Note on Quasi-Numerically Positive Log Canonical Divisors

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.

1. Introduction

In this note, every algebraic variety is defined over the field of complex numbers. We follow the terminology and notation in [1].

Definition 1. Let be a -Cartier -divisor on a projective variety . The divisor is numerically positive (nup, for short), if for every curve on . The divisor is quasi-numerically positive (quasi-nup, for short), if it is nef and if there exists a union of at most countably many prime divisors on such that for every curve (i.e., if is nef and if for every very general curve ).

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 , 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 , 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 -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 -divisor, then it is semiample.

There is another approach to the semiampleness Conjecture 4. Let be a klt pair whose log canonical divisor is quasi-nup. Hacon and McKernan (Lazic [14], Theorem A.6) considered embedding into some log canonical pair so that and , that the log canonical divisor is nef and big, that , and that is endowed with the birational contraction morphism that contracts the prime divisor 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 Appendix A, we survey the celebrated extension theorem [15] which is recently proven by Demailly-Hacon-Păun.

In Appendix B, 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 be a Kawamata log terminal pair such that is projective. If the log canonical divisor 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 -trivial curve that generates a -extremal ray for some other klt pair 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 -factoriality of .

Definition 5. One defines and for a -Cartier -divisor on .

Subconjecture 1. Let be a Kawamata log terminal pair such that is projective. Suppose that the log canonical divisor is not ample but quasi-nup. Then there exists an effective -Cartier divisor such that the intersection number for some class .

Procedure 1. Let be a Kawamata log terminal pair such that is projective. Suppose that the log canonical divisor is not ample but quasi-nup. Assume the existence of an effective -Cartier divisor on and a member of such that the intersection number . Let be a sufficiently small positive rational number. We can write this class in the form , where and are distinct -extremal rays. Then , because is nef and . We consider the birational contraction morphism of the -extremal ray . Put . We note that the Picard number , that , that is Kawamata log terminal, and that 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 -factoriality of .

Procedure 1 relates Subconjecture 1 to Conjecture 4. The following is the main result of this section.

Theorem 6. Subconjecture 1 implies Conjecture 4.

Proof. Let be a Kawamata log terminal pair such that is projective and the log canonical divisor is quasi-nup. If Subconjecture 1 is true, then, by repeating Procedure 1, we obtain a Kawamata log terminal pair with the birational morphism such that and is ample, because the Picard numbers decrease 1 by 1 in the process of contraction of extremal rays.

Corollary 7. Subconjecture 1 and the termination conjecture for log flips imply the log abundance conjecture for klt pairs.

Proof. See Remark 3 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 Subconjecture 1 is a part of Conjecture 4.

Lemma 9. Let be a Kawamata log terminal pair such that is projective. Suppose that is not ample but quasi-nup and semiample. Then there exists an effective -Cartier divisor such that the intersection number for some class .

Proof. Consider the surjective morphism induced by the linear system for a sufficiently large and divisible integer . This morphism becomes birational, because of the Stein factorisation theorem and the fact that the pullbacks of ample divisors by finite morphisms are ample. Then for an ample divisor on . By the Kodaira Lemma, if is sufficiently large and divisible, then for some ample divisor and some effective divisor . For every -exceptional curve , we obtain the inequality that , because and . Here the class belongs to .

Proposition 10. Conjecture 4 implies Subconjecture 1.

Proof. Lemma 9 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. Let be a birational morphism between normal projective varieties of dimension such that is a prime divisor and let and be divisorial log terminal pairs. Assume that is nup.

Proposition 12. Under Assumption 11, the divisor is nef for every small number .

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 -extremal rays. By using the argument in [21], we have is an extremal rational curve for . Thus is nef.

Assumption 13. Furthermore assume that is -Cartier, that is -ample, and that, in the case where is not a point, the divisor is ample.

Remark 14. If is -factorial, then the condition that is -ample in Assumption 13 is automatically satisfied, under Assumption 11 (cf. Kollár and Mori [22], Lemma 2.62).

Definition 15. Under Assumptions 11 and 13, one defines the number by the equation . Then , because is nup.

Proposition 16. Under Assumptions 11 and 13, the divisor is big.

Proof. Assume that is not big for every small number . Thus its self-intersection number is zero for every from Proposition 12. Therefore . This contradicts the -ampleness of . Consequently is big for every small number and so is .

Proposition 17. Under Assumptions 11 and 13, the divisor is ample.

