ON THE EXISTENCE OF FLIPS

CHRISTOPHER D. HACON AND JAMES M^{C}KERNAN

Abstract. Using the techniques of [17], [7], [13] and [15], we prove that flips exist in dimension n, if one assumes the termination of real flips in dimension n − 1.

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, [14], the termination of real flips in dimension three, we get a new proof of the following result of Shokurov [15]:

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.

Date: December 31, 2005.

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, Paolo Cascini and Alessio Corti for many useful comments on the contents of this paper. We would also like to thank Florin Ambro and Osamu Fujino, whose extensive comments helped improve the exposition of this paper, especially §6.

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 ℚ-Cartier divisor such that −D is relatively ample the flip of D, if it exists at all, is a commutative diagram

X  ^{}X′

[figure: f; f′]

Z,

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} + Δ 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, Δ) be a kawamata log terminal ℚ-factorial pair of dimension n, where Δ is a K-divisor. Let f: X ⟶ Z be a flipping contraction, so that −(K_{X} + Δ) 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, Δ) be a kawamata log terminal ℚ-factorial pair of dimension n, where Δ is a K-divisor.

Then there is no infinite sequence of (K_{X} + Δ)-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).

Mori, in a landmark paper [12], proved the existence of 3-fold flips, when X is terminal and Δ is empty. Later on Shokurov [13] and Kollár [9] proved the existence of 3-fold flips for kawamata log terminal pairs (X, Δ), that is they proved (1.4)_{3}. Much more recently [15], Shokurov proved (1.4)_{4}.

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, [14], (1.5)_{ℝ,3}. Further, Shokurov proved, [16], that (1.5)_{ℝ,n} follows from two conjectures on the behaviour of the log discrepancy of pairs (X, Δ) of dimension n (namely acc for the set of log discrepancies, whenever the coefficients of Δ 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} + Δ 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, Δ) be a kawamata log terminal pair, where K_{X} + Δ is ℚ-Cartier, and let π: X ⟶ Z be a proper morphism, where Z is affine and normal.

If K_{X} + Δ 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 [15]. The first key step was already established in [13], 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

(K_{X} + S + B)|_{S} = K_{S} + B′,

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

H^{0}(X, O_{X}(m(K_{X} + S + B))) ⟶ H^{0}(X, O_{S}(m(K_{S} + B′))),

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 [17] 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

iΘ_{i} + jΘ_{j} ≤ (i + j)Θ_{i+j},

and it suffices to prove that this sequence stabilises, that is

Θ_{m} = Θ,

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 mobile 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 P_{m} of some fixed multiple ml(K_{T} + Θ_{m}) is in fact free, and that the limit of P_{m}/m is semiample.

The second way in 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 [17] 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 [13]. Moreover, we use many of the results and ideas contained in [15], 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, Takagi 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.

2. 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 ℚdivisor A and an effective ℚ-divisor B, such that D ∼_{ℚ} A + B. If T is a subvariety of X, not contained in the base locus of |D|, then |D|_{T} denotes the image of the linear system |D| under restriction to T.

A log pair (X, Δ) is a normal variety X and an effective ℚ-Weil divisor Δ such that K_{X} + Δ is ℚ-Cartier. We say that a log pair (X, Δ) is log smooth, if X is smooth and the support of Δ is a divisor with global normal crossings. A projective morphism g: Y ⟶ X is a log resolution of the pair (X, Δ) if Y is smooth and g^{−1}(Δ)∪{ exceptional set of g } is a divisor with normal crossings support. We write g^{∗}(K_{X}+Δ) = 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, Δ), denoted LCS(X, Δ), is equal to the image of those components of Γ of coefficient at least one (equivalently log discrepancy at most zero). The pair (X, Δ) 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, Δ) is kawamata log terminal if the locus of log canonical singularities is empty. We say that the pair (X, Δ) is purely log terminal if the log discrepancy of any exceptional divisor is greater than zero.

We will also write

K_{Y} + Γ = g^{∗}(K_{X} + Δ) + E,

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. 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,

∥D − D′∥ < δ.

If A and B are two ℝ-divisors, then we let (A, B] denote the line segment

{ λA + μB | λ + μ = 1, λ ≥ 0, μ > 0 },

and max(A, B) denotes the divisor obtained by taking the maximum coefficient by coefficient.

We recall some definitions involving divisors with real coefficients:

Definition 2.1. Let X be a variety.

(1) An ℝ-Weil divisor D is an ℝ-linear combination of prime divisors.

(2) Two ℝ-divisors D and D′ are ℝ-linearly equivalent if their difference is an ℝ-linear combination of principal divisors.

(3) An ℝ-Cartier divisor D is an ℝ-linear combination of Cartier divisors.

(4) An ℝ-Cartier divisor D is ample if it is ℝ-linearly equivalent to a positive linear combination of ample divisors (in the usual sense).

(5) An ℝ-divisor D is effective if it is a positive real linear combination of prime divisors.

(6) An ℝ-Cartier divisor D is big if it is ℝ-linearly equivalent to the sum of an ample divisor and an effective divisor.

(7) An ℝ-Cartier divisor D is semiample if there is a contraction π: X ⟶ Y such that D is ℝ-linearly equivalent to the pullback of an ample 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.

