APPLICATIONS
OSAMU FUJINO AND KENTA HASHIZUME
Abstract. The main purpose of this paper is to establish some useful partial resolutions of singularities for pairs from the minimal model theoretic viewpoint. We first establish the existence of log canonical modifications of normal pairs under some suitable assumptions.
It recovers Kawakita’s inversion of adjunction on log canonicity in full generality. We also discuss the existence of semi-log canonical modifications for demi-normal pairs and construct dlt blow-ups with several extra good properties. As applications, we study lengths of extremal rational curves and so on.
Contents
1. Introduction 1
2. Preliminaries 5
3. Proof of Theorems 1.1, 1.2, and 1.5 8
4. On semi-log canonical modifications of demi-normal pairs 11
5. On inversion of adjunction on log canonicity 15
6. Proof of Theorem 1.6 17
7. Quick review of quasi-log schemes 21
8. Proof of Theorems 1.7 and 1.8 22
References 23
1. Introduction
The main purpose of this paper is to establish some useful partial resolutions of singu- larities for pairs from the minimal model theoretic viewpoint.
Let us start with an elementary example. Let X be a normal surface. Then it is well known that there exists a unique minimal resolution of singularities f: Y → X of X. It plays a crucial role for the study of singularities ofX. Let g: Z →X be any resolution of singularities of X. Then we can see f: Y →X as a relative minimal model of Z over X.
When X is a higher-dimensional quasi-projective variety andg: Z →X is a resolution of singularities ofX, we can always construct a relative minimal modelf: Y →X ofZ over X by running a minimal model program (see [BCHM]). Unfortunately, however, Y may be singular. In general,Y has Q-factorial terminal singularities. Since the singularities of Y is milder than those of X, f: Y → X sometimes plays an important role as a partial resolution of singularities ofX.
In the recent study of higher-dimensional algebraic varieties, we know that it is natural and useful to treat pairs. Let us consider a quasi-projective log canonical pair (X,∆).
Based on [BCHM], Hacon constructed a projective birational morphism f: (Y,∆Y) → (X,∆) from aQ-factorial divisorial log terminal pair (Y,∆Y) withKY+ ∆Y =f∗(KX+ ∆)
Date: 2021/2/20, version 0.19.
2010 Mathematics Subject Classification. Primary 14E30; Secondary 14J45, 32Q45.
Key words and phrases. minimal model program, dlt blow-ups, log canonical modifications, inversion of adjunction, lengths of extremal rational curves, cone theorem, Mori hyperbolicity, quasi-log schemes.
1
(see [Fujin1] and [KK]). We usually callf: (Y,∆Y)→(X,∆) a dlt blow-up of (X,∆). By dlt blow-ups, many problems on log canonical pairs can be reduced to those onQ-factorial divisorial log terminal pairs. We can see f: (Y,∆Y) → (X,∆) as a partial resolution of singularities of (X,∆) from the minimal model theoretic viewpoint.
In this paper, we are mainly interested in pairs whose singularities are not necessarily log canonical. We first establish the existence of log canonical modifications, which is a kind of partial resolution of singularities of pairs from the minimal model theoretic viewpoint.
Theorem 1.1 (Log canonical modifications). Let X be a normal variety and let ∆ be an effective R-divisor onX such thatKX+ ∆ isR-Cartier. Let B be anR-divisor on X such that the coefficients of B belong to [0,1], ∆−B is effective and SuppB = Supp ∆. Then there exists a log canonical modification of X and B, that is, a log canonical pair (Y, BY) and a projective birational morphism f:Y →X such that
• the divisor BY is the sum off∗−1B and the reduced f-exceptional divisor E, that is, E =∑
jEj where Ej are thef-exceptional prime divisors on Y, and
• the divisor KY +BY is f-ample.
The following special case of Theorem 1.1 is important for applications of this paper.
Theorem 1.2. Let X be a normal variety and let ∆ be an effective R-divisor on X such thatKX + ∆ is R-Cartier. We put
B = ∆<1+ Supp ∆≥1. Then there exists a log canonical modification of X and B.
Theorem 1.1 is a generalization of [OX, Theorem 1.2]. Odaka and Xu proved Theorem 1.1 under the extra assumption that ∆ = B and B is a Q-divisor. By Theorem 1.2, we can recover Kawakita’s inversion of adjunction on log canonicity (see Corollary5.5).
Theorem 1.3 (see [Ka] and Corollary 5.5). Let (X, S+B) be a normal pair such that S is a reduced divisor, B is effective, and S and B have no common irreducible components.
Let ν: Sν →S be the normalization of S. We put KSν +BSν =ν∗(KX +S +B). Then (X, S+B) is log canonical near S if and only if (Sν, BSν) is log canonical.
