Corollary 1.2 follows from Theorem 1.1 easily (see the end of this section). We divide the proof of Theorem 1.1 into two cases. One case is when every lc centre of (X/Z,B) is vertical/Z (the vertical case) and the other case is when some lc centre is horizontal/Z (the horizontal case). Each case needs a different kind of argument. Here we give a brief account of the main ideas (a very similar line of thought is used to prove Theorem1.4).
Suppose that (X/Z,B+A) is as in the statement of Theorem 1.1. By taking a Q-factorial dlt blowup we can assume that(X/Z,B)isQ-factorial dlt. First note that if
A intersects the generic fibre of X→Z, then KX+B is not pseudo-effective/Z and the result follows from [9]. So, we could assume that A is vertical/Z.
Now assume that every lc centre of(X/Z,B)is vertical/Z which means thatB is vertical/Z. By Theorem 4.1, the termination statement in Theorem1.1(3) holds for (X/Z,B) if statement (1) holds. Thus, we only need to construct a good log minimal model of (X/Z,B). Pick a sequence t1>t2>· · · of sufficiently small positive rational numbers such that limi→∞ti =0. Then, (X/Z,B−tiB ) is a klt pair for each i. So, by Theorem 5.2, for each i, we get a good log minimal model(Yi/Z,BYi −tiBYi )of (X/Z,B−tiB )by running some LMMP/Z on KX+B−tiB .
Assume that Yi=Yi+1 for every i0 and let Y be this common model. Then, a simple calculation on log discrepancies show that(Y/Z,BY)is actually a weak lc model of(X/Z,B)from which we get a log minimal model as in Corollary3.7. In general the Yi
may be different but at least we could assume that they are all isomorphic in codimension one. In particular, if BY1 =0, then we could replace each Yi with Y1 and proceed as before. We may then assume thatBY1 =0.
Since KY1+BY1−t1BY1 is semi-ampleness/Z, KY1+BY1−t1BY1 ∼QCY1/Z
for some CY1 ≥0 so that KY1 +BY1+CY1 is lc. If ti=t1−tti i and i=1, then KY1+BY1+tiCY1 ∼Q
1+ti
KY1+BY1−tiBY1
/Z
We can make sure that ti≤1 for every i>1. Next we run an LMMP/Z on KY2+BY2with scaling of t2CY2. We use special termination (see Lemma6.1) to show that the LMMP terminates with a model Y . We can proceed as before by replacing each Yi with Y, for i0 (see Proposition6.2for more details). Finally, we will use Theorem1.7to show that the log minimal model we have constructed is actually good (see Proposition6.3).
Now the horizontal case: assume that some lc centre of(X/Z,B)is horizontal/Z, that is, some component ofB is horizontal/Z. The above arguments do not work since KX+B−tiB is not pseudo-effective/Z so we do not have a log minimal model of (X/Z,B−tiB ). However, (X/Z,B−tiB )has a Mori fibre space. Assume that for some i we already have a Mori fibre structure on X, that is, we have a(KX+B−tiB )-negative extremal contraction g:X→T/Z which is of fibre type, that is, dim X>dim T.
The condition KX+B+A∼Q 0/Z and the pseudo-effectivity of KX+B ensures that KX+B∼Q0/T, in particular, KX+B∼Qg∗M/Z for someQ-Cartier divisor M on T.
Now there is a component ofB , say S, which maps onto T sinceB is ample/T. By applying induction to
KS+BS+AS:=(KX+B+A)|S
we deduce that (S/Z,BS) has a good log minimal model hence the algebra R(S/Z, KS+BS) is finitely generated over OZ. It turns out that this implies that R(T/Z,M)
is finitely generated (see Lemma6.4) henceR(X/Z,KX+B)is also finitely generated.
Next, we can apply Theorem5.1(see Lemma6.5). In general, there may not be any Mori fibre structure on X but there is such a structure on some birational model of X which can be used in a somewhat similar way (see Proposition6.6and Lemma6.7).
The vertical case. — In this subsection, we deal with Theorem 1.1 in the vertical case, i.e. when every lc centre of(X/Z,B)is vertical/Z, in particular, when X→Z is bi-rational. First we prove a kind of special termination which will enable us to do induction on dimension. It is helpful to recall Remarks2.9and2.10before reading the proof of the next result.
