aa r X i v : . [ m a t h . AG ] F e b CONE THEOREM AND MORI HYPERBOLICITY
OSAMU FUJINO
Abstract.
We discuss the cone theorem for quasi-log schemes and the Mori hyperbol-icity. In particular, we establish that the log canonical divisor of a Mori hyperbolicprojective normal pair is nef if it is nef when restricted to the non-lc locus. This answersSvaldi’s question completely. We also treat the uniruledness of the degenerate locus of anextremal contraction morphism for quasi-log schemes. Furthermore, we prove that everyfiber of a relative quasi-log Fano scheme is rationally chain connected modulo the non-qlclocus.
Contents
1. Introduction 12. Preliminaries 92.1. Basic definitions 92.2. Uniruledness, rationally connectedness, and rationally chain connectedness 113. On normal pairs 123.1. Singularities of pairs 123.2. Dlt blow-ups revisited 144. On quasi-log schemes 154.1. Definitions and basic properties of quasi-log schemes 164.2. Kleiman–Mori cones 204.3. Lemmas on quasi-log schemes 215. Proof of Theorem 1.9 256. On basic slc-trivial fibrations 277. On normal quasi-log schemes 298. Proof of Theorem 1.10 319. Proof of Theorem 1.8 3410. Proof of Theorems 1.4, 1.5, and 1.6 3711. Ampleness criterion for quasi-log schemes 3912. Proof of Theorems 1.12 and 1.13 4113. Proof of Theorem 1.14 4414. Towards Conjecture 1.15 47References 531.
Introduction
This paper gives not only new results around the cone theorem and Mori hyperbolic-ity of quasi-log schemes but also a new framework and some techniques to treat higher-dimensional complex algebraic varieties based on the theory of mixed Hodge structures. It
Date : 2021/2/20, version 0.25.2010
Mathematics Subject Classification.
Primary 14E30; Secondary 32Q45, 14J45.
Key words and phrases. cone theorem, Mori hyperbolic, extremal rational curves, quasi-log schemes,adjunction, subadjunction, rationally chain connected, uniruled. also shows that the theory of quasi-log schemes is very powerful even for the study of logcanonical pairs. We note that this paper heavily depends on [F11, Chapter 6] and [F14].In his epoch-making paper [Mo], Shigefumi Mori established the following cone theoremfor smooth projective varieties.
Theorem 1.1 (Cone theorem for smooth projective varieties) . Let X be a smooth projec-tive variety defined over an algebraically closed field. (i) There are countably many ( possibly singular ) rational curves C j ⊂ X such that < − ( C j · K X ) ≤ dim X + 1 and N E ( X ) = N E ( X ) K X ≥ + X j R ≥ [ C j ] . (ii) For any ε > and any ample Cartier divisor H on X , N E ( X ) = N E ( X ) ( K X + εH ) ≥ + X finite R ≥ [ C j ] . In particular, we have:
Theorem 1.2.
Let X be a smooth projective variety defined over an algebraically closedfield. Assume that there are no rational curves on X . Then K X is nef. Precisely speaking, Mori proved the existence of rational curves on X under the as-sumption that K X is not nef (see Theorem 1.2) by his ingenious method of bend and break .Then he obtained the above cone theorem for smooth projective varieties (see Theorem1.1). For the details, see [Mo], [KM, Sections 1.1, 1.2, and 1.3], [D], [Ko1], [Ma, Chapter10], and so on.From now on, we will work over C , the complex number field. Our arguments in thispaper heavily depend on Hironaka’s resolution of singularities and its generalizations andseveral Kodaira type vanishing theorems. Hence they do not work over a field of charac-teristic p >
0. Let us recall the notion of
Mori hyperbolicity following [LZ] and [S].
Definition 1.3 (Mori hyperbolicity) . Let ( X, ∆) be a normal pair such that ∆ is effective.This means that X is a normal variety and ∆ is an effective R -divisor on X such that K X + ∆ is R -Cartier. Let W be an lc stratum of ( X, ∆). We put U := W \ ( ( W ∩ Nlc( X, ∆)) ∪ [ W ′ W ′ ) , where W ′ runs over lc centers of ( X, ∆) strictly contained in W and Nlc( X, ∆) denotesthe non-lc locus of ( X, ∆), and call it the open lc stratum of ( X, ∆) associated to W . Wesay that ( X, ∆) is Mori hyperbolic if there is no non-constant morphism f : A −→ U for any open lc stratum U of ( X, ∆).The following theorem is a generalization of Theorem 1.2 for normal pairs and is ananswer to [S, Question 6.6]. Theorem 1.4.
Let X be a normal projective variety and let ∆ be an effective R -divisoron X such that K X + ∆ is R -Cartier. Assume that ( X, ∆) is Mori hyperbolic and that K X + ∆ is nef when restricted to Nlc( X, ∆) . Then K X + ∆ is nef. Theorem 1.4 follows from the following cone theorem for normal pairs. We can see it asa generalization of Theorem 1.1 for normal pairs.
ONE THEOREM AND MORI HYPERBOLICITY 3
Theorem 1.5 (Cone theorem for normal pairs) . Let ( X, ∆) be a normal pair such that ∆ is effective and let π : X → S be a projective morphism between schemes. (i) Then
N E ( X/S ) =
N E ( X/S ) ( K X +∆) ≥ + N E ( X/S ) −∞ + X j R j holds, where R j ’s are the ( K X + ∆) -negative extremal rays of N E ( X/S ) that arerational and relatively ample at infinity. In particular, each R j is spanned by anintegral curve C j on X such that π ( C j ) is a point. (ii) Let H be a π -ample R -divisor on X . Then N E ( X/S ) =
N E ( X/S ) ( K X +∆+ H ) ≥ + N E ( X/S ) −∞ + X finite R j holds. (iii) For each ( K X + ∆) -negative extremal ray R j of N E ( X/S ) that are rational andrelatively ample at infinity, there are an open lc stratum U j of ( X, ∆) and a non-constant morphism f j : A −→ U j such that C j , the closure of f j ( A ) in X , spans R j in N ( X/S ) with < − ( K X + ∆) · C j ≤ U j . More generally, we establish the following cone theorem for quasi-log schemes. We notethat Theorem 1.5 is a very special case of Theorem 1.6.
Theorem 1.6 (Cone theorem for quasi-log schemes) . Let [ X, ω ] be a quasi-log scheme andlet π : X → S be a projective morphism between schemes. (i) Then
N E ( X/S ) =
N E ( X/S ) ω ≥ + N E ( X/S ) −∞ + X j R j holds, where R j ’s are the ω -negative extremal rays of N E ( X/S ) that are rationaland relatively ample at infinity. In particular, each R j is spanned by an integralcurve C j on X such that π ( C j ) is a point. (ii) Let H be a π -ample R -line bundle on X . Then N E ( X/S ) =
N E ( X/S ) ( ω + H ) ≥ + N E ( X/S ) −∞ + X finite R j holds. (iii) For each ω -negative extremal ray R j of N E ( X/S ) that are rational and relativelyample at infinity, there are an open qlc stratum U j of [ X, ω ] and a non-constantmorphism f j : A −→ U j such that C j , the closure of f j ( A ) in X , spans R j in N ( X/S ) with < − ω · C j ≤ U j . We make a remark on U j in Theorem 1.6. Remark 1.7.
In Theorem 1.6 (iii), let ϕ R j be the extremal contraction morphism associ-ated to R j . Then the proof of Theorem 1.6 shows that U j is any open qlc stratum of [ X, ω ]such that ϕ R j : U j → ϕ R j ( U j ) is not finite and that ϕ R j : W † → ϕ R j ( W † ) is finite for everyqlc center W † of [ X, ω ] with W † ( U j , where U j is the closure of U j in X .The main ingredients of the proof of Theorem 1.6 are the following three results. OSAMU FUJINO
Theorem 1.8.
Let X be a normal variety and let ∆ be an effective R -divisor on X suchthat K X + ∆ is R -Cartier. Let π : X → S be a projective morphism onto a scheme S .Assume that ( K X + ∆) | Nklt( X, ∆) is nef over S , where Nklt( X, ∆) denotes the non-klt locusof ( X, ∆) , and that K X + ∆ is not nef over S . Then there exists a non-constant morphism f : A −→ X \ Nklt( X, ∆) such that π ◦ f ( A ) is a point and that the curve C , the closure of f ( A ) in X , is a ( possiblysingular ) rational curve with < − ( K X + ∆) · C ≤ X. We prove Theorem 1.8 with the aid of the minimal model theory for higher-dimensionalalgebraic varieties mainly due to [BCHM]. Theorem 1.9 is a slight generalization of [FLh,Theorem 1.1], where [
X, ω ] is a quasi-log canonical pair. In Theorem 1.9, [
X, ω ] is notnecessarily quasi-log canonical.
Theorem 1.9.
Let [ X, ω ] be a quasi-log scheme such that X is irreducible. Let ν : Z → X be the normalization. Then there exists a proper surjective morphism f ′ : ( Y ′ , B Y ′ ) → Z from a quasi-projective globally embedded simple normal crossing pair ( Y ′ , B Y ′ ) such thatevery stratum of Y ′ is dominant onto Z and that ( Z, ν ∗ ω, f ′ : ( Y ′ , B Y ′ ) → Z ) naturally becomes a quasi-log scheme with Nqklt(
Z, ν ∗ ω ) = ν − Nqklt(
X, ω ) . More pre-cisely, the following equality ν ∗ I Nqklt(
Z,ν ∗ ω ) = I Nqklt(
X,ω ) holds, where I Nqklt(
X,ω ) and I Nqklt(
Z,ν ∗ ω ) are the defining ideal sheaves of Nqklt(
X, ω ) and Nqklt(
Z, ν ∗ ω ) respectively. Theorem 1.10 is similar to [F15, Theorem 1.1]. The proof of Theorem 1.10 needs somedeep results on basic slc-trivial fibrations obtained in [F14] and [FFL]. Therefore, Theorem1.10 depends on the theory of variations of mixed Hodge structure (see [FF] and [FFS]).
Theorem 1.10.
Let [ X, ω ] be a quasi-log scheme such that X is a normal quasi-projectivevariety. Let H be an ample R -divisor on X . Then there exists an effective R -divisor ∆ on X such that K X + ∆ ∼ R ω + H and that Nklt( X, ∆) = Nqklt( X, ω ) holds set theoretically, where Nklt( X, ∆) denotes the non-klt locus of ( X, ∆) . Furthermore,if [ X, ω ] has a Q -structure and H is an ample Q -divisor on X , then we can make ∆ a Q -divisor on X such that K X + ∆ ∼ Q ω + H holds.When X is a smooth curve, we can take an effective R -divisor ∆ on X such that K X + ∆ ∼ R ω and that Nklt( X, ∆) = Nqklt( X, ω ) holds set theoretically. Of course, if we further assume that [ X, ω ] has a Q -structure, thenwe can make ∆ an effective Q -divisor on X such that K X + ∆ ∼ Q ω holds. ONE THEOREM AND MORI HYPERBOLICITY 5
Let us briefly explain the idea of the proof of Theorem 1.6 (iii), which is one of themain results of this paper. We take an ω -negative extremal ray R j of N E ( X/S ) that arerational and relatively ample at infinity. Then, by the contraction theorem, there exists acontraction morphism ϕ := ϕ R j : X → Y over S associated to R j . We take a qlc stratum W of [ X, ω ] such that ϕ : W → ϕ ( W ) is not finite and that ϕ : W † → ϕ ( W † ) is finite for everyqlc center W † with W † ( W . By adjunction, W ′ := W ∪ Nqlc(
X, ω ) with ω | W ′ becomes aquasi-log scheme. Hence we can replace [ X, ω ] with [ W ′ , ω | W ′ ]. By using Theorem 1.9, wecan reduce the problem to the case where X is a normal variety. By Theorem 1.10, we seethat it is sufficient to treat normal pairs. For normal pairs, by Theorem 1.8, we can finda non-constant morphism f j : A −→ X \ Nqklt(
X, ω )with the desired properties.We also treat an ampleness criterion for Mori hyperbolic normal pairs. It is a general-ization of [S, Theorem 7.5].
Theorem 1.11 (Ampleness criterion for Mori hyperbolic normal pairs) . Let X be a normalprojective variety and let ∆ be an effective R -divisor on X such that K X + ∆ is R -Cartier.Assume that ( X, ∆) is Mori hyperbolic, ( K X + ∆) | Nlc( X, ∆) is ample, and K X + ∆ is log bigwith respect to ( X, ∆) . Then K X + ∆ is ample. Theorem 1.11 is a very special case of the ampleness criterion for quasi-log schemes(see Theorem 11.1). We omit the precise statement of Theorem 11.1 here since it lookstechnical. We note that K X + ∆ is nef by Theorem 1.4 since ( X, ∆) is Mori hyperbolicand ( K X + ∆) | Nlc( X, ∆) is ample. Therefore, K X + ∆ is nef and log big with respect to( X, ∆) in Theorem 1.11. Hence we can see that K X + ∆ is semi-ample with the aid of thebasepoint-free theorem of Reid–Fukuda type (see [F10]). Then we prove that K X + ∆ isample.By using the method established for the proof of Theorem 1.6, we can prove the followingtheorems. Note that Theorems 1.12, 1.13, and 1.14 are free from the theory of minimalmodels. Theorem 1.12 is a generalization of Kawamata’s famous theorem (see [Ka]). Theorem 1.12.
Let [ X, ω ] be a quasi-log scheme and let ϕ : X → W be a projectivemorphism between schemes such that − ω is ϕ -ample. Let P be an arbitrary closed pointof W . Let E be any positive-dimensional irreducible component of ϕ − ( P ) such that E X −∞ . Then E is covered by ( possibly singular ) rational curves ℓ with < − ω · ℓ ≤ E. In particular, E is uniruled. For the reader’s convenience, let us explain the idea of the proof of Theorem 1.12. Wetake an effective R -Cartier divisor B on W passing through P such that E is a qlc stratumof [ X, ω + ϕ ∗ B ]. Let ν : E → E be the normalization. By adjunction for quasi-log schemes,Theorems 1.9, 1.10, and so on, for any ample R -divisor H on E , we obtain an effective R -divisor ∆ E,H on E such that ν ∗ ω + H ∼ R K E + ∆ E,H holds. This implies that C · K E < C on E . Thus, it isnot difficult to see that E is covered by rational curves (see [MM]). Our approach isdifferent from Kawamata’s original one, which uses a relative Kawamata–Viehweg vanish-ing theorem for projective bimeromorphic morphisms between complex analytic spaces.Kawamata’s approach does not work for our setting.As a direct consequence of Theorem 1.12, we have: OSAMU FUJINO
Theorem 1.13 (Lengths of extremal rational curves) . Let [ X, ω ] be a quasi-log scheme andlet π : X → S be a projective morphism between schemes. Let R be an ω -negative extremalray of N E ( X/S ) that are rational and relatively ample at infinity. Let ϕ R : X → W be thecontraction morphism over S associated to R . We put d = min E dim E, where E runs over positive-dimensional irreducible components of ϕ − R ( P ) for all P ∈ W .Then R is spanned by a ( possibly singular ) rational curve ℓ with < − ω · ℓ ≤ d. If ( X, ∆) is a log canonical pair, then [ X, K X +∆] naturally becomes a quasi-log canonicalpair. Hence we can apply Theorems 1.12 and 1.13 to log canonical pairs. Note thatTheorems 1.12 and 1.13 are new even for log canonical pairs (see also Corollary 12.3). Wecan prove the following result on rationally chain connectedness for relative quasi-log Fanoschemes. Theorem 1.14 (Rationally chain connectedness) . Let [ X, ω ] be a quasi-log scheme andlet π : X → S be a projective morphism between schemes with π ∗ O X ≃ O S . Assume that − ω is ample over S . Then π − ( P ) is rationally chain connected modulo π − ( P ) ∩ X −∞ for every closed point P ∈ S . In particular, if further π − ( P ) ∩ X −∞ = ∅ holds, that is, [ X, ω ] is quasi-log canonical in a neighborhood of π − ( P ) , then π − ( P ) is rationally chainconnected. Let us see the idea of the proof of Theorem 1.14. We assume that π − ( P ) ∩ X −∞ = ∅ for simplicity. By using the framework of quasi-log schemes, we construct a good finiteincreasing sequence of closed subschemes Z − := Nqlc( X, ω ) ⊂ Z ( Z ( · · · ( Z k of X such that π − ( P ) ⊂ Z k after shrinking X around π − ( P ). It is well known thatif ( V, ∆) is a projective normal pair such that ∆ is effective and that − ( K V + ∆) isample then V is rationally chain connected modulo Nklt( V, ∆) (see [HM] and [BP]). Bythis fact, adjunction for quasi-log schemes, Theorems 1.9, 1.10, and so on, we prove that Z i +1 ∩ π − ( P ) is rationally chain connected modulo Z i ∩ π − ( P ) for every − ≤ i ≤ k − Z k ∩ π − ( P ) = π − ( P ) and Z − ∩ π − ( P ) = π − ( P ) ∩ X −∞ , we obtain that π − ( P )is rationally chain connected modulo π − ( P ) ∩ X −∞ .Theorems 1.6, 1.12, and 1.14 are closely related one another. Let us see these theoremsfor extremal birational contraction morphisms of log canonical pairs. Let ( X, ∆) be aprojective log canonical pair and let R be a ( K X + ∆)-negative extremal ray of N E ( X ).Assume that the contraction morphism ϕ R : X → W associated to R is birational. We takea closed point P of W such that dim ϕ − R ( P ) >
0. Then Theorem 1.14 says that ϕ − R ( P )is rationally chain connected. However, Theorem 1.14 gives no informations on degreesof rational curves on ϕ − R ( P ) with respect to − ( K X + ∆). On the other hand, Theorem1.12 shows that every irreducible component of ϕ − R ( P ) is covered by rational curves ℓ with 0 < − ( K X + ∆) · ℓ ≤ ϕ − R ( P ). In particular, every irreducible component ofthe exceptional locus of ϕ R is uniruled. Note that the rationally chain connectedness of ϕ − ( P ) does not directly follow from Theorem 1.12. Theorem 1.6 (see also Theorem 1.5)shows that there exist a rational curve C on X and an open lc stratum U of ( X, ∆) suchthat ϕ R ( C ) is a point and that the normalization of C ∩ U contains A .We pose a conjecture related to [LZ, Theorem 3.1]. ONE THEOREM AND MORI HYPERBOLICITY 7
Conjecture 1.15.
Let [ X, ω ] be a quasi-log scheme and let π : X → S be a projectivemorphism between schemes such that − ω is π -ample and that π : Nqklt( X, ω ) → π (Nqklt( X, ω )) is finite. Let P be a closed point of S such that there exists a curve C † ⊂ π − ( P ) with Nqklt(
X, ω ) ∩ C † = ∅ . Then there exists a non-constant morphism f : A −→ ( X \ Nqklt(
X, ω )) ∩ π − ( P ) such that C , the closure of f ( A ) in X , satisfies C ∩ Nqklt(
X, ω ) = ∅ with < − ω · C ≤ . In this paper, we solve Conjecture 1.15 under the assumption that any sequence of kltflips terminates.
Theorem 1.16 (see Theorem 14.2) . Assume that any sequence of klt flips terminates afterfinitely many steps. Then Conjecture 1.15 holds true.
For the precise statement of Theorem 1.16, see Theorem 14.2. In a joint paper withKenta Hashizume (see [FH]), we will prove the following theorem, which is a very specialcase of Conjecture 1.15, by using some deep results in the theory of minimal models forlog canonical pairs obtained in [H2].
Theorem 1.17 (see [FH]) . Let X be a normal variety and let ∆ be an effective R -divisoron X such that K X + ∆ is R -Cartier. Let π : X → S be a projective morphism onto ascheme S such that − ( K X + ∆) is π -ample. We assume that π : Nklt( X, ∆) → π (Nklt( X, ∆)) is finite. Let P be a closed point of S such that there exists a curve C † ⊂ π − ( P ) with Nklt( X, ∆) ∩ C † = ∅ . Then there exists a non-constant morphism f : A −→ ( X \ Nklt( X, ∆)) ∩ π − ( P ) such that the curve C , the closure of f ( A ) in X , is a ( possibly singular ) rational curvesatisfying C ∩ Nklt( X, ∆) = ∅ with < − ( K X + ∆) · C ≤ . Although Theorem 1.17 looks very similar to Theorem 1.8, the proof of Theorem 1.17is much harder. By using Theorem 1.17, we will establish:
Theorem 1.18 (see [FH]) . Conjecture 1.15 holds true.
As an application of Theorem 1.18, we will prove the following statement in [FH], whichsupplements Theorem 1.6 (iii).
Theorem 1.19 (see [FH]) . Let [ X, ω ] be a quasi-log scheme and let π : X → S be aprojective morphism between schemes. Let R j be an ω -negative extremal ray of N E ( X/S ) that are rational and relatively ample at infinity and let ϕ R j be the contraction morphismassociated to R j . Let U j be any open qlc stratum of [ X, ω ] such that ϕ R j : U j → ϕ R j ( U j ) is not finite and that ϕ R j : W † → ϕ R j ( W † ) is finite for every qlc center W † of [ X, ω ] with W † ( U j , where U j is the closure of U j in X . Let P be a closed point of ϕ R j ( U j ) . Ifthere exists a curve C † such that ϕ R j ( C † ) = P , C † U j , and C † ⊂ U j , then there exists anon-constant morphism f j : A −→ U j ∩ ϕ − R j ( P ) such that C j , the closure of f j ( A ) in X , spans R j in N ( X/S ) and satisfies C j U j with < − ω · C j ≤ . OSAMU FUJINO
We note that Theorem 1.19 is a generalization of [LZ, Theorem 3.1]. In this paper,we prove the following simpler statement for dlt pairs for the reader’s convenience sinceTheorems 1.17, 1.18, and 1.19 are difficult. Theorem 1.20 is much weaker than Theorem1.19. However, it contains a generalization of [LZ, Theorem 3.1].
