ON KAWAMATA’S THEOREMOSAMU FUJINO
Abstract. We give an alternate proof to the main theorem of
Kawamata’s paper: Pluricanonical systems on minimal algebraic
varieties. It is the first rigorous proof of Kawamata’s theorem since
the original argument contains a mistake. Our proof also works forvarieties in class C. We note that our proof is completely different
from Kawamata’s.

Contents
1. Introduction
2. Proof of Kawamata’s theorem
3. Appendix: Quick review of Ambro’s formula
4. Kawamata’s theorem for varieties in class C
5. Comments for the coming generation
References

1
3
6
7
10
11

1. Introduction
One of the main purposes of this paper is to cut off a chain of mistakes
and troubles caused by [Ka, Theorem 4.3]. We give an alternate proof
to the following famous theorem, which we call Kawamata’s theorem inthis paper. This theorem is indispensable for the abundance conjecture.
Theorem 1.1 (cf. [KMM, Theorem 6-1-11]). Let (X, B) be a klt pair,let π : X → S be a proper surjective morphism of normal varieties.
Assume the following conditions:(a) H is a π-nef Q-Cartier divisor on X,(b) H − (KX + B) is π-nef and π-abundant, and(c) κ(Xη , (aH − (KX + B))η ) ≥ 0 and ν(Xη , (aH − (KX + B))η ) =ν(Xη , (H − (KX + B))η ) for some a ∈ Q with a > 1, where η isthe generic point of S.
Then H is π-semi-ample.
Date: 2009/1/5.
2000 Mathematics Subject Classification. 14C20.
1

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OSAMU FUJINO

It was first proved in [Ka] on the assumption that S is a point.
Kawamata’s proof heavily depends on a very technical generalizationof Kollár’s injectivity theorem on generalized normal crossing varieties(see [Ka, Section 4]). Once we adopt this difficult injectivity theorem,
X-method works and the proof is essentially the same as the one of
Kawamata–Shokurov base point free theorem. Unfortunately, there isan ambiguity in the proof of [Ka, Theorem 4.3] (see [F2, Remark 3.10.3]
and 5.1 below). Our proof is completely different from Kawamata’s.
His proof relies on the theory of mixed Hodge structures for reducible
varieties. Our proof grew out from the theory of variation of Hodge
structures, especially, Deligne’s canonical extensions of Hodge bundles.
We note that our method saves Kawamata’s theorem but does notrecover the results in [Ka, Section 4]. They are completely generalizedin [F4, Chapter 2] for embedded simple normal crossing pairs. However,[F4] does not recover [Ka, Theorem 4.3]. Compare the arguments in[F4, Chapter 2] with Kawamata’s ones. The reader can find a slight
generalization of Kawamata’s theorem and some other applications ofour methods in [F3] and [F5].
We summarize the contents of this paper. In Section 2, we will givean alternate proof of Kawamata’s theorem. By using Ambro’s formula, we will reduce Kawamata’s theorem to a reformulated versionof Kawamata–Shokurov base point free theorem. Section 3 is an appendix, where we will quickly review Ambro’s formula for the reader’sconvenience. In Section 4, we will prove Kawamata’s theorem for varieties in class C, which is [N2, Theorem 5.5]. We separate this section
from Section 2 in order not to make needless confusion. In the final
section: Section 5, we will make some comments on the topics related
to Kawamata’s theorem for the coming generation.

Acknowledgments. I was partially supported by The Sumitomo Foundation, The Inamori Foundation, and by the Grant-in-Aid for Young
Scientists (A) ♯20684001 from JSPS. I would like to thank Professors
Yujiro Kawamata and Noboru Nakayama for answering my questions.
I thank Professors Daisuke Matsushita and Shigefumi Mori for encouraging me during the preparation of this paper.

We will work over an algebraically closed field k of characteristic
zero throughout this paper. We adopt the language of b-divisors anduse the standard notation of the log minimal model program. See, forexample, [C].

