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On the existence of flips

We prove the existence of flips in dimension n, contingent on the termination of real flips in dimension n-1.


Introduction
The main result of this paper is: Theorem 1.1. Assume the real MMP in dimension n − 1.
Then flips exist in dimension n.
Here are two consequences of this result: Corollary 1.2. Assume termination of real flips in dimension n − 1 and termination of flips in dimension n.
Then the MMP exists in dimension n.
As Shokurov has proved, [13], the termination of real flips in dimension three, we get a new proof of the following result of Shokurov [14]: Corollary 1.3. Flips exist in dimension four.
Given a proper variety, it is natural to search for a good birational model. An obvious, albeit hard, first step is to pick a smooth projective model. Unfortunately there are far too many such models; indeed given any such, we can construct infinitely many more, simply by virtue of successively blowing up smooth subvarieties. To construct a unique model, or at least cut down the examples to a manageable number, we have to impose some sort of minimality on the birational model.
The choice of such a model depends on the global geometry of X. One possibility is that we can find a model on which the canonical divisor K X is nef, so that its intersection with any curve is non-negative.
The first author was partially supported by NSF research grant no: 0456363 and by a grant from the Sloan Foundation. We would like to thank Adebisi Agboola, Florin Ambro, Paolo Cascini and Alessio Corti for many useful comments on the contents of this paper.
Conjecturally this is equivalent to the condition that the Kodaira dimension of X is non-negative, that is there are global pluricanonical forms. Another possibility is that through any point of X there passes a rational curve. In this case the canonical divisor is certainly negative on such a curve, and the best one can hope for is that there is a fibration on which the anticanonical divisor is relatively ample. In other words we are searching for a birational model X such that either (1) K X is nef, in which case X is called a minimal model, or (2) there is a fibration X −→ Z of relative Picard number one, such that −K X is relatively ample; we call this a Mori fibre space. The minimal model program is an attempt to construct such a model step by step. We start with a smooth projective model X. If K X is nef, then we have case (1). Otherwise the cone theorem guarantees the existence of a contraction morphism f : X −→ Z of relative Picard number one, such that −K X is relatively ample. If the dimension of Z is less than the dimension of X, then we have case (2). Otherwise f is birational. If f is divisorial (that is the exceptional locus is a divisor), then we are free to replace X by Z and continue this process. Even though Z may be singular, it is not hard to prove that it is Q-factorial, so that any Weil divisor is Q-Cartier (some multiple is Cartier), and that X has terminal singularities. In particular it still makes sense to ask whether K X is nef, and the cone theorem still applies at this level of generality. The tricky case is when f is not divisorial, since in this case it is not hard to show that no multiple of K Z is Cartier, and it no longer even makes sense to ask if K Z is nef. At this stage we have to construct the flip.
Let f : X −→ Z be a small projective morphism of normal varieties, so that f is birational but does not contract any divisors. If D is any integral Weil divisor such that −D is relatively ample the flip of D, if it exists at all, is a commutative diagram where X X ′ is birational, and the strict transform D ′ of D is relatively ample. Note that f ′ is unique, if it exists at all; indeed if we set R = R(X, D) = n∈N f * O X (nD), then X ′ = Proj Z R. In particular the existence of the flip is equivalent to finite generation of the ring R.
It is too much to expect the existence of general flips; we do however expect that flips exist if D = K X + ∆ is kawamata log terminal. Supposing that the flip of D = K X exists, we replace X by X ′ and continue. Unfortunately this raises another issue, how do we know that this process terminates? It is clear that we cannot construct an infinite sequence of divisorial contractions, since the Picard number drops by one after every divisorial contraction, and the Picard number of X is finite. In other words, to establish the existence of the MMP, it suffices to prove the existence and termination of flips. Thus (1.1) reduces the existence of the MMP in dimension n, to termination of flips in dimension n.
In the following two conjectures, we work with either the field K = Q or R. Conjecture 1.4 (Existence of Flips). Let (X, ∆) be a kawamata log terminal Q-factorial pair of dimension n, where ∆ is a K-divisor. Let f : X −→ Z be a flipping contraction, so that −(K X + ∆) is relatively ample, and f is a small contraction of relative Picard number one.
Then the flip of f exists.
Conjecture 1.5 (Termination of Flips). Let (X, ∆) be a kawamata log terminal Q-factorial pair of dimension n, where ∆ is a K-divisor. Then there is no infinite sequence of (K X + ∆)-flips.
For us, the statement "assuming the (real) MMP in dimension n" means precisely assuming (1.4) Q,n and (1.5) R,n . In fact it is straightforward to see that (1.4) Q,n implies (1.4) R,n , see for example the proof of (7.2).
Kawamata, [6], proved the termination of any sequence of threefold, kawamata log terminal flips, that is he proved (1.5) Q,3 . As previously pointed out, Shokurov proved, [13], (1.5) R,3 . Further, Shokurov proved, [15], that (1.5) R,n , follows from two conjectures on the behaviour of the log discrepancy of pairs (X, ∆) of dimension n (namely acc for the set of log discrepancies, whenever the coefficients of ∆ are confined to belong to a set of real numbers which satisfies dcc, and semicontinuity of the log discrepancy). Finally, Birkar, in a very recent preprint, [1], has reduced (1.5) R,n , in the case when K X + ∆ has non-negative Kodaira dimension, to acc for the log canonical threshold and the existence of the MMP in dimension n − 1.
