N be a positive integer such that Nh′^{∗}(L_{Y′/C′}^{log,ss}) and bNΞ^{v} are Weil divisors. Then for each P, there exist u_{P} , v_{P} ∈ ℤ_{>0} such that 0 < v_{P} ≤ bN and s_{P} = (bNu_{P} − v_{P})/(Nu_{P}).

Proof. First, (i) is obvious by the formula (3). Similarly, (ii) follows because (g′)_{∗}(B_{−}^{Ξ}) = 0 by the equidimensionality of f′.

By (ii) and the conditions on Ξ, the following holds for all i > 0:

H^{0}(X, ⌊ib(K_{X} + Δ)⌋) = H^{0}(Y′, ⌊ib(K_{Y′} + Ξ)⌋)
= H^{0}(Y′, ⌊ib(K_{Y′} + Ξ) + iB_{−}^{Ξ}⌋).

By the log-canonical bundle formula and then by (i), we have

H^{0}(Y′, ⌊ib(K_{Y′} + Ξ) + iB_{−}^{Ξ}⌋)
= H^{0}(Y′, ⌊i(h′)^{∗}(bK_{C′} + L_{(Y′,Ξ)/C′}^{ss} + ∑ s_{P}^{Ξ}P) + iB_{+}^{Ξ}⌋)
= H^{0}(C′, ⌊ibK_{C′} + iL_{Y′/C′}^{log,ss} + ∑ i s_{P}^{Ξ}P ⌋).

Thus (iii) is settled. The property (iv) will be settled by 4.8, and (v) at the end of this section. q.e.d.

Proposition 4.6. Under the notation and the assumptions of Theorem 4.5, L_{(Y′,Ξ)/C′}^{ss} does not depend on the choice of Ξ. In particular, L_{(Y′,Θ′)/C′}^{ss} = L_{(Y′,Θ′_{+})/C′}^{ss} = L_{(Y′,Ξ)/C′}^{ss}.

Proof. Let H ⊂ Y′ be an effective horizontal ℚ-divisor which is exceptional over X. If H is added to Ξ, then B^{Ξ} increases by bH and s_{P}^{Ξ} stays the same. Hence L_{(Y′,Ξ)/C′}^{ss} stays the same in this case.

Assume now that A := Ξ − Θ′ is exceptional over X and vertical, and let

α_{P} := max{t ∈ ℝ | (B^{Θ′})^{v} + bA^{v} − α(h′)^{∗}P is effective over η_{P} },

where η_{P} is the generic point of P.

Then it is easy to see that B^{Ξ} = B^{Θ′} + bA − ∑ α_{P} (h′)^{∗}P and ∑ s_{P}^{Ξ}P = ∑(s_{P}^{Θ′} + α_{P})P. Hence L_{(Y′,Ξ)/C′}^{ss} = L_{(Y′,Θ′)/C′}^{ss}. q.e.d.

Proposition 4.7. Assume that f : X → C is as in 2.1. Then (X, 0) is klt, and the definitions in 4.3 for f : (X, 0) → C are compatible with the corresponding ones for f : X → C in Section 2.

In other words, B^{0} = B (in 2.2), L_{(X,0)/C} ∼ L_{X/C} (in 2.3) and s_{P}^{0} = s_{P} (in 2.8) and hence L_{(X,0)/C}^{ss} ∼ L_{X/C}^{ss} (in 2.5).

Proof. First, B^{0} = B and L_{(X,0)/C} ∼ L_{X/C} are obvious because the definitions for X and (X, 0) coincide. During the proof, we will denote B by B_{X} to avoid confusion.

Since L_{X/C}^{ss} = L_{X/C} − ∑ s_{P} P and L_{(X,0)/C}^{ss} = L_{(X,0)/C} − ∑ s_{P}^{0} P, it is enough to show s_{P}^{0} = s_{P}. Let Σ ⊂ C be the reduced divisor given in 4.4. Then s_{P}^{0} = s_{P} = 0 for P ⊄ Σ by 2.8 and 4.3. Thus it is sufficient to prove s_{P}^{0} = s_{P} for every P ⊂ Σ.

