Weak Zariski decompositions and log terminal models for generalized polarized pairs
aa r X i v : . [ m a t h . AG ] J a n WEAK ZARISKI DECOMPOSITIONS AND LOGTERMINAL MODELS FOR GENERALIZED POLARIZEDPAIRS
JINGJUN HAN AND ZHAN LI
Abstract.
We show that the existence of a birational weak Zariski de-composition for a pseudo-effective generalized polarized lc pair is equiv-alent to the existence of a generalized polarized log terminal model.
Contents
1. Introduction 12. Preliminaries 42.1. Generalized polarized pairs 42.2. G-log minimal models and g-log terminal model 62.3. Weak Zariski Decompositions 72.4. Nef Q -Cartier combinations (NQC) 83. MMP for generalized polarized pairs 83.1. MMP for generalized polarized pairs 83.2. Length of extremal rays 133.3. Shokurov’s polytope 143.4. G-MMP with scaling of an NQC divisor 183.5. Lifting the sequence of flips 194. Special termination for g-MMP with scaling 205. Proofs of the main results 26References 331. Introduction
Throughout the paper, we work over the complex numbers.The minimal model conjecture is one of the main problems in birationalgeometry.
Conjecture 1.1 (Minimal model conjecture) . For a Q -factorial dlt pair ( X, B ) , if K X + B is pseudo-effective, then ( X, B ) has a log terminal model. Conjecture 1.1 is known when dim X ≤
4, [3,22], or when (
X, B ) is a Q -factorial klt pair with a big boundary B , [8]. But when dim X ≥ Birkar related the minimal model conjecture to the existence of sectionsfor pseudo-effective adjoint divisors (i.e. the non-vanishing conjecture), [2–4]. The minimal model conjecture is also related to the existence of Zariskidecompositions for adjoint divisors, [9], or even weaker, to the existence of weak
Zariski decompositions for adjoint divisors, [6]. The later means that K X + B ≡ P + N where P is a nef divisor and N is an effective divisor. Notethat if K X + B is numerically equivalent to an effective divisor N , then itautomatically admits a weak Zariski decomposition with P = 0.In [9], Birkar and Hu asked that if the existence of weak Zariski decom-positions for adjoint pairs implies the minimal model conjecture. This isour starting point of the current paper. However, it seems that the naturalsetting of Birkar-Hu’s question lies in a larger category – the generalizedpolarized pairs (g-pairs), and this observation leads to the analogous con-jecture on the existence of minimal models for g-pairs (see Conjecture 1.2below). A g-pair ( X, B + M ) consists of an “ordinary” log pair ( X, B ) plusan auxiliary nef part M . A rudimentary model of such pairs had emergedin [6,9] but the actual definition only appeared in [10] (although they camefrom different sources: the former came from the weak Zariski decompositionwhile the later came from the canonical bundle formula).A pair (resp. g-pair) is said to be pseudo-effective if K X + B (resp. K X + B + M ) is pseudo-effective. We use “ /Z ” to denote a pair relative tothe variety Z , and the abbreviation “NQC” below stands for “nef Q -Cartiercombinations” (see Definition 2.13). Also see Definition 2.13 for the meaningof a birational NQC weak Zariski decomposition. The NQC pairs are thosebehave well in the Minimal Model Program (MMP). Hence we state thefollowing conjectures for such pairs. Conjecture 1.2 (Minimal model conjecture for g-pairs) . For a Q -factorialNQC g-dlt pair ( X/Z, B + M ) , if K X + B + M is pseudo-effective /Z , then ( X/Z, B + M ) has a g-log terminal model /Z . Conjecture 1.3 (Birational weak Zariski decomposition conjecture for g–pairs) . Let ( X/Z, B + M ) be a Q -factorial NQC g-dlt pair. If K X + B + M is pseudo-effective /Z , then it admits a birational NQC weak Zariskidecomposition /Z . Remark 1.4.
By Proposition 5.1, Conjecture 1.2 implies Conjecture 1.3.
Under the NQC assumption, we can answer the aforementioned questionof Birkar-Hu in the generalized polarized categories.
Theorem 1.5.
The birational weak Zariski decomposition conjecture forg-pairs (Conjecture 1.3) is equivalent to the minimal model conjecture forg-pairs (Conjecture 1.2).
Besides, for Conjecture 1.2, we show that it is a consequence of the ter-mination conjecture for ordinary log pairs.
EAK ZARISKI DECOMPOSITIONS AND LOG TERMINAL MODELS 3
Theorem 1.6.
Assume the termination of flips for Q -factorial dlt pairs /Z ,then any pseudo-effective Q -factorial NQC g-dlt pair ( X/Z, B + M ) has ag-log terminal model. As for the termination of g-MMP, we show the following result.
Theorem 1.7.
Assume the birational weak Zariski decomposition conjecture(Conjecture 1.3). Let ( X/Z, B + M ) be a pseudo-effective Q -factorial NQCg-dlt pair. Then any sequence of g-MMP on ( K X + B + M ) with scaling ofan ample divisor /Z terminates. We briefly describe the idea of the proof of Theorem 1.5. The nontrivialpart is to show that Conjecture 1.3 implies Conjecture 1.2. Suppose that K X + B + M ≡ P + N is a weak Zariski decomposition with P nef and N ≥
0. As it was pointed out in [9, § N ⊆ Supp ⌊ B ⌋ . In this case, we run a special g-MMP on ( K X + B + M )with scaling of a divisor (not necessarily effective) such that the g-MMP is N -negative and in each step, N i + ν i P i is nef, where ν i := inf { t ≥ | N i + tP i is nef } is the nef threshold. Notice that once ν i = 1, then K X i + B i + M i ≡ P i + N i is nef, and we are done. Thus the idea is to decrease ν i until it reaches 1.In fact, as long as ν i >
1, we can get a smaller ν i +1 . After this, we use thespecial termination and the induction on dimensions to conclude that thesequence { v i } cannot be infinite. The special termination still holds for thissetting (see Theorem 4.5) by an appropriate adaptation of the argument in[5]. Finally, by putting these together, we can prove the result. Remark 1.8.
By different approaches, Hacon and Moraga independentlyobtain some of the above results in certain general forms, [13]. They useideas from [2], while our proof is based on ideas in [4,5,9]. Furthermore,our results are about boundaries with real coefficients, while results in [13]require the boundaries with rational coefficients.
The paper is organized as following. In Section 2, we collect the relevantdefinitions. In Section 3, we first elaborate on the MMP for g-pairs (g-MMP)developed in [10], and then prove some standard results in this setting. Wealso introduce the g-MMP with scaling of an NQC divisor. In Section 4, weestablish the special termination result for g-MMP with scaling. The proofsof the theorems are given in Section 5.
Acknowledgements . We would like to thank Caucher Birkar, the paper isdeeply influenced by his ideas. We thank Chen Jiang for showing us a sim-ple proof of the length of extremal rays for the g-pairs. We also thank theparticipants of the Birational Geometry Seminar at BICMR/Peking Uni-versity for their interests in the work, especially Yifei Chen and ChuyuZhou. J. H. thanks Caucher Birkar for the invitation to the University ofCambridge where the paper was written, the thanks also go to his advisors
JINGJUN HAN AND ZHAN LI
Gang Tian and Chenyang Xu for constant support and encouragement. Z.L. thanks Keiji Oguiso for the invitation to the University of Tokyo wheresome key ideas are conceived. This work is partially supported by NSFCGrant No.11601015. 2.
Preliminaries
Generalized polarized pairs.Definition 2.1 (Generalized polarized pair) . A generalized polarized pair(g-pair) over Z consists of a normal variety X equipped with projective mor-phisms ˜ X f −→ X → Z, where f is birational and ˜ X is normal, an R -boundary B ≥ on X , andan R -Cartier divisor ˜ M on ˜ X which is nef /Z such that K X + B + M is R -Cartier, where M := f ∗ ˜ M . We say that B is the boundary part and M isthe nef part. For our convenience, when the base Z , the boundary part and the nefpart are clear from the context, we will just say that ( X, B + M ) is a g-pair.Notice that, in contrast to [10], we denote the generalized polarized pair by( X, B + M ) instead of ( X ′ , B ′ + M ′ ).Let g : X ′ → ˜ X be a birational morphism, such that X ′ → X is a logresolution of ( X, B ). Let K X ′ + B ′ + M ′ = g ∗ ( K X + B + M ) , where M ′ = g ∗ ˜ M . We say that ( X ′ , B ′ + M ′ ) → X is a log resolution of( X, B + M ). By replacing ( ˜ X, ˜ B + ˜ M ) with ( X ′ , B ′ + M ′ ), we may assumethat ˜ X → X is a log resolution of ( X, B + M ). In the same fashion, ˜ X can be chosen as a sufficiently high model of X . In particular, if thereexists a variety Y birational to X , we can always assume that there existsa morphism from ˜ X to Y which commutes with X Y .Many definitions/notions for ordinary log pairs have counterparts for gen-eralized polarized pairs. For convenience, we use the prefix “g-” to denotethe corresponding notions. For example, one can define the generalized logdiscrepancy (g-log discrepancy) of a prime divisor E over X : let ˜ X be ahigh enough model which contains E , and let K ˜ X + ˜ B + ˜ M = f ∗ ( K X + B + M ) . Then the g-log discrepancy of E is defined as (see [10] Definition 4.1) a ( E, X, B + M ) = 1 − mult E ˜ B. A g-lc place is a divisor E on a birational model of X , such that a ( E ; X, B + M ) = 0. A g-lc center is the image of a g-lc place, and the g-lc locus is theunion of all the g-lc centers. EAK ZARISKI DECOMPOSITIONS AND LOG TERMINAL MODELS 5
We say that (
X, B + M ) is generalized lc (g-lc) (resp. generalized klt(g-klt)) if the g-log discrepancy of any prime divisor is ≥ > M is a nef divisor, if M is R -Cartier, by the negativitylemma (see [16, Lemma 3.39]), f ∗ M = ˜ M + E with E ≥ K X + B is R -Cartier, then the logdiscrepancy of a divisor E with respect to ( X, B ) is greater than or equalto the g-log discrepancy of E with respect to ( X, B + M ).The definition of generalized dlt (g-dlt) is subtle. Definition 2.2 (G-dlt) . Let ( X, B + M ) be a g-pair. We say that ( X, B + M ) is g-dlt if it is g-lc and there is a closed subset V ⊂ X ( V can be equal to X ) such that(1) X \ V is smooth and B | X \ V is a simple normal crossing divisor,(2) if a ( E, X, B + M ) = 0 , then the center of E satisfies Center X ( E ) V and Center X ( E ) \ V is a lc center of ( X \ V, B \ V ) . Remark 2.3. If ( X, B + M ) is a Q -factorial dlt pair, then X is klt. Our definition of g-dlt is slightly different from the definition in [7, page13]. We will show that our definition of g-dlt is preserved under adjunctionsand running MMPs.
Remark 2.4.
Another possible definition of g-dlt is as follows:A g-pair ( X, B + M ) is g-dlt if there exists a log resolution π : X ′ → X of ( X, B + M ) , such that a ( E, X, B + M ) > for every exceptional divisor E ⊂ Y .For the ordinary log pairs (i.e. ˜ M = 0 ), the above two definitions areequivalent (see [16, Theorem 2.44]). However, for g-pairs, it is not knownwhether the two definitions are the same or not. The adjunction formula for g-lc pairs is given in [10, Definition 4.7].