We also recall the abundance conjecture, Conjecture 1.6 (Abundance). Let (X, ∆) be a kawamata log terminal pair, where K X + ∆ is Q-Cartier, and let π : X −→ Z be a proper morphism, where Z is affine and normal.
If K X + ∆ is nef, then it is semiample.
Note that the three conjectures, existence and termination of flips, and abundance, are the three most important conjectures in the MMP. For example, Kawamata proved, [5], that these three results imply additivity of Kodaira dimension.
Our proof of (1.1) follows the general strategy of [14]. The first key step was already established in [12], see also [9] and [3]. In fact it suffices to prove the existence of the flip for D = K X + S + B, see (4.2), where S has coefficient one, and K X + S + B is purely log terminal.
The key point is that this allows us to restrict to S, and we can try to apply induction. By adjunction we may write where B ′ is effective and K S + B ′ is kawamata log terminal. In fact, since we are trying to prove finite generation of the ring R, the key point is to consider the restriction maps see (3.5). Here and often elsewhere, we will assume that Z is affine, so that we can replace f * by H 0 . Now if these maps were surjective, we would be done by induction. Unfortunately this is too much to expect. However we are able to prove, after changing models, that something close to this does happen.
The starting point is to use the extension result proved in [4], which in turn builds on the work of Siu [16] and Kawamata [7]. To apply this result, we need to improve how S sits inside X. To this end, we pass to a resolution Y −→ X. Let T be the strict transform of S and let Γ be those divisors of log discrepancy less than one. Then by a generalisation of (3.17) of [4], we can extend sections from T to Y , provided that we can find G ∈ |m(K Y + Γ)| which does not contain any log canonical centre of K Y + Γ .
In fact if we blow up more, we can separate all of the components of Γ, except the intersections with T . In this case, the condition on G becomes that it does not share any components with Γ. Thus, for each m we are able to cancel common components, and lift sections. Putting all of this together, see §5 and §6 for more details, we get a sequence of divisors Θ • on T , such that and it suffices to prove that this sequence stabilises, that is is constant for m sufficiently large and divisible. To this end, we take the limit Θ of this sequence. Then K T + Θ is kawamata log terminal, but in general since Θ is a limit, it has real coefficients, rather than just rational. Now there are two ways in which the sequence Θ • might vary. By assumption each K T + Θ m is big and so there is some model T m −→ T on which the moving part of mk(K T + Θ m ) becomes semiample. The problem is that the model T m depends on m. This is obviously an issue of birational geometry, and can only occur in dimensions two and higher. To get around this, we need to run the real MMP, see §7. Thus replacing T by a higher model, we may assume that the mobile part of some fixed multple of K T + Θ m is in fact free, and the positive part of K X + Θ is semiample.
The second way in which which the sequence Θ • might vary is that we might get freeness of the mobile part of mk(K T + Θ m ) on the same model, but Θ m is still not constant. There are plenty of such examples, even on the curve P 1 . Fortunately Shokurov has already proved that this cannot happen, since the sequence Θ • satisfies a subtle asymptoptic saturation property, see §8 and §9.
Hopefully it is clear, from what we just said, the great debt our proof of (1.1) owes to the work of Kawamata, Siu and especially Shokurov. The material in §5 was inspired by the work of Siu [16] and Kawamata [7] on deformation invariance of plurigenera, and lifting sections using multiplier ideas sheaves. On the other hand, a key step is to use the reduction to pl flips, due to Shokurov contained in [12]. Moreover, we use many of the results and ideas contained in [14], especially the notion of a saturated algebra.
Since the proof of (1.1) is not very long, we have erred on the side of making the proofs as complete as possible. We also owe a great debt to the work of Ambro, Fujino and especially Corti, who did such a good job of making the work of Shokurov more accessible. In particular much of the material contained in §3 and §7-9 is due to Shokurov, as well as some of the material in the other sections, and we have followed the exposition of [2] and [3] quite closely.

Notation and conventions
We work over the field of complex numbers C. A Q-Cartier divisor D on a normal variety X is nef if D · C ≥ 0 for any curve C ⊂ X. We say that two Q-divisors D 1 , D 2 are Q-linearly equivalent (D 1 ∼ Q D 2 ) if there exists an integer m > 0 such that mD i are linearly equivalent. We say that a Q-Weil divisor D is big if we may find an ample divisor A and an effective divisor B, such that D ∼ Q A + B.
A log pair (X, ∆) is a normal variety X and an effective Q-Weil divisor ∆ such that K X + ∆ is Q-Cartier. We say that a log pair (X, ∆) is log smooth, if X is smooth and the support of ∆ is a divisor with global normal crossings. A projective morphism g : Y −→ X is a log resolution of the pair (X, ∆) if Y is smooth and g −1 (∆)∪{ exceptional set of g } is a divisor with normal crossings support. We write g * (K X +∆) = K Y +Γ and Γ = a i Γ i where Γ i are distinct reduced irreducible divisors. The log discrepancy of Γ i is 1 − a i . The locus of log canonical singularities of the pair (X, ∆), denoted LCS(X, ∆), is equal to the image of those components of Γ of coefficient at least one (equivalently log discrepancy at most zero). The pair (X, ∆) is kawamata log terminal if for every (equivalently for one) log resolution g : Y −→ X as above, the coefficients of Γ are strictly less than one, that is a i < 1 for all i. Equivalently, the pair (X, ∆) is kawamata log terminal if the locus of log canonical singularities is empty. We say that the pair (X, ∆) is purely log terminal if the log discrepancy of any exceptional divisor is greater than zero.
We will also write where Γ and E are effective, with no common components, and E is g-exceptional. Note that this decomposition is unique. Note that the group of Weil divisors with rational or real coefficients forms a vector space, with a canonical basis given by the prime divisors. If A and B are two R-divisors, then we let (A, B] denote the line segment Given an R-divisor, D denotes the sup norm with respect to this basis. We say that D ′ is sufficiently close to D if there is a finite dimensional vector space V such that D and D ′ ∈ V and D ′ belongs to a sufficiently small ball of radius δ > 0 about D, We recall some definitions involving divisors with real coefficients: cone of curves minus the origin.
divisor and an effective divisor.
Note that we may pullback R-Cartier divisors, so that we may define the various flavours of log terminal and log canonical in the obvious way.

Generalities on Finite generation
In this section we give some of the basic definitions and results concerning finite generation; we only include the proofs for completeness.
We fix some notation. Let f : X −→ Z be a projective morphism of normal varieties, where Z is affine. Let A be the coordinate ring of Z.
for a positive integer d.
Then R is finitely generated iff there is a positive integer d such that R (d) is finitely generated.
Proof. Suppose that R is finitely generated. The cyclic group Z d acts in an obvious way on R, and under this action R (d) is the algebra of invariants. Thus R (d) is finitely generated by Noether's Theorem, which says that the ring of invariants of a finitely generated ring, under the action of a finite group, is finitely generated. Now suppose that R (d) is finitely generated. Let f ∈ R. Then f is a root of the monic polynomial In particular R is integral over R (d) and the result follows by Noether's Theorem on the finiteness of the integral closure.
We are interested in finite generation of the following algebras: In particular we have: Lemma 3.3. Let X be a normal variety and let R and R ′ be two divisorial algebras associated to divisors D and D ′ .
If aD ∼ a ′ D ′ then R is finitely generated iff R ′ is finitely generated.
Proof. Clear, since R and R ′ have the same truncation.
We want to restrict a divisorial algebra to a prime divisor S: under the obvious restriction map.
Lemma 3.5. If the algebra R is finitely generated then so is the restricted algebra. Conversely, if S is linearly equivalent to a multiple of B, where B is an effective divisor which does not contain S and the restricted algebra is finitely generated, then so is R.
Proof. Since by definition there is a surjective homomorphism it follows that if R is finitely generated then so is R S . Now suppose that S ∼ bB. Passing to a truncation, we may assume that b = 1. We can identify the space of sections of with rational functions g, such that (g) + mB ≥ 0.
Let g 1 ∈ R 1 be the rational function such that Suppose we have g ∈ R m , with φ(g) = 0. Then the support of (g) + mB, contains S, so that we may write But this says exactly that g/g 1 ∈ R m−1 , so that the kernel of φ is precisely the principal ideal generated by g 1 . But then if R S is finitely generated, it is clear that R is finitely generated.
and we say that it is bounded if there is a divisor B such that Since the maps in (3.4) are not in general surjective, the restricted algebra is not necessarily divisorial. However we will be able to show that it is of the following form: where B • is an additive sequence of integral Weil divisors, will be called geometric.
We are interested in giving necessary and sufficient conditions for a divisorial or more generally a geometric algebra to be finitely generated.
is called the characteristic sequence.
We say that R is free if M m is base point free, for every m.
Clearly the mobile sequence is additive and the characteristic sequence is convex. The key point is that finite generation of a divisorial algebra only depends on the mobile part in each degree, even up to a birational map: Lemma 3.10. Let R be a geometric algebra asociated to the convex sequence B • . Let g : Y −→ X be any birational morphism and let R ′ be the geometric algebra on Y associated to a convex sequence B ′ • . If the mobile part of g * B i is equal to the mobile part of B ′ i then R is finitely generated iff R ′ is finitely generated.
Lemma 3.11. Let R be a free geometric algebra and let D be the limit of the characteristic sequence.
If D = D k for some positive integer k then R is finitely generated.
Proof. Passing to a truncation, we may assume that D = D 1 . But then and so D = D m , for all positive integers m. Let h : X −→ W be the contraction over Z associated to M 1 , so that M 1 = h * H, for some very ample divisor on W . We have g * M m = mg * M 1 = h * (mH) and so the algebra R is nothing more than the coordinate ring of W under the embedding of W in P n given by H, which is easily seen to be finitely generated by Serre vanishing.

