On numerically effective log canonical divisors

PII. S0161171202012450 http://ijmms.hindawi.com © Hindawi Publishing Corp.

ON NUMERICALLY EFFECTIVE LOG CANONICAL DIVISORS

SHIGETAKA FUKUDA

Received 18 March 2001 and in revised form 9 October 2001

Let(X,∆)be a 4-dimensional log variety which is proper over the field of complex numbers and with only divisorial log terminal singularities. The log canonical divisorKX+∆is semiample, if it is numerically effective (NEF) and the Iitaka dimensionκ(X, KX+∆)is strictly positive. For the proof, we use Fujino’s abundance theorem for semi-log canonical threefolds.

2000 Mathematics Subject Classification: 14E30, 14J35, 14C20.

1. Introduction. In this paper, every variety is proper over the fieldCof complex numbers. We follow the notation and terminology of [11].

LetXbe a normal algebraic variety and let∆=di∆ibe aQ-divisor with 0≤di≤1

onXsuch that the log canonical divisorKX+∆isQ-Cartier. We call(X,∆)a log pair.

LetDbe a numerically effective (NEF)Q-CartierQ-divisor onX. We define the nu-merical Iitaka dimensionν(X, D):=max{e;(De_{, S) >0 for some subvariety S of}

di-mensioneonX}. The divisorDisabundant if the Iitaka dimensionκ(X, D)equals ν(X, D). If, for some positive integerm, the divisormDis Cartier and the linear sys-tem|mD|is free from base points,Dis said to besemiample.

For a birational morphismf :Y →X between normal algebraic varieties and for a divisor E onX, the symbol f−1

∗ E expresses the strict transform of E by f, and

f−1_{(E)expresses the set-theoretical inverse image. A resolutionµ:Y →Xis said to} be alog resolutionof the log pair(X,∆)if the support of the divisorµ−1

∗ ∆+{E;E

is aµ-exceptional prime divisor}is with only simple normal crossings. The log pair (X,∆)islog terminalif there exists a log resolutionµ:Y →Xsuch thatKY+µ_∗−1∆= µ∗(KX+∆)+aiEiwithai>−1. Moreover, if the exceptional locus Exc(µ)consists

of divisors,(X,∆)is said to bedivisorial log terminal (DLT). Szabó [16] proved that the notions of DLT and wklt in [15] are equivalent. In the case where(X,∆)is log terminal and∆ =0, we say that(X,∆)isKawamata log terminal (KLT).

Log abundance conjecture1.1(cf. [10]). Assume thatXis projective and(X,∆)

is DLT. IfKX+∆is NEF, thenKX+∆is semiample.

This conjecture claims that the concept of “log minimal” (i.e., the log canonical divisor is NEF) should be not only numerical but also geometric. Kawamata [7] and Fujita [5] proved the conjecture in dimX=2 and Keel, Matsuki, and McKernan [10] proved it in dimX=3. (The assumption concerning singularities in their papers is that(X,∆)is log canonical, which is more general than DLT.) Moreover, Fujino proved the following theorem.

Theorem1.2(see [3, Theorem 3.1]). Assume that(X,∆)is DLT anddimX=4. If

KX+∆is NEF and big, thenKX+∆is semiample.

The following two theorems, due to Kawamata, are helpful to deal with the conjec-ture.

Theorem_{1.3(see [8, Theorem 6.1])}. _{Assume that(X,}∆_{)is KLT andKX}+∆_{is NEF.} IfKX+∆is abundant, then it is semiample.

Theorem1.4(see [8, Theorem 7.3], cf. [10, Lemma 5.6]). Assume that(X,∆)is KLT

andKX+∆is NEF. Ifκ(X, KX+∆) >0and the log minimal model and the log abundance conjectures hold in dimensiondimX−κ(X, KX+∆), thenKX+∆is semiample.

In this paper, we try to generalize the above-mentioned theorems and obtain the following main theorem.

Theorem 1.5. Assume that (X,∆) is DLT and dimX =4. If K_X+∆ is NEF and κ(X, KX+∆) >0, thenKX+∆is semiample.

We prove the main theorem (Theorem 1.5) along the lines in the proofs of Theorems

1.2and1.3, using Fujino’s abundance theorem for semi-log canonical threefolds which are not necessarily irreducible. (For the definition of the concept “SDLT” appearing below, seeDefinition 2.7inSection 2.)

