Shokurov polytope

130 4. MINIMAL MODEL PROGRAM

C^†⊂SuppF, thenf(C^†)⊂Nlc(X,∆). This is a contradiction because π◦f(C^†) is a point. Thus, C^†6⊂SuppF. Since

−(K_Y + ∆_Y) = −(K_Y +D) +F,

we can see thatR⁰ is a (K_Y + ∆_Y)-negative extremal ray of N E(Y /S).

Therefore, we can find a rational curveC⁰ onY such that C⁰ spans R⁰ and that

0<−(K_Y + ∆_Y)·C⁰ ≤2 dimX

by Theorem 4.6.2. By the above argument, we can easily see that C⁰ 6⊂SuppF. Therefore, we obtain

0<−(K_Y +D)·C⁰ =−(K_Y + ∆_Y)·C⁰−F ·C⁰

≤ −(K_Y + ∆_Y)·C⁰ ≤2 dimX.

SinceK_Y +D=f^∗(K_X+ ∆),C =f(C⁰) is a rational curve onX such that π(C) is a point and 0<−(KX + ∆)·C≤2 dimX.

Remark 4.6.8. In Theorem 4.6.7, we can prove 0< −(K_X + ∆)· C ≤dimX+ 1 when dimX ≤2. For the details, see [F29, Proposition 3.7].

4.7. SHOKUROV POLYTOPE 131

by Theorem 4.6.2. Let Γ be an extremal curve generatingR. Then we have

−(K_X + ∆)·Γ

H·Γ = −(K_X + ∆)·C H·C . Therefore,

−(K_X + ∆)·Γ = (−(K_X + ∆)·C)· H·Γ

H·C ≤2 dimX.

LetF be a reduced divisor onX. We consider the finite dimensional real vector space V = ⊕

kRF_k where F = ∑

kF_k is the irreducible decomposition. We have already seen that

L={D∈V |(X, D) is log canonical}

is a rational polytope inV, that is, it is the convex hull of finitely many rational points in V (see Lemma 4.6.1).

LetD₁,· · ·, D_r be the vertices of L and letm be a positive integer such that m(K_X +D_j) is Cartier for every j. We take an R-divisor

∆ ∈ L. Then we can find non-negative real numbers a₁,· · ·, a_r such that ∆ =∑

ja_jD_j, ∑

ja_j = 1, and (X, D_j) is log canonical for every j (see Lemma 4.6.1). For every curve C on X, the intersection number

−(K_X + ∆)·C can be written as

∑

j

aj

n_j m

such that n_j ∈ Z for every j. If C is an extremal curve, then we can see thatn_j ≤2mdimX for every j by the above arguments.

On the real vector space V, we consider the following norm

||∆||= max

j {|bj|}, where ∆ =∑

jb_jF_j.

We explain Shokurov’s important results (cf. [Sh3]) following [Bir3, Proposition 3.2].

Theorem 4.7.2. We use the same notation as in 4.7.1. We fix an R-divisor ∆ ∈ L. Then we can find positive real numbers α and δ, which depend on (X,∆) and F, with the following properties.

(1) If Γ is any extremal curve over S and(KX + ∆)·Γ>0, then (KX + ∆)·Γ> α.

(2) IfD ∈ L, ||D−∆||< δ, and (K_X+D)·R≤0 for an extremal curve Γ, then (K_X + ∆)·Γ≤0.

132 4. MINIMAL MODEL PROGRAM

(3) Let {R_t}t∈T be any set of extremal rays of N E(X/S). Then NT ={D∈ L |(K_X +D)·R_t ≥0 for every t∈T}

is a rational polytope in V.