Proof. The divisor on is ample for every small number   (cf. [22], Proposition 1.45). We also recall that is nef by Proposition 12. Thus is ample, from the inequality .

We state the main result of this section.

Theorem 18. Under Assumptions 11 and 13, the divisor is ample if and only if for every minimal log canonical (i.e., minimal non-klt) center with respect to the pair such that .

For proof, we cite the following ampleness result.

Proposition 19 (see [23]). Let be a divisorial log terminal pair which is not Kawamata log terminal such that is projective. Assume that the log canonical divisor is nup and that for every minimal log canonical (i.e., minimal non-klt) center with respect to the pair . Then is ample.

Proof of Theorem 18. The “only if” part is trivial. So we prove the “if” part.
For every minimal log canonical center with respect to such that , we have that from Ambro (see [24], Proposition 3.3) because is a log canonical center with respect to .
Thus for every minimal log canonical center with respect to the pair by Proposition 17.
Consequently Proposition 19 implies that is ample.

Example 20. Let be a projective space with homogeneous coordinate and hyperplane . We consider the hypersurface () defined by the irreducible homogeneous equation . We note that is normal and that is Cartier. Blow up at the subspace and obtain the morphism and the exceptional divisor . Let be the strict transform of by . We note that is nonsingular. We have . Thus . Then is nef, because the linear system is base-point-free. Consequently is nup because is -ample. Let be the restriction of a general member of to . We put and . Then is a smooth prime divisor and is -ample. We note that is nup and that is ample. When (i.e., ), the divisor is ample. Lastly Theorem 18 implies that is ample.

Appendices

A. A Survey of the Demailly-Hacon-Păun Extension Theorem [15]

In this appendix, we survey the celebrated extension theorem due to Demailly-Hacon-Pun.

Proposition A.1 (see [15]). Let be a projective purely log terminal pair with a prime divisor such that . Assume that the log canonical divisor is nef and that there exists an effective -divisor which is -linearly equivalent to with . Then the restriction map is surjective for all sufficiently large and divisible integers .

Let be a projective Kawamata log terminal pair whose log canonical divisor is nef.

Conjecture A.2 (log abundance conjecture). The (nef) log canonical divisor is semiample.

Subconjecture 2. There exists an effective divisor on such that is purely log terminal and is linearly equivalent to some multiple of .

Proposition A.3. Log Abundance Conjecture A.2 implies Subconjecture 2.

Proof. If the logarithmic Kodaira dimension , then we are done (letting ). So we may assume that .
For a sufficiently large and divisible integer , the linear system is base-point-free and gives the algebraic fiber space . Then is linearly equivalent to for some hyperplane section of . Consider a log resolution of such that the morphism is projective, that the exceptional locus is divisorial, and that the locus is with only simple normal crossings. For a general member of the linear system , the divisor is a disjoint union of a finite number of smooth prime divisors . Thus the divisor satisfies the required condition.

We consider the converse statement for Proposition A.3.

Claim 1. Under Subconjecture 2, if and is an irreducible component of , then we have the following properties:(1)The prime divisor is a connected component of .(2)The pair , is Kawamata log terminal.(3)The log canonical divisor is nef.(4)The prime divisor is -Cartier and nef.(5)The restriction map is surjective for all sufficiently large and divisible integers .

Proof. (1) and (2) are the elementary facts of purely log terminal pairs. (3) is trivial.
Because is -Cartier and nef, we have (4) from the fact that is a disjoint union of prime divisors .
Thus is a nef -Cartier divisor and . For a sufficiently small rational number , the pair is purely log terminal and . We note that there exists an effective -divisor which is -linearly equivalent to such that . Because , we have that . So we get the following commutative diagram from Proposition A.1 [15]:

Theorem A.4. Subconjecture 2 in dimension implies Log Abundance Conjecture A.2.

Proof. If , we are done. So we may assume that , where are distinct prime divisors. We follow the notation in Claim 1. By induction on dimension, the log canonical divisor is semiample. Therefore Claim 1 (5) implies that the base locus is disjoint from for a sufficiently large and divisible integer . Thus is disjoint from . From the assumption that is -linearly equivalent to some multiple of , the log canonical divisor is semiample.

Conjecture A.5 (smooth abundance conjecture). Assume that is smooth and . The (nef) canonical divisor is semiample.

Subconjecture 3. Assume that is smooth and . There exists an effective divisor such that is log smooth and purely log terminal and that is linearly equivalent to some multiple of .