Theorem_{1.6(see [2])}. _Let(S,Θ_{)be an SDLT threefold. IfKS}+Θ_{is NEF, thenKS}+Θ is semiample.

Remark1.7. If the log minimal model and the log abundance conjectures hold in

dimension less than or equal ton−1, andTheorem 1.6holds in dimensionn−1, then

Theorem 1.5holds in dimensionn.

2. Preliminaries. In this section, we state notions and results needed in the proof ofTheorem 1.5. The next two propositions are from the theories of the Kodaira-Iitaka dimension and the minimal model, respectively.

Proposition_{2.1(see [6, Theorem 10.3])}. _{LetDbe an effective divisor on a smooth}

varietyY. Suppose that the rational mapΦ|D|:Y→Zis a morphism between algebraic

varieties and that the rational function fieldRat(Φ|mD|(Y ))is isomorphic toRat(Z)for

all positive integerm. ThenRat(Z)is algebraically closed inRat(Y )andκ(W , D|W)=0

Proposition 2.2(see [9, Section 5-1]). Assume that(X_lm,∆_lm)is a log minimal model for aQ-factorial, DLT projective variety(X,∆). Then, every common resolution X←g Y h→Xlmsatisfies the condition that

KY+g−_∗1∆+E≥g∗KX+∆≥h∗KXlm+∆lm

, (2.1)

whereEis the reduced divisor composed of theg-exceptional prime divisors.

The following is a vanishing theorem of Kollár-type.

Theorem2.3(see [12, Theorem 10.13], [8, Theorem 3.2], [1, Section 3.5]). Letf:

X→Ybe a surjective morphism from a smooth projective varietyXto a normal variety Y. LetLbe a divisor onXandDan effective divisor onXsuch thatf (D)≠Y. Assume that(X,∆)is KLT andL−D−(KX+∆)isQ-linearly equivalent tof∗MwhereM is a NEF and bigQ-CartierQ-divisor onY. Then the homomorphismsHi_(X,ᏻ

X(L−D))→ Hi_(X,ᏻ

X(L))are injective for alli.

When we work on the non-KLT locus∆of a log terminal pair(X,∆), we need the following lemma.

Lemma2.4(cf. [6, Proposition 1.43]). LetSbe a reduced scheme andᏲan invertible

sheaf onS. Then, the restriction mapH0_(S,Ᏺ)→H0_{(U ,Ᏺ)is injective for all open dense} subsetUofS.

The following lemma is used to manage cases whereTheorem 2.3cannot be applied (see [10, Section 7]).

Lemma2.5(cf. [4, Section 1.20]). Letf :S→Zbe a surjective morphism between

normal varieties andHZ a Cartier divisor onZ. Iff∗HZ is semiample, then so isHZ.

The set Strata(D)defined below is the set of non-KLT centers for a smooth pair(Y , D).

Definition2.6. LetD=l_i=1D_ibe a reduced simple normal crossing divisor on a

smooth varietyY. We set Strata(D):= {Γ; 1≤i1< i2<···< ik≤l,Γ is an irreducible

component ofDi1∩Di2∩···∩Dik≠∅}.

When we manage the non-KLT locus∆of a DLT pair(X,∆), we need the following notion.

Definition2.7(see [2, Definition 1.1]). LetS be a reduced S2 scheme which is

puren-dimensional and normal crossing in dimension 1. LetΘbe an effectiveQ-Weil divisor such thatKS+ΘisQ-Cartier. LetS=Sibe the decomposition into irreducible

components. The pair(S,Θ)issemi-divisorial log terminal (SDLT)ifSiis normal and (Si,Θ|Si)is DLT for alli.

Proposition_{2.8(see [2, Remark 1.2.(3)], [15, Section 3.2.3], [13, Corollary 5.52])}.

If(X,∆)is DLT, then(∆,Diff(∆−∆))is SDLT.

Proposition3.1(see [8, Theorem 7.3], [10, Lemma 5.6]). Let(X,∆)be a variety

with only log canonical singularities such thatKX+∆is NEF andκ(X, KX+∆) >0. If the log minimal model and the log abundance conjectures hold in dimensiondimX− κ(X, KX+∆), thenκ(X, KX+∆)=ν(X, KX+∆).

