Minimal model program
4.8. MMP for lc pairs
138 4. MINIMAL MODEL PROGRAM
Proof. By the same argument as Lemma4.3.2, we havea(E, X,∆)≤ a(E, X0,∆0) for every prime divisorE overX. We take a common res- olution
W
q
!!B
BB BB BB
p B
~~}}}}}}}}
X
πAAAAAA AA
φ _ _ _//
_ _ _
_ X0
π0
}}||||||||
S
of X and X0. We put ∆W = p−∗1∆ +E, where E is the sum of all p-exceptional divisors. Then we can write
KW + ∆W =p∗(KX + ∆) +F and
KW + ∆W =q∗(KX0 + ∆0) +G
whereF is effective andp-exceptional, andGis effective andq-exceptional.
Therefore,
X0 = ProjS⊕
m≥0
π∗0OX0(bm(KX0 + ∆0)c)
= ProjS⊕
m≥0
(π0◦q)∗OW(bm(KW + ∆W)c)
= ProjS⊕
m≥0
π∗OX(bm(KX + ∆)c).
This is the desired description ofX0.
Lemma 4.8.3. Let (X,∆) be a log canonical pair and let π:X →S be a proper morphism onto a variety. Let(Xm,∆m)be a minimal model of (X,∆) overS and let (Xlc,∆lc) be a log canonical model of (X,∆) over S. Then there is a natural morphism α:Xm →Xlc such that
KXm+ ∆m =α∗(KXlc+ ∆lc).
In particular, KXm+ ∆m is semi-ample overS, that is, (Xm,∆m) is a good minimal model of (X,∆) over S.
Proof. We take a common resolution X oo p W r //
q
Xlc
Xm
4.8. MMP FOR LC PAIRS 139
ofX,Xm, and Xlc. LetE be the sum of allp-exceptional divisors. We put ∆W =p−∗1∆ +E. Then we have
KW + ∆W =q∗(KXm + ∆m) +F and
KW + ∆W =r∗(KXlc+ ∆lc) +G
whereF is effective andq-exceptional andGis effective andr-exceptional by the negativity lemma (see Lemma 2.3.26). Therefore, we obtain
q∗(KXm + ∆m) +F =r∗(KXlc+ ∆lc) +G.
Note that q∗(G−F) is effective and −(G−F) is q-nef. This implies G−F ≥ 0 by the negativity lemma (see Lemma 2.3.26). Similarly, r∗(F −G) is effective and−(F −G) isr-nef. This impliesF −G≥0 by the negativity lemma (see Lemma 2.3.26) again. Therefore,F =G.
So, we have
q∗(KXm+ ∆m) = r∗(KXlc+ ∆lc).
We assume that r ◦q−1 : Xm 99K Xlc is not a morphism. Then we can find a curve C on W such that q(C) is a point and that r(C) is a curve. In this case,
0 =C·q∗(KXm+ ∆m) = C·r∗(KXlc+ ∆lc)>0.
This is a contradiction. Therefore, α : r ◦q−1 : Xm 99K Xlc is a morphism and KXm+ ∆m =α∗(KXlc+ ∆lc).
By the proof of Lemma 4.8.3, we have:
Corollary 4.8.4. Let (X,∆) be a log canonical pair and let π : X →Sbe a proper morphism onto a variety. Let(X1,∆1)and(X2,∆2) be log canonical models of(X,∆) overS. Then (X1,∆1)is isomorphic to (X2,∆2) over S. Therefore, the log canonical model of (X,∆) over S is unique.
In order to discuss the minimal model program for log canonical pairs, it is convenient to use the following definitions of minimal models and Mori fiber spaces due to Birkar–Shokurov.
