FINITE GENERATION OF THE LOG CANONICALRING IN DIMENSION FOUROSAMU FUJINO
Dedicated to the memory of Professor Masayoshi Nagata
Abstract. We treat three different topics on the log minimalmodel program, especially for four-dimensional log canonical pairs.
(a) Finite generation of the log canonical ring in dimension four.
(b) Abundance theorem for irregular fourfolds.
(c) Invariance of log plurigenera for threefolds.
We obtain (a) as a direct consequence of Birkar’s existence offour-dimensional log minimal models by using Fukuda’s theorem on
the four-dimensional log abundance conjecture. We can prove (b)only by using traditional arguments. More precisely, we prove theabundance conjecture for irregular (n + 1)-folds on the assumptionthat the minimal model conjecture holds in dimension ≤ n. The
invariance of log plurigenera for threefolds (c) is a consequenceof the semi-stable log minimal model program for fourfolds and ageneralization of Kollár’s torsion-free theorem.

Contents
1. Introduction
2. Preliminaries
3. Log canonical ring
3.1. Appendix
4. Abundance theorem for irregular varieties
5. Invariance of log plurigenera
References

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1. Introduction
In this paper, we treat three different topics on the log minimalmodel program, especially for four-dimensional log canonical pairs. We
Date: 2009/4/23.
2000 Mathematics Subject Classification. Primary 14J35; Secondary 14E30.
Key words and phrases. log canonical ring, log abundance conjecture.
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will freely use the results on the three-dimensional log minimal model
program (cf. [FA], [KeMM], etc.). Sorry, we do not always refer to theoriginal papers since the results are scattered in various places.
1 (Finite generation of the log canonical ring in dimensionfour). The following theorem is the main result of Section 3 (cf. [F8,Section 3.1]).
Theorem 1.1 (Finite generation of the log canonical ring in dimensionfour). Let π : X → Z be a proper surjective morphism from a smooth
fourfold X. Let B be a boundary Q-divisor on X such that SuppB is
a simple normal crossing divisor on X. Then the relative log canonicalring
M
R(X/Z, KX + B) =π∗ OX (xm(KX + B)y)m≥0

is a finitely generated OZ -algebra.

It is easy to see that Theorem 1.1 is equivalent to Theorem 1.2.
Theorem 1.2. Let π : X → Z be a proper surjective morphism from afour-dimensional log canonical pair (X, B) such that B is an effective
Q-divisor. Then the relative log canonical ringM
R(X/Z, KX + B) =π∗ OX (xm(KX + B)y)m≥0

is a finitely generated OZ -algebra.

In Section 3, we give a proof of Theorem 1.1 by using Birkar’s existence theorem of four-dimensional log minimal models (cf. [B]) andFukuda’s result on the log abundance conjecture for fourfolds (cf. [Fk]).
A key point of Fukuda’s result is the abundance theorem for semi log
canonical threefolds in [F1].
2 (Abundance theorem for irregular fourfolds). In Section 4,
we prove the abundance theorem for irregular (n + 1)-folds on theassumption that the minimal model conjecture holds in dimension ≤n (see Theorem 4.3). By this result, we know that the abundance
conjecture for irregular varieties is the problem for lower dimensionalvarieties. Since the minimal model conjecture holds in dimension ≤ 3,
we obtain the next theorem (see Corollary 4.5).
Theorem 1.3 (Abundance theorem for irregular fourfolds). Let X bea normal complete fourfold with only canonical singularities. Assume
that KX is nef and the irregularity q(X) is not zero. Then KX is
semi-ample.

