# Note on Quasi-Numerically Positive Log Canonical Divisors

Authors: Shigetaka Fukuda
Journal: Journal of Mathematics
 (2015)

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## Abstract

We propose a subconjecture that implies the semiampleness conjecture for quasi-numerically positive log canonical divisors and prove the ampleness in some elementary cases.


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## Body

## 1. Introduction

In this note, every algebraic variety is defined over the fieldC of complex numbers. We follow the terminology and notation in [1].Definition 1.
LetD be a Q-Cartier Q-divisor on a projective variety X. The divisor D isnumerically positive (nup, for short), if (
D
,
C
)
>
0 for every curve C on X. The divisor D isquasi-numerically positive (quasi-nup, for short), if it is nef and if there exists a union F of at most countably many prime divisors on X such that (
D
,
C
)
>
0 for every curve C
⊈
F (i.e., if D is nef and if (
D
,
C
)
>
0 for every very general curve C).Remark 2.
The quasi-nup divisors are the divisors “of maximal nef dimension” in the terminology of the “Eight Authors” [2].Ambro [3] and Birkar et al. [4] reduced the famous log abundance conjecture to the termination conjecture for log flips and the semiampleness conjecture (Conjecture 4) for quasi-nup log canonical divisors K
X
+
Δ, in the category of Kawamata log terminal (klt, for short) pairs. In Section 2 we propose a subconjecture (Subconjecture 1) that implies the semiampleness Conjecture 4.Remark 3.
We state the history in detail (c.f. [5]). In the category of klt pairs (
X
,
Δ
), Fukuda [6] (2002) reduced the log abundance to the existence and termination of log flips, the existence of log canonical bundle formula, and the semiampleness of quasi-nup log canonical divisors, by using the numerically trivial fibrations (see [7]; see also [2]) due to Tsuji and the semiampleness criterion (see [8, 9]; see also Fujino [10]) for log canonical divisors due to Kawamata-Nakayama. Ambro [3] gave and proved the celebrated log canonical bundle formula. The existence of log flips is now the theorem [4] due to Birkar et al. This history is along the line of Reid’s philosophy stated in the famous Pagoda paper [11].
We also note two relevant theorems. In Fukuda [12] (base point free theorem of Reid type, 1999), we proved that if the log canonical divisor on a Q-factorial divisorial log terminal variety is nef and log big, then it is semiample. In Fukuda [13] (2011), we proved that if the log canonical divisor on a klt variety is numerically equivalent to some semiample Q-divisor, then it is semiample.There is another approach to the semiampleness Conjecture4. Let (
X
,
Δ
) be a klt pair whose log canonical divisor K
X
+
Δ is quasi-nup. Hacon and McKernan (Lazic [14], Theorem A.6) considered embedding (
X
,
Δ
) into some log canonical pair (
X
¯
,
Δ
¯
) so that d
i
m
X
¯
=
d
i
m
X
+
1 and Δ
¯
≥
X, that the log canonical divisor K
X
¯
+
Δ
¯ is nef and big, that (
K
X
¯
+
Δ
¯
)
X
=
K
X
+
Δ, and that X
¯ is endowed with the birational contraction morphism ϕ
:
X
¯
→
Y
¯ that contracts the prime divisor X
(
=
E
x
c
(
ϕ
)
) to some point. In Section 3, motivated by this consideration, we prove the ampleness (Theorem 18) for log canonical pairs in some elementary cases.In AppendixA, we survey the celebrated extension theorem [15] which is recently proven by Demailly-Hacon-Păun.In AppendixB, we give a straightforward proof to the theorem due to Boucksom et al. [16] and Birkar and Hu [17] and Cacciola [18] that, for every divisorial log terminal pair whose log canonical divisor is strongly log big, the log canonical ring is finitely generated.