In the literature (see [8, Theorem 7.3]), this is proved for KLT pairs. However, the proof is valid for log canonical pairs also. Thus, in the proof below we note only the parts where we have to be careful in reading [8, Proof of Theorem 7.3].

Proof. (See [8, Proof of Theorem 7.3].) ByProposition 2.1, we have a diagram

X←µ Y f→Z (3.1)

with the following properties:

(a) Y and Z are smooth projective varieties. Moreover, Y is a log resolution of (X,∆);

(b) µis birational andf is surjective. The morphismfsatisfies that dimZ=κ(X, KX+∆)andf∗ᏻY=ᏻZ;

(c) KY+µ−_∗1∆+E=µ∗(KX+∆)+Eµ, whereE is the reduced divisor composed of

theµ-exceptional prime divisors andEµis an effectiveQ-divisor;

(d) for a general fiberW=Yzoff,KY|W=KW andκ(W , KW+(µ−_∗1∆+E)|W)=0.

We note that W is smooth and Supp((µ−1

∗ ∆+E)|W) is with only simple normal

crossings.

We apply the log minimal model program to(W , (µ−1

∗ ∆+E)|W)and obtain a log

min-imal model(Wlm,∆lm), whereKWlm+∆lm∼Q0 from the log abundance. We consider a

common resolutionW←ρW σ→WlmofWandWlmsuch thatWis projective. From Proposition 2.2,

ρ∗KW+µ_∗−1∆+E|W=σ∗KW_lm+∆lm+Eσ∼QEσ (3.2)

for someσ-exceptional effectiveQ-divisorEσ. Thus, we have the relation

ρ∗µ∗KX+∆

|W=ρ∗KW+

µ_∗−1∆+E|W−Eµ|W∼QEσ−ρ∗

Eµ|W. (3.3)

We putE+−E−:=Eσ−ρ∗(Eµ|W), whereE+andE−are effectiveQ-divisors that have

no common irreducible components. Here,E₊isσ-exceptional.

This paragraph is due to an argument in Miyaoka [14, Proposition IV 2.4]. Pute:=

dimWandc:=the codimension ofσ (E+)inWlm. We take general membersA1, A2, . . . , Ae−c∈ |A|andH1, H2, . . . , Hc−2∈ |H|whereAandHare very ample divisors onWlm

andW, respectively. Set

S=

e−c

i=1 σ−1Ai

∩

c−2

i=1 Hi

. (3.4)

Taking into account the argument above, we proceed along the lines in [8, Proof of Theorem 7.3]. Then we have the fact thatρ∗(µ∗(KX+∆)|W)isQ-linearly trivial and

In the following, we cope with the base points that lie on the non-KLT locus∆.

Proposition3.2. Let(X,∆)be a log terminal variety andHa NEFQ-CartierQ

-divisor such thatH−(KX+∆)is NEF and abundant. Assume thatν(X, aH−(KX+∆))= ν(X, H−(KX+∆))andκ(X, aH−(KX+∆))≥0for somea∈Qwitha >1. IfH|∆

is semiample, then Bs|mH| ∩ ∆ = ∅ for some positive integer mwith mH being Cartier. (HereBs|mH|denotes the base locus of|mH|.)

Proof. From an argument in [8, Proof of Theorem 6.1] we have a diagram

X←µ Y f→Z (3.5)

with the following properties:

(a) Y and Z are smooth projective varieties. Moreover, Y is a log resolution of (X,∆);

(b) µis birational andf is surjective with the property thatf∗ᏻY=ᏻZ;

(c) µ∗(H−(KX+∆))∼Qf∗M0for some NEF and bigQ-divisorM0; (d) µ∗H∼Qf∗H0for some NEFQ-divisorH0.

We define rational numbersaibyKY=µ∗(KX+∆)+

aiEi. We may assume thatH0

andHare Cartier. We put

S:=∆, E:= ai>0

ai

Ei, S:=

ai=−1

Ei. (3.6)

We note that,mµ∗H+E−S−(KY+{−ai}Ei)=(m−1)µ∗H+µ∗(H−(KX+∆)),

which isQ-linearly equivalent to the inverse image of a NEF and bigQ-divisor onZ. There are two cases.