ON KAWAMATA’S THEOREM

3

2. Proof of Kawamata’s theorem
The following theorem is a reformulation of Kawamata–Shokurov
base point free theorem. The original proof works without any changes(cf. [KMM, Theorem 3-1-1]).
Theorem 2.1 (Base Point Free Theorem). Let (X, B) be a sub klt pair,let π : X → S be a proper surjective morphism of normal varieties andD a π-nef Cartier divisor on X. Assume the following conditions:(1) rD − (KX + B) is nef and big over S for some positive integerr, and(2) π∗ OX (pA(X, B)q + jD) ⊆ π∗ OX (jD) for any positive integerj, where A(X, B) is the discrepancy Q-b-divisor and D is the
Cartier closure of D (see [C, Example 2.3.12 (1) (3)]).
Then mD is π-generated for m ≫ 0, that is, there exists a positive
integer m0 such that for any m ≥ m0 the natural homomorphismπ ∗ π∗ OX (mD) → OX (mD) is surjective.
Before we go to the proof of Theorem 1.1, let us recall the definitionof abundant divisors, which are called good divisors in [Ka]. See [KMM,§6-1].
Definition 2.2 (Abundant divisor). Let X be a complete normal variety and D a Q-Cartier nef divisor on X. We define the numerical
Iitaka dimension to be
ν(X, D) = max{e; D e 6≡ 0}.
′This means that D e · S = 0 for any e′ -dimensional subvarieties S of X
with e′ > e and there exists an e-dimensional subvariety T of X suchthat D e · T > 0. Then it is easy to see that κ(X, D) ≤ ν(X, D), whereκ(X, D) denotes Iitaka’s D-dimension. A nef Q-divisor D is said to beabundant if the equality κ(X, D) = ν(X, D) holds. Let π : X → S bea proper surjective morphism of normal varieties and D a Q-Cartierdivisor on X. Then D is said to be π-abundant if D|Xη is abundant,
where Xη is the generic fiber of π.
Proof of Theorem 1.1. If H − (KX + B) is π-big, then the statement
follows from the original Kawamata–Shokurov base point free theorem.
Thus, from now on, we assume that H − (KX + B) is not π-big. Then
there exists a diagramf
Y −−−→ Z



ϕ
µy
y
X −−−→ Sπ

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OSAMU FUJINO

which satisfies the following conditions (see [KMM, Proposition 6-1-3
and Remark 6-1-4] or [N1, Lemma 6]):(i) µ, f and ϕ are projective morphism,(ii) Y and Z are non-singular varieties,(iii) µ is a birational morphism and f is a surjective morphism having connected fibers,(iv) there exists a ϕ-nef and ϕ-big Q-divisor M0 on Z such thatµ∗ (H − (KX + B)) ∼Q f ∗ M0 ,and(v) there is a ϕ-nef Q-divisor D on Z such thatµ∗ H ∼Q f ∗ D.
Note that f : Y → Z is the Iitaka fibration with respect to H − (KX +B) over S. We put KY + BY = µ∗ (KX + B) and HY = µ∗ H. We
note that (Y, BY ) is not necessarily klt but sub klt. Thus, we have
HY − (KY + BY ) ∼Q f ∗ M0 (resp. HY ∼Q f ∗ D), where M0 (resp. D)
is a ϕ-nef and ϕ-big (resp. ϕ-nef) Q-divisor as we saw in (iv) and (v).
Furthermore, we can assume that D and H are Cartier divisors andHY ∼ f ∗ D by replacing D and H by sufficiently divisible multiples. If
we need, we modify Y and Z birationally and can assume the following
conditions:(1) KY + BY ∼Q f ∗ (KZ + BZ + M), where BZ is the discriminant
Q-divisor of (Y, BY ) on Z and M is the moduli Q-divisor on Z,(2) (Z, BZ ) is a sub klt pair,(3) M is a ϕ-nef Q-divisor on Z,(4) ϕ∗ OZ (pA(Z, BZ )q+jD) ⊆ ϕ∗ OZ (jD) for every positive integerj, and(5) D − (KZ + BZ ) is ϕ-nef and ϕ-big.
Indeed, let P ⊂ Z be a prime divisor. Let aP be the largest real number
t such that (Y, BY + tf ∗ P ) is sub lc over the generic point of P . It is
obvious that aP = 1 for all but finitely many prime divisors P of Z. We
note that aP is a positive rational number for any P . The discriminant
Q-divisor on Z defined by the following formulaX
BZ =(1 − aP )P.
P

