EXISTENCE OF MINIMAL MODELS FOR VARIETIES OF LOG GENERAL TYPE

CAUCHER BIRKAR, PAOLO CASCINI, CHRISTOPHER D. HACON, AND JAMES M^{C}KERNAN

Abstract. We prove that the canonical ring of a smooth projective variety is finitely generated.

Contents

1. Introduction  2
1.1. Minimal models  5
1.2. Moduli spaces  8
1.3. Fano varieties  9
1.4. Birational geometry  10
2. Description of the proof  12
2.1. Sketch of the proof  14
2.2. Standard conjectures of the MMP  23
3. Preliminary results  24
3.1. Notation and conventions  24
3.2. Preliminaries  26
3.3. Nakayama-Zariski decomposition  27
3.4. Adjunction  28
3.5. Stable base locus  28
3.6. Types of model  32

Date: August 5, 2009.

The first author was partially supported by EPSRC grant GR/S92854/02, the second author was partially supported by NSF research grant no: 0801258, the third author was partially supported by NSF research grant no: 0456363 and an AMS Centennial fellowship and the fourth author was partially supported by NSA grant no: H98230-06-1-0059 and NSF grant no: 0701101. The last author would like to thank Sogang University and Professor Yongnam Lee for their generous hospitality, where some of the work for this paper was completed. We would like to thank Dan Abramovich, Valery Alexeev, Florin Ambro, Tommaso de Fernex, Stephane Dreul, Seán Keel, Kalle Karu, János Kollár, Sándor Kovács, Michael McQuillan, Shigefumi Mori, Martin Olsson, Genia Tevelev, Burt Totaro, Angelo Vistoli and Chengyang Xu for answering many of our questions and pointing out some errors in an earlier version of this paper. We would also like to thank the referee for some useful comments.

3.7. Convex geometry and Diophantine approximation  38
3.8. Rational curves of low degree  43
3.9. Effective base point free theorem  45
3.10. The MMP with scaling  46
3.11. Shokurov’s polytopes  51
4. Special finiteness  53
5. Log terminal models  58
6. Non-vanishing  62
7. Finiteness of models  70
8. Finite generation  73
9. Proof of Theorems  74
10. Proof of Corollaries  74
References  81

1. Introduction

The purpose of this paper is to prove the following result in birational algebraic geometry:

Theorem 1.1. Let (X, Δ) be a projective kawamata log terminal pair.

If Δ is big and K_{X} + Δ is pseudo-effective then K_{X} + Δ has a log terminal model.

In particular, it follows that if K_{X} + Δ is big then it has a log canonical model and the canonical ring is finitely generated. It also follows that if X is a smooth projective variety, then the ring

R(X, K_{X}) = ⨁_{m∈ℕ}H^{0}(X, O_{X}(mK_{X})),

is finitely generated.

The birational classification of complex projective surfaces was understood by the Italian Algebraic Geometers in the early 20th century: If X is a smooth complex projective surface of non-negative Kodaira dimension, that is κ(X, K_{X}) ≥ 0, then there is a unique smooth surface Y birational to X such that the canonical class K_{Y} is nef (that is K_{Y} · C ≥ 0 for any curve C ⊂ Y). Y is obtained from X simply by contracting all −1-curves, that is all smooth rational curves E with K_{X} · E = −1. If, on the other hand, κ(X, K_{X}) = −∞, then X is birational to either ℙ^{2} or a ruled surface over a curve of genus g > 0.

The minimal model program aims to generalise the classification of complex projective surfaces to higher dimensional varieties. The main goal of this program is to show that given any n-dimensional smooth complex projective variety X, we have:

• If κ(X, K_{X}) ≥ 0, then there exists a minimal model, that is a variety Y birational to X such that K_{Y} is nef.

• If κ(X, K_{X}) = −∞, then there is a variety Y birational to X which admits a Fano fibration, that is a morphism Y ⟶ Z whose general fibres F have ample anticanonical class −K_{F}.

It is possible to exhibit 3-folds which have no smooth minimal model, see for example (16.17) of [37], and so one must allow varieties X with singularities. However, these singularities cannot be arbitrary. At the very minimum, we must still be able to compute K_{X} · C for any curve C ⊂ X. So, we insist that K_{X} is ℚ-Cartier (or sometimes we require the stronger property that X is ℚ-factorial). We also require that X and the minimal model Y have the same pluricanonical forms. This condition is essentially equivalent to requiring that the induced birational map ϕ: X −−→ Y is K_{X}-non-positive.

There are two natural ways to construct the minimal model (it turns out that if one can construct a minimal model for pseudo-effective K_{X}, then we can construct Mori fibre spaces whenever K_{X} is not pseudoeffective). Since one of the main ideas of this paper is to blend the techniques of both methods, we describe both methods.

The first method is to use the ideas behind finite generation. If the canonical ring

R(X, K_{X}) = ⨁_{m∈ℕ}H^{0}(X, O_{X}(mK_{X})),

is finitely generated and K_{X} is big, then the canonical model Y is nothing more than the Proj of R(X, K_{X}). It is then automatic that the induced rational map ϕ: X −−→ Y is K_{X}-negative.

The other natural way to ensure that ϕ is K_{X}-negative is to factor ϕ into a sequence of elementary steps all of which are K_{X}-negative. We now explain one way to achieve this factorisation.

If K_{X} is not nef, then, by the cone theorem, there is a rational curve C ⊂ X such that K_{X} · C < 0 and a morphism f: X ⟶ Z which is surjective, with connected fibres, on to a normal projective variety and which contracts an irreducible curve D if and only if [D] ∈ ℝ^{+}[C] ⊂ N_{1}(X). Note that ρ(X/Z) = 1 and −K_{X} is f-ample. We have the following possibilities:

• If dim Z < dim X, this is the required Fano fibration.

• If dim Z = dim X and f contracts a divisor, then we say that f is a divisorial contraction and we replace X by Z.

• If dim Z = dim X and f does not contract a divisor, then we say that f is a small contraction. In this case K_{Z} is not ℚ-Cartier, so that we can not replace X by Z. Instead, we would like to replace f: X ⟶ Z by its flip f^{+} : X^{+} ⟶ Z where X^{+} is isomorphic to X in codimension 1 and K_{X^{+}} is f^{+}-ample. In other words, we wish to replace some K_{X}-negative curves by K_{X^{+}}-positive curves.

The idea is to simply repeat the above procedure until we obtain either a minimal model or a Fano fibration. For this procedure to succeed, we must show that flips always exist and that they eventually terminate. Since the Picard number ρ(X) drops by one after each divisorial contraction and is unchanged after each flip, there can be at most finitely many divisorial contractions. So we must show that there is no infinite sequence of flips.