Definition 2.5 (Adjunction formula for g-pairs) . Let ( X/Z, B + M ) be ag-dlt pair with data ˜ X f −→ X → Z and ˜ M . Let S be a component of ⌊ B ⌋ and ˜ S its birational transform on ˜ X . We may assume that f is a log resolutionof ( X, B + M ) . Write K ˜ X + ˜ B + ˜ M = f ∗ ( K X + B + M ) , then K ˜ S + B ˜ S + M ˜ S := ( K ˜ X + ˜ B + ˜ M ) | ˜ S where B ˜ S = ( ˜ B − ˜ S ) | ˜ S and M ˜ S = ˜ M | ˜ S . Let g be the induced morphism ˜ S → S . Set B S = g ∗ B ˜ S and M S = g ∗ M ˜ S . Then we get the equality K S + B S + M S = ( K X + B + M ) | S , which is referred as generalized adjunction formula . JINGJUN HAN AND ZHAN LI
Suppose that ˜ M = P µ i ˜ M i , where ˜ M i is a nef /Z Cartier divisor for each i ,and B = P b j B j the prime decomposition of the R -divisor B . Let b = { b j } , µ = { µ i } be the coefficient sets. For a set of real numbers Γ, set(1) S (Γ) := { − m + X j r j γ j m ≤ | m ∈ Z > , r j ∈ Z ≥ , γ j ∈ Γ } ∪ { } . Then the coefficients of B S belong to the set S ( b , µ ) := S ( b ∪ µ ) (see [10,Proposition 4.9]). Lemma 2.6.
Let ( X/Z, B + M ) be a g-dlt pair with data ˜ X f −→ X → Z and ˜ M . Let S be a component of ⌊ B ⌋ , and B S , M S be the divisors in theg-adjunction formula (see Definition 2.5 ). Then ( S, B S + M S ) is still g-dlt.Proof. We use the notation in Definition 2.5. Let V be the closed subset V ⊂ X in Definition 2.2, and V S = V ∩ S . It is clear that S \ V S is smoothand B | S \ V S is a simple normal crossing divisor.If a ( E, S, B S + M S ) = 0, then Center ˜ S ( E ) is a stratum of ( ˜ S, ˜ B ˜ S ), andthus a stratum of ( ˜ X, ˜ B ). Let E ′ be a g-lc place of ( X, B + M ), such thatCenter ˜ S ( E ) = Center ˜ X ( E ′ ). Since ( X, B + M ) is g-dlt, Center X ( E ′ ) V and Center X ( E ′ ) \ V is a lc center of ( X \ V, B \ V ). Thus, Center S ( E ) \ V S isa lc center of ( S \ V S , B S | S \ V S ). (cid:3) Remark 2.7.
In general, K S + B S may not be R -Cartier and thus ( S, B S ) may not be dlt. In particular, ( S, B S + M S ) may not be g-dlt in the sense of[7]. This is the main reason that we do not use the definition of g-dlt as [7]. The following proposition is similar to [11, Proposition 3.9.2].
Proposition 2.8.
Let ( X/Z, B + M ) be a g-dlt pair with data ˜ X f −→ X → Z and ˜ M . Suppose that ˜ M = P µ i ˜ M i , where ˜ M i is a nef /Z Cartier divisorfor every i , and let B = P b j B j be the prime decomposition of an R -divisor B . Let V be a g-lc center of ( X, B + M ) . Then there exists a g-dlt pair ( V, B V + M V ) such that ( K X + B + M ) | V = K V + B V + M V , where M V is the push forward of ˜ M | ˜ V on V , and ˜ V is the birational trans-form of V on ˜ X . Moreover, the coefficients of B V belong to the set S ( b , µ ) .Proof. Let k be the codimension of V . By definition of g-dlt, V is an irre-ducible component of V ∩ V ∩ . . . ∩ V k for some V i ⊆ ⌊ B ⌋ . Under the notationof (1), a straightforward computation shows that S ( b , µ ) = S ( S ( b , µ )) (forexample, see [12, Proposition 3.4.1]). Then the claim follows from applyingLemma 2.6 k times. (cid:3) G-log minimal models and g-log terminal model.
The notionsof log minimal/terminal models still make sense in the generalized polarized
EAK ZARISKI DECOMPOSITIONS AND LOG TERMINAL MODELS 7 setting. Following Shokurov [20], certain extractions of g-lc places are al-lowed for g-log minimal models. First, if f : X Y is a birational map,and B is an effective divisor on X , we define(2) B Y := f ∗ B + E, where f ∗ B is the birational transform of B on Y , and E is the sum of reducedexceptional divisors of f − . Definition 2.9 (G-log minimal model & g-log terminal model) . Let ( X/Z, B + M ) be a g-pair with data ˜ X → X and nef part M . Then a g-pair ( Y /Z, B Y + M Y ) is called a g-log minimal model of ( X/Z, B + M ) , if(1) there is a birational map X Y ,(2) B Y is the same as (2) , and M Y is the pushforward of ˜ M (we canalways assume that there exists a morphism ˜ X → Y ),(3) K Y + B Y + M Y is nef,(4) ( Y /Z, B Y + M Y ) is Q -factorial g-dlt with data ˜ X → Y and nef part M Y ,(5) a ( D, X, B + M ) < a ( D, Y, B Y + M Y ) for any divisor D on X whichis exceptional over Y .Furthermore, if X Y is a birational contraction (i.e. there is no divisoron Y which is exceptional over X ), we say that ( Y /Z, B Y + M Y ) is a g-logterminal model of ( X/Z, B + M ) . Remark 2.10.
Just as the case for log pairs, a g-log minimal model canonly extracts g-lc places. That is, a divisor E ⊂ Y is exceptional over X satisfies a ( E, X, B + M ) = 0 . Weak Zariski Decompositions.
On a normal variety X over Z (wewrite this by X/Z ), an R -Cartier divisor D is said to admit a weak Zariskidecomposition if D ≡ N + P/Z, with N ≥ P a nef /Z R -Cartier divisor (see [6, Definition 1.3]). UnlikeZariski decompositions, weak Zariski decompositions may not be unique. Definition 2.11.
Let
X/Z be a variety and D be a Cartier divisor. We saythat D admits a birational Zariski decomposition if there exists a birationalmorphism f : Y → X from a normal variety Y , such that f ∗ D admits aweak Zariski decomposition. Notice that birational weak Zariski decomposition is called weak Zariskidecomposition in [6, Definition 1.3]. For a lc pair ( X, ∆), the non-vanishingconjecture asserts that K X + ∆ ∼ R N for some effective divisor N . Thisimplies that K X + ∆ admits a weak Zariski decomposition by taking P = 0.For (weak) Zariski decompositions, the most important case is when D equals to the adjoint divisor K X + B (or K X + B + M ). In the sequel, whensaying “existence a (weak) Zariski decomposition” without referring to adivisor, it should be understood that the decomposition is for the adjointdivisor. JINGJUN HAN AND ZHAN LI
Remark 2.12.
Zariski proved that on a smooth projective surface, an effec-tive divisor D can be decomposed as a sum of a nef divisor and an effectivedivisor with some extra properties, [24], which is known as the Zariski de-composition. There are various generalizations to higher dimensions. See[19] for more details. For an arbitrary divisor, it may not always admit aweak Zariski decomposition, [18]. But for the adjoint divisor K X + B , theexistence of a weak Zariski decomposition is a consequence of the existenceof a minimal model, which is highly plausible. Nef Q -Cartier combinations (NQC). We need a technical assump-tion to guarantee that certain g-MMP on g-pairs behave as the ordinary logpairs (see Section 3.2). Here the abbreviation “NQC” stands for “nef Q -Cartier combinations”. Definition 2.13.
We have following definitions on decompositions of nef /Z R -Cartier divisors in various settings.(1) We say that an R -Cartier divisor M is NQC over Z , if M ≡ X i r i M i /Z, where r i ∈ R > and M i are Q -Cartier nef /Z divisors.(2) A g-pair ( X/Z, B + M ) with data ˜ X f −→ X → Z and ˜ M is said to bean NQC g-pair , if ˜ M is NQC.(3) We define NQC g-lc, NQC g-klt , etc. if an NQC g-pair is g-lc, g-klt,etc.(4) We say that a g-pair ( X/Z, B + M ) admits a birational NQC weakZariski decomposition /Z , if there exists a birational morphism g : Y → X/Z such that g ∗ ( K X + B + M ) ≡ P + N/Z , where N ≥ and P is NQC over Z . We will avoid repeating “over Z ” if the base Z is clear in the context. Bydefinition, the NQC property is preserved under g-MMPs and generalizedadjunctions.The NQC assumption excludes the pathological phenomenon incurred bythe nef part. In [10], Birkar proved ACC for g-lc thresholds and GlobalACC for NQC pairs (though the name was not given there). In the currentpaper, we need the g-lc pairs to be NQC in order to run some kind of specialg-MMPs.In Proposition 5.1, we show that the existence of a g-log minimal modelfor an NQC g-lc pair implies the existence of a birational NQC weak Zariskidecomposition for this g-pair.3. MMP for generalized polarized pairs
MMP for generalized polarized pairs.
For g-lc pairs, the conetheorem, the existence of flips and the termination of flips are still expectedto hold true.
EAK ZARISKI DECOMPOSITIONS AND LOG TERMINAL MODELS 9
Conjecture 3.1 (Cone theorem for g-lc pairs) . Let ( X, B + M ) be a g-lcpair. We have the following claims.(1) There are countably many curves C j ⊂ X such that < − ( K X + B + M ) · C j ≤ X , and NE( X ) = NE( X ) ( K X + B + M ) ≥ + X R ≥ [ C j ] . (2) Let F ⊂ NE be a ( K X + B + M ) -negative extremal face. There therea unique morphism cont F : X → Y to a projective variety such that (cont F ) ∗ O X = O Y , and an irreducible curve C ⊂ X is mapped to apoint by cont F if and only if [ C ] ∈ F .(3) Let F and cont F : X → Y be as in (2) . Let L be a line bundle on X such that L · C = 0 for every curve C with [ C ] ∈ F . Then there is aline bundle L Y on Y such that L ≃ cont ∗ F L Y . Definition 3.2 (Flips for g-pairs) . Let ( X, B + M ) be a g-lc pair. A ( K X + B + M )-flipping contraction is a proper birational morphism f : X → Y such that Exc( f ) has codimension at least two in X , − ( K X + B + M ) is f -ample and the relative Picard group has rank ρ ( X/Y ) = 1 .A g-lc pair ( X + , B + + M + ) together with a proper birational morphism f + : X + → Y is called a ( K X + B + M )-flip of f if(1) B + , M + are the birational transforms of B, M on X + , respectively,(2) K X + + B + + M + is f + -ample, and(3) Exc( f + ) has codimension at least two in X + .For convenience, We call the induced birational map, X X + , a ( K X + B + M ) -flip. Conjecture 3.3 (Existence of flips for g-lc pairs) . For a g-lc pair, the flipof a flipping contraction always exists.
Conjecture 3.4 (Termination of flips for g-lc pairs) . There is no infinitesequence of flips for g-lc pairs.
Although the MMP for g-pairs is not established in full generality, someimportant cases could be derived from the standard MMP. We elaboratethese results which are developed in [10, § X/Z, B + M ) be a Q -factorial g-lc pair, and A be a general ample /Z divisor. ( ⋆ ) Suppose that for any 0 < ǫ ≪
1, there exists a boundary∆ ǫ ∼ R B + M + ǫA/Z , such that ( X, ∆ ǫ ) is klt.Under the assumption ( ⋆ ), we can run a g-MMP /Z on ( K X + B + M ),although the termination is not known. In fact, let R be an extremal ray /Z ,such that ( K X + B + M ) · R <
0. For 0 < ǫ ≪
1, we have ( K X + B + M + ǫA ) · R <
0. By assumption, there exists ∆ ǫ ∼ R B + M + ǫA/Z , such that ( X, ∆ ǫ )is klt, and ( K X + ∆ ǫ ) · R <
0. Now, R can be contracted and its flip exists ifthe contraction is a flipping contraction. If we obtain a g-log minimal modelor a g-Mori fiber space, we stop the process. Otherwise, let X Y be the divisorial contraction or the flip, then ( Y, B Y + M Y ) is naturally a g-lc pair.Moreover, for any 0 < ǫ ≪
1, ( Y, ∆ ǫ,Y ) is klt. Repeating this process givesthe g-MMP /Z .The usual notion of g-MMP with scaling of the general divisor A alsomakes sense under assumption ( ⋆ ) (see [10]).The following lemma shows that assumption ( ⋆ ) is satisfied in two cases.As a result, we may run a g-MMP for such g-pairs. Lemma 3.5.