To the best of the authors’ knowledge, it is the first time to recover Kawakita’s inversion of adjunction on log canonicity in full generality by the minimal model theory. We note that B is an effective R-divisor but is not necessarily a boundary R-divisor in Theorem 1.3.
For equidimensional reduced and reducible schemes, Koll´ar and Shepherd-Barron con- structed minimal semi-resolutions of surfaces (see [KSB, Proposition 4.10]). As a higher- dimensional generalization, Fujita (see [Fujit]) established semi-terminal modifications of demi-normal pairs. Here, we note that ademi-normal scheme means an equidimensional reduced scheme which satisfies Serre’sS2 condition and is normal crossing in codimension one. On the other hand, Odaka and Xu treated semi-log canonical modifications of demi- normal pairs in [OX, Corollary 1.2]. The following theorem is a generalization of [OX, Corollary 1.2] and Theorem 1.2 for Q-divisors.
Theorem 1.4 (see Theorem 4.4). Let X be a demi-normal scheme, and let ∆ be an effective Q-divisor on X such that Supp ∆does not contain any codimension one singular locus and KX + ∆ isQ-Cartier. We put
B = ∆<1+ Supp ∆≥1.
ThenX equipped withB has a semi-log canonical modification, that is, a semi-log canonical pair (Y, BY) and a projective birational morphism f: Y →X such that
• f is an isomorphism over the generic point of any codimension one singular locus,
• BY is the sum of the birational transform of B on Y and the reducedf-exceptional divisor, and
• KY +BY is f-ample.
We remark thatKX+B in Theorem1.4is not necessarilyQ-Cartier. As in [OX, Example 3.1], there is an example of demi-normal pairs having no semi-log canonical modifications.
In our proof of Theorem 1.4, the R-Cartier property of KX + ∆ is crucial to apply the gluing theory of Koll´ar. For the details, see Remark 4.5.
By combining the idea of the proof of Theorem 1.1 with the minimal model theory for Q-factorial divisorial log terminal pairs, we obtain Theorem 1.5, which is a generalization of [Fujin3, Lemma 3.10]. Note that the morphismg: (Y,∆Y)→(X,∆) in Theorem 1.5 is a kind of dlt blow-up with some extra good properties. Here, we call it aspecial crepant modelof (X,∆).
Theorem 1.5 (Special crepant models). Let X be a normal quasi-projective variety and let ∆ an effective R-divisor onX such that KX + ∆ is R-Cartier. Then we can construct a crepant modelg: (Y,∆Y)→(X,∆), that is, a projective birational morphism g: Y →X from a normal Q-factorial variety Y and an effective R-divisor ∆Y on Y such that
(i) KY + ∆Y =g∗(KX + ∆),
(ii) the pair (Y,∆′Y)is dlt, where ∆′Y = ∆<1Y + Supp ∆≥Y1, such thatKY + ∆′Y isg-semi- ample,
(iii) every g-exceptional prime divisor is a component of (∆′Y)=1,
(iv) g−1(Nklt(X,∆)) coincides with Nklt(Y,∆Y) and Nklt(Y,∆′Y) set theoretically, (v) g−1(Nlc(X,∆)) coincides with Nlc(Y,∆Y) and Supp ∆>1Y set theoretically, and (vi) there is an effective R-divisor ΓY on Y such that
(a) Supp ΓY =g−1(Nklt(X,∆)) = Supp ∆≥1Y set theoretically, (b) −ΓY is g-semi-ample, and
(c) ∆Y −ΓY is effective and (Y,∆Y −ΓY) is klt.
We note that we only need the minimal model program at the level of [BCHM] for the proof of [Fujin3, Lemma 3.10]. On the other hand, the proof of Theorems 1.1 and 1.5 is much harder because it heavily depends on the minimal model theory for log canonical pairs discussed in [Has].
Although Theorem 1.5 may look artificial, it is very powerful and useful. As an appli- cation of Theorem1.5, we prove:
Theorem 1.6. Let X be a normal variety and let ∆ be an effective R-divisor on X such thatKX+ ∆ isR-Cartier. Let π: X →S be a projective morphism onto a scheme S such that−(KX + ∆) isπ-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−1(P) with Nklt(X,∆)∩C†̸=∅. Then there exists a non-constant morphism
f: A1 −→(X\Nklt(X,∆))∩π−1(P)
such that the curve C, the closure of f(A1) in X, is a (possibly singular) rational curve satisfying C∩Nklt(X,∆) ̸=∅ with
0<−(KX + ∆)·C≤1.