Definition 4.8.5 (Minimal models, see [Bir4, Definition 2.1]). Let (X,∆) be a log canonical pair and let π :X → S be a projective morphism onto a variety S. let φ :X 99KY be a birational map over S. We put ∆Y =∆ +e E where ∆ is the birational transform of ∆ one Y and E is the reduced exceptional divisor of φ−1, that is,E =∑
jEj whereEj is a prime divisor on Y which is exceptional overX for every j. We assume that
(i) (Y,∆Y) is a Q-factorial dlt pair and Y is projective over S,
140 4. MINIMAL MODEL PROGRAM
(ii) KY + ∆Y is nef over S, and
(iii) for any prime divisorE onX which is exceptional overY, we have
a(E, X,∆)< a(E, Y,∆Y).
Then (Y,∆Y) is called aminimal model of(X,∆)overS. Furthermore, ifKY + ∆Y is semi-ample overS, then (Y,∆Y) is called agood minimal model of (X,∆) over S.
Remark 4.8.6. By the same argument as Lemma 4.3.2, we can prove that a(E, X,∆)≤a(E, Y,∆Y) for every prime divisorE overX in Definition 4.8.5. Therefore, if (X,∆) is plt in Definition 4.8.5, then (Y,∆Y) is plt and φ−1 has no exceptional divisors.
Definition 4.8.7 (Mori fiber spaces, see [Bir4, Definition 2.2]). Let (X,∆) be a log canonical pair and let π :X → S be a projective morphism onto a variety S. let φ :X 99KY be a birational map over S. We put ∆Y =∆ +e E where ∆ is the birational transform of ∆ one Y and E is the reduced exceptional divisor of φ−1, that is,E =∑
jEj whereEj is a prime divisor on Y which is exceptional overX for every j. We assume that
(i) (Y,∆Y) is a Q-factorial dlt pair and Y is projective over S, (ii) there is a (KY+∆Y)-negative extremal contractionϕ :Y →Z,
that is,−(KY+∆Y) isϕ-ample,ρ(Y /Z) = 1, andϕ∗OY ' OZ, overS with dimY >dimZ, and
(iii) we have
a(E, X,∆)≤a(E, Y,∆Y).
for any prime divisorE overX and strct inequality holds ifE is on X and φ contracts E.
Then (Y,∆Y) is called a Mori fiber space of (X,∆) over S.
Let us quickly recall some results in [Bir4] and [HaX1]. For the details, see the original papers [Bir4] and [HaX1].
Theorem 4.8.8 (cf. [Bir4, Theorem 1.1] and [HaX1, Theorem 1.6]). Let (X,∆) be a Q-factorial log canonical pair such that ∆ is a Q-divisor and let π : X → S be a projective morphism between quasi- projective varieties. Assume that there is an effective Q-divisor ∆0 on X such that (X,∆ + ∆0) is log canonical and KX + ∆ + ∆0 ∼Q,π 0.
Then (X,∆) has a Mori fiber space or a good minimal model over S.
As a direct consequence of Theorem4.8.8, we have:
Corollary 4.8.9. In Theorem4.8.8, we further assume thatKX+
∆ is π-big. Then (X,∆) has a log canonical model over S.
4.8. MMP FOR LC PAIRS 141
Proof. Let (Y,∆Y) be a good minimal model of (X,∆) over S.
Then ⊕
m≥0
π∗OX(bm(KX + ∆)c)'⊕
m≥0
πY∗OY(bm(KY + ∆Y)c) as OS-algebras (see Lemma 4.8.2), where πY :Y →S, and
⊕
m≥0
πY∗OY(bm(KY + ∆Y)c)
is a finitely generated OS-algebra since KY + ∆Y is a πY-semi-ample Q-divisor. We put
X0 = ProjS⊕
m≥0
π∗OX(bm(KX + ∆)c).
Then (X0,∆0), where ∆0 is the strict transform of ∆ on X0, is a log canonical model of (X,∆) over S by Lemma 4.8.2.
Corollary4.8.10 (cf. [Bir4, Corollary 1.2] and [HaX1, Corollary 1.8]). Let ϕ : (X,∆) → W be a log canonical flipping contraction associated to a(KX+∆)-negative extremal ray. Then the(KX+∆)-flip of ϕ: (X,∆)→W exists.