3. 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.

Definition-Lemma 3.1. Let ℜ be any graded A-algebra. A truncation of ℜ is any A-algebra of the form

ℜ_{(d)} = ⨁_{m∈ℕ}ℜ_{md},

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

x^{d} − f^{d} ∈ ℜ_{(d)}[x].

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:

Definition 3.2. Let B be an integral Weil divisor on X. We call any O_{Z}-algebra of the form

⨁_{m∈ℕ}f_{∗}O_{X}(mB),

divisorial.

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′, where a and a′ are positive integers, 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:

Definition 3.4. Let ℜ be the divisorial algebra associated to the divisor B. The restricted algebra ℜ_{S} is the image

⨁_{m∈ℕ}f_{∗}O_{X}(mB) ⟶ ⨁_{m∈ℕ}f_{∗}O_{S}(mB),

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

ϕ: ℜ ⟶ ℜ_{S},

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

ℜ_{m} = f_{∗}O_{X}(mB)

with rational functions g, such that

(g) + mB ≥ 0.

Let g_{1} ∈ ℜ_{1} be a rational function such that

(g_{1}) + B = S.

Suppose we have g ∈ ℜ_{m}, with ϕ(g) = 0. Then the support of

(g) + mB,

contains S, so that we may write

(g) + mB = S + S′,

where S′ is effective. But then

(g) + mB = (g_{1}) + B + S′,

so that

(g/g_{1}) + (m − 1)B = S′.

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. □

Definition 3.6. We say that a sequence of ℝ-divisors B_{•} is additive if

B_{i} + B_{j} ≤ B_{i+j},

we say that it is convex if

\frac{i}{i + j} B_{i} + \frac{j}{i + j} B_{j} ≤ B_{i+j},

and we say that it is bounded if there is a divisor B such that

B_{i} ≤ B.

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:

Definition 3.7. Any O_{Z}-algebra of the form

⨁_{m∈ℕ}f_{∗}O_{X}(B_{m}),

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.

Definition 3.8. Let B be an integral divisor on X, such that h^{0}(X, O_{X}(B)) ≠ 0. Let F be the fixed part of the linear system |B|, and set M = B−F. We may write

|B| = |M| + F.

We call M = Mob B the mobile part of B and we call B = M + F the decomposition of B into its mobile and fixed part. We say that a divisor is mobile if the fixed part is empty.

Note that M is not necessarily effective, but that h^{0}(X, O_{X}(M)) ≠ 0.

Definition 3.9. Let ℜ be the geometric algebra associated to the additive sequence B_{•}. Let

B_{m} = M_{m} + F_{m},

be the decomposition of B_{m} into its mobile and fixed parts. The sequence of divisors M_{•} is called the mobile sequence and the sequence of ℚdivisors D_{•} given by

D_{i} = \frac{M_{i}}{i} ,

is called the characteristic sequence.

We say that ℜ is semiample if D is semiample, where D is the limit of the characteristic sequence.

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 additive sequence B_{•}, where each B_{i} is ℚ-Cartier. Let g: Y ⟶ X be any birational morphism and let ℜ′ be the geometric algebra on Y associated to an additive sequence B′_{•}.

Then ℜ is isomorphic to ℜ′.

Proof. Clear. □

Lemma 3.11. Let ℜ be a semiample 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

mD = mD_{1} = mM_{1} ≤ M_{m} = mD_{m} ≤ mD,

and so D = D_{m}, for all positive integers m. Possibly passing to another truncation, we may then assume that M_{m} is free for all 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 M_{m} = mM_{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. □

4. Reduction to pl flips and finite generation

We recall the definition of a pl flipping contraction:

Definition 4.1. We call a morphism f: X ⟶ Z of normal varieties, where Z is affine, a pl flipping contraction if

(1) f is a small birational contraction of relative Picard number one,

(2) X is ℚ-factorial,

(3) K_{X} + Δ is purely log terminal, where S = ⌞Δ⌟ is irreducible, and

(4) −(K_{X} + Δ) and −S are ample.

Shokurov, [13], see also [9] and [3], has shown:

Theorem 4.2 (Shokurov). To prove (1.1) it suffices to construct the flip of a pl flipping contraction.

The aim of the rest of the paper is to prove:

Theorem 4.3. Let (X, Δ) 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} +Δ) is Cartier, and let ℜ be the divisorial algebra associated to D. Assume that

(1) K_{X} + Δ is purely log terminal,

(2) S = ⌞Δ⌟ is irreducible,

(3) there is a divisor G ∈ |D|, such that S is not contained in the support of G,

(4) Δ−S ∼_{ℚ} A+B, where A is ample and B is an effective divisor, whose support does not contain S, and

(5) −(K_{X} + Δ) 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} + Δ) is ample implies that κ(X, K_{X} + Δ) = −∞.

We note that to prove (1.1), it is sufficient to prove (4.3):

Lemma 4.4. (4.3)_{n} implies (1.1)_{n}.