Theorem1.6is a kind of generalization of [Fujin3, Theorem 1.8]. By combining Theorem 1.6 with the powerful framework of quasi-log schemes discussed in [Fujin3], we have:
Theorem 1.7. Let[X, ω]be a quasi-log scheme and letπ:X →S be a projective morphism 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−1(P) with Nqklt(X, ω)∩C†̸=∅. Then there exists a non-constant morphism
f: A1 −→(X\Nqklt(X, ω))∩π−1(P)
such that C, the closure of f(A1) in X, satisfies C∩Nqklt(X, ω)̸=∅ with 0<−ω·C ≤1.
Theorem 1.7 completely solves the first author’s conjecture (see [Fujin3, Conjecture 1.15]). As an easy direct consequence of Theorem1.7, we establish:
Theorem 1.8(Lengths of extremal rational curves for quasi-log schemes). Let[X, ω]be a quasi-log scheme and let π: X →S be a projective morphism between schemes. Let Rj be an ω-negative extremal ray of N E(X/S) that are rational and relatively ample at infinity and let φRj be the contraction morphism associated to Rj. Let Uj be any open qlc stratum of [X, ω] such that φRj: Uj →φRj(Uj) is not finite and that φRj: W† →φRj(W†) is finite for every qlc center W† of [X, ω] with W†⊊Uj, where Uj is the closure of Uj in X. Let P be a closed point of φRj(Uj). If there exists a curve C† such that φRj(C†) = P, C† ̸⊂Uj, and C† ⊂Uj, then there exists a non-constant morphism
fj: A1 −→Uj ∩φ−R1
j(P)
such that Cj, the closure of fj(A1) in X, spans Rj in N1(X/S) and satisfies Cj ̸⊂Uj with 0<−ω·Cj ≤1.
Note that Theorem 1.8 supplements [Fujin3, Theorem 1.6 (iii)]. We also note that the above results generalize [LZ, Theorem 3.1] completely. The following example may help the reader understand Theorem1.8.
Example 1.9. This example shows that the conditionC†̸⊂Uj is necessary for the estimate of the length of Cj in Theorem 1.8. Let H1, . . . , Hn be general hyperplanes on X = Pn. We put ∆ =∑n
i=1Hi and ∆′ =∑n−1
i=1 Hi. Let us consider the structure morphismπ: X → S= SpecC. We note that (X,∆) and (X,∆′) are log canonical and that −(KX+ ∆) and
−(KX + ∆′) areπ-ample. Since the Picard number of X is one, π: X →S is an extremal contraction. Let C be any one-dimensional lc center of (X,∆). Then it is easy to see that C≃P1,−(KX+∆)·C = 1, and the open lc center associated toCis isomorphic toA1. On the other hand, there are no zero-dimensional lc centers of (X,∆′) and−(KX+ ∆′)·C′ ≥2 holds for every curveC′ on X.
We summarize the contents of this paper. In Section2, we collect some basic definitions and results for the reader’s convenience. Section 3 is the main part of this paper. We prove Theorems 1.1, 1.2, and 1.5 by using the minimal model theory for log canonical pairs. The main ingredient of this section is the second author’s theorem: Theorem 3.1, which was obtained in [Has]. In Section 4, we discuss semi-log canonical modifications for demi-normal pairs. We prove Theorem 1.4 by using Theorem 1.2 and Koll´ar’s gluing theory in [Ko]. If the reader is interested only in normal pairs, then he or she can skip this section. In Section 5, we treat inversion of adjunction on log canonicity. We first prove a slight generalization of Hacon’s inversion of adjunction on log canonicity for log canonical centers. Then we recover Kawakita’s inversion of adjunction in full generality (see Theorem 1.3) as a special case. Section 6 is devoted to the proof of Theorem 1.6, which heavily depends on the minimal model program for normal pairs. In Section 7, we
quickly review some basic definitions in the theory of quasi-log schemes. In Section8, we prove Theorems1.7and1.8 by using Theorem1.6and the powerful framework of quasi-log schemes. We note that we need quasi-log schemes only in Sections 7and 8.
Acknowledgments. The first author was partially supported by JSPS KAKENHI Grant Numbers JP16H03925, JP16H06337. The second author was partially supported by JSPS KAKENHI Grant Numbers JP16J05875, JP19J00046. The authors thank Christopher Hacon very much for answering their question.
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 overC.
2. Preliminaries
In this paper, we use the theory of minimal models for higher-dimensional log canonical pairs. Here we collect some definitions and results for the reader’s convenience. For the details, see [Fujin1], [Fujin2], [Ko], and [KM].