Lemma6.1. — Let(X/Z,B+A)be a lc pair of dimension d as in Theorem 1.1such that (X/Z,B)satisfies the assumptions of Lemma5.5. Then, assuming Theorem1.1in dimension d−1, there is an LMMP/Z on KX+B with scaling of cC which terminates.
Proof. — By Lemma5.5, there is an LMMP/Z on KX+B with scaling of cC such that λ=limi→∞λi=0. We can assume that the LMMP consists of only log flips. Let (X/Z,B)be aQ-factorial dlt blowup of(X/Z,B)and let Cbe the birational transform of C. By Remark2.9(1), (2), we can lift the above LMMP/Z on KX+B to an LMMP/Z on KX +Bwith scaling of cC. We could assume that the latter LMMP consists of only log flips.
Let S be a component of B and let T be the normalisation of the image of S in Z. Put KS+BS:=(KX+B)|S and let(S,BS)be aQ-factorial dlt blowup of(S,BS).
By construction,
KS +BS+AS ∼Q0/T
where AS is the pullback of A. By Remark 2.10, we may assume that the LMMP on KX+Binduces an LMMP/T on KS+BS with scaling of cCS where CS is the pullback of C.
Sinceλ=limi→∞λi=0, KS +BS is pseudo-effective/T which means that AS is vertical/T. By induction,(S/T,BS)has a log minimal model. Thus, by Theorem4.1, the above LMMP/T on KS+BS terminates. Thus, the LMMP on KX+B terminates near S. The same argument applied to each component ofB shows that the LMMP on KX+Bterminates nearB .
By assumptions, KX+B∼QP+C/Z where P≥0 and Supp P=SuppB . More-over, SuppB contains all the lc centres of (X/Z,B). Thus, there is a Q-divisor P≥0 with Supp P=SuppB such that KX +B∼QP+C/Z. So, each extremal ray con-tracted by the LMMP on KX +B intersects P. But the LMMP terminates near B which in turn implies that it terminates near Ptherefore it terminates everywhere.
Next we use the above special termination to derive parts (1)and (3) of Theo-rem1.1, in the vertical case, from Theorem1.1in lower dimensions. The semi-ampleness statement in part(2)will be proved afterwards using Theorem1.7.
Proposition6.2. — Theorem1.1in dimension d−1 implies Theorem1.1(1), (3) in dimension d in the vertical case, i.e. when every lc centre of(X/Z,B)is vertical/Z.
Proof.
Step 1. After taking a Q-factorial dlt blowup using Corollary 3.6 we may assume that (X/Z,B)isQ-factorial dlt, in particular,B is vertical/Z. Run an LMMP/Z on KX+B with scaling of an ample/Z divisor. If KX+B is not pseudo-effective/Z, then the LMMP ends up with a Mori fibre space by [9]. So, from now on we assume that KX+B is pseudo-effective/Z. By Theorem 4.1, the LMMP terminates if we show that (X/Z,B) has a log minimal model. Since KX+B∼Q−A/Z, A|F=0 where F is the generic fibre of f : X→Z. Thus, A is vertical/Z and(KX+B)|F∼Q0.
Step 2. Let t1 > t2 >· · · be a sequence of sufficiently small rational numbers with limi→∞ti=0. Each(X/Z,B−tiB )is klt and(KX+B−tiB )|F∼Q0 hence by The-orem5.2each(X/Z,B−tiB )has a good log minimal model(Yi/Z,BYi−tiBYi )so that YiX does not contract divisors. Moreover, SuppBYi contains all the lc cen-tres of(Yi/Z,BYi)because(Yi/Z,BYi−tiBYi )is klt which means that(Yi/Z,BYi)is klt outside SuppBYi . Now, since A andB are vertical/Z, there are vertical/ZQ-divisors M,N≥0 such that
KX+B∼Q−A∼QM/Z and − B ∼QN/Z and
KX+B−tiB ∼QM+tiN/Z
So, any prime divisor contracted by XYi is a component of M+N hence after replacing the sequence with a subsequence we can assume that the maps XYi
contract the same divisors, i.e. Yi are isomorphic in codimension one.