Case1(f (S)≠Z). In this case we use Fujino’s argument [3, Section 2]. ByTheorem 2.3we have an injection

H1Y ,ᏻY

mµ∗H+E−S →H1Y ,ᏻY

mµ∗H+E. (3.7)

Then we consider the commutative diagram

H0_{Y ,ᏻ}

Ymµ∗H+E

surjective

H0_S,ᏻSmµ∗_{H+E 0}

H0_{Y ,ᏻ}

Ymµ∗H

H0_S,ᏻSmµ∗_H

i

H0_X,ᏻ

X(mH)

s

H0_S,ᏻ

S(mH). j

(3.8)

The homomorphismiis injective fromLemma 2.4and the fact thatEandShave no common irreducible component. The homomorphismjis injective fromLemma 2.4

Case2(f (S)=Z). In this case we use an argument in [10, Section 7]. There exists

an irreducible componentS ofSsuch thatf (S)=Z. BecauseH|S is semiample and µ∗H∼Qf∗H0, f∗H0|S is semiample. Consequently, theQ-divisor H0 also is semiample fromLemma 2.5.

We generalize Kawamata’s result [8, Theorem 6.1] (see alsoTheorem 1.3) concerning the semiampleness for KLT pairs to the case of log terminal pairs in the following form.

Proposition 3.3. Assume that (X,∆) is log terminal. LetH be a NEFQ-Cartier Q-divisor onXwith the following properties:

(1) H−(KX+∆)is NEF and abundant;

(2) ν(X, aH−(KX+∆))=ν(X, H−(KX+∆))andκ(X, aH−(KX+∆))≥0for some a∈Qwitha >1.

If, for some positive integerp1, the divisorp1His Cartier andBs|p1H|∩∆ = ∅, then His semiample.

In the proof below we proceed along the lines in [8, Proof of Theorem 6.1] and thus omit the parts which are parallel. However, we have to be very delicate in dealing with the non-KLT locus∆.

Proof. _{From [8, Theorem 6.1], we may assume that}∆≠_{0. Therefore, the}

con-dition that Bs|p1H| ∩ ∆ = ∅ implies that Bs|p1H|≠X. Thus, |p1tH|≠∅for all t∈N>0(whereN>0denotes the set of all positive integers).

We have smooth projective varietiesY andZ and morphismsX←µY f→Z with the following properties:

(1) µis birational andf is surjective; (2) f∗ᏻY=ᏻZ;

(3) µ∗(H−(KX+∆))∼Qf∗M0for some NEF and bigQ-divisorM0(where the sym-bol∼Qexpresses theQ-linear equivalence);

(4) µ∗H∼Qf∗H0for some NEFQ-divisorH0.

We may assume thatH0andHare Cartier andf∗H0andµ∗Hare linearly equivalent. PuttingΛ(m):=Bs|mH|, we may assume thatΛ(p1)≠∅(otherwise we immedi-ately obtain the assertion). By repetition of blowing-ups overY, we may replaceY and get a simple normal crossing divisorF=i∈IFionY such that

(5) µ∗|p1H| = |L|+i∈IriFiand|L|is base point free.

Then, by replacingZandYwe haveL∼Qf∗L0for someQ-divisorL0, because

νY , µ∗aH−KX+∆

≥ν

Y ,

_(a−1)

p1

L+µ∗H−KX+∆

≥νY , µ∗H−KX+∆

(3.9)

from the argument in [8, Proof of Proposition 2.1]. We note that

Λp1=µ

ri≠0 Fi

We have an effective divisorM1such thatM0−δM1 is ample for allδ∈Q

with 0< δ1. By further repetition of blowing-ups overY, we may replaceY and get the following properties:

(6) KY=µ∗(KX+∆)+i∈IaiFi;

(7) f∗M1=i∈IbiFi.

