In this section, we give the proof of Theorem1.4which is parallel to the arguments of section 6 with some small changes.
The vertical case. — First, we deal with Theorem1.4in the vertical case. It is helpful to recall Remarks2.9and2.10before reading the proof of the next special termination result.
Lemma 7.1. — Let (X/Z,B) be a lc pair of dimension d as in Theorem 1.4 such that it satisfies the assumptions of Lemma5.5. Then, assuming Conjecture1.3and Theorem1.4in 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 Remark 2.10, we may assume that the LMMP on KX +B induces an LMMP/T on KS+BS with scaling of cCS where CS is the pullback of C. On the other hand, by assumptions, there is a non-empty open subset U⊆Z such that KX+B∼Q0 over U and
such that the generic point of each lc centre of (X/Z,B)is mapped into U. Let V⊆T be the inverse image of U under T→Z. Then, KS+BS∼Q0 over V and the generic point of each lc centre of(S/T,BS)is mapped into V.
By induction,(S/T,BS)has a log minimal model. Sinceλ=limi→∞λi=0, by Theorem 4.1, the above LMMP/T on KS +BS terminates. Thus, the LMMP/Z on KX +B terminates near S. The same argument applied to each component of B shows that the LMMP/Z on KX+Bterminates nearB . By assumptions, KX+B∼Q
P+C/Z where P≥0 and Supp P=SuppB . Moreover, SuppB contains all the lc centres of(X/Z,B). Thus, there is aQ-divisor P≥0 with Supp P=SuppB such that KX +B∼QP+C/Z. So, each extremal ray contracted by the LMMP/Z on KX+B intersects P. But the LMMP terminates nearB which in turn implies that it terminates
near Ptherefore it terminates everywhere.
Proposition7.2. — Theorem1.4in dimension d−1 implies Theorem1.4(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. By Theorem4.1, the LMMP terminates if we prove that(X/Z,B)has a log minimal model. By assumptions, KX+B∼Q0 over some non-empty open subset U⊆Z and ifηis the generic point of any lc centre of(X/Z,B), then f(η)∈U.
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 where F is the generic fibre of f . Hence by Theorem5.2each(X/Z,B−tiB )has a good log minimal model(Yi/Z,BYi−tiBYi )so that YiX does not contract divisors. Since we are as-suming Conjecture1.3in dimension d, we can assume that each(Yi/Z,BYi)is lc. More-over, SuppBYi contains all the lc centres of(Yi/Z,BYi)because(Yi/Z,BYi −tiBYi ) is klt which means that(Yi/Z,BYi)is klt outside SuppBYi . Since KX+B∼Q0 over U and sinceB is vertical/Z, there are vertical/ZQ-divisors M,N≥0 such that
KX+B∼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+1for i0, and let Y be this common model. Since KY+ 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
Step 5. We will modify the situation so that if η is the generic point of any lc centre of (Y2/Z,BY2), thenηis mapped into U. Put PY2 =t1BY2 . Then,
KY2+BY2∼QPY2 +CY2/Z
By Lemma5.5, we can run an LMMP/Z on KY2 +BY2 with scaling of t2CY2 so that if λk are the numbers appearing in the LMMP, thenλ=limk→∞λk=0. Since Y2Yi
is an isomorphism in codimension 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. Now, by replacing the ti with a subsequence, we
can assume that for each i>2 there is k such that λk ≥ti ≥λk+1. If T is the variety corresponding toλk+1, then KT+BT+tiCTis nef/Z. By replacing Yiwith T, we could assume that Yi occurs in some step of the LMMP. Thus, we can assume that every Yi
occurs in some step of the LMMP if i>1.
After finitely many steps, the LMMP does not contract any lc centres. So, perhaps af-ter replacing (Y2/Z,BY2)with some (Yj/Z,BYj), with j0, we may assume that the
where the equality follows from the fact that Y2Yiis an isomorphism nearη. Then, 0=a(E,Y2,BY2)= lim
i→∞a
E,Y2,BY2−tiBY2
≥a(E,X,B) >0
which is a contradiction. So, from now on we can assume that ifηis the generic point of any lc centre of(Y2/Z,BY2), then η is mapped into U. Moreover, by construction, R-divisors. So, the LMMP does not contract any divisors, i.e. YYi is 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.