By the same argument as in the proofs of Proposition A.3 and Theorem A.4, we have the following two results.

Proposition A.6. Smooth Abundance Conjecture A.5 implies Subconjecture 3.

Theorem A.7. Subconjecture 3 in dimension implies Smooth Abundance Conjecture A.5.

B. Strong Log Bigness

Let be a projective variety over the field of complex numbers and an effective -divisor on where the pair is dlt (i.e., divisorial log terminal).

Definition A.8. A -Cartier -divisor is strongly log big on if, for some integer , the following three conditions are satisfied:(i)The -Cartier -divisor is a Cartier divisor.(ii)The base locus does not contain any generic point of the log canonical centers of .(iii)The rational map is birational to its image and, furthermore, is isomorphic onto its image in some neighborhood of every generic point of the log canonical centers of .

Remark A.9. Boucksom et al. [16] proved that, for a big divisor , the strong log bigness of is equivalent to the condition that the augmented base locus does not contain any generic point of the log canonical centers.

Theorem A.10 (see [16–18]). If the log canonical divisor is strongly log big on the dlt pair , then the log canonical ring is finitely generated over the field .

From Remark A.9, the theorem above is a reduction of Birkar and Hu [17] or Cacciola [18]. But we give a straightforward proof to the theorem.

Proof. We follow the notation in Definition A.8 for the -Cartier -divisor . From the assumption and the divisorial log terminal theorem (Szabó [25]), there exists some nonempty Zariski-open subset of with the following properties:(i) contains all the generic points of log canonical centers of .(ii).(iii)The rational map is isomorphic onto its image.(iv)The pair is a nonsingular variety with a reduced simply normal crossing divisor on .
We set .
From the resolution lemma [25] due to Szabó, there exists a log resolution : of the pair such that is isomorphic and that is divisorial.
Here the exceptional locus denotes the locus where the morphism is not isomorphic.
From the Hironaka resolution theorem, by the repetitions of blowups along smooth subvarieties included in the singular locus of , we have a resolution : of singularities such that is isomorphic and that there exists some -antiample effective divisor whose support coincides with the exceptional locus .
We consider the rational map . Then we obtain the commutative diagram: We take the elimination of indeterminacy for the rational map : Note that the morphism is isomorphic.
Because the variety (, resp.) is -factorial, there exists some -antiample (-antiample, resp.) effective divisor whose support coincides with (, resp.).
We put . Then we have the commutative diagram: We have the relation between complete linear systems where is a hyperplane section of and is an effective divisor on with the property that .
We consider the -divisor .
We set , which is purely codimension in .
Consider the Zariski-open subset .
We note that and that .
From the resolution lemma [25] due to Szabó, we have a projective morphism : which satisfies the following four conditions:(a) is a composition of blowups along smooth subvarieties.(b) is isomorphic.(c) is nonsingular.(d) is a divisor with only simple normal crossings.
Putting and , we have the diagram and have the property that the loci and are divisorial. We define the -divisors , , and by the following relations:(i).(ii) (i.e., ).(iii).(iv) and have no common irreducible component.
Then we have the properties that is a reduced divisor with only simple normal crossings, that + + is disjoint from , and that .
There exists some -antiample (-antiample, -antiample, resp.) effective divisor whose support is (, , resp.). Thus the -divisors are ample for some effective -divisors , , and with the property that , , and . We write where and note that . Then where .
Here . Thus does not include any log canonical center of the smooth pair . For a sufficiently small rational number , the -divisor is ample. Therefore, for a sufficiently large and divisible integer , the divisor is very ample and linearly equivalent to some prime divisor such that is with only simple normal crossings and that does not include any log canonical center of the smooth pair . We have the following relation and the klt (i.e., Kawamata log terminal) pair for a sufficiently small rational number : From the Birkar-Cascini-Hacon-McKernan theorem [4], the log canonical ring for a klt pair is finitely generated.
Consequently the equivalence between the finite generation of the log canonical ring and that of some truncation of this ring implies the assertion.

Disclosure

The content of Remark 3 (History) (see [5]) was presented in the short communications at ICM 2014 (Seoul) on August 16 in the year 2014. Remark 3 corrects a chronological typo and a chronological mistake in Fukuda [5].

Conflict of Interests

The author declares that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

In this series of researches, the author was supported by the research grant of Gifu Shotoku Gakuen University in the years 2011 and 2014. The author would like to thank the referee who carefully read the paper and gave the advice to improve the explanation.