Definition 2.1 (Singularities of pairs). Let X be a variety and let E be a prime divisor onY for some birational morphism f: Y →X from a normal varietyY. ThenE is called a divisor over X. A normal pair (X,∆) consists of a normal variety X and an R-divisor onX such that KX + ∆ is R-Cartier. Let (X,∆) be a normal pair and let f:Y →X be a projective birational morphism from a normal varietyY. Then we can write
KY =f∗(KX + ∆) +∑
E
a(E, X,∆)E with
f∗ (∑
E
a(E, X,∆)E )
=−∆,
where E runs over prime divisors on Y. We call a(E, X,∆) the discrepancy of E with respect to (X,∆). Note that we can define the discrepancy a(E, X,∆) for any prime divisor E over X by taking a suitable resolution of singularities of X. If a(E, X,∆)≥ −1 (resp. > −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 calledlog canonical(lc, for short) andkawamata log terminal (klt, for short) if it is sub log canonical 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 varietyY such that both Exc(f), the exceptional locus off, and Exc(f)∪Suppf∗−1∆ are simple normal crossing divisors on Y and that a(E, X,∆) >−1 holds for every f-exceptional divisor E onY, then (X,∆) is called divisorial log terminal (dlt, for short).
Definition 2.2 (Non-klt loci, non-lc loci, and lc centers). 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 canonical in a neighborhood of the generic point off(E) and thata(E, X,∆) =−1, thenf(E) is called alog canonical center (an lc center, for short) of (X,∆).
From now on, we further assume that ∆ is effective. Then the non-klt locus of (X,∆), denoted by Nklt(X,∆), is the smallest closed subset Z of X whose complement (X \ Z,∆|X\Z) is a klt pair. Similarly, the non-lc locus of (X,∆), denoted by Nlc(X,∆), is the smallest closed subset Z of X such that the complement (X\Z,∆|X\Z) is log canonical.
We can define natural scheme structures on Nklt(X,∆) and Nlc(X,∆) by the multiplier
ideal sheaf of (X,∆) and the non-lc ideal sheaf of (X,∆), respectively. However, we omit them here since we do not need them in this paper.
Definition 2.3. Let X be an equidimensional reduced scheme and let D = ∑
idiDi be an R-divisor on X such that di is a real number and Di is an irreducible reduced closed subscheme ofX of pure codimension one for everyi with Di ̸=Dj. We put
D<1 = ∑
di<1
diDi, D≤1 = ∑
di≤1
diDi, D=1 = ∑
di=1
diDi, D≥1 = ∑
di≥1
diDi, and D>1 =∑
di>1
diDi. We also put
⌊D⌋=∑
i
⌊di⌋Di, ⌈D⌉=−⌊−D⌋, and {D}=D− ⌊D⌋,
where⌊di⌋is the integer defined bydi−1<⌊di⌋ ≤di. We say thatDis aboundary divisor if D is effective andD=D≤1. We say that D is a reduced divisor if D=D=1.
Notation 2.4. Let f: X 99K X′ be a birational map of normal varieties and let D be an R-divisor on X. If there is no risk of confusion, DX′ denotes the the sum of f∗D and the reduced f−1-exceptional divisor E on X′, that is, E = ∑
jEj where Ej are the f−1-exceptional prime divisors on X′.
Definition 2.5. Letp: V →W be a projective surjective morphism from a normal variety V to a varietyW and letD1 andD2 beR-Cartier divisors onV. Then D1 ∼R,W D2 means that there exists an R-Cartier divisor D on W such that D1−D2 ∼R p∗D. We say that D1 is R-linearly equivalent toD2 overW when D1 ∼R,W D2.
In this paper, we adopt the following definition of models.
Definition 2.6 (Models). Let (X,∆) be a log canonical pair and X → Z a projective morphism to a variety Z. Let X′ → Z be a projective morphism from a normal variety and let ϕ: X 99K X′ be a birational map over Z. Let E be the reduced ϕ−1-exceptional divisor onX′, that is, E =∑
jEj where Ej are the ϕ−1-exceptional prime divisors on X′. Put ∆′ =ϕ∗∆ +E. IfKX′+ ∆′ isR-Cartier, then the pair (X′,∆′) is called alog birational modelof (X,∆) over Z. A log birational model (X′,∆′) of (X,∆) over Z is called a good minimal modelif
• X′ is Q-factorial,
• KX′ + ∆′ is semi-ample over Z, and
• for any prime divisor D onX which is exceptional over X′, we have a(D, X,∆)< a(D, X′,∆′).