We note that xBZ y ≤ 0. By the properties (iv) and (v), we can write
KY + BY ∼Q f ∗ (M1 )ON KAWAMATA’S THEOREM

5

for a Q-Cartier divisor M1 on Z. We define M = M1 − (KZ + BZ )and call it the moduli Q-divisor on Z, where BZ is the discriminant Qdivisor defined above. Note that M is called the log-semistable part in[FM, Section 4]. So, the condition (1) obviously holds by the definitionsof the discriminant Q-divisor BZ and the moduli Q-divisor M. If wetake birational modifications of Y and Z suitably, we have that M is
ϕ-nef and (Z, BZ ) is sub klt. Thus we obtain (2) and (3). For thedetails, see [A1, Theorems 0.2 and 2.7] or Theorem 3.2 below. We notethe following lemma (cf. [A1, Lemma 6.2]), which we need to apply[A1, Theorems 0.2 and 2.7] or Theorem 3.2 to f : Y → Z (see thecondition (2) in 3.1).
Lemma 2.3. We have rankf∗ OY (pA(Y, BY )q) = 1.
Proof of the lemma. Since OZ ≃ f∗ OY ⊆ f∗ OY (pA(Y, BY )q), we know
rankf∗ OY (pA(Y, BY )q) ≥ 1. Without loss of generality, we can shrink
S and assume that S is affine. Let A be a ϕ-very ample divisor suchthat f∗ OY (pA(Y, BY )q) ⊗ OZ (A) is ϕ-generated. Since M0 is a ϕ-big
Q-divisor on Z, we have OZ (A) ⊂ OZ (mM0 ) for a sufficiently large anddivisible integer m. We note that π∗ µ∗ OY (pA(Y, BY )q + f ∗ (mM0 )) ≃π∗ µ∗ OY (f ∗ (mM0 )), where f ∗ (mM0 ) is the Cartier closure of f ∗ (mM0 )(see [C, Example 2.3.12 (1)]). It is because µ∗ (H − (KX + B)) =HY − (KY + BY ) ∼Q f ∗ M0 . Therefore,ϕ∗ (f∗ OY (pA(Y, BY )q) ⊗ OZ (A))⊆ ϕ∗ (f∗ OY (pA(Y, BY )q) ⊗ OZ (mM0 ))≃ ϕ∗ OZ (mM0 ).

So, rankf∗ OY (pA(Y, BY )q) ≤ 1. We finish the proof of the lemma. 

We know the following lemma by Lemma 9.2.2 and Proposition 9.2.3
in [A2] (see also Theorem 3.2 (a) below).
Lemma 2.4. We have OZ (pA(Z, BZ )q + jD) ⊆ f∗ OY (pA(Y, BY )q +jHY ) for any integer j.

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OSAMU FUJINO

Pushing forward it by ϕ, we obtain thatϕ∗ OZ (pA(Z, BZ )q + jD) ⊆ ϕ∗ f∗ OY (pA(Y, BY )q + jHY )≃ π∗ µ∗ OY (pA(Y, BY )q + jHY )
≃
≃
≃
≃
≃