Replacing C by a general hyperplane-section H, Σ by Σ ∩ H and X by f^{∗}H, we can further assume that C is a curve. Note that we are free to enlarge Σ.

Let t_{P} := 1 − s_{P} /b for all P. Then we have the following equality.

f^{∗}(K_{C} + \frac{1}{b}L_{X/C}^{ss} + Σ) = K_{X} − \frac{1}{b}B_{X} + ∑_{P ∈Σ} t_{P} f^{∗}P. (4)

By the semistable reduction theorem, there is a finite surjective morphism π : C′ → C from a smooth curve C′ such that the normalization X′ of X ×_{C} C′ has a semistable resolution over C′ as in 2.5. Let β : X′ → X and f′ : X′ → C′ be the induced morphisms as in the following commutative diagram,

X ⟵^{β} X′
f↓  ↓^{f′}
C ⟵_{π} C′.

Enlarging Σ, we may assume that π is étale on C′ \ Σ′, where Σ′ := π^{−1}(Σ). In particular, s_{P′} = 0 for P′ ∈ Σ′ by 2.8.(ii), and the formula (4) for f′ reduces to

(f′)^{∗}(K_{C′} + \frac{1}{b}L_{X′/C′}^{ss} + Σ′) = K_{X′} − \frac{1}{b}B_{X′} + ∑_{P′∈Σ′}(f′)^{∗}P′. (5)

Since π^{∗}L_{X/C}^{ss} = L_{X′/C′}^{ss} by 2.5 and f^{∗}(K_{C} + Σ) = K_{C′} + Σ′, π pulls back the formula (4) to (5) and we have the following equality.

β^{∗}(K_{X} − \frac{1}{b}B_{X} + ∑_{P ∈Σ} t_{P} f^{∗}P) = K_{X′} − \frac{1}{b}B_{X′} + ∑_{P′∈Σ′}(f′)^{∗}P′. (6)

Since t_{P}^{0} = 1 − s_{P}^{0} /b by 4.3, it is enough to prove t_{P} = t_{P}^{0} , the log-canonical threshold of f^{∗}P with respect to (X, −B_{X}/b). Since β is finite and dominating, it is enough to prove that the right hand side of the formula (6) is sub lc and not sub klt on a neighborhood U_{P′} of (f′)^{−1}(P′) for every P′ ∈ Σ′ (see [13, (20.3)]).

Indeed, the right hand side is sub lc by the semistable resolution of f′ and not sub klt on U_{P′} by B_{X′} ⊅ (f′)^{−1}(P′). q.e.d.

Theorem 4.8. The log-semistable part L_{Y′/C′}^{log,ss} is nef.

Proof. By 4.6, we can assume that Ξ = Θ′_{+}. By the definition of B_{+}^{Ξ}, h′_{∗}O_{Y′}(⌈B_{+}^{Ξ}/b⌉) = O_{C′} holds. Therefore, the condition (3) in 4.10 is satisfied. The condition (1) is also satisfied because h′(Supp Ξ^{v}) ⊂ Supp Σ′ = Supp π^{∗}Σ by 4.4.(i). Applying 4.10 to (Y′, Ξ − B^{Ξ}/b), we see that L_{Y′/C′}^{log,ss} is nef. q.e.d.

4.9. We recall Kawamata’s positivity theorem [8, Theorem 2] for the reader’s convenience (see also [2]).

Theorem 4.10. Let g : Y → T be a surjective morphism of smooth projective varieties with connected fibers. Let P = ∑_{j} P_{j} and Q = ∑_{l} Q_{l} be normal crossing divisors on Y and T, respectively, such that g^{−1}(Q) ⊂ P and g is smooth over T \Q. Let D = ∑_{j} d_{j}P_{j} be a ℚ-divisor on Y (d_{j}’s may be negative) such that:

(1) g : Supp(D^{h}) → T is relatively normal crossing over T \ Q, and g(Supp(D^{v})) ⊂ Q.