A log birational model (X′,∆′) of (X,∆) over Z is called a Mori fiber space if X′ is Q-factorial and there is a contraction X′ →W overZ with dimW <dimX′ such that
• the relative Picard number ρ(X′/W) is one and−(KX′+ ∆′) is ample overW, and
• for any prime divisor D over X, we have
a(D, X,∆) ≤a(D, X′,∆′)
and strict inequality holds if D is a divisor onX and is exceptional over X′. We make two important remarks on the minimal model program for log canonical pairs.
Remark 2.7. Let (X,∆) be a Q-factorial dlt pair and π: X →Z a projective morphism from a normal quasi-projective varietyX to a quasi-projective variety Z. If (X,∆) has a good minimal model or a Mori fiber space overZ as in Definition2.6, then all (KX + ∆)- minimal model programs over Z with scaling of an ample divisor terminate (see [B2, Theorem 4.1]).
Remark 2.8. Let π: X → Z be a projective morphism from a normal quasi-projective variety X to a quasi-projective variety Z. Let (X,∆) and (X,∆′) be two Q-factorial dlt pairs such thatKX+ ∆′ ∼R,Z t(KX+ ∆) for a positive real numbert. Suppose that (X,∆) has a good minimal model overZ. By Remark2.7, there exists a (KX+ ∆)-minimal model program overZ with scaling of an ample divisor that terminates after finitely many steps.
Because any (KX+ ∆)-minimal model program overZ with scaling of an ample divisor is also a (KX + ∆′)-minimal model program over Z with scaling of an ample divisor, we see that there is a (KX+ ∆′)-minimal model program overZ terminating with a good minimal model. Thus, we see that (X,∆′) has a good minimal model overZ.
Definition 2.9 (Log canonical modifications). LetX be a normal variety and let B be a boundaryR-divisor onX. Then, alog canonical modification ofX andB is a log canonical pair (Y, BY) and a projective birational morphism f: Y →X such that
• the divisor BY is the sum of f∗−1B and the reduced f-exceptional divisor E, that is, E =∑
jEj where Ej are the f-exceptional prime divisors on Y, and
• the divisor KY +BY is f-ample.
In this paper, if there is no risk of confusion, then the notationf: (Y, BY)→(X, B) denotes the structure of the log canonical modification when there is a log canonical modification of X and B.
In this paper, we will freely use the existence of dlt blow-ups, which was obtained in [Fujin3].
Theorem 2.10 (Dlt blow-ups, see [Fujin3, Theorem 3.9]). Let X be a normal quasi- projective variety and let ∆ = ∑
idi∆i be an effective R-divisor on X such that KX + ∆ is R-Cartier. In this case, we can construct a projective birational morphism f: Y → X from a normal quasi-projective variety Y with the following properties.
(i) Y is Q-factorial.
(ii) a(E, X,∆)≤ −1 for every f-exceptional divisor E on Y. (iii) We put
∆†:= ∑
0<di<1
dif∗−1∆i+∑
di≥1
f∗−1∆i+ ∑
E:f-exceptional
E.
Then (Y,∆†) is dlt and the following equality KY + ∆† =f∗(KX + ∆) + ∑
a(E,X,∆)<−1
(a(E, X,∆) + 1)E holds.
Note that ∆ is not necessarily a boundary divisor in Theorem 2.10. We close this section with an important remark on Theorem2.10.
Remark 2.11. Let us recall how to construct f: Y → X in Theorem 2.10. In the proof of [Fujin3, Theorem 3.9], we first take a suitable resolution of singularities of X and then run a minimal model program overX. After finitely many flips and divisorial contractions over X, we get a desired birational map f: Y → X. Hence we may further assume that f: Y →X is the identity map over some nonempty Zariski open subset ofX in Theorem 2.10.
3. Proof of Theorems 1.1, 1.2, and 1.5
In this section, we prove Theorems 1.1, 1.2, and 1.5. One of the main ingredients of this section is the second author’s following result on the minimal model program for log canonical pairs.
Theorem 3.1 ([Has, Corollary 3.6]). Let π: X →Z be a projective morphism of normal quasi-projective varieties and let (X, B) be a log canonical pair. Suppose that there is an effective R-divisor D on X such that
(a) −(KX +B+D) is nef over Z, and
(b) (X, B+aD) is log canonical for some positive real number a.
Then, (X, B) has a good minimal model or a Mori fiber space over Z.
Before we prove Theorem 1.1, we prepare an elementary lemma.
Lemma 3.2. Let X be a normal variety and B a boundary R-divisor on X. Suppose that there are two log canonical modifications f: (Y, BY)→(X, B) and f′: (Y′, BY′)→(X, B) of X and B. Then the induced birational map ϕ:=f′−1◦f: Y 99KY′ is an isomorphism and ϕ∗BY =BY′.