Proof. (1) If ∆ is aQ-divisor, then the claim is obvious even if Γ is not extremal. We assume that ∆ is not a Q-divisor. Then we can write KX + ∆ = ∑

jaj(KX +Dj) as in 4.7.1. Then (KX + ∆)·Γ =

∑

ja_j(K_X +D_j)·Γ. If (K_X + ∆)·Γ<1, then

−2 dimX ≤(K_X +D_j0)·Γ< 1

a_j0{−∑

j6=j0

a_j(K_X +D_j)·Γ + 1}

≤ 2 dimX+ 1 a_j0

for a_j0 6= 0. This is because (K_X +D_j) ·Γ ≥ −2 dimX for every j. Thus there are only finitely many possibilities of the intersection numbers (KX +Dj)·Γ for aj 6= 0 when (KX + ∆)·Γ<1. Therefore, the existence of α is obvious.

(2) If we take δ sufficiently small, then, for every D ∈ L with

||D−∆||< δ, we can always find D⁰ ∈ L such that K_X +D= (1−s)(K_X + ∆) +s(K_X +D⁰) with

0≤s≤ α

α+ 2 dimX.

Since Γ is extremal, we have (K_X+D⁰)·Γ≥ −2 dimX for everyD⁰ ∈ L. We assume that (K_X + ∆)·Γ > 0. Then (K_X + ∆)·Γ > α by (1).

Therefore,

(K_X +D)·Γ = (1−s)(K_X + ∆)·Γ +s(K_X +D⁰)·Γ

>(1−s)α+s(−2 dimX)≥0.

This is a contradiction. Therefore, we obtain (K_X + ∆)·Γ ≤ 0. We complete the proof of (2).

(3) For every t ∈ T, we may assume that there is some D_t ∈ L such that (K_X +D_t)·R_t < 0. We note that (K_X +D)·R_t < 0 for some D ∈ L implies (K_X +D_j)·R_t < 0 for some j. Therefore, we may assume that T is contained in N. This is because there are only countably many (KX +Dj)-negative extremal rays for every j by the cone theorem (see Theorem 4.5.2). We note thatNT is a closed convex subset ofL by definition. If T is a finite set, then the claim is obvious.

Thus, we may assume that T =N. By (2) and by the compactness of

4.7. SHOKUROV POLYTOPE 133

NT, we can take ∆1,· · · ,∆n∈ NT and δ1,· · ·, δn >0 such that NT is covered by

Bi ={D∈ L | ||D−∆i||< δi}

and that ifD∈ Bi with (K_X+D)·R_t<0 for somet, then (K_X+ ∆_i)· R_t = 0. If we put

T_i ={t∈T |(K_X +D)·R_t<0 for someD∈ Bi},

then (K_X + ∆_i)·R_t = 0 for every t ∈ T_i by the above construction.

Since {Bi}ⁿi=1 gives an open covering ofNT, we haveNT =∩

1≤i≤nNTi

by the following claim.

Claim. NT =∩

1≤i≤nNTi.

Proof of Claim. We note thatNT ⊂∩

1≤i≤nNTi is obvious. We assume that NT ( ∩

1≤i≤nNTi. We take D ∈ ∩

1≤i≤nNTi \ NT which is very close to NT. Since NT is covered by {Bi}ⁿi=1, there is some i₀ such that D ∈ Bi0. Since D 6∈ NT, there is some t0 ∈ T such that (KX +D)·Rt0 < 0. Thus, t0 ∈ Ti0. This is a contradiction because D∈ NTi0. Therefore, NT =∩

1≤i≤nNTi.

So, it is sufficient to see that each NTi is a rational polytope in V. By replacing T with T_i, we may assume that there is some D ∈ NT

such that (K_X +D)·R_t = 0 for everyt∈T.

If dim_RL = 1, then this already implies the claim. We assume dim_RL > 1. Let L¹,· · · ,L^p be the proper faces of L. Then N_Tⁱ = NT ∩ Lⁱ is a rational polytope by induction on dimension. Moreover, for each D⁰⁰ ∈ NT which is not D, there is D⁰ on some proper face of L such that D⁰⁰ is on the line segment determined byD and D⁰. Note that (KX +D)·Rt = 0 for every t ∈ T. Therefore, if D⁰ ∈ Lⁱ, then D⁰ ∈ NTⁱ. Thus, NT is the convex hull of D and all the NTⁱ. So there is a finite subset T⁰ ⊂T such that

∪

i

NTⁱ =NT⁰∩(∪

i

Lⁱ).