## 2. Subconjecture for klt Pairs

Conjecture 4.
Let(
X
,
Δ
) be a Kawamata log terminal pair such that X is projective. If the log canonical divisor K
X
+
Δ is quasi-nup, then it is semiample.We give an approach towards the above-mentioned semiampleness conjecture in this section. The approach repeats the process of finding some(
K
X
+
Δ
)-trivial curve that generates a (
K
X
+
Γ
)-extremal ray for some other klt pair (
X
,
Γ
) and contracting this extremal ray. The process would terminate at the ample log canonical divisor. To run the process, it is important not to require the Q-factoriality of X.Definition 5.
One definesN
E
¯
D
=
0
(
X
)
≔
{
l
∈
N
E
¯
(
X
)
∣
t
h
e
i
n
t
e
r
s
e
c
t
i
o
n
n
u
m
b
e
r
(
D
,
l
)
=
0
} and N
E
¯
D
≥
0
(
X
)
≔
{
l
∈
N
E
¯
(
X
)
∣
(
D
,
l
)
≥
0
} for a Q-Cartier Q-divisor D on X.Subconjecture 1.
Let(
X
,
Δ
) be a Kawamata log terminal pair such that X is projective. Suppose that the log canonical divisor K
X
+
Δ is not ample but quasi-nup. Then there exists an effective Q-Cartier divisor E such that the intersection number (
E
,
l
)
<
0 for some class l
∈
N
E
¯
K
X
+
Δ
=
0
(
X
).Procedure 1.
Let(
X
,
Δ
) be a Kawamata log terminal pair such that X is projective. Suppose that the log canonical divisor K
X
+
Δ is not ample but quasi-nup. Assume the existence of an effective Q-Cartier divisor E on X and a member l of N
E
¯
K
X
+
Δ
=
0
(
X
) such that the intersection number (
E
,
l
)
<
0. Let ϵ be a sufficiently small positive rational number. We can write this class l in the form l
=
l
0
+
l
1
+
l
2
+
⋯
+
l
p
(
p
≥
1
), where l
0
∈
N
E
¯
K
X
+
Δ
+
ϵ
E
≥
0
(
X
) and R
+
l
i
(
i
≥
1
) are distinct (
K
X
+
Δ
+
ϵ
E
)-extremal rays. Then (
K
X
+
Δ
,
l
1
)
=
0, because K
X
+
Δ is nef and (
K
X
+
Δ
,
l
)
=
0. We consider the birational contraction morphism ϕ
:
X
→
X
1 of the (
K
X
+
Δ
+
ϵ
E
)-extremal ray R
+
l
1. Put Δ
1
≔
ϕ
∗
(
Δ
). We note that the Picard number ρ
(
X
1
)
=
ρ
(
X
)
-
1, that K
X
+
Δ
=
ϕ
∗
(
K
X
1
+
Δ
1
), that (
X
1
,
Δ
1
) is Kawamata log terminal, and that K
X
1
+
Δ
1 is quasi-nup. Remark that we can permit each of the divisorial-contraction case and the small-contraction case, because we do not require the Q-factoriality of X
1.Procedure1 relates Subconjecture 1 to Conjecture 4. The following is the main result of this section.Theorem 6.
Subconjecture1 implies Conjecture 4.Proof.
Let(
X
,
Δ
) be a Kawamata log terminal pair such that X is projective and the log canonical divisor K
X
+
Δ is quasi-nup. If Subconjecture 1 is true, then, by repeating Procedure 1, we obtain a Kawamata log terminal pair (
X
′
,
Δ
′
) with the birational morphism ψ
:
X
→
X
′ such that K
X
+
Δ
=
ψ
∗
(
K
X
′
+
Δ
′
) and K
X
′
+
Δ
′ is ample, because the Picard numbers decrease 1 by 1 in the process of contraction of extremal rays.Corollary 7.
Subconjecture1 and the termination conjecture for log flips imply the log abundance conjecture for klt pairs.Proof.
See Remark3 and Theorem 6.Remark 8.
From the corollary above and the existence theorem [19] for extremal rational curves by Kawamata, we can say that the log abundance conjecture is the existence problem for some kind of rational curves, modulo the termination of log flips.We show that Subconjecture1 is a part of Conjecture 4.Lemma 9.
Let(
X
,
Δ
) be a Kawamata log terminal pair such that X is projective. Suppose that K
X
+
Δ is not ample but quasi-nup and semiample. Then there exists an effective Q-Cartier divisor E such that the intersection number (
E
,
l
)
<
0 for some class l
∈
N
E
¯
K
X
+
Δ
=
0
(
X
).Proof.
Consider the surjective morphismϕ
:
X
→
Y
(
=
Φ
|
k
(
K
X
+
Δ
)
|
(
X
)
) induced by the linear system |
k
(
K
X
+
Δ
)
| for a sufficiently large and divisible integer k. This morphism ϕ becomes birational, because of the Stein factorisation theorem and the fact that the pullbacks of ample divisors by finite morphisms are ample. Then k
(
K
X
+
Δ
)
=
ϕ
∗
H for an ample divisor H on Y. By the Kodaira Lemma, if m is sufficiently large and divisible, then m
ϕ
∗
H
=
A
+
E for some ample divisor A and some effective divisor E. For every ϕ-exceptional curve C, we obtain the inequality that (
E
,
C
)
<
0, because (
m
ϕ
∗
H
,
C
)
=
0 and (
A
,
C
)
>
0. Here the class [
C
] belongs to N
E
¯
K
X
+
Δ
=
0
(
X
).Proposition 10.
Conjecture4 implies Subconjecture 1.Proof.
Lemma9 gives the assertion.