LOG CANONICAL RING IN DIMENSION FOUR

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We also prove that there exists a good minimal model for any smoothprojective irregular fourfold (see Theorem 4.6).
Theorem 1.4 (Good minimal models of irregular fourfolds). Let X bea smooth projective irregular fourfold. If X is not uni-ruled, then there
exists a normal projective variety X 0 such that X 0 has only Q-factorial
terminal singularities, X 0 is birationally equivalent to X, and KX 0 issemi-ample.
3 (Invariance of log plurigenera for threefolds). In the final section: Section 4, we treat the invariance of log plurigenera (see Corollary5.5). It is a direct generalization of Nakayama’s result (cf. [N]).
Theorem 1.5 (Invariance of log plurigenera). Let f : X → C bea smooth projective morphism onto a non-singular curve C with connected fibers. Let D be a simple normal crossing divisor on X suchthat D is relatively normal crossing over C. If dim X ≤ 4, thendim H 0 (Xs , OXs (m(KXs + Ds ))),
where Ds = D|Xs , is constant for any s ∈ C and any non-negative
integer m. In particular, the logarithmic Kodaira dimension of Xs \Ds ,that is, κ(Xs , KXs + Ds ), is constant for any s ∈ C.
We obtain this theorem as a consequence of the semi-stable log
minimal model program for fourfolds and a generalization of Kollár’s
torsion-free theorem. We give an important remark. If we establish
the log minimal model program and the log abundance conjecture indimension ≤ n, then the arguments in Section 4 works without any
changes in dimension ≤ n. For the details, see Theorem 5.3.
We note that each section can be read independently.
Acknowledgments. The author was partially supported by the Grantin-Aid for Young Scientists (A) ]20684001 from JSPS. He was also
supported by the Inamori Foundation.
We will work over C, the complex number field, throughout thispaper. We will freely use the notation in [KMM] and [KM]. Note that
we do not use R-divisors.
2. Preliminaries
In this section, we collect basic definitions.
Definition 2.1 (Divisors,
Pr Q-divisors). Let X be a normal variety. For
a Q-Weil divisor D = j=1 dj Dj on X such that Di 6= Dj for i 6= j,Pr
we define the round-up pDq =j=1 pdj qDj (resp. the round-down

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xDy = rj=1 xdj yDj ), where for any rational 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.
We call D a boundary Q-divisor if 0 ≤ dj ≤ 1 for any j.
We note that ∼Q denotes the Q-linear equivalence of Q-divisors.
We call X Q-factorial if and only if every Weil divisor on X is QCartier.
Definition 2.2 (Exceptional locus). For a proper birational morphism
f : X → Y , the exceptional locus Exc(f ) ⊂ X is the locus where f is
not an isomorphism.
Let us quickly recall the definitions of singularities of pairs.
Definition 2.3 (Singularities of pairs). Let X be a normal variety andB an effective Q-divisor on X such that KX + B is Q-Cartier. Let
f : Y → X be a resolution such that Exc(f ) ∪ f∗−1 B has a simple
normal crossing support, where f∗−1 B is the strict transform of B on
Y . We write
X
KY = f ∗ (KX + B) +a i Eiiand a(Ei , X, B) = ai . We say that (X, B) is lc (resp. klt) if and only
if ai ≥ −1 (resp. ai > −1) for any i. We note that lc (resp. klt) is anabbreviation of log canonical (resp. Kawamata log terminal). We also
note that the discrepancy a(E, X, B) ∈ Q can be defined for any prime
divisor E over X.
In the above notation, if B = 0 and ai > 0 (resp. ai ≥ 0) for any i,then we say that X has only terminal (resp. canonical) singularities.

Definition 2.4 (Divisorial log terminal pair). Let X be a normal variety and B a boundary Q-divisor such that KX + B is Q-Cartier. If
there exists a resolution f : Y → X such that(i) both Exc(f ) and Exc(f )∪Supp(f∗−1 B) are simple normal crossing divisors on Y , and(ii) a(E, X, B) > −1 for every exceptional divisor E ⊂ Y ,then (X, B) is called divisorial log terminal (dlt, for short).
For the details of singularities of pairs, see, for example, [KM] and[F5].
Definition 2.5 (Center, lc center). Let E be a prime divisor over X.
The closure of the image of E on X is denoted by cX (E) and called
the center of E on X.
Let (X, B) be an lc pair. If a(E, X, B) = −1, cX (E) is called an lc
center of (X, B).