Let ( X/Z, B + M ) be a g-lc pair, and A be an ample /Z divisor.Suppose that either(i) ( X, B + M ) is g-klt, or(ii) there exists a boundary C , such that ( X, C ) is klt.Then, there exists a boundary ∆ ∼ R B + M + A/Z , such that ( X, ∆) isklt.Moreover, if X is Q -factorial, we may run a g-MMP on K X + B + M .Proof. Suppose that (
X, B + M ) is g-klt. We have K ˜ X + ˜ B + ˜ M + f ∗ ( A ) = f ∗ ( K X + B + M + A ) , where ˜ M + f ∗ ( A ) is big and nef. Hence for k ∈ N , there exists an ampledivisor /Z H k ≥
0, and an effective divisor E , such that ˜ M + f ∗ ( A ) ∼ R H k + k E . For general H k and k ≫ K ˜ X + ˜ B + H k + k E is sub-klt. Let∆ := f ∗ ( ˜ B + H k + 1 k E ) ∼ R B + M + A. Since K ˜ X + ˜ B + ˜ M + f ∗ ( A ) ∼ R K ˜ X + ˜ B + H k + 1 k E, we have K ˜ X + ˜ B + H k + 1 k E = f ∗ ( K X + ∆) , and ( X, ∆) is klt.Suppose that (ii) holds. By assumption, B + M − C is R -Cartier, andthere exists 0 < ǫ ≪
1, such that ǫ ( B − C + M ) + A/ H be ageneral ample divisor, such that ǫH ∼ R ǫ ( B − C + M )+ A/
2, and (
X, C + H )is klt. Thus ( X, ( ǫC + (1 − ǫ ) B ) + ( ǫH + (1 − ǫ ) M )) is g-klt with boundarypart ǫC + (1 − ǫ ) B and nef part ǫH + (1 − ǫ ) M . Besides, K X + ( ǫC + (1 − ǫ ) B ) + ( ǫH + (1 − ǫ ) M ) ∼ R ǫ ( K X + C + H ) + (1 − ǫ )( K X + B + M ) ∼ R K X + B + M + A/ . Apply (i) to ( X, ( ǫC + (1 − ǫ ) B ) + ( ǫH + (1 − ǫ ) M )) with A/ (cid:3) Remark 3.6.
As a simple corollary, suppose that X is Q -factorial klt and ( X/Z, B + M ) is g-lc, then there are countably many extremal rays R/Z ,such that ( K X + B + M ) · R < . A g-MMP for a g-dlt pair preserves g-dltness.
EAK ZARISKI DECOMPOSITIONS AND LOG TERMINAL MODELS 11
Lemma 3.7.
Let ( X/Z, B + M ) be a Q -factorial g-dlt pair. Let g : X Y /Z be either a divisorial contraction of a ( K X + B + M ) -negative extremalray or a ( K X + B + M ) -flip. Let B Y = g ∗ ( B ) , M Y = g ∗ ( M ) , then ( Y, B Y + M Y ) is also Q -factorial g-dlt.Proof. Fix a general ample /Z divisor H . As X is klt (see Remark 2.3), byLemma 3.5, there exist 0 < ǫ ≪ ǫ , such that ∆ ǫ ∼ R B + M + ǫH , and( X, ∆ ǫ ) is Q -factorial klt. Moreover, g is also either a divisorial contractionof a ( K X +∆ ǫ )-negative extremal ray or a ( K X +∆ ǫ )-flip. By [16, Proposition3.36, 3.37], Y is Q -factorial.To show that the g-dltness is preserved, we use a similar argument as[16, Lemma 3.44]. Let V be the closed subset V ⊂ X in Definition 2.2. When g is a divisorial contraction, set V Y = g ( V ∪ Exc( g )). Then X \ ( V ∪ Exc( g )) ≃ Y \ V Y , and thus Y \ V Y is smooth and B | Y \ V Y is a simple normal crossingdivisor. By the negativity lemma, for any divisor E , we have a ( E, Y, B Y + M Y ) ≥ a ( E, X, B + M ) ≥
0, and a ( E, Y, B Y + M Y ) > a ( E, X, B + M ) whenCenter X ( E ) ⊂ Exc( g ) (see [16, Lemma 3.38]). When a ( E, Y, B Y + M Y ) = 0,we have a ( E, X, B + M ) = 0 and Center X ( E ) Exc( g ). Thus Center X ( E ) V ∪ Exc( g ) by the definition of g-dlt. Hence Center Y ( E ) V Y . This showsthat ( Y, B Y + M Y ) is g-dlt.When g is a flip, let c : X → W/Z, c : Y → W/Z be the correspond-ing contractions. Let L , L be the contraction loci of c , c respectively.Set V X = V ∪ L ∪ c − ( c ( L )) and V Y = c − ( c ( V )) ∪ L ∪ c − ( c ( L )).Then X \ V X ≃ Y \ V Y , and thus Y \ V Y is smooth and B | Y \ V Y is a simplenormal crossing divisor. By the negativity lemma, for any divisor E , wehave a ( E, Y, B Y + M Y ) ≥ a ( E, X, B + M ) ≥
0, and a ( E, Y, B Y + M Y ) >a ( E, X, B + M ) when Center X ( E ) ⊂ L or Center Y ( E ) ⊂ L (see [16,Lemma 3.38, 3.44]). When a ( E, Y, B Y + M Y ) = 0, we have a ( E, X, B + M ) =0 and Center Y ( E ) L ∪ c − ( c ( L )). Besides, by the definition of g-dlt,Center X ( E ) V . Thus Center Y ( E ) V Y . This shows that ( Y, B Y + M Y )is g-dlt. (cid:3) Let f : X → Y be a projective morphism of varieties and D be an R -Cartier divisor on X , then D is called a very exceptional divisor of f ([21, Definition 3.2]) if (1) f ( D ) ( Y , and (2) for any prime divisor P on Y there is a prime divisor Q on X which is not a component of D but f ( Q ) = P . Notice that if f is birational, then any exceptional divisor isvery exceptional. The point is that the negativity lemma also holds forvery exceptional divisors ([5, Lemma 3.3]). The following Proposition isan easy consequence of the negativity lemma, which is a generalization of[5, Theorem 3.5] in the setting of g-lc pairs. Proposition 3.8.
Let ( X/Z, B + M ) be a g-lc pair with X klt such that K X + B + M ≡ D − D , where D ≥ , D ≥ have no common components.Suppose that D is a very exceptional divisor over Z . Then any g-MMP /Z on K X + B + M with scaling of an ample divisor /Z either terminates to a Mori fiber space or contracts D after finite steps. Moreover, if D = 0 ,then the g-MMP terminates to a model Y such that K Y + B Y + M Y ≡ /Z .Proof. Let A be an ample /Z divisor, by Lemma 3.5, we can run a g-MMP /Z with scaling of A . Let ν i = inf { t ∈ R ≥ | K X i + B i + M i + tA i is nef over Z } be the nef threshold in each step of the scaling. Set µ = lim ν i .If µ >
0, then the g-MMP is also a g-MMP on K X + B + M + µA . ByLemma 3.5, there exists a boundary ∆, such that K X + B + M + µA ∼ R K X +∆, and ( X, ∆) is klt with ∆ big. Then the ( K X +∆)-MMP with scalingterminates by [8, Corollary 1.4.2]. In this case, without loss of generality,we can assume that K X + B + M is nef /Z .If µ = 0, after finite steps, we can assume that the g-MMP only consistsof flips. Thus K X + B + M is a limit of movable /Z R -Cartier divisors.In either case, for any prime divisor S on X , and a very general curve C ⊂ S , we have ( K X + B + M ) · C = ( D − D ) · C ≥
0. Since D is avery exceptional divisor over Z , by the negativity lemma ([5, Lemma 3.3]), D − D ≥
0, which implies that D = 0. Hence the g-MMP contracts D after finite steps. When D = 0, on a model Y such that D is contracted,we have K Y + B Y + M Y ≡ /Z . (cid:3) The proof of [10, Lemma 4.5] gives the existence of g-dlt modifications.
Proposition 3.9 (G-dlt modification [10, Lemma 4.5]) . Let ( X, B + M ) bea g-lc pair with data ˜ X f −→ X → Z and ˜ M . Then perhaps after replacing f with a higher resolution, there exist a Q -factorial g-dlt pair ( X ′ , B ′ + M ′ ) with data ˜ X g −→ X ′ → Z and ˜ M , and a projective birational morphism φ : X ′ → X such that K X ′ + B ′ + M ′ = φ ∗ ( K X + B + M ) . Moreover, eachexceptional divisor of φ is a component of ⌊ B ′ ⌋ . We call ( X ′ , B ′ + M ′ ) ag-dlt modification of ( X, B + M ) .Proof. We may assume that f is a log resolution of ( X, B + M ). Let E i bean irreducible exceptional divisor of f . We have K ˜ X + ˜ B + E + ˜ M = f ∗ ( K X + B + M ) + E ≡ E/X, where E = P a i E i ≥
0, and a i = a ( E i , X, B + M ) is the g-log discrepancy.We can run a g-MMP /X on ( K ˜ X + ˜ B + E + ˜ M ) with scaling of an ample divi-sor. By Proposition 3.8, the g-MMP terminates to X ′ , and E is contracted.Thus K X ′ + B ′ + M ′ = φ ∗ ( K X + B + M ). As ( ˜ X, ˜ B + E + ˜ M ) is g-dlt, byLemma 3.7, ( X ′ , B ′ + M ′ ) is also Q -factorial g-dlt. By the construction of E , we see that each exceptional divisor of φ is a component of ⌊ B ′ ⌋ (cid:3) Although the MMP is expected to hold for g-pairs, abundance conjecture,finite generations of canonical rings and non-vanishing conjecture all fail forg-pairs (see [9, §
3] for discussions). However, as for non-vanishing conjecture,one can still ask under the numerical sense. In general, abundance conjecturedoes not hold under the numerical sense.
EAK ZARISKI DECOMPOSITIONS AND LOG TERMINAL MODELS 13
Example 3.10.
Let X be P blown up 9 points in sufficiently general posi-tions, M = − K X . Then K X + M = − K X is nef, but there is no semiampledivisor N such that K X + M ≡ N , [1]. We ask the following question.
Conjecture 3.11 (Weak non-vanishing & weak abundance for g-pairs) . Let ( X/Z, B + M ) be a Q -factorial NQC g-dlt pair. Suppose that K X + B + M is nef, then(1) there exists an effective R -divisor N , such that K X + B + M ≡ N ,(2) there exists ≤ t ≤ , and a semi-ample R -divisor D , such that K X + B + tM ≡ D . Length of extremal rays.
The bound on the length of extremal raysalso holds for g-pairs. Following [22, Definition 1], we define extremal curves.
Definition 3.12 (Extremal curve) . An irreducible curve C on X/Z is called extremal if it generates an extremal ray R = R + [ C ] of the Kleiman-Moricone NE(
X/Z ) , and has the minimal degree for this ray (with respect to anyample divisor). By definition, there exists an ample /Z divisor H , such that H · C = min { H · Γ | [Γ] ∈ R } . We have the following result on the length of extremal rays. We thankChen Jiang for showing us the following simple proof.
Proposition 3.13 (The length of extremal rays for g-pairs) . Let X be a Q -factorial klt variety, and ( X/Z, B + M ) be a g-lc pair. Then for any ( K X + B + M ) -negative extremal ray R/Z , there exists a curve C , such that [ C ] ∈ R , and < − ( K X + B + M ) · C ≤ X. Proof.