Theorem 1.20.
Let ( X, ∆) be a dlt pair and let π : X → S be a projective morphismbetween schemes. Let R j be a ( K X + ∆) -negative extremal ray of N E ( X/S ) and let ϕ R j be the contraction morphism associated to R j . Let U j be any open lc stratum of ( X, ∆) such that ϕ R j : U j → ϕ R j ( U j ) is not finite and that ϕ R j : W † → ϕ R j ( W † ) is finite for everylc center W † of ( X, ∆) with W † ( U j , where U j is the closure of U j in X . If there existsa curve C † such that ϕ R j ( C † ) is a point, C † U j , and C † ⊂ U j , then there exists anon-constant morphism f j : A −→ U j such that C j , the closure of f j ( A ) in X , spans R j in N ( X/S ) and satisfies C j U j with < − ω · C j ≤ . Although we need some deep results on the minimal model program for log canonicalpairs in [H1] in the proof of Theorem 1.20, the proof of Theorem 1.20 is much simpler thanthat of Theorems 1.17, 1.18 and 1.19 in [FH] and will help the reader understand [FH].Finally, we make a conjecture on lengths of extremal rational curves (see [Ma, Remark-Question 10-3-6]).
Conjecture 1.21. If ϕ R j : U j → ϕ R j ( U j ) is proper in Theorem 1.6 (iii), where ϕ R j isthe contraction morphism associated to R j , then there exists a ( possibly singular ) rationalcurve C j ⊂ U j which spans R j in N ( X/S ) and satisfies < − ω · C j ≤ d j + 1 with d j = min E dim E, where E runs over positive-dimensional irreducible components of ( ϕ R j | U j ) − ( P ) for all P ∈ ϕ R j ( U j ) . The following remark on Conjecture 1.21 is obvious.
Remark 1.22.
We use the same notation as in Conjecture 1.21. If ϕ R j : U j → ϕ R j ( U j ) isproper in Theorem 1.6 (iii), we can make C j satisfy0 < − ω · C j ≤ d j by Theorem 1.12.Of course, we hope that the following sharper estimate0 < − ω · ℓ ≤ dim E + 1should hold true in Theorem 1.12.We briefly look at the organization of this paper. In Section 2, we recall some basicdefinitions and results. Then we treat the notion of uniruledness, rationally connectedness,and rationally chain connectedness. In Section 3, we treat some basic definitions and resultson normal pairs and then discuss dlt blow-ups for quasi-projective normal pairs. In Section4, we briefly review the theory of quasi-log schemes and prepare some useful and importantlemmas. In Section 5, we give a detailed proof of Theorem 1.9. Theorem 1.9 plays a crucialrole since a quasi-log scheme is not necessarily normal even when it is a variety. In Section6, we quickly explain basic slc-trivial fibrations. The results in [F14] make the theory ofquasi-log schemes very powerful. In Section 7, we prove a very important result on normal ONE THEOREM AND MORI HYPERBOLICITY 9 quasi-log schemes, which is a slight generalization of [F14, Theorem 1.7]. In Section 8, weprove Theorem 1.10 by using the result explained in Section 7. Hence Theorem 1.10 heavilydepends on some deep results on the theory of variations of mixed Hodge structure. InSection 9, we prove Theorem 1.8. Note that Theorem 1.8 was essentially obtained in [LZ]and [S] under some extra assumptions. In Section 10, we prove Theorems 1.4, 1.5, and 1.6.We note that Theorem 1.5 is a special case of Theorem 1.6. In Section 11, we discuss anampleness criterion for quasi-log schemes. As a very special case, we prove Theorem 1.11.In Section 12, we treat Theorems 1.12 and 1.13. They are generalizations of Kawamata’sfamous result for quasi-log schemes. In Section 13, we prove Theorem 1.14, which is wellknown for normal pairs. In Section 14, we discuss several results related to Conjecture1.15.
Acknowledgments.
The author was partially supported by JSPS KAKENHI GrantNumbers JP16H03925, JP16H06337. He thanks Kenta Hashizume very much for manyuseful comments and suggestions. 2.
Preliminaries
We will work over C , the complex number field, throughout this paper. In this paper, a scheme means a separated scheme of finite type over C . A variety means an integral scheme,that is, an irreducible and reduced separated scheme of finite type over C . Note that Z , Q ,and R denote the set of integers , rational numbers , and real numbers , respectively. We alsonote that Q > and R > are the set of positive rational numbers and positive real numbers ,respectively.2.1. Basic definitions.
We collect some basic definitions and several useful results. Letus start with the definition of Q -line bundles and R -line bundles . Definition 2.1 ( Q -line bundles and R -line bundles) . Let X be a scheme and let Pic( X ) bethe group of line bundles on X , that is, the Picard group of X . An element of Pic( X ) ⊗ Z R (resp. Pic( X ) ⊗ Z Q ) is called an R -line bundle (resp. a Q -line bundle ) on X .In this paper, we write the group law of Pic( X ) ⊗ Z R additively for simplicity of notation.The notion of R -Cartier divisors and Q -Cartier divisors also plays a crucial role for thestudy of higher-dimensional algebraic varieties. Definition 2.2 ( Q -Cartier divisors and R -Cartier divisors) . Let X be a scheme and letDiv( X ) be the group of Cartier divisors on X . An element of Div( X ) ⊗ Z R (resp. Div( X ) ⊗ Z Q ) is called an R -Cartier divisor (resp. a Q -Cartier divisor ) on X . Let ∆ and ∆ be R -Cartier (resp. Q -Cartier) divisors on X . Then ∆ ∼ R ∆ (resp. ∆ ∼ Q ∆ ) means that ∆ is R -linearly (resp. Q -linearly) equivalent to ∆ . Let f : X → Y be a morphism betweenschemes and let D be an R -Cartier divisor on X . Then D ∼ R ,f R -Cartier divisor G on Y such that D ∼ R f ∗ G .The following remark is very important. Remark 2.3 (see [F11, Remark 6.2.3]) . Let X be a scheme. We have the following grouphomomorphism Div( X ) → Pic( X )given by A
7→ O X ( A ), where A is a Cartier divisor on X . Hence it induces a homomorphism δ X : Div( X ) ⊗ Z R → Pic( X ) ⊗ Z R . Note that Div( X ) → Pic( X ) is not always surjective. We write A + L ∼ R B + M for A, B ∈ Div( X ) ⊗ Z R and L , M ∈
Pic( X ) ⊗ Z R . This means that δ X ( A ) + L = δ X ( B ) + M holds in Pic( X ) ⊗ Z R . We usually use this type of abuse of notation, that is, the confusionof R -line bundles with R -Cartier divisors. In the theory of minimal models for higher-dimensional algebraic varieties, we sometimes use R -Cartier divisors for ease of notationeven when they should be R -line bundles.On normal varieties or equidimensional reduced schemes, we often treat R -divisors and Q -divisors. Definition 2.4 (Operations for Q -divisors and R -divisors) . Let X be an equidimensionalreduced scheme. Note that X is not necessarily regular in codimension one. Let D be an R -divisor (resp. a Q -divisor), that is, D is a finite formal sum P i d i D i , where D i is anirreducible reduced closed subscheme of X of pure codimension one and d i is a real number(resp. a rational number) for every i such that D i = D j for i = j . We put D
In the theory of minimal models, we are mainly interested in normal quasi-projective varieties. Let X be a normal variety. Then, for K = Z , Q , and R , the homo-morphism α : Div( X ) ⊗ Z K → Pic( X ) ⊗ Z K is surjective and the homomorphism β : Div( X ) ⊗ Z K → Weil( X ) ⊗ Z K is injective, where Weil( X ) is the abelian group generated by Weil divisors on X . Weusually use the surjection α and the injection β implicitly. In this paper, however, wefrequently treat highly singular schemes X . Hence we have to be careful when we consider α : Div( X ) ⊗ Z K → Pic( X ) ⊗ Z K and β : Div( X ) ⊗ Z K → Weil( X ) ⊗ Z K .Let us recall the following standard notation for the sake of completeness. ONE THEOREM AND MORI HYPERBOLICITY 11
Definition 2.6 ( N ( X/S ), N ( X/S ), ρ ( X/S ), and so on) . Let π : X → S be a propermorphism between schemes. Let Z ( X/S ) be the free abelian group generated by integralcomplete curves which are mapped to points on S by π . Then we obtain a bilinear form · : Pic( X ) × Z ( X/S ) → Z , which is induced by the intersection pairing. We have the notion of numerical equivalence both in Z ( X/S ) and in Pic( X ), which is denoted by ≡ , and we obtain a perfect pairing N ( X/S ) × N ( X/S ) → R , where N ( X/S ) = { Pic( X ) / ≡} ⊗ Z R and N ( X/S ) = { Z ( X/S ) / ≡} ⊗ Z R . It is well known that dim R N ( X/S ) = dim R N ( X/S ) < ∞ . We write ρ ( X/S ) = dim R N ( X/S ) = dim R N ( X/S )and call it the relative Picard number of X over S . When S = Spec C , we usually drop / Spec C from the notation, for example, we simply write N ( X ) instead of N ( X/ Spec C ).We will freely use the following useful lemma without mentioning it explicitly in thesubsequent sections. Lemma 2.7 (Relative real Nakai–Moishezon ampleness criterion) . Let π : X → S be aproper morphism between schemes and let L be an R -line bundle on X . Then L is π -ampleif and only if L dim Z · Z > for every positive-dimensional closed integral subscheme Z ⊂ X such that π ( Z ) is a point. For the details of Lemma 2.7, see [FM]. In the theory of quasi-log schemes, we mainlytreat highly singular reducible schemes. Hence Lemma 2.7 is very useful in order to checkthe ampleness of R -line bundles.2.2. Uniruledness, rationally connectedness, and rationally chain connected-ness.
In this subsection, we quickly recall the notion of uniruledness, rationally connect-edness, rationally chain connectedness, and so on. We need it for Theorems 1.12, 1.13, and1.14. For the details, see [Ko1, Chapter IV.]. We note that a scheme means a separatedscheme of finite type over C in this paper. Let us start with the definition of uniruledvarieties. Definition 2.8 (Uniruledness, see [Ko1, Chapter IV. 1.1 Definition]) . Let X be a variety.We say that X is uniruled if there exist a variety Y of dimension dim X − P × Y X. Although the notion of rationally connectedness is dispensable for Theorem 1.14, weexplain it for the reader’s convenience.
Definition 2.9 (Rationally connectedness, see [Ko1, Chapter IV. 3.6 Proposition]) . Let X be a projective variety. We say that X is rationally connected if for general closed points x , x ∈ X there exists an irreducible rational curve C which contains x and x .The following lemma is almost obvious by definition. Lemma 2.10.
Let X X ′ be a generically finite dominant rational map between vari-eties. If X is uniruled, then X ′ is also uniruled. Furthermore, we assume that X X ′ isa birational map between projective varieties. Then X is rationally connected if and onlyif X ′ is rationally connected. Let us define rationally chain connectedness for projective schemes.
Definition 2.11 (Rationally chain connectedness, see [Ko1, Chapter IV. 3.5 Corollaryand 3.6 Proposition]) . Let X be a projective scheme. We say that X is rationally chainconnected if for arbitrary closed points x , x ∈ X there is a connected curve C whichcontains x and x such that every irreducible component of C is rational.Note that X may be reducible in Definition 2.11. For projective varieties, we have: Lemma 2.12.
Let X be a projective variety. If X is rationally connected, then X isrationally chain connected.Proof. This follows from [Ko1, Chapter IV. 3.6 Proposition]. (cid:3)
We need the following definition for Theorem 1.14.
Definition 2.13 ([HM, Definition 1.1]) . Let X be a projective scheme and let V be anyclosed subset. We say that X is rationally chain connected modulo V if(1) either V = ∅ and X is rationally chain connected, or(2) V = ∅ and, for every P ∈ X , there is a connected pointed curve 0 , ∞ ∈ C withrational irreducible components and a morphism h P : C → X such that h P (0) = P and h P ( ∞ ) ∈ V .We close this subsection with a small remark. Remark 2.14.
Let X be a singular normal projective rationally chain connected variety.Then the resolution of X is not always rationally chain connected. Hence the notion ofrationally chain connectedness is more subtle than that of uniruledness and rationallyconnectedness (see Lemma 2.10). 3. On normal pairs
In this section, we collect some basic definitions and then discuss dlt blow-ups for normalpairs. Note that the results on dlt blow-ups discussed in Subsection 3.2 are new. For thedetails of normal pairs, see [BCHM], [F6], and [F11]. Let us start with the definition ofnormal pairs in this paper.
Definition 3.1 (Normal pairs) . A normal pair ( X, ∆) consists of a normal variety X andan R -divisor ∆ on X such that K X + ∆ is R -Cartier. Here we do not always assume that∆ is effective.We note the following definition of exceptional loci of birational morphisms betweenvarieties. Definition 3.2 (Exceptional loci) . Let f : X → Y be a birational morphism betweenvarieties. Then the exceptional locus Exc( f ) of f : X → Y is the set { x ∈ X | f is not biregular at x } . Singularities of pairs.
Let us explain singularities of pairs and some related defini-tions.
Definition 3.3.
Let X be a variety and let E be a prime divisor on Y for some birationalmorphism f : Y → X from a normal variety Y . Then E is called a divisor over X . Definition 3.4 (Singularities of pairs) . Let ( X, ∆) be a normal pair and let f : Y → X be a projective birational morphism from a normal variety Y . Then we can write K Y = f ∗ ( K X + ∆) + X E a ( E, X, ∆) E ONE THEOREM AND MORI HYPERBOLICITY 13 with f ∗ X E a ( E, X, ∆) E ! = − ∆ , where E runs over prime divisors on Y . We call a ( E, X, ∆) the discrepancy of E withrespect to ( X, ∆). Note that we can define the discrepancy a ( E, X, ∆) for any primedivisor E over X by taking a suitable resolution of singularities of X . If a ( E, X, ∆) ≥ − > −
1) for every prime divisor E over X , then ( X, ∆) is called sub log canonical (resp. sub kawamata log terminal ). We further assume that ∆ is effective. Then ( X, ∆)is called log canonical and kawamata log terminal ( lc and klt , for short) if it is sub logcanonical and sub kawamata log terminal, respectively.Let ( X, ∆) be a log canonical pair. If there exists a projective birational morphism f : Y → X from a smooth variety Y such that both Exc( f ) and Exc( f ) ∪ Supp f − ∗ ∆are simple normal crossing divisors on Y and that a ( E, X, ∆) > − f -exceptional divisor E on Y , then ( X, ∆) is called divisorial log terminal ( dlt , for short).Let ( X, ∆) be a normal pair. If there exist a projective birational morphism f : Y → X from a normal variety Y and a prime divisor E on Y such that ( X, ∆) is sub log canonicalin a neighborhood of the generic point of f ( E ) and that a ( E, X, ∆) = −
1, then f ( E ) iscalled a log canonical center (an lc center , for short) of ( X, ∆). A closed subvariety W of X is called a log canonical stratum (an lc stratum , for short) of ( X, ∆) if W is a logcanonical center of ( X, ∆) or W is X itself.Although it is well known, we recall the notion of multiplier ideal sheaves here for thereader’s convenience. Definition 3.5 (Multiplier ideal sheaves and non-lc ideal sheaves) . Let X be a normalvariety and let ∆ be an effective R -divisor on X such that K X + ∆ is R -Cartier. Let f : Y → X be a resolution with K Y + ∆ Y = f ∗ ( K X + ∆)such that Supp ∆ Y is a simple normal crossing divisor on Y . We put J ( X, ∆) = f ∗ O Y ( −⌊ ∆ Y ⌋ ) . Then J ( X, ∆) is an ideal sheaf on X and is known as the multiplier ideal sheaf associatedto the pair ( X, ∆). It is independent of the resolution f : Y → X . The closed subschemeNklt( X, ∆) defined by J ( X, ∆) is called the non-klt locus of ( X, ∆). It is obvious that( X, ∆) is kawamata log terminal if and only if J ( X, ∆) = O X . Similarly, we put J NLC ( X, ∆) = f ∗ O X ( −⌊ ∆ Y ⌋ + ∆ =1 Y )and call it the non-lc ideal sheaf associated to the pair ( X, ∆). We can check that it isindependent of the resolution f : Y → X . The closed subscheme Nlc( X, ∆) defined by J NLC ( X, ∆) is called the non-lc locus of ( X, ∆). It is obvious that ( X, ∆) is log canonicalif and only if J NLC ( X, ∆) = O X .By definition, the natural inclusion J ( X, ∆) ⊂ J NLC ( X, ∆)always holds. Therefore, we haveNlc( X, ∆) ⊂ Nklt( X, ∆) . For the details of J ( X, ∆) and J NLC ( X, ∆), see [F4], [F6, Section 7], and [L, Chapter9]. In this paper, we need the notion of open lc strata . Definition 3.6 (Open lc strata) . Let ( X, ∆) be a normal pair such that ∆ is effective.Let W be an lc stratum of ( X, ∆). We put U := W \ ( ( W ∩ Nlc( X, ∆)) ∪ [ W ′ W ′ ) , where W ′ runs over lc centers of ( X, ∆) strictly contained in W , and call it the open lcstratum of ( X, ∆) associated to W .3.2. Dlt blow-ups revisited.
Let us discuss dlt blow-ups. We give a slight generalizationof [F11, Theorem 4.4.21]. Here we use the theory of minimal models mainly due to [BCHM].Let us start with the definition of movable divisors.