Reduction to pl flips and finite generation
We recall the definition of a pl flipping contraction: We call a morphism f : X −→ Z or normal varieties, where Z is affine, a pl flipping contraction if (1) f is a small birational contraction of relative Picard number one, and (4) −(K X + ∆) and −S are ample.
Shokurov, [12], see also [9] and [3], has shown: The aim of the rest of the paper is to prove: Let (X, ∆) be a log pair of dimension n and let f : X −→ Z be a morphism, where Z is affine and normal. Let k be a positive integer such that D = k(K X + ∆) is Cartier, and let R be the divisorial algebra associated to D. Assume that (1) K X + ∆ is purely log terminal, there is a divisor G ∈ |D|, such that S is not contained in the support of G, (4) ∆−S ∼ Q A+B, where A is ample and B is an effective divisor, whose support does not contain S, and (5) −(K X + ∆) is ample. If the real MMP holds in dimension n − 1 then the restricted algebra R S is finitely generated.
Note that the only interesting case of (4.3) is when f is birational, since otherwise the condition that −(K X + ∆) is ample implies that κ(X, K X + ∆) = −∞.
We note that to prove (1.1), it is sufficient to prove (4.3): Proof. By (4.2) it suffices to prove the existence of pl flips. Since Z is affine and f is small, it follows that S is mobile. By (3.5) it follows that it suffices to prove that the restricted algebra is finitely generated.
Hence it suffices to prove that a pl flip satisfies the hypothesis of (4.3). Properties (1-2) and (5) are automatic and (3) follows as S is mobile.
∆ is automatically big, as f is birational, and so ∆ ∼ Q A + B, where A if ample, and B is effective. As S is mobile, we may assume that B does not contain S.

Extending sections
The key idea of the proof of (1.1) is to use the main result of [4] to lift sections. In this section, we show that we can improve this result, if we add some hypotheses. We recall some of the basic results about multiplier ideal sheaves.
Definition 5.1. Let (X, ∆) be a log pair, where X is smooth and let µ : Y −→ X be a log resolution. Suppose that we write The multiplier ideal sheaf of the log pair (X, ∆) is defined as Note that the pair (X, ∆) is kawamata log terminal iff the multiplier ideal sheaf is equal to O X . Another key property of a multiplier ideal sheaf is that it is independent of the log resolution. Multiplier ideal sheaves have the following basic property, see (2.2.1) of [17]: Lemma 5.2. Let (X, ∆) be a kawamata log terminal pair, where X is a smooth variety, and let D be any divisor. Let f : X −→ Z be any projective morphism, where Z is affine and normal. Let σ ∈ H 0 (X, L) be any section of a line bundle L, with zero locus S ⊂ X.
Proof. Let g : Y −→ X be a log resolution of the pair (X, D + ∆). As S is integral g * D − g * S ≤ g * ∆ , and as the pair (X, ∆) is kawamata log terminal, Pushing forward via g, we get σ ∈ H 0 (X, L ⊗ J (D)).
We also have the following important vanishing result, which is an easy consequence of Kawamata-Viehweg vanishing: Theorem 5.3. (Nadel Vanishing) Let X be a smooth variety, let ∆ be an effective divisor. Let f : X −→ Z be any projective morphism and let N be any integral divisor such that N − ∆ is relatively big and nef. Then Here is the main result of this section: Theorem 5.4. Let (Y, Γ) be a smooth log pair and let π : Y −→ Z be a projective morphism, where Z is normal and affine. Let m be a positive integer, and let L be any line bundle on X, such that Suppose that there is an effective divisor H, which does not contain T , such that for every sufficiently divisible positive integer l, the natural homomorphism Then the natural restriction homomorphism is surjective.
Proof. As K Y + T + (1 − ǫ)Γ + ǫA + ǫB is purely log terminal for any ǫ > 0 sufficiently small, replacing A by ǫA and B by ǫB + (1 − ǫ)Γ, we may assume that K Y + Γ = K Y + T + A + B is purely log terminal. We let primes denote restriction to T , so that, for example, H ′ = H| T . Fix a non-zero section σ ∈ H 0 (T, L ′ ).
Let S be the zero locus of σ. By assumption, we may find a divisor G l ∼ lc 1 (L) + H, such that is ample for l sufficiently large, it follows that, Since (Y, T + A + B) is purely log terminal, (T, B ′ ) is kawamata log terminal, and so is kawamata log terminal for l sufficiently large. But then σ ∈ H 0 (T, L ′ ⊗ J (Θ ′ )) ⊂ H 0 (T, L ′ ⊗ J (Θ)), by (5.2).

Limiting algebras
To state the main result of this section, we need a: Definition 6.1. We say that a geometric algebra R, given by an additive sequence B • , is limiting, if there are Q-divisors ∆ m and a positive integer k such that (1) B m = mk(K X + ∆ m ), (2) the limit ∆ of the convex sequence ∆ • exists, and (3) K X + ∆ is kawamata log terminal.