This program was successfully completed for 3-folds in the 1980’s by the work of Kawamata, Kollár, Mori, Reid, Shokurov and others. In particular, the existence of 3-fold flips was proved by Mori in [26].

Naturally, one would hope to extend these results to dimension 4 and higher by induction on the dimension.

Recently, Shokurov has shown the existence of flips in dimension 4 [34] and Hacon and M^{c}Kernan [8] have shown that assuming the minimal model program in dimension n − 1 (or even better simply finiteness of minimal models in dimension n − 1), then flips exist in dimension n. Thus we get an inductive approach to finite generation.

Unfortunately the problem of showing termination of an arbitrary sequence of flips seems to be a very difficult problem and in dimension ≥ 4 only some partial answers are available. Kawamata, Matsuda and Matsuki proved [18] the termination of terminal 4-fold flips, Matsuki has shown [25] the termination of terminal 4-fold flops and Fujino has shown [5] the termination of canonical 4-fold (log) flips. Alexeev, Hacon and Kawamata [1] have shown the termination of kawamata log terminal 4-fold flips when the Kodaira dimension of −(K_{X} +Δ) is nonnegative and the existence of minimal models of kawamata log terminal 4-folds when either Δ or K_{X} +Δ is big by showing the termination of a certain sequence of flips (those that appear in the MMP with scaling). However, it is known that termination of flips follows from two natural conjectures on the behaviour of the log discrepancies of n-dimensional pairs (namely the ascending chain condition for minimal log discrepancies and semicontinuity of log discrepancies, cf. [35]). Moreover, if κ(X, K_{X} +Δ) ≥ 0, Birkar has shown [2] that it suffices to establish acc for log canonical thresholds and the MMP in dimension one less.

We now turn to the main result of the paper:

Theorem 1.2. Let (X, Δ) be a kawamata log terminal pair, where K_{X} + Δ is ℝ-Cartier. Let π: X ⟶ U be a projective morphism of quasi-projective varieties.

If either Δ is π-big and K_{X} + Δ is π-pseudo-effective or K_{X} + Δ is π-big, then

(1) K_{X} + Δ has a log terminal model over U,

(2) if K_{X} + Δ is π-big then K_{X} + Δ has a log canonical model over U, and

(3) if K_{X} + Δ is ℚ-Cartier, then the O_{U}-algebra

ℜ(π, Δ) = ⨁_{m∈ℕ}π_{∗}O_{X}(⌞m(K_{X} + Δ)⌟),

is finitely generated.

We now present some consequences of (1.2), most of which are known to follow from the MMP. Even though we do not prove termination of flips, we are able to derive many of the consequences of the existence of the MMP. In many cases we do not state the strongest results possible; anyone interested in further applications is directed to the references. We group these consequences under different headings.

1.1. Minimal models. An immediate consequence of (1.2) is:

Corollary 1.1.1. Let X be a smooth projective variety of general type. Then

(1) X has a minimal model,

(2) X has a canonical model,

(3) the ring

R(X, K_{X}) = ⨁_{m∈ℕ}H^{0}(X, O_{X}(mK_{X})),

is finitely generated, and

(4) X has a model with a Kähler-Einstein metric.

Note that (4) follows from (2) and Theorem D of [4]. Note that Siu has announced a proof of finite generation for varieties of general type, using analytic methods, see [36].

Corollary 1.1.2. Let (X, Δ) be a projective kawamata log terminal pair, where K_{X} + Δ is ℚ-Cartier.

Then the ring

R(X, K_{X} + Δ) = ⨁_{m∈ℕ}H^{0}(X, O_{X}(⌞m(K_{X} + Δ)⌟)),

is finitely generated.

Let us emphasize that in (1.1.2) we make no assumption about K_{X} + Δ or Δ being big. Indeed Fujino and Mori, [6], proved that (1.1.2) follows from the case when K_{X} + Δ is big.

We will now turn our attention to the geography of minimal models. It is well known that log terminal models are not unique. The first natural question about log terminal models is to understand how any two are related. In fact there is a very simple connection:

Corollary 1.1.3. Let π: X ⟶ U be a projective morphism of normal quasi-projective varieties. Suppose that K_{X} + Δ is kawamata log terminal and Δ is big over U. Let ϕ_{i}: X −−→ Y_{i}, i = 1 and 2 be two log terminal models of (X, Δ) over U. Let Γ_{i} = ϕ_{i∗}Δ.

Then the birational map Y_{1} −−→ Y_{2} is the composition of a sequence of (K_{Y_{1}} + Γ_{1})-flops over U.

Note that (1.1.3) has been generalised recently to the case when Δ is not assumed big, [17]. The next natural problem is to understand how many different models there are. Even if log terminal models are not unique, in many important contexts, there are only finitely many. In fact Shokurov realised that much more is true. He realised that the dependence on Δ is well-behaved. To explain this, we need some definitions:

Definition 1.1.4. Let π: X ⟶ U be a projective morphism of normal quasi-projective varieties, and let V be a finite dimensional affine subspace of the real vector space WDiv_{ℝ}(X) of Weil divisors on X. Fix an ℝ-divisor A ≥ 0 and define

V_{A} = { Δ | Δ = A + B, B ∈ V },
L_{A}(V) = { Δ = A + B ∈ V_{A} | K_{X} + Δ is log canonical and B ≥ 0 },
E_{A,π}(V) = { Δ ∈ L_{A}(V) | K_{X} + Δ is pseudo-effective over U },
N_{A,π}(V) = { Δ ∈ L_{A}(V) | K_{X} + Δ is nef over U }.

Given a birational contraction ϕ: X −−→ Y over U, define

W_{ϕ,A,π}(V) = { Δ ∈ E_{A,π}(V) | ϕ is a weak log canonical model for (X, Δ) over U },

and given a rational map ψ: X −−→ Z over U, define

A_{ψ,A,π}(V) = { Δ ∈ E_{A,π}(V) | ψ is the ample model for (X, Δ) over U },

(cf. (3.6.7) and (3.6.5) for the definitions of weak log canonical model and ample model for (X, Δ) over U).

We will adopt the convention that L(V) = L_{0}(V). If the support of A has no components in common with any element of V then the condition that B ≥ 0 is vacuous. In many applications, A will be an ample ℚ-divisor over U. In this case, we often assume that A is general in the sense that we fix a positive integer such that kA is very ample over U, and we assume that A = \frac{1}{k}A′, where A′ ∼_{U} kA is very general. With this choice of A, we have

N_{A,π}(V) ⊂ E_{A,π}(V) ⊂ L_{A}(V) = L(V) + A ⊂ V_{A} = V + A,

and the condition that the support of A has no common components with any element of V is then automatic. The following result was first proved by Shokurov [32] assuming the existence and termination of flips:

Corollary 1.1.5. Let π: X ⟶ U be a projective morphism of normal quasi-projective varieties. Let V be a finite dimensional affine subspace of WDiv_{ℝ}(X) which is defined over the rationals. Suppose there is a divisor Δ_{0} ∈ V such that K_{X} +Δ_{0} is kawamata log terminal. Let A be a general ample ℚ-divisor over U, which has no components in common with any element of V.