Proposition7.3. — Theorem1.4in dimension d −1 implies Theorem1.4in dimension d in the vertical case, i.e. when every lc centre of(X/Z,B)is vertical/Z.
Proof. — By Proposition 7.2, statements (1) and (3) of Theorem 1.4 hold for (X/Z,B), in particular, (X/Z,B)has a log minimal model(Y/Z,BY). By assumptions, KX+B∼Q0 over some non-empty open subset U⊆Z and if ηis the generic point of any lc centre of(X/Z,B)then f(η)∈U. This implies that KY+BY∼Q 0 over U and that ifη is the generic point of any lc centre of(Y/Z,BY)then ηis mapped into U. By replacing(X/Z,B)with(Y/Z,BY)we may assume that(X/Z,B)is Q-factorial dlt and 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. By [6, Proposition 3.2], KX+B is numer-ically trivial on each step of this LMMP. SinceB is vertical/Z,(KX+B−B )|F∼Q0 where F is the generic fibre of f . So, by Theorem5.2, the LMMP terminates on a model X on which KX +BX and KX +BX −BX are both nef/Z. Another application of Theorem5.2shows that KX+BX−δBX is semi-ample/Z for anyδ∈(0, ]. 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). By construction, KY+BY∼Q0 over U and ifηis the generic point of any lc centre of(Y/Z,BY)thenηis mapped into U.
Let S be a component of T:= BY and put KS+BS :=(KY+BY)|S. If Q is the normalisation of the image of S in Z and if V⊆Q is the inverse image of U, then KS+BS ∼Q0 over V and if η is the generic point of any lc centre of (S/Q,BS)then η is mapped into V. So, by induction, KS+BS is semi-ample/Q hence 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.
In the proof of Proposition7.3, it is also possible to compactify X and Z first before running the LMMP on KX+B−B ; in this way we would need Theorem1.7only when X,Z are projective.
The horizontal case. — In this subsection, we deal with the horizontal case of Theo-rem1.4.
Lemma7.4. — Assume Conjecture1.3and Theorem1.4in dimension d−1. Let(X/Z,B) be of dimension d as in Theorem1.4. 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 assumptions, KX+B∼Q 0 over some non-empty open subset U⊆Z and if η is the generic point of any lc centre of(X/Z,B)then f(η)∈U. By adjunction define KS+BS:=(KX+B)|S. Then, KS+BS∼Q0 over U and ifηis the generic point of any lc centre of(S/Z,BS)then f(η)∈U. Moreover, KS+BS∼Q0/T.
Since we are assuming Conjecture 1.3 and Theorem 1.4 in 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 di-visor L on T. So, I(KS +BS)∼e∗L where e: S→T is the induced morphism. Now, by Lemma6.4, 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
Theo-rem5.1,(X/Z,B)has a good log minimal model.
Proposition 7.5. — Assume Theorem 1.4in dimension d−1. Then, Theorem1.4holds 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. Moreover, KX+ B∼Q 0 over some non-empty open subset U⊆Z and ifη is the generic point of any lc centre of(X/Z,B)thenηis mapped into U. Run an LMMP/Z on KX+B with scaling of some ample/Z divisor. 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 contraction which defines the Mori fibre space structure, and let R be the corresponding extremal ray.
By construction, KY+BY∼Q 0 over U hence(KY+BY)·R=0 which in turn implies that KY+BY∼Q0/T. Since KY+BY−BY is numerically negative/T, BY =0.
As we are assuming Conjecture 1.3 in dimension d, we can assume that (Y/Z,BY) is lc.
Let(W/Z,BW)be a log smooth model of(X/Z,B)of type (2) as in Definition2.3 such that W dominates Y, say by a morphism h: W→Y. In particular, we may assume that ifηis the generic point of any lc centre of(W/Z,BW), thenηis mapped into U. 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 Theorem3.5, we reach a model Xon which GX ≤0. Moreover,
KX+BX−GX ∼Q0/T as it is the pullback of KY+BY.