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. Δ − S is automatically big, as f is birational, and so Δ − S ∼_{ℚ} A + B, where A is ample, and B is effective. As S is mobile, possibly replacing B by a ℚ-linearly equivalent divisor, we may assume that B does not contain S. □

5. 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, Δ) be a log pair, where X is smooth and let g: Y ⟶ X be a log resolution. Suppose that we write

K_{Y} + Γ = g^{∗}(K_{X} + Δ).

The multiplier ideal sheaf of the log pair (X, Δ) is defined as

J (X, Δ) = J (Δ) = g_{∗}O_{Y} (−⌞Γ⌟).

Note that the pair (X, Δ) 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 [18]:

Lemma 5.2. Let (X, Δ) be a kawamata log terminal pair, where X is a smooth variety, and let D be any ℚ-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.

If D − S ≤ Δ then σ ∈ H^{0}(X, L ⊗ J (D)).

Proof. Let g: Y ⟶ X be a log resolution of the pair (X, D + Δ). As S is integral

⌞g^{∗}D⌟ − g^{∗}S ≤ ⌞g^{∗}Δ⌟,

and as the pair (X, Δ) is kawamata log terminal,

K_{Y/X} − ⌞g^{∗}Δ⌟ = −⌞Γ⌟ ≥ 0.

Thus

g^{∗}σ ∈ H^{0}(Y, g^{∗}L(−g^{∗}S))
⊂ H^{0}(Y, g^{∗}L(−g^{∗}S + K_{Y/X} − ⌞g^{∗}Δ⌟))
⊂ H^{0}(Y, g^{∗}L(K_{Y/X} − ⌞g^{∗}D⌟)).

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 Δ be an effective ℚ-divisor. Let f: X ⟶ Z be any projective morphism and let N be any integral divisor such that N − Δ is relatively big and nef.

Then

R^{i}f_{∗}(O_{X}(K_{X} + N) ⊗ J (Δ)) = 0, for i > 0.

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 c_{1}(L) ∼_{ℚ} m(K_{Y} + Γ). Assume that

(1) (Y, Γ) is purely log terminal,

(2) T = ⌞Γ⌟ is irreducible,

(3) Γ−T ∼_{ℚ} A+B, where A is ample and B is an effective divisor, which does not contain T.

Let Δ = (Γ − T)|_{T}, so that

(K_{Y} + Γ)|_{T} = K_{T} + Δ.

Suppose that there is an effective divisor H, which does not contain T, such that for every sufficiently divisible positive integer s, the image of the natural homomorphism

H^{0}(Y, L^{⊗s}(H)) ⟶ H^{0}(T, L^{⊗s}(H)|_{T}),

contains the image of H^{0}(T, L^{⊗s}|_{T}), considered as a subspace of H^{0}(T, L^{⊗s}(H)|_{T}) by the inclusion induced by H.

Then the natural restriction homomorphism

H^{0}(Y, L) ⟶ H^{0}(T, L|_{T}),

is surjective.

Proof. As K_{Y} +T +(1−ϵ)(Γ−T)+ϵA+ϵB is purely log terminal for any ϵ > 0 sufficiently small, replacing A by ϵA and B by ϵB+(1−ϵ)(Γ−T), we may assume that K_{Y} + Γ = K_{Y} + T + A + B.

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_{s} ∼ sc_{1}(L) + H, such that

G′_{s} = sS + H′.

If we set

N = c_{1}(L) − K_{Y} − T and Θ = \frac{m − 1}{ms}G_{s} + B,

then

N ∼_{ℚ} (m − 1)(K_{Y} + Γ) + A + B.

Since

N − Θ ∼_{ℚ} A + B − \frac{m − 1}{ms}H − B = A − \frac{m − 1}{ms}H,

is ample for s sufficiently large, it follows that,

H^{1}(Y, L(−T) ⊗ J (Θ)) = H^{1}(Y, O_{Y} (K_{Y} + N) ⊗ J (Θ)) = 0,

by Nadel vanishing (5.3). By (9.5.1) of [11] or (2.4.2) of [18], there is a short exact sequence

0 ⟶ J (Y, Θ)(−T) ⟶ J ⟶ J (T, Θ′) ⟶ 0,

where J is an ideal sheaf. Then

(5.4.1)  H^{0}(Y, L ⊗ J) ⟶ H^{0}(T, L′ ⊗ J (T, Θ′)),

is surjective. Now

Θ′ − S = B′ + \frac{m − 1}{ms}(sS + H′) − S

≤ B′ + \frac{m − 1}{ms}H′.

Since (Y, T + A + B) is purely log terminal, (T, B′) is kawamata log terminal, and so

(T, B′ + \frac{m − 1}{ms}H′),

is kawamata log terminal for s sufficiently large. But then

σ ∈ H^{0}(T, L′ ⊗ J (Θ′))

by (5.2). Thus we may lift σ, using (5.4.1). □

6. 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 Δ_{m} and a positive integer k such that

(1) B_{m} = mk(K_{X} + Δ_{m}),

(2) the limit Δ of the convex sequence Δ_{•} exists, and

(3) K_{X} + Δ is kawamata log terminal.

Theorem 6.2. Let (X, Δ) 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} + Δ) is Cartier. Assume that

(1) K_{X} + Δ is purely log terminal,

(2) S = ⌞Δ⌟ is irreducible,

(3) 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) Δ−S ∼_{ℚ} A+B, where A is ample and B is an effective divisor, whose support does not contain S.