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The following definitions are now classical.
Definition 2.6 (Iitaka’s D-dimension and numerical D-dimension).
Let X be a normal complete variety and D a Q-Cartier Q-divisor.
Assume that mD is Cartier for a positive integer m. Let
Φ|tmD| : X 99K Pdim |tmD|
be rational mappings given by linear systems |tmD| for positive integers
t. We define Iitaka’s D-dimension(
max dim Φ|tmD| (X), if |tmD| 6= ∅ for some t,t>0κ(X, D) =−∞,otherwise.
In case D is nef, we can also define the numerical D-dimension
ν(X, D) = max{ e | D e 6≡ 0},
where ≡ denotes numerical equivalence. We note that ν(X, D) ≥κ(X, D) holds.
Definition 2.7 (Nef and abundant divisors). Let X be a normal complete variety and D a Q-Cartier Q-divisor on X. Assume that D
is nef. The nef Q-divisor D is said to be abundant if the equality
κ(X, D) = ν(X, D) holds. Let π : X → Z be a proper surjective morphism of normal varieties and D a π-nef Q-divisor on X. Then D is
said to be π-abundant if Dη is abundant, where Dη = D|Xη and Xη isthe generic fiber of π.
Definition 2.8 (Irregularity). Let X be a normal complete variety
with only rational singularities. We put
q(X) = h0 (X, OX ) = dim H 0 (X, OX ) < ∞and call it the irregularity of X.
Let X be as above. If q(X) 6= 0, then we call X irregular.
If X 0 is a normal complete variety with only rational singularities
such that X 0 is birationally equivalent to X, then it is easy to see thatq(X) = q(X 0 ).
3. Log canonical ring
In this section, we prove the following theorem: Theorem 1.1.
Theorem 3.1 (Finite generation of the four-dimensional log canonicalring). Let π : X → Z be a proper surjective morphism from a smooth
fourfold X. Let B be a boundary Q-divisor on X such that SuppB is

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a simple normal crossing divisor on X. Then the relative log canonicalring
M
R(X/Z, KX + B) =π∗ OX (xm(KX + B)y)m≥0

is a finitely generated OZ -algebra.

Let us start the proof of Theorem 1.1. The next proposition is wellknown and a slight generalization of [K4, Theorem 7.3].
Proposition 3.2. Let (X, B) be a proper log canonical fourfold suchthat KX + B is nef and κ(X, KX + B) > 0. Then KX + B is abundant,
that is, κ(X, KX + B) = ν(X, KX + B).
Proof. See, for example, [Fk, Proposition 3.1]. We note that we need
the three-dimensional log minimal model program and log abundancetheorem here (see [FA], [KeMM], and [KeMM2]).

Let us recall Fukuda’s result in [Fk]. We will generalize this in Theorem 3.8.
Theorem 3.3 (cf. [Fk, Theorem 1.5]). Let (X, B) be a proper dlt fourfold. Assume that KX + B is nef and that κ(X, KX + B) > 0. Then
KX + B is semi-ample.
Proof. By Proposition 3.2, κ(X, KX + B) = ν(X, KX + B). We put
S = xBy and KS + BS = (KX + B)|S . Then the pair (S, BS ) is semi
divisorial log terminal and KS + BS is semi-ample by [F1, Theorem
0.1]. Finally, by [F7, Corollary 6.7], we obtain that KX + B is semiample.
Remark 3.4. The proof of [Fk, Proposition 3.3] depends on [K4, Theorem 5.1]. It requires [K4, Theorem 4.3] whose proof contains a nontrivial gap. See [F5, Remark 3.10.3] and [F9]. So, we adopted [F7,Corollary 6.7] in the proof of Theorem 3.3.
Let us go to the proof of Theorem 1.1. Here, we adopt Birkar’s
definition of the log minimal model (see [B, Definition 2.4]), which isa slightly different from [KM, Definition 3.50]. See Remark 3.6 andExample 3.7 below.
Definition 3.5 (cf. [B, Definition 2.4]). Let (X, B) be a log canonicalpair over Z. A log minimal model (Y /Z, BY + E) of (X/Z,
PB) consistsof a birational map φ : X 99K Y /Z, BY = φ∗ B, and E = j Ej , where
Ej is a prime divisor on Y and φ−1 -exceptional for any i, and satisfies
the following conditions:(1) Y is Q-factorial and (Y, BY + E) is dlt,