π∗ OX (pA(X, B)q + jH)
π∗ OX (jH)π∗ µ∗ OY (jHY )ϕ∗ f∗ OY (jHY )ϕ∗ OZ (jD)

for any integer j. Thus, we have (4). The relation HY − (KY + BY ) ∼Qf ∗ (D − (KZ + BZ + M)) implies that D − (KZ + BZ + M) is ϕ-nef
and ϕ-big. By (3), M is ϕ-nef. Therefore, D − (KZ + BZ ) = D −(KZ + BZ + M) + M is ϕ-nef and ϕ-big. It is the condition (5). Apply
Theorem 2.1 to D on (Z, BZ ). Then we obtain that D is ϕ-semi-ample.
This implies that H is π-semi-ample. We finish the proof.
The following corollaries are obvious by Theorem 1.1.
Corollary 2.5. Let (X, B) be a klt pair, let π : X → S be a propersurjective morphism of normal varieties. Assume that KX + B is π-nef
and π-abundant. Then KX + B is π-semi-ample.
Corollary 2.6. Let X be a complete normal variety such that KX ∼Q
0. Assume that X has only klt singularities. Let H be a nef andabundant Q-Cartier divisor on X. Then H is semi-ample.
3. Appendix: Quick review of Ambro’s formula
In this appendix, we quickly review Ambro’s formula. For the details,
see the original paper [A1] or Kollár’s survey article [Ko].
3.1. Let f : X → Y be a proper surjective morphism of normal varieties. Let p : Y → S be a proper morphism onto a variety S. Assume
the following conditions:(1) KX +B is Q-Cartier and (X, B) is sub klt over the generic pointof Y ,(2) rankf∗ OX (pA(X, B)q) = 1, and(3) KX + B ∼Q,f 0.
By (3), we can write KX + B ∼Q f ∗ D for some Q-Cartier divisor D on
Y . Let BY be the discriminant Q-divisor on Y . For the definition, see
the proof of Theorem 1.1. We put MY = D−(KY +BY ) and call MY themoduli Q-divisor on Y . Then we have KX + B ∼Q f ∗ (KY + BY + MY ).

ON KAWAMATA’S THEOREM

7

Let σ : Y ′ → Y be a proper birational morphism from a normal variety
Y ′ . Then we obtain the following commutative diagram:µX ←−−− X ′



f ′
fy
y
Y ←−−− Y ′σsuch that(i) µ is birational morphism from a normal variety X ′ ,(ii) we put KX ′ +B ′ = µ∗ (KX +B). Then we can write KX ′ +B ′ ∼Q
f ′ ∗ (KY ′ + BY ′ + MY ′ ), where BY ′ is the discriminant Q-divisor
on Y ′ associated to f ′ : X ′ → Y ′ .
Ambro’s theorem [A1, Theorems 0.2 and 2.7] says
Theorem 3.2. If we choose Y ′ appropriately, then we have the following properties for every proper birational morphism ν : Y ′′ → Y ′ froma normal variety Y ′′ .
(a) KY ′ + BY ′ is Q-Cartier and ν ∗ (KY ′ + BY ′ ) = KY ′′ + BY ′′ . In
particular, A(Y ′ , BY ′ )Y ′′ = −BY ′′ .
(b) The moduli Q-divisor MY ′ is nef over S and ν ∗ (MY ′ ) = MY ′′ .
We note that the nefness of the moduli Q-divisor follows from Fujita–
Kawamata’s semi-positivity theorem. It is a consequence of the theoryof variation of Hodge structures. For the details, see, for example, [M,Section 5], [F1, Section 5], or [Ko].
4. Kawamata’s theorem for varieties in class C
In this section, we treat Nakayama’s theorem: [N2, Theorem 5.5],
which is Kawamata’s theorem for varieties in class C. First, let us
recall the definition of the varieties in class C.
Definition 4.1 (Class C). A compact complex variety in class C is avariety which is dominated by a compact Kähler manifold. It is known
that X is in class C if and only if X is bimeromorphically equivalent
to a compact Kähler manifold.
Next, we recall the definitions of the Kähler cone and the nef line
bundles on a compact Kähler manifold.
Definition 4.2 (Kähler cone). Let Y be a d-dimensional compactKähler manifold. We define the Kähler cone KC(Y ) of Y to be theset{[ω] ∈ H 1,1 (Y, R); ω is a Kähler form on Y.},