(1) There are finitely many birational contractions ϕ_{i}: X −−→ Y_{i} over U, 1 ≤ i ≤ p such that

E_{A,π}(V) = ⋃_{i=1}^{p} W_{i},

where each W_{i} = W_{ϕ_{i},A,π}(V) is a rational polytope. Moreover, if ϕ: X −−→ Y is a log terminal model of (X, Δ) over U, for some Δ ∈ E_{A,π}(V), then ϕ = ϕ_{i}, for some 1 ≤ i ≤ p.

(2) There are finitely many rational maps ψ_{j}: X −−→ Z_{j} over U, 1 ≤ j ≤ q which partition E_{A,π}(V) into the subsets A_{j} = A_{ψ_{j},A,π}(V).

(3) For every 1 ≤ i ≤ p there is a 1 ≤ j ≤ q and a morphism f_{i,j}: Y_{i} ⟶ Z_{j} such that W_{i} ⊂ Ā_{j}.

In particular E_{A,π}(V) is a rational polytope and each Ā_{j} is a finite union of rational polytopes.

Definition 1.1.6. Let (X, Δ) be a kawamata log terminal pair and let D be a big divisor. Suppose that K_{X} + Δ is not pseudo-effective. The effective log threshold is

σ(X, Δ, D) = sup{ t ∈ ℝ | D + t(K_{X} + Δ) is pseudo-effective }.

The Kodaira energy is the reciprocal of the effective log threshold.

Following ideas of Batyrev, one can easily show that:

Corollary 1.1.7. Let (X, Δ) be a projective kawamata log terminal pair and let D be an ample divisor. Suppose that K_{X} +Δ is not pseudoeffective.

If both K_{X} + Δ and D are ℚ-Cartier then the effective log threshold and the Kodaira energy are rational.

Definition 1.1.8. Let π: X ⟶ U be a projective morphism of normal quasi-projective varieties. Let D^{•} = (D_{1}, D_{2}, …, D_{k}) be a sequence of ℚ-divisors on X. The sheaf of O_{U}-modules

ℜ(π, D^{•}) = ⨁_{m∈ℕ^{k}} π_{∗}O_{X}(⌞∑ m_{i}D_{i}⌟),

is called the Cox ring associated to D^{•}.

Using (1.1.5) one can show that adjoint Cox rings are finitely generated:

Corollary 1.1.9. Let π: X ⟶ U be a projective morphism of normal quasi-projective varieties. Fix A ≥ 0 an ample ℚ-divisor over U. Let Δ_{i} = A + B_{i}, for some ℚ-divisors B_{1}, B_{2}, …, B_{k} ≥ 0. Assume that D_{i} = K_{X} + Δ_{i} is divisorially log terminal and ℚ-Cartier. Then the Cox ring

ℜ(π, D^{•}) = ⨁_{m∈ℕ^{k}} π_{∗}O_{X}(⌞∑ m_{i}D_{i}⌟),

is a finitely generated O_{U}-module.

1.2. Moduli spaces. At first sight (1.1.5) might seem a hard result to digest. For this reason, we would like to give a concrete, but non-trivial example. The moduli spaces \overline{M}_{g,n} of n-pointed stable curves of genus g are probably the most intensively studied moduli spaces. In particular the problem of trying to understand the related log canonical models via the theory of moduli has attracted a lot of attention (e.g. see [7],

[24] and [11]).

Corollary 1.2.1. Let X = \overline{M}_{g,n} the moduli space of stable curves of genus g with n marked points and let Δ_{i}, 1 ≤ i ≤ k denote the boundary divisors.

Let Δ = ∑_{i} a_{i}Δ_{i} be a boundary. Then K_{X} + Δ is log canonical and if K_{X} +Δ is big then there is a log canonical model X −−→ Y. Moreover if we fix a positive rational number δ and require that the coefficient a_{i} of Δ_{i} is at least δ for each i then the set of all log canonical models obtained this way is finite.

1.3. Fano varieties. The next set of applications is to Fano varieties. The key observation is that given any divisor D, a small multiple of D is linearly equivalent to a divisor of the form K_{X} + Δ, where Δ is big and K_{X} + Δ is kawamata log terminal.

Definition 1.3.1. Let π: X ⟶ U be a projective morphism of normal varieties, where U is affine.

We say that X is a Mori dream space if h^{1}(X, O_{X}) = 0 and the Cox ring is finitely generated over the coordinate ring of U.

Corollary 1.3.2. Let π: X ⟶ U be a projective morphism of normal varieties, where U is affine. Suppose that X is ℚ-factorial, K_{X} + Δ is divisorially log terminal and −(K_{X} + Δ) is ample over U.

Then X is a Mori dream space.

There are many reasons why Mori dream spaces are interesting. As the name might suggest, they behave very well with respect to the minimal model program. Given any divisor D, one can run the D-MMP, and this ends with either a nef model, or a fibration, for which −D is relatively ample, and in fact any sequence of D-flips terminates.

(1.3.2) was conjectured in [12] where it is also shown that Mori dream spaces are GIT quotients of affine varieties by a torus. Moreover the decomposition given in (1.1.5) is induced by all the possible ways of taking GIT quotients, as one varies the linearisation.

Finally, it was shown in [12] that if one has a Mori dream space, then the Cox Ring is finitely generated.

We next prove a result that naturally complements (1.2). We show that if K_{X} + Δ is not pseudo-effective, then we can run the MMP with scaling to get a Mori fibre space:

Corollary 1.3.3. Let (X, Δ) be a ℚ-factorial kawamata log terminal pair. Let π: X ⟶ U be a projective morphism of normal quasiprojective varieties. Suppose that K_{X} + Δ is not π-pseudo-effective.

Then we may run f: X −−→ Y a (K_{X} + Δ)-MMP over U and end with a Mori fibre space g: Y ⟶ W over U.

Note that we do not claim in (1.3.3) that however we run the (K_{X} + Δ)-MMP over U, we always end with a Mori fibre space, that is we do not claim that every sequence of flips terminates.

Finally we are able to prove a conjecture of Batyrev on the closed cone of nef curves for a Fano pair.

Definition 1.3.4. Let X be a projective variety. A curve Σ is called nef if B·Σ ≥ 0 for all Cartier divisors B ≥ 0. NF(X) denotes the cone of nef curves sitting inside H_{2}(X, ℝ) and \overline{NF}(X) denotes its closure.