Then there exists a birational morphism T ⟶ S with the following properties:

For each m > 0, there exists a resolution g_{m}: Y_{m} ⟶ X, such that, if we write,

K_{Y_{m}} + Γ′_{m} = g_{m}^{∗}(K_{X} + Δ) + E_{m},

where Γ′_{m} and E_{m} are effective with no common components, and we let, (a) N_{m} be the mobile part of mk(K_{Y_{m}} + Γ′_{m}),

(b) mk(K_{Y_{m}} + Γ_{m}) be the log mobile part, see (6.4), of mk(K_{Y_{m}} + Γ′_{m}).

(c) Z_{m} = E_{m} − Γ′_{m} + T,

(d) Θ_{m} = (Γ_{m} − T)|_{T}, and

(e) M_{m} be the mobile part of mk(K_{T} + Θ_{m}), then:

(i) the strict transform of S is isomorphic, over S, to T, and Z_{m}|_{T} is independent of m. Further for every m and n, there are birational morphisms Y_{m,n} ⟶ Y_{m} and Y_{m,n} ⟶ Y_{n}, where Y_{m,n} is smooth and the strict transform of T is isomorphic to T.

(ii) N_{m}|_{T} = M_{m} and |N_{m}|_{T} = |M_{m}|.

(iii) Θ_{•} is convex, up to truncation.

(iv) Let ℜ be the limiting algebra associated to Θ_{•}. Then the restricted algebra ℜ_{S} is finitely generated iff ℜ is finitely generated.

(v) If there is a positive integer s, such that the mobile part P_{m} of ml(K_{T} + Θ_{m}) is free, where l = ks, then the mobile part Q_{m} of ml(K_{Y_{m}} + Γ_{m}) is free.

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:

Theorem 6.3. Let (Y, Γ) be a smooth log pair, and let Y ⟶ Z be a projective morphism, where Z is normal and affine. Let H = lA, where l is a sufficiently large and divisible positive integer and A is very ample. Let k be any positive integer such that k(K_{Y} + Γ) is Cartier, and let m be any positive integer. Let L be the line bundle O_{Y} (mk(K_{Y} + Γ)). Assume that

(1) Γ contains T with coefficient one,

(2) (Y, Γ) is log canonical, and

(3) there is a positive integer m_{0} and a divisor G_{0} ∈ |m_{0}k(K_{Y} +Γ)|

which does not contain any log canonical centre of K_{Y} + ⌜Γ⌝. Let Θ = (Γ − T)|_{T}, so that

(K_{Y} + Γ)|_{T} = K_{T} + Θ.

Suppose that H does not contain T. Then the image of the natural homomorphism

H^{0}(Y, L(H)) ⟶ H^{0}(T, L(H)|_{T}),

contains the image of H^{0}(T, L|_{T}), where H^{0}(T, L|_{T}) is considered as a subspace of H^{0}(T, L(H)|_{T}) by the inclusion induced by H|_{T}.

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 Δ = ∑_{i} a_{i}Δ_{i}, we set

⟨Δ⟩ = ∑_{i} b_{i}Δ_{i} where b_{i} = \begin{cases}a_{i} & if 0 < a_{i} < 1 \\ 0 & otherwise. \end{cases}

Definition 6.4. Let (X, Δ) be a log pair and let k be a positive integer such that k(K_{X} + Δ) is an integral divisor. Let m be a positive integer and let mk(K_{X} + Δ) = M_{m} + F_{m} be the decomposition of mk(K_{X} + Δ) into its mobile M_{m} and fixed F_{m} part.

The log mobile part mk(K_{X} + Δ_{m}) of mk(K_{X} + Δ) = M_{m} + F_{m} is defined by

Δ_{m} = max(Δ − \frac{F_{m}}{mk} , 0),

(recall that max(A, B) is defined coefficient by coefficient). We say that mk(K_{X} + Δ) is log mobile, if Δ = Δ_{m}

The sequence Δ_{•} is called the log mobile sequence.

Note that a log mobile divisor is not necessarily mobile. However, we do have:

Lemma 6.5. Let (X, Δ) be a log pair, and let k be a positive integer such that k(K_{X} + Δ) is integral. Let m be a positive integer and let mk(K_{X} + Δ) = M_{m} + F_{m} be the decomposition of mk(K_{X} + Δ) into its mobile M_{m} and fixed F_{m} part.

Then

(1) 0 ≤ Δ_{m} ≤ Δ.

(2) M_{m} is the mobile part of mk(K_{X} + Δ_{m}).

(3) No component of Δ_{m} is a fixed component.

(4) Possibly replacing k by a multiple, the sequence Δ_{•} is convex.

Proof. (1-3) are easy to check.