Proof. Let h: W → Y and h′: W → Y′ be a common resolution of ϕ. We define an R-divisor E on W by
E :=h∗(KY +BY)−h′∗(KY′ +BY′).
Since ϕ is a birational map over X, every component D of E is exceptional over X. If a component D of E is not h-exceptional, then h∗D is exceptional over X. Thus we have a(D, Y, BY) = −coeffh∗D(BY) = −1. On the other hand, we have a(D, Y′, BY′) ≥ −1 because (Y′, BY′) is log canonical. So, we obtain coeffD(E)≥0. Applying the negativity lemma ([BCHM, Lemma 3.6.2 (2)]) toh: W →X and E, we have E ≥ 0. We apply the same argument to −E, then we obtain −E ≥ 0. Therefore, it follows that E = 0. Since KY +BY andKY′+BY′ are both ample overX,ϕis an isomorphism andϕ∗BY =BY′. □ Remark 3.3. Let X be a smooth projective variety and let g: X → X be any auto- morphism of X. Then g: X → X is a log canonical modification of X and B = 0 by definition.
In some geometric applications, we implicitly require that a log canonical modification f: (Y, BY) → (X, B) satisfies the extra assumption that f is the identity morphism over some nonempty Zariski open subset of X. Under this extra assumption, by Lemma 3.2, the log canonical modificationf: (Y, BY)→(X, B) of X and B is unique if it exists.
Let us prove Theorem 1.1.
Proof of Theorem 1.1. In Step 1, we will prove Theorem 1.1 under the extra assumption that X is quasi-projective. Then, in Step 2, we will treat the general case.
Step 1. In this step, we assume thatX is quasi-projective. We take a dlt blow-upg: Z → X with KZ + ∆Z = g∗(KX + ∆) as in Theorem 2.10, that is, g is a projective birational morphism such that every g-exceptional prime divisor F satisfies a(F, X,∆) ≤ −1 and that (Z,∆<1Z + Supp ∆≥Z1) is aQ-factorial dlt pair. Note that we may further assume that g is the identity morphism over some nonempty Zariski open subset ofX by Remark2.11.
We define an R-divisor BZ on Z to be the sum of g∗−1B and the reduced g-exceptional divisor (Notation2.4). Then the relations
BZ ≥0 and (
∆<1Z + Supp ∆≥Z1)
−BZ ≥0
hold since the coefficients ofB belong to [0,1] and ∆−B is effective. This implies that the pair (Z, BZ) is a Q-factorial dlt pair. We prove that (Z, BZ) has a good minimal model
overX. We put
DZ = ∆Z −BZ. We have ∆Z−(
∆<1Z + Supp ∆≥Z1)
≥0 by construction, so DZ = ∆Z−BZ ≥(
∆<1Z + Supp ∆≥Z1)
−BZ ≥0,
from which DZ is an effective R-divisor on Z. Furthermore, recalling SuppB = Supp ∆ and that BZ is the sum of g−1∗ B and the reduced g-exceptional divisor, it follows that Supp ∆Z = SuppBZ. Thus, we see that
SuppDZ ⊂Supp ∆Z = SuppBZ.
We can find a real number t > 0 such that BZ−tDZ ≥ 0. Then the pair (Z, BZ −tDZ) is dlt because (Z, BZ) is a dlt pair andDZ is effective. SinceKZ+ ∆Z =g∗(KX + ∆), we have
KZ+BZ =KZ+ ∆Z−DZ ∼R,X −DZ. By this relation, we obtain
KZ+BZ−tDZ ∼R,X −(1 +t)DZ ∼R,X (1 +t)(KZ+BZ).
By Remark2.8, the existence of a good minimal model of (Z, BZ) overX follows from the existence of a good minimal model of (Z, BZ−tDZ) over X. We put
B˜Z =BZ −tDZ.
ThenKZ+ ˜BZ+ (1 +t)DZ ∼R,X 0 and (Z,B˜Z+tDZ) is dlt. By Theorem3.1, (Z,B˜Z) has a good minimal model overX. Therefore, (Z, BZ) also has a good minimal model over X.
By running a minimal model program over X, we get a good minimal model (Z′, BZ′) of (Z, BZ) over X (see Remark 2.7). Let Z′ → Y be the contraction over X induced by KZ′ +BZ′. We define BY to be the birational transform of BZ′ on Y. Then it is easy to check that (Y, BY) is a log canonical pair and the induced morphism f: Y → X is the desired birational morphism. By construction, we may assume thatf: Y → X is the identity morphism over some nonempty Zarsiki open subset ofX.