Now suppose that (X, Δ) is a log pair. A (K_{X} + Δ)-co-extremal ray is an extremal ray F of the closed cone of nef curves \overline{NF}(X) on which K_{X} + Δ is negative.

Corollary 1.3.5. Let (X, Δ) be a projective ℚ-factorial kawamata log terminal pair such that −(K_{X} + Δ) is ample.

Then \overline{NF}(X) is a rational polyhedron. If F = F_{i} is a (K_{X} + Δ)-co-extremal ray then there exists an ℝ-divisor Θ such that the pair (X, Θ) is kawamata log terminal and the (K_{X} + Θ)-MMP π: X −−→ Y ends with a Mori fibre space f: Y ⟶ Z such that F is spanned by the pullback to X of the class of any curve Σ which is contracted by f.

1.4. Birational geometry. Another immediate consequence of (1.2) is the existence of flips:

Corollary 1.4.1. Let (X, Δ) be a kawamata log terminal pair and let π: X ⟶ Z be a small (K_{X} + Δ)-extremal contraction.

Then the flip of π exists.

As already noted, we are unable to prove the termination of flips in general. However, using (1.1.5), we can show that any sequence of flips for the MMP with scaling terminates:

Corollary 1.4.2. Let π: X ⟶ U be a projective morphism of normal quasi-projective varieties. Let (X, Δ) be a ℚ-factorial kawamata log terminal pair, where K_{X} + Δ is ℝ-Cartier and Δ is π-big. Let C ≥ 0 be an ℝ-divisor.

If K_{X} + Δ + C is kawamata log terminal and π-nef, then we may run the (K_{X} + Δ)-MMP over U with scaling of C.

Another application of (1.2) is the existence of log terminal models which extract certain divisors:

Corollary 1.4.3. Let (X, Δ) be a log canonical pair and let f: W ⟶ X be a log resolution. Suppose that there is a divisor Δ_{0} such that K_{X} + Δ_{0} is kawamata log terminal. Let 𝔈 be any set of valuations of f-exceptional divisors which satisfies the following two properties:

(1) 𝔈 contains only valuations of log discrepancy at most one, and (2) the centre of every valuation of log discrepancy one in 𝔈 does

not contain any non kawamata log terminal centres. Then we may find a birational morphism π: Y ⟶ X, such that Y is

ℚ-factorial and the exceptional divisors of π correspond to the elements of 𝔈.

For example, if we assume that (X, Δ) is kawamata log terminal and we let 𝔈 be the set of all exceptional divisors with log discrepancy at most one, then the birational morphism π: Y ⟶ X defined in (1.4.3) above is a terminal model of (X, Δ). In particular there is an ℝdivisor Γ ≥ 0 on W such that K_{Y} + Γ = π^{∗}(K_{X} + Δ) and the pair (Y, Γ) is terminal.

If instead we assume that (X, Δ) is kawamata log terminal but 𝔈 is empty, then the birational morphism π: Y ⟶ X defined in (1.4.3) above is a log terminal model. In particular π is small, Y is ℚ-factorial and there is an ℝ-divisor Γ ≥ 0 on Y such that K_{Y} + Γ = π^{∗}(K_{X} + Δ).

We are able to prove that every log pair admits a birational model with ℚ-factorial singularities such that the non kawamata log terminal locus is a divisor:

Corollary 1.4.4. Let (X, Δ) be a log pair.

Then there is a birational morphism π: Y ⟶ X, where Y is ℚfactorial, such that if we write

K_{Y} + Γ = K_{Y} + Γ_{1} + Γ_{2} = π^{∗}(K_{X} + Δ),

where every component of Γ_{1} has coefficient less than one and every component of Γ_{2} has coefficient at least one, then K_{Y} +Γ_{1} is kawamata log terminal and nef over X and no component of Γ_{1} is exceptional.

Even thought the result in (1.4.3) is not optimal as it does not fully address the log canonical case, nevertheless, we are able to prove the following result (cf. [31], [21], [13]):

Corollary 1.4.5 (Inversion of adjunction). Let (X, Δ) be a log pair and let ν: S ⟶ S′ be the normalisation of a component S′ of Δ of coefficient one.

If we define Θ by adjunction,

ν^{∗}(K_{X} + Δ) = K_{S} + Θ,

then the log discrepancy of K_{S} + Θ is equal to the minimum of the log discrepancy with respect to K_{X} + Δ of any valuation whose centre on X is of codimension at least two and intersects S.

One of the most compelling reasons to enlarge the category of varieties to the category of algebraic spaces (equivalently Moishezon spaces, at least in the proper case) is to allow the possibility of cut and paste operations, such as one can perform in topology. Unfortunately, it is then all too easy to construct proper smooth algebraic spaces over ℂ, which are not projective. In fact the appendix to [10] has two very well known examples due to Hironaka. In both examples, one exploits the fact that for two curves in a threefold which intersect in a node, the order in which one blows up the curves is important (in fact the resulting threefolds are connected by a flop).

It is then natural to wonder if this is the only way to construct such examples, in the sense that if a proper algebraic space is not projective then it must contain a rational curve. Kollár dealt with the case when X is a terminal threefold with Picard number one, see [19]. In a slightly different but related direction, it is conjectured that if a complex Kähler manifold M does not contain any rational curves then K_{M} is nef (see for example [30]), which would extend some of Mori’s famous results from the projective case. Kollár also has some unpublished proofs of some related results.

The following result, which was proved by Shokurov assuming the existence and termination of flips, cf. [33], gives an affirmative answer to the first conjecture and at the same time connects the two conjectures:

Corollary 1.4.6. Let π: X ⟶ U be a proper map of normal algebraic spaces, where X is analytically ℚ-factorial.

If K_{X} + Δ is divisorially log terminal and π does not contract any rational curves then π is a log terminal model. In particular π is projective and K_{X} + Δ is π-nef.

2. Description of the proof

Theorem A (Existence of pl-flips). Let f: X ⟶ Z be a pl-flipping contraction for an n-dimensional purely log terminal pair (X, Δ).

Then the flip f^{+} : X^{+} ⟶ Z of f exists.

Theorem B (Special finiteness). Let π: X ⟶ U be a projective morphism of normal quasi-projective varieties, where X is ℚ-factorial of dimension n. Let V be a finite dimensional affine subspace of WDiv_{ℝ}(X), which is defined over the rationals, let S be the sum of finitely many prime divisors and let A be a general ample ℚ-divisor over U. Let (X, Δ_{0}) be a divisorially log terminal pair such that S ≤ Δ_{0}. Fix a finite set 𝔈 of prime divisors on X.