Let C be any extremal curve such that [ C ] ∈ R . By definition, thereexists an ample /Z divisor H , such that H · C = min { H · Γ | [Γ] ∈ R } . By Lemma 3.5, for any 1 ≫ ǫ >
0, there exists a klt pair ( X, ∆ ǫ ) with K X + ∆ ǫ ∼ R K X + B + M + ǫH , such that R is also a ( K X + ∆ ǫ )-negativeextremal ray. By Kawamata’s length of extremal rays, [14], there exists arational curve Γ ǫ , such that [Γ ǫ ] ∈ R , and0 < − ( K X + B + M + ǫH ) · Γ ǫ ≤ X. By the definition of extremal curve, we have H · CH · Γ ǫ ≤
1, thus − ( K X + B + M + ǫH ) · C = − (( K X + B + M + ǫH ) · Γ ǫ )( H · CH · Γ ǫ ) ≤ X. Let ǫ →
0, we finish the proof. (cid:3)
Shokurov’s polytope.
In this subsection, we have the following se-tups. Let X be a Q -factorial klt variety, ( X/Z, B + M ) be an NQC g-lcpolarized pair with data ˜ X f −→ X → Z and ˜ M . Suppose ˜ M = P µ j ˜ M j ,where ˜ M i is a nef Q -Cartier divisor and µ i ≥
0. Let M j be the pushforwardof ˜ M j on X . Let B = P b i B i be the prime decomposition.Consider the vector space V := ( ⊕ i R B i ) ⊕ ( ⊕ j R M j ) . If x i , y j ≥
0, then (
X/Z, P i x i B i + P j y j M j ) is a g-pair with data ˜ X → X and a nef divisor P j y j ˜ M j . Let ∆ = P d i B i , N = P ν j M j , and set theEuclidean norm || B + M − ∆ − N || = ( X i ( b i − d i ) + X j ( µ j − ν j ) ) / . The set of NQC g-lc pairs L ( B, M ) := { X i x i B i + X j y j M j ∈ V | ( X, X i x i B i + X j y j M j ) is g-lc } is a rational polytope (may be unbounded). In fact, we may assume that f : ˜ X → X is a log resolution of ( X/Z, B + M ). Given any ∆ + N ∈ V with∆ ∈ ( ⊕ i R ≥ B i ) and N ∈ ( ⊕ j R ≥ M j ), if we write K ˜ X + ˜∆ + ˜ N = f ∗ ( K X + ∆ + N ) , then the coefficients of ˜∆ are rational affine linear functions of the coefficientsof ∆ and N . The g-lc condition is the same as that such coefficients arechosen from [0 , Lemma 3.14.
Under the above notation. Let X be a Q -factorial klt variety,and ( X/Z, B + M ) be an NQC g-lc pair.(1) Then there exists a positive real number α > , such that for anyextremal curve Γ /Z , if ( K X + B + M ) · Γ > , then ( K X + B + M ) · Γ > α .(2) There exists a positive number δ > , such that if ∆ + N ∈ L ( B, M ) , || (∆ − B ) + ( N − M ) || < δ , and ( K X + ∆ + N ) · R ≤ for an extremal ray R/Z , then ( K X + B + M ) · R ≤ .Proof. (1) Because L ( B, M ) is a rational polytope, there exist r, m ∈ N , a j ∈ R ≥ and D j ∈ ( ⊕ i Q ≥ B i ) , N j ∈ ( ⊕ j Q ≥ M j ) such that P rj =1 a j = 1,( K X + B + M ) = r X j =1 a j ( K X + D j + N j ) , ( X/Z, D j + N j ) is NQC g-lc, and m ( K X + D j + N j ) is Cartier. By Proposition3.13 and the definition of the extremal curve (see Definition 3.12), if ( K X + D j + N j ) · Γ <
0, then ( K X + D j + N j ) · Γ ≥ − X . Hence, if ( K X + D j + N j ) · Γ ≤
1, then there are only finitely many possibilities for the
EAK ZARISKI DECOMPOSITIONS AND LOG TERMINAL MODELS 15 intersection numbers ( K X + D j + N j ) · Γ. Therefore there are also finitemany intersection numbers for ( K X + B + M ) · Γ <
1, and the existence of α is clear.(2) Let B ( B + M, ⊂ V be a ball centered at B + M with radius 1.Because L ( B, M ) is a rational polytope which may be unbounded, we choosea bounded rational polytope L ′ ( B, M ) ⊃ L ( B, M ) ∩ B ( B + M, δ <
1. Let ∆ ′ + N ′ be the intersection point of the line connecting B + M and ∆ + N on the boundary of L ′ ( B, M ), such that ∆ + N lies insidethe interval between ∆ ′ + N ′ and B + M (if this line lies on the boundary of L ′ ( B, M ), we choose ∆ ′ + N ′ to be the furthest intersection point). Thereexist r, s ∈ R ≥ , such that r + s = 1, and ∆ + N = r ( B + M ) + s (∆ ′ + N ′ ).Suppose that there is an extremal ray R/Z such that ( K X + ∆ + N ) · R ≤ K X + B + M ) · R >
0. Let Γ be an extremal curve of R . By (1) thereexists α >
0, such that ( K X + B + M ) · Γ > α . If r > s dim Xα , or equivalently r > X X + α , then by the proof of Proposition 3.13, we have( K X + ∆ + N ) · Γ = r ( K X + B + M ) · Γ + s ( K X + ∆ ′ + N ′ ) · Γ >rα − s dim X > , which is a contradiction.From the above discussion, we can construct the desired δ as follows. Let l > B + M to the boundary of L ′ ( B, M ). We choose 0 < δ ≪ l − δl > X X + α . Then by the choice of l , we see that r = l ′ − δl ′ ≥ l − δl , where l ′ is the distancefrom B + M to ∆ ′ + N ′ . This δ satisfies the requirement. (cid:3) Example 3.15.
Without the NQC assumption, Lemma 3.14(1) no longerholds true. Indeed, let E be a general elliptic curve, X = E × E . Fix a point P ∈ E , consider the divisor classes f = [ { P } × E ] , f = [ E × { P } ] , δ = [∆] ,where ∆ ⊂ E × E is the diagonal. According to [17, Lemma 1.5.4], N = f + √ f + ( √ − δ is nef. It is not hard to show that for any ǫ > , thereexists a curve C , such that N · C < ǫ (see [23]).
Let dim( ⊕ i R B i ) ⊕ ( ⊕ j R M j ) = d . For k ∈ Q , let[0 , k ] d := [0 , k ] × · · · × [0 , k ] ⊂ R d . If { R t } t ∈ T is a family of extremal rays of NE( X/Z ), set N T = { ∆ + N ∈ L ( B, M ) | ( K X + ∆ + N ) · R t ≥ ∀ t ∈ T } . By the same argument as [4] (cf. the original proof in [22]), we have thefollowing result for NQC g-lc pairs.
Proposition 3.16.
Under the above notation. Let X be a Q -factorial kltvariety, and ( X/Z, B + M ) be an NQC g-lc pair. Then the set N T ∩ [0 , k ] d is a rational polytope for any k ∈ Q . In particular, if K X + B + M is nef /Z ,then there exist NQC g-lc pairs ( X, D i + N i ) with nef part N i and boundarypart D i , and a i ∈ R > , such that K X + B + M = X i a i ( K X + D i + N i ) with P i a i = 1 . Moreover, there exists m ∈ N , such that m ( K X + D i + N i ) is a nef /Z and Cartier divisor for each i .Proof. By definition, N T ∩ [0 , k ] d is just { ∆ + N ∈ L ( B, M ) ∩ [0 , k ] d | ( K X + ∆ + N ) · R t ≥ ∀ t ∈ T } , and L ( B, M ) ∩ [0 , k ] d is a bounded rational polytope.Since N T ∩ [0 , k ] d is compact, by Lemma 3.14(2), there are(∆ + N ) , . . . , (∆ n + N n ) ∈ N T ∩ [0 , k ] d , and δ , . . . , δ n >
0, such that N T ∩ [0 , k ] d is covered by B i = { ∆ + N ∈ N T ∩ [0 , k ] d | || ∆ + N − (∆ i + N i ) || < δ i } , ≤ i ≤ n. Moreover, if ∆ + N ∈ B i with ( K X + ∆ + N ) · R t < t ∈ T , then( K X + ∆ i + N i ) · R t = 0. Let T i = { t ∈ T | ( K X + ∆ + N ) · R t < N ∈ B i } . Then, ( K X + ∆ i + N i ) · R t = 0 for each t ∈ T i . Moreover, we have N T ∩ [0 , k ] d = n \ i =1 ( N T i ∩ [0 , k ] d ) . It suffices to show that N T i ∩ [0 , k ] d is a rational polytope.By replacing T with T i and ∆ + N with ∆ i + N i , we may assume thatthere exists ∆ + N such that ( K X + ∆ + N ) · R t = 0 for any t ∈ T . Because { ∆ + N ∈ L ( B, M ) | ( K X + ∆ + N ) · R t = 0 ∀ t ∈ T } is a rational polytope, we can further assume that ∆ + N is a rational point.We do induction on dimensions of polytopes. If dim( L ( B, M ) ∩ [0 , k ] d ) =1, then the statement is straightforward to verify. If dim( L ( B, M ) ∩ [0 , k ] d ) >
1, let L , . . . , L p be the proper faces of L ( B, M ) ∩ [0 , k ] d . By induction ondimensions, N iT := N T ∩ L i is a rational polytope. If ∆ ′ + N ′ ∈ N T ∩ [0 , k ] d ,then there exists a line connecting with ∆ ′ + N ′ and ∆ + N which intersectssome L i on ∆ ′′ + N ′′ . Moreover, we can assume that ∆ ′ + N ′ lies inside theline segment between ∆ + N and ∆ ′′ + N ′′ . Because ( K X + ∆ + N ) · R t = 0,we have ( K X ′′ + ∆ ′′ + N ′′ ) · R t ≥ t ∈ T . Thus ∆ ′′ + N ′′ ∈ N iT .This shows that N T ∩ [0 , k ] d is the convex hull of ∆ + N and all N iT , whichis also a rational polytope. (cid:3) EAK ZARISKI DECOMPOSITIONS AND LOG TERMINAL MODELS 17
Let D be a divisor on X , we say that a divisorial/flipping contraction f : X → Y is D -trivial , if for any contraction curve C , we have D · C = 0. Lemma 3.17.
Let ( X/Z, ( B + A ) + M ) be a Q -factorial NQC g-lc pair withboundary part B + A and nef part M . Suppose that X is klt, ( X/Z, B + M ) is g-lc and K X + B + M is nef. Then there exists δ > , such that forany δ ∈ (0 , δ ) , any sequence of g-MMP /Z on ( K X + B + δA + M ) is ( K X + B + M ) -trivial.Proof. By Proposition 3.16, there exist NQC g-lc pairs ( K X + B ′ k + M ′ k ),such that K X + B + M = P a k ( K X + B ′ k + M ′ k ) with P a k = 1 , a k > K X + B ′ k + M ′ k is a nef /Z Q -Cartier divisor for each k . Thus thereexists m ∈ N , such that m ( K X + B ′ k + M ′ k ) is Cartier.Let α := min k { a k m } and δ = α X + α . Choose δ ∈ (0 , δ ), then for any extremal curve C of ( K X + B + M + δA ),if ( K X + B + M ) · C >
0, we have ( K X + B + M ) · C ≥ α . By the length ofextremal rays (Proposition 3.13),( K X + B + M + δA ) · C = δ ( K X + B + M + A ) · C + (1 − δ )( K X + B + M ) · C ≥ − δ dim X + (1 − δ ) α > . This is a contradiction. Thus, any ( K X + B + M + δA )-flip or divisorialcontraction, f : X Y /Z , is ( K X + B + M )-trivial. As K X + B ′ k + M ′ k is nef, f is also ( K X + B ′ k + M ′ k )-trivial, and thus m ( K Y + B ′ Y,k + M ′ Y,k ) := mf ∗ ( K X + B ′ k + M ′ k ) is nef and Cartier. We can repeat the above argumenton Y . This proves the claim. (cid:3) The dual of the above result is the following lemma.