## 3. Log Canonical Pairs in Some Elementary Cases

We prove the ampleness for log canonical pairs in some elementary cases.Assumption 11.
Letf
:
X
→
Y be a birational morphism between normal projective varieties of dimension n such that E
≔
E
x
c
(
f
) is a prime divisor and let (
X
,
Δ
) and (
X
,
Δ
+
E
) be divisorial log terminal pairs. Assume that K
X
+
Δ
+
E is nup.Proposition 12.
Under Assumption11, the divisor K
X
+
Δ
+
(
1
-
ϵ
)
E is nef for every small number ϵ
>
0.Proof.
The result [19] of Kawamata for klt pairs and its variant (see [20], Proposition 1) of Shokurov for dlt pairs give the boundedness of the length of (
K
X
+
Δ
)-extremal rays. By using the argument in [21], we have ν
≔
i
n
f
{
(
K
X
+
Δ
+
E
,
C
)
/
-
(
K
X
+
Δ
,
C
)
∣
C is an extremal rational curve for K
X
+
Δ
}
>
0. Thus K
X
+
Δ
+
E
+
ν
(
K
X
+
Δ
) is nef.Assumption 13.
Furthermore assume thatK
Y
+
f
∗
Δ is Q-Cartier, that -
E is f-ample, and that, in the case where f
(
E
) is not a point, the divisor K
Y
+
f
∗
Δ
f
(
E
) is ample.Remark 14.
IfY is Q-factorial, then the condition that -
E is f-ample in Assumption 13 is automatically satisfied, under Assumption 11 (cf. Kollár and Mori [22], Lemma 2.62).Definition 15.
Under Assumptions11 and 13, one defines the number λ by the equation K
X
+
Δ
+
E
=
f
∗
(
K
Y
+
f
∗
Δ
)
+
(
1
+
λ
)
E. Then 1
+
λ
<
0, because K
X
+
Δ
+
E is nup.Proposition 16.
Under Assumptions11 and 13, the divisor K
X
+
Δ
+
E is big.Proof.
Assume thatK
X
+
Δ
+
(
1
-
ϵ
)
E
=
f
∗
(
K
Y
+
f
∗
Δ
)
+
(
1
+
λ
-
ϵ
)
E is not big for every small number ϵ
>
0. Thus its self-intersection number is zero for every ϵ from Proposition 12. Therefore (
-
E
)
d
i
m
E
-
d
i
m
f
(
E
)
·
(
f
∗
(
K
Y
+
f
∗
Δ
)
d
i
m
f
(
E
)
·
E
)
=
0. This contradicts the f-ampleness of -
E. Consequently K
X
+
Δ
+
(
1
-
ϵ
)
E is big for every small number ϵ
>
0 and so is K
X
+
Δ
+
(
1
-
ϵ
)
E
+
ϵ
E.Proposition 17.
Under Assumptions11 and 13, the divisor (
K
X
+
Δ
+
E
)
|
E is ample.Proof.
The divisorf
∗
(
K
Y
+
f
∗
Δ
)
|
E
-
ϵ
E
|
E on E is ample for every small number ϵ
>
0  (cf. [22], Proposition 1.45). We also recall that f
∗
(
K
Y
+
f
∗
Δ
)
+
(
1
+
λ
-
ϵ
)
E
=
K
X
+
Δ
+
(
1
-
ϵ
)
E is nef by Proposition 12. Thus (
K
X
+
Δ
+
E
)
|
E
=
(
f
∗
(
K
Y
+
f
∗
Δ
)
+
(
1
+
λ
)
E
)
|
E is ample, from the inequality -
ϵ
>
1
+
λ
>
1
+
λ
-
ϵ.We state the main result of this section.Theorem 18.
Under Assumptions11 and 13, the divisor K
X
+
Δ
+
E is ample if and only if (
(
K
X
+
Δ
+
E
)
|
Γ
)
d
i
m
Γ
>
0 for every minimal log canonical (i.e., minimal non-klt) center Γ with respect to the pair (