Therefore, the convex hull of D and ∪

iNTⁱ is just NT⁰. We complete

the proof of (3).

By Theorem4.7.2 (3), Lemma 2.6 in [Bir2] holds for log canonical pairs. It may be useful for the minimal model program with scaling.

Theorem4.7.3 (cf. [Bir2, Lemma 2.6]). Let(X,∆) be a log canon- ical pair, let ∆ be an R-divisor, and let π : X → S be a projective morphism between algebraic varieties. Let H be an effective R-Cartier R-divisor on X such that K_X + ∆ +H is π-nef and(X,∆ +H) is log

134 4. MINIMAL MODEL PROGRAM

canonical. Then, either KX + ∆ is also π-nef or there is a (KX + ∆)- negative extremal ray R such that (K_X + ∆ +λH)·R= 0, where

λ:= inf{t≥0|K_X + ∆ +tH is π-nef}. Of course, K_X + ∆ +λH is π-nef.

Note that Theorem 4.7.3is nothing but Theorem 4.5.2 (6).

Proof. Assume that K_X + ∆ is not π-nef. Let{R_j} be the set of (K_X+ ∆)-negative extremal rays over S. LetC_j be an extremal curve spanning Rj for every j. We put µ= sup

j {µj}, where µ_j = −(K_X + ∆)·C_j

H·C_j .

Obviously, λ = µ and 0 < µ ≤ 1. So, it is sufficient to prove that µ = µ_j0 for some j₀. There are positive real numbers r₁,· · · , r_l such that ∑

ir_i = 1 and a positive integer m, which are independent of j, such that

−(K_X + ∆)·C_j =

∑l i=1

r_in_ij m >0

(see Lemma4.6.1, Theorem4.6.2, and4.7.1). Since Cj is extremal, nij

is an integer with nij ≤ 2mdimX for every i and j. If (KX + ∆ + H)·R_j0 = 0 for some j₀, then there are nothing to prove since λ = 1 and (K_X + ∆ + H) ·R = 0 with R = R_j0. Thus, we assume that (K_X + ∆ +H)·R_j > 0 for every j. We put F = Supp(∆ +H). Let F =∑

kF_k be the irreducible decomposition. We put V =⊕

kRF_k, L ={D∈V |(X, D) is log canonical},

and

N ={D∈ L |(K_X +D)·R_j ≥0 for every j}.

Then N is a rational polytope in V by Theorem4.7.2 (3) and ∆ +H is in the relative interior ofN by the above assumption. Therefore, we can write

K_X + ∆ +H =

∑q p=1

r⁰_p(K_X +D_p), wherer₁⁰,· · · , r⁰_qare positive real numbers such that∑

pr⁰_p = 1, (X, D_p) is log canonical for every p, m⁰(KX +Dp) is Cartier for some positive integer m⁰ and every p, and (K_X +D_p)·C_j >0 for every p and j. So, we obtain

(KX + ∆ +H)·Cj =

∑q p=1

r_p⁰n⁰_pj m⁰

4.7. SHOKUROV POLYTOPE 135

with 0 < n⁰_pj = m⁰(KX +Dp)· Cj ∈ Z. Note that m⁰ and r_p⁰ are independent of j for every p. We also note that

1

µ_j = H·C_j

−(K_X + ∆)·C_j = (K_X + ∆ +H)·C_j

−(K_X + ∆)·C_j + 1

= m∑_q

p=1r_p⁰n⁰_pj m⁰∑_l

i=1r_jn_ij + 1.

Since

∑l i=1

r_in_ij m >0

for every j and n_ij ≤ 2mdimX with n_ij ∈ Z for every i and j, the number of the set {n_ij}i,j is finite. Thus,

infj

{ 1 µ_j

}

= 1 µ_j0

for some j₀. Therefore, we obtain µ=µ_j0. We finish the proof.