Lemma 3.18.
Let ( X/Z, B + M ) be a g-lc pair with X a Q -factorial kltvariety. Suppose that P is an NQC divisor /Z . Then for any β ≫ , anysequence of g-MMP /Z on ( K X + B + M + βP ) is P -trivial.Proof. Since P is NQC, there exists α >
0, such that for any curve
C/Z , if P · C = 0, then P · C > α . Set d = dim X and choose β > dα . Suppose that C is an extremal curve such that ( K X + B + M + βP ) · C <
0. If P · C = 0,then by the length of extremal rays (Proposition 3.13), we have( K X + B + M + βP ) · C = ( K X + B + M ) · C + βP · C ≥ − d + β · α > . This is a contradiction, and thus P · C = 0. Just as Lemma 3.17, by the P -triviality, we can continue this process and α is independent of this g-MMP. (cid:3) G-MMP with scaling of an NQC divisor.
In this subsection, wewill define a g-MMP with scaling of a divisor Q = E + P , where E is aneffective divisor and P is the pushforward of an NQC divisor. Notice that P may not be an effective divisor. To emphasize this special property, wecoin the name “g-MMP with scaling of an NQC divisor”, although therealso exists an effective part (i.e. E ) in Q . Lemma 3.19.
Let X be a Q -factorial klt variety, and ( X/Z, B + M ) be anNQC g-lc pair with boundary part B and nef part M . Let E be an effectivedivisor on X and P be a pushforward of an NQC divisor from a sufficientlyhigh model. Set Q = E + P . Suppose that ( X/Z, ( B + E ) + ( M + P )) is NQCg-lc with boundary part B + E and nef part M + P , and K X +( B + E )+( M + P ) is nef /Z . Then, either K X + B + M is nef /Z , or there is an extremal ray R/Z such that ( K X + B + M ) · R < , ( K X + B + M + νQ ) · R = 0 , where ν := inf { t ≥ | K X + B + M + tQ is nef /Z } . In particular, K X + B + M + νQ is nef /Z ,Proof. Suppose that K X + B + M is not nef /Z . Let { R i } i ∈I be the setof ( K X + B + M )-negative extremal rays /Z , and Γ i be an extremal curveof R i . As L ( B, M ) is a rational polytope, by Proposition 3.13, there are r , . . . , r s ∈ R > and m ∈ N , such that − X ≤ ( K X + B + M ) · Γ i = s X j =1 r j n i,j m < , where − m (dim X ) ≤ n i,j ∈ Z . By Proposition 3.16, there are r ′ , . . . , r ′ t ∈ R > and m ∈ N (after changing the above m by a sufficiently divisiblemultiple), such that( K X + B + E + M + P ) · Γ i = t X k =1 r ′ k n ′ i,k m , where n ′ i,k ∈ Z ≥ .Since n i,j is bounded above, { n i,j } is a finite set, and so is { P j r j n i,j } .Moreover, P k r ′ k n ′ i,k belongs to a DCC set, where DCC stands for descendingchain condition. Let ν i := − ( K X + B + M ) · Γ i Q · Γ i . Thus, 1 ν i = P k r ′ k n ′ i,k − P j r j n i,j + 1belongs to a DCC set. Hence there exists a maximal element ν = ν s in theset { ν i } i ∈I . Then, ( K X + B + M + νQ ) · Γ i ≥ i ∈ I . For the extremal curve Γ s , ( K X + B + M + νQ ) · Γ s = 0. (cid:3) EAK ZARISKI DECOMPOSITIONS AND LOG TERMINAL MODELS 19
Definition 3.20 (G-MMP with scaling of an NQC divisor) . Under the sameassumptions and notation of Lemma 3.19, we define the g-MMP with scalingof an NQC divisor as follows.(1) If K X + B + M is nef /Z , we stop.(2) If K X + B + M is not nef /Z , there exists an extremal ray R as inLemma 3.19. By Lemma 3.5, we can contract R . • If the contraction is a Mori fiber space, we stop. • If the contraction is a divisorial (resp. flipping) contraction, let X X be the corresponding contraction (resp. flip). Let ( X , B + M + νQ ) be the birational transform of ( X, B + M + νQ ) . We cancontinue the previous process on ( X , B + M + νQ ) in place of ( X, B + M + Q ) . In fact, by ( X, B + M + νQ ) -triviality, ( X , B + M + νQ ) is nef /Z .By doing this, we obtain a sequence (may be infinite) of varieties X i andcorresponding nef thresholds ν i := inf { t ≥ | K X i + B i + M i + tQ i is nef /Z } . Remark 3.21.
By definition, the nef thresholds ν i ≥ ν i +1 for each i . Lifting the sequence of flips.
We use the same notation as Section3.4. Suppose that a sequence of g-MMP /Z with scaling of Q = E + P on K X + B + M only consists of flips, X i X i +1 /Z i , where X i → Z i is theflipping contraction. Let h : ( X ′ /Z, B ′ + M ′ ) → X be a g-dlt modificationof ( X , B + M ) (see Proposition 3.9). Pick an ample /Z divisor H ≥ K X + B + M + H ∼ R /Z , and ( X , B + M + H ) is g-lc. By Lemma 3.5, ( X , ∆ ) is klt for someboundary ∆ ∼ R B + M + ǫH/Z . According to the proof of Lemma 3.5,we may choose ∆ ≥
0, such that h ∗ ( K X + ∆ ) = K X ′ + ∆ ′ for someeffective divisor ∆ ′ , and ( X ′ , ∆ ′ ) is klt. Now run an MMP /Z on K X ′ + ∆ ′ with scaling of h ∗ ( H ). By [8, Corollary 1.4.2], the MMP terminates with alog terminal model, X ′ X ′ . By construction, we have(1 − ǫ )( K X ′ + B ′ + M ′ ) ∼ R (1 − ǫ ) h ∗ ( K X + B + M ) ∼ R h ∗ ( K X + B + M + ǫH ) /Z ∼ R K X ′ + ∆ ′ /Z . Thus this MMP is also a g-MMP /Z on K X ′ + B ′ + M ′ , and thus K X ′ + B ′ + M ′ is nef /Z . We define Q ′ = h ∗ ( Q ) as follows. Suppose that W p −→ X ′ h −→ X is a sufficiently high log resolution such that P = ( h ◦ p ) ∗ P W for an NQCdivisor P W on W . Then by the negativity lemma, P W + F = ( h ◦ p ) ∗ P with F ≥
0. Set(3) E ′ := h ∗ E + p ∗ F and P ′ := p ∗ P W , and(4) Q ′ := E ′ + P ′ = h ∗ ( E ) + p ∗ ( p ∗ ◦ h ∗ ( P )) = h ∗ ( E + P ) = h ∗ ( Q ) . Because ρ ( X /Z ) = 1, Q ≡ aH/Z for some a >
0. Thus the g-MMP/ Z X ′ i X ′ i +1 is also a g-MMP /Z on K X ′ + B ′ + M ′ with scaling of Q ′ .Because X , X are isomorphic in codimension 1 and K X + B + M is ample /Z , ( X , B + M ) is a g-log canonical model of ( X , B + M )over Z (here g-log canonical model means a g-log terminal model with K X + B + M ample /Z ). Thus there exists a morphism h : X ′ → X suchthat K X ′ + B ′ + M ′ = h ∗ ( K X + B + M ), which is also a g-dlt modificationof ( X , B + M ). We can continue the above process for X , X ′ , etc. inplaces of X , X ′ , etc.From the above, we have a sequence of g-MMP /Z on ( K X ′ + B ′ + M ′ ) withscaling of Q ′ . The reason is as follows. A priori, the g-MMP X ′ i X ′ i +1 with scaling of Q ′ is over Z rather than over Z . We denote this g-MMP /Z by X ′ i = Y Y · · · Y k = X ′ i +1 , and let ν ′ j , ≤ j ≤ k be the corresponding nef thresholds /Z . By K X ′ i + B ′ i + M ′ i + ν i Q ′ i ≡ /Z , we have ν ′ j = ν i for 0 ≤ j ≤ k −
1. Thus ν ′ j = inf { t | K Y j + B Y j + M Y j + tQ Y j is nef over Z } , for 0 ≤ j ≤ k −
1. This shows that the g-MMP with scaling of Q ′ i is alsoover Z .By doing above, we lift the original g-MMP with scaling to a new g-MMPwith scaling. The advantage is that each ( X ′ i , B ′ i + M ′ i ) becomes Q -factorialand g-dlt.4. Special termination for g-MMP with scaling
It is crucial to observe that some termination results still hold for g-MMPwith scaling of an NQC divisor. The following is a variation of [5, Theorem1.9].
Theorem 4.1.
Under the assumptions and notation of Definition 3.20.Suppose that there is a g-MMP with scaling of Q . Let µ = lim j →∞ ν j .If µ = ν j for any j , and ( X/Z, ( B + µE ) + ( M + µP )) has a g-log minimalmodel, then the g-MMP terminates. This theorem is proved in several steps.
Proposition 4.2.
Under the above notation, Theorem 4.1 holds if there isa birational map φ : X Y /Z between Q -factorial varieties satisfying:(1) the induced map X i Y is isomorphic in codimension one forevery i ,(2) ( Y /Z, ( B Y + µE Y )+( M Y + µP Y )) is a g-log minimal model of ( X/Z, ( B + µE ) + ( M + µP )) , EAK ZARISKI DECOMPOSITIONS AND LOG TERMINAL MODELS 21 (3) there is a reduced divisor A ≥ on X , whose components are mov-able divisors and generate N ( X/Z ) ,(4) there exists ǫ > , such that ( X/Z, ( B + E + ǫA ) + ( M + P )) is g-dltwith boundary part ( B + E + ǫA ) and nef part ( M + P ) ,(5) there exists δ > , such that ( Y /Z, ( B Y + ( µ + δ ) E Y + ǫA Y ) + ( M Y +( µ + δ ) P Y )) is g-dlt with boundary part ( B Y + ( µ + δ ) E Y + ǫA Y ) andnef part ( M Y + ( µ + δ ) P Y )) .Proof. Suppose that the g-MMP does not terminate. Pick j ≫
1, so that ν j − > ν j . Then X X j is a partial g-MMP/ Z on K X + B + M + ν j − Q .It is also a partial g-MMP/ Z on K X + B + M + ν j − Q + ǫA after replacing ǫ with a smaller number. In particular, ( X j /Z, ( B j + ǫA j ) + M j + ν j − Q ) is Q -factorial g-dlt, where A j is the birational transform of A on X j . As j ≫ /Z starting with ( X /Z, B + M ) = ( X/Z, B + M ). Moreover, by replacing B + M with B + M + µQ , we may assume that µ = 0.Possibly by choosing a smaller ǫ again, by Lemma 3.17, we may assumethat any sequence of g-MMP /Z on ( K X j + ( B j + ν j − E j + ǫA j ) + ( M j + ν j − P j )) is a sequence of ( K X j + ( B j + ν j − E j ) + ( M j + ν j − P j ))-flop. Byassumption, K X j +( B j + ν j − E j + ǫA j )+( M j + ν j − P j ) is a limit of movable /Z R -divisors. Since the components of A j generate N ( X j /Z ), there exists ageneral ample /Z divisor H and an effective divisor H ′ < A j , such that A j ∼ R H + H ′ , and ( X j /Z, ( B j + ν j − E j + ǫH ′ + ǫH ) + ( M j + ν j − P j )) isg-dlt. By Lemma 3.5, there exists a klt pair ( X j , ∆ j ) such that K X j + ∆ j ∼ R K X j + ( B j + ν j − E j + ǫH ′ + ǫH ) + ( M j + ν j − P j ) . By [8], we may run an MMP /Z with scaling of an ample divisor on K X j +∆ j ,which is the same as an MMP /Z on ( K X j + ( B j + ν j − E j + ǫA j ) + ( M j + ν j − P j )). It terminates with a g-log minimal model ( T /Z, ( B T + ν j − E T + ǫA T ) + ( M T + ν j − P T )). Notice that X j , T are isomorphic in codimension1, and ( K T + ( B T + ν j − E T ) + ( M T + ν j − P T )) is nef /Z . Again, since thecomponents of A T generate N ( T /Z ), we can choose 0 < D T ≤ A T such that K T + B T + M T + ν j − Q T + ǫD T is ample. Moreover, K T + B T + M T + ν j − Q T is nef /Z by the choice of ǫ .For the same reason, possibly by choosing smaller ν j and ǫ , we can run ag-MMP /Z on ( K Y + B Y + M Y + ν j − Q Y + ǫD Y ) with scaling of an ampledivisor, and get a g-log minimal model, ( Y ′ , B Y ′ + M Y ′ + ν j − Q Y ′ + ǫD Y ′ ),such that both K Y ′ + B Y ′ + M Y ′ + ν j − Q Y ′ + ǫD Y ′ and K Y ′ + B Y ′ + M Y ′ are nef (see Lemma 3.17). Because Y, Y ′ are Q -factorial varieties which areisomorphic in codimension 1 and K T + B T + M T + ν j − Q T + ǫD T is ample /Z ,we have Y ′ = T . Hence, both K T + B T + M T + ν j − Q T and K T + B T + M T are nef /Z . By ν j − > ν j > µ = 0, K T + B T + M T + ν j Q T is nef /Z . Let r : U → X j , s : U → T be a common log resolution. By the negativitylemma, we have r ∗ ( K X j + B j + M j ) >s ∗ ( K T + B T + M T ) ,r ∗ ( K X j + B j + M j + ν j − Q j ) = s ∗ ( K T + B T + M T + ν j − Q T ) ,r ∗ ( K X j + B j + M j + ν j Q j ) = s ∗ ( K T + B T + M T + ν j Q T ) . This is a contradiction. (cid:3)
Proof of Theorem 4.1.