Then there are finitely 1 ≤ i ≤ k many birational maps ϕ_{i}: X −−→ Y_{i} over U such that if ϕ: X −−→ Y is any ℚ-factorial weak log canonical model over U of K_{X} + Δ where Δ ∈ L_{S+A}(V) which only contracts elements of 𝔈 and which does not contract every component of S, then there is an index 1 ≤ i ≤ k such that the induced birational map ξ: Y_{i} −−→ Y is an isomorphism in a neighbourhood of the strict transforms of S.

Theorem C (Existence of log terminal models). Let π: X ⟶ U be a projective morphism of normal quasi-projective varieties, where X has dimension n. Suppose that K_{X} + Δ is kawamata log terminal, where Δ is big over U.

If there exists an ℝ-divisor D such that K_{X} + Δ ∼_{ℝ,U} D ≥ 0, then K_{X} + Δ has a log terminal model over U.

Theorem D (Non-vanishing theorem). Let π: X ⟶ U be a projective morphism of normal quasi-projective varieties, where X has dimension n. Suppose that K_{X} + Δ is kawamata log terminal, where Δ is big over U.

If K_{X} + Δ is π-pseudo-effective, then there exists an ℝ-divisor D such that K_{X} + Δ ∼_{ℝ,U} D ≥ 0.

Theorem E (Finiteness of models). Let π: X ⟶ U be a projective morphism of normal quasi-projective varieties, where X has dimension n. Fix a general ample ℚ-divisor A ≥ 0 over U. Let V be a finite dimensional affine subspace of WDiv_{ℝ}(X) which is defined over the rationals. Suppose that there is a kawamata log terminal pair (X, Δ_{0}).

Then there are finitely many birational maps ψ_{j}: X −−→ Z_{j} over U, 1 ≤ j ≤ l such that if ψ: X −−→ Z is a weak log canonical model of K_{X} + Δ over U, for some Δ ∈ L_{A}(V) then there is an index 1 ≤ j ≤ l and an isomorphism ξ: Z_{j} ⟶ Z such that ψ = ξ ∘ ψ_{j}.

Theorem F (Finite generation). Let π: X ⟶ Z be a projective morphism to a normal affine variety. Let (X, Δ = A + B) be a kawamata log terminal pair of dimension n, where A ≥ 0 is an ample ℚ-divisor and B ≥ 0. If K_{X} + Δ is pseudo-effective, then

(1) The pair (X, Δ) has a log terminal model μ: X −−→ Y. In particular if K_{X} + Δ is ℚ-Cartier then the log canonical ring

R(X, K_{X} + Δ) = ⨁_{m∈ℕ}H^{0}(X, O_{X}(⌞m(K_{X} + Δ)⌟)),

is finitely generated.

(2) Let V ⊂ WDiv_{ℝ}(X) be the vector space spanned by the components of Δ. Then there is a constant δ > 0 such that if G is a prime divisor contained in the stable base locus of K_{X} + Δ and Ξ ∈ L_{A}(V) such that ∥Ξ − Δ∥ < δ, then G is contained in the stable base locus of K_{X} + Ξ.

(3) Let W ⊂ V be the smallest affine subspace of WDiv_{ℝ}(X) containing Δ, which is defined over the rationals. Then there is a constant η > 0 and a positive integer r > 0 such that if Ξ ∈ W is any divisor and k is any positive integer such that ∥Ξ − Δ∥ < η and k(K_{X} + Ξ)/r is Cartier, then every component of Fix(k(K_{X} + Ξ)) is a component of the stable base locus of K_{X} + Δ.

The proof of Theorem A, Theorem B, Theorem C, Theorem D, Theorem E and Theorem F proceeds by induction:

• Theorem F_{n−1} implies Theorem A_{n}, see the main result of [9].

• Theorem E_{n−1} implies Theorem B_{n}, cf. (4.4).

• Theorem A_{n} and Theorem B_{n} imply Theorem C_{n}, cf. (5.6).

• Theorem D_{n−1}, Theorem B_{n} and Theorem C_{n} imply Theorem D_{n}, cf. (6.6).

• Theorem C_{n} and Theorem D_{n} imply Theorem E_{n}, cf. (7.3).

• Theorem C_{n} and Theorem D_{n} imply Theorem F_{n}, cf. (8.1).

2.1. Sketch of the proof. To help the reader navigate through the technical problems which naturally arise when trying to prove (1.2), we review a natural approach to proving that the canonical ring

R(X, K_{X}) = ⨁_{m∈ℕ}H^{0}(X, O_{X}(mK_{X})),

of a smooth projective variety X of general type is finitely generated. Even though we do not directly follow this method to prove the existence of log terminal models, instead using ideas from the MMP, many of the difficulties which arise in our approach are mirrored in trying to prove finite generation directly.

A very natural way to proceed is to pick a divisor D ∈ |kK_{X}|, whose existence is guaranteed as we are assuming that K_{X} is big, and then to restrict to D. One obtains an exact sequence

0 ⟶ H^{0}(X, O_{X}((l−k)K_{X})) ⟶ H^{0}(X, O_{X}(lK_{X})) ⟶ H^{0}(D, O_{D}(mK_{D})),

where l = m(1 + k) is divisible by k + 1, and it is easy to see that it suffices to prove that the restricted algebra, given by the image of the maps

H^{0}(X, O_{X}(m(1 + k)K_{X})) ⟶ H^{0}(D, O_{D}(mK_{D})),

is finitely generated. Various problems arise at this point. First D is neither smooth nor even reduced (which, for example, means that the symbol K_{D} is only formally defined; strictly speaking we ought to work with the dualising sheaf ω_{D}). It is natural then to pass to a log resolution, so that the support S of D has simple normal crossings, and to replace D by S. The second problem is that the kernel of the map

H^{0}(X, O_{X}(m(1 + k)K_{X})) ⟶ H^{0}(S, O_{S}(mK_{S})),

no longer has any obvious connection with

H^{0}(X, O_{X}((m(1 + k) − k)K_{X})),

so that even if we knew that the new restricted algebra were finitely generated, it is not immediate that this is enough. Another significant problem is to identify the restricted algebra as a subalgebra of

⨁_{m∈ℕ}H^{0}(S, O_{S}(mK_{S})),

since it is only the latter that we can handle by induction. Yet another problem is that if C is a component of S, it is no longer the case that C is of general type, so that we need a more general induction. In this case the most significant problem to deal with is that even if K_{C} is pseudo-effective, it is not clear that the linear system |kK_{C}| is nonempty for any k > 0. Finally, even though this aspect of the problem may not be apparent from the description above, in practise it seems as though we need to work with infinitely many different values of k and hence D = D_{k}, which entails working with infinitely many different birational models of X (since for every different value of k, one needs to resolve the singularities of D).