Definition 3.7 (Movable divisors and movable cones, see [F11, Definition 2.4.4]) . Let f : X → Y be a projective morphism from a normal variety X onto a variety Y . A Cartierdivisor D on X is called f -movable or movable over Y if f ∗ O X ( D ) = 0 and if the cokernelof the natural homomorphism f ∗ f ∗ O X ( D ) → O X ( D )has a support of codimension ≥ X/Y ) as the closure of the convex cone in N ( X/Y ) generated by the nu-merical equivalence classes of f -movable Cartier divisors. We call Mov( X/Y ) the movablecone of f : X → Y .The following lemma is a very minor generalization of [F11, Lemma 2.4.5]. Lemma 3.8 (Negativity lemma) . Let f : X → Y be a projective birational morphismbetween normal varieties. Let E be an R -Cartier R -divisor on X such that − f ∗ E is effectiveand E ∈ Mov(
X/Y ) . Then − E is effective.Proof. We take a resolution of singularities of X . Then we may assume that X is smooth.We write E = E + − E − such that E + and E − are effective R -divisors and have no commonirreducible components. By assumption, E + is f -exceptional. Hence the proof of [F11,Lemma 2.4.5] works without any changes. Therefore, we obtain that E + = 0, equivalently, − E is effective. (cid:3) By Lemma 3.8, we can prove the existence of dlt blow-ups for quasi-projective normalpairs. We note that ∆ is assumed to be a boundary R -divisor in [F11, Theorem 4.4.21]. Theorem 3.9 (Dlt blow-ups) . Let X be a normal quasi-projective variety and let ∆ = P i d i ∆ i be an effective R -divisor on X such that K X + ∆ is R -Cartier. In this case, wecan construct a projective birational morphism f : Y → X from a normal quasi-projectivevariety Y with the following properties. (i) Y is Q -factorial. (ii) a ( E, X, ∆) ≤ − for every f -exceptional divisor E on Y . (iii) We put ∆ † = X ONE THEOREM AND MORI HYPERBOLICITY 15 Sketch of Proof of Theorem 3.9. Let g : Z → X be a resolution such that Exc( g ) ∪ Supp g − ∗ ∆is a simple normal crossing divisor on X and g is projective. We write K Z + e ∆ = g ∗ ( K X + ∆) + F, where e ∆ = X Let X be a normal quasi-projective variety and let ∆ be an effective R -divisor on X such that K X + ∆ is R -Cartier. Then we can construct a projective birationalmorphism g : Y → X from a normal Q -factorial variety Y with the following properties. (i) K Y + ∆ Y := g ∗ ( K X + ∆) , (ii) the pair Y, ∆ ′ Y := X d i < d i D i + X d i ≥ D i ! is dlt, where ∆ Y = P i d i D i is the irreducible decomposition of ∆ Y , (iii) every g -exceptional prime divisor is a component of (∆ ′ Y ) =1 , and (iv) g − Nklt( X, ∆) coincides with Nklt( Y, ∆ Y ) and Nklt( Y, ∆ ′ Y ) set theoretically. By Theorem 3.9, the proof of [S, Theorem 3.4] works without any changes even when ∆is not a boundary R -divisor. We give a proof for the sake of completeness. Proof of Lemma 3.10. There exists a dlt blow-up α : Z → X with K Z + ∆ Z := α ∗ ( K X + ∆)satisfying (i), (ii), and (iii) by Theorem 3.9. Note that ( Z, ∆ < Z ) is a Q -factorial kawamatalog terminal pair. We take a minimal model ( Z ′ , ∆ < Z ′ ) of ( Z, ∆ < Z ) over X by [BCHM]. Z α ❅❅❅❅❅❅❅❅ ϕ / / ❴❴❴❴❴❴❴ Z ′ α ′ ~ ~ ⑥⑥⑥⑥⑥⑥⑥⑥ X Then K Z ′ + ∆ < Z ′ ∼ R − ∆ ≥ Z ′ + α ′∗ ( K X + ∆) is nef over X . Of course, we put ∆ Z ′ = ϕ ∗ ∆ Z .We take a dlt blow-up β : Y → Z ′ of ( Z ′ , ∆ < Z ′ + Supp ∆ ≥ Z ′ ) again by Theorem 3.9 (or [F11,Theorem 4.4.21]) and put g := α ′ ◦ β : Y → X . It is not difficult to see that this birationalmorphism g : Y → X with K Y + ∆ Y := g ∗ ( K X + ∆) satisfies the desired properties. It isobvious that g − Nklt( X, ∆) contains the support of β ∗ ∆ ≥ Z ′ . Since − β ∗ ∆ ≥ Z ′ is nef over X ,we see that β ∗ ∆ ≥ Z ′ coincides with g − Nklt( X, ∆) set theoretically. (cid:3) For the details of the proof of Lemma 3.10, see [S, Theorem 3.4]. In [FH], Theorem 3.9and Lemma 3.10 will be generalized completely by using the minimal model program forlog canonical pairs established in [H2].4. On quasi-log schemes In this section, we explain some basic definitions and results on quasi-log schemes. Forthe details of the theory of quasi-log schemes, we recommend the reader to see [F11,Chapter 6]. Definitions and basic properties of quasi-log schemes. The notion of quasi-logschemes was first introduced by Florin Ambro (see [A]) in order to establish the cone andcontraction theorem for ( X, ∆), where X is a normal variety and ∆ is an effective R -divisoron X such that K X + ∆ is R -Cartier. Here we use the formulation in [F11, Chapter 6],which is slightly different from Ambro’s original one. We recommend the interested readerto see [F12, Appendix A] for the difference between our definition of quasi-log schemes andAmbro’s one.In order to define quasi-log schemes, we use the notion of globally embedded simplenormal crossing pairs . Definition 4.1 (Globally embedded simple normal crossing pairs, see [F11, Definition6.2.1]) . Let Y be a simple normal crossing divisor on a smooth variety M and let B bean R -divisor on M such that Supp( B + Y ) is a simple normal crossing divisor on M andthat B and Y have no common irreducible components. We put B Y = B | Y and considerthe pair ( Y, B Y ). We call ( Y, B Y ) a globally embedded simple normal crossing pair and M the ambient space of ( Y, B Y ). A stratum of ( Y, B Y ) is a log canonical center of ( M, Y + B )that is contained in Y .Let us recall the definition of quasi-log schemes . Definition 4.2 (Quasi-log schemes, see [F11, Definition 6.2.2]) . A quasi-log scheme is ascheme X endowed with an R -Cartier divisor (or R -line bundle) ω on X , a closed subscheme X −∞ ( X , and a finite collection { C } of reduced and irreducible subschemes of X suchthat there is a proper morphism f : ( Y, B Y ) → X from a globally embedded simple normalcrossing pair satisfying the following properties:(1) f ∗ ω ∼ R K Y + B Y .(2) The natural map O X → f ∗ O Y ( ⌈− ( B < Y ) ⌉ ) induces an isomorphism I X −∞ ≃ −→ f ∗ O Y ( ⌈− ( B < Y ) ⌉ − ⌊ B > Y ⌋ ) , where I X −∞ is the defining ideal sheaf of X −∞ .(3) The collection of reduced and irreducible subschemes { C } coincides with the imagesof the strata of ( Y, B Y ) that are not included in X −∞ .We simply write [ X, ω ] to denote the above data( X, ω, f : ( Y, B Y ) → X )if there is no risk of confusion. Note that a quasi-log scheme [ X, ω ] is the union of { C } and X −∞ . The reduced and irreducible subschemes C are called the qlc strata of [ X, ω ], X −∞ is called the non-qlc locus of [ X, ω ], and f : ( Y, B Y ) → X is called a quasi-log resolution of[ X, ω ]. We sometimes use Nqlc( X, ω ) orNqlc( X, ω, f : ( Y, B Y ) → X )to denote X −∞ . If a qlc stratum C of [ X, ω ] is not an irreducible component of X , then itis called a qlc center of [ X, ω ].We say that ( X, ω, f : ( Y, B Y ) → X ) or [ X, ω ] has a Q -structure if B Y is a Q -divisor, ω isa Q -Cartier divisor (or Q -line bundle), and f ∗ ω ∼ Q K Y + B Y holds in the above definition.In this paper, the notion of open qlc strata is indispensable. Definition 4.3 (Open qlc strata) . Let W be a qlc stratum of a quasi-log scheme [ X, ω ].We put U := W \ ( ( W ∩ Nqlc( X, ω )) ∪ [ W ′ W ′ ) , ONE THEOREM AND MORI HYPERBOLICITY 17 where W ′ runs over qlc centers of [ X, ω ] strictly contained in W , and call it the open qlcstratum of [ X, ω ] associated to W .In Section 11, we need the notion of log bigness . For the details of relatively big R -divisors, see [F11, Section 2.1]. Definition 4.4 (Log bigness) . Let [ X, ω ] be a quasi-log scheme and let π : X → S be aproper morphism between schemes. Let D be an R -Cartier divisor (or R -line bundle) on X . We say that D is log big over S with respect to [ X, ω ] if D | W is big over π ( W ) for everyqlc stratum W of [ X, ω ].We collect some basic and important properties of quasi-log schemes for the reader’sconvenience. Theorem 4.5 ([F11, Theorem 6.3.4]) . In Definition 4.2, we may assume that the ambientspace M of the globally embedded simple normal crossing pair ( Y, B Y ) is quasi-projective.In particular, Y is quasi-projective and f : Y → X is projective. For the details of Theorem 4.5, see the proof of [F11, Theorem 6.3.4]. In the theory ofquasi-log schemes, we sometimes need the projectivity of f in order to use the theory ofvariations of mixed Hodge structure (see [F14] and [FFL]). Hence Theorem 4.5 plays acrucial role. The most important result in the theory of quasi-log schemes is as follows. Theorem 4.6 ([F11, Theorem 6.3.5]) . Let [ X, ω ] be a quasi-log scheme and let X ′ be theunion of X −∞ with a ( possibly empty ) union of some qlc strata of [ X, ω ] . Then we havethe following properties. (i) (Adjunction). Assume that X ′ = X −∞ . Then X ′ naturally becomes a quasi-logscheme with ω ′ = ω | X ′ and X ′−∞ = X −∞ . Moreover, the qlc strata of [ X ′ , ω ′ ] areexactly the qlc strata of [ X, ω ] that are included in X ′ . (ii) (Vanishing theorem). Assume that π : X → S is a proper morphism between schemes.Let L be a Cartier divisor on X such that L − ω is nef and log big over S withrespect to [ X, ω ] . Then R i π ∗ ( I X ′ ⊗ O X ( L )) = 0 for every i > , where I X ′ is thedefining ideal sheaf of X ′ on X . In this paper, we will repeatedly use adjunction for quasi-log schemes in Theorem 4.6(i). We strongly recommend the reader to see the proof of [F11, Theorem 6.3.5]. Here, weonly explain the main idea of the proof of Theorem 4.6 (i) for the reader’s convenience. Idea of Proof of Theorem 4.6 (i). By definition, X ′ is the union of X −∞ with a union ofsome qlc strata of [ X, ω ] set theoretically. We assume that X ′ = X −∞ holds. By [F11,Proposition 6.3.1], we may assume that the union of all strata of ( Y, B Y ) mapped to X ′ by f , which is denoted by Y ′ , is a union of some irreducible components of Y . We put Y ′′ = Y − Y ′ , K Y ′′ + B Y ′′ = ( K Y + B Y ) | Y ′′ , and K Y ′ + B Y ′ = ( K Y + B Y ) | Y ′ . We set f ′′ = f | Y ′′ and f ′ = f | Y ′ . Then we claim that( X ′ , ω ′ , f ′ : ( Y ′ , B Y ′ ) → X ′ )becomes a quasi-log scheme satisfying the desired properties. Let us consider the followingshort exact sequence:0 → O Y ′′ ( ⌈− ( B < Y ′′ ) ⌉ − ⌊ B > Y ′′ ⌋ − Y ′ | Y ′′ ) → O Y ( ⌈− ( B < Y ) ⌉ − ⌊ B > Y ⌋ ) → O Y ′ ( ⌈− ( B < Y ′ ) ⌉ − ⌊ B > Y ′ ⌋ ) → , which is induced by 0 → O Y ′′ ( − Y ′ | Y ′′ ) → O Y → O Y ′ → . We take the associated long exact sequence. Then we can check that the connectinghomomorphism δ : f ′∗ O Y ′ ( ⌈− ( B < Y ′ ) ⌉ − ⌊ B > Y ′ ⌋ ) → R f ′′∗ O Y ′′ ( ⌈− ( B < Y ′′ ) ⌉ − ⌊ B > Y ′′ ⌋ − Y ′ | Y ′′ )is zero by using a generalization of Koll´ar’s torsion-freeness based on the theory of mixedHodge structures on cohomology with compact support (see [F11, Chapter 5]). We put I X ′ := f ′′∗ O Y ′′ ( ⌈− ( B < Y ′′ ) ⌉ − ⌊ B > Y ′′ ⌋ − Y ′ | Y ′′ ) , which is an ideal sheaf on X since I X ′ ⊂ I X −∞ , and define a scheme structure on X ′ by I X ′ . Then we obtain the following big commutative diagram:0 (cid:15) (cid:15) (cid:15) (cid:15) / / f ′′∗ O Y ′′ ( ⌈− ( B < Y ′′ ) ⌉ − ⌊ B > Y ′′ ⌋ − Y ′ | Y ′′ ) = / / (cid:15) (cid:15) I X ′ (cid:15) (cid:15) / / f ∗ O Y ( ⌈− ( B < Y ) ⌉ − ⌊ B > Y ⌋ ) = I X −∞ / / (cid:15) (cid:15) O X (cid:15) (cid:15) / / O X −∞ / / / / f ′∗ O Y ′ ( ⌈− ( B < Y ′ ) ⌉ − ⌊ B > Y ′ ⌋ ) = I X ′−∞ (cid:15) (cid:15) / / O X ′ (cid:15) (cid:15) / / O X ′−∞ / / 00 0by the above arguments. More precisely, by the above big commutative diagram, I X ′−∞ = f ′∗ O Y ′ ( ⌈− ( B < Y ′ ) ⌉ − ⌊ B > Y ′ ⌋ )is an ideal sheaf on X ′ such that O X / I X −∞ = O X ′ / I X ′−∞ . Thus we obtain that( X ′ , ω ′ , f ′ : ( Y ′ , B Y ′ ) → X ′ )is a quasi-log scheme satisfying the desired properties. (cid:3) As an obvious corollary, we have: Corollary 4.7 ([F11, Notation 6.3.10]) . Let [ X, ω ] be a quasi-log scheme. The union of X −∞ with all qlc centers of [ X, ω ] is denoted by Nqklt( X, ω ) , or, more precisely, Nqklt( X, ω, f : ( Y, B Y ) → X ) . If Nqklt( X, ω ) = X −∞ , then [Nqklt( X, ω ) , ω | Nqklt( X,ω ) ] naturally becomes a quasi-log scheme by adjunction. In the framework of quasi-log schemes, Nqklt( X, ω ) plays an important role by inductionon dimension. When Nqklt( X, ω ) = ∅ , we have the following lemma. Lemma 4.8 ([F11, Lemma 6.3.9]) . Let [ X, ω ] be a quasi-log scheme with X −∞ = ∅ .Assume that every qlc stratum of [ X, ω ] is an irreducible component of X , equivalently, Nqklt( X, ω ) = ∅ . Then X is normal. For the proof of Lemma 4.8, see [F11, Lemma 6.3.9]. It is convenient to introduce thenotion of quasi-log canonical pairs . ONE THEOREM AND MORI HYPERBOLICITY 19 Definition 4.9 (Quasi-log canonical pairs, see [F11, Definition 6.2.9]) . Let( X, ω, f : ( Y, B Y ) → X )be a quasi-log scheme. If X −∞ = ∅ , then it is called a quasi-log canonical pair ( qlc pair ,for short).By using adjunction, we can prove: Theorem 4.10 ([F11, Theorem 6.3.11 (i)]) . Let [ X, ω ] be a quasi-log canonical pair. Thenthe intersection of two qlc strata is a union of qlc strata. The following example is very important. Example 4.11 shows that we can treat logcanonical pairs as quasi-log canonical pairs. In some sense, Ambro introduced the notionof quasi-log schemes in order to treat the following example (see [A]). Example 4.11 ([F11, 6.4.1]) . Let ( X, ∆) be a normal pair such that ∆ is effective. Let f : Y → X be a resolution of singularities such that K Y + B Y = f ∗ ( K X + ∆)and that Supp B Y is a simple normal crossing divisor on Y . We put ω = K X + ∆. Then K Y + B Y ∼ R f ∗ ω holds. Since ∆ is effective, ⌈− ( B < Y ) ⌉ is effective and f -exceptional.Therefore, the natural map O X → f ∗ O Y ( ⌈− ( B < Y ) ⌉ )is an isomorphism. We put I X −∞ := J NLC ( X, ∆) = f ∗ O Y ( ⌈− ( B < Y ) ⌉ − ⌊ B > Y ⌋ ) , where J NLC ( X, ∆) is the non-lc ideal sheaf associated to ( X, ∆) in Definition 3.5. We put M = Y × C and D = B Y × C . Then ( Y, B Y ) ≃ ( Y × { } , B Y × { } ) is a globally embeddedsimple normal crossing pair. Thus( X, ω, f : ( Y, B Y ) → X )becomes a quasi-log scheme. By construction, ( X, ∆) is log canonical if and only if [ X, ω ]is quasi-log canonical. We note that C is a log canonical center of ( X, B ) if and only if C is a qlc center of [ X, ω ]. We also note that X itself is a qlc stratum of [ X, ω ].Example 4.11 shows that [ X, K X + ∆] has a natural quasi-log scheme structure. Ingeneral, however, [ X, K X + ∆] has many different quasi-log scheme structures. Remark 4.12. In Example 4.11, we take an effective R -divisor ∆ ′ on X such that K X +∆ ∼ R K X + ∆ ′ . Let f ′ : Y ′ → X be a resolution of singularities such that K Y ′ + B Y ′ = ( f ′ ) ∗ ( K X + ∆ ′ )and that Supp B Y ′ is a simple normal crossing divisor on Y ′ . Then( X, ω, f ′ : ( Y ′ , B Y ′ ) → X )is also a quasi-log scheme since K Y ′ + B Y ′ ∼ R ( f ′ ) ∗ ω . In this case, there is no correspon-dence between qlc strata of ( X, ω, f ′ : ( Y ′ , B Y ′ ) → X ) and lc strata of ( X, ∆).By combining Theorem 4.10 with Example 4.11, we have: Corollary 4.13 ([F6, Theorem 9.1 (2)]) . Let ( X, ∆) be a log canonical pair. Then theintersection of two lc centers is a union of lc centers. For the basic properties of quasi-log schemes, see [F11, Chapter 6]. We also recommendthe reader to see [F5], which is a gentle introduction to the theory of quasi-log schemes.In [F8], we establish that every quasi-projective semi-log canonical pair naturally becomesa quasi-log canonical pair. Hence we can use the theory of quasi-log schemes for the studyof semi-log canonical pairs. For the details, see [F8]. Kleiman–Mori cones. In this subsection, we discuss basic definitions and resultsaround Kleiman–Mori cones of quasi-log schemes. Let us start with the definition ofKleiman–Mori cones. Definition 4.14 (Kleiman–Mori cones) . Let π : X → S be a proper morphism betweenschemes. Let N E ( X/S ) be the convex cone in N ( X/S ) generated by effective 1-cycles on X mapped to points by π . Let N E ( X/S ) be the closure of N E ( X/S ) in N ( X/S ). Wecall it the Kleiman–Mori cone of π : X → S . As usual, we drop / Spec C from the notationwhen S = Spec C .Let us explain some basic definitions. Definition 4.15 ([F11, Definition 6.7.1]) . Let [ X, ω ] be a quasi-log scheme with the non-qlc locus X −∞ . Let π : X → S be a projective morphism between schemes. We put N E ( X/S ) −∞ = Im (cid:0) N E ( X −∞ /S ) → N E ( X/S ) (cid:1) . We sometimes use N E ( X/S ) Nqlc( X/S ) to denote N E ( X/S ) −∞ . For an R -Cartier divisor(or R -line bundle) D , we define D ≥ = { z ∈ N ( X/S ) | D · z ≥ } . Similarly, we can define D > , D ≤ , and D < . We also define D ⊥ = { z ∈ N ( X/S ) | D · z = 0 } . We use the following notation N E ( X/S ) D ≥ = N E ( X/S ) ∩ D ≥ , and similarly for > ≤ 0, and < Definition 4.16 ([F11, Definition 6.7.2]) . An extremal face of N E ( X/S ) is a non-zerosubcone F ⊂ N E ( X/S ) such that z, z ′ ∈ F and z + z ′ ∈ F imply that z, z ′ ∈ F . Equiva-lently, F = N E ( X/S ) ∩ H ⊥ for some π -nef R -divisor (or π -nef R -line bundle) H , which iscalled a support function of F . An extremal ray is a one-dimensional extremal face.(1) An extremal face F is called ω -negative if F ∩ N E ( X/S ) ω ≥ = { } .(2) An extremal face F is called rational if we can choose a π -nef Q -divisor (or Q -linebundle) H as a support function of F .