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OSAMU FUJINO

where H 1,1(Y, R) := H 2 (Y, R) ∩ H 1,1(Y, C). Then KC(Y ) is an open
convex cone in H 1,1 (Y, R). KC(Y ) is the closure of KC(Y ) in H 1,1 (Y, R).
Finally, we recall the definitions of the quasi-nef line bundles, thehomological Kodaira dimension, and the quasi-nef and abundant line
bundles, which were introduced in [N2].
Definition 4.3 (cf. [N2, Definition 2.4]). Let L be a line bundle on acompact Kähler manifold Y . L is said to be nef if the real first Chernclass c1 (L) is contained in KC(Y ).
Remark 4.4. For a new numerical characterization of the Kähler coneof a compact Kähler manifold, see [DP, Main Theorem 0.1]. A nef line
bundle on a compact Kähler manifold can be characterized numericallyby [DP, Corollaries 0.3 and 0.4].
Definition 4.5 (cf. [N2, Definition 2.6]). Let X be a compact complex
variety in class C. A line bundle L on X is called quasi-nef if there
exists a bimeromorphic morphism µ : Y → X from a compact Kähler
manifold Y such that µ∗ L is nef.
Definition 4.6 (cf. [N2, Definition 2.9]). Let L be a quasi-nef line
bundle on X ∈ C. Take a bimeromorphic morphism µ : Y → X froma compact Kähler manifold Y such that µ∗ L is nef. Then we defineκhom (L) := max{l ≥ 0; 0 6= c1 (µ∗ L)l ∈ H l,l (Y, R)}and call it the homological Kodaira dimension of L. It is well-defined,
because it is independent of the choice of Y .
Definition 4.7 (cf. [N2, Definition 2.11]). Let L be a line bundle on acompact complex variety X in class C. L is said to be big if κ(X, L) =dim X. If L is quasi-nef and κ(X, L) = κhom (L), then L is called
abundant.
Now, we sate the main theorem of this section. It is nothing but[N2, Theorem 5.5]. The reader can find some applications of Theorem
4.8 in [COP].
Theorem 4.8 (cf. [N2, Theorem 5.5]). Let X be a compact normal
complex variety in class C, B an effective Q-divisor on X, and H a
Q-Cartier divisor on X. Then H is semi-ample under the following
conditions:(1) (X, B) is klt,
(2) H is quasi-nef,(3) H − (KX + B) is quasi-nef and abundant, andON KAWAMATA’S THEOREM

9(4) κhom (aH − (KX + B)) = κhom (H − (KX + B)) and κ(X, aH −(KX + B)) ≥ 0 for some a ∈ Q with a > 1.
Sketch of the proof. First, we recall Nakamaya’s result.
Lemma 4.9 ([N2, Proposition 2.14 and Corollary 2.16]). There exists
the following diagramµ