LOG CANONICAL RING IN DIMENSION FOUR

7(2) KY + BY + E is nef over Z, and(3) for any prime divisor D on X which is exceptional over Y , wehavea(D, X, B) < a(D, Y, BY + E),
where a(D, X, B) (resp. a(D, Y, BY + E)) denotes the discrepancy of D with respect to (X, B) (resp. (Y, BY + E)).
Remark 3.6. In [KM, Definition 3.50], it is required that φ−1 has no
exceptional divisors.
Example 3.7. Let X = P2 and DX the complement of the big torus.
Then KX + DX is dlt and KX + DX ∼ 0. Let Y = PP1 (OP1 ⊕ OP1 (−1))
and DY the complement of the big torus. Then (Y, DY ) is a log minimalmodel of (X, DX ) in the sense of Definition 3.5 (cf. [B, Definition 2.4]).
Of course, KY + DY is dlt and KY + DY ∼ 0. On the other hand,(Y, DY ) is not a log minimal model of (X, DX ) in the sense of [KM,
Definition 3.50].
Proof of Theorem 1.1. We can assume that the fiber of π is connected.
First, if κ(Xη , KXη +Bη ) = −∞, where η is the generic point of Z, thenthe statement is trivial. We note that the statement is obvious when Z
is a point and κ(X, KX + B) = 0. So, we can assume that κ(Xη , KXη +Bη ) ≥ 0 and that κ(X, KX + B) ≥ 1 when Z is a point. Since theproblem is local, we can assume that Z is affine. By compactifying
Z and X and taking a resolution of X, we can assume that X andZ are projective and that SuppB is a simple normal crossing divisor.
By the assumption, we can find an effective Q-divisor M on X suchthat KX + B ∼Q,π M , that is, there exists a Q-divisor N on Z suchthat KX + B ∼Q M + π ∗ N . We take a log minimal model of (X, B)
by using the arguments in [B, Section 3]. Then we obtainP a projective
surjective morphism πY : Y → Z such that (Y /Z, BY + j Ej ) is a log
minimal model of (X/Z, B), where BY is the pushforward of B on Y
by φ : X 99K Y and Ej is exceptional over X and is a prime divisor on
Y for any i. Let A be a sufficiently ample
P general Cartier divisor on Z.
Then (Y, BY + E + πY∗ A), where E = j Ej , is a log minimal model of(X, B +π ∗ A) by Kawamata’s bound of the length of extremal rays (see,for example, [F8, Subsection 3.1.3]). Since κ(Y, KY + BY + E + πY∗ A) ≥1, KY + BY + E + πY∗ A is semi-ample by Theorem 3.3. In particular,
KY + BY + E = KY + BY + E + πY∗ A − πY∗ A
is πY -semi-ample. Thus,M
πY ∗ OY (xm(KY + BY + E)y)m≥0

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is a finitely generated OZ -algebra. Therefore,M
π∗ OX (xm(KX + B)y)m≥0

is a finitely generated OZ -algebra. We finish the proof.