Step 3. Assume that Yi=Yi+1 for i0, and let Y be this common model. Since KY+ BY−tiBY is nef/Z for each i0, KY+BYis also nef/Z. For any prime divisor D on X and each i0, we have
a(D,X,B)≤a
D,X,B−tiB
≤a
D,Y,BY−tiBY
which implies that
a(D,X,B)≤ lim
i→∞a
D,Y,BY−tiBY
=a(D,Y,BY)
Therefore, (Y/Z,BY) is a weak lc model of (X/Z,B), and by Corollary 3.7, we can construct a log minimal model of(X/Z,B)as required.
Step 4. Assume thatBY1 =0. Then, KY1 +BY1 −tiBY1 is nef/Z for each i, and since Y1Yi is an isomorphism in codimension one, we can replace each Yi with Y1and then apply Step 3.
From now on we assume thatBY1 =0. Since(Y1/Z,BY1−t1BY1 )is klt and KY1 + BY1 −t1BY1 is nef/Z, KY1+BY1−t1BY1 is semi-ample/Z by Theorem5.2. Thus,
KY1+BY1−t1BY1 ∼QCY1/Z
for some CY1 ≥0 such that KY1 +BY1 +CY1 is lc, in particular, Supp CY1 does not contain any lc centre of(Y1/Z,BY1). For each rational number t we have
KY1+BY1+tCY1 ∼Q(1+t) KY1+BY1− tt1
1+tBY1
Z In particular, if ti=t1−tti i and i=1, then
KY1+BY1+tiCY1 ∼Q
1+ti
KY1+BY1−tiBY1
/Z
We may assume that 2ti ≤ t1 for each i > 1 which implies that ti ∈ [0,1]. So, (Yi/Z,BYi +tiCYi)is a weak lc model of(Y1/Z,BY1 +tiCY1)for each i>1. Note that (Y1/Z,BY1 +tiCY1)is lc because ti≤1.
Step 5. By construction,
KY2+BY2+AY2 ∼Q0/Z
and every lc centre of(Y2/Z,BY2)is contained inBY2 because(Y2/Z,BY2 −t2BY2 ) is klt. On the other hand, since A is vertical/Z, AY2 is also vertical/Z, and if we put PY2 =t1BY2 , then by construction
KY2+BY2∼QPY2 +CY2/Z
Now KY2 +BY2 +t2CY2 is nef/Z, so by Lemma 6.1, we can run an LMMP/Z on KY2+BY2with scaling of t2CY2which terminates on a model Y on which KY+BY+δCY
is nef/Z for any sufficiently smallδ≥0. Since Y2Yiis an isomorphism in codimen-sion one, KYi +BYi +tiCYi is nef/Z, and limi→∞ti =0, we deduce that KY2 +BY2 is (numerically) a limit of movable/Z R-divisors. So, the LMMP does not contract any divisors, i.e. YYiis an isomorphism in codimension one.
For any i0, KY+BY+tiCY is nef/Z which in turn implies that KY+BY−tiBY
is also nef/Z. Since Y Yi is an isomorphism in codimension one and since (Yi/Z,BYi−tiBYi )is a log minimal model of(X/Z,B−tiB ),(Y/Z,BY−tiBY )is a log minimal model of(X/Z,B−tiB ), for every i0. By Step 3, we are done.
Proposition6.3. — Theorem1.1in dimension d −1 implies Theorem1.1in dimension d in the vertical case, i.e. when every lc centre of(X/Z,B)is vertical/Z.
Proof. — After taking aQ-factorial dlt blowup using Corollary3.6we may assume that(X/Z,B)isQ-factorial dlt, in particular, B is vertical/Z. By Proposition6.2, any LMMP/Z on KX+B with scaling of an ample/Z divisor ends up with a Mori fibre space or a log minimal model(Y/Z,BY). Assume that (Y/Z,BY)is a log minimal model. Let g: W→X and h: W→Y be a common resolution, and let AY:=h∗g∗A. Then,
KY+BY+AY=h∗g∗(KX+B+A)∼Q0/Z
where we use the fact that h∗g∗(KX+B)=KY+BYwhich in turn follows from the fact that g∗(KX+B)−h∗(KY+BY)is exceptional/Y, by Remark2.6. Also,(Y/Z,BY+AY) is lc because
h∗(KY+BY+AY)=g∗(KX+B+A)
So, by replacing(X/Z,B+A)with(Y/Z,BY+AY)we may assume that KX+B is nef/Z.