We set

c:=min

r_i≠0

ai+1−δbi ri

. (3.11)

Note that, ifai= −1 thenµ(Fi)⊂ ∆and that ifµ(Fi)⊂ ∆thenri=0 from the

assumption of the theorem. Thus, by takingδsmall enough, we may assume thatc >0 and that, ifFi⊂µ−1(∆), thenai+1−δbi>0 (even ifbi≠0). SetI0:= {i∈I; ai+

1−δbi=cri, ri≠0}and {Zα}:= {f (Γ);Γ ∈Strata(

i∈I0Fi)}. LetZ1be a minimal element of{Zα}with respect to the inclusion relation. We note that,Z1≠Z. Because M0−δM1is ample, for someq∈N>0, there exists a memberM2∈ |q(M0−δM1)|such thatZ1⊂M2andZα⊂M2for allα≠1.

We would like to show that we may assume that Supp(f∗M2)⊂F. Then, we inves-tigate the variation of the numbersai+1−δbiand the setI0under the blowing-up σ:Y→Ywith permissible smooth centerCwith respect toF. We get a simple normal crossing divisorF=i∈IFionY(whereI=I∪{0}) with the following properties:

F₀=σ−1_(C), KY=σ∗µ∗

KX+∆

+ i∈I

aiFi,

σ∗

i∈I riFi

= i∈I

r_iF_i,

σ∗f∗M1= i∈I

biFi.

(3.12)

We setI0:= {i∈I; ai+1−δbi=cri, ri≠0}. LetFi1, . . . , Fiu be the irreducible

com-ponents ofF that containC. LetF_i

j be the strict transform ofFijbyσ. We note that

σ∗

KY− u

j=1 aijFij

=KY−

codimYC−1

F0− u

j=1 aij

Fi_j+F0

. (3.13)

Thus,a0=(codimYC−1)+uj=1ai_j. Therefore,

a0+1≥ u

j=1

aij+1

, (3.14)

where the equality holds if and only ifu=codimYC. We note also thatr0=

u

j=1rij

andb0=

u

j=1bij.

Proof ofClaim3.4. First we note the inequality

a0+1−δb0≥ u

j=1

aij+1−δbij

, (3.15)

where the equality holds if and only if codimYC=u. BecauseFij⊂µ−

1_{(∆), we have} aij+1−δbij>0. Here ifrij≠0 thenaij+1−δbij≥crij, from the definition ofc. On

the other hand, ifri_j=0 thenai_j+1−δbi_j> cri_j. Now we note the inequality

u

j=1

aij+1−δbij

≥ u

j=1

crij, (3.16)

where the equality holds if and only ifrij≠0 andaij+1−δbij=crij(i.e.,ij∈I0) for

allj. Hereuj=1crij=cr0.

Claim3.5. Ifi_j∈I0for alljandC∈Strata(u_j=1F_i

j), thenI

0=I0∪{0}. Otherwise, I0=I0.

Proof ofClaim_3.5. _{Note that, codimYC}=_{uif and only ifC}∈_Strata(u_j=1Fi j).

Thus,Claim 3.4implies the assertion, because ifF0⊂(µσ )−1(∆)thenr0=0.

Claim3.6. We have an equation

min

r_i≠0

a_i+1−δb_i

r_i =c. (3.17)

Proof ofClaim3.6. In the case wherer₀≠0, we haveF₀⊂(µσ )−1(∆). Thus, Claim 3.4implies the assertion.

Claim_3.7. _IfF0⊂_{(µσ )}−1₍∆_{), thena0}+₁−_δb0>0.

Proof ofClaim3.7. In this case,a₀+1>0. Ifb₀≠0, thenC⊂f∗M1, sou≠ 0.

Thus,a0+1−δb0≥

u

j=1(aij+1−δbij) >0 because allFij⊂µ− 1_(∆).

By virtue of Claims3.5,3.6, and3.7, we may assume thatf∗M2=i∈IsiFiwhere F=i∈IFiis a simple normal crossing divisor. We put

c:= min

µ(F_i)⊂∆

ai+1−δbi ri+δsi

(3.18)

andI1:= {i∈I;ai+1−δbi=c(ri+δsi), µ(Fi)⊂ ∆}, for a rational numberδwith

0< δδ.

Claim_3.8. _{We have a relationI1}⊂_I0.

Proof ofClaim3.8. Because ifµ(F_i)⊂ ∆thena_i+1−δb_i>0, in the case where ri=0 the divisorFidoes not attain the minimum in (3.18).