The final theorem in this section is a generalization of Fukuda’s theorem (see Theorem 3.3).
Theorem 3.8 (A special case of the log abundance theorem). Let
π : X → Z be a proper surjective morphism from a four-dimensionallog canonical pair (X, B) such that B is an effective Q-divisor andthat KX + B is π-nef. When Z is a point, we further assume that
κ(X, KX + B) > 0. Then KX + B is π-semi-ample.
Proof. Without loss of generality, we can assume that π has connected
fibers. By Proposition 3.2 and the log abundance theorem in dimension≤ 3, KXη + Bη isL
nef and abundant, where η is the generic point of Z.
By Theorem 1.2, m≥0 π∗ OX (xm(KX +B)y) is a finitely generated OZ algebra. Therefore, KX +B is π-semi-ample by Lemma 3.10 below. The next lemma is well known. We omit the proof for the reader’s
exercise.

Lemma 3.9 (cf. [L, Theorem 2.3.15]). Let π : X → Z be a projective
surjective morphism from a smooth variety X to a Lnormal variety Z
and M a π-nef and π-big Cartier divisor on X. Then m≥0 π∗ OX (mM )is a finitely generated OZ -algebra if and only if M is π-semi-ample.
By [KMM, Proposition 6-1-3], we can reduce Lemma 3.10 to Lemma
3.9.

Lemma 3.10. Let π : X → Z be a proper surjective morphism between
normalLvarieties and M a π-nef and π-abundant Cartier divisor on X.
Then m≥0 π∗ OX (mM ) is a finitely generated OZ -algebra if and only
if M is π-semi-ample.
3.1. Appendix. In this appendix, we explicitly state the results indimension ≤ 3 because we can find no good references for the relative
statements (cf. [Ft], [KeMM], and [KeMM2]).

Theorem 3.11. Let π : X → Z be a proper surjective morphism
between normal varieties. Assume that (X, B) is log canonical with
dim X ≤ 3 and that B is an effective Q-divisor. ThenM
π∗ OX (xm(KX + B)y)m≥0

is a finitely generated OZ -algebra.

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Proof. When Z is a point, this theorem is well known (cf. [Ft], [KeMM],and [KeMM2]). So, we assume that dim Z ≥ 1.
LBy the arguments inthe proof of Theorem 1.1, we can prove thatm≥0 π∗ OX (xm(KX +B)y) is a finitely generated OZ -algebra.

Theorem 3.12. Let π : X → Z be a proper surjective morphism suchthat (X, B) is log canonical with dim X ≤ 3. Assume that KX + B is
π-nef and B is an effective Q-divisor. Then KX + B is π-semi-ample.
Proof. When Z is a point, this theorem is well known (cf. [Ft], [KeMM],and [KeMM2]). So, we assume that dim Z ≥ 1. Without loss of generality, we can assume that π has connected fibers. It is well known
that KX + B is π-nef and π-abundant by the log abundance theorem
in dimension ≤ 2. By Theorem 3.11 and Lemma 3.10, KX + B is
π-semi-ample.

We close this appendix with a remark.
Remark 3.13. Let π : X → Z be a proper surjective morphism between normal varieties. Assume that (X, B) is klt and that B is aneffective Q-divisor. ThenM
π∗ OX (xm(KX + B)y)m≥0

is a finitely generated OZ -algebra by [BCHM]. Therefore, by Lemma
3.10, KX + B is π-semi-ample if and only if KX + B is π-nef andπ-abundant by Lemma 3.10.
Of course, we know that KX + B is π-semi-ample if and only if
KX + B is π-nef and π-abundant without appealing [BCHM]. See, forexample, [F9]. It is known as Kawamata’s theorem (cf. [K4]).
4. Abundance theorem for irregular varieties
In this section, we treat the abundance conjecture for irregular varieties. Let us recall the following minimal model conjecture.
Conjecture 4.1 (Minimal model conjecture (weak form)). Let X be asmooth projective variety. Assume that KX is pseudo-effective. Then
there exists a normal projective variety X 0 which satisfies the following
conditions:(i) X 0 is birationally equivalent to X.
(ii) X 0 has at most canonical singularities.
(iii) KX 0 is semi-ample.
In particular, the Kodaira dimension κ(X) = κ(X, KX ) is non-negative.
We sometimes call X 0 a good minimal model of X.