Let (
Y /Z, ( B Y + µE Y ) + ( M Y + µP Y )) be the g-logminimal model of ( X/Z, ( B + µE ) + ( M + µP )) with corresponding map φ : X Y . As in Proposition 4.2, we may assume that µ = 0, andthe g-MMP /Z only consists of flips, X i X i +1 . Because there are finitemany g-lc centers, we can assume that no g-lc centers are contracted in thesequence. Moreover, choose ν i − > ν i , then for any birational morphism f : W → X i , we can write(5) f ∗ Q i = f ∗ ( E i + P i ) = ˜ E i + P W,i + Θ
W,i , with ˜ E i the birational transform of E i . The meanings of P W,i , Θ W,i areas follows (cf. (3)). By definition, we can assume that P i = q ∗ P ′ where q : W ′ → X i is a sufficiently high model and P ′ is an NQC divisor. By takinga common log resolution, we can assume that there also exists a morphism p : W ′ → W . Then we set P W,i = p ∗ P ′ . By E i ≥ W,i ≥ f -exceptional divisor. By ( X i , B i + M i + ν i − Q i ) isg-lc, there is no g-lc place of ( X i , B i + M i + ν i Q i ) which is contained inSupp Θ W,i . We can replace (
X/Z, B + M ) with ( X i /Z, B i + M i ) and Q with ν i Q i .Step 1. Let f : W → X and g : W → Y be a sufficiently high commonlog resolution of ( X/Z, ( B + E ) + ( M + P )) and ( Y /Z, B Y + M Y + Q Y ). Wehave F := f ∗ ( K X + B + M ) − g ∗ ( K Y + B Y + M Y ) , and F ′ := K W + B W + M W − f ∗ ( K X + B + M ) , (6)where B W is defined as (2). Then F, F ′ are effective exceptional divisorsover Y, X respectively. By the definition of g-log minimal model, F ′ is alsoexceptional over Y .Let E W be the birational transform of E on W , and P W be the nef /Z divisor corresponding to P on W . Set Q W = P W + E W . We have K W + B W + M W ≡ F + F ′ /Y. (7)By Proposition 3.8, we can run a g-MMP /Y on ( K W + B W + M W ) withscaling of an ample divisor, and it terminates with a model Y ′ , such that F + F ′ is contracted. Thus ( Y ′ , B Y ′ + M Y ′ ) is a g-dlt modification of ( Y, B Y + M Y ).Step 2. We prove that φ : X Y does not contract any divisor. Other-wise, let D be a prime divisor on X which is contracted by φ , and D W be the EAK ZARISKI DECOMPOSITIONS AND LOG TERMINAL MODELS 23 birational transform of D on W . Since a ( D, X, B + M ) < a ( D, Y, B Y + M Y ), D W is a component of F . In Step 1, the g-MMP /Y on ( K W + B W + M W )contracts D W . We will get a contradiction as follows. Let ν i be sufficientlysmall so that W Y ′ is a partial g-MMP /Z on K W + B W + M W + ν i Q W .Since ( X/Z, B + M + ν i Q ) is g-lc, K W + B W + M W + ν i Q W − f ∗ ( K X + B + M + ν i Q )is effective and exceptional over X . On the other hand, X X i is a partialg-MMP /Z on ( K X + B + M + ν i Q ), we have f ∗ ( K X + B + M + ν i Q ) ≥ N, where N = p ∗ q ∗ ( K X i + B i + M i + ν i Q i )for some common log resolution p : W ′ → W, q : W ′ → X i . Since K X i + B i + M i + ν i Q i is nef /Z , N is a pushforward of a nef divisor. In particular, N is a limit of movable /Z divisors. We have K W + B W + M W + ν i Q W = N + G, where G ≥ X . Here we use the fact that X and X i are isomorphic in codimension one. Since G is exceptional /X , D W is not acomponent of G . For the g-MMP in Step 1, if D W were contracted by anextremal contraction of a curve C , we have ( K W + B W + M W + ν i Q W ) · C < N · C ≥ G · C ≥
0. Thus D W cannot be contracted. This is acontradiction.Step 3. From W , we construct a g-dlt modification of ( X, B + M ). Let F ′′ := K W + ( B W + E W ) + ( M W + P W ) − f ∗ ( K X + ( B + E ) + ( M + P )) , which is effective and exceptional over X . We run a g-MMP /X on K W +( B W + E W ) + ( M W + P W ) which terminates with a model h : X ′ → X andcontracts F ′′ . This h is a g-dlt modification of ( X, ( B + E ) + ( M + P )). Let K X ′ + ( B ′ + E ′ ) + ( M ′ + P ′ ) = h ∗ ( K X + ( B + E ) + ( M + P )) , where E ′ is the strict transform of E and P ′ is the pushforward of P W . Byassumption (see the paragraph before Step 1) that for Q ′ := h ∗ ( E + P ) = E ′ + P ′ + Θ ′ as in (5), there is no g-lc place of ( X, B + M ) which is contained in Θ ′ . ThusΘ ′ = 0. Hence h is also a g-dlt modification of ( X, B + M ), that is K X ′ + B ′ + M ′ = h ∗ ( K X + B + M ) . In particular, h extracts all the g-lc places of ( X, B + M ) on W . Because φ − : Y X can only extract g-lc places of ( X, B + M ) (see Remark 2.10),we see that these divisors are all on X ′ .Step 4. By Subsection 3.5, we can lift the sequence X i X i +1 /Z i to ag-MMP /Z on K X ′ + B ′ + M ′ with scaling of Q ′ . Hence, each ( X ′ i , B ′ i + M ′ i )is Q -factorial and g-dlt. Step 5. Possibly by replacing X ′ with X ′ i for i ≫
1, we show that X ′ , Y ′ are also isomorphic in codimension 1, and ( Y ′ /Z, B ′ + M ′ ) is a g-log minimalmodel of ( X/Z, B + M ).First, We show that Y ′ X ′ does not contract any divisor. Supposethat D ⊂ Y ′ is a prime divisor which is exceptional over X ′ . If D is on Y ,then a ( D, X, B + M ) = 0 as D is exceptional over X . Thus, by Step 3, D is on X ′ , a contradiction. If D is exceptional over Y , as ( Y ′ , B Y ′ + M Y ′ ) isa g-dlt modification of ( Y, B Y + M Y ), we have a ( D, Y, B Y + M Y ) = 0. Thisimplies that a ( D, X, B + M ) = 0, and again we get a contradiction fromStep 3.Next, We show that X ′ Y ′ does not contract any divisor. Possiblyby replacing X ′ with X ′ i for i ≫
1, we may assume that the g-MMP /Z on ( K X ′ + B ′ + M ′ ) with scaling of Q ′ only consists of flips. By using thesame method as Step 2, it suffices to show that ( Y ′ /Z, B Y ′ + M Y ′ ) is a g-log minimal model of ( X/Z, B + M ). Thus we only need to compare g-logdiscrepancies. Suppose that D ⊂ X ′ is a prime divisor which is exceptionalover Y ′ . Since X, Y are isomorphic in codimension 1, D is exceptional over X . Hence a ( D, X ′ , B ′ + M ′ ) = a ( D, X, B + M ) = 0. If a ( D, Y ′ , B Y ′ + M Y ′ ) =0, then a ( D, Y, B Y + M Y ) = 0. Thus the birational transform of D cannotbe a component of F + F ′ in (6), and it can not be contracted over Y ′ . Thisis a contradiction. Therefore, a ( D, Y ′ , B Y ′ + M Y ′ ) >
0, which implies that( Y ′ /Z, B ′ + M ′ ) is a g-log minimal model of ( X/Z, B + M ).Step 6. Let A ≥ W whose components aregeneral ample /Z divisors such that they generate N ( W/Z ). Since X ′ isobtained by running some g-MMP on K W + B W + M W + Q W , this g-MMPis also a partial g-MMP on K W + B W + M W + Q W + ǫA for any 1 ≫ ǫ > X ′ /Z, ( B ′ + E ′ + ǫA ′ ) + ( M ′ + P ′ )) is g-dlt, where A ′ is thebirational transform of A . For similar reasons, we can choose 1 ≫ ǫ, δ > Y ′ /Z, ( B Y ′ + δE Y ′ + ǫA Y ′ ) + M Y ′ + δP Y ′ ) is also g-dlt.Now, by Proposition 4.2, the g-MMP /Z , X ′ i X ′ i +1 , terminates. Thisimplies that the original g-MMP /Z , X i X i +1 , also terminates. Thisfinishes the proof. (cid:3) We introduce the notion of difficulty for g-pairs before proving the specialtermination.
Definition 4.3 (Difficulty for g-pairs) . Let ( X, B + M ) be a Q -factorialg-dlt pair with data ˜ X f −→ X → Z and ˜ M . For R -divisors B, ˜ M , assumethat B = P b j B j is the prime decomposition of B , and ˜ M = P µ i ˜ M i with ˜ M i a nef /Z Cartier divisor for each i . Let b = { b j } , µ = { µ i } . Recall that S ( b , µ ) = { − m + X j r j b j m + X i s i µ i m ≤ | m ∈ Z > , r j , s i ∈ Z ≥ } ∪ { } . Let S be a g-lc center of ( X, B + M ) , then K S + B S + M S = ( K X + B + M ) | S EAK ZARISKI DECOMPOSITIONS AND LOG TERMINAL MODELS 25 is defined in Proposition 2.8. The difficulty of the g-pair ( X, B + M ) isdefined to be d b , µ ( S, B S + M S )= X α ∈ S ( b , µ ) { E | a ( E, B S + M S ) < − α, Center S ( E ) * ⌊ B S ⌋} + X α ∈ S ( b , µ ) { E | a ( E, B S + M S ) ≤ − α, Center S ( E ) * ⌊ B S ⌋} . Remark 4.4.