(3) An extremal face F is called relatively ample at infinity if F ∩ N E ( X/S ) −∞ = { } .Equivalently, H | X −∞ is π | X −∞ -ample for any supporting function H of F .The contraction theorem for quasi-log schemes plays an important role in this paper. Theorem 4.17 (Contraction theorem, see [F11, Theorem 6.7.3]) . Let [ X, ω ] be a quasi-logscheme and let π : X → S be a projective morphism between schemes. Let R be an ω -negative extremal ray of N E ( X/S ) that is rational and relatively ample at infinity. Thenthere exists a projective morphism ϕ R : X → Y over S with the following properties. (i) Let C be an integral curve on X such that π ( C ) is a point. Then ϕ R ( C ) is a pointif and only if [ C ] ∈ R , where [ C ] denotes the numerical equivalence class of C in N ( X/S ) . (ii) O Y ≃ ( ϕ R ) ∗ O X . (iii) Let L be a line bundle on X such that L · C = 0 for every curve C with [ C ] ∈ R .Then there is a line bundle L Y on Y such that L ≃ ϕ ∗ R L Y .Proof. Since R is relatively ample at infinity, ϕ R : X −∞ → ϕ R ( X −∞ ) is finite. Hence L ⊗ m | X −∞ is ϕ R | X −∞ -generated for every m ≥ 0. Therefore, this theorem is a special caseof [F11, Theorem 6.7.3]. (cid:3) ONE THEOREM AND MORI HYPERBOLICITY 21 Theorem 4.17 is a generalization of the famous Kawamata–Shokurov basepoint-free the-orem.4.3. Lemmas on quasi-log schemes. In this subsection, we treat useful lemmas onquasi-log schemes. The first two lemmas were already proved in [F10]. We will repeatedlyuse Lemma 4.19 throughout this paper. Lemma 4.18 ([F10, Lemma 3.12]) . Let ( X, ω, f : ( Y, B Y ) → X ) be a quasi-log scheme.Then we can construct a proper morphism f ′ : ( Y ′ , B Y ′ ) → X from a globally embeddedsimple normal crossing pair ( Y ′ , B Y ′ ) such that (i) f ′ : ( Y ′ , B Y ′ ) → X gives the same quasi-log scheme structure as one given by f : ( Y, B Y ) → X , and (ii) every irreducible component of Y ′ is mapped by f ′ to X \ X −∞ , the closure of X \ X −∞ in X . We give the proof for the sake of completeness. Proof. Let M be the ambient space of ( Y, B Y ). By taking some blow-ups of M , we mayassume that the union of all strata of ( Y, B Y ) that are not mapped to X \ X −∞ , whichis denoted by Y ′′ , is a union of some irreducible components of Y (see [F11, Proposition6.3.1]). We put Y ′ = Y − Y ′′ and K Y ′′ + B Y ′′ = ( K Y + B Y ) | Y ′′ . We may further assume thatthe union of all strata of ( Y, B Y ) mapped to X \ X −∞ ∩ X −∞ is a union of some irreduciblecomponents of Y by [F11, Proposition 6.3.1]. We consider the short exact sequence0 → O Y ′′ ( − Y ′ ) → O Y → O Y ′ → . We put A = ⌈− ( B < Y ) ⌉ and N = ⌊ B > Y ⌋ . By applying ⊗O Y ( A − N ), we have0 → O Y ′′ ( A − N − Y ′ ) → O Y ( A − N ) → O Y ′ ( A − N ) → . By taking R i f ∗ , we obtain0 → f ∗ O Y ′′ ( A − N − Y ′ ) → f ∗ O Y ( A − N ) → f ∗ O Y ′ ( A − N ) → R f ∗ O Y ′′ ( A − N − Y ′ ) → · · · . By [F11, Theorem 5.6.2], no associated prime of R f ∗ O Y ′′ ( A − N − Y ′ ) is contained in f ( Y ′ ) ∩ X −∞ . Note that( A − N − Y ′ ) | Y ′′ − ( K Y ′′ + { B Y ′′ } + B =1 Y ′′ − Y ′ | Y ′′ ) = − ( K Y ′′ + B Y ′′ ) ∼ R − ( f ∗ ω ) | Y ′′ . Therefore, the connecting homomorphism δ : f ∗ O Y ′ ( A − N ) → R f ∗ O Y ′′ ( A − N − Y ′ )is zero. This implies that0 → f ∗ O Y ′′ ( A − N − Y ′ ) → I X −∞ → f ∗ O Y ′ ( A − N ) → J = f ∗ O Y ′′ ( A − N − Y ′ ) is zero when it is restricted to X −∞ be-cause J ⊂ I X −∞ . On the other hand, J is zero on X \ X −∞ because f ( Y ′′ ) ⊂ X −∞ . There-fore, we obtain J = 0. Thus we have I X −∞ = f ∗ O Y ′ ( A − N ). So f ′ = f | Y ′ : ( Y ′ , B Y ′ ) → X ,where K Y ′ + B Y ′ = ( K Y + B Y ) | Y ′ , gives the same quasi-log scheme structure as one givenby f : ( Y, B Y ) → X with the property (ii). (cid:3) By using Lemma 4.18, we establish the following very useful lemma. Lemma 4.19 ([F10, Lemma 3.14]) . Let [ X, ω ] be a quasi-log scheme. Let us consider X † = X \ X −∞ , the closure in X , with the reduced scheme structure. Then [ X † , ω † ] , where ω † = ω | X † , has a natural quasi-log scheme structure induced by [ X, ω ] . This means that (i) C is a qlc stratum of [ X, ω ] if and only if C is a qlc stratum of [ X † , ω † ] , and (ii) I Nqlc( X,ω ) = I Nqlc( X † ,ω † ) holds. We include the proof for the benefit of the reader. Proof. Let I X † be the defining ideal sheaf of X † on X . Let f ′ : ( Y ′ , B Y ′ ) → X be thequasi-log resolution constructed in the proof of Lemma 4.18. By construction, f ′ : Y ′ → X factors through X † . Note that I X −∞ ≃ f ′∗ O Y ′ ( A − N ) ≃ f ′∗ O Y ′ ( − N )and that f ′ ( N ) = X −∞ ∩ f ′ ( Y ′ ) = X −∞ ∩ X † set theoretically, where A = ⌈− ( B < Y ′ ) ⌉ and N = ⌊ B > Y ′ ⌋ (see [F11, Remark 6.2.10]). There-fore, we obtain I X † ∩ I X −∞ = I X † ∩ f ′∗ O Y ′ ( − N ′ ) = { } . Thus we can construct the following big commutative diagram.0 (cid:15) (cid:15) (cid:15) (cid:15) f ′∗ O Y ′ ( A − N ) (cid:15) (cid:15) f ′∗ O Y ′ ( A − N ) (cid:15) (cid:15) / / I X † / / O X / / (cid:15) (cid:15) O X † / / (cid:15) (cid:15) / / I X † / / O X −∞ / / (cid:15) (cid:15) O X †−∞ (cid:15) (cid:15) / / 00 0Hence f ′ : ( Y ′ , B Y ′ ) → X † gives the desired quasi-log scheme structure on [ X † , ω † ]. (cid:3) By Lemmas 4.18 and 4.19, we can abandon unnecessary components from f : ( Y, B Y ) → X . Lemma 4.20 is almost obvious by definition. Lemma 4.20. Let ( X, ω, f : ( Y, B Y ) → X ) be a quasi-log scheme and let B be an effective R -Cartier divisor on X , that is, a finite R > -linear combination of effective Cartier divisors on X . Let X ′ be the union of Nqlc( X, ω ) and all qlc centers of [ X, ω ] contained in Supp B . Assume that the union of all strataof ( Y, B Y ) mapped to X ′ by f , which is denoted by Y ′ , is a union of some irreduciblecomponents of Y . We put Y ′′ = Y − Y ′ , K Y ′′ + B Y ′′ = ( K Y + B Y ) | Y ′′ , and f ′′ = f | Y ′′ . Wefurther assume that ( Y ′′ , B Y ′′ + ( f ′′ ) ∗ B ) is a globally embedded simple normal crossing pair. Then ( X, ω + B, f ′′ : ( Y ′′ , B Y ′′ + ( f ′′ ) ∗ B ) → X ) is a quasi-log scheme.Proof. Since K Y + B Y ∼ R f ∗ ω , we have K Y ′′ + B Y ′′ ∼ R ( f ′′ ) ∗ ω . Therefore, K Y ′′ + B Y ′′ +( f ′′ ) ∗ B ∼ R ( f ′′ ) ∗ ( ω + B ) holds true. By the proof of adjunction (see the idea of the proofof Theorem 4.6 (i) and the proof of [F11, Theorem 6.3.5 (i)]), we have I X ′ = f ′′∗ O X ′′ ( ⌈− ( B < Y ′′ ) ⌉ − ⌊ B > Y ′′ ⌋ − Y ′ | Y ′′ ) , ONE THEOREM AND MORI HYPERBOLICITY 23 where I X ′ is the defining ideal sheaf of X ′ on X . Note that the following key inequality ⌈− ( B Y ′′ + ( f ′′ ) ∗ B ) < ⌉ − ⌊ ( B Y ′′ + ( f ′′ ) ∗ B ) > ⌋ ≤ ⌈− ( B < Y ′′ ) ⌉ − ⌊ B > Y ′′ ⌋ − Y ′ | Y ′′ holds. Therefore, we put I Nqlc( X,ω + B ) := f ′′∗ O Y ′′ ( ⌈− ( B Y ′′ + ( f ′′ ) ∗ B ) < ⌉ − ⌊ ( B Y ′′ + ( f ′′ ) ∗ B ) > ⌋ ) ⊂ I X ′ ⊂ O X and define the closed subscheme Nqlc( X, ω + B ) of X by I Nqlc( X,ω + B ) . Then( X, ω + B, f ′′ : ( Y ′′ , B Y ′′ + ( f ′′ ) ∗ B ) → X )is a quasi-log scheme. Let W be a reduced and irreducible subscheme of X . As usual, wesay that W is a qlc stratum of [ X, ω + B ] when W is not contained in Nqlc( X, ω + B )and is the f ′′ -image of a stratum of ( Y ′′ , B Y ′′ + ( f ′′ ) ∗ B ). By construction, we have X ′ ⊂ Nqlc( X, ω + B ). We note that ( X, ω + B, f ′′ : ( Y ′′ , B Y ′′ + ( f ′′ ) ∗ B ) → X ) coincides with( X, ω, f : ( Y, B Y ) → X ) outside Supp B . (cid:3) By using Lemma 4.20, we can prove the following lemma. Lemma 4.21. Let [ X, ω ] be a quasi-log scheme and let G be an effective R -Cartier divisoron X , that is, a finite R > -linear combination of effective Cartier divisors on X . Then, forevery < ε ≪ , [ X, ω + εG ] naturally becomes a quasi-log scheme such that Nqklt( X, ω + εG ) = Nqklt( X, ω ) holds. More precisely, I Nqklt( X,ω + εG ) = I Nqklt( X,ω ) holds. Note that Lemma 4.21 is almost obvious for normal pairs by the definition of multiplierideal sheaves. Proof of Lemma 4.21. Let f : ( Y, B Y ) → X be a proper morphism from a globally em-bedded simple normal crossing pair ( Y, B Y ) as in Definition 4.2. Let X ′ be the union ofNqlc( X, ω ) and all qlc centers of [ X, ω ] contained in Supp G . By [F11, Proposition 6.3.1]and [Ko2, Theorem 3.35], we may assume that the union of all strata of ( Y, B Y ) mapped to X ′ by f , which is denoted by Y ′ , is a union of some irreducible components of Y . By [F11,Proposition 6.3.1] and [Ko2, Theorem 3.35] again, we may further assume that the union ofall strata of ( Y, B Y ) mapped to Nqklt( X, ω ) by f , which is denoted by Z ′ , is a union of someirreducible components of Y . By construction, Y ′ ≤ Z ′ obviously holds. As in Lemma4.20, we put Y ′′ = Y − Y ′ , K Y ′′ + B Y ′′ = ( K Y + B Y ) | Y ′′ , and f ′′ = f | Y ′′ . By [F11, Propo-sition 6.3.1] and [Ko2, Theorem 3.35], we further assume that ( Y ′′ , ( f ′′ ) ∗ G + Supp B Y ′′ ) isa globally embedded simple normal crossing pair. By Lemma 4.20, we know that( X, ω + εG, f ′′ : ( Y ′′ , B Y ′′ + ε ( f ′′ ) ∗ G ) → X )is a quasi-log scheme for every ε > 0. We put Z ′′ = Y − Z ′ , K Z ′′ + B Z ′′ = ( K Y + B Y ) | Z ′′ ,and h = f | Z ′′ . Thus, by the proof of adjunction (see the idea of the proof of Theorem 4.6(i) and the proof of [F11, Theorem 6.3.5 (i)]), we have I Nqklt( X,ω ) = h ∗ O Z ′′ ( ⌈− ( B < Z ′′ ) ⌉ − ⌊ B > Z ′′ ⌋ − Z ′ | Z ′′ ) . We note that ⌈− ( B < Z ′′ ) ⌉ − ⌊ B > Z ′′ ⌋ − Z ′ | Z ′′ = ⌊ B Z ′′ ⌋ holds by definition. On the other hand, by the proof of adjunction again (see the idea ofthe proof of Theorem 4.6 (i) and the proof of [F11, Theorem 6.3.5 (i)]), I Nqklt( X,ω + εG ) = h ∗ O Z ′′ ( ⌈− ( B Z ′′ + εh ∗ G ) < ⌉ − ⌊ ( B Z ′′ + εh ∗ G ) > ⌋ − ( Z ′ − Y ′ ) | Z ′′ )for every 0 < ε ≪ 1. By direct calculation, for 0 < ε ≪ ⌈− ( B Z ′′ + εh ∗ G ) < ⌉ − ⌊ ( B Z ′′ + εh ∗ G ) > ⌋ − ( Z ′ − Y ′ ) | Z ′′ = −⌊ B Z ′′ ⌋ = ⌈− ( B < Z ′′ ) ⌉ − ⌊ B > Z ′′ ⌋ − Z ′ | Z ′′ . Hence we obtain I Nqklt( X,ω + εG ) = I Nqklt( X,ω ) . This means that ( X, ω + εG, f ′′ : ( Y ′′ , B Y ′′ + ε ( f ′′ ) ∗ G ) → X )is a quasi-log scheme with Nqklt( X, ω + εG ) = Nqklt( X, ω )for 0 < ε ≪ 1. We finish the proof of Lemma 4.21. (cid:3) We need the following lemma in order to reduce some problems to the case where quasi-log schemes have Q -structures. Lemma 4.22. Let ( X, ω, f : ( Y, B Y ) → X ) be a quasi-log scheme. Then we obtain a Q -divisor D i on Y , a Q -line bundle ω i on X , and a positive real number r i for ≤ i ≤ k such that (i) P ki =1 r i = 1 , (ii) Supp D i = Supp B Y , D =1 i = B =1 Y , ⌊ D > i ⌋ = ⌊ B > Y ⌋ , and ⌈− ( D < i ) ⌉ = ⌈− ( B < Y ) ⌉ forevery i , (iii) ω = P ki =1 r i ω i and B Y = P ki =1 r i D i , and (iv) ( X, ω i , f : ( Y, D i ) → X ) is a quasi-log scheme with K Y + D i ∼ Q f ∗ ω i for every i .We note that Nqlc( X, ω i ) = Nqlc( X, ω ) holds for every i . We also note that W is a qlc stratum of [ X, ω ] if and only if W is a qlcstratum of [ X, ω i ] for every i .Proof. Without loss of generality, we may assume that ω is an R -line bundle. We put B Y = P j b j B j , where B j is a simple normal crossing divisor on Y for every j , b j = b j for j = j , and Supp B j and Supp B j have no common irreducible components for j = j .We may assume that b j ∈ R \ Q for 1 ≤ j ≤ l and b j ∈ Q for j ≥ l + 1. We put ω = P mp =1 a p ω p , where a p ∈ R and ω p is a line bundle on X for every p . We can write K Y + B Y = m X p =1 a p f ∗ ω p in Pic( Y ) ⊗ Z R . We consider the following linear map ψ : R l + m −→ Pic( Y ) ⊗ Z R defined by ψ ( x , . . . , x l + m ) = m X α =1 x α f ∗ ω α − l X β =1 x m + β B β . We note that ψ is defined over Q . By construction, A := ψ − K Y + X j ≥ l +1 b j B j ! is a nonempty affine subspace of R l + m defined over Q . We put P := ( a , . . . , a m , b , . . . , b l ) ∈ A . We can take P , . . . , P k ∈ A ∩ Q l + m and r , . . . , r k ∈ R > such that P ki =1 r i = 1 and P ki =1 r i P i = P in A . Note that we can make P i arbitrary close to P for every i . So wemay assume that P i is sufficiently close to P for every i . For each P i , we obtain(4.1) K Y + D i ∼ Q f ∗ ω i ONE THEOREM AND MORI HYPERBOLICITY 25 which satisfies (ii) by using ψ . By construction, (i) and (iii) hold. By (4.1) and (ii),( X, ω i , f : ( Y, D i ) → X )is a quasi-log scheme with the desired properties for every i . Therefore, we get (iv). (cid:3) Proof of Theorem 1.9 In this section, we prove Theorem 1.9. In some sense, Theorem 1.9 is a generalizationof [FLh, Theorem 1.1]. Proof of Theorem 1.9. Let f : ( Y, B Y ) → X be a proper surjective morphism from a quasi-projective globally embedded simple normal crossing pair ( Y, B Y ) as in Definition 4.2 (seeTheorem 4.5). By [F11, Proposition 6.3.1], we may assume that Y is quasi-projectiveand that the union of all strata of ( Y, B Y ) mapped to Nqklt( X, ω ), which is denotedby Y ′′ , is a union of some irreducible components of Y . We put Y ′ = Y − Y ′′ and K Y ′ + B Y ′ = ( K Y + B Y ) | Y ′ . Then we obtain the following commutative diagram: Y ′ f ′ (cid:15) (cid:15) (cid:31) (cid:127) ι / / Y f (cid:15) (cid:15) V p / / X where ι : Y ′ → Y is a natural closed immersion and Y ′ f ′ / / V p / / X is the Stein factorization of f ◦ ι : Y ′ → X . By construction, ι : Y ′ → Y is an isomorphismover the generic point of X . By construction again, the natural map O V → f ′∗ O Y ′ is anisomorphism and every stratum of Y ′ is dominant onto V . Therefore, p is birational. Claim 1. V is normal.Proof of Claim 1. Let π : V n → V be the normalization. Since every stratum of Y ′ isdominant onto V , there exists a closed subset Σ of Y ′ such that codim Y ′ Σ ≥ π − ◦ f ′ : Y ′ V n is a morphism on Y ′ \ Σ. Let e Y be the graph of π − ◦ f ′ : Y ′ V n .Then we have the following commutative diagram: e Y e f (cid:15) (cid:15) q / / Y ′ f ′ (cid:15) (cid:15) V n π / / V where q and e f are natural projections. Note that q : e Y → Y ′ is an isomorphism over Y \ Σ by construction. Since Y ′ is a simple normal crossing divisor on a smooth varietyand codim Y ′ Σ ≥ 2, the natural map O Y ′ → q ∗ O e Y is an isomorphism. Therefore, thecomposition O V → π ∗ O V n → π ∗ e f ∗ O e Y = f ′∗ q ∗ O e Y ≃ O V is an isomorphism. Thus we have O V ≃ π ∗ O V n . This implies that V is normal. (cid:3) Therefore, p : V → X is nothing but the normalization ν : Z → X . So we have thefollowing commutative diagram. Y ′ f ′ (cid:15) (cid:15) (cid:31) (cid:127) ι / / Y f (cid:15) (cid:15) Z ν / / X Claim 2. The natural map α : O Z → f ′∗ O Y ′ ( ⌈− ( B < Y ′ ) ⌉ ) is an isomorphism outside ν − Nqlc( X, ω ) .Proof of Claim 2. Note that ν : Z → X is an isomorphism over X \ Nqklt( X, ω ) by Lemma4.8. Moreover, f ′ : Y ′ → Z is nothing but f : Y → X over Z \ ν − Nqklt( X, ω ) byconstruction. Therefore, α is an isomorphism outside ν − Nqklt( X, ω ). By replacing X with X \ Nqlc( X, ω ), we may assume that Nqlc( X, ω ) = ∅ . Hence the natural map O X → f ∗ O Y ( ⌈− ( B < Y ) ⌉ ) is an isomorphism. Therefore, we have f ∗ O Y ≃ O X . Since Z isnormal and f ′∗ O Y ′ ( ⌈− ( B < Y ′ ) ⌉ ) is torsion-free, it is sufficient to see that α is an isomorphismin codimension one. Let P be any prime divisor on Z such that P ⊂ ν − Nqklt( X, ω ).We note that every fiber of f is connected by f ∗ O Y ≃ O X . Then, by construction, thereexists an irreducible component of B =1 Y ′ which maps onto P . Therefore, the effective divisor ⌈− ( B < Y ′ ) ⌉ does not contain the whole fiber of f ′ over the generic point of P . Thus, α is anisomorphism at the generic point of P . This means that α is an isomorphism. (cid:3) We put S := f ′∗ O Y ′ ( ⌈− ( B < Y ′ ) ⌉ − ⌊ B > Y ′ ⌋ − Y ′′ | Y ′ ). Then we have: Claim 3. S is an ideal sheaf on Z .Proof of Claim 3. By definition, S is a torsion-free coherent sheaf on Z . By the proof of[F11, Theorem 6.3.5 (i)] (see also the idea of the proof of Theorem 4.6 (i)), we have ν ∗ S = f ∗ O Y ′ ( ⌈− ( B < Y ′ ) ⌉ − ⌊ B > Y ′ ⌋ − Y ′′ | Y ′ ) = I Nqklt( X,ω ) ⊂ O X . Since ν is finite, ν ∗ ν ∗ S → S is surjective. This implies that S is an ideal sheaf on Z . (cid:3) We put T := f ′∗ O Y ′ ( ⌈− ( B < Y ′ ) ⌉ − ⌊ B > Y ′ ⌋ ). Then we have: Claim 4. T is an ideal sheaf on Z .Proof of Claim 4. Outside ν − Nqlc( X, ω ), it is obvious that T = f ′∗ O Y ′ ( ⌈− ( B < Y ′ ) ⌉ ) holds.Therefore, we obtain T = O Z outside ν − Nqlc( X, ω ) by Claim 2. Since T is torsion-freeand Z is normal, it is sufficient to show that T is an ideal sheaf in codimension one. Let Q be any prime divisor on X such that Q ⊂ Nqlc( X, ω ). We take a prime divisor P on Z such that ν ( P ) = Q .If ⌈− ( B < Y ′ ) ⌉ does not contain the whole fiber of f ′ over the generic point of P , then thenatural map α : O Z → f ′∗ O Y ′ ( ⌈− ( B < Y ′ ) ⌉ )is an isomorphism at the generic point of P since the natural map O Z → f ′∗ O Y ′ is anisomorphism by construction. Then f ′∗ O Y ′ ( ⌈− ( B < Y ′ ) ⌉ − ⌊ B > Y ′ ⌋ ) is an ideal sheaf at thegeneric point of P .If ⌈− ( B < Y ′ ) ⌉ contains the whole fiber of f ′ over the generic point of P , then S = T holdsover the generic point of P because ⌈− ( B < Y ′ ) ⌉ and Y ′′ | Y ′ have no common irreduciblecomponents. Therefore, T is an ideal sheaf at the generic point of P by Claim 3Hence T is an ideal sheaf on Z . This is what we wanted. (cid:3) By construction, K Y ′ + B Y ′ ∼ R f ′∗ ν ∗ ω obviously holds. We can define Nqlc( Z, ν ∗ ω ) by the ideal sheaf f ′∗ O Y ′ ( ⌈− ( B < Y ′ ) ⌉ − ⌊ B > Y ′ ⌋ )(see Claim 4). Hence ( Z, ν ∗ ω, f ′ : ( Y ′ , B Y ′ ) → Z ) ONE THEOREM AND MORI HYPERBOLICITY 27 naturally becomes a quasi-log scheme. By Claim 3 and its proof and [F11, Propositions6.3.1 and 6.3.2], I Nqklt( Z,ν ∗ ω ) = f ′∗ O Y ′ ( ⌈− ( B < Y ′ ) ⌉ − ⌊ B > Y ′ ⌋ − Y ′′ | Y ′ )satisfies ν ∗ I Nqklt( Z,ν ∗ ω ) = I Nqklt( X,ω ) . Hence ( Z, ν ∗ ω, f ′ : ( Y ′ , B Y ′ ) → Z )is a quasi-log scheme with the desired properties. (cid:3) On basic slc-trivial fibrations In this section, we quickly explain basic slc-trivial fibrations . For the details, see [F14]and [FFL]. Let us start with the definition of potentially nef divisors . Definition 6.1 (Potentially nef divisors, see [F14, Definition 2.5]) . Let X be a normalvariety and let D be a divisor on X . If there exist a completion X † of