Step 2.Let us treat the general case, that is,Xis not necessarily quasi-projective. We take a finite affine open coveringX =∪
iUi. By Step 1, there exist log canonical modifications fi: (Vi, BVi) → (Ui, B|Ui) of Ui and B|Ui such that fi is the identity morphism over some nonempty Zariski open subset of Ui for all i. By Lemma 3.2 (see also Remark 3.3), fi: (Vi, BVi)→ (Ui, B|Ui) coincides with fj: (Vj, BVj) → (Uj, B|Uj) over Ui ∩Uj for every j ̸=i. Therefore, we get a log canonical modification of X and B by gluing them.
We finish the proof of Theorem 1.1. □
Proof of Theorem 1.2. It is a special case of Theorem 1.1. □ The following remark easily follows from the definition of log canonical modifications.
It is very useful for various geometric applications.
Remark 3.4. Let X be a normal variety and let ∆ be an effective R-divisor on X such that KX + ∆ is R-Cartier. We put B = ∆<1+ Supp ∆≥1. Let f: (Y, BY)→ (X, B) be a log canonical modification of X and B. We give two important remarks.
We put U = X\f(Exc(f)). Then, any point x of X is contained in U if and only if KX +B is R-Cartier and (X, B) is log canonical near x.
We define ∆Y byKY + ∆Y =f∗(KX+ ∆), and we put ΓY = ∆Y −BY. Then, it follows that ΓY is effective, −ΓY is ample over X, and we have Exc(f)⊂Supp ΓY = Supp ∆>1Y .
By the same argument as in the proof of Theorem 1.1, we obtain:
Lemma 3.5 (Good dlt blow-ups). Let X be a normal quasi-projective variety and let
∆ =∑
idi∆i be an effective R-divisor on X such that KX + ∆ is R-Cartier. Then there exists a projective birational morphismf: Y →X as in Theorem 2.10 such that KY + ∆† in Theorem 2.10 is f-semi-ample.
Proof. We put KY + ∆Y =f∗(KX + ∆). Then
∆†= ∆<1Y + Supp ∆≥Y1
holds. Therefore, as in the proof of Theorem 1.1 (put B = ∆<1+ Supp ∆≥1 in the proof of Theorem 1.1), by Theorem 3.1, the dlt pair
(Y,∆†) = (Y,∆<1Y + Supp ∆≥Y1)
has a good minimal model overX. Hence, after finitely many flips and divisorial contrac-
tions, we can makeKY + ∆† f-semi-ample. □
Remark 3.6. As in Remark 2.11, by construction, we may further assume that f is the identity morphism over some nonempty Zariski open subset of X in Theorems 1.1, 1.2, and Lemma3.5.
Let us prove Theorem 1.5.
Proof of Theorem 1.5. Let f: (Z,∆Z) →(X,∆) be a good dlt blow-up as in Lemma 3.5.
This means thatf: Z →X is a projective birational morphism from a normalQ-factorial varietyZ satisfying (i)–(iii). If necessary, then we may further assume thatf is the identity morphism over some nonempty Zariski open subset ofX (see Remark 3.6).
Step 1. We can always take ϵ > 0 such that, for any (KZ + ∆<1Z + (1−ϵ) Supp ∆≥Z1)- minimal model program over X, the divisor KZ+ ∆<1Z + Supp ∆≥1Z is numerically trivial over all extremal contractions of the steps of the minimal model program. This fact follows from the well-known estimate of lengths of extremal rational curves (see, for example, [B1, Proposition 3.2]). Since (Z,∆<1Z + (1− ϵ) Supp ∆≥1Z ) is klt, we can run a (KZ +
∆<1Z + (1−ϵ) Supp ∆≥Z1)-minimal model program overX and finally obtain a good minimal model (Z′,∆<1Z′ + (1−ϵ) Supp ∆≥Z1′) over X by [BCHM]. By the choice of ϵ, the divisor KZ′ + ∆<1Z′ + Supp ∆≥Z1′ is semi-ample over X.
Therefore, for any u∈[0, ϵ], the divisor
KZ′ + ∆<1Z′ + (1−u) Supp ∆≥Z′1
is semi-ample overX. Note that the divisor−(∆≥Z1′ −(1−ϵ) Supp ∆≥Z1′) is also semi-ample overX because
KZ′ + ∆<1Z′ + (1−ϵ) Supp ∆≥Z1′ ∼R,X −(∆≥Z1′ −(1−ϵ) Supp ∆≥Z1′)
holds. By the above construction of (Z′,∆Z′), the pair (Z′,∆<1Z′ + Supp ∆≥Z1′) is lc and Nklt(
Z′,∆<1Z′ + Supp ∆≥Z1′)
= Supp ∆≥Z′1 holds set theoretically.