Letπ be the given morphism W→X. By definition of log smooth models, we can write
KW+BW=π∗(KX+B)+E
where E is effective and exceptional/X. On the other hand, since KX+B∼Q 0 and KY+BY∼Q0 over U, we have π∗(KX+B)=h∗(KY+BY)over U which means that E=G over U. Therefore, every component of G with negative coefficient is mapped into Z\U hence GX is also mapped into Z\U. In particular, this means that KX+BX∼Q0 over U. Moreover, if η is the generic point of any lc centre of (X/Z,BX) then η is mapped into U because the same holds for(W/Z,BW). Let UT⊆T be the inverse image of U. Then, KX +BX ∼Q 0 over UT and if η is the generic point of any lc centre of (X/Z,BX)thenηis mapped into UT.
By construction,BY is ample/T hence in particular some of its components are horizontal/T. This implies that some component ofBX is horizontal/T becauseBY
is the pushdown ofBX . By Lemma7.6below, we can run an LMMP/T on KX+BX
which ends up with a model X on which KX+BX is semi-ample/T (more precisely, we should take X,Y,Z,B in the lemma to be our X,Y,T,BX respectively). Over UT, XX is an isomorphism. So, if X→T/T is the contraction associated to KX+ BX, then T→T is birational and some component ofBX is horizontal/T.
By construction, KX +BX∼Q0/T, and KX+BX∼Q0 over U. Moreover, if η is the generic point of any lc centre of (X/Z,BX) then η is mapped into U. Now we can apply Lemma7.4 (by taking X,Z,T,B in the lemma to be our X,Z,T,BX
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.
Lemma7.6. — Assume Conjecture1.3in dimension d and Theorem1.4in dimension d−1.
Let(X/Z,B)be of dimension d as in Theorem1.4. Moreover, assume that
• (X/Z,B)isQ-factorial dlt,
• there exist contractions e: X→Y and g:Y→Z,
• e is birational and Y isQ-factorial,
• KY+BYis lc, D:=KY+BY−BY is klt, and g is a D-negative extremal contraction for some >0 where 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 assumptions, KX+B∼Q0 over some non-empty open subset U⊆Z
and if η is the generic point of any lc centre of (X/Z,B) then f(η)∈U. Let UY⊆Y be the inverse image of U. Then, KX+B∼Q 0 over UY and ifη is the generic point of any lc centre of(X/Z,B)then e(η)∈UY. Since e is birational, by Proposition 7.3we 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 g is an extremal contraction and since KY+BY∼Q0 over U, KY+BY∼Q 0/Z. Moreover, since KX+B is ample/Y, there is an e-exceptionalQ-divisor A≥0 such that
KX+B+A=e∗(KY+BY)
and KX+B+A∼Q0/Z, and since KY+BYis lc, KX+B+A is lc too. Also, since−A is ample/Y, Supp A contains all the prime exceptional/Y divisors on X. Now we can add a small multiple of A to B and assume that Supp B contains all the prime exceptional/Y divisors on X. Note that A ∼Q 0 over UY hence A =0 over UY, so we still have KX+B∼Q0 over U after adding a small multiple of A to B. Now,
Supp e∗BY ⊆
SuppB ∪Supp A
⊆Supp B
Replacing(X/Z,B)with aQ-factorial dlt blowup, we can again assume that(X/Z,B)is Q-factorial dlt; the property Supp e∗BY ⊆Supp B is preserved.
By assumptions,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 of B hence (X/Z, B−τe∗BY )is klt and we can indeed findwith the required properties. By Theorem 5.2, 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 Theorem1.4. — This follows from Propositions7.3and7.5.
Acknowledgements
This work was carried out while the author was visiting Institut de Mathématiques de Jussieu (Paris VI) during the 2010–2011 academic year. The visit was financially and administratively supported by the Fondation Sciences Mathématiques de Paris. He would
like to thank these organisations for their incredible support and hospitality, and would like to thank Professor Claire Voisin who made the visit possible. He is grateful to the referee for the comments and suggestions that helped to improve this paper consider-ably.
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C. B.
DPMMS, Centre for Mathematical Sciences, Cambridge University,
Wilberforce Road,
Cambridge, CB3 0WB, UK c.birkar@dpmms.cam.ac.uk
Manuscrit reçu le 7 mai 2011 Version révisée le 12 février 2012 Manuscrit accepté le 4 avril 2012 publié en ligne le 3 mai 2012.