X , that is, X † isa complete normal variety and contains X as a dense Zariski open set, and a nef divisor D † on X † such that D = D † | X , then D is called a potentially nef divisor on X . A finite Q > -linear (resp. R > -linear) combination of potentially nef divisors is called a potentiallynef Q -divisor (resp. R -divisor).It is convenient to use b-divisors to explain several results on basic slc-trivial fibrations.Here we do not repeat the definition of b-divisors. For the details, see [C, 2.3.2 b-divisors]and [F14, Section 2]. Definition 6.2 (Canonical b-divisors) . Let X be a normal variety and let ω be a toprational differential form of X . Then ( ω ) defines a b-divisor K . We call K the canonicalb-divisor of X . Definition 6.3 ( Q -Cartier closures) . The Q -Cartier closure of a Q -Cartier Q -divisor D on a normal variety X is the Q -b-divisor D with trace D Y = f ∗ D, where f : Y → X is a proper birational morphism from a normal variety Y .We use the following definition in order to state the main result of [F14]. Definition 6.4 ([F14, Definition 2.12]) . Let X be a normal variety. A Q -b-divisor D of X is b-potentially nef (resp. b-semi-ample ) if there exists a proper birational morphism X ′ → X from a normal variety X ′ such that D = D X ′ , that is, D is the Q -Cartier closureof D X ′ , and that D X ′ is potentially nef (resp. semi-ample). A Q -b-divisor D of X is Q -b-Cartier if there is a proper birational morphism X ′ → X from a normal variety X ′ suchthat D = D X ′ .Roughly speaking, a basic slc-trivial fibration is a canonical bundle formula for simplenormal crossing pairs. Definition 6.5 (Simple normal crossing pairs) . We say that the pair ( X, B ) is a simplenormal crossing pair if ( X, B ) is Zariski locally a globally embedded simple normal crossingpair at any point x ∈ X .We note that a globally embedded simple normal crossing pair is obviously a simplenormal crossing pair by definition. We introduce the notion of basic slc-trivial fibrations. Definition 6.6 (Basic slc-trivial fibrations, see [F14, Definition 4.1]) . A pre-basic slc-trivial fibration f : ( X, B ) → Y consists of a projective surjective morphism f : X → Y and a simple normal crossing pair ( X, B ) satisfying the following properties:(1) Y is a normal variety,(2) every stratum of X is dominant onto Y and f ∗ O X ≃ O Y ,(3) B is a Q -divisor such that B = B ≤ holds over the generic point of Y , and(4) there exists a Q -Cartier Q -divisor D on Y such that K X + B ∼ Q f ∗ D. If a pre-basic slc-trivial fibration f : ( X, B ) → Y also satisfies(5) rank f ∗ O X ( ⌈− B < ⌉ ) = 1,then it is called a basic slc-trivial fibration .If X is irreducible and ( X, B ) is sub kawamata log terminal (resp. sub log canonical)over the generic point of Y in Definition 6.6, then it is a klt-trivial fibration (resp. anlc-trivial fibration). For the details of lc-trivial fibrations, see [F9], [FG2], and so on.In order to define discriminant Q -b-divisors and moduli Q -b-divisors for basic slc-trivialfibrations, we need the notion of induced ( pre- ) basic slc-trivial fibrations . Definition 6.7 (Induced (pre-)basic slc-trivial fibrations, see [F14, 4.3]) . Let f : ( X, B ) → Y be a (pre-)basic slc-trivial fibration and let σ : Y ′ → Y be a generically finite surjec-tive morphism from a normal variety Y ′ . Then we have an induced ( pre- ) basic slc-trivialfibration f ′ : ( X ′ , B X ′ ) → Y ′ , where B X ′ is defined by µ ∗ ( K X + B ) = K X ′ + B X ′ , with thefollowing commutative diagram: ( X ′ , B X ′ ) µ / / f ′ (cid:15) (cid:15) ( X, B ) f (cid:15) (cid:15) Y ′ σ / / Y, where X ′ coincides with X × Y Y ′ over a nonempty Zariski open set of Y ′ . More precisely,( X ′ , B X ′ ) is a simple normal crossing pair with a morphism X ′ → X × Y Y ′ that is anisomorphism over a nonempty Zariski open set of Y ′ such that X ′ is projective over Y ′ andthat every stratum of X ′ is dominant onto Y ′ .Now we are ready to define discriminant Q -b-divisors and moduli Q -b-divisors for basicslc-trivial fibrations. Definition 6.8 (Discriminant and moduli Q -b-divisors, see [F14, 4.5]) . Let f : ( X, B ) → Y be a (pre-)basic slc-trivial fibration as in Definition 6.6. Let P be a prime divisor on Y .By shrinking Y around the generic point of P , we assume that P is Cartier. We set b P = max (cid:26) t ∈ Q (cid:12)(cid:12)(cid:12)(cid:12) ( X ν , Θ + tν ∗ f ∗ P ) is sub log canonicalover the generic point of P (cid:27) , where ν : X ν → X is the normalization and K X ν + Θ = ν ∗ ( K X + B ), that is, Θ is the sumof the inverse images of B and the singular locus of X , and set B Y = X P (1 − b P ) P, where P runs over prime divisors on Y . Then it is easy to see that B Y is a well-defined Q -divisor on Y and is called the discriminant Q -divisor of f : ( X, B ) → Y . We set M Y = D − K Y − B Y ONE THEOREM AND MORI HYPERBOLICITY 29 and call M Y the moduli Q -divisor of f : ( X, B ) → Y . By definition, we have K X + B ∼ Q f ∗ ( K Y + B Y + M Y ) . Let σ : Y ′ → Y be a proper birational morphism from a normal variety Y ′ and let f ′ : ( X ′ , B X ′ ) → Y ′ be an induced (pre-)basic slc-trivial fibration by σ : Y ′ → Y . We candefine B Y ′ , K Y ′ and M Y ′ such that σ ∗ D = K Y ′ + B Y ′ + M Y ′ , σ ∗ B Y ′ = B Y , σ ∗ K Y ′ = K Y and σ ∗ M Y ′ = M Y . We note that B Y ′ is independent of the choice of ( X ′ , B X ′ ), that is, B Y ′ is well defined. Hence there exist a unique Q -b-divisor B such that B Y ′ = B Y ′ for every σ : Y ′ → Y and a unique Q -b-divisor M such that M Y ′ = M Y ′ for every σ : Y ′ → Y . Notethat B is called the discriminant Q -b-divisor and that M is called the moduli Q -b-divisor associated to f : ( X, B ) → Y . We sometimes simply say that M is the moduli part of f : ( X, B ) → Y .Let us see the main result of [F14]. Theorem 6.9 ([F14, Theorem 1.2]) . Let f : ( X, B ) → Y be a basic slc-trivial fibration andlet B and M be the induced discriminant and moduli Q -b-divisors of Y respectively. Thenwe have the following properties: (i) K + B is Q -b-Cartier, where K is the canonical b-divisor of Y , and (ii) M is b-potentially nef, that is, there exists a proper birational morphism σ : Y ′ → Y from a normal variety Y ′ such that M Y ′ is a potentially nef Q -divisor on Y ′ andthat M = M Y ′ . When dim Y = 1 in Theorem 6.9, we have: Theorem 6.10 ([FFL, Corollary 1.4]) . In Theorem 6.9, we further assume that dim Y = 1 .Then the moduli Q -divisor M Y of f : ( X, B ) → Y is semi-ample. The proof of Theorems 6.9 and 6.10 heavily depends on the theory of variations of mixedHodge structure discussed in [FF] (see also [FFS]). For some related topics, see [F2], [F9],[FG2], and so on. 7. On normal quasi-log schemes In this section, we treat the following deep result on the structure of normal quasi-logschemes. It is a generalization of [F14, Theorem 1.7]. The proof of Theorem 7.1 usesTheorems 6.9 and 6.10. Theorem 7.1. Let [ X, ω ] be a quasi-log scheme such that X is a normal variety. Thenthere exists a projective birational morphism p : X ′ → X from a smooth quasi-projectivevariety X ′ such that K X ′ + B X ′ + M X ′ = p ∗ ω, where B X ′ is an R -divisor such that Supp B X ′ is a simple normal crossing divisor and that B < X ′ is p -exceptional, and M X ′ is a potentially nef R -divisor on X ′ . Furthermore, we canmake B X ′ satisfy p ( B ≥ X ′ ) = Nqklt( X, ω ) set theoretically. When X is a curve, we can make M X ′ semi-ample in the above statement.We further assume that [ X, ω ] has a Q -structure. Then we can make B X ′ and M X ′ Q -divisors in the above statement. Let us prove Theorem 7.1. Proof of Theorem 7.1. We divide the proof into several steps. Step 1. Although this step is essentially the same as the proof of Theorem 1.9, we explain itagain with some remarks on Nqlc( X, ω ) for the reader’s convenience. Let f : ( Y, B Y ) → X be a proper surjective morphism from a quasi-projective globally embedded simple normal crossing pair ( Y, B Y ) as in Definition 4.2 (see Theorem 4.5). By [F11, Proposition 6.3.1],we may assume that the union of all strata of ( Y, B Y ) mapped to Nqklt( X, ω ), which isdenoted by Y ′′ , is a union of some irreducible components of Y . We put Y ′ = Y − Y ′′ and K Y ′ + B Y ′ = ( K Y + B Y ) | Y ′ . By the proof of Theorem 1.9, we obtain the followingcommutative diagram: Y ′ f ′ (cid:15) (cid:15) (cid:31) (cid:127) ι / / Y f (cid:15) (cid:15) X X where ι : Y ′ → Y is a natural closed immersion such that the natural map O V → f ′∗ O Y ′ isan isomorphism and that every stratum of Y ′ is dominant onto X . By Theorem 1.9 andits proof, ( X, ω, f ′ : ( Y ′ , B Y ′ ) → X )is a quasi-log scheme with I Nqklt( X,ω,f ′ : ( Y ′ ,B Y ′ ) → X ) = I Nqklt( X,ω,f : ( Y,B Y ) → X ) . We note that if ( X, ω, f : ( Y, B Y ) → X )has a Q -structure then it is obvious that( X, ω, f ′ : ( Y ′ , B Y ′ ) → X )also has a Q -structure by construction. Therefore, by replacing f : ( Y, B Y ) → X with f ′ : ( Y ′ , B Y ′ ) → X , we may assume that every stratum of Y is mapped onto X by f . Byconstruction, we can easily see thatNqlc( X, ω, f : ( Y ′ , B Y ′ ) → X ) ⊂ Nqlc( X, ω, f : ( Y, B Y ) → X )holds set theoretically. However, the relationship between Nqlc( X, ω, f : ( Y ′ , B Y ′ ) → X )and Nqlc( X, ω, f : ( Y, B Y ) → X ) is not clear. We note that all we need in this proof is thefact that Nqklt( X, ω, f : ( Y ′ , B Y ′ ) → X ) = Nqklt( X, ω, f : ( Y, B Y ) → X )holds set theoretically. Step 2. By Step 1, we may assume that f : ( Y, B Y ) → X is a projective surjective mor-phism from a simple normal crossing pair ( Y, B Y ) such that every stratum of Y is dominantonto X . By taking some more blow-ups, we may further assume that ( B hY ) =1 is Cartier andthat every stratum of ( Y, ( B hY ) =1 ) is dominant onto X (see, for example, [BVP, Theorem1.4 and Section 8] and [F13, Lemma 2.11]). Step 3. In this step, we treat the case where [ X, ω ] has a Q -structure. We note that O X → f ∗ O Y ( ⌈− ( B < Y ) ⌉ )is an isomorphism outside Nqlc( X, ω ). Hence rank f ∗ O Y ( ⌈− ( B < Y ) ⌉ ) = 1 holds. There-fore, we can check that f : ( Y, B Y ) → X is a basic slc-trivial fibration (see Definition6.6). Let B be the discriminant Q -b-divisor and let M be the moduli Q -b-divisor as-sociated to f : ( Y, B Y ) → X . By Lemma [F14, Lemma 11.2], we obtain that B X is aneffective Q -divisor on X . By definition, we have f (( B vY ) ≥ ) = Nqklt( X, ω ). We take a pro-jective birational morphism p : X ′ → X from a smooth quasi-projective variety X ′ . Let ONE THEOREM AND MORI HYPERBOLICITY 31 f ′ : ( Y ′ , B Y ′ ) → X ′ be an induced basic slc-trivial fibration with the following commutativediagram. ( Y, B Y ) f (cid:15) (cid:15) ( Y ′ , B Y ′ ) f ′ (cid:15) (cid:15) q o o X X ′ p o o By Theorem 6.9, we may assume that there exists a simple normal crossing divisor Σ X ′ on X ′ such that M = M X ′ , Supp M X ′ and Supp B X ′ are contained in Σ X ′ , and that everystratum of ( Y ′ , Supp B hY ′ ) is smooth over X ′ \ Σ X ′ . Of course, we may assume that M X ′ := M X ′ is potentially nef by Theorem 6.9. When X is a curve, we may further assume that M X ′ is semi-ample by Theorem 6.10. We may assume that every irreducible component of q − ∗ (cid:0) ( B vY ) ≥ (cid:1) is mapped onto a prime divisor in Σ X ′ with the aid of the flattening theorem(see [RG, Th´eor`eme (5.2.2)]). We put B X ′ := B X ′ . In the above setup, f ′ ( q − ∗ ( B vY ) ≥ ) ⊂ B ≥ X ′ by the definition of B . Thus, we get Nqklt( X, ω ) ⊂ p ( B ≥ X ′ ). On the other hand, wecan easily see that p ( B ≥ X ′ ) ⊂ Nqklt( X, ω ) by definition. Therefore, p ( B ≥ X ′ ) = Nqklt( X, ω )holds. Since p ∗ B X ′ = B X and B X is effective, B < X ′ is p -exceptional. Hence, B X ′ and M X ′ satisfy the desired properties. We note that B X ′ and M X ′ are obviously Q -divisors byconstruction. Step 4. In this step, we treat the general case. We first use Lemma 4.22 and get positivereal numbers r i and ( X, ω i , f : ( Y, D i ) → X ) for 1 ≤ i ≤ k with the properties in Lemma4.22. Then we apply the argument in Step 3 to( X, ω i , f : ( Y, D i ) → X )for every i . By Theorem 6.9, we can take a projective birational morphism p : X ′ → X from a smooth quasi-projective variety X ′ which works for( X, ω i , f : ( Y, D i ) → X )for every i . By summing them up with weight r i , we get R -divisors B X ′ and M X ′ with thedesired properties.We finish the proof of Theorem 7.1. (cid:3) Proof of Theorem 1.10 In this section, we prove Theorem 1.10 as an application of Theorem 7.1. Then, by usingTheorem 1.10, we prove Corollary 8.1 and Lemma 8.2, which will play an important rolein Section 9. Let us start the proof of Theorem 1.10. Proof of Theorem 1.10. By Theorem 7.1, there is a projective birational morphism p : X ′ → X from a smooth quasi-projective variety X ′ such that K X ′ + B X ′ + M X ′ = p ∗ ω, where B X ′ is an R -divisor on X ′ whose support is a simple normal crossing divisor, B < X ′ is p -exceptional, M X ′ is a potentially nef R -divisor on X ′ , and p ( B ≥ X ′ ) = Nqklt( X, ω ). Bytaking some more blow-ups, we may further assume that there is an effective p -exceptionaldivisor F on X ′ such that − F is p -ample and that Supp F ∪ Supp B X ′ is contained ina simple normal crossing divisor on X ′ . Then p ∗ H − εF + M X ′ is semi-ample for any0 < ε ≪ 1. We take a general effective R -divisor G on X ′ such that G ∼ R p ∗ H − εF + M X ′ with 0 < ε ≪ 1, Supp G ∪ Supp B X ′ ∪ Supp F is contained in a simple normal crossingdivisor on X ′ , and ( B X ′ + εF + G ) ≥ = B ≥ X ′ holds set theoretically. Then we have K X ′ + B X ′ + M X ′ + p ∗ H = K X ′ + B X ′ + εF + p ∗ H − εF + M X ′ ∼ R K X ′ + B X ′ + εF + G. We put ∆ := p ∗ ( B X ′ + εF + G ). By construction, K X + ∆ ∼ R ω + H . By constructionagain, we haveNklt( X, ∆) = p (cid:0) ( B X ′ + εF + G ) ≥ (cid:1) = p (cid:0) B ≥ X ′ (cid:1) = Nqklt( X, ω )set theoretically.When [ X, ω ] has a Q -structure, we can make B X ′ and M X ′ Q -divisors by Theorem 7.1.Then it is easy to see that we can make ∆ a Q -divisor on X such that K X + ∆ ∼ Q ω + H when H is an ample Q -divisor and [ X, ω ] has a Q -structure by the above construction of∆.Finally, if X is a curve in the above argument, then p : X ′ → X is an isomorphism and M X ′ is semi-ample (see Theorem 7.1). Hence we can take ∆ such that K X + ∆ ∼ R ω with the desired properties. (cid:3) For some related results, see [FG1], [F15], and so on. By applying Theorem 1.10 tonormal pairs, we have the following useful result. Corollary 8.1. Let X be a normal variety and let ∆ be an effective R -divisor on X suchthat K X + ∆ is R -Cartier. Let C be a log canonical center of ( X, ∆) such that C is asmooth curve. Then ( K X + ∆) | C ∼ R K C + ∆ C holds for some effective R -divisor ∆ C such that Supp ∆ ≥ C = C ∩ Nlc( X, ∆) ∪ [ C W W ! , where W runs over lc centers of ( X, ∆) which do not contain C , holds set theoretically.When K X + ∆ is Q -Cartier, we can make ∆ C a Q -divisor such that ( K X + ∆) | C ∼ Q K C + ∆ C in the above statement.Proof. As we saw in Example 4.11, [ X, K X + ∆] naturally becomes a quasi-log scheme.By construction, Nqlc( X, K X + ∆) = Nlc( X, ∆), W is a qlc center of [ X, K X + ∆] if andonly if W is a log canonical center of ( X, ∆). Hence we can see that C is a qlc center of[ X, K X + ∆]. Therefore, by adjunction (see Theorem 4.6 (i) and [F11, Theorem 6.3.5 (i)]),[ C ′ , ( K X + ∆) | C ′ ] is a quasi-log scheme, where C ′ = C ∪ Nlc( X, ∆). By Lemma 4.19, wesee that [ C, ( K X + ∆) | C ] is also a quasi-log scheme such thatNqklt( C, ( K X + ∆) | C ) = Nqklt( C ′ , ( K X + ∆) | C ′ ) ∩ C holds set theoretically. By construction, we can easily see thatNqklt( C ′ , ( K X + ∆) | C ′ ) ∩ C = C ∩ Nlc( X, ∆) ∪ [ C W W ! , where W runs over lc centers of ( X, ∆) which do not contain C , holds set theoretically(see Theorem 4.10 and Corollary 4.13). By applying Theorem 1.10 to [ C, ( K X + ∆) | C ], wecan find an effective R -divisor ∆ C on C such that( K X + ∆) | C ∼ R K C + ∆ C with Supp ∆ ≥ C = Nqklt( C, ( K X + ∆) | C ) = C ∩ Nlc( X, ∆) ∪ [ C W W ! . ONE THEOREM AND MORI HYPERBOLICITY 33 Of course, if K X + ∆ is Q -Cartier, then we can make ∆ C a Q -divisor such that( K X + ∆) | C ∼ Q K C + ∆ C in the above statement. (cid:3) We will use the following lemma in Section 9. Lemma 8.2. Let ϕ : X → Y be a proper surjective morphism between normal varietiessuch that R ϕ ∗ O X = 0 and that dim ϕ − ( y ) ≤ holds for every closed point y ∈ Y . Let C be a projective curve on X such that ϕ ( C ) is a point. Then C ≃ P . Let ∆ be an effective R -divisor on X such that K X + ∆ is R -Cartier. If C Nlc( X, ∆) and C ∩ Nlc( X, ∆) ∪ [ C W W ! = ∅ , where W runs over lc centers of ( X, ∆) which do not contain C , then the following in-equality − ( K X + ∆) · C ≤ holds.Proof. In Step 1, we will prove that C ≃ P holds. In Step 2, we will prove that − ( K X +∆) · C ≤ Step 1. Although the argument in this step is well known, we will explain it in detail forthe reader’s convenience. Let us consider the following short exact sequence0 → I C → O X → O C → , where I C is the defining ideal sheaf of C on X . Since dim ϕ − ( y ) ≤ y ∈ Y byassumption, R ϕ ∗ I C = 0 holds. Therefore, we get the following surjection R ϕ ∗ O X → R ϕ ∗ O C → . By assumption, R ϕ ∗ O X = 0. Hence R ϕ ∗ O C = 0 holds. Since ϕ ( C ) is a point byassumption, H ( C, O C ) = 0 holds. This means that C ≃ P . Step 2. By shrinking Y around ϕ ( C ), we may assume that Y is quasi-projective. Let B , . . . , B n +1 be general very ample Cartier divisors on Y passing through ϕ ( C ) with n = dim X . Then it is well known that X, ∆ + n +1 X i =1 ϕ ∗ B i ! is not log canonical at any point of C (see, for example, [F6, Lemma 13.2]) such thatNklt X, ∆ + (1 − ε ) n +1 X i =1 ϕ ∗ B i ! = Nklt( X, ∆)holds outside ϕ − ( ϕ ( C )) for every 0 < ε ≤ 1. Hence we can take 0 ≤ c < C isa log canonical center of ( X, ∆ + ϕ ∗ B ), where B = c P n +1 i =1 B i . Since B is effective, we seethat C ∩ Nlc( X, ∆ + ϕ ∗ B ) ∪ [ C W W ! = ∅ , where W runs over lc centers of ( X, ∆ + ϕ ∗ B ) which do not contain C . By Corollary 8.1,we can take an effective R -divisor ∆ C on C such that( K X + ∆) | C ∼ R ( K X + ∆ + ϕ ∗ B ) | C ∼ R K C + ∆ C and that Supp ∆ ≥ C = C ∩ Nlc( X, ∆ + ϕ ∗ B ) ∪ [ C W W ! = ∅ holds. This implies that − ( K X + ∆) · C = − deg( K C + ∆ C ) = 2 − deg ∆ C ≤ . We finish the proof of Lemma 8.2. (cid:3) Proof of Theorem 1.8 In this section, we prove Theorem 1.8. Let us start with the following proposition, whichis a consequence of the cone and contraction theorem for normal pairs (see [F6, Theorem1.1]) with the aid of Lemma 8.2. This is essentially due to [S, Proposition 5.2]. Proposition 9.1 ([S, Proposition 5.2] and [F15, Proposition 7.1]) . Let π : X → S be aprojective morphism from a normal Q -factorial variety X onto a scheme S . Let ∆ = P i d i ∆ i be an effective R -divisor on X , where the ∆ i ’s are the distinct prime componentsof ∆ for all i , such that X, ∆ ′ := X d i < d i ∆ i + X d i ≥ ∆ i ! is dlt. Assume that ( K X + ∆) | Nklt( X, ∆) is nef over S . Then K X + ∆ is nef over S or thereexists a non-constant morphism f : A −→ X \ Nklt( X, ∆) such that π ◦ f ( A ) is a point. More precisely, the curve C , the closure of f ( A ) in X , isa ( possibly singular ) rational curve with < − ( K X + ∆) · C ≤ X. Moreover, if C ∩ Nklt( X, ∆) = ∅ , then we can make C satisfy a sharper estimate < − ( K X + ∆) · C ≤ . Proof. We note that Nklt( X, ∆) coincides with (∆ ′ ) =1 = ⌊ ∆ ′ ⌋ , ∆ ≥ , and ⌊ ∆ ⌋ set theoret-ically because ( X, ∆ ′ ) is dlt by assumption. It is sufficient to construct a non-constantmorphism f : A −→ X \ Nklt( X, ∆)such that π ◦ f ( A ) is a point with the desired properties when K X + ∆ is not nef over S .When ( X, ∆) is kawamata log terminal, that is, ⌊ ∆ ⌋ = 0, the statement is well known (see,for example, [F6, Theorem 1.1], Theorem 1.12, or Corollary 12.3 below). Therefore, wemay assume that ( X, ∆) is not kawamata log terminal. By shrinking S suitably, we mayassume that S and X are both quasi-projective. By the cone and contraction theorem fornormal pairs (see [F6, Theorem 1.1]), we can take a ( K X + ∆)-negative extremal ray R of N E ( X/S ) and the associated extremal contraction morphism ϕ := ϕ R : X → Y over S since ( K X + ∆) | Nklt( X, ∆) is nef over S . Note that ( K X + ∆ < ) · R < K X + ∆ ′ ) · R < K X +∆) | Nklt( X, ∆) is nef over S . Since ( X, ∆ < ) is kawamata log terminal and ONE THEOREM AND MORI HYPERBOLICITY 35 − ( K X + ∆ < ) is ϕ -ample, we get R i ϕ ∗ O X = 0 for every i > ϕ : Nklt( X, ∆) → ϕ (Nklt( X, ∆)) is finite. We have the following short exact sequence0 → O X ( −⌊ ∆ ′ ⌋ ) → O X → O ⌊ ∆ ′ ⌋ → . Since −⌊ ∆ ′ ⌋ − ( K X + { ∆ ′ } ) = − ( K X + ∆ ′ ) is ϕ -ample and ( X, { ∆ ′ } ) is kawamata logterminal, R i ϕ ∗ O X ( −⌊ ∆ ′ ⌋ ) = 0 holds for every i > → ϕ ∗ O X ( −⌊ ∆ ′ ⌋ ) → O Y → ϕ ∗ O ⌊ ∆ ′ ⌋ → ⌊ ∆ ′ ⌋ = Supp ∆ ≥ is connected in a neighborhood of anyfiber of ϕ . Case 1. Assume that ϕ is a Fano contraction, that is, dim Y < dim X . Then we see that∆ ≥ is ϕ -ample and that dim Y = dim X − 1. Note that Supp ∆ ≥ is finite over Y sinceno curves in Supp ∆ ≥ are contracted by ϕ .Assume that there exists a closed subvariety Σ on X with dim Σ ≥ ϕ (Σ) isa point. Then dim (cid:0) Σ ∩ Supp ∆ ≥ (cid:1) ≥ ≥ is ϕ -ample. This is a contradiction because Supp ∆ ≥ is finite over Y .Hence we obtain that dim ϕ − ( y ) = 1 for every closed point y ∈ Y .Let C be any projective curve on X such that ϕ ( C ) is a point. Then ( X, ∆) is logcanonical at the generic point of C , equivalently, C Nlc( X, ∆), since Supp ∆ ≥ is finiteover Y . More precisely, since Supp ∆ ≥ = Nklt( X, ∆) is finite over Y , C Nklt( X, ∆)and Supp ∆ ≥ = Nklt( X, ∆) = Nlc( X, ∆) ∪ [ C W W ! holds, where W runs over lc centers of ( X, ∆) which do not contain C . On the other hand, C ∩ Supp ∆ ≥ = ∅ because ∆ ≥ is ϕ -ample. Hence, by Lemma 8.2, we obtain that C ≃ P and that − ( K X +∆) · C ≤ ≥ discussed above, C ∩ Supp ∆ ≥ is a point. Therefore,we can find a non-constant morphism f : A −→ X \ Nklt( X, ∆)such that π ◦ f ( A ) is a point and that 0 < − ( K X + ∆) · C ≤ C is theclosure of f ( A ) in X . Case 2. Assume that ϕ is a birational contraction and that the exceptional locus Exc( ϕ ) of ϕ is disjoint from Nklt( X, ∆). In this situation, we can find a rational curve C in a fiber of ϕ with 0 < − ( K X + ∆) · C ≤ X by the cone theorem for kawamata log terminal pairs(see [F6, Theorem 1.1], Theorem 1.12, or Corollary 12.3 below). It is obviously disjointfrom Nklt( X, ∆). Therefore, we can take a non-constant morphism f : A −→ X \ Nklt( X, ∆)such that the closure of f ( A ) is C . Case 3. Assume that ϕ is a birational contraction and that Exc( ϕ ) ∩ Nklt( X, ∆) = ∅ .In this situation, as in Case 1, we see that ∆ ≥ is ϕ -ample and that dim ϕ − ( y ) ≤ y ∈ Y . Let C be any projective curve C on X such that ϕ ( C ) is a point. Then, C ∩ Supp ∆ ≥ = ∅ holds since ∆ ≥ is ϕ -ample, and C ∩ Supp ∆ ≥ is a point by theconnectedness of Supp ∆ ≥ discussed above. In particular, we obtain C Nklt( X, ∆) and C ∩ Supp ∆ ≥ = ∅ , and Supp ∆ ≥ = Nklt( X, ∆) = Nlc( X, ∆) ∪ [ C W W ! , where W runs over lc centers of ( X, ∆) which do not contain C . Hence, by Lemma 8.2, C ≃ P with − ( K X + ∆) · C ≤ 1. Since C ∩ Supp ∆ ≥ is a point, we get a non-constantmorphism f : A −→ X \ Nklt( X, ∆)such that f ( A ) = C ∩ ( X \ Nklt( X, ∆)).Therefore, we get the desired statement. (cid:3) Let us prove Theorem 1.8 as an application of Proposition 9.1. Proof of Theorem 1.8. By shrinking S suitably, we may assume that X and S are bothquasi-projective. By Lemma 3.10, we can construct a projective birational morphism g : Y → X from a normal Q -factorial variety Y satisfying (i), (ii), and (iv) in Lemma 3.10.Let us consider π ◦ g : Y → S . Note that K Y + ∆ Y is not nef over S since K Y + ∆ Y = g ∗ ( K X + ∆) holds. It is obvious that ( K Y + ∆ Y ) | Nklt( Y, ∆ Y ) is nef over S by (iv) because sois ( K X + ∆) | Nklt( X, ∆) . Therefore, by Proposition 9.1, we have a non-constant morphism h : A −→ Y \ Nklt( Y, ∆ Y )such that ( π ◦ g ) ◦ h ( A ) is a point and that0 < − ( K Y + ∆ Y ) · C Y ≤ Y = 2 dim X holds, where C Y is the closure of h ( A ) in Y . Since K Y + ∆ Y = h ∗ ( K X + ∆) holds, g doesnot contract C Y to a pont. This implies that f := g ◦ h : A −→ X \ Nklt( X, ∆)is a desired non-constant morphism such that π ◦ f ( A ) is a point by (iv). (cid:3) For the proof of Theorem 1.6, we prepare the following somewhat artificial statement asan application of Theorem 1.8. Theorem 9.2. Let π : X → S be a proper surjective morphism from a normal quasi-projective variety X onto a scheme S . Let P be an R -Cartier divisor on X and let H bean ample Cartier divisor on X . Let Σ be a closed subset of X . Assume that π is not finite, −P is π -ample, and π : Σ → π (Σ) is finite. We further assume • { ε i } ∞ i =1 is a set of positive real numbers with ε i ց for i ր ∞ , and • for every i , there exists an effective R -divisor ∆ i on X such that P + ε i H ∼ R K X + ∆ i and that Σ = Nklt( X, ∆ i ) holds set theoretically.Then there exists a non-constant morphism f : A −→ X \ Σ such that π ◦ f ( A ) is a point and that the curve C , the closure of f ( A ) in X , is a rationalcurve with < −P · C ≤ X. ONE THEOREM AND MORI HYPERBOLICITY 37 Proof. We take an ample Q -divisor A on X such that − ( P + A ) is π -ample. Without lossof generality, we may assume that − ( P + A + ε i H ) is π -ample for every i because ε i ց i ր ∞ . By assumption, P + ε i H ∼ R K X + ∆ i with Nklt( X, ∆ i ) = Σfor every i . Hence, by Theorem 1.8, there is a non-constant morphism f i : A −→ X \ Σsuch that π ◦ f i ( A ) is a point and that0 < − ( K X + ∆ i ) · C i = − ( P + ε i H ) · C i ≤ X, where C i is the closure of f i ( A ) in X . We note that0 < A · C i = (( P + ε i H + A ) − ( P + ε i H )) < X. It follows that the curves C i belong to a bounded family. Thus, possibly passing to asubsequence, we may assume that f i and C i are constant, that is, there is a non-constantmorphism f : A −→ X \ Σsuch that C i = C for every i , where C is the closure of f ( A ) in X . Therefore, we get0 < −P · C = lim i →∞ − ( P + ε i H ) · C = lim i →∞ − ( P + ε i H ) · C i ≤ X. We finish the proof of Theorem 9.2. (cid:3) Proof of Theorems 1.4, 1.5, and 1.6 In this section, we prove Theorems 1.4, 1.5 and 1.6. Since Theorem 1.4 is an easyconsequence of Theorem 1.5 and Theorem 1.5 can be seen as a very special case of Theorem1.6 by Example 4.11, it is sufficient to prove Theorem 1.6. Let us start with the proof ofTheorem 1.6. Proof of Theorem 1.6. We note that (i) and (ii) were already established in [F11, Theorem6.7.4]. Therefore, it is sufficient to prove (iii). From Step 1 to Step 4, we will reduce theproblem to the case where X is a normal variety. Then, in Step 5, we will obtain a desirednon-constant morphism from A by Theorem 9.2. Step 1. Let ϕ R j : X → Y be the extremal contraction associated to R j (see Theorem 4.17and [F11, Theorems 6.7.3 and 6.7.4]). We note that ϕ R j : Nqlc( X, ω ) → ϕ R j (Nqlc( X, ω ))is finite. By replacing π : X → S with ϕ R j : X → Y , we may assume that − ω is π -ampleand that N E ( X/S ) −∞ = ∅ . Step 2. We take a qlc stratum W of [ X, ω ] such that π : W → π ( W ) is not finite andthat π : W † → π ( W † ) is finite for every qlc center W † of [ X, ω ] with W † ( W . We put W ′ = W ∪ Nqlc( X, ω ). Then, by adjunction (see Theorem 4.6 (i) and [F11, Theorem 6.3.5(i)]), [ W ′ , ω | W ′ ] naturally becomes a quasi-log scheme. By replacing [ X, ω ] with [ W ′ , ω | W ′ ],we may further assume that X \ X −∞ is irreducible and that π : Nqklt( X, ω ) → π (Nqklt( X, ω ))is finite. Step 3. By Lemma 4.19, we may replace X with X \ X −∞ and assume that X is a variety.We note that the finiteness of π : Nqklt( X, ω ) → π (Nqklt( X, ω ))still holds. Step 4. Let ν : Z → X be the normalization. Then [ Z, ν ∗ ω ] naturally becomes a quasi-logscheme by Theorem 1.9. Since Nqklt( Z, ν ∗ ω ) = ν − Nqklt( X, ω ) by Theorem 1.9, we mayassume that X is normal by replacing [ X, ω ] with [ Z, ν ∗ ω ]. Step 5. By shrinking S suitably, we may further assume that X and S are both quasi-projective. Hence we have the following properties:(a) π : X → S is a projective morphism from a normal quasi-projective variety X to ascheme S ,(b) − ω is π -ample, and(c) π : Σ → π (Σ) is finite, where Σ := Nqklt( X, ω ).Let H be an ample Cartier divisor on X and let { ε i } ∞ i =1 be a set of positive real numberssuch that ε i ց i ր ∞ . Then, by Theorem 1.10, we have:(d) there exists an effective R -divisor ∆ i on X such that K X + ∆ i ∼ R ω + ε i H with Nklt( X, ∆ i ) = Σfor every i .Thus, by Theorem 9.2, we have a desired non-constant morphism f : A −→ X \ Nqklt( X, ω ) . We finish the proof of Theorem 1.6. (cid:3) As we already mentioned above, Theorem 1.5 is a very special case of Theorem 1.6. Proof of Theorem 1.5. By Example 4.11, [ X, K X +∆] naturally becomes a quasi-log scheme.Then, by Theorem 1.6, the desired cone theorem holds for ( X, ∆). (cid:3) Theorem 1.4 easily follows from Theorem 1.5. Proof of Theorem 1.4. Since ( X, ∆) is Mori hyperbolic by assumption, there is no ( K X +∆)-negative extremal ray of N E ( X ) that is rational and relatively ample at infinity (seeTheorem 1.5). By assumption, ( K X + ∆) | Nlc( X, ∆) is nef. Hence the subcone N E ( X ) −∞ isincluded in N E ( X ) ( K X +∆) ≥ . This implies that N E ( X ) = N E ( X ) ( K X +∆) ≥ holds by Theorem 1.5. Thus K X + ∆ is nef. (cid:3) The author thinks that the proof of Theorems 1.4, 1.5 and 1.6 shows that the frameworkof quasi-log schemes established in [F11, Chapter 6] and [F14] is very powerful and usefuleven for the study of normal pairs. ONE THEOREM AND MORI HYPERBOLICITY 39 Ampleness criterion for quasi-log schemes The main purpose of this section is to establish the following ampleness criterion forquasi-log schemes. Then we will see that Theorem 1.11 is a very special case of Theorem11.1. Theorem 11.1 (Ampleness criterion for quasi-log schemes) . Let [ X, ω ] be a quasi-logscheme and let π : X → S be a projective morphism between schemes. Assume that ω | Nqlc( X,ω ) is ample over S and that ω is log big over S with respect to [ X, ω ] . We furtherassume that there is no non-constant morphism f : A −→ U such that π ◦ f ( A ) is a point, where U is any open qlc stratum of [ X, ω ] . Then ω is ampleover S . Let us treat a special case of Theorem 11.1. Theorem 11.2. Let [ X, ω ] be a quasi-log scheme such that X is a normal variety. Let π : X → S be a projective morphism onto a scheme S . Assume that ω | Nqklt( X,ω ) is ampleover S and that there is no non-constant morphism f : A −→ X \ Nqklt( X, ω ) such that π ◦ f ( A ) is a point. We further assume that ω is big over S . Then ω is ampleover S .Proof. We divide the proof into several small steps. Step 1. By Lemma 4.22, we can obtain quasi-log schemes( X, ω i , f : ( Y, D i ) → X )for 1 ≤ i ≤ k with the following properties:(a) [ X, ω i ] has a Q -structure for every i ,(b) Nqlc( X, ω i ) = Nqlc( X, ω ) holds for every i ,(c) W is an qlc stratum of [ X, ω ] if and only if W is a qlc stratum of [ X, ω i ] for every i , and(d) there exist positive real numbers r i for 1 ≤ i ≤ k such that ω = P ki =1 r i ω i with P ki =1 r i = 1.By construction, we can make ω i sufficiently close to ω (see the proof of Lemma 4.22).Therefore, we may assume that ω i | Nqklt( X,ω i ) is ample over S for every i by (b) and (c).Thus [ X, ω i ] satisfies all the assumptions for [ X, ω ] in Theorem 11.2. Hence, by replacing[ X, ω ] with [ X, ω i ], it is sufficient to prove the ampleness of ω under the extra assumptionthat [ X, ω ] has a Q -structure by (a) and (d). Step 2. By assumption and Theorem 1.6 (iii), ω is nef over S . Since ω | Nqklt( X,ω ) is ampleover S by assumption, ω is nef and log big over S with respect to [ X, ω ]. Therefore, by[F10, Theorem 1.1], we obtain that ω is semi-ample over S . Hence mω gives a birationalcontraction morphism Φ : X → Y between normal varieties over S , where m is a sufficientlylarge and divisible positive integer. Step 3. In this step, we will get a contradiction under the assumption that Φ is not anisomorphism.By shrinking S , we may assume that S , X , and Y are quasi-projective. By construction,Φ : Nqklt( X, ω ) → Φ(Nqklt( X, ω )) is finite. Since Φ is birational and Y is quasi-projective, we can take an effective Cartierdivisor G on X such that − G is Φ-ample. By Lemma 4.21, for 0 < ε ≪ 1, [ X, ω + εG ] isa quasi-log scheme such thatNqklt( X, ω + εG ) = Nqklt( X, ω )holds. By the cone theorem (see Theorem 1.6 (iii)), we can find a non-constant morphism f : A −→ X \ Nqklt( X, ω + εG ) = X \ Nqklt( X, ω )such that π ◦ f ( A ) is a point and that 0 < − ( ω + εG ) · C ≤ X holds, where C is theclosure of f ( A ) in X . This is a contradiction.Hence Φ is an isomorphism. Therefore, we obtain that ω is ample over S . This is whatwe wanted. (cid:3) Once we know Theorem 11.2, it is not difficult to prove Theorem 11.1. Proof of Theorem 11.1. By Theorem 1.6 (iii), ω is nef and log big over S with respect to[ X, ω ]. We put [ X , ω ] := [ X, ω ]and [ X i +1 , ω i +1 ] := [Nqklt( X i , ω i ) , ω i | Nqklt( X i ,ω i ) ]for i ≥ 0. Then there exists k ≥ X k , ω k ) = Nqlc( X k , ω k ) = Nqlc( X, ω ) . We note that Nqlc( X, ω ) may be empty. By assumption, ω k | Nqklt( X k ,ω k ) is ample over S . Wewant to show by inverse induction on i that ω i is ample over S . Therefore, it is sufficientto prove the following claim. Claim. Let [ X, ω ] be a quasi-log scheme and let π : X → S be a projective morphismbetween schemes such that ω | Nqklt( X,ω ) is ample over S and that ω is nef and log big over S with respect to [ X, ω ] . Then ω is ample over S .Proof of Claim. By adjunction (see Theorem 4.6 (i) and [F11, Theorem 6.3.5 (i)]), we mayassume that X \ X −∞ is irreducible. By Lemma 4.19, we may further assume that X isirreducible. Then, by Theorem 1.9, we can reduce the problem to the case where X is anormal variety. Hence ω is ample over S by Theorem 11.2. (cid:3) As we have already mentioned above, by applying Claim inductively, we obtain thedesired relative ampleness of ω = ω . (cid:3) We close this section with the proof of Theorem 1.11. Proof of Theorem 1.11. By Example 4.11, [ X, K X + ∆] naturally becomes a quasi-logscheme. We apply Theorem 11.1 to [ X, K X + ∆]. Then we obtain that K X + ∆ is ample.This is what we wanted. (cid:3) The author knows no proof of Theorem 1.11 that does not use the framework of quasi-logschemes. Note that a similar result for dlt pairs was already established in [F7, Theorem5.1], whose proof is much easier than that of Theorem 1.11 and depends on the basepoint-free theorem of Reid–Fukuda type for dlt pairs (see [F1, Theorem 0.1]). We recommendthe interested reader to see [F7, Theorem 5.1] and [F1, Theorem 0.1]. ONE THEOREM AND MORI HYPERBOLICITY 41 Proof of Theorems 1.12 and 1.13 In this section, we prove Theorems 1.12 and 1.13, and explain an application for normalpairs. For the basic properties of uniruled varieties, see [Ko1, Chapter IV. 1]. Let us startwith the following lemma, which is a generalization of [Ka, Lemma]. Lemma 12.1. Let [ X, ω ] be a quasi-log scheme and let ϕ : X → W be a