It remains to prove that KX+B is semi-ample/Z.
Run an LMMP/Z on KX+B−B with scaling of some ample/Z divisor, for some sufficiently small rational number >0. Since KX+B is nef/Z and KX+B+A∼Q
0/Z, (KX+B)|F ∼Q 0 where F is the generic fibre of X→Z. Moreover, since B is vertical/Z,
KX+B−B
F∼Q0
So, by Theorem5.2, the LMMP terminates on a model Xon which KX+BX−BX
is semi-ample/Z. On the other hand, by [6, Proposition 3.2], KX+B is numerically trivial on each extremal ray contracted by the LMMP. Therefore, KX+BXis also nef/Z.
Another application of Theorem5.2shows that KX+BX−δBX is semi-ample/Z for anyδ∈(0, ]. In addition, XXis an isomorphism over the generic point of Z.
The pair (X/Z,BX) is Q-factorial and lc and (X/Z,BX −BX ) is klt. If BX =0 we are done so we can assume thatBX =0. Let(Y/Z,BY)be aQ-factorial dlt blowup of(X/Z,BX)and let S be a component of T:= BY . Let AYbe the pullback of AX. Then, from
KX+BX+AX∼Q0/Z we get
KY+BY+AY∼Q0/Z which implies that
KS+BS+AS∼Q0/Z
where KS +BS :=(KY+BY)|S and AS :=AY|S. Now, by induction, KS+BS is semi-ample/Z. On the other hand, if P:=e∗BX and ifδ >0 is any sufficiently small rational number, where e is the morphism Y→X, then
KY+BY−δP=e∗
KX +BX −δBX
is semi-ample/Z. Now, by Theorem1.7, KY+BYis semi-ample/Z hence KX+B is also
semi-ample/Z.
The horizontal case. — In this subsection, we deal with the horizontal case of The-orem1.1, that is, when some lc centre of (X/Z,B)is horizontal/Z. First, we need the following result on finite generation of algebras.
Lemma6.4. — Let f : X→Y/Z be a surjective morphism of normal varieties, projective over an affine variety Z=Spec R, and L a Cartier divisor on Y. If R(X/Z,f∗L)is a finitely generated R-algebra, then R(Y/Z,L)is also a finitely generated R-algebra.
The lemma can be easily derived from the deep fact thatOYsplits f∗OX. However, a simpler proof (suggested by the referee and independently by Kollár and Totaro) uses only basic commutative algebra: R(X/Z,f∗L)is integral over R(Y/Z,L)so the latter is finitely generated since the former is assumed to be finitely generated (cf. [4, Proposi-tion 7.8]). Okawa [27, Theorem 4.1] proves a more general statement.
We will reduce the horizontal case to the next lemma by finding a suitable Mori fibre space.
Lemma6.5. — Assume Theorem1.1in dimension d−1. Let(X/Z,B+A)be of dimension d as in Theorem1.1 such that (KX+B)|F ∼Q 0 for the generic fibre F of f : X→Z. Moreover, assume that there is a contraction g: X→T/Z such that
(1) KX+B∼Q0/T,
(2) some lc centre of(X/Z,B)is horizontal over T.
Then,(X/Z,B)has a good log minimal model.
Proof. — By replacing(X/Z,B)with aQ-factorial dlt blowup, we can assume that (X/Z,B)isQ-factorial dlt and that there is a component S ofB which is horizontal/T.
Run an LMMP/Z on KX+B with scaling of some ample/Z divisor. Since termination and semi-ampleness/Z are local on Z, we can assume that Z is affine, say Spec R. By Theorem 4.1, the LMMP terminates with a good log minimal model if we prove that (X/Z,B)has a good log minimal model.
By adjunction define KS+BS:=(KX+B)|S and let AS:=A|S. Then, KS+BS+AS=(KX+B+A)|S∼Q0/Z
and KS+BS∼Q0/T. Moreover, (KS+BS)|H∼Q0 where H is the generic fibre of the induced morphism h:S→Z.