X
,
Δ
+
E
) such that Γ
∩
E
=
∅.For proof, we cite the following ampleness result.Proposition 19 (see [23]).
Let(
M
,
S
) be a divisorial log terminal pair which is not Kawamata log terminal such that M is projective. Assume that the log canonical divisor K
M
+
S is nup and that (
(
K
M
+
S
)
|
Γ
)
d
i
m
Γ
>
0 for every minimal log canonical (i.e., minimal non-klt) center Γ with respect to the pair (
M
,
S
). Then K
M
+
S is ample.Proof of Theorem18.
The “only if” part is trivial. So we prove the “if” part.
For every minimal log canonical centerΓ with respect to (
X
,
Δ
+
E
) such that Γ
∩
E
≠
∅, we have that Γ
⊂
E from Ambro (see [24], Proposition 3.3) because E is a log canonical center with respect to (
X
,
Δ
+
E
).
Thus(
(
K
X
+
Δ
+
E
)
|
Γ
)
d
i
m
Γ
>
0 for every minimal log canonical center Γ with respect to the pair (
X
,
Δ
+
E
) by Proposition 17.
Consequently Proposition19 implies that K
X
+
Δ
+
E is ample.Example 20.
LetP
n
(
n
≥
3
) be a projective space with homogeneous coordinate (
x
0
:
x
1
:
x
2
:
⋯
:
x
n
) and hyperplane G. We consider the hypersurface Y (⊂
P
n) defined by the irreducible homogeneous equation x
1
l
+
x
2
l
+
x
3
l
+
⋯
+
x
m
l
=
0
(
3
≤
m
≤
n
)
(
l
≥
n
+
1
). We note that Y is normal and that K
Y
=
(
-
(
n
+
1
)
G
+
l
G
)
|
Y
=
(
l
-
(
n
+
1
)
)
G
|
Y is Cartier. Blow up P
n at the subspace {
x
1
=
x
2
=
⋯
=
x
m
=
0
} and obtain the morphism ϕ
:
P
′
→
P
n and the exceptional divisor F. Let X be the strict transform of Y by ϕ. We note that X is nonsingular. We have K
P
′
=
ϕ
∗
(
-
(
n
+
1
)
G
)
+
(
m
-
1
)
F. Thus K
X
=
(
K
P
′
+
X
)
|
X
=
(
ϕ
∗
(
-
(
n
+
1
)
G
)
+
(
m
-
1
)
F
+
ϕ
∗
(
l
G
)
-
l
F
)
|
X
=
(
ϕ
∗
(
l
-
(
n
+
1
)
)
G
-
(
l
-
(
m
-
1
)
)
F
)
|
X. Then K
X
+
(
n
-
m
+
1
)
(
ϕ
∗
G
)
|
X
+
F
|
X
=
(
l
-
m
)
(
ϕ
∗
G
-
F
)
|
X is nef, because the linear system |
ϕ
∗
G
-
F
| is base-point-free. Consequently K
X
+
(
ϕ
∗
(
n
-
m
+
2
)
G
)
|
X
+
F
|
X
=
(
(
l
-
m
)
ϕ
∗
G
-
(
l
-
m
)
F
+
ϕ
∗
G
)
|
X is nup because -
F is ϕ-ample. Let H be the restriction of a general member of |
(
n
-
m
+
2
)
G
| to Y. We put f
≔
ϕ
|
X and Δ
≔
f
∗
H. Then E
≔
E
x
c
(
f
)
=
F
|
X is a smooth prime divisor and -
E
=
-
F
|
X is f-ample. We note that K
X
+
Δ
+
E is nup and that (
K
Y
+
f
∗
Δ
)
|
f
(
E
)
=
(
l
-
(
n
+
1
)
+
(
n
-
m
+
2
)
)
G
|
f
(
E
) is ample. When Δ
∩
E
=
∅ (i.e., n
=
m), the divisor (
K
X
+
Δ
+
E
)
|
Δ
=
(
l
-
(
n
+
1
)
+
2
)
ϕ
∗
G
|
Δ is ample. Lastly Theorem 18 implies that K
X
+
Δ
+
E is ample.



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*Source: 841746-2015-05-31.xml*