Theorem 6.2. Let (X, ∆) be a log pair of dimension n and let f : X −→ Z be a morphism, where Z is affine and normal. Let k be any positive integer such that D = k(K X + ∆) is Cartier. Assume that there is a positive integer m 0 and a divisor G 0 ∈ |m 0 D|, such that S is not contained in the support of G 0 , and (4) ∆−S ∼ Q A+B, where A is ample and B is an effective divisor, whose support does not contain S. Then there is a log resolution g : Y −→ X with the following properties. Suppose that we write where Γ and E are effective, with no common components and E is exceptional. Let T be the strict transform of S and let π the composition of f and g. Let R be the divisorial algebra associated to G = k(K Y +Γ).
Then the restricted algebra R T is a limiting algebra, given by a convex sequence Θ • .
Note that the hypotheses of (6.2) are simply those of (4.3), excluding (5) of (4.3), and the hypothesis that the MMP holds. To prove (6.2), we are going to apply (5.4). The idea will be to start with the main result (3.17) of [4], which we state in a convenient form: Now to apply (6.3), the main point will be to change models and alter Γ, so that property (3) holds. To this end, we will need some results concerning manipulation of log pairs. Given a divisor ∆ = i a i ∆ i , we set Lemma 6.4. Let (Y, Γ) be a smooth log pair, and let π : Y −→ Z be a projective morphism, where Z is normal and affine. Let m be any positive integer such that m( no two components of Γ intersect, and (4) there is a divisor G ∈ |m(K Y + Γ)| such that G and Γ have no common components.
Then we may find a Q-divisor 0 ≤ Θ ≤ Θ ′ such that the image of the natural homomorphism Proof. Since no two components of Γ intersect, and T is the only component of coefficient one, the only possible log canonical centres of K Y + Γ contained in G, are the components of T ∩ Γ .
It follows that there is a resolution h : Y ′ −→ Y of the base locus of m(K Y + Γ), which is a sequence of smooth blow ups with centres equal to the irreducible components of T ∩ Γ , with the following property.
We may write where Γ ′ and E are effective, with no common components, and E is exceptional. Note that m(K Y ′ + Γ ′ ) and mE are integral and that be the decomposition of m(K Y ′ + Γ ′ ) into its moving and fixed parts. Then the base locus of N m does not contain any log canonical centre of K Y ′ + Γ ′ .
Cancelling common components of G m and Γ ′ , we may therefore find divisors G ′ m and Γ ′ m , with no common components, such that into its moving and fixed parts. Let T ′ be the strict transform of T . Since h is a composition of blow ups, with smooth centres, which intersect T in a divisor, in fact h| T ′ : ). Possibly replacing kA by a linearly equivalent divisor, we may assume that g * A and the strict transform of A are equal. Since A is ample, there is an effective and exceptional divisor F such that g * A − F is ample. In this case we are thus free to replace the pair (Y, L) by (Y ′ , L ′ ), so that, letting Θ = (Γ ′ m − T ′ )| T ′ , the result follows by (6.3) and (5.4). Lemma 6.5. Let (X, ∆) be a log pair. We may find a birational projective morphism g : Y −→ X, with the following properties. Suppose that we write where Γ and E are effective, with no common components, and E is exceptional.
Then no two components of Γ intersect.
Proof. Passing to a log resolution, we may assume that the pair (X, ∆) has global normal crossings. We will construct g as a sequence of blow ups of irreducible components of the intersection of a collection of components of ∆ . Now if a collection of components of ∆ intersect, then certainly no irreducible component of their intersection is contained in ∆ . Since (X, ∆) has global normal crossings, it follows that we may as well replace ∆ by ∆ . Thus we may assume that the coefficients of the components of ∆ are all less than one, so that the pair (X, ∆) is kawamata log terminal, and our aim is to find g, so that no two components of Γ intersect.
We proceed by induction on the maximum number k of components of ∆ which intersect. Since (X, ∆) has normal crossings, k ≤ n = dim X, and it suffices to decrease k. We now proceed by induction on the maximum sum s of the coefficients of k components which intersect.
If we pick r such that r∆ is integral, then s is at least k/r and rs is an integer, so it suffices to decrease s. We further proceed by induction on the number l of subvarieties V which are the components of the intersection of k components of ∆ whose coefficients sum to s. We aim to decrease l by blowing up.
Suppose that we blow up g : Y −→ X along the intersection V of k components ∆ 1 , ∆ 2 , . . . , ∆ k of ∆, with coefficients a 1 , a 2 , . . . , a k . A simple calculation, see for example (2.29) of [10], gives that the discrepancy of the exceptional divisor E is (k − 1) − s, so that . . , Γ k be the components of Γ ′ which are the strict transforms of ∆ 1 , ∆ 2 , . . . , ∆ k . Then there are k subvarieties of Y which dominate V , which are the intersection of k components of Γ, namely the intersection with E of all but one of Γ 1 , Γ 2 , . . . , Γ k . If a k is the smallest coefficient, then the maximum sum of the coefficients of these intersections is and so we have decreased l by one.