Claim_3.9. _{There exists a memberj}∈_{I0such thatsj>0.}

Proof ofClaim3.9. The condition that Z1⊂M2 implies that, for somej ∈I,

sj> 0 andFjcontains an elementΓ∈Strata(

Claim3.10. We have an inequalitys_i>0for alli∈I1.

Proof ofClaim_3.10. _{Claims 3.8 and 3.9 and the formula (3.18) imply the}

assertion.

Claim3.11. We have a relationf (Γ)=Z1for allΓ∈Strata(_i∈I

1Fi).

Proof ofClaim_3.11. _{FromClaim 3.10,f (}Γ₎⊂_{M2. The condition thatZα}⊂_M2

for allα≠1 implies the fact that f (Γ)≠Zα for allα≠1. Thus, f (Γ)=Z1 from Claim 3.8.

Now we setN:=mµ∗H+i∈I(−c(ri+δsi)+ai−δbi)Fi−KY for an integerm≥ cp1+1. Then,

N=c

− i∈I

riFi

+mµ∗H−µ∗KX+∆−δ

i∈I

biFi−cδ

i∈I siFi

∼QcL−p1µ∗H+mµ∗H−µ∗H+f∗M0−δM1−cδ

i∈I siFi

∼Qcf∗L0+m−cp1+1µ∗H+1−cδqf∗M0−δM1.

(3.19)

Becauseµ∗Handf∗H0are linearly equivalent,NisQ-linearly equivalent to the pull-back of an ampleQ-divisor onZ. We put

A:=

i∈I\I1, µ(Fi)⊂∆

−cri+δsi

+ai−δbi

Fi,

B1:= i∈I1

Fi,

C:= µ(F_i)⊂∆

−cri+δsi

+ai−δbi

Fi.

(3.20)

Then, i∈I(−c(ri+δsi)+ai−δbi)Fi=A−B1+C. We express C:= −B2+B3in

effective divisorsB2andB3without common irreducible components. Here, we note thatf (B1+B2)≠Z, fromClaim 3.11and from the fact that the locusf−1_(Bs|p1H0|)= µ−1_{(Λ(p1))≠∅and the locusµ}−1_{(∆)are mutually disjoint. Note also that,Aand} B3areµ-exceptional effective divisors because ifai>0 thenFiisµ-exceptional.

ByTheorem 2.3, the homomorphism

H1

Y ,ᏻY

mµ∗H+ i∈I

−cri+δsi

+ai−δbi

Fi

→H1Y ,ᏻYmµ∗H+A+B3

(3.21)

is injective becausef (B1+B2)≠Z. Hence

H0Y ,ᏻY

mµ∗H+A+B3

→H0_B1,ᏻ B1

mµ∗H+A+B3⊕H0_B2,ᏻ B2

is surjective, becauseB1∩B2= ∅fromClaim 3.8. Here

H0_B1,ᏻ B1

mµ∗H+A+B3H0_B1,ᏻ B1

mµ∗H+A (3.23)

becauseB1∩B3= ∅fromClaim 3.8. We note that Supp(A|B1) is with only simple normal crossings andA|B1 is effective. Becausemµ∗H|B1+A|B1−KB1 =N|B1, we obtain a positive integerp2such that

H0_B1,ᏻ B1

p2tµ∗H+A≠0 (3.24)

for allt0, fromClaim 3.11and [8, Theorem 5.1]. Consequently, the assertion of

Proposition 3.3follows.

Proof ofTheorem_1.5. _{Because κ(X, KX} +∆_{) > 0, we have κ(X, KX} +∆₎ = ν(X, KX+∆)from the log minimal model and the log abundance theorems in

dimen-sion less than or equal 3 (see [10,15] andProposition 3.1). We note that,(KX+∆)|∆is

semiample fromProposition 2.8andTheorem 1.6. Thus,Proposition 3.2implies that Bs|m(KX+∆)| ∩ ∆ = ∅for somem∈N>0 withm(KX+∆) being Cartier.

Conse-quently,Proposition 3.3gives the assertion.

Acknowledgment. _{The author would like to thank the referees for their valuable}

advice concerning the presentation and the quotations.

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Shigetaka Fukuda: Faculty of Education, Gifu Shotoku Gakuen University, Yanaizu-Cho, Gifu_501-6194, Japan