Step 2. We denote Z′ →X byα′. Then we take a dlt blow-up β: Y →Z′ of (Z′,∆<1Z′ + Supp ∆≥Z1′) such thata(E, Z′,∆<1Z′ + Supp ∆≥Z′1) =−1 holds for everyβ-exceptional divisor EonY (see Theorem2.10). We setg =α′◦βand define ∆Y byKY+∆Y =β∗α′∗(KX+∆).
By construction,
KY + ∆′Y =β∗(KZ′ + ∆<1Z′ + Supp ∆≥Z1′) holds. Therefore, KY + ∆′Y is semi-ample over X.
From now on, we will check that g: (Y,∆Y)→(X,∆) satisfies (i)–(iv). It is not difficult to see that g: (Y,∆Y) → (X,∆) satisfies (i)–(iii) by construction. Here we only outline how to check (iv). We put
EZ′ = ∆≥Z′1−(1−ϵ) Supp ∆≥Z′1.
Then SuppEZ′ = Supp ∆≥Z1′, and −EZ′ is semi-ample over Z by Step 1. We can also see that
Suppβ∗EZ′ = Supp ∆≥Y1 = Nklt(Y,∆Y)
holds and −β∗EZ′ is semi-ample over X. Now g−1(Nklt(X,∆)) ⊃ Supp ∆≥Y1 is obvi- ous and it is easy to check that g(Supp ∆≥Y1) = Nklt(X,∆) holds set theoretically. If g−1(Nklt(X,∆)) ⊋ Supp ∆≥Y1, then there is a curve C ⊂ Y such that g(C) ∈ Nklt(X,∆) and (C·β∗EZ′)>0. This is a contradiction because −β∗EZ′ is nef over X. Hence we see that g−1(Nklt(X,∆)) = Supp ∆≥Y1 holds. This means that g: (Y,∆Y) → (X,∆) satisfies (iv).
For (v), we note that ∆Y −∆′Y is effective and that −(∆Y −∆′Y) ∼R.X KY + ∆′Y is g-semi-ample. By the definition of ∆′Y, Supp(∆Y −∆′Y) = Supp ∆>1Y holds. By the same argument as in the proof of (iv) above, we can check that
g−1(Nlc(X,∆)) = Supp ∆>1Y = Nlc(Y,∆Y) holds set theoretically.
Finally, we will construct ΓY as in (vi). Since the pair (Z′,∆<1Z′ + (1−u) Supp ∆≥Z1′) is klt and KZ′ + ∆<1Z′ + (1−u) Supp ∆≥Z′1 is semi-ample over X for every u ∈ (0, ϵ], by the construction ofβ: Y →Z′, we can find a positive real number usuch that if we set ∆uY by
KY + ∆uY =β∗(KZ′ + ∆<1Z′ + (1−u) Supp ∆≥Z′1),
then ∆uY is effective, (Y,∆uY) is klt, and KY + ∆uY is semi-ample over X. Fix such u > 0 and put
ΓY := ∆Y −∆uY =β∗(
∆Z′ −(∆<1Z′ + (1−u) Supp ∆≥Z′1)) .
Note that ΓY = (KY+∆Y)−(KY+∆uY)∼R,X −(KY+∆uY) holds. Hence−ΓY is semi-ample overX. It is clear that (Y,∆Y −ΓY) is klt because ∆Y −ΓY = ∆uY. Since
Supp(
∆Z′−(∆<1Z′ + (1−u) Supp ∆≥Z1′))
= Supp ∆≥Z1′ = SuppEZ′, we have Supp ΓY = Suppβ∗EZ′, thus
Supp ΓY = Supp ∆≥Y1 =g−1(Nklt(X,∆))
holds. In this way, we see that ΓY satisfies all the desired conditions in (vi).
We complete the proof of Theorem 1.5. □
Remark 3.7. By construction (see also Remarks 2.11 and 3.6), we may further assume that g: (Y,∆Y) →(X,∆) in Theorem 1.5 is the identity morphism over some nonempty Zariski open subset of X. Hence we can see g: (Y,∆Y) → (X,∆) as a partial resolution of singularities of the pair (X,∆).
4. On semi-log canonical modifications of demi-normal pairs
Ademi-normal schemeXis a reduced and equidimensional scheme which satisfies Serre’s S2condition and is normal crossing in codimension one. For basic definitions and properties of demi-normal pairs and semi-log canonical pairs, see [Ko, Sections 5.1 and 5.2]. In this section, we prove the existence of semi-log canonical modifications of demi-normal pairs (see Theorem 4.4). Let us start with the following lemma.