Since we are assuming Theorem1.1in dimension d−1,(S/Z,BS)has a good log minimal model. Therefore, if I(KS+BS)is Cartier for some I∈N, then R(S/Z,I(KS+ BS))is a finitely generated R-algebra (cf. [7]). We can choose I such that I(KX+B) is Cartier and such that I(KX+B)∼g∗L for some Cartier divisor L on T. So, I(KS+BS)∼ e∗L where e: S →T is the induced morphism. Now, by Lemma 6.4, R(T/Z,L) is a finitely generated R-algebra which in turn implies that R(X/Z,I(KX+B))is a finitely generated R-algebra since X→T is a contraction. Therefore, R(X/Z,KX+B) is a finitely generated R-algebra and according to Theorem 5.1, (X/Z,B) has a good log
minimal model.
Proposition 6.6. — Assume Theorem1.1in dimension d −1, and assume Theorem1.1in dimension d in the vertical case. Then, Theorem 1.1holds in dimension d in the horizontal case, i.e.
when some lc centre of(X/Z,B)is horizontal over Z.
Proof. — We can assume that(X/Z,B)is Q-factorial dlt and that f : X→Z is a contraction. By assumptions, some component ofB is horizontal/Z. Run an LMMP/Z on KX+B with scaling of some ample/Z divisor. If A is not vertical/Z, then KX+B is not pseudo-effective/Z hence the LMMP terminates with a Mori fibre space by [9]. So, we can assume that A is vertical/Z hence(KX+B)|F∼Q0 where F is the generic fibre of f : X→Z. By Theorem4.1, the LMMP terminates with a good log minimal model if we show that(X/Z,B)has a good log minimal model.
If > 0 is a sufficiently small rational number, then (X/Z,B −B ) is klt and KX +B −B is not pseudo-effective/Z. Thus, by [9] there is a Mori fibre space (Y/Z,BY−BY ) for (X/Z,B−B ) obtained by running an LMMP/Z on KX+B−B . Let g: Y→T/Z be the KY+BY−BY -negative extremal contrac-tion which defines the Mori fibre space structure, and let R be the corresponding extremal ray. Since A is vertical/Z, AYis vertical over T. So, from
KY+BY+AY∼Q0/Z
we deduce that (KY+BY)·R=0 which in turn implies that KY+BY∼Q0/T. Since KY+BY−BY is numerically negative/T, BY =0. We could apply Lemma 6.5 to get a good log minimal model of (Y/Z,BY)but this does not give a good log mini-mal model of(X/Z,B)because we cannot easily compare the singularities of(Y/Z,BY) and(X/Z,B). We need to find a model which is closely related to both(Y/Z,BY)and (X/Z,B).
Let(W/Z,BW)be a log smooth model of(X/Z,B)of type (1) as in Definition2.3 such that W dominates Y, say by a morphism h: W→Y. We can write
KW+BW=h∗(KY+BY)+G
where G is exceptional/Y. Run the LMMP/Y on KW+BWwith scaling of some ample/Y divisor. By Theorem 3.5, at some step of this LMMP we reach a model X on which GX ≤0. Also,
KX+BX−GX ∼Q0/T
as it is the pullback of KY+BY. Since KX+B is pseudo-effective/Z, KW+BW and KX+BX are pseudo-effective over both Z and T. Thus, GX is vertical/T hence KX +BX ∼Q 0 over the generic point of T. By construction,BY is ample/T hence in particular some of its components are horizontal/T. This implies that some compo-nent ofBX is horizontal/T becauseBY is the pushdown ofBX .
Now by Lemma6.7below, we can run an LMMP/T on KX+BX which ends up with a model X on which KX+BX is semi-ample/T (to be more precise, in place of X,Y,Z,B,A in the lemma use X, Y, T, BX,−GX respectively). Since KX +BX∼Q0 over the generic point of T, XXis an isomorphism over the generic point of T. So, if X→T/T is the contraction associated to KX+BX, then T→T is birational and some component ofBX is horizontal/T.