(2) d_{j} < 1 if P_{j} is horizontal.

(3) dim_{k(η)} g_{∗}O_{Y} (⌈−D⌉) ⊗_{O_{T}} k(η) = 1 for the generic point η of T.

(4) K_{Y} + D ∼_{ℚ} g^{∗}(K_{T} + L) for some ℚ-divisor L on T.

Let

g^{∗}Q_{l} = ∑_{j} w_{lj}P_{j},

\overline{d}_{j} = \frac{d_{j} + w_{lj} − 1}{w_{lj}} if g(P_{j}) = Q_{l},
δ_{l} = max \{\overline{d}_{j} | g(P_{j}) = Q_{l}\},
Δ_{0} = ∑_{l} δ_{l}Q_{l}, and
M = L − Δ_{0}.

Then M is nef.

Remark 4.11. (i) It can be checked easily that 1 − δ_{l} is the logcanonical threshold of g^{∗}Q_{l} with respect to (Y, D) over the generic point η_{Q_{l}}.

(ii) In [8, Theorem 2], it is assumed that d_{j} < 1 for all P_{j}. However, the assumption that d_{j} < 1 for vertical P_{j} was not used in the proof (see the proof of [8, Theorem 2]).

We start with a lemma to prove 4.5.(v).

Lemma 4.12. Under the notation and the assumptions of 4.5.(v), assume that C′ is a curve. Then the following holds.

b(K_{Y′/C′} + Ξ + ((h′)^{−1}Σ′)_{red}) ≻ (h′)^{∗}(L_{Y′/C′}^{log,ss} + bΣ′).

Proof. Note that B_{−}^{Ξ} = 0 since C is a curve. Then we have

K_{Y′/C′} + Ξ + ((h′)^{−1}Σ′)_{red} ≻ K_{Y′/C′} + Ξ − B_{+}^{Ξ}/b + ∑_{P ∈Σ′} t_{P}^{Ξ}(h′)^{∗}P
= \frac{1}{b}(h′)^{∗}L_{(Y′,Ξ)/C′}^{ss} + (h′)^{∗}Σ′,

where the first relation follows from Ξ ≻ 0 and the definitions of Σ′ and t_{P}^{Ξ}, and the second from the log-canonical bundle formula and the relation s_{P}^{Ξ}/b + t_{P}^{Ξ} = 1 for all P. q.e.d.

Proof of 4.5.(v). Replacing C′ by a general hyperplane-section H and Y′ by (h′)^{∗}(H), we can immediately reduce to the case where C′ is a curve. For simplicity, Ξ in B^{Ξ}, s_{P}^{Ξ} and t_{P}^{Ξ} will be suppressed during the proof. We note that B is effective.

By the hypothesis, the vertical part D of the Weil divisor

bN(K_{Y′/C′} + Ξ^{v}) − (h′)^{∗}NL_{Y′/C′}^{log,ss} = N ∑_{P} s_{P} (h′)^{∗}P + NB − bNΞ^{h}

is a Weil divisor. We note that

D = N ∑_{P} s_{P} (h′)^{∗}P + NB^{v} = bN(K_{Y′/C′} + Ξ) − (h′)^{∗}NL_{Y′/C′}^{log,ss} − NB^{h}.

By 4.12, we have

bN(K_{Y′/C′} + Ξ) − (h′)^{∗}NL_{Y′/C′}^{log,ss} + bN ((h′)^{−1}Σ′)_{red} ≻ (h′)^{∗}bNΣ′.