Notice that d b , µ ( S, B S + M S ) is slightly different from [11,Definition 4.2.9] (cf. [15, 7.5.1 Definition]): in the second summand, wealso includes E whose g-log discrepancy equals 1 − α . By doing this, thestandard argument can be simplified (cf. [11, Proposition 4.2.14] and theargument below). Just as for log pairs, d b , µ ( S, B S + M S ) < + ∞ (cf. [15,4.12.2 Lemma]). Theorem 4.5.
Under the assumptions and notation of Definition 3.20. Werun a g-MMP /Z with scaling of Q on K X + B + M . Assume the existenceof g-log minimal models for pseudo-effective NQC g-lc pairs in dimensions ≤ dim X − . Suppose that ν i > µ for µ = lim ν i (in particular, the g-MMPis an infinite sequence). Then, after finitely many steps, the flipping locusis disjoint from the birational transform of ⌊ B ⌋ .Proof. We follow the proof of [11, Theorem 4.2.1].Because the number of g-lc centers of a g-lc pair is finite, we may assumethat after finitely many steps, the flipping locus contains no g-lc centers.Thus φ i : X i X i +1 induces an isomorphism of 0-dimensional g-lc centersfor every i .We show that φ i induces an isomorphism of every g-lc center by inductionon dimensions d of g-lc centers. Then the theorem follows from d = dim X −
1. Now, for each d ≤ k −
1, we assume that φ i induces an isomorphism forevery d -dimensional g-lc centers.Let S be a k -dimensional g-lc center of ( X, B + M ), and S i be the bi-rational transform of S on X i . By adjunction formula (Proposition 2.8),( K X i + B i + M i ) | S i = K S i + B S i + M S i , and the coefficients of B S i belongto the set S ( b , µ ). By the induction hypothesis, after finitely many flips, φ i induces an isomorphism on ⌊ B S i ⌋ , and thus Center S i E ⊆ ⌊ B S i ⌋ if and onlyif Center S i +1 E ⊆ ⌊ B S i +1 ⌋ . By the negativity lemma, a ( E, S i , B S i + M S i ) ≤ a ( E, S i +1 , B S i +1 + M S i ). Hence, d b , µ ( S i , B S i + M S i ) ≥ d b , µ ( S i +1 , B S i +1 + M S i +1 ) . Moreover, if S i and S i +1 are not isomorphic in codimension 1, then the aboveinequality is strict. In fact, if there exists a divisor E ⊂ S i which is not on S i +1 , then E is counted by the second summand in d b , µ ( S i , B S i + M S i ),while not counted in d b , µ ( S i +1 , B S i +1 + M S i +1 ). Similarly for the case that E is on S i +1 but not on S i . For i ≫
1, we can assume that the difficulties are constant, and thus S i and S i +1 are isomorphic in codimension 1. This isthe advantage of introducing the above difficulty: in [11, Proposition 4.2.14],the case that S i → T i is a divisorial contraction but S i +1 → T i is a smallcontraction cannot be excluded by the difficulty therein.Let T be the normalization of the image of S (hence the image of any S i ) in Z , and T i be the normalization of the image of S i in Z i . In general, S i S i +1 /T i may not be a ( K S i + B S i + M S i )-flip /T . However, we can usethe same argument as Subsection 3.5 to construct a sequence of g-MMP /T with scaling of an NQC divisor over some g-dlt modifications, S ′ i S ′ i +1 .For simplicity, we just sketch the argument below.Because K X + B + M + ν Q ≡ /Z , we have K S + B S + M S + ν Q S ≡ /T , where Q S = Q | S is defined inductively as follows (cf. (3)(4)). Supposethat π : ˜ X → X is a model of X such that P = π ∗ ˜ P with ˜ P an NQCdivisor. Then we have π ∗ P = ˜ P + F with F ≥
0. Notice that S is anirreducible component of V ∩ · · · ∩ V n − k , where V i ⊂ ⌊ B i ⌋ (see Proposition2.8). We first define Q V . Let π also denote the induced morphism ˜ V → V .Then π ∗ ( P | V ) = ˜ P | ˜ V + F | ˜ V . Here ˜ P | ˜ V is still an NQC divisor, and F | ˜ V is an effective divisor. Because ν >
0, no component of E is contained in ⌊ B ⌋ , and thus E | V ≥
0. Now set E V = E | V + π ∗ ( F | ˜ V ) and P V = π ∗ ( ˜ P | ˜ V ) , and Q V := E V + P V = π ∗ ( π ∗ ( E | V + P | V ) = Q | V . We can repeat the above process to define Q S , P S . Notice that P S is apushforward of an NQC divisor.Let K S ′ + B S ′ + M S ′ = h ∗ i ( K S + B S + M S ) be a Q -factorial g-dltmodification of ( S , B S + M S ). By the same argument as Subsection 3.5,we can run a g-MMP /T on K S ′ + B S ′ + M S ′ with scaling of Q ′ . It terminateswith ( S ′ , B S ′ + M S ′ ) which is a g-dlt modification of ( S , B S + M S ). Wecan continue this process on K S ′ + B S ′ + M S ′ . This gives a sequence of g-MMP /T with scaling of Q S ′ . If this sequence does not terminate. Then bythe assumption, there exists a g-log minimal model /T for K S ′ + B S ′ + M S ′ + µQ S ′ . By Theorem 4.1, the g-MMP terminates, and this is a contradiction.Hence the g-MMP /T terminates, that is, ( S ′ i , B S ′ i + M S ′ i ) = ( S ′ i +1 , B S ′ i +1 + M S ′ i +1 ) for i ≫
1. This implies that S i ≃ S i +1 by [11, Lemma 4.2.16]. (cid:3) Proofs of the main results
A birational NQC weak Zariski decomposition can be obtained from ag-log minimal model.
EAK ZARISKI DECOMPOSITIONS AND LOG TERMINAL MODELS 27
Proposition 5.1.
Let ( X/Z, B + M ) be an NQC g-lc pair. Suppose that ( X/Z, B + M ) has a g-log minimal model, then ( X/Z, B + M ) admits abirational NQC weak Zariski decomposition.Proof. Let (
Y /Z, B Y + M Y ) be a g-log minimal model of ( X/Z, B + M ). ByProposition 3.16, there exist Q -Cartier nef /Z divisors M i , and µ i ∈ R > ,such that K Y + B Y + M Y = X µ i M i . Let p : W → X, q : W → Y be a common resolution of X Y , then p ∗ ( K X + B + M ) = q ∗ ( K Y + B Y + M Y ) + E = X µ i q ∗ ( M i ) + E, with E ≥
0. This is a birational NQC weak Zariski decomposition of(
X/Z, B + M ). (cid:3) This shows one direction of Theorem 1.5. For the other direction, we firstshow the existence of g-log minimal models instead of g-log terminal models(see Definition 2.9).
Definition 5.2 ([6] Definition 2.1) . For a g-pair ( X/Z, B + M ) with bound-ary part B and nef part M . Let f : W → X be a projective birationalmorphism from a normal variety, and N be an effective R -divisor on W .Let f ∗ N = P i ∈ I a i N i be a prime decompostion. We define θ ( X/Z, B + M, N ) := { i ∈ I | N i is not a component of ⌊ B ⌋} . Definition 5.3.
A g-pair ( X/Z, B + M ) is called log smooth if X is smooth,with data X id −→ X → Z and M (in particular, M is nef /Z ), and Supp( B ) [ Supp( M ) is a simple normal crossing divisor. Theorem 5.4.
Suppose that Conjecture 1.3 holds in dimensions ≤ d . Theng-log minimal models exist for pseudo-effective NQC g-lc pairs of dimensions ≤ d .Proof. Step 1. It is enough to show Theorem 5.4 in the log smooth case (cf.[3, Remark 2.6] or [5, Remark 2.8]). In fact, let (
X/Z, B + M ) be an NQCg-lc pair. Let π : ( W/Z, B W + M W ) → X be a log resolution of ( X, B + M ),where B W is defined as (2), and M W is an NQC divisor. Thus K W + B W + M W = π ∗ ( K X + B + M ) + F, with F ≥ Y /Z, B Y + M Y ) be a g-log mini-mal model of ( W/Z, B W + M W ) and D be a prime divisor on X which iscontracted over Y . Then, a ( D, X, B + M ) = a ( D, W, B W + M W ) < a ( D, Y, B Y + M Y ) . This implies that (
Y /Z, B Y + M Y ) is also a g-log minimal model of ( X/Z, B + M ) (see Definition 2.9).Assume that π : W → X is a sufficiently high model such that π ∗ ( K X + B + M ) = N + P/Z is an NQC weak Zariski decomposition, where P is anNQC divisor and N is effective. Then K W + B W + M W = π ∗ ( K X + B + M ) + F = ( N + F ) + P/Z.
This is an NQC weak Zariski decomposition /Z for K W + B W + M W . More-over, θ ( X/Z, B + M, N ) = θ ( W/Z, B W + M W , N ) = θ ( W/Z, B W + M W , N + F ) . Thus we may assume that (
X, B + M ) is log smooth with M an NQCdivisor, and K X + B + M ≡ P + N/Z is an NQC weak Zariski decomposition.Moreover, by induction on dimensions, we can assume that Theorem 5.4holds in dimensions ≤ d − θ ( X, B + M, N ).Step 2. When θ ( X, B + M, N ) = 0, we show Theorem 5.4.By definition, θ ( X, B + M, N ) = 0 implies that Supp ⌊ B ⌋ ⊃ Supp N . ByLemma 3.5, for any β >
0, we can run a g-MMP /Z on ( K X + B + M + β P )with scaling of an ample divisor. By proposition 3.18, for β ≫
1, suchg-MMP /Z is P -trivial. Thus it is also a g-MMP /Z on ( K X + B + M ).Moreover, by K X + B + M ≡ P + N/Z and P is nef /Z , the contractinglocus belongs to the birational transform of Supp N ⊂ Supp B . Because M + β P is NQC, the above g-MMP /Z terminates by Theorem 4.5. Let( X , B + M + βP ) be the corresponding g-log minimal model with K X + B + M ≡ N + P /Z . Moreover, P is nef /Z .Next, we run a special kind of g-MMP /Z on ( K X + B + M ) with scalingof P as follows.Suppose that we have constructed ( X i , B i + M i ). Let ν i = inf { t ≥ | K X i + B i + M i + tP i is nef /Z } . (i) If ν i = 0. Then ( X i , B i + M i ) is a g-log minimal model, and we havedone.(ii) If 0 < ν i < ν i − (we set ν = + ∞ ). By Lemma 3.19, there exists anextremal ray R such that ( K X i + B i + M i ) · R < K X i + B i + M i + ν i P i ) · R = 0. We contract R and get a divisorial contraction or a flippingcontraction. Let X i X i +1 be the corresponding divisorial contraction orflip.(iii) If ν i = ν i − >
0. Choose β i < ν i sufficiently close to ν i ( β i canbe determined from the discussion later), we run a g-MMP /Z with scalingof an ample /Z divisor H on ( K X i + B i + M i + β i P i ). We claim that thisg-MMP /Z is also a g-MMP /Z with scaling of P i , and it terminates. Let( X i +1 , B i +1 + M i +1 ) be the resulting g-pair. In particular, ν i +1 ≤ β i < ν i . EAK ZARISKI DECOMPOSITIONS AND LOG TERMINAL MODELS 29
Proof of the Claim in (iii).