Proof. Since ρ(X/W) = 1, we may assume that ∆ is a Q-divisor by perturbing ∆ slightly. By taking an affine cover of W, we may assume that W is affine. Then we can find an effective Q-divisor ∆0 onX such that KX+ ∆ + ∆0 ∼Q,ϕ 0. Then (X,∆) has a log canonical model over W. It is nothing but a flip of ϕ: (X,∆)→W. Remark 4.8.11. By Corollary 4.8.10, log canonical flips always exist. On the other hand, log canonical flops do not always exist. For the details, see [F38, Section 7], where Koll´ar’s examples are described in details.
4.8.12 (MMP for Q-factorial log canonical pairs). Let (X,∆) be a Q-factorial log canonical pair and let f : X → S be a projective morphism onto a variety S. By Theorem 4.5.2 and Corollary 4.8.10, we can run the minimal model program for (X,∆) overS. This means that the minimal model program discussed in 4.3.5works by replacing dltwithlog canonical. Moreover, by Theorem4.5.2(6), we can run the minimal model program with scaling discussed in4.4.11forQ-factorial log canonical pairs. Note that the termination of the above minimal model programs is an important open problem of the minimal model theory (see Conjecture 4.3.6 and Lemma4.9.3 below).
We note the following well-known lemma.
142 4. MINIMAL MODEL PROGRAM
Lemma 4.8.13 (see [KoMo, Corollary 3.44]). Let (X,∆) be a dlt (resp. klt or lc)pair. Let g :X 99KX0 be either a divisorial contraction of a (KX + ∆)-negative extremal ray or a (KX + ∆)-flip. We put
∆0 =g∗∆. Then (X0,∆0) is also dlt (resp. klt or lc).
Proof. This lemma easily follows from Lemma2.3.27when (X,∆) is klt or lc. From now on, we treat the case when (X,∆) is dlt. Let Z ⊂ X be as in Proposition 2.3.20. We put Z0 = g(Z)∪Exc(g−1) such that Exc(g−1) is the closed subset of X0 where g−1 is not an isomorphism. Then X0\Z0 is isomorphic to an open subset of X\Z.
Therefore,X0\Z0 is smooth and ∆0|X0\Z0 has a simple normal crossing support. LetE be an exceptional divisor overX0 such thatcX0(E), the center of E onX0, is contained in Z0. Then cX(E), the center of E on X, is contained inZ ∪Exc(g), where Exc(g) is the closed subset of X where g is not an isomorphism. We have
a(E, X0,∆0)≥a(E, X,∆) ≥ −1
by Lemma2.3.27. IfcX(E) is contained inZ, then the second inequality is strict by the definition of dlt pairs. If cX(E) is contained in Exc(g), then the first inequality is strict by Lemma 2.3.27. Anyway, (X0,∆0)
is dlt by Proposition 2.3.20.
We also note the following easy two propositions.
Proposition 4.8.14 (cf. [KoMo, Proposition 3.36]). Let (X,∆) be a Q-factorial log canonical pair and let π : X → S be a projective morphism. Let ϕR :X →Y be the contraction of a (KX+ ∆)-negative extremal ray R of N E(X/S). Assume that ϕR is either a divisorial or a Fano contraction. Then we have
(i) Y is Q-factorial, and (ii) ρ(Y /S) = ρ(X/S)−1.
Proof. This proposition directly follows from Corollary 4.5.3 and
Corollary 4.5.4.
Remark 4.8.15. If ϕR: X → Y is a Fano contraction in Proposi- tion4.8.14, then we know thatY has only log canonical singularities by [F38]. We further assume that X has only log terminal singularities.
Then Y has only log terminal singularities. For the details and some related topics, see [F38].
Proposition 4.8.16 (cf. [KoMo, Proposition 3.37]). Let (X,∆) be a Q-factorial log canonical pair and let π : X → S be a projective
4.8. MMP FOR LC PAIRS 143
morphism. LetϕR:X →W be the flipping contraction of a(KX+ ∆)- negative extremal rayR ofN E(X/S)and let ϕ+R :X+→W be the flip.