If M_{m} forms an additive sequence, then B_{m} = Δ−F_{m}/mk is a convex sequence. In this case, the only way that the sequence max(B_{•}, 0) is not convex, is when the coefficient of a component of B_{m} transitions from being negative to positive. But since there are only finitely many components of Δ, this can only happen finitely many times. □

Lemma 6.6. Let (Y, Γ) be a smooth log pair, and let π: Y ⟶ Z be a projective morphism, where Z is normal and affine. Let k be any positive integer such that k(K_{Y} + Γ) is integral, and let m be a positive integer. Assume that

(1) (Y, Γ) is purely log terminal,

(2) T = ⌞Γ⌟ is irreducible,

(3) Γ − T ∼_{ℚ} A + B, where A is ample and B is effective, not containing T,

(4) no two components of ⟨Γ⟩ intersect, and

(5) the fixed part of mk(K_{Y} + Γ) and the support of Γ has global normal crossings.

Then there is a sequence of blow ups h_{m}: Y_{m} ⟶ Y along smooth centres, which are strata of the support of the fixed locus of mk(K_{Y} +Γ) and the support of Γ and which in a neighbourhood of T are smooth divisors in T, with the following property:

Suppose that we write

K_{Y_{m}} + Γ′_{m} = h_{m}^{∗}(K_{Y} + Γ) + F_{m}.

Set mk(K_{Y_{m}} + Γ_{m}) to be the log mobile part of mk(K_{Y_{m}} + Γ′_{m}) and Θ_{m} = (Γ_{m} − T)|_{T}. Let N_{m} be the mobile part of mk(K_{Y} + Γ_{m}) and let M_{m} be the mobile part of mk(K_{T} + Θ_{m}). Then

N_{m}|_{T} = M_{m}  and  |N_{m}|_{T} = |M_{m}|.

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 the base locus of |mk(K_{Y} + Γ)|, are the components of T ∩ ⟨Γ⟩. It follows that we may choose a sequence of blow ups h_{m}: Y_{m} ⟶ Y, as above, so that the base locus of N_{m} contains no log canonical centre of K_{Y_{m}} + ⌜Γ′_{m}⌝. But then the base locus of mk(K_{Y_{m}} + Γ_{m}) does not contain any log canonical centre of K_{Y_{m}} + ⌜Γ_{m}⌝.

Possibly replacing kA by a linearly equivalent divisor, we may assume that h_{m}^{∗}A and the strict transform of A are equal. Since A is ample, there is an effective and exceptional divisor F such that A_{m} = h_{m}^{∗}A−F is ample. In this case

Γ_{m} − T ∼_{ℚ} (h_{m}^{∗}A − F) + (Γ_{m} − T − h_{m}^{∗}A + F) ∼_{ℚ} A_{m} + B_{m}.

It follows by (6.3) and (5.4), that

H^{0}(Y_{m}, O_{Y_{m}}(mk(K_{Y} + Γ_{m}))) ⟶ H^{0}(T, O_{T} (mk(K_{T} + Θ_{m}))),

is surjective. It follows that N_{m}|_{T} = M_{m} + L_{m}, where L_{m} is contained in some fixed normal crossings divisor. Possibly blowing up more along the components of L_{m}, we may assume that L_{m} is empty. □

Lemma 6.7. Let (X, Δ) be a log pair. We may find a birational projective morphism

g: Y ⟶ X,

with the following properties. Suppose that we write

K_{Y} + Γ = g^{∗}(K_{X} + Δ) + E,

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, Δ) 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 ⟨Δ⟩. Now if a collection of components of ⟨Δ⟩ intersect, then certainly no irreducible component of their intersection is contained in ⌞Δ⌟. Since (X, Δ) has global normal crossings, it follows that we may as well replace Δ by ⟨Δ⟩. Thus we may assume that the coefficients of the components of Δ are all less than one, so that the pair (X, Δ) 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 Δ which intersect. Since (X, Δ) 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 rΔ 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 Δ 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 Δ_{1}, Δ_{2}, …, Δ_{k} of Δ, 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

K_{Y} + Γ = g^{∗}(K_{X} + Δ) + (k − 1 − s)E.

If k − 1 − s ≥ 0, then E is not a component of Γ. Otherwise E is a component of Γ, with coefficient s + 1 − k. Let Γ_{1}, Γ_{2}, …, Γ_{k} be the components of Γ′ which are the strict transforms of Δ_{1}, Δ_{2}, …, Δ_{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

(s − a_{k}) + (s + 1 − k) = s + [(s − a_{k}) − (k − 1)] < s,

and so we have decreased l by one. □

Proof of (6.2). Let g: Y ⟶ X be any morphism, whose existence is guaranteed by (6.7). We may write

K_{Y} + Γ = g^{∗}(K_{X} + Δ) + E,

where Γ and E are effective, with no common components and E is exceptional. Since k(K_{X} +Δ) is Cartier, k(K_{Y} +Γ) and kE are integral. Let T be the strict transform of S. Possibly blowing up more, and replacing k by a multiple, we may assume that the fixed part of |k(K_{Y} + Γ)| and the support of Γ has normal crossings.

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 A′ = g^{∗}A − F is ample. In this case

Γ − T ∼_{ℚ} (g^{∗}A − F) + (Γ − T − g^{∗}A + F) ∼_{ℚ} A′ + B′.