Whence

D + NB^{h} + bN ((h′)^{−1}Σ′)_{red} ≻ (h′)^{∗}bNΣ′. (7)

Let D_{P} and B_{P}^{v} be the parts of D and B^{v} lying over P. Let (h′)^{∗}P = ∑_{k} a_{k}F_{k} be the irreducible decomposition. Then D_{P} −NB_{P}^{v} = Ns_{P} (h′)^{∗}P and Supp(D_{P} − Ns_{P} (h′)^{∗}P) ⊅ F_{c} for some c by the definition of B_{P}^{v}. In particular Ns_{P} a_{c} ∈ ℤ. Furthermore, comparing the coefficients of F_{c} in the formula (7), we obtain Ns_{P} a_{c} + bN ≥ bNa_{c}, that is, Na_{c}s_{P} ≥ bN(a_{C} − 1). Since (Y′, Ξ) is klt, we have t_{P} > 0 and hence s_{P} < b. Hence u_{P} := a_{c} works. q.e.d.

5. Log-canonical rings

In this section, we consider an application of the log-canonical bundle formula 4.5 to the log-canonical ring of a pair (X, Δ):

R(X, K_{X} + Δ) = ⊕_{i≥0}H^{0}(X, ⌊iK_{X} + iΔ⌋).

We recall a well-known proposition on graded rings.

Proposition 5.1. Let R = ⊕_{i≥0}R_{i} be a graded integral domain with degree i part R_{i} such that R_{0} = ℂ. For m ∈ ℤ_{>0}, let R^{(m)} := ⊕_{i≥0}R_{im} be the graded ring whose degree i part is R_{im}. Then R is finitely generated over R_{0} iff so is R^{(m)} over R_{0}.

Sketch of the proof. If R is finitely generated, we can let ℤ/(m) act on R so that R^{(m)} is the invariant part. Hence R^{(m)} is finitely generated. If R^{(m)} is finitely generated, then R is contained in the integral closure of R^{(m)} in the quotient field of R. Hence R is a finite R^{(m)}-module and thus finitely generated. q.e.d.

Theorem 5.2. Let (X, Δ) be a proper klt pair with

κ(X, K_{X} + Δ) = l ≥ 0.

Then there exist an l-dimensional klt pair (C′, Δ′) with κ(C′, K_{C′} + Δ′) = l, two e, e′ ∈ ℤ_{>0} and an isomorphism

R(X, K_{X} + Δ)^{(e)} ≅ R(C′, K_{C′} + Δ′)^{(e′)}

of graded rings.

Proof. Let f : X −−→ C be the Iitaka fibering with respect to K_{X}+Δ. By replacing X and C, we assume that the following conditions hold:

1. X and C are projective smooth varieties and f : X → C is a proper surjective morphism with connected fibers.

2. Supp Δ is an snc divisor and ⌊Δ⌋ = 0.

3. There exists an effective divisor Σ ⊂ C such that f is smooth and Supp Δ^{h} is relatively normal crossing over C \ Σ and such that f(Supp(Δ^{v})) ⊂ Σ.

Note that adding effective exceptional ℚ-divisors does not change the log-canonical ring.

We use the same notation as 4.5. By the hypothesis, bK_{C′} +L_{Y′/C′}^{log,ss} + ∑_{P} s_{P} P is big on C′. Thus let A an ample ℚ-divisor and G an effective ℚ-divisor on C′ such that

bK_{C′} + L_{Y′/C′}^{log,ss} + ∑_{P} s_{P} P ∼_{ℚ} A + G.

Then by 4.5.(v), there exists a sufficiently small rational number ε > 0 such that the pair

(C′, ∑_{P}(s_{P} /b)P + (ε/b)G)

is klt. Since L_{Y′/C′}^{log,ss} is nef, L_{Y′/C′}^{log,ss} + εA is ample and it is ℚ-linearly equivalent to δH for some rational δ > 0 and a very ample divisor H. We can furthermore assume that

(C′, ∑_{P}(s_{P} /b)P + (ε/b)G + (δ/b)H)

is klt.