Let us consider one special case of the considerations above, which will hopefully throw some more light on the problem of finite generation. Suppose that to resolve the singularities of D we need to blow up a subvariety V. The corresponding divisor C will typically fibre over V and if V has codimension two then C will be close to a ℙ^{1}- bundle over V. In the best case, the projection π: C ⟶ V will be a ℙ^{1}-bundle with two disjoint sections (this is the toroidal case) and sections of tensor powers of a line bundle on C will give sections of an algebra on V which is graded by ℕ^{2}, rather than just ℕ. Let us consider then the simplest possible algebras over ℂ which are graded by ℕ^{2}. If we are given a submonoid M ⊂ ℕ^{2} (that is a subset of ℕ^{2} which contains the origin and is closed under addition) then we get a subalgebra R ⊂ ℂ^{2}[x, y] spanned by the monomials

{ x^{i}y^{j} | (i, j) ∈ M }.

The basic observation is that R is finitely generated iff M is a finitely generated monoid. There are two obvious cases when M is not finitely generated,

M = \{ (i, j) ∈ ℕ^{2} | j > 0 \} ∪ \{(0, 0)\} and \{ (i, j) ∈ ℕ^{2} | i ≥ \sqrt{2}j \}.

In fact, if C ⊂ ℝ^{2} is the convex hull of the set M, then M is finitely generated iff C is a rational polytope. In the general case, we will be given a convex subset C of a finite dimensional vector space of Weil divisors on X and a key part of the proof is to show that the set C is in fact a rational polytope. As naive as these examples are, hopefully they indicate why it is central to the proof of finite generation to

• consider divisors with real coefficients, and

• prove a non-vanishing result.

We now review our approach to the proof of (1.2). As is clear from the plan of the proof given in the previous subsection, the proof of (1.2) is by induction on the dimension and the proof is split into various parts. Instead of proving directly that the canonical ring is finitely generated, we try to construct a log terminal model for X. The first part is to prove the existence of pl-flips. This is proved by induction in [8], and we will not talk about the proof of this result here, since the methods used to prove this result are very different from the methods we use here. Granted the existence of pl-flips, the main issue is to prove that some MMP terminates, which means that we must show that we only need finitely many flips.

As in the scheme of the proof of finite generation sketched above, the first step is to pick D ∈ |kK_{X}|, and to pass to a log resolution of the support S of D. By way of induction we want to work with K_{X} + S rather than K_{X}. As before this is tricky since a log terminal model for K_{X} + S is not the same as a log terminal model for K_{X}. In other words having added S, we really want to subtract it as well. The trick however is to first add S, construct a log terminal model for K_{X} + S and then subtract S (almost literally component by component). This is one of the key steps, to show that Theorem A_{n} and Theorem B_{n} imply Theorem C_{n}. This part of the proof splits naturally into two parts. First we have to prove that we may run the relevant minimal model programs, see §4 and the beginning of §5, then we have to prove this does indeed construct a log terminal model for K_{X}, see §5.

To gain intuition for how this part of the proof works, let us first consider a simplified case. Suppose that D = S is irreducible. In this case it is clear that S is of general type and K_{X} is nef if and only if K_{X} + S is nef and in fact a log terminal model for K_{X} is the same as a log terminal model for K_{X} + S. Consider running the (K_{X} + S)-MMP. Then every step of this MMP is a step of the K_{X}-MMP and vice-versa. Suppose that we have a (K_{X} + S)-extremal ray R. Let π: X ⟶ Z be the corresponding contraction. Then S · R < 0, so that every curve Σ contracted by π must be contained in S. In particular π cannot be a divisorial contraction, as S is not uniruled. Hence π is a pl-flip and by Theorem A_{n}, we can construct the flip of π, ϕ: X −−→ Y. Consider the restriction ψ: S −−→ T of ϕ to S, where T is the strict transform of S. Since log discrepancies increase under flips and S is irreducible, ψ is a birational contraction. After finitely many flips, we may therefore assume that ψ does not contract any divisors, since the Picard number of S cannot keep dropping. Consider what happens if we restrict to S. By adjunction, we have

(K_{X} + S)|_{S} = K_{S}.

Thus ψ: S −−→ T is K_{S}-negative. We have to show that this cannot happen infinitely often. If we knew that every sequence of flips on S terminates, then we would be done. In fact this is how special termination works. Unfortunately we cannot prove that every sequence of flips terminates on S, so that we have to do something slightly different. Instead we throw in an auxiliary ample divisor H on X, and consider K_{X} + S + tH, where t is a positive real number. If t is large enough then K_{X} + S + tH is ample. Decreasing t, we may assume that there is an extremal ray R such that (K_{X} + S + tH) · R = 0. If t = 0, then K_{X} +S is nef and we are done. Otherwise (K_{X} +S)·R < 0, so that we are still running a (K_{X} + S)-MMP, but with the additional restriction that K_{X} + S + tH is nef and trivial on any ray we contract. This is the (K_{X} + S)-MMP with scaling of H. Let G = H|_{S}. Then K_{S} + tG is nef and so is K_{T} + tG′, where G′ = ψ_{∗}G. In this case K_{T} + tG′ is a weak log canonical model for K_{S} + tG (it is not a log terminal model, both because ϕ might contract divisors on which K_{S} +tG is trivial and more importantly because T need not be ℚ-factorial). In this case we are then done, by finiteness of weak log canonical models for (S, tG), where t ∈ [0, 1] (cf. Theorem E_{n−1}).

We now turn to the general case. The idea is similar. First we want to use finiteness of log terminal models on S to conclude that there are only finitely many log terminal models in a neighbourhood of S. Secondly we use this to prove the existence of a very special MMP and construct log terminal models using this MMP. The intuitive idea is that if ϕ: X −−→ Y is K_{X}-negative then K_{X} is bigger than K_{Y} (the difference is an effective divisor on a common resolution) so that we can never return to the same neighbourhood of S. As already pointed out, in the general case we need to work with ℝ-divisors. This poses no significant problem at this stage of the proof, but it does make some of the proofs a little more technical. By way of induction, suppose that we have a log pair K_{X} + Δ = K_{X} + S + A + B, where S is a sum of prime divisors, A is an ample divisor (with rational coefficients) and the coefficients of B are real numbers between zero and one. We are also given a divisor D ≥ 0 such that K_{X} + Δ ∼_{ℝ} D. The construction of log terminal models is similar to the one sketched above and breaks into two parts.