Let h_{m}: Y_{m} ⟶ Y be the birational morphism, whose existence is guaranteed by (6.6). Let g_{m} = h_{m} ∘ g: Y_{m} ⟶ X. Then the restriction of h_{m} to the strict transform of T is an isomorphism, and an easy log discrepancy calculation shows that Z_{m}|_{T} is independent of m. Thus (i) holds. (ii) holds by (6.6). Note that the restriction of mk(K_{Y} + Γ_{m}) to T is equal to the restriction of the log mobile part of mk(K_{Y} + Γ) to T. (iii) holds by (4) of (6.5). (iv) holds by (3.10).

Suppose that P_{m} is free. As mk(K_{Y_{m}} + Γ_{m}) is log mobile, it is clear that ml(K_{Y_{m}} + Γ_{m}) is log mobile as well. Arguing as above, and possibly blowing up more, we may assume that Q_{m}|_{T} = P_{m} and that |Q_{m}|_{T} = |P_{m}|. In this case, Q_{m} is free in a neighbourhood of T, and so possibly blowing up further, we may assume that (v) holds. □

7. Real versus rational

Most of the ideas and a significant part of the proofs of the results in this section are contained in [14]. 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, Δ) be a ℚ-factorial kawamata log terminal pair, where Δ 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} + Δ) 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

D − (K_{X} + Δ) = A + E,

where A is ample and E is effective. Thus

D − (K_{X} + Δ + ϵE),

is ample for all ϵ > 0. Since the pair (X, Δ + ϵE) is kawamata log terminal for ϵ small enough, replacing Δ by Δ + ϵE, we may assume that

D − (K_{X} + Δ),

is ample. Perturbing Δ, we may therefore assume that K_{X} + Δ 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} + Δ is negative on F. Let H be any ample divisor. For every ray R = ℝ^{+}α contained in F, there is an ϵ > 0 such that K_{X} + Δ + ϵH is negative on R. By compactness of a slice, it follows that there is an ϵ > 0, such that K_{X} + Δ + ϵH 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_{l} be the prime components of D. Consider the convex subset P of

{ B = ∑_{i} d_{i}D_{i} | d_{i} ∈ ℝ },

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} +Δ) is also ample. Now if B_{i} = (B_{i}−(K_{X} +Δ))+ (K_{X} + Δ) is not nef, then it must be negative on a (K_{X} + Δ)-extremal ray. As the extremal rays of K_{X} + Δ 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, Δ) be a kawamata log terminal pair of dimension n, such that K_{X} + Δ 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 projective morphism, where Z is normal and affine. Fix a finite dimensional vector subspace V of the space of ℝ-divisors containing Δ.

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 Δ, 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 one,

(2) W_{i} is ℚ-factorial, and

(3) K_{W_{i}} + ψ_{i∗}Θ is semiample.

Further there is a positive integer s such that

(4) if k(K_{X} + Θ) is integral then sk(K_{W_{i}} + ψ_{i∗}Θ) is base point free.

Proof. As the property of being big is an open condition, we may asssume that for any Θ ∈ V sufficiently close to Δ, K_{X} + Θ is big.

Suppose that we have established (1-3). As W_{i} is ℚ-factorial and W_{i} has rational singularities, 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 k(K_{X} + Θ) is integral, then s_{i}k(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}k(K_{W_{i}} + ψ_{i∗}Θ) is base point free. If we set s to be the least common multiple of the Ms_{i}, then this is (4). Thus it suffices to prove (1-3).

As K_{X} + Θ is big, if Θ is sufficiently close to Δ, by (7.1), we may replace (3) by the weaker condition,

(3′) K_{W_{i}} + ψ_{i∗}Θ is nef.

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} + Δ is not nef. Let R be an extremal ray for K_{X} + Δ. R is necessarily (K_{X} + Θ)-negative, for any ℚ-divisor Θ close enough to Δ. 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} + Δ is big. If ψ is divisorial (that is the exceptional locus is a divisor) then we replace the pair (X, Δ) by the pair (X′, ψ_{∗}Δ). 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} + Δ, and so we can replace the pair (X, Δ) 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} + Δ is nef.

By (7.1) it follows that K_{X}+Δ 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} + Δ = ψ^{∗}H. Thus if Θ is sufficiently close to Δ and K_{X} + Θ is relatively nef over W, then

K_{X} + Θ = K_{X} + Δ + (Θ − Δ) = ψ^{∗}H + (Θ − Δ),

is nef.

Note that we may replace Z by W, and use the fact that a Cartier 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} + Δ is ℝ-linearly equivalent to zero. Let B be the closure in V of a ball with radius δ centred at Δ. 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} + Δ is ℝ-linearly equivalent to zero, note that for every curve C,

(K_{X} + Θ) · C < 0 iff (K_{X} + Θ′) · C < 0, ∀Θ′ ∈ (Δ, Θ].

In particular every step of the (K_{X} +Θ)-MMP is a step of (K_{X} +Θ′)- MMP, for every Θ′ ∈ (Δ, 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′ ∈ (Δ, 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 Δ 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. □

Here is the key consequence of (7.2):

Corollary 7.3. Assume the real MMP in dimension n. Let (X, Δ) be a kawamata log terminal ℚ-factorial pair of dimension n, where K_{X} + Δ is an ℝ-divisor. Let f: X ⟶ Z be a contraction morphism. Let k be a positive integer. Fix a finite dimensional vector subspace V of the space of ℝ-divisors containing Δ.