projective mor-phism between schemes. Let P be an arbitrary closed point of W . Let E be a positive-dimensional irreducible component of ϕ − ( P ) such that E X −∞ and let ν : E → E be the normalization. Then, for every ample R -divisor H on E , there exists an effective R -divisor ∆ E,H on E such that ν ∗ ω + H ∼ R K E + ∆ E,H holds. Therefore, A dim E − · ω · E ≥ ( ν ∗ A ) dim E − · K E holds for every ϕ -ample line bundle A on X .In the above statement, if [ X, ω ] has a Q -structure and H is an ample Q -divisor on E ,then we can make ∆ E,H an effective Q -divisor on E with ν ∗ ω + H ∼ Q K E + ∆ E,H . Proof. Our approach is different from Kawamata’s in [Ka]. A key ingredient of this proofis Theorem 1.10. Step 1. If E is a qlc stratum of [ X, ω ], then we put B = 0 and go to Step 3. Step 2. By Step 1, we may assume that E is not a qlc stratum of [ X, ω ]. Withoutloss of generality, we may assume that W is quasi-projective by shrinking W around P .Let B , . . . , B n +1 be general very ample Cartier divisors on W passing through P with n = dim X . Let f : ( Y, B Y ) → X be a proper morphism from a globally embedded simplenormal crossing pair ( Y, B Y ) as in Definition 4.2. Let X ′ be the union of X −∞ = Nqlc( X, ω )and all qlc strata of [ X, ω ] mapped to P by ϕ . By [F11, Proposition 6.3.1] and [Ko2,Theorem 3.35], we may assume that the union of all strata of ( Y, B Y ) mapped to X ′ by f ,which is denoted by Y ′ , is a union of some irreducible components of Y . As usual, we put Y ′′ = Y − Y ′ , K Y ′′ + B Y ′′ = ( K Y + B Y ) | Y ′′ , and f ′′ = f | Y ′′ . By [F11, Proposition 6.3.1]and [Ko2, Theorem 3.35] again, we may further assume that Y ′′ , ( f ′′ ) ∗ ϕ ∗ n +1 X i =1 B i + Supp B Y ′′ ! is a globally embedded simple normal crossing pair. By [F11, Lemma 6.3.13], we can take0 < c < f ′′ B Y ′′ + c ( f ′′ ) ∗ ϕ ∗ n +1 X i =1 B i ! > E and that there exists an irreducible component G of B Y ′′ + c ( f ′′ ) ∗ ϕ ∗ n +1 X i =1 B i ! =1 with f ′′ ( G ) = E . By Lemma 4.20, we obtain that( X, ω + B, f ′′ : ( Y ′′ , B Y ′′ + ( f ′′ ) ∗ B ) → X ) , where B = ϕ ∗ (cid:0) c P n +1 i =1 B (cid:1) , is a quasi-log scheme such that E is a qlc stratum of thisquasi-log scheme. Step 3. We put E ′ = E ∪ Nqlc( X, ω + B ). Then, by adjunction (see Theorem 4.6 (i)and [F11, Theorem 6.3.5 (i)]), [ E ′ , ( ω + B ) | E ′ ] is a quasi-log scheme. By Lemma 4.19,[ E, ( ω + B ) | E ] is also a quasi-log scheme. We note that ( ω + B ) | E ∼ R ω | E since ϕ ( E ) = P .Hence [ E, ω | E ] is a quasi-log scheme. By Theorem 1.9, [ E, ν ∗ ω ] naturally becomes a quasi-log scheme. By Theorem 1.10, there exists an effective R -divisor ∆ E,H on E such that ν ∗ ω + H ∼ R K E + ∆ E,H . This implies that( ν ∗ A ) dim E − · ( ν ∗ ω + H ) · E = ( ν ∗ A ) dim E − ( K E + ∆ E,H ) ≥ ( ν ∗ A ) dim E − · K E . Since the above inequality holds for every ample R -divisor H on E , we obtain A dim E − · ω · E = ( ν ∗ A ) dim E − · ν ∗ ω · E ≥ ( ν ∗ A ) dim E − · K E . This is what we wanted. By the above proof, it is easy to see that we can make ∆ E,H aneffective Q -divisor on E if [ X, ω ] has a Q -structure and H is an ample Q -divisor on E .We finish the proof of Theorem 12.1. (cid:3) Remark 12.2. In the proof of [Ka, Lemma], Kawamata uses a relative Kawamata–Viehweg vanishing theorem for projective bimeromorphic morphisms between complexanalytic spaces. His argument does not work for quasi-log schemes.Let us prove Theorem 1.12. Proof of Theorem 1.12. In this proof, we will freely use the notation of Lemma 12.1. Case 1. We will treat the case where dim E = 1.We take an ample Q -divisor H on E such that − ( ν ∗ ω + H ) is still ample. Then, byLemma 12.1, − K E is ample since ∆ E,H is effective. This means that E ≃ P . By Lemma12.1 again, we have 0 < − ω · E ≤ − deg K E = 2 . Case 2. We will treat the case where dim E ≥ ϕ -ample line bundle A such that ν ∗ A is very ample. We put C = D ∩· · · ∩ D dim E − , where D i is a general member of | ν ∗ A| for every i . Then C is a smoothirreducible curve on E such that C lies in the smooth locus of E . By Lemma 12.1, weobtain C · K E ≤ A dim E − · ω · E < − ω is ϕ -ample. We note that0 > ν ∗ ω · C = ν ∗ ω · ( ν ∗ A ) dim E − · E = ω · A dim E − · E ≥ ( ν ∗ A ) dim E − · K E = C · K E . Therefore, for any given point x ∈ C , there exists a rational curve Γ on E passing through x with 0 < − ν ∗ ω · Γ ≤ E · − ν ∗ ω · C − K E · C ≤ E. This is essentially due to Miyaoka–Mori (see [MM]). We note that E is not always smoothbut it is smooth in a neighborhood of C . Hence we can use the argument of [MM]. For the ONE THEOREM AND MORI HYPERBOLICITY 43 details, see [Ko1, Chapter II. 5.8 Theorem]. Thus, E is covered by rational curves ℓ := ν ∗ Γwith 0 < − ω · ℓ ≤ E. Hence, by [Ko1, Chapter IV. 1.4 Proposition–Definition], E is uniruled. We finish theproof of Theorem 1.12. (cid:3) We prove Theorem 1.13. Proof of Theorem 1.13. Since ϕ R is the contraction morphism associated to R , ϕ R : Nqlc( X, ω ) → ϕ R (Nqlc( X, ω ))is finite. We apply Theorem 1.12 to ϕ R : X → W , we can take a rational curve ℓ on X such that ϕ R ( ℓ ) is a point with 0 < − ω · ℓ ≤ d. We finish the proof of Theorem 1.13. (cid:3) We explain an application of Theorems 1.12 and 1.13 for normal pairs, which is a gen-eralization of [Ka, Theorem 1]. Corollary 12.3. Let X be a normal variety and let ∆ be an effective R -divisor on X suchthat K X + ∆ is R -Cartier. Let π : X → S be a projective morphism between schemes. Let R be a ( K X + ∆) -negative extremal ray of N E ( X/S ) that are rational and relatively ampleat infinity. Let ϕ R : X → W be the contraction morphism over S associated to R . We put d = min E dim E, where E runs over positive-dimensional irreducible components of ϕ − R ( P ) for all P ∈ W .Then R is spanned by a ( possibly singular ) rational curve ℓ with < − ( K X + ∆) · ℓ ≤ d. Furthermore, if ϕ R is birational and ( X, ∆) is kawamata log terminal, then R is spannedby a ( possibly singular ) rational curve ℓ with < − ( K X + ∆) · ℓ < d. Let V be an irreducible component of the degenerate locus { x ∈ X | ϕ R is not an isomorphism at x } of ϕ R . Then V is uniruled.Proof. We divide the proof into three small steps. Step 1. By Example 4.11, [ X, K X +∆] naturally becomes a quasi-log scheme. By applyingTheorem 1.13 to [ X, K X + ∆], we see that R is spanned by a rational curve ℓ with0 < − ( K X + ∆) · ℓ ≤ d. Step 2. When ( X, ∆) is kawamata log terminal and ϕ R is a birational contraction, wetake a d -dimensional irreducible component E of ϕ − R ( P ) for some P ∈ W . By shrinking W around P , we may assume that W is affine. Since ϕ R is birational, there exists aneffective Q -divisor G on X such that ( X, ∆ + G ) is kawamata log terminal and that − G is ϕ R -ample. By applying Theorem 1.12 to [ X, K X + ∆ + G ], we see that E is covered byrational curves ℓ with 0 < − ( K X + ∆ + G ) · ℓ ≤ d. Since − G is ϕ R -ample, we have 0 < − ( K X + ∆) · ℓ < d. This implies that R is spanned by a rational curve ℓ with0 < − ( K X + ∆) · ℓ < d when ( X, ∆) is kawamata log terminal and ϕ R is birational. Step 3. From now on, we will check that V is uniruled. We shrink W around the genericpoint of ϕ R ( V ) and assume that W is quasi-projective. By replacing − ( K X + ∆) with − ( K X + ∆) + ϕ ∗ R H for some sufficiently ample general Cartier divisor H , we may assumethat − ( K X + ∆) is ample. By Theorem 1.12, V ∩ ϕ − R ( P ) is covered by rational curves ℓ of − ( K X + ∆)-degree at most 2 dim V for every P ∈ ϕ R ( V ) ⊂ W . We take a suitableprojective completion X of X and apply [Ko1, Chapter IV. 1.4 Proposition–Definition].Then we obtain that V is uniruled.We finish the proof of Corollary 12.3. (cid:3) Proof of Theorem 1.14 In this section, we prove Theorem 1.14. Let us start with the following definition. Definition 13.1 ([F11, Definition 6.8.1]) . Let [ X, ω ] be a quasi-log scheme and let π : X → S be a projective morphism between schemes. If − ω is ample over S , then [ X, ω ] is calleda relative quasi-log Fano scheme over S . When S is a point, we simply say that [ X, ω ] isa quasi-log Fano scheme .We recall an easy consequence of the vanishing theorem (see Theorem 4.6 (ii)), which ismissing in [F11, Section 6.8]. Lemma 13.2. Let [ X, ω ] be a quasi-log scheme and let π : X → S be a proper morphismbetween schemes with π ∗ O X ≃ O S . Assume that − ω is nef and log big over S with respectto [ X, ω ] . Then X −∞ ∩ π − ( P ) is connected for every closed point P ∈ S .Proof. By Theorem 4.6 (ii), R π ∗ I X −∞ = 0. Therefore, the restriction map O S ≃ π ∗ O X → π ∗ O X −∞ is surjective. This implies that X −∞ ∩ π − ( P ) is connected for every closed point P ∈ S . (cid:3) Lemma 13.2 should have been stated in [F11, Lemma 6.8.3], which plays an importantrole in this section. The main ingredient of the proof of Theorem 1.14 is the followingtheorem. Theorem 13.3 ([Z, Theorem 1], [HM], and [BP, Corollary 1.4]) . Let X be a normal projec-tive variety and let ∆ be an effective R -divisor on X such that K X +∆ is R -Cartier. Assumethat − ( K X + ∆) is ample. Then X is rationally chain connected modulo Nklt( X, ∆) .Proof. We take an effective Q -divisor ∆ ′ on X such that K X + ∆ ′ is Q -Cartier, − ( K X + ∆ ′ )is ample, and Nklt( X, ∆ ′ ) = Nklt( X, ∆) holds. If Nklt( X, ∆ ′ ) = ∅ , that is, ( X, ∆ ′ ) iskawamata log terminal, then X is rationally connected by [Z, Theorem 1]. In particular, X is rationally chain connected by Lemma 2.12. When Nklt( X, ∆ ′ ) = ∅ , by applying[BP, Corollary 1.4] to ( X, ∆ ′ ), we obtain that for any general point x of X there exists arational curve R x passing through x and intersecting Nklt( X, ∆ ′ ). By [Ko1, Chapter II. 2.4Corollary], for every x ∈ X , we can find a chain of rational curves R x such that x ∈ R x and R x ∩ Nklt( X, ∆ ′ ) = ∅ . Hence X is rationally chain connected modulo Nklt( X, ∆). Wenote that if − ( K X + ∆) is an ample Q -divisor then the proof of [BP, Theorem 1.2 andCorollary 1.4] becomes much simpler than the general case. Hence we obtain that X isrationally chain connected modulo Nklt( X, ∆). (cid:3) We prepare one useful lemma. ONE THEOREM AND MORI HYPERBOLICITY 45 Lemma 13.4. Let [ X, ω ] be a projective quasi-log canonical pair such that Nqklt( X, ω ) = ∅ , − ω is ample, and X is connected. Then X is rationally connected. Hence X is rationallychain connected.Proof. By Lemma 4.8 and [F11, Theorem 6.3.11 (i)], X is a normal variety. By Theorem1.10, we can find an effective R -divisor ∆ on X such that − ( K X + ∆) is ample withNklt( X, ∆) = ∅ . Hence X is rationally connected by [Z, Theorem 1] (see the proof ofTheorem 13.3). (cid:3) By the framework of quasi-log schemes, we can prove the following lemma as a gen-eralization of Theorem 13.3 without any difficulties. We note that if Nqlc( X, ω ) = ∅ inLemma 13.5 then it is nothing but [F15, Theorem 1.7]. For semi-log canonical Fano pairs,we recommend the reader to see [FLw]. Lemma 13.5. Let [ X, ω ] be a quasi-log scheme such that X is connected. Assume that − ω is ample. Then X is rationally chain connected modulo X −∞ .Proof. As in the proof of Theorem 11.1, we put[ X , ω ] := [ X, ω ]and [ X i +1 , ω i +1 ] := [Nqklt( X i , ω i ) , ω i | Nqklt( X i ,ω i ) ]for i ≥ 0. Then there exists k ≥ X k , ω k ) = Nqlc( X k , ω k ) = Nqlc( X, ω ) = X −∞ . We note that if X −∞ = ∅ , that is, [ X, ω ] is quasi-log canonical, then X k is the uniqueminimal qlc stratum of [ X, ω ] by [F11, Theorem 6.8.3 (ii)]. By applying Lemma 13.4 to[ X k , ω k ], we obtain that X k is rationally connected when X −∞ = ∅ . We want to show byinverse induction on i that X i +1 is rationally chain connected modulo X −∞ = Nqlc( X, ω ).Note that we want to show that X is rationally chain connected modulo X k when X −∞ = ∅ .We also note that X i is connected by Lemma 13.2 and [F11, Theorem 6.8.3]. Hence it issufficient to prove the following claim. Claim. Let [ X, ω ] be a quasi-log scheme such that X is connected, Nqklt( X, ω ) = ∅ , and − ω is ample. Then X is rationally chain connected modulo Nqklt( X, ω ) .Proof of Claim. By adjunction (see Theorem 4.6 (i) and [F11, Theorem 6.3.5 (i)]) and[F11, Theorem 6.8.3], we may assume that X \ X −∞ is irreducible. We note that every qlcstratum of [ X, ω ] intersects with Nqklt( X, ω ) (see [F11, Theorem 6.8.3]). By Lemma 4.19,we may further assume that X itself is irreducible. Then, by Theorem 1.9, we can furtherreduce the problem to the case where X is a normal variety. Then, by Theorem 1.10, wecan take an ample R -divisor H on X such that − ( ω + H ) is still ample and that K X + ∆ ∼ R ω + H holds for some effective R -divisor ∆ on X withNklt( X, ∆) = Nqklt( X, ω ) . By applying Theorem 13.3 to ( X, ∆), we obtain that X is rationally chain connectedmodulo Nqklt( X, ω ). We finish the proof of Claim. (cid:3) By using Claim inductively, we can check that X is rationally chain connected modulo X −∞ = Nqlc( X, ω ). (cid:3) Let us prove Theorem 1.14. Proof of Theorem 1.14. When π − ( P ) ∩ X −∞ = ∅ , we may assume that X −∞ = ∅ byshrinking X around π − ( P ). We divide the proof into several steps. Step 1. Let X be the union of X −∞ and all qlc strata of [ X, ω ] contained in π − ( P ). ByLemma 13.2 and [F11, Theorem 6.8.3], X ∩ π − ( P ) is connected. Case 1. If X = X −∞ , then [ X , ω ], where ω = ω | X , is a quasi-log scheme by adjunction(see Theorem 4.6 (i) and [F11, Theorem 6.3.5 (i)]). Let us consider X † = X \ Nqlc( X , ω ).Then [ X † , ω † ], where ω † = ω | X † , is a quasi-log scheme by Lemma 4.19. By construction, − ω † is ample since π ( X † ) = P . Therefore, by Lemma 13.5, X † is rationally chain connectedmodulo Nqlc( X † , ω † ). This means that X ∩ π − ( P ) is rationally chain connected modulo π − ( P ) ∩ X −∞ . Case 2. If X = X −∞ , that is, there is no qlc stratum of [ X, ω ] contained in π − ( P ),then X ∩ π − ( P ) is obviously rationally chain connected modulo π − ( P ) ∩ X −∞ because X ∩ π − ( P ) = π − ( P ) ∩ X −∞ . Note that X = X −∞ = ∅ may happen.Hence π − ( P ) is rationally chain connected modulo π − ( P ) ∩ X −∞ when π − ( P ) ⊂ X .Thus, from now on, we may assume that π − ( P ) X . Step 2. Without loss of generality, we may assume that S is quasi-projective by shrinking S around P . We take general very ample Cartier divisors B , . . . , B n +1 passing through P with n = dim X . Let f : ( Y, B Y ) → X be a proper morphism from a globally embeddedsimple normal crossing pair ( Y, B Y ) as in Definition 4.2. By [F11, Proposition 6.3.1] and[Ko2, Theorem 3.35], we may assume that the union of all strata of ( Y, B Y ) mapped to X by f , which is denoted by Y ′ , is a union of some irreducible components of Y . As usual,we put Y ′′ = Y − Y ′ , K Y ′′ + B Y ′′ = ( K Y + B Y ) | Y ′′ , and f ′′ = f | Y ′′ . By [F11, Proposition6.3.1] and [Ko2, Theorem 3.35] again, we may further assume that Y ′′ , ( f ′′ ) ∗ π ∗ n +1 X i =1 B i + Supp B Y ′′ ! is a globally embedded simple normal crossing pair. By [F11, Lemma 6.3.13], we can take0 < c < f ′′ B Y ′′ + c ( f ′′ ) ∗ π ∗ n +1 X i =1 B i ! > = X holds set theoretically and that there exists an irreducible component G of B Y ′′ + c ( f ′′ ) ∗ π ∗ n +1 X i =1 B i ! =1 with f ′′ ( G ) X . By Lemma 4.20,( X, ω + c B, f ′′ : ( Y ′′ , B Y ′′ + c ( f ′′ ) ∗ B ) → X ) , where B = π ∗ (cid:0)P n +1 i =1 B (cid:1) , is a quasi-log scheme.Let X be the union of Nqlc( X, ω + c B ) and all qlc strata of [ X, ω + c B ] containedin π − ( P ). By construction, Nqlc( X, ω + c B ) = X holds set theoretically. Therefore, byCase 1 in Step 1, X ∩ π − ( P ) is rationally chain connected modulo X ∩ π − ( P ). We notethat by Step 1 X ∩ π − ( P ) is rationally chain connected modulo π − ( P ) ∩ X −∞ . Thismeans that X ∩ π − ( P ) is rationally chain connected modulo π − ( P ) ∩ X −∞ . Step 3. By repeating the argument in Step 2, we can construct a finite increasing sequenceof positive real numbers 0 < c < · · · < c k < X ( · · · ( X k ONE THEOREM AND MORI HYPERBOLICITY 47 of X with the following properties:(a) [ X i , ω i ] is a quasi-log scheme, where ω i = ( ω + c i B ) | X i , for every i ,(b) Nqlc( X i +1 , ω i +1 ) = X i holds set theoretically for every i ,(c) π − ( P ) ⊂ X k holds, and(d) X i +1 ∩ π − ( P ) is rationally chain connected modulo X i ∩ π − ( P ) for every i .Hence we obtain that π − ( P ) = π − ( P ) ∩ X k is rationally chain connected modulo π − ( P ) ∩ X −∞ .We finish the proof of Theorem 1.14. (cid:3) Towards Conjecture 1.15 In this final section, we treat several results related to Conjecture 1.15. This sectionneeds some deep results on the theory of minimal models for higher-dimensional algebraicvarieties. Let us start with the following special case of the flip conjecture II. Conjecture 14.1 (Termination of flips) . Let ( X, ∆) be a Q -factorial klt pair and let π : X → S be a projective surjective morphism between normal quasi-projective varietiessuch that K X + ∆ is not pseudo-effective over S . Let ( X, ∆) =: ( X , ∆ ) ( X , ∆ ) · · · ( X i , ∆ i ) · · · be a sequence of flips over S starting from ( X, ∆) . Then it terminates after finitely manysteps. In this section, we establish the following theorem, which is a precise version of Theorem1.16. Theorem 14.2 (see Theorem 1.16) . Assume that Conjecture 14.1 holds true in dimensionat most dim π − ( P ) . Then Conjecture 1.15 holds true. For the proof of Theorem 14.2, we prepare a variant of Theorem 1.8. We need thetermination of flips in this theorem. Theorem 14.3. Let X be a normal variety and let ∆ be an effective R -divisor on X such that K X + ∆ is R -Cartier. Assume that Conjecture 14.1 holds true in dim X . Let π : X → S be a projective morphism onto a scheme S such that − ( K X + ∆) is π -amplewith dim S < dim X . We assume that Nklt( X, ∆) is not empty such that π : Nklt( X, ∆) → π (Nklt( X, ∆)) is finite. Then there exists a non-constant morphism f : A −→ X \ Nklt( X, ∆) such that π ◦ f ( A ) is a point and that the curve C , the closure of f ( A ) in X , is a ( possiblysingular ) rational curve satisfying C ∩ Nklt( X, ∆) = ∅ with < − ( K X + ∆) · C ≤ . Proof. By shrinking S suitably, we may assume that X and S are both quasi-projective.By Lemma 3.10, we can construct a projective birational morphism g : Y → X from anormal Q -factorial variety satisfying (i), (ii), and (iv) in Lemma 3.10. Since K Y + ∆ Y = g ∗ ( K X + ∆), ( K Y + ∆ Y ) | Nklt( Y, ∆ Y ) is nef over S by Lemma 3.10 (iv). Let us consider π ◦ g : Y → S . By construction, ( Y, ∆ < Y ) is a Q -factorial klt pair. Since − ( K X + ∆) is π -ample by assumption, K Y + ∆ Y is not pseudo-effective over S . Hence K Y + ∆ < Y is notpseudo-effective over S . Since ( K Y + ∆ Y ) | Nklt( Y, ∆ Y ) is nef over S , the cone theorem N E ( Y /S ) = N E ( Y /S ) ( K Y +∆ Y ) ≥ + X j R j holds by [F6, Theorem 1.1] (see also Theorem 1.5 (i)). Since K Y +∆ Y is not pseudo-effectiveover S , K Y + ∆ Y is not nef over S . Hence we have a ( K Y + ∆ Y )-negative extremal ray R of N E ( Y /S ). Then we consider the contraction morphism ϕ R : Y → W over S associatedto R (see [F6, Theorem 1.1] and Theorem 4.17). We note that − ( K Y + ∆ < Y ) · R > K Y + ∆ Y ) | Nklt( Y, ∆ Y ) is nef over S . If ϕ R is an isomorphism in a neighborhood ofNklt( Y, ∆ Y ), then we can run a minimal model program with respect to K Y + ∆ Y over S by [BCHM]. Thus we run a minimal model program with respect to K Y + ∆ Y over S .Then we have a sequence of flips and divisorial contractions Y =: Y φ / / ❴❴❴ Y φ / / ❴❴❴ · · · φ i − / / ❴❴❴ Y i φ i / / ❴❴❴ · · · over S . As usual, we put ( Y , ∆ Y ) := ( Y, ∆ Y ) and ∆ Y i +1 = φ i ∗ ∆ Y i for every i . Case 1. We assume that φ i is an isomorphism in a neighborhood of Nklt( Y i , ∆ Y i ) forevery i . Then this minimal model program is a minimal model program with respect to K Y + ∆ < Y . Hence, by Conjecture 14.1, we finally get the following diagram Y =: Y φ / / ❴❴❴ π ◦ g (cid:26) (cid:26) ✻✻✻✻✻✻✻✻✻✻✻✻✻✻✻✻ Y φ / / ❴❴❴ · · · φ k − / / ❴❴❴ Y kp (cid:15) (cid:15) Z v v ♠♠♠♠♠♠♠♠♠♠♠♠♠♠♠♠ S where φ i is a flip or a divisorial contraction for every i and p : Y k → Z is a Fano contractionover S . We note that ( K Y k + ∆ Y k ) | Nklt( Y k , ∆ Yk ) is nef over S . By Case 1 in the proofof Proposition 9.1, we can find a curve C Y k ≃ P on Y k such that p ( C Y k ) is a point, C Y k ∩ Nklt( Y k , ∆ Y k ) is a point, and 0 < − ( K Y k + ∆ Y k ) · C Y k ≤ − ( K Y + ∆ Y ) · C Y ≤ − ( K Y k + ∆ Y k ) · C Y k ≤ C Y is the strict transform of C Y k on Y . Note that C Y ∩ Nklt( Y, ∆ Y ) is apoint since φ i is an isomorphism in a neighborhood of Nklt( Y i , ∆ Y i ) for every i . Therefore, C = g ( C Y ) is a curve on X such that C ∩ Nklt( X, ∆) is a point by Lemma 3.10 (iv) with0 < − ( K X + ∆) · C ≤ 1. Hence we can construct a morphism f : A −→ X \ Nklt( X, ∆)such that f ( A ) = C ∩ ( X \ Nklt( X, ∆)). This is a desired morphism. Case 2. We assume that there exists i such that φ i is an isomorphism in a neighborhoodof Nklt( Y i , ∆ Y i ) for 0 ≤ i < i and φ i is not an isomorphism in a neighborhood ofNklt( Y i , ∆ Y i ). Then, by using Case 3 in the proof of Proposition 9.1, we can find acurve C Y i ≃ P on Y i such that C Y i ∩ Nklt( Y i , ∆ Y i ) is a point, C Y i is mapped to a pointon S , and 0 < − ( K Y i + ∆ Y i ) · C Y i ≤ f : A −→ X \ Nklt( X, ∆) . We finish the proof of Theorem 14.3. (cid:3) By Theorem 14.3, we have: Theorem 14.4. In Theorem 9.2, we further assume that dim S < dim X and that Σ = ∅ .If Conjecture 14.1 holds true in dim X , then there exists a non-constant morphism f : A −→ X \ Σ ONE THEOREM AND MORI HYPERBOLICITY 49 such that π ◦ f ( A ) is a point and that the curve C , the closure of f ( A ) in X , is a rationalcurve satisfying C ∩ Σ = ∅ with < −P · C ≤ . Proof. We use Theorem 14.3 instead of Theorem 1.8. Then the proof of Theorem 9.2implies the existence of f : A −→ X \ Σwith the desired properties. (cid:3) For the proof of Theorem 14.2, we establish the following somewhat technical lemma. Lemma 14.5. Let π : X → S be a projective surjective morphism between normal quasi-projective varieties with π ∗ O X ≃ O S and dim S > and let [ X, ω ] be a quasi-log schemesuch that π : Nqklt( X, ω ) → π (Nqklt( X, ω )) is finite. Let P be a closed point of S such that dim π − ( P ) > . Then there exists aneffective R -Cartier divisor B on S such that [ X, ω + π ∗ B ] is a quasi-log scheme with thefollowing properties: (i) Nqklt( X, ω ) ⊂ Nqklt( X, ω + π ∗ B ) , (ii) Nqklt( X, ω ) = Nqklt( X, ω + π ∗ B ) holds outside π − ( P ) , (iii) π : Nqlc( X, ω + π ∗ B ) → π (Nqlc( X, ω + π ∗ B )) is finite, and (iv) there exists a positive-dimensional qlc center of [ X, ω + π ∗ B ] in π − ( P ) .We further assume that − ω is π -ample. Let W be a positive-dimensional qlc center of [ X, ω + π ∗ B ] with π ( W ) = P . Let ν : W ν → W be the normalization. Then [ W ν , ν ∗ ω ] naturally becomes a quasi-log Fano scheme such that ν − (cid:0) Nqklt( X, ω ) ∩ π − ( P ) (cid:1) ⊂ Nqklt( W ν , ν ∗ ω ) holds set theoretically.Proof. Let B , . . . , B n +1 be general very ample Cartier divisors on S passing through P with n = dim X . Let f : ( Y, B Y ) → X be a proper morphism from a globally embeddedsimple normal crossing pair ( Y, B Y ) as in Definition 4.2. Let X ′ be the union of Nqlc( X, ω )and all qlc centers of [ X, ω ] mapped to P by π . By [F11, Proposition 6.3.1] and [Ko2,Theorem 3.35], we may assume that the union of all strata of ( Y, B Y ) mapped to X ′ by f ,which is denoted by Y ′ , is a union of some irreducible components of Y . As usual, we put Y ′′ = Y − Y ′ , K Y ′′ + B Y ′′ = ( K Y + B Y ) | Y ′′ , and f ′′ = f | Y ′′ . By [F11, Proposition 6.3.1]and [Ko2, Theorem 3.35] again, we may further assume that Y ′′ , ( f ′′ ) ∗ π ∗ n +1 X i =1 B i + Supp B Y ′′ ! is a globally embedded simple normal crossing pair. By [F11, Lemma 6.3.13], we can take0 < c < f ′′ B Y ′′ + c ( f ′′ ) ∗ π ∗ n +1 X i =1 B i ! > ∩ π − ( P ) = ∅ or dim f ′′ B Y ′′ + c ( f ′′ ) ∗ π ∗ n +1 X i =1 B i ! > ∩ π − ( P ) = 0 , and (b) the following inequalitydim f ′′ B Y ′′ + c ( f ′′ ) ∗ π ∗ n +1 X i =1 B i ! =1 ∩ π − ( P ) ≥ X, ω + π ∗ B, f ′′ : ( Y ′′ , B Y ′′ + ( f ′′ ) ∗ π ∗ B ) → X ) , where B = c P n +1 i =1 B i , is a quasi-log scheme. By construction, we see that (i) holds trueand Nqklt( X, ω + π ∗ B ) coincides with Nqklt( X, ω ) outside π − ( P ) since B , . . . , B n +1 aregeneral very ample Cartier divisors on S . Hence we have (ii). Therefore, (iii) and (iv)follow from (a) and (b), respectively.From now on, we further assume that − ω is π -ample. As usual, we put W ′ = W ∪ Nqlc( X, ω + π ∗ B ) . By adjunction (see Theorem 4.6 (i) and [F11, Theorem 6.3.5 (i)]), [ W ′ , ( ω + π ∗ B ) | W ′ ] is aquasi-log scheme. By Lemma 4.19, [ W, ( ω + π ∗ B ) | W ] naturally becomes a quasi-log scheme.We note that ( π ∗ B ) | W ∼ R π ( W ) = P . Therefore, by replacing ( ω + π ∗ B ) | W with ω | W , we see that [ W, ω | W ] is a quasi-log scheme. By Theorem 1.9, [ W ν , ν ∗ ω ] becomes aquasi-log scheme. Note that − ν ∗ ω is ample since π ◦ ν ( W ν ) = P . Claim. We have Nqklt( X, ω ) ∩ π − ( P ) ⊂ Nqklt( W ′ , ( ω + π ∗ B ) | W ′ ) ∩ π − ( P )= Nqklt( W, ( ω + π ∗ B ) | W )= Nqklt( W, ω | W ) set theoretically.Proof of Claim. We divide the proof into several steps. Step 1. By (iii) and Lemma 13.2, Nqlc( X, ω + π ∗ B ) ∩ π − ( P ) is empty or a point. By [F11,Theorem 6.8.3 (i)], every qlc center of [ X, ω + π ∗ B ] in π − ( P ) contains Nqlc( X, ω + π ∗ B ) ∩ π − ( P ) when Nqlc( X, ω + π ∗ B ) ∩ π − ( P ) = ∅ . When Nqlc( X, ω + π ∗ B ) ∩ π − ( P ) = ∅ , theset of all qlc centers intersecting π − ( P ) has a unique minimal element with respect to theinclusion by [F11, Theorem 6.8.3 (ii)]. Step 2. In this step, we will check thatNqklt( X, ω ) ∩ π − ( P ) ⊂ Nqklt( W ′ , ( ω + π ∗ B ) | W ′ ) ∩ π − ( P )holds set theoretically.If Nqklt( X, ω ) ∩ π − ( P ) = ∅ , then it is obvious. Hence we may assume that Nqklt( X, ω ) ∩ π − ( P ) = ∅ . By assumption, Nqklt( X, ω ) ∩ π − ( P ) is zero-dimensional. We take Q ∈ Nqklt( X, ω ) ∩ π − ( P ). If Q is a qlc center of [ X, ω ] or Q ∈ Nqlc( X, ω ), then Q ∈ Nqlc( X, ω + π ∗ B ) by the construction of the quasi-log scheme structure of [ X, ω + π ∗ B ]. Then we have Q ∈ Nqlc( W ′ , ( ω + π ∗ B ) | W ′ ) ⊂ Nqklt( W ′ , ( ω + π ∗ B ) | W ′ ) . Therefore, we have Q ∈ Nqklt( W ′ , ( ω + π ∗ B ) | W ′ ) ∩ π − ( P ) . From now on, we assume that Q is not a qlc center of [ X, ω ] and that Q Nqlc( X, ω ).Then, there exists a positive-dimensional qlc center V of [ X, ω ] such that Q ∈ V ∩ π − ( P ).Since Nqklt( X, ω ) = Nqklt( X, ω + π ∗ B ) holds outside π − ( P ) (see (ii)), V is also a qlc ONE THEOREM AND MORI HYPERBOLICITY 51 center of [ X, ω + π ∗ B ]. If Nqlc( X, ω + π ∗ B ) ∩ π − ( P ) = ∅ , then Nqlc( X, ω + π ∗ B ) ∩ π − ( P )is a point by (iii) and Lemma 13.2. In this case, by [F11, Theorem 6.8.3 (i)], we have Q ∈ V ∩ π − ( P ) ∩ Nqlc( X, ω + π ∗ B ). Hence Q ∈ Nqlc( W ′ , ( ω + π ∗ B ) | W ′ ) ∩ π − ( P ).This implies that Q ∈ Nqklt( W ′ , ( ω + π ∗ B ) | W ′ ) ∩ π − ( P ). Thus we further assume thatNqlc( X, ω + π ∗ B ) ∩ π − ( P ) = ∅ . By shrinking X around π − ( P ), we may assume that[ X, ω + π ∗ B ] is quasi-log canonical. Then Q ∈ V ∩ W ∩ π − ( P ) by Step 1 (see also [F11,Theorem 6.8.3 (ii)]). Hence Q ∈ Nqklt( W ′ , ( ω + π ∗ B ) | W ′ ) ∩ π − ( P ) by [F11, Theorem6.3.11 (i)]. More precisely, Q is a qlc center of [ W ′ , ( ω + π ∗ B ) | W ′ ].In any case, we obtainNqklt( X, ω ) ∩ π − ( P ) ⊂ Nqklt( W ′ , ( ω + π ∗ B ) | W ′ ) ∩ π − ( P )set theoretically. Step 3. By Step 1 and Lemma 4.19,Nqklt( W ′ , ( ω + π ∗ B ) | W ′ ) ∩ π − ( P ) = Nqklt( W, ( ω + π ∗ B ) | W )holds set theoretically. By the definition of the quasi-log scheme structure of [ W, ω | W ],Nqklt( W, ( ω + π ∗ B ) | W ) = Nqklt( W, ω | W )obviously holds.We finish the proof of Lemma 14.5. (cid:3) Hence by Claim ν − (cid:0) Nqklt( X, ω ) ∩ π − ( P ) (cid:1) ⊂ Nqklt( W ν , ν ∗ ω )holds set theoretically since ν − (Nqklt( W, ω | W )) = Nqklt( W ν , ν ∗ ω ) by Theorem 1.9. (cid:3) Let us prove Theorem 14.2, which is stronger than Theorem 1.16. Proof of Theorem 14.2. We first use the reduction as in Steps 2, 3, and 4 in the proof ofTheorem 1.6. Let us explain it more precisely for the reader’s convenience. Step 1. We take an irreducible component W of X such that C † ⊂ W . We put X ′ = W ∪ Nqlc( X, ω ). Then, by adjunction (see Theorem 4.6 (i) and [F11, Theorem 6.3.5 (i)]),[ X ′ , ω ′ = ω | X ′ ] is a quasi-log scheme. By replacing [ X, ω ] with [ X ′ , ω ′ ], we may assume that X \ X −∞ is irreducible. By Lemma 4.19, we may replace X with X \ X −∞ and assumethat X is a variety. Then, by taking the normalization, we may further assume that X isa normal variety (see Theorem 1.9). Step 2. By taking the Stein factorization, we may further assume that π ∗ O X ≃ O S . Weput Σ = Nqklt( X, ω ). It is sufficient to find a non-constant morphism f : A −→ ( X \ Σ) ∩ π − ( P )such that the curve C , the closure of f ( A ) in X , is a (possibly singular) rational curvesatisfying C ∩ Σ = ∅ with 0 < − ω · C ≤ . Without loss of generality, we may assume that X and S are quasi-projective by shrinking S suitably. Step 3. By assumption, dim π − ( P ) > π − ( P ) ∩ Σ = ∅ . When dim S > 0, byLemma 14.5, we take an effective R -Cartier divisor B on S such that [ X, ω + π ∗ B ] is aquasi-log scheme satisfying the properties (i), (ii), (iii), and (iv) in Lemma 14.5. Thenwe take a positive-dimensional qlc center W of [ X, ω + π ∗ B ] in π − ( P ) such that there isno positive-dimensional qlc center W † ( W of [ X, ω + π ∗ B ]. By Lemma 14.5, [ W ν , ν ∗ ω ]naturally becomes a quasi-log Fano scheme, where ν : W ν → W is the normalization. When dim S = 0, it is sufficient to put B = 0 and W = X . By construction, Nqklt( W ν , ν ∗ ω )is finite. On the other hand, Nqklt( W ν , ν ∗ ω ) is connected (see Lemma 13.2 and [F11,Theorem 6.8.3]). By Lemma 14.5, we obtain ∅ 6 = ν − (cid:0) Σ ∩ π − ( P ) (cid:1) ⊂ Nqklt( W ν , ν ∗ ω ) . Hence Nqklt( W ν , ν ∗ ω ) is a point such that ν − (Σ ∩ π − ( P )) = Nqklt( W ν , ν ∗ ω ) holds settheoretically. By applying Theorems 1.10 and 14.4 to [ W ν , ν ∗ ω ] as in Step 5 in the proofof Theorem 1.6, we obtain a non-constant morphism h : A −→ W ν \ Nqklt( W ν , ν ∗ ω )such that C ′ , the closure of h ( A ) in W ν , is a (possibly singular) rational curve satisfying C ′ ∩ Nqklt( W ν , ν ∗ ω ) = ∅ with 0 < − ν ∗ ω · C ′ ≤ 1. Then f := ι ◦ ν ◦ h : A −→ ( X \ Σ) ∩ π − ( P ) , where ι : W ֒ → X is a natural inclusion, is a desired morphism.We finish the proof of Theorem 14.2. (cid:3) For the proof of Theorem 1.20, we prepare the following theorem. The proof of Theorem14.6 uses a deep result on the existence problem of minimal models in [H1]. Theorem 14.6. Let ( X, ∆) be a dlt pair and let π : X → S be a projective morphismbetween normal varieties such that − ( K X + ∆) is π -ample. We assume that Nklt( X, ∆) is not empty such that π : Nklt( X, ∆) → π (Nklt( X, ∆)) is finite and that there exists acurve C † on X such that π ( C † ) is a point with C † ∩ Nklt( X, ∆) = ∅ . Then there exists anon-constant morphism f : A −→ X \ Nklt( X, ∆) such that π ◦ f ( A ) is a point and that the curve C , the closure of f ( A ) in X , is a ( possiblysingular ) rational curve satisfying C ∩ Nklt( X, ∆) = ∅ with < − ( K X + ∆) · C ≤ . Proof. By shrinking S suitably, we may assume that X and S are both quasi-projective.By Lemma 3.10, we can construct a projective birational morphism g : Y → X from anormal Q -factorial variety satisfying (i), (ii), and (iv) in Lemma 3.10. Since K Y + ∆ Y = g ∗ ( K X + ∆), ( K Y + ∆ Y ) | Nklt( Y, ∆ Y ) is nef over S by Lemma 3.10 (iv). Let us consider π ◦ g : Y → S . By construction, ( Y, ∆ Y ) is a Q -factorial dlt pair.If dim S = dim X , then K Y + ∆ Y is pseudo-effective over S . In this case, we can takean effective R -divisor A on Y such that K Y + ∆ Y + A ∼ R ,π ◦ g Y, ∆ Y + A ) isdlt since − ( K Y + ∆ Y ) = − g ∗ ( K X + ∆) is ( π ◦ g )-semi-ample. Hence ( Y, ∆ Y ) has a goodminimal model over S by [H1, Theorem 1.1] and any ( K Y + ∆ Y )-minimal model programover S with scaling of an ample divisor terminates (see [H1, Theorem 2.11]).If dim S < dim X , then K Y + ∆ Y is not pseudo-effective over S since − ( K X + ∆) isample over S . In this case, we have a ( K Y +∆ Y )-minimal model program which terminatesat a Mori fiber space by [BCHM].Therefore, we have a finite sequence of flips and divisorial contractions Y =: Y φ / / ❴❴❴ Y φ / / ❴❴❴ · · · φ i − / / ❴❴❴ Y i φ i / / ❴❴❴ · · · φ k − / / ❴❴❴ Y k starting from ( Y , ∆ Y ) := ( Y, ∆ Y ) over S such that ( Y k , ∆ Y k ) is a good minimal modelof ( Y, ∆ Y ) over S or p : Y k → Z is a Mori fiber space with respect to K Y k + ∆ Y k over S ,where ∆ Y i +1 = φ i ∗ ∆ Y i for every i . By assumption, we can take a curve C ′ on Y such that − ( K Y + ∆ Y ) · C ′ > C ′ ∩ Nklt( Y, ∆ Y ) = ∅ . If φ i is an isomorphism in a neighborhoodof Nklt( Y i , ∆ Y i ) for 0 ≤ i < l , then(14.1) 0 < − ( K Y + ∆ Y ) · C ′ ≤ − ( K Y l + ∆ Y l ) · C ′ Y l ONE THEOREM AND MORI HYPERBOLICITY 53 holds by the negativity lemma, where C ′ Y l is the strict transform of C ′ on Y l . Case 1. We assume that φ i is an isomorphism in a neighborhood of Nklt( Y i , ∆ Y i ) for every i . Then, by (14.1), the final model Y k has a Mori fiber space structure p : Y k → Z over S .We note that ( K Y k + ∆ Y k ) | Nklt( Y k , ∆ Yk ) is nef over S . Hence the argument in Case 1 in theproof of Theorem 14.3 works without any changes. Then we get a non-constant morphism f : A −→ X \ Nklt( X, ∆)with the desired properties. Case 2. By Case 1, we may assume that there exists i such that φ i is an isomorphismin a neighborhood of Nklt( Y i , ∆ Y i ) for 0 ≤ i < i and φ i is not an isomorphism in aneighborhood of Nklt( Y i , ∆ Y i ). The argument in Case 2 in the proof of Theorem 14.3works without any changes. Then we get a non-constant morphism f : A −→ X \ Nklt( X, ∆)with the desired properties.We finish the proof of Theorem 14.6. (cid:3) We close this section with the proof of Theorem 1.20. Since adjunction works well fordlt pairs, Theorem 1.20 directly follows from Theorem 14.6. Proof of Theorem 1.20. We put W = U j . Then W is an lc stratum of ( X, ∆). By adjunc-tion, it is well known that we have( K X + ∆) | W = K W + ∆ W such that ( W, ∆ W ) is dlt and that the lc centers of ( W, ∆ W ) are exactly the lc centersof ( X, ∆) that are strictly included in W (see, for example, [F3, Proposition 3.9.2]). Byreplacing π : X → S with the Stein factorization of ϕ R j : W → ϕ R j ( W ), we may assumethat π : Nklt( X, ∆) → π (Nklt( X, ∆)) is finite and that there exists a curve C † on X suchthat π ( C † ) is a point with C † ∩ Nklt( X, ∆) = ∅ . By Theorem 14.6, we obtain a desirednon-constant morphism f : A → X \ Nklt( X, ∆)with the desired properties. (cid:3) As we have already mentioned, we will completely prove Conjecture 1.15 in a joint paperwith Kenta Hashizume (see [FH]), where we use some deep results on the minimal modelprogram for log canonical pairs. We strongly recommend the interested reader to see [FH]. References [A] F. Ambro, Quasi-log varieties, Tr. Mat. Inst. Steklova (2003), Biratsion. Geom. Line˘ın.Sist. 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