X
ϕR
A
AA AA AA A
π
00
0000 0000 0000 0
φ_ _ _//
_ _ _
_ X+
ϕ+R
}}zzzzzzzz
W
S Then we have
(i) X+ is Q-factorial, and (ii) ρ(X+/S) =ρ(X/S).
Proof. By perturbing ∆ slightly, we may assume that ∆ is a Q- divisor. Sine φ : X 99K X+ is an isomorphism in codimension one, it induces a natural isomorphism between the group of Weil divisors on X and the group of Weil divisors onX+. Let D+ be a Weil divisor on X+ and let D be the strict transform of D+ on X. Then there is a rational number r such that
(((D+r(KX + ∆))·R) = 0.
We take a positive integer m such thatm(D+r(KX + ∆)) is Cartier.
By Theorem 4.5.2 (4), there is a Cartier divisor DW on W such that m(D+r(KX + ∆))∼ϕ∗RDW. Thus we obtain that
mD+ =mφ∗D∼(ϕ+R)∗DW −(mr)(KX+ + ∆+)
is Q-Cartier. This means that X+ is Q-factorial. It is easy to see that ρ(X/S) =ρ(X+/S) by the above argument.
4.8.17 (Conjectures concerning MMP for lc pairs). The following conjecture is one of the most important open problems of the minimal model program for log canonical pairs.
Conjecture 4.8.18. Let (X,∆) be a projective log canonical pair such that ∆ is a Q-divisor on X. Then the log canonical ring
R(X,∆) =⊕
m≥0
H0(X,OX(bm(KX + ∆)c)) is a finitely generated C-algebra.
It is known that Conjecture 4.8.18 holds when dimX ≤ 4. When dimX ≥5, Conjecture 4.8.18 is still an open problem.
144 4. MINIMAL MODEL PROGRAM
Theorem 4.8.19 (cf. [F22, Theorem 1.2]). Let (X,∆) be a pro- jective log canonical pair such that ∆ is a Q-divisor with dimX ≤ 4.
Then the log canonical ring
R(X,∆) =⊕
m≥0
H0(X,OX(bm(KX + ∆)c)) is a finitely generated C-algebra.
Let us recall the good minimal model conjecture.
Conjecture4.8.20 (Good minimal model conjecture). Let(X,∆) be aQ-factorial projective dlt pair and let∆be anR-divisor. IfKX+∆
is pseudo-effective, then (X,∆) has a good minimal model.
In [FG2], we obtained:
Theorem 4.8.21. Conjecture 4.8.18 with dimX = n and Conjec- ture 4.8.20 with dimX ≤n−1 are equivalent.
Moreover, in [FG2], we proved:
Theorem 4.8.22. Conjecture 4.8.20 with dimX ≤n−1 is equiv- alent to Conjecture 4.8.23 with dimX =n.
Conjecture 4.8.23. Let(X,∆)be a Q-factorial projective plt pair such that∆is aQ-divisor onX and thatb∆cis irreducible. We further assume that KX + ∆ is big. Then the log canonical ring
R(X,∆) =⊕
m≥0
H0(X,OX(bm(KX + ∆)c)) is a finitely generated C-algebra.
Therefore, Conjecture 4.8.18 is equivalent to Conjecture 4.8.23 by Theorems 4.8.21 and 4.8.22.
Let us recall some related conjectures. For the details, see [FG1]
and [FG2].
Conjecture4.8.24 (Non-vanishing conjecture).LetXbe a smooth projective variety. If KX is pseudo-effective, then there exists some ef- fective Q-divisor D such that KX ∼Q D.
Remark 4.8.25. Let X be a smooth projective variety. Then KX is pseudo-effective if and only ifX is not uniruled by [BDPP]. For the proof, see, for example, [La2, Corollary 11.4.20].
Conjecture 4.8.26 (DLT extension conjecture, see [DHP, Con- jecture 1.3] and [FG2, Conjecture G]). Let (X,∆) be a projective