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We know that Conjecture 4.1 holds in dimension ≤ 3 (cf. [KMM],[FA], etc.).
Remark 4.2. Let X be a smooth projective variety. Then X is uniruled if and only if KX is not pseudo-effective by [BDPP].
The next theorem is the main theorem of this section.
Theorem 4.3 (Abundance theorem for irregular (n + 1)-folds). Assume that Conjecture 4.1 holds in dimension ≤ n. Let X be a normal
complete (n + 1)-fold with only canonical singularities. If KX is nef
and q(X) 6= 0, then KX is semi-ample.
Proof. Let π : X → X be a resolution and α : X → A = Alb(X) the
Albanese mapping. By the assumption, we have dim A ≥ 1. Since X
has only rational singularities, β = α ◦ π −1 : X → A is a morphism(cf. [R, Proposition 2.3], [BS, Lemma 2.4.1])
Claim 1. We have κ(X, KX ) = κ(X, KX ) ≥ 0.
Proof of Claim 1. Let f : X → S be the Stein factorization of α andF a general fiber of f . Then, by [K5, Corollary 1.2], we haveκ(X, KX ) ≥ κ(F, KF ) + κ(S, KS ),where S is a resolution of S. We note that κ(S, KS ) ≥ 0 because
S → β(X) ⊂ A is generically finite (see, for example, [U, Theorem
6.10, Lemma 10.1]). We also note that κ(F, KF ) = κ(G, KG ) ≥ 0
since dim G ≤ n, G has only canonical singularities, and KG is nef,
where G = π(F ). Here, we used Conjecture 4.1 in dimension dim G =dim F ≤ n. Therefore, we obtain κ(X, KX ) ≥ 0.
Claim 2. If κ(X, KX ) = 0, then ν(X, KX ) = 0.
Proof of Claim 2. By Kawamata’s theorem (cf. [K3, Theorem 1]), β
is surjective and β∗ OX ' OA . Let G be a general fiber of β. Then
κ(G, KG ) = 0 by
0 = κ(X, KX ) ≥ κ(G, KG ) + κ(A, KA ) = κ(G, KG )as in Claim 1 and κ(G, KG ) ≥ 0 by Conjecture 4.1 in dim G ≤ n since
KG is nef. We note that X and G have only canonical singularities.
By Remark 3.13 and the assumption: Conjecture 4.1 in dimension ≤ n,
KX is β-semi-ample. Therefore, β : X → A can be written as follows:fg

β : X −→ S −→ A,
where KX ∼Q f ∗ D for some g-ample Q-Cartier Q-divisor D on S,
g : S → A is a birational morphism, and S is a normal variety. Since

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κ(X, KX ) = 0, we obtain κ(S, D) = 0. So, it is sufficient to prove that
D ∼Q 0. By [A, Theorem 0.2], we can write D ∼Q KS + ∆S such that(S, ∆S ) is klt. In particular, ∆S is effective. By Lemma 4.4 below, we
obtain that g is an isomorphism. Therefore, D ∼Q 0 since κ(S, D) = 0
and S = A is an Abelian variety.

By Claim 1 and Claim 2, ν(X, KX ) > 0 implies κ(X, KX ) > 0. In
this case, we obtain κ(X, KX ) = ν(X, KX ) by Kawamata’s argument
and the assumption: Conjecture 4.1 in dimension ≤ n (see the proof of[K4, Theorem 7.3]). Therefore, KX is semi-ample by Remark 3.13. 
We already used the following lemma in the proof of Claim 2.
Lemma 4.4. Let g : S → A be a projective birational morphism froma klt pair (S, ∆S ) to an Abelian variety A. Assume that KS + ∆S is
g-nef and κ(S, KS + ∆S ) = 0. Then g is an isomorphism.
Proof. By replacing S with its small projective Q-factorialization (cf. [BCHM]),
we can assume that S is Q-factorial. We note that KS = E, where E
is effective and SuppE = Exc(g) since A is an Abelian variety. If
B = g∗ ∆S 6= 0, then g ∗ B ≤ m(KS + ∆S ) for some m > 0. In this case,1 ≤ κ(A, B) = κ(S, g ∗ B) ≤ κ(S, KS + ∆S ) = 0.
It is a contradiction. Therefore, B = 0. This means that ∆S is gexceptional. Thus, KS + ∆S is effective, g-exceptional, and Exc(g) =Supp(KS + ∆S ). By the assumption, KS + ∆S is g-nef. So, g is an
isomorphism by the negativity lemma.