Since for any ν i > β i >
0, we have ν i ν i − β i ( K X i + B i + M i + β i P i )=( K X i + B i + M i ) + β i ν i − β i ( K X i + B i + M i + ν i P i ) . (8)If β i is sufficiently close to ν i , then β i ν i − β i is sufficiently large. For a g-MMP /Z with scaling of an ample /Z divisor on ( K X i + B i + M i + β i P i ), by (8), it is alsoa g-MMP /Z on ( K X i + B i + M i )+ β i ν i − β i ( K X i + B i + M i + ν i P i ). By Proposition3.16, K X i + B i + M i + ν i P i is an NQC divisor /Z . Hence by Proposition 3.18,for a sufficiently large β i ν i − β i , this g-MMP /Z is ( K X i + B i + M i + ν i P i )-trivial.By β i < ν i , it is also a g-MMP /Z on ( K X i + B i + M i ) with scaling of P i .In fact, if Y is an intermediate variety in this g-MMP /Z , for a contractingcurve Γ,( K Y + B Y + M Y + β i P Y ) · Γ < K X Y + B Y + M Y + ν i P Y ) · Γ = 0 , thus ( K X Y + B Y + M Y ) · Γ < P Y · Γ > /Z terminates. Because K X i + B i + M i = N i + P i = 11 + ν i ( N i + (1 + ν i ) P i ) + ν i ν i N i = 11 + ν i ( K X i + B i + M i + ν i P i ) + ν i ν i N i , a flipping curve intersects the birational transform of N i negatively. Thusthe flipping locus is contained in the birational transform of Supp N i ⊆ Supp ⌊ B i ⌋ . Suppose that the g-MMP /Z does not terminate. Because the g-MMP /Z is a scaling of an ample /Z divisor, the corresponding nef thresholds ν ′ j satisfies ν ′ j = lim ν ′ j . Otherwise, µ ′ = lim ν ′ j >
0, then the g-MMP /Z canbe viewed as a g-MMP /Z on K X i + B i + M i + β i P i + µ ′ H , then it terminatesby Lemma 3.5 and [8, Corollary 1.4.2]. However, by Theorem 4.5, the aboveg-MMP /Z terminates. This is a contradiction. (cid:3) By applying (i)-(iii), we obtain a g-MMP /Z on K X i + B i + M i with scalingof P , X i = Y i · · · Y k i = X i +1 . Let ˜ ν j be the corresponding nef thresholds. Then either the g-MMP /Z terminates or lim ˜ ν j = lim ν i > ˜ ν j . Moreover, as K Y ji + B Y ji + M Y ji = N Y ji + P Y ji and P Y ji · Γ > N Y ji · Γ <
0. Thus the flipping locus is contained in thebirational transform of Supp N i ⊆ Supp ⌊ B i ⌋ . By Theorem 4.5 again, theg-MMP /Z terminates. This finishes the proof of the θ ( X, B + M, N ) = 0case.
Step 3. Next we show the induction step. The argument is identical to[6, Proof of Theorem 1.5], except that we deal with the g-pairs.First, for a divisor D = P i d i D i , we write D ≤ := P i min { d i , } D i .Suppose that Theorem 5.4 does not hold. We assume that θ ( X, B + M, N ) ≥ K X + B + M does not have a log minimalmodel. By Step 1, we can assume that ( X, B + M ) is log smooth. By θ ( X, B + M, N ) ≥ α := min { t > | ⌊ ( B + tN ) ≤ ⌋ 6 = ⌊ B ⌋} is a finite number. Let C be the divisor such that ( B + αN ) ≤ = B + C .Thus Supp C ⊆ Supp N , and(9) θ ( X, B + M, N ) = { components of C } . Let A ≥ αN = C + A , then Supp A ⊆ Supp ⌊ B ⌋ , and A = ( B + αN ) − ( B + αN ) ≤ .Because θ ( X, ( B + C ) + M, N + C ) < θ ( X, B + M, N ), by the inductionhypothesis, ( X, ( B + C ) + M ) has a log minimal model, ( Y, ( B + C ) Y + M Y ).Notice that ( X, ( B + C ) + M ) is a g-lc pair with boundary part B + C andnef part M . Let g : U → X, h : U → Y be a sufficiently high log resolution,then(10) g ∗ ( K X + ( B + C ) + M ) = h ∗ ( K Y + ( B + C ) Y + M Y ) + N ′ , with N ′ ≥ h -exceptional. Let P ′ := h ∗ ( K Y + ( B + C ) Y + M Y ) , then it is nef /Z and NQC by Proposition 3.16. Thus P ′ + N ′ is an NQCweak Zariski decomposition /Z for g ∗ ( K X + ( B + C ) + M ). We have g ∗ ( N + C ) − N ′ = h ∗ ( K Y + ( B + C ) Y + M Y ) − g ∗ P. Since h ∗ ( K Y +( B + C ) Y + M Y ) − g ∗ P is anti-nef /Y , by the negativity lemma, N ′ ≤ g ∗ ( N + C ). As Supp C ⊆ Supp N , we have Supp N ′ ⊆ Supp g ∗ N .By above, we have(1 + α ) g ∗ ( K X + B + M ) = g ∗ ( K X + B + M ) + αg ∗ P + αg ∗ N = g ∗ ( K X + B + M ) + αg ∗ P + g ∗ C + g ∗ A = P ′ + N ′ + αg ∗ P + g ∗ A. Thus, g ∗ ( K X + B + M ) = 11 + α ( P ′ + αg ∗ P ) + 11 + α ( N ′ + g ∗ A ) . Set P ′′ := 11 + α ( P ′ + αg ∗ P ) and N ′′ := 11 + α ( N ′ + g ∗ A ) , then g ∗ ( K X + B + M ) = P ′′ + N ′′ is a birational NQC weak Zariski decomposition /Z for K X + B + M . Since αN = C + A , we have Supp N ′′ ⊆ Supp g ∗ N , and thus Supp g ∗ N ′′ ⊆ Supp N . EAK ZARISKI DECOMPOSITIONS AND LOG TERMINAL MODELS 31
Because θ ( X, B + M, N ) is minimal, θ ( X, B + M, N ) = θ ( X, B + M, g ∗ N ′′ ),and every component of C is also a component of g ∗ N ′′ according to (9). Be-cause A, C do not have common components, we have Supp C ⊆ Supp g ∗ N ′ ,and thus C is exceptional over Y by (10). Hence by definition ( B + C ) Y = B Y , and P ′ = h ∗ ( K Y + B Y + M Y ). We will compare the g-log discrepanciesbelow.Let G ≥ G ≤ g ∗ C and G ≤ N ′ . Set˜ C = g ∗ C − G, ˜ N ′ = N ′ − G . By (10), we have(11) g ∗ ( K X + B + M ) + ˜ C = P ′ + ˜ N ′ . (i) If ˜ C is exceptional over X , then because g ∗ ( K X + B + M ) − P ′ = ˜ N ′ − ˜ C is anti-nef over X , by the negativity lemma, ˜ N ′ − ˜ C ≥
0, which implies ˜ C = 0as ˜ C and ˜ N ′ have no common components. From (11), g ∗ ( K X + B + M ) = P ′ + ˜ N ′ is a birational NQC-weak Zariski decomposition /Z for K X + B + M . More-over, g ∗ ( K X + B + M ) − h ∗ ( K Y + B Y + M Y )= X D ( a ( D ; Y, B Y + M Y ) − a ( D ; X, B + M )) D = ˜ N ′ , where D runs over the prime divisors on U .(ia) Suppose that Supp g ∗ ˜ N ′ = Supp g ∗ N ′ . Then by (10), Supp ˜ N ′ con-tains the birational transform of all the prime exceptional /Y divisors on X . Hence ( Y, B Y + M Y ) is also a g-log minimal model of ( X, B + M ), acontradiction.(ib) Hence we can assume Supp g ∗ ˜ N ′ ( Supp g ∗ N ′ . Thus,Supp( g ∗ N ′ − g ∗ G ) = Supp g ∗ ˜ N ′ ( Supp g ∗ N ′ ⊆ Supp N. Since G is the largest divisor such that G ≤ g ∗ C and G ≤ N ′ , some compo-nent of C is not a component of g ∗ ˜ N ′ . By (9), we have θ ( X/Z, B + M, ˜ N ′ ) < θ ( X/Z, B + M, N ) , which contradicts the minimality of θ ( X/Z, B + M, N ).(ii) Hence ˜ C is not exceptional over X . Let β > A := βg ∗ N − ˜ C and g ∗ ˜ A ≥ . Then there exists a component D of g ∗ ˜ C which is not a component of g ∗ ˜ A .We have(1 + β ) g ∗ ( K X + B + M ) = g ∗ ( K X + B + M ) + ˜ C + ˜ A + βg ∗ P = P ′ + βg ∗ P + ˜ N ′ + ˜ A. By the negativity lemma, we have ˜ N ′ + ˜ A ≥
0. Let P ′′′ := 11 + β ( P ′ + βg ∗ P ) and N ′′′ := 11 + β ( ˜ N ′ + ˜ A ) , then g ∗ ( K X + B + M ) = P ′′′ + N ′′′ is a birational NQC weak Zariski decomposition of K X + B + M . Byconstruction, D is a component of Supp g ∗ ˜ C ⊆ Supp C ⊆ Supp N . SinceSupp ˜ C ∩ Supp ˜ N ′ = ∅ , D is not a component of g ∗ ˜ N ′ . Thus, D is not acomponent of Supp( g ∗ N ′′′ ) = Supp( g ∗ ˜ N ′ ) ∪ Supp( g ∗ ˜ A ) . Hence θ ( X, B + M, N ) > θ ( X, B + M, N ′′′ ) , which still contradicts the minimality of θ ( X/Z, B + M, N ). (cid:3) Proof of Theorem 1.7.
Let ν i be the nef threshold in the g-MMP /Z withscaling of an ample /Z divisor A (see Definition 3.20). By Lemma 3.5, forany ǫ >
0, there exists a klt pair ( X, ∆ ǫ ), such that ∆ ǫ ∼ R B + M + ǫA/Z .If lim ν i = µ >
0, then this g-MMP /Z is also a MMP /Z on K X + ∆ µ ∼ R B + M + µA/Z . By [8, Corollary 1.4.2], it terminates. Hence we have µ = 0.If this g-MMP /Z does not terminate, we have ν i > µ = 0 for all i . ByTheorem 5.4, ( X/Z, B + M ) has a g-log minimal model ( Y /Z, B Y + M Y ).Hence the g-MMP terminates by Theorem 4.1. (cid:3) Proof of Theorem 1.5.
The g-minimal model conjecture (Conjecture 1.2) im-plies the birational weak Zariski decomposition conjecture (Conjecture 1.3)by Proposition 5.1. For the other direction, Theorem 1.7 implies that anyg-MMP /Z with scaling of an ample divisor terminates. The resulting modelis a g-log terminal model as a g-MMP /Z does not extract divisors. (cid:3) Proof of Theorem 1.6.
First, we show that (
X/Z, B + M ) has a g-log minimalmodel (see Definition 2.9). By Step 1 of the proof of Theorem 5.4, we canassume that ( X/Z, B + M ) is log smooth. By Lemma 3.18, there exists a β ≫
1, such that a g-MMP /Z with scaling of an ample divisor on ( K X + B + M + βM ) is M -trivial. Thus, this g-MMP /Z is also a MMP /Z on the dlt pair( K X + B ). By assumption, it terminates with a model Y . Since ( X/Z, B + M ) is pseudo-effective, X Y /Z is birational and K Y + B Y + ( β + 1) M Y is nef. By Lemma 3.19, we can run a g-MMP /Z on ( K Y + B Y + M Y )with scaling of M Y . This g-MMP is also a MMP /Z on ( K Y + B Y ). Byassumption, it terminates with Y ′ . Because the sequence X Y Y ′ is also a g-MMP /Z on K X + B + M , ( Y ′ /Z, B Y ′ + M Y ′ ) is a desired g-logminimal model.From a g-log minimal model to a g-log terminal model, we use the sameargument as Theorem 1.5. In fact, the existence of g-log minimal model im-plies the existence of birational weak Zariski decomposition by Proposition EAK ZARISKI DECOMPOSITIONS AND LOG TERMINAL MODELS 33 /Z with scaling of an ample divisorterminates. The resulting model is a g-log terminal model. (cid:3) References [1] T. Bauer and T. Peternell. Nef reduction and anticanonical bundles.
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Beijing International Center for Mathematical Research, Peking Univer-sity, Beijing 100871, China
E-mail address : [email protected] Department of Mathematics, Johns Hopkins University, Baltimore, MD 21218,USA
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