If K_{X} + Δ is relatively big, then there is a birational model g: Y ⟶ X and a positive integer s, such that if π: Y ⟶ Z is the composition of f and g, then for every divisor Θ ∈ V sufficiently close to Δ, (1) if k(K_{X} +Θ) is integral, then the mobile part of g^{∗}(ks(K_{X} +Θ))

is base point free, where l = ks.

(2) If Θ_{•} is a convex sequence of divisors with limit Θ, such that mk(K_{X} + Θ_{m}) is integral, then the limit D of the characteristic sequence D_{•} associated to B_{m} = g^{∗}(ml(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

[figure: g; ϕ_{i}]

X  ^{ψ_{i}  ^{}}W_{i}.

Let Θ be sufficiently close to Δ. Then for some i, K_{W_{i}} + ψ_{i∗}Θ is semiample. Suppressing the index i, we may write

g^{∗}(K_{X} + Θ) = ϕ^{∗}(K_{W} + ψ_{∗}Θ) + E + F,

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 k(K_{X} + Θ) is integral. By (7.2) there is a fixed s such that l(K_{W} +ψ_{∗}Θ) is free. In this case the mobile part of g^{∗}(l(K_{X} +Θ)) is equal to the mobile part of ϕ^{∗}(l(K_{W} + ψ_{∗}Θ)), which is free. This is (1).

Now suppose that Θ_{•} is a convex sequence with limit Θ, such that mk(K_{X} + Θ_{m}) is integral. Let M_{m} be the mobile part of g^{∗}(ml(K_{X} + Θ_{m})). Then, by what we have already said, M_{m} is also the mobile part of ϕ_{i}^{∗}(ml(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

D_{m} = \frac{M_{m}}{m} ,

is

ϕ^{∗}(l(K_{W} + ψ_{∗}Θ)),

so that D is nef. It follows that D is semiample by (7.2) (or indeed (7.1)). □

8. Diophantine Approximation

All of the results in this section are implicit in the work of Shokurov [15], 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.

Lemma 8.1 (Diophantine Approximation). Let Y be a smooth variety and let π: Y ⟶ Z be a projective morphism, where Z is affine and normal. Let D be a semiample divisor on Y. Let ϵ > 0 be a positive rational number.

Then there is an integral divisor M and a positive integer m such that

(1) M is base point free,

(2) ∥mD − M∥ < ϵ, and

(3) If mD ≥ M then mD = M.

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

P = ∑ ℝ_{+}[P_{k}] ⊂ ℝ_{+}[G_{k}] = G.

Let v ∈ N_{ℝ} be the vector corresponding to D. Let A be the cyclic subgroup of the torus

\frac{N_{ℝ}}{N_{ℤ}},

generated by the image of v. Let Ā be the closure of A and let A_{0} be the connected component of the identity of Ā. 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 ϵ > 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

• ∥mv − w∥ < ϵ, whilst

• mv − w ∉ G.

Note that if ϵ > 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. □

Definition 8.2. 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 integers i and j,

Mob(⌜jD_{i} + F ⌝) ≤ M_{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 semiample geometric algebra on Y.

Then ℜ is finitely generated.

Proof. Let D_{•} be the characteristic sequence, with limit D. Let G be the support of D, and pick ϵ > 0 such that ⌜F − ϵG⌝ ≥ 0. By diophantine approximation, we know that there is a positive integer m and an integral divisor M such that

(1) M is mobile,

(2) ∥mD − M∥ < ϵ, and

(3) If mD ≥ M then mD = M.

But then

mD + F = M + (mD − M) + F
≥ M + F − ϵG,

so that

Mob(⌜mD + F ⌝) ≥ M.

On the other hand, by definition of saturation we have

Mob(⌜mD_{i} + F ⌝) ≤ M_{m} = mD_{m}.

Letting i go to infinity we have

M ≤ Mob(⌜mD + F ⌝) ≤ mD_{m} ≤ mD.

By (3) above, it follows that the sequence of inequalities must in fact be equalities, so that we have

D = D_{m},

for some m and we may apply (3.11). □

9. Saturation on T

We fix some notation for this section. Let (X, Δ) 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 = ⌞Δ⌟ is irreducible. Let g_{m}: Y_{m} ⟶ X be the birational morphisms whose existence is guaranteed by (6.2). Then we may write

K_{Y_{m}} + Γ′_{m} = g_{m}^{∗}(K_{X} + Δ) + E_{m},

where Γ′_{m} and E_{m} are effective, with no common components and E_{m} is g_{m}-exceptional. Let k be a positive integer such that k(K_{X} + Δ) is Cartier, and assume that the fixed part of k(K_{X} + Δ) does not contain S. Then both k(K_{Y_{m}} + Γ′_{m}) and kE_{m} are integral. Let mk(K_{Y_{m}} + Γ_{m}) be the log mobile part of mk(K_{Y_{m}} + Γ′_{m}). Set T = ⌞Γ_{m}⌟ the strict transform of S, Z_{m} = E_{m} − Γ′_{m} + T and Z = Z_{m}|_{T} (note that T and Z do not depend on m, by (6.2)). As K_{X} + Δ is purely log terminal, ⌜Z_{m}⌝ is effective and exceptional. Let Θ_{m} = (Γ_{m} −T)|_{T} and let M_{m} be the mobile part of mk(K_{T} + Θ_{m}). Our aim is to prove that the algebra R associated to mk(K_{T} + Θ_{m}) is finitely generated.