Now choose a ∈ ℤ_{>0} such that aε, aδ ∈ ℤ and aL_{Y′/C′}^{log,ss} +aεA ∼ aδH. Then we have

(a + aε)(bK_{C′} + L_{Y′/C′}^{log,ss} + ∑ s_{P} P)

∼ abK_{C′} + aδH + a(∑ s_{P} P + εG)

= ab(K_{C′} + ∑ \frac{s_{P}}{b} P + \frac{ε}{b}G + \frac{δ}{b} H).

Thus we can take e = (a + aε)b, e′ = ab and

Δ′ = ∑_{P}(s_{P} /b)P + (ε/b)G + (δ/b)H.

q.e.d.

The following corollary contains a generalization of Moriwaki’s result [16, Theorem (3.1)].

Corollary 5.3. Let (X, Δ) be a proper klt pair such that κ(X, K_{X} + Δ) ≤ 3. Then the log-canonical ring of (X, Δ) is finitely generated.

Proof. By 5.1 and 5.2, our problem is reduced to the case dim X = κ(X, K_{X} + Δ) ≤ 3. The case dim X = 2 is settled by [4, (1.5)] and the case dim X = 3 by [11, Corollary]. q.e.d.

The following theorem is a generalization of [18, Corollary] and [17, (3.7) Proposition].

Theorem 5.4. Let (X, Δ) be a proper klt pair. Assume that the log-canonical ring R(X, K_{X} +Δ) is finitely generated. Then there exists an effective ℚ-divisor Ξ on S := Proj R(X, K_{X} + Δ) such that (S, Ξ) is klt. Especially, S has only rational singularities.

Proof. By 5.2, we can assume that dim X = κ(X, K_{X} + Δ). By changing the model birationally, we can further assume that X is nonsingular and Δ is an snc divisor. Then it is settled by [17, (3.7) Proposition]. q.e.d.

6. Pluricanonical systems

Theorem 6.1. For arbitrary b, k ∈ ℤ_{>0}, there exists an effectively computable natural number M = M(b, k) with the following property.

Let X be a nonsingular projective variety of dimension m + 1 and Kodaira dimension κ(X) = 1, C a nonsingular projective curve and f : X → C a surjective morphism with connected fibers. Assume that its generic fiber F has κ(F) = 0, P_{b}(F) = 1 and the m-th Betti number B_{m}(F) ≤ k. Then |M(b, k)K_{X}| induces the rational map f : X −−→ C.

Since one can change the smooth model birationally so that B_{2}(F) ≤ 22 and |12K_{F} | ≠ ∅ for threefolds X, we need no extra conditions for 3-folds.

Corollary 6.2. There exists an effectively computable natural number M such that, for every algebraic 3-fold X with Kodaira dimension κ(X) = 1, |MK_{X}| induces the Iitaka fibering of X.

Proof of 6.1. By 2.3 and 2.8, we have the following for all n ≥ 0.

H^{0}(X, nbK_{X}) = H^{0}(C, ⌊n(bK_{C} + L_{X/C}^{ss} + ∑_{P} s_{P} P)⌋).

We note that deg(bK_{C} + L_{X/C}^{ss}) + ∑_{P} s_{P} > 0 by κ(X) = 1. Let g be the genus of C and let N = N(k). We treat the problem in 3 cases.

Case 1 (g ≥ 2). Setting n = iN in the above, we obtain the divisor

iNbK_{C} + iNL_{X/C}^{ss} + ⌊iNb(∑_{P} s_{P} P)⌋

on C. We note that NL_{X/C}^{ss} is nef Cartier divisor by 2.5 and 3.1 and that ⌊iNb(∑_{P} s_{P} P)⌋ ≻ 0 by 2.8. Hence the divisor is very ample if iNb ≥ 3 because deg ≥ 3 deg K_{C} ≥ 2g + 1. So M(b, k) = 3Nb works.