Proof of (6.2). Let g : Y −→ X be any morphism, whose existence is guaranteed by (6.5). We may write where Γ and E are effective, with no common components and E is exceptional. Since k(K X +∆) is Cartier, k(K Y +Γ) and kE are integral. Let T be the strict transform of S and let π the composition of f and g. Let be the decomposition of mk(K Y + Γ) into its moving and fixed parts. By assumption, T is not a component of G m . Possibly replacing k by a multiple, we may assume that kA is very ample. Possibly replacing kA by a linearly equivalent divisor, we may assume that g * A and the strict transform of A are equal. Cancelling common components of Γ and G m , we may find divisors T + g * A ≤ Γ m ≤ Γ and G ′ m , with no common components, such that ). By(6.4), there is a divisor Θ m ≤ Θ ′ m , such that the image of the natural is equal to H 0 (T, O T (m(K T +Θ m ))), considered as a subspace of H 0 (T, L| T ) by the inclusion induced by m(Θ ′ m − Θ m ). On the other hand, as Θ m ≤ Θ, the limit Θ ′ of the sequence Θ • exists and K T + Θ ′ is kawamata log terminal.

Real versus rational
Most of the ideas and a significant part of the proofs of the results in this section are contained in [13]. We have only restated these results at the level of generality we need to prove (1.1).
We will need a generalisation of the base point free theorem to the case of real divisors: Theorem 7.1 (Base Point Free Theorem). Let (X, ∆) be a Q-factorial kawamata log terminal pair, where ∆ is a R-divisor. Let f : X −→ Z be a projective morphism, where Z is affine and normal, and let D be a nef R-divisor, such that aD − (K X + ∆) is nef and big, for some positive real number a.
Then D is semiample.
Proof. Replacing D by aD we may assume that a = 1. By assumption we may write where A is ample and E is effective. Thus is ample for all ǫ > 0. Since the pair (X, ∆ + ǫE) is kawamata log terminal for ǫ small enough, replacing ∆ by ∆ + ǫE, we may assume that is ample. Perturbing ∆, we may therefore assume that K X + ∆ is Q-Cartier. Let F be the set of all elements α of the closed cone of curves on which D is zero. Then K X + ∆ is negative on F . Let H be any ample divisor. For every ray R = R + α contained in F , there is an ǫ > 0 such that K X + ∆ + ǫH is negative on R. By compactness of a slice, it follows that there is an ǫ > 0, such that K X + ∆ + ǫH is negative on the whole of F . It follows by the cone theorem that F is the span of finitely many extremal rays R 1 , R 2 , . . . , R k , where each extremal ray R i is spanned by an integral curve C i . Let D 1 , D 2 , . . . , D k be the prime components of D. Consider the convex subset P of consisting of all divisors B such that B is zero on F . Then P is a closed rational polyhedral cone.
In particular D ∈ P is a convex linear combination of divisors B i ∈ P ∩N Q ∩U, where U is any neighbourhood of D. But if U is sufficiently small, then B i −(K X +∆) is also ample. Now if B i = (B i −(K X +∆))+ (K X + ∆) is not nef, then it must be negative on a (K X + ∆)-extremal ray. As the extremal rays of K X + ∆ are discrete in a neighbourhood of F , it follows that B i is also nef if U is sufficiently small. By the base point free theorem, it follows that each B i is semiample, so that D is semiample.
Theorem 7.2. Assume the real MMP in dimension n. Let (X, ∆) be a kawamata log terminal pair of dimension n, such that K X + ∆ is R-Cartier and big. Assume that there is a divisor Ψ such that K X + Ψ is Q-Cartier and kawamata log terminal. Let f : X −→ Z be any proper morphism, where Z is normal and affine. Fix a finite dimensional vector subspace V of the space of R-divisors containing ∆.
Then there are finitely many birational maps ψ i : X W i , 1 ≤ i ≤ l over Z, such that for every divisor Θ ∈ V sufficiently close to ∆, there is an 1 ≤ i ≤ l with the following properties: (1) ψ i is the composition of a sequence of (K X + Θ)-negative divisorial contractions and birational maps, which are isomorphisms in codimension two, Proof. As the property of being big is an open condition, we may asssume that for any Θ ∈ V sufficiently close to ∆, K X + Θ is big. Suppose that we have established (1)(2)(3). As W i is Q-factorial, it follows that the group of Weil divisors modulo Cartier divisors is a finite group. Thus there is a fixed positive integer s i such that if r(K X + Θ) is integral, then s i r(K W i + ψ i * Θ) is Cartier. By Kollár's effective base point free theorem, [8], there is then a positive integer M such that Ms i r(K W i + ψ i * Θ) is base point free. If we set k to be Ms, where s is the maximum of the s i , then this is (4). Thus it suffices to prove (1)(2)(3). Now if Θ is sufficiently close to ∆, then K X + Θ is big, so that by (7.1) we may replace (3) by the weaker condition, Thus it suffices to establish (1), (2) and (3 ′ ).