By (7.3), we may assume that there is a birational model h: T′ ⟶ T of T and a positive integer s, such that the mobile part of h^{∗}(ml(K_{T} + Θ_{m})) is free. Possibly blowing up more, we may therefore assume that the mobile part P_{m} of ml(K_{T} + Θ_{m}) is free, where l = ks. Let Q_{m} be the mobile part of ml(K_{Y} + Γ_{m}). By (7.3) we may assume that Q_{m} is free. Note that

sM_{m} ≤ P_{m} ≤ M_{ms}.

Let D_{•} be the characteristic sequence of M_{•}, let J_{•} be the characteristic sequence of Q_{•}, and let I_{•} be the characteristic sequence of P_{•}.

Lemma 9.1. For every pair of positive integers i and j,

Mov(⌜jJ_{i} + Z_{i}⌝)|_{T} ≤ M_{js}.

Proof. We work on the common resolution Y_{i,js} of Y_{i} and Y_{js}, whose existence is guaranteed by (1) of (6.2). We have

Mov(⌜jJ_{i} + Z_{i}⌝) ≤ Mov(⌜jl(K_{Y_{i}} + Γ_{i}) + Z_{i}⌝)
≤ Mov(⌜jl(K_{Y_{i}} + Γ′_{i}) + Z_{i}⌝)
= Mov(jl(K_{Y_{i}} + Γ′_{i}) + ⌜Z_{i}⌝)
= Mov(jlg_{i}^{∗}(K_{X} + Δ) + jlE_{i} + ⌜Z_{i}⌝)
= Mov(jlg_{i}^{∗}(K_{X} + Δ))
≤ Mov(jlg_{js}^{∗}(K_{X} + Δ) + jlE_{js})
= Mov((js)k(K_{Y_{js}} + Γ′_{js})) = N_{js},

where we used the fact that jlE_{i} + ⌜Z_{i}⌝ is effective and g_{i}-exceptional. Restricting to T we get

Mov(⌜jJ_{i} + Z_{i}⌝)|_{T} ≤ N_{js}|_{T} = M_{js}.
□

Lemma 9.2. The natural restriction map,

H^{0}(Y_{i}, O_{Y_{i}}(⌜jJ_{i} + Z_{i}⌝)) ⟶ H^{0}(T, O_{T} (⌜jI_{i} + Z⌝)),

is surjective, for any positive integers i and j.

Proof. Considering the restriction exact sequence,

0 ⟶ O_{Y_{i}}(⌜jJ_{i} + Z_{i}⌝−T) ⟶ O_{Y_{i}}(⌜jJ_{i} + Z_{i}⌝) ⟶ O_{T} (⌜jI_{i} + Z⌝) ⟶ 0,

it follows that the obstruction to surjectivity of the restriction map above is given by,

H^{1}(Y_{i}, O_{Y_{i}}(⌜jJ_{i} + (Z_{i} − T)⌝))

= H^{1}(Y_{i}, O_{Y_{i}}(K_{Y_{i}} + ⌜g_{i}^{∗}(−(K_{X} + Δ) + jJ_{i})⌝)),

which vanishes by Kawamata-Viehweg vanishing, as

g_{i}^{∗}(−(K_{X} + Δ)),

is big and nef and jJ_{i} is nef. □

Lemma 9.3. Suppose that −(K_{X} + Δ) is nef and big. Then the truncation of algebra ℜ, given by s, is saturated with respect to Z.

Proof. By assumption

⌜Z⌝ ≥ 0.

On the other hand, by (9.2),

Mov(⌜jI_{i} + Z⌝) ≤ Mov(⌜jJ_{i} + Z_{i}⌝)|_{T} .

Thus

Mov(⌜(js)D_{i} + Z⌝) ≤ Mov(⌜jI_{i} + Z⌝)
≤ M_{js} = jsD_{js},

where we used (9.1), and the fact that sM_{i} ≤ P_{i}. If we replace i by is and then pass from the sequence D_{i} to the sequence D_{is}, this has the effect of replacing js by j, and the result follows. □

Proof of (4.3). By (iv) of (6.2), it suffices to prove that ℜ is finitely generated. We want to apply (8.3). By (9.3), we may assume that ℜ is saturated. By (7.3), the limit I of the characteristic sequence I_{•} is semiample. On the other hand, as

sD_{i} ≤ I_{i} ≤ sD_{si},

I = sD, so that D is also semiample, and we may apply (8.3). □

Proof of (1.1). Immediate from (4.3) and (4.4). □

Proof of (1.2). Clear. □

Proof of (1.3). Follows from (5.1.3) of [14]. □

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Department of Mathematics, University of Utah, 155 South 1400 E, JWB 233, Salt Lake City, UT 84112, USA

  E-mail address: hacon@math.utah.edu

Department of Mathematics, University of California at Santa Barbara, Santa Barbara, CA 93106, USA

  E-mail address: mckernan@math.ucsb.edu