On the other hand, if we denote X→Y by e, then we can write KX+BX+AX=e∗(KY+BY+AY)
by taking AX := −GX+e∗AY. Thus, KX+BX+AX∼Q0/Z and
KX+BX+AX∼Q0/Z
where AX is the birational transform of AX. Since KX +BX is pseudo-effective/Z, AX is vertical/Z and KX +BX ∼Q 0 over the generic point of Z. Now we can apply Lemma6.5(by taking X,Z,T,B,A in the lemma to be X,Z,T,BX,AX respectively) to get a good log minimal model of (X/Z,BX)which would give a good log minimal model of(W/Z,BW)hence of(X/Z,B), by Remark2.8.
Lemma6.7. — Assume Theorem1.1in dimension d in the vertical case. Let(X/Z,B+A) be of dimension d as in Theorem1.1such that(KX+B)|F∼Q0 for the generic fibre F of f :X→Z.
Moreover, assume that
• (X/Z,B)isQ-factorial dlt,
• there exist contractions e: X→Y and g:Y→Z,
• e is birational, A is exceptional/Y, and Y isQ-factorial,
• D:=KY+BY−BY is klt and g is a D-negative extremal contraction where >0 and BY:=e∗B.
Then, any LMMP/Z on KX+B with scaling of an ample/Z divisor terminates with a good log minimal model of(X/Z,B).
Proof. — By Theorem 4.1, it is enough to prove that (X/Z,B) has a good log minimal model. By adding a small multiple of A to B we can assume that Supp A⊆ Supp B. Since e is birational, by Theorem 1.1 in dimension d in the vertical case, we can assume that KX+B is actually semi-ample/Y. By replacing X with the lc model of (X/Y,B)we can assume that KX+B is ample/Y (we may loose theQ-factorial dlt property of(X/Z,B)but we will recover it later). Since−A is ample/Y, Supp A contains all the prime exceptional/Y divisors on X. In particular,
Supp e∗BY ⊆
SuppB ∪Supp A
⊆Supp B
By replacing(X/Z,B)with aQ-factorial dlt blowup, we can again assume that(X/Z,B) isQ-factorial dlt; note that this preserves the property Supp e∗BY ⊆Supp B.
Since A is exceptional/Y and KX+B+A∼Q0/Z, KY+BY∼Q0/Z. So,BY is ample/Z hence e∗BY is semi-ample/Z. Thus, for a small rational numberτ >0 we can write
KX+B=KX+B−τe∗BY +τe∗BY ∼QKX+/Z
where is some rational boundary such that(X/Z, )is klt: since (Y/Z,BY−BY ) is klt,BY contains all the lc centres of(Y/Z,BY), in particular, the image of all the lc centres of(X/Z,B); so, Supp e∗BY contains all the components ofB hence(X/Z,B− τe∗BY )is klt and we can indeed findwith the required properties. By Theorem5.2, we can run an LMMP/Z on KX+which ends up with a good log minimal model of
(X/Z, )hence a good log minimal model of(X/Z,B).
Proof of Theorem 1.1. — We argue by induction so in particular we may assume that Theorem1.1holds in dimension d −1. By Proposition6.3, Theorem1.1 holds in dimension d in the vertical case. On the other hand, by Proposition 6.6, Theorem 1.1
also holds in dimension d in the horizontal case.
Proof of Corollary 1.2. — First assume that B has rational coefficients. Since
−(KX+B)is ample/Y, we can find A∼Q−(KX+B)/Y such that A≥0,(X/Z,B+A)is lc, and
KX+B+A∼Q0/Y
Now by Theorem1.1,(X/Y,B)has a good log minimal model and its lc model gives the (KX+B)-flip. When B is not rational, we can find rational lc divisors KX+Bi and real numbers ri>0 such that
KX+B=
ri(KX+Bi) and ri=1
Moreover, we can assume that−(KX+Bi)is ample/Y for every i. For each i, j there is a rational number bi,j such that
KX+Bi≡bi,j(KX+Bj)/Y
By the cone theorem for lc pairs proved by Ambro [2] and Fujino [14], KX+Bi∼Qbi,j(KX+Bj)/Y
Therefore, there is a morphism X+→Y which gives the flip of KX+Bi for every i. The
morphism also gives the(KX+B)-flip.