Since we are assuming existence and termination of flips for Qdivisors, we may construct a log terminal model of (X, Ψ). As (X, Ψ) is kawamata log terminal, the log terminal model is small over X. Thus passing to a log terminal model of (X, Ψ), we may assume that X is Q-factorial and that f is projective.
Suppose that K X + ∆ is not nef. Let R be an extremal ray for K X + ∆. R is necessarily (K X + Θ)-negative, for any Q-divisor Θ close enough to ∆. By the cone and contraction theorems applied to K X +Θ, we can contract R, ψ : X −→ X ′ . ψ must be birational, as K X + ∆ is big. If ψ is divisorial (that is the exceptional locus is a divisor) then we replace the pair (X, ∆) by the pair (X ′ , ψ * ∆). If ψ is small, then using (1.4) Q,n , we know the flip of K X +Θ exists. But then this is also the flip of K X + ∆, and so we can replace the pair (X, ∆) by the flip. Since we are assuming (1.5) R,n , and we can only make finitely many divisorial contractions, we must eventually arrive at the case when K X + ∆ is nef. By (7.1) it follows that K X +∆ is relatively semiample. Let ψ : X −→ W be the corresponding contraction over Z. Then there is an ample R-divisor H on W such that K X + ∆ = ψ * H. Thus if Θ is sufficiently close to ∆ and K X + Θ is relatively nef over W , then Note that we may replace Z by W , and use the fact that a divisor is relatively generated iff it is locally base point free. Thus replacing Z by an open affine subset of W , we may assume that f is birational and K X + ∆ is R-linearly equivalent to zero. Let B be the closure in V of a ball with radius δ centred at ∆. If δ is sufficiently small, then for every Θ ∈ B, K X + Θ is kawamata log terminal. Pick Θ a point of the boundary of B. Since K X + ∆ is R-linearly equivalent to zero, note that for every curve C, In particular every step of the (K X +Θ)-MMP is a step of (K X +Θ ′ )-MMP, for every Θ ′ ∈ (∆, Θ]. Since we are assuming existence and termination of flips, we have a birational map ψ : X W over Z, such that K W + ψ * Θ is nef, and it is clear that K W + ψ * Θ ′ is nef, for every Θ ′ ∈ (∆, Θ]. At this point we want to proceed by induction on the dimension of B. To this end, note that as B is compact and ∆ is arbitrary, our result is equivalent to proving that (3 ′ ) holds in B. By what we just said, this is equivalent to proving that (3 ′ ) holds on the boundary of B, which is a compact polyhedral cone (since we are working in the sup norm) and we are done by induction on the dimension of B.
The key consequence of (7.2) is: Assume the real MMP in dimension n. Let (X, ∆) be a kawamata log terminal Q-factorial pair of dimension n, where K X + ∆ is an R-divisor. Let f : X −→ Z be a contraction morphism. Let r be a positive integer.
If K X + ∆ is relatively big, then there is a birational model g : Y −→ X and a positive integer k, such that if π : Y −→ Z is the composition of f and g, then for every divisor Θ sufficiently close to ∆, (1) if r(K X +Θ) is integral, then the moving part of g * (rk(K X +Θ)) is base point free.
(2) If Θ • is a convex sequence of divisors with limit Θ, such that mr(K X + Θ m ) is integral then the limit D of the characteristic sequence D • associated to B m = g * (mrk(K X + Θ m )) is semiample.
Proof. Let ψ i : X W i be the models, whose existence is guaranteed by (7.2), and let g : Y −→ X be any birational morphism which resolves the indeterminancy of ψ i , 1 ≤ i ≤ l. Let φ i : Y −→ W i be the induced birational morphisms, so that we have commutative diagrams Y Let Θ be sufficiently close to ∆. Then for some i, K W i + ψ i * Θ is semiample. Suppressing the index i, we may write where for E we sum over the common exceptional divisors of g and φ, and for F we sum over the exceptional divisors of φ which are not exceptional for g (by assumption there are no exceptional divisors of g which are not also exceptional for φ). (1) of (7.2) implies that F is effective. But then by negativity of contraction, see (2.19) of [9], E is also effective. Suppose that r(K X + Θ) is integral. Then the moving part of g * (rk(K X + Θ)) is equal to the moving part of φ * (rk(K W + ψ * Θ)), and we can apply (7.2) to conclude that there is a fixed k such that the moving part of g * (rk(K X + Θ)) is base point free. This is (1). Now suppose that Θ • is a convex sequence with limit Θ, such that mr(K X + Θ m ) is integral. Let M m be the mobile part of g * (mrk(K X + Θ m )). Then, by what we have already said, M m is also the mobile part of φ * i (mrk(K W + ψ i * Θ m )). Possibly passing to a subsequence, we may assume that i is constant, and in this case we suppress it. It is then clear that the limit D of so that D is nef. It follows that D is semiample by (7.2) (or indeed (7.1)).