Case 2 (g = 1). We have deg L_{X/C}^{ss} + ∑_{P} s_{P} > 0. It is enough to find one i ∈ ℤ_{>0} such that the divisor ⌊iN(L_{X/C}^{ss} + ∑_{P} s_{P} P)⌋ has deg ≥ 3, because such a divisor is very ample on C. Hence this case is reduced to 6.3 with a = 0 and c = bN.

As pointed out by the referee, the Case 1 above is in [6, §3] and the Case 2 can probably be deduced as well. Thus the following is the key case.

Case 3 (g = 0). We have deg(bK_{C} + L_{X/C}^{ss}) + ∑_{P} s_{P} > 0 and need to one i ∈ ℤ_{>0} such that deg⌊iN(L_{X/C}^{ss} + ∑_{P} s_{P} P)⌋ − 2biN > 0. Thus this case is again reduced to 6.3 with a = 2bN and c = bN.

q.e.d.

Thus it remains to prove the following.

Proposition 6.3. Let a ∈ ℤ and c ∈ ℤ_{>0}. Let S(a, c) be the set of

ξ = (r; m_{1}, ⋯, m_{r}; b_{1}, ⋯, b_{r})

where r ∈ ℤ_{≥0}, m_{1}, ⋯, m_{r} ∈ ℤ_{>0} and b_{1}, ⋯, b_{r} ∈ ℤ_{≥0} satisfy the conditions b_{i} < min{m_{i}, c + 1} for all i and

−a + ∑_{i=1}^{r} \frac{m_{i} − b_{i}}{m_{i}} > 0.

For n ∈ ℤ_{>0}, let

f(n, ξ) = −an + ∑_{i=1}^{r}⌊n\frac{m_{i} − b_{i}}{m_{i}} ⌋.

Then there exists n = n(a, c) ∈ ℤ_{>0} such that f(νn, ξ) ≥ ν for all ξ ∈ S(a, c) and ν ∈ ℤ_{>0}.

Proof. We note that f(nm, ξ) ≥ mf(n, ξ) for n, m ∈ ℤ_{>0}. Hence it is enough to get f(n, ξ) ≥ 1. For each ξ ∈ S(a, c), let

I_{1}(ξ) = {i ∈ [1, r] | m_{i} ≤ 2c}, I_{2}(ξ) = [1, r] \ I_{1}(ξ),

and J(ξ) = (m_{i})_{i∈[1,r]}. It is easy to see that if |I_{2}(ξ)| ≥ 2a + 1 then f(2, ξ) ≥ 1. Thus we may assume that |I_{2}(ξ)| ≤ 2a. It is also easy to see that if α(ξ) := ∑_{i∈I_{1}(ξ)}(m_{i} − b_{i})/m_{i} > a, then (2c)!α(ξ) ∈ ℤ and

f((2c)!, ξ) = (2c)! (α(ξ) − a) + ∑_{i∈I_{2}(ξ)}⌊(2c)! · \frac{m_{i} − b_{i}}{m_{i}} ⌋ > 0.

Thus we may also assume that α(ξ) ≤ a. Hence by m_{i} ≤ 2c (i ∈ I_{1}(ξ)), we have |I_{1}(ξ)| ≤ 2cα(ξ) ≤ 2ac.

Under the conditions r = |I_{1}(ξ)| + |I_{2}(ξ)| ≤ 2ac + 2a and b_{i} ≤ c, the set T = {(r, b_{1}, ⋯, b_{r})} is finite. Hence for each η ∈ T, we work on the subset S(a, c)_{η} of S(a, c) with the fixed r and b_{i}’s.

We note that f(n, ξ) ≤ f(n, ξ′) if ξ, ξ′ ∈ S(a, c)_{η} satisfy J(ξ) ≤ J(ξ′) (coordinatewise). Hence we are done by the following lemma. q.e.d.

Lemma 6.4. Let r ∈ ℤ_{>0} and S ⊂ ℤ_{>0}^{r}. With respect to the total order, S contains only a finite number of minimal elements.