As a special case of Theorem 4.3, we obtain the abundance theorem
for irregular fourfolds.
Corollary 4.5 (Abundance theorem for irregular fourfolds). Let X bea normal complete fourfold with only canonical singularities. Assume
that KX is nef and the irregularity q(X) is not zero. Then KX is
semi-ample.
Proof. It is obvious by Theorem 4.3 because Conjecture 4.1 holds forthreefolds (cf. [FA], [KM], etc.).

We close this section with the following theorem.
Theorem 4.6 (Good minimal models of irregular fourfolds). Let X bea smooth projective irregular fourfold. If X is not uni-ruled, then X has
a good minimal model. More precisely, there exists a normal projective
variety X 0 such that X 0 has only Q-factorial terminal singularities, X 0
is birationally equivalent to X, and KX 0 is semi-ample.

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Proof. We run the minimal model program. Then we obtain a minimalmodel X 0 of X since KX is pseudo-effective by the assumption. Here,
we used the existence and the termination of four-dimensional terminal
flips (cf. [KMM, Theorem 5-1-15], [S], and [HM, Corollary 5.1.2]). We
note that q(X 0 ) = h0 (X 0 , OX 0 ) = q(X) 6= 0. Therefore, by Theorem
4.3, we obtain that KX 0 is semi-ample.

5. Invariance of log plurigenera
Let us recall the definition of dlt morphisms.
Definition 5.1 ([KM, Definition 7.1]). Let X be a normal variety, B
an effective Q-divisor on X and f : X → C a non-constant morphism
to a non-singular curve C. We say that f : (X, B) → C is dlt if(X, B + f ∗ P ) is dlt for every closed point P ∈ C.
The following theorem is an easy consequence of the generalized version of the torsion-free theorem (see, for example, [F8, Theorem 2.39(i)]).
Theorem 5.2. Let f : (X, B) → C be a proper dlt morphism. Assume
that KX + B is Q-Cartier and that KX + B is f -semi-ample. ThenRi f∗ OX (m(KX + B)) is locally free for any i ≥ 0, where m is any
positive integer such that m(KX + B) is integral. In particular,
dim H 0 (Xs , OXs (m(KXs + Bs )))is constant for any s ∈ C, where Bs = B|Xs .
Proof. Let g : Y → X be a resolution such that g is an isomorphism
over the generic point of any lc center of (X, B), KY = g ∗ (KX +B)+E,
and that SuppE is a simple normal crossing divisor on Y . We put
F = xf∗−1 By. Thenpg ∗ (m(KX + B)) + E + F q− (KY + {−(g ∗ (m(KX + B)) + E + F )} + F ) = g ∗ ((m − 1)(KX + B)).
By the vanishing theorem of Reid–Fukuda type (see, for example, [F8,Lemma 4.10]), we obtain thatRi g∗ OY (pg ∗ (m(KX + B)) + E + F q) = 0
for any i > 0. On the other hand, it is easy to see thatg∗ OY (pg ∗ (m(KX + B)) + E + F q) ' OX (m(KX + B)).
By the generalized version of the torsion-free theorem (see, for example,[F8, Theorem 2.39 (i)]), we have thatRi (f ◦ g)∗ OY (pg ∗ (m(KX + B)) + E + F q)

LOG CANONICAL RING IN DIMENSION FOUR

13

is torsion-free for any i. We note that any lc center of the pair(Y, {−(g ∗ (m(KX + B)) + E + F )} + F )is dominant onto C by f . Therefore,
Ri f∗ OX (m(KX + B)) ' Ri (f ◦ g)∗ OY (pg ∗ (m(KX + B)) + E + F q)
is locally free for every i. We finish the proof by the base change
theorem.