In the first part, for simplicitly of exposition we assume that S is a prime divisor and K_{X} + Δ is purely log terminal. We fix S and A but we allow B to vary and we want to show that finiteness of log terminal models for S implies finiteness of log terminal models in a neighbourhood of S. We are free to pass to a log resolution, so we may assume that (X, Δ) is log smooth and if B = ∑ b_{i}B_{i} the coefficients (b_{1}, b_{2}, …, b_{k}) of B lie in [0, 1]^{k}. Let Θ = (Δ − S)|_{S} so that (K_{X} + Δ)|_{S} = K_{S} + Θ.

Suppose that f: X −−→ Y is a log terminal model of (X, Δ). There are three problems that arise, two of which are quite closely related. Suppose that g: S −−→ T is the restriction of f to S, where T is the strict transform of S. The first problem is that g need not be a birational contraction. For example, suppose that X is a threefold and f flips a curve Σ intersecting S, which is not contained in S. Then S · Σ > 0 so that T · E < 0, where E is the flipped curve. In this case E ⊂ T so that the induced birational map S −−→ T extracts the curve E. The basic observation is that E must have log discrepancy less than one with respect to (S, Θ). Since the pair (X, Δ) is purely log terminal if we replace (X, Δ) by a fixed model which is high enough then we can ensure that the pair (S, Θ) is terminal, so that there are no such divisors E and g is then always a birational contraction. The second problem is that if E is a divisor intersecting S which is contracted to a divisor lying in T, then E ∩ S is not contracted by S. For this reason, g is not necessarily a weak log canonical model of (S, Θ). However we can construct a divisor 0 ≤ Ξ ≤ Θ such that g is a weak log canonical model for (S, Ξ). Suppose that we start with a smooth threefold Y and a smooth surface T ⊂ Y which contains a −2-curve Σ, such that K_{Y} + T is nef. Let f: X ⟶ Y be the blow up of Y along Σ with exceptional divisor E and let S be the strict transform of T. Then f is a step of the (K_{X} +S +eE)-MMP for any e > 0 and f is a log terminal model of K_{X} + S + eE. The restriction of f to S, g: S ⟶ T is the identity, but g is not a log terminal model for K_{S} + eΣ, since K_{S} + eΣ is negative along Σ. It is a weak log canonical model for K_{T}, so that in this case Ξ = 0. The details of the construction of Ξ are contained in (4.1).

The third problem is that the birational contraction g does not determine f. This is most transparent in the case when X is a surface and S is a curve, since in this case g is always an isomorphism. To remedy this particular part of the third problem we use the different, which is defined by adjunction,

(K_{Y} + T)|_{T} = K_{T} + Φ.

The other parts of the third problem only occurs in dimension three or more. For example, suppose that Z is the cone over a smooth quadric in ℙ^{3} and p: X ⟶ Z and q: Y ⟶ Z are the two small resolutions, so that the induced birational map f: X −−→ Y is the standard flop. Let π: W ⟶ Z blow up the maximal ideal, so that the exceptional divisor E is a copy of ℙ^{1}×ℙ^{1}. Pick a surface R which intersects E along a diagonal curve Σ. If S and T are the strict transforms of R in X and Y then the induced birational map g: S ⟶ T is an isomorphism (both S and T are isomorphic to R). To get around this problem, one can perturb Δ so that f is the ample model, and one can distinguish between X and Y by using the fact that g is the ample model of (S, Ξ). Finally it is not hard to write down examples of flops which fix Ξ, but switch the individual components of Ξ. In this case one needs to keep track not only of Ξ but the individual pieces (g_{∗}B_{i})|_{T}, 1 ≤ i ≤ k. We prove that an ample model f is determined in a neighbourhood of T by g, the different Φ and (f_{∗}B_{i})|_{T}, see (4.3). To finish this part, by induction we assume that there are finitely many possibilities for g and it is easy to see that there are then only finitely many possibilities for the different Φ and the divisors (f_{∗}B_{i})|_{T} and this shows that there are only finitely many possibilities for f. This explains the implication Theorem E_{n−1} implies Theorem B_{n}. The details are contained in §4.

The second part consists of using finiteness of models in a neighbourhood of S to run a sequence of minimal model programs to construct a log terminal model. We may assume that X is smooth and the support of Δ + D has normal crossings.

Suppose that there is a divisor C such that

(∗)  K_{X} + Δ ∼_{ℝ,U} D + αC,

where K_{X} + Δ + C is divisorially log terminal and nef and the support of D is contained in S. If R is an extremal ray which is (K_{X} + Δ)- negative, then D · R < 0, so that S_{i} · R < 0 for some component S_{i} of S. As before this guarantees the existence of flips. It is easy to see that the corresponding step of the (K_{X} +Δ)-MMP is not an isomorphism in a neighbourhood of S. Therefore the (K_{X} +Δ)-MMP with scaling of C must terminate with a log terminal model for K_{X} + Δ. To summarise, whenever the conditions above hold, we can always construct a log terminal model of K_{X} + Δ.

We now explain how to construct log terminal models in the general case. We may write D = D_{1} + D_{2}, where every component of D_{1} is a component of S and no component of D_{2} is a component of S. If D_{2} is empty, that is every component of D is a component of S then we take C to be a sufficiently ample divisor, and the argument in the previous paragraph implies that K_{X} + Δ has a log terminal model. If D_{2} ≠ 0 then instead of constructing a log terminal model, we argue that we can construct a neutral model, which is exactly the same as a log terminal model, except that we drop the hypothesis on negativity. Consider (X, Θ = Δ + λD_{2}), where λ is the largest real number so that the coefficients of Θ are at most one. Then more components of ⌞Θ⌟ are components of D. By induction (X, Θ) has a neutral model, f: X −−→ Y. It is then easy to check the conditions in the paragraph above apply, and we can construct a log terminal model g: Y −−→ Z for K_{X}+g_{∗}Δ. It is then automatic that the composition h = g∘f: X −−→ Z is a neutral model of K_{X} + Δ (since f is not (K_{X} + Δ)-negative it is not true in general that h is a log terminal model of K_{X} + Δ). However g is automatically a log terminal model provided we only contract components of the stable base locus of K_{X}+Δ. For this reason, we pick D so that we may write D = M +F where every component of M is semiample and every component of F is a component of the stable base locus. This explains the implication Theorem A_{n} and Theorem B_{n} imply Theorem C_{n}. The details are contained in §5.

Now we explain how prove that if K_{X} + Δ = K_{X} + A + B is pseudoeffective, then K_{X} +Δ ∼_{ℝ} D ≥ 0. The idea is to mimic the proof of the non-vanishing theorem. As in the proof of the non-vanishing theorem and following the work of Nakayama, there are two cases. In the first case, for any ample divisor H,

h^{0}(X, O_{X}(⌞m(K_{X} + Δ)⌟ + H)),

is a bounded function of m. In this case it follows that K_{X} + Δ is numerically equivalent to the divisor N_{σ}(K_{X} + Δ) ≥ 0. It is then not hard to prove that Theorem C_{n} implies that K_{X} +Δ has a log terminal model and we are done by the base point free theorem.