f

X ←−−− Y −−−→ Z,
where(a) Y is a compact Kähler manifold and µ is a bimeromorphic morphism,(b) Z is a smooth projective variety,(c) f is surjective and has connected fibers,(d) there exists a nef and big Q-divisor M0 on Z such thatµ∗ (H − (KX + B)) ∼Q f ∗ M0 ,and(e) there is a nef Q-divisor D on Z such thatµ∗ H ∼Q f ∗ D.
We note that Z is a smooth projective variety.
Let f : Y → Z be the proper surjective morphism from a compactKähler manifold Y to a normal projective variety Z obtained in Lemma
4.9. Let BY be a Q-divisor on Y such that KY + BY = µ∗ (KX + B).
Then we have the following properties:(1) KY + BY is Q-Cartier and (Yz , Bz ) is sub klt for general z ∈ Z,
where Yz = f −1 (z) and Bz = BY |Yz ,(2) rankf∗ OY (pA(Y, BY )q) = 1, and(3) KY + BY ∼Q,f 0.
We note that (1) is obvious by the definition of BY , (2) follows fromthe proof of Lemma 2.3, and (3) is also obvious by Lemma 4.9. Under
these conditions (1), (2), and (3), Ambro’s theorem (see [A1, Theorems
0.2 and 2.7] or Theorem 3.2) holds if we adopt [N3, 3.7. Theorem (4)]in the proof of Ambro’s theorem. Note that it is not difficult to modify
the arguments in [A1] for our setting. More explicitly, let σ : Z ′ → Z
be a proper birational morphism from a normal projective variety Z ′ .
If we choose Z ′ appropriately, then we have the following properties
for every proper birational morphism ν : Z ′′ → Z ′ from a normal
projective variety Z ′′ .
(a) KZ ′ + BZ ′ is Q-Cartier and ν ∗ (KZ ′ + BZ ′ ) = KZ ′′ + BZ ′′ , where
BZ ′ and BZ ′′ are the discriminant Q-divisors. In particular,A(Z ′ , BZ ′ )Z ′′ = −BZ ′′ .

10

OSAMU FUJINO(b) The moduli Q-divisor MZ ′ is nef and ν ∗ (MZ ′ ) = MZ ′′ .
For the details and the notation, see Section 3.
By applying Ambro’s theorem to f : Y → Z, the proof of Theorem
1.1 works without any modifications. We note that Z is a projective
variety. Thus, we obtain the semi-ampleness of H.

5. Comments for the coming generation
The results in [Ka] had already been used in various papers. We
think that almost all the papers just used the main results of [Ka],that is, Theorems 1.1 and 6.1 in [Ka]. Therefore, by this paper, agreat part of the trouble caused by [Ka, Theorem 4.3] were removed.
Unfortunately, some authors used the arguments in [Ka]. We give somecomments for the coming generation.
5.1. As we pointed out in [F2, Remark 3.10.3], the proof of [Ka, Theorem 4.3] is not completed (see also [KMM, Theorem 6-1-6]). We recall
the trouble in [Ka] here for the reader’s convenience.
We use the same notation as in the proof of Theorem 4.3 in [Ka].
By [Ka, Theorem 3.2], ′E1p,q → ′′E1p,q are zero for all p and q. It does
not directly say that
H i (X, OX (−pLq)) → H i (D, OD (−pLq))
are zero for all i. So, the proofs of Theorems 4.4, 4.5, 5.1, and 6.1 in[Ka] do not work. It is because everything depends on Theorem 4.3
in [Ka]. Thus, we have no rigorous proofs for [KMM, Theorems 6-1-8,
6-1-9].
If someone corrects the proof of [Ka, Theorem 4.3], then the following
comments are unnecessary.
5.2. In [N1], Nakayama obtained the relative version of Kawamata’s
theorem. The proof given there heavily depends on Kawamata’s original proof. So, it does not work by the trouble in [Ka, Theorem 4.3].
Of course, [N1, Theorem 5] is true by our main theorem: Theorem 1.1.
5.3. Section 5 in [N2] contains the same trouble. It is because it depends on Kawamata’s paper [Ka]. In Section 4, we give a rigorous
proof to [N2, Theorem 5.5].
5.4. In [Fk], Fukuda obtained a slight generalization of Kawamata’s
theorem. See [Fk, Proposition 3.3]. In the final step of the proof of [Fk,
Proposition 3.3], Fukuda used [Ka, Theorem 5.1]. So, Fukuda’s original
proof also has some troubles by [Ka, Theorem 4.3]. Fortunately, we canprove a slight generalization of [Fk, Proposition 3.3] in [F3, Section 6].

ON KAWAMATA’S THEOREM

11

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N. Nakayama, Invariance of the plurigenera of algebraic varieties underminimal model conjectures, Topology 25 (1986), no. 2, 237–251.
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Department of Mathematics, Kyoto University
E-mail address: fujino@math.kyoto-u.ac.jp