Diophantine Approximation
All of the results in this section are implicit in the work of Shokurov [14], and we claim no originality. In fact we have only taken Corti's excellent introduction to Shokurov's work on the existence of flips and restated those results without the use of b-divisors. Proof. If D is rational, then pick m such that mD is integral and set M = mD. Thus we may suppose that D is not rational.
Let N Z be the lattice spanned by the components G j of D, and let N Q and N R be the corresponding vector spaces. Since D is semiample and π is projective, we may pick a basis {P k } of N Q , where each P k is base point free, and D belongs to the cone Let v ∈ N R be the vector corresponding to D. Let A be the cyclic subgroup of the torus N R N Z , generated by the image of v. LetĀ be the closure of A and let A 0 be the connected component of the identity ofĀ. Let V ⊂ N R be the inverse image of A 0 . Then A 0 is a Lie group and so V is a linear subspace. As we are assuming that D is not rational, A is infinite and so A 0 and V are both positive dimensional. In particular V is not contained in G. But then for every ǫ > 0, we can find a positive multiple mv of v, and a vector w ∈ N Z , which is an integral linear combination of the divisors P k , such that Note that if ǫ > 0 is sufficiently small then w ∈ P since it is integral and close to mv ∈ P. Thus if M is the divisor corresponding to w, then M is base point free, and the rest is clear.
Let π : Y −→ Z be a projective morphism of normal varieties, where Z is affine. Let R be the geometric algebra associated to the additive sequence M • of mobile divisors, with characteristic sequence D • . We say that R is saturated if there is a Q-divisor F , such that (1) F ≥ 0, and (2) for every pair of positive integers i and j, Theorem 8.3. Let Y be a smooth variety and π : Y −→ Z a projective morphism, where Z is affine and normal. Let R be a saturated and free geometric ring on Y whose characteristic sequence tends to a semiample limit. Then R is finitely generated.
Proof. Let D • be the characteristic sequence, with limit D. Let G be the support of D, and pick ǫ > 0 such that F − ǫG ≥ 0. By diophantine approximation, we know that there is a positive integer m and an integral divisor M such that (1) M is mobile, Letting i go to infinity we have M ≤ Mov( mD + F ) ≤ mD m ≤ mD.
By (3) above, it follows that the sequence of inequalities must in fact be equalities, so that we have for some m and we may apply (3.11).

Saturation of the restricted algebra
We fix some notation for this section. Let (X, ∆) be a purely log terminal pair and let f : X −→ Z be a projective morphism of normal varieties, where Z is affine. We assume that S = ∆ is irreducible. Let g : Y −→ X be any log resolution of the pair (X, ∆). Then we may write K Y + Γ = g * (K X + ∆) + E, where Γ and E are effective, with no common components and E is g-exceptional. We set T = Γ the strict transform of S, and F = E − Γ + T . We suppose that K Y + Γ is purely log terminal, so that F ≥ 0 is effective and exceptional. Fix a positive integer k such that k(K X + ∆) is Cartier, so that both G = k(K Y + Γ) and kE are integral. Let π : Y −→ Z be the composition of f and g. Let N m + G m be the decomposition of mG into its moving and fixed parts, and let M m be the restriction of N m to T . Finally let so that D • is the characteristic sequence of the restricted algebra R T . where we used the fact that jkE + F is g-exceptional.
Lemma 9.2. Suppose that −(K X + ∆) is nef and big and that M m is free.
Then the restricted algebra R T is saturated with respect to F | T .
Proof. By assumption F | T ≥ 0. Proof of claim. Fix i. Since M i is free, it follows that N i is free in a neighbourhood of T . Then there is a model h i : Y i −→ Y , on which N i becomes free, which is an isomorphism in a neighbourhood of T . Thus, replacing Y by Y i , we may assume that N i is free. Considering the restriction exact sequence, it follows that the obstruction to the surjectivity of the restriction map above is given by, which vanishes by Kawamata-Viehweg vanishing, as g * (−(K X + ∆)), is big and nef and (j/i)N i is nef.
The result is now an easy consequence of (9.1) and the claim.
Proof of (4.3). Let g : Y −→ X be the log resolution of (X, ∆), whose existence is guaranteed by (6.2). We may write where Γ and E are effective, with no common components, and E is exceptional. It follows that k(K Y + Γ) and kE are integral and mk(K Y + Γ) and mkg * (K X + ∆) have the same moving parts. Let T be the strict transform of S and let R T be the restricted algebra associated to the divisor k(K Y + Γ). By (3.10), it follows that R T is finitely generated iff R S is finitely generated. We may suppose that R T is a limiting algebra, given by the convex sequence Θ • . By (7.3) it follows that there is a birational model h : T ′ −→ T , where the mobile parts of mkh * (K T + Θ m ) are base point free, and that the limit of the characteristic sequence is semiample. We may assume that h is induced by a birational morphism h ′ : Y ′ −→ Y . Replacing Y by Y ′ , we may assume that the mobile parts of mk(K T + Θ m ) are base point free and that the limit of the characteristic sequence is semiample.
It follows by (9.2) that the characteristic sequence is saturated. But then the restricted algebra R T is finitely generated by (8.3), and as we have already observed this implies that R S is finitely generated.