Proof. We use the induction on r. If r = 1, this is obvious. Fix any s = (s_{1}, ⋯, s_{r}) ∈ S and let m := max{s_{i}}. For i ∈ [1, r] and n ∈ [1, m], let T_{n,i} = {(t_{1}, ⋯, t_{r}) ∈ S | t_{i} = n}. Since each T_{n,i} has only a finite number of minimal elements by the induction hypothesis, so does the finite union ∪_{n,i}T_{n,i}. If t ∈ S \ ∪_{n,i}T_{n,i}, then t > s. q.e.d.

References

[1] E. Bierstone & P. Milman, Canonical desingularization in characteristic zero by blowing up the maximum strata of a local invariant, Invent. Math. 128 (1997) 207–302.

[2] O. Fujino, Applications of Kawamata’s positivity theorem, Proc. Japan Acad. Ser. A Math. Sci. 75 (1999) 75–79.

[3] T. Fujita, On Kähler fiber spaces over curves, J. Math. Soc. Japan 30 (1978) 779–794.

[4] , Fractionally logarithmic canonical rings of algebraic surfaces, J. Fac. Sci. Univ. Tokyo Sect. IA Math. 30 (1984) 685–696.

[5] , Zariski decomposition and canonical rings of elliptic threefolds, J. Math. Soc. Japan 38 (1986) 20–37.

[6] Y. Kawamata, Kodaira dimension of algebraic fiber spaces over curves, Invent. Math. 66 (1982) 57–71.

[7] , Kodaira dimension of certain algebraic fiber spaces, J. Fac. Sci. Univ. Tokyo Sect.IA Math. 30 (1983) 1–24.

[8] , Subadjunction of log canonical divisors. II, Amer. J. Math. 120 (1998) 893–899.

[9] , Deformations of canonical singularities, J. Amer. Math. Soc. 12 (1999) 85–92.

[10] , The theory of algebraic varieties, (in Japanese), Kyoritsu Shuppan Co., Ltd., Tokyo, 1997.

[11] S. Keel, K. Matsuki & J. McKernan, Log abundance theorem for threefolds, Duke Math. J. 75 (1994) 99–119.

[12] K. Kodaira, On the structure of compact complex analytic surfaces. I, Amer. J. Math. 86 (1964) 751–798.

[13] J. Kollár, et al, Flips and abundance for algebraic threefolds, Seminar Proc. (Utah Aug. 1991), Asterisque 211.

[14] J. Kollár & S. Mori, Birational geometry of algebraic varieties, Cambridge Univ. Press, Cambridge, 1998.

[15] S. Mori, Classification of higher-dimensional varieties, Proc. Symp. Pure Math. 46 (1987) 269–331.

[16] A. Moriwaki, Semiampleness of the numerically effective part of Zariski decomposition, J. Math.Kyoto Univ. 26 (1986) 465–481.

[17] , Semi-ampleness of the numerically effective part of Zariski decomposition. II, Algebraic geometry and commutative algebra, Kinokuniya, Tokyo, 1988, Vol. I, 289–311.

[18] N. Nakayama, The singularity of the canonical model of compact Kähler manifolds, Math. Ann. 280 (1988) 509–512.

[19] M. Reid, Canonical 3-folds, Algebraic geometry Angers, 1979, (ed. by A. Beauville), Sijthoff & Noordhoff, 1980, 273–310.

[20] Jan Stevens, On canonical singularities as total spaces of deformations, Abh. Math. Sem. Univ. Hamburg, 58 (1988) 275–283.

[21] E. Szabó, Divisorial log terminal singularities, J. Math. Sci. Univ. Tokyo 1 (1994) 631–639.

[22] E. Viehweg, Weak positivity and the additivity of the Kodaira dimension for certain fibre spaces, Algebraic varieties and analytic varieties, Tokyo, 1981, Adv. Stud. Pure Math., North-Holland, Amsterdam-New York, 1 (1983) 329–353.

Kyoto University, Japan