Let us recall the abundance conjecture, which is one of the most im- portant conjectures in the minimal model theory for higher-dimensional algebraic varieties.

4.7.4 (Abundance conjecture). We treat some applications of The- orem 4.7.2 (3) to the abundance conjecture for R-divisors (see [Sh3, 2.7. Theorem on log semi-ampleness for 3-folds]).

Conjecture 4.7.5 (Abundance conjecture). Let (X,∆) be a log canonical pair and let f : X → Y be a projective morphism between varieties. If KX + ∆ is f-nef, then KX + ∆ is f-semi-ample.

For the recent developments of the abundance conjecture, see, for example, [FG1].

The following proposition is a useful application of Theorem 4.7.2 (see [Sh3, 2.7]).

Proposition 4.7.6. Let f : X → Y be a projective morphism between algebraic varieties. Let ∆ be an effective R-divisor on X such that(X,∆)is log canonical and that K_X+ ∆isf-nef. Assume that the abundance conjecture holds for Q-divisors. More precisely, we assume thatK_X+Dis f-semi-ample if D∈ L, D is a Q-divisor, and K_X+D is f-nef, where

L ={D∈V |(X, D) is log canonical}, V = ⊕

kRF_k, and ∑

kF_k is the irreducible decomposition of Supp ∆.

Then K_X + ∆ is f-semi-ample.

136 4. MINIMAL MODEL PROGRAM

Proof. Let {R_t}t∈T be the set of all extremal rays ofN E(X/Y).

We consider NT as in Theorem4.7.2 (3). Then NT is a rational poly- tope in L by Theorem 4.7.2 (3). We can easily see that

NT ={D∈ L |KX +D is f-nef}.

By assumption, ∆∈ NT. Let F be the minimal face ofNT containing

∆. Then we can find Q-divisors D1,· · · , Dl on X such that Di is in the relative interior of F,

KX + ∆ =∑

i

di(KX +Di),

where d_i is a positive real number for every i and ∑

id_i = 1. By assumption, K_X +D_i is f-semi-ample for every i. Therefore, K_X + ∆

is f-semi-ample.

Remark4.7.7 (Stability of Iitaka fibrations). In the proof of Propo- sition 4.7.6, we note the following property. If C is a curve on X such that f(C) is a point and (K_X +D_i0)·C = 0 for some i₀, then (K_X+D_i)·C = 0 for everyi. This is because we can find ∆⁰ ∈ F such that (K_X+∆⁰)·C <0 if (K_X+D_i)·C > 0 for somei6=i₀. This is a con- tradiction. Therefore, there exists a contraction morphism g :X →Z over Y and h-ample Q-divisors A₁,· · · , A_l on Z, where h : Z → Y, such that KX +Di ∼Q g^∗Ai for every i. In particular,

K_X + ∆∼_Rg^∗(∑

i

d_iA_i).

Note that ∑

id_iA_i is h-ample. Roughly speaking, the Iitaka fibration of KX + ∆ is the same as that of KX +Di for every i.

Corollary 4.7.8. Let f : X → Y be a projective morphism be- tween algebraic varieties. Assume that (X,∆) is log canonical and that K_X + ∆ isf-nef. We further assume one of the following conditions.

(i) dimX ≤3.

(ii) dimX = 4 and dimY ≥1.

Then K_X + ∆ is f-semi-ample.

Proof. It is obvious by Proposition 4.7.6 and the log abundance theorems for threefolds and fourfolds (see, for example, [KeMM, 1.1. The-

orem] and [F22, Theorem 3.10]).

Corollary 4.7.9. Let f : X → Y be a projective morphism be- tween algebraic varieties. Assume that (X,∆) is klt and K_X + ∆ is f-nef. We further assume that dimX−dimY ≤ 3. Then K_X + ∆ is f-semi-ample.

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