The next result is the invariance of log plurigenera in dim ≤ 3. It isa direct generalization of [N]. For related topics, see [K1] and [K2].
Theorem 5.3. Let f : (X, B) → C be a projective dlt morphism suchthat KX + B is Q-Cartier and dim X ≤ 4. Assume that all the fibersof f are irreducible. Thendim H 0 (Xs , OXs (m(KXs + Bs )))is constant for any s ∈ C, where Bs = B|Xs and m is any positive
integer such that m(KX + B) is integral.
Proof. First, by the existence of log flips (see [S] and [HM]) and thespecial termination in dimension four (see [S] and [F6]), there existse B)e → (X, B) such thata small projective Q-factorialization ϕ : (X,e B)e is dlt and K e + B
e = ϕ∗ (KX + B). By replacing (X, B) with
(X,
X
e B),
e we can assume that X is Q-factorial. Apply the log minimal(X,model program to (X, B) over C. By the existence of log flips andthe termination of four-dimensional semi-stable log flips (see [F4] andRemark 5.4 below), we obtain a sequence of semi-stable log flips anddivisorial contractions(X, B) = (X0 , B0 ) 99K · · · 99K (Xi , Bi ) 99K · · · 99K (Xk , Bk ) = (X 0 , B 0 )such that(1) KX 0 + B 0 is f 0 -nef, where f 0 : X 0 → C,or(2) there exists a contraction ψ : X 0 → Z over C such that dim Z <dim X 0 and that −(KX 0 + B 0 ) is ψ-ample.
In (1), KX 0 + B 0 is f 0 -semi-ample by the abundance theorem (see Theorem 3.8). Therefore,
dim H 0 (Xs0 , OXs0 (m(KXs0 + Bs0 ))) = dim H 0 (Xs , OXs (m(KXs + Bs )))is constant for any s ∈ C by Theorem 5.2, where Bs0 = B 0 |Xs0 . In (2),dim H 0 (Xs , OXs (m(KXs + Bs )))= dim H 0 (Xs0 , OXs0 (m(KXs0 + Bs0 ))) = 0

14

OSAMU FUJINO

for any s ∈ C. Thus, we finish the proof.

Remark 5.4. In [F2], [F3], and [F4], we assumed that X is projective.
However, we do not need this assumption there. Since the exceptionalloci of (2, 1)-flips are always projective (see the proof of [AHK, Lemma3.1]).
The following corollary is a very special case of Theorem 5.3.
Corollary 5.5 (Invariance of log plurigenera). Let f : X → C bea smooth projective morphism onto a non-singular curve C with connected fibers. Let D be a simple normal crossing divisor on X suchthat D is relatively normal crossing over C. If dim X ≤ 4, thendim H 0 (Xs , OXs (m(KXs + Ds ))),
where Ds = D|Xs , is constant for any s ∈ C and any non-negative
integer m. In particular, the logarithmic Kodaira dimension of Xs \Ds ,that is, κ(Xs , KXs + Ds ), is constant for any s ∈ C.
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[U]

OSAMU FUJINO

K. Ueno, Classification theory of algebraic varieties and compact complex
spaces, Notes written in collaboration with P. Cherenack. Lecture Notes
in Mathematics, Vol. 439. Springer-Verlag, Berlin-New York, 1975.

Department of Mathematics, Faculty of Science, Kyoto University,
Kyoto 606-8502 Japan
E-mail address: fujino@math.kyoto-u.ac.jp