In the second case we construct a non kawamata log terminal centre for

m(K_{X} + Δ) + H,

when m is sufficiently large. Passing to a log resolution, and using standard arguments, we are reduced to the case when

K_{X} + Δ = K_{X} + S + A + B,

where S is irreducible and (K_{X} + Δ)|_{S} is pseudo-effective, and the support of Δ has global normal crossings. Suppose first that K_{X} + Δ is ℚ-Cartier. We may write

(K_{X} + S + A + B)|_{S} = K_{S} + C + D,

where C is ample and D ≥ 0. By induction we know that there is a positive integer m such that h^{0}(S, O_{S}(m(K_{S} + C + D))) > 0. To lift sections, we need to know that h^{1}(X, O_{X}(m(K_{X} +S+A+B)−S)) = 0. Now

m(K_{X} + Δ) − (K_{X} + B) − S = (m − 1)(K_{X} + Δ) + A

= (m − 1)(K_{X} + Δ + \frac{1}{m − 1}A).

As K_{X} + Δ + A/(m − 1) is big, we can construct a log terminal model ϕ: X −−→ Y for K_{X} +Δ+A/(m−1), and running this argument on Y, the required vanishing holds by Kawamata-Viehweg vanishing. In the general case, K_{X} + S + A + B is an ℝ-divisor. The argument is now a little more delicate as h^{0}(S, O_{S}(m(K_{S} + C + D))) does not make sense. We need to approximate K_{S} + C + D by rational divisors, which we can do by induction. But then it is not so clear how to choose m. In practise we need to prove that the log terminal model Y constructed above does not depend on m, at least locally in a neighbourhood of T, the strict transform of S, and then the result follows by Diophantine approximation. This explains the implication Theorem D_{n−1}, Theorem B_{n} and Theorem C_{n} imply Theorem D_{n}. The details are in §6.

Finally, in terms of induction, we need to prove finiteness of weak log canonical models. We fix an ample divisor A and work with divisors of the form K_{X} + Δ = K_{X} + A + B, where the coefficients of B are variable. For ease of exposition, we assume that the support of A and B have global normal crossings, so that K_{X} + Δ = K_{X} + A + ∑ b_{i}B_{i} is log canonical if and only if 0 ≤ b_{i} ≤ 1 for all i. The key point is that we allow the coefficients of B to be real numbers, so that the set of all possible choices of coefficients [0, 1]^{k} is a compact subset of ℝ^{k}. Thus we may check finiteness locally. In fact since A is ample, we can always perturb the coefficients of B so that none of the coefficients are equal to one or zero and so we may even assume that K_{X} + Δ is kawamata log terminal.

Observe that we are certainly free to add components to B (formally we add components with coefficient zero and then perturb so that their coefficients are non-zero). In particular we may assume that B is the support of an ample divisor and so working on the weak log canonical model, we may assume that we have a log canonical model for a perturbed divisor. Thus it suffices to prove that there are only finitely many log canonical models. Since the log canonical model is determined by any log terminal model, it suffices to prove that we can find a cover of [0, 1]^{k} by finitely many log terminal models. By compactness, it suffices to do this locally.

So pick b ∈ [0, 1]^{k}. There are two cases. If K_{X} + Δ is not pseudoeffective, then K_{X} + A + B′ is not pseudo-effective, for B′ in a neighbourhood of B, and there are no weak log canonical models at all. Otherwise we may assume that K_{X} + Δ is pseudo-effective. By induction we know that K_{X} + Δ ∼_{ℝ} D ≥ 0. Then we know that there is a log terminal model ϕ: X −−→ Y. Replacing (X, Δ) by (Y, Γ = ϕ_{∗}Δ), we may assume that K_{X} + Δ is nef. By the base point free theorem it is semiample. Let X ⟶ Z be the corresponding morphism. The key observation is that locally about Δ, any log terminal model over Z is an absolute log terminal model. Working over Z, we may assume that K_{X} + Δ is numerically trivial. In this case the problem of finding a log terminal model for K_{X} + Δ′ only depends on the line segment spanned by Δ and Δ′. Working in a small box about Δ, we are then reduced to finding a log terminal model on the boundary of the box and we are done by induction on the dimension of the affine space containing B. Note that in practise, we need to work in slightly more generality than we have indicated; first we need to work in the relative setting and secondly we need to work with an arbitrary affine space containing B (and not just the space spanned by the components of B). This poses no significant problem. This explains the implication Theorem C_{n} and Theorem D_{n} imply Theorem E_{n}. The details are contained in §7.

The implication Theorem C_{n}, Theorem D_{n} and Theorem E_{n} imply Theorem F_{n} is straightforward. The details are contained in §8.

Let us end the sketch of the proof by pointing out some of the technical advantages with working with kawamata log terminal pairs (X, Δ), where Δ is big. The first observation is that since the kawamata log terminal condition is open, it is straightforward to show that Δ is ℚlinearly equivalent to A + B where A is an ample ℚ-divisor, B ≥ 0 and K_{X} +A+B. The presence of the ample divisor A is very convenient for a number of reasons, two of which we have already seen in the sketch of the proof.

Firstly the restriction of an ample divisor to any divisor S is ample, so that if B does not contain S in its support then the restriction of A + B to S is big. This is very useful for induction.

Secondly, as we vary the coefficients of B, the closure of the set of kawamata log terminal pairs is the set of log canonical pairs. However, we can use a small piece of A to perturb the coefficients of B so that they are bounded away from zero and K_{X} + A + B is always kawamata log terminal.

Finally, if (X, Δ) is divisorially log terminal and f: X ⟶ Y is a (K_{X} +Δ)-trivial contraction then K_{Y} +Γ = K_{X} +f_{∗}Δ is not necessarily divisorially log terminal, only log canonical. For example, suppose that Y is a surface with a simple elliptic singularity and f: X ⟶ Y is the blow up with exceptional divisor E. Then f is a weak log canonical model of K_{X} +E, but Y is not log terminal as it does not have rational singularities. On the other hand, if Δ = A + B where A is ample then K_{Y} + Γ is always divisorially log terminal.

2.2. Standard conjectures of the MMP. Having sketched the proof of (1.2), we should point out the main obstruction to extending these ideas to the case when X is not of general type. The main issue seems to be the implication K_{X} pseudo-effective implies κ(X, K_{X}) ≥ 0. In other words we need:

Conjecture 2.1. Let (X, Δ) be a projective kawamata log terminal pair.

If K_{X} + Δ is pseudo-effective then κ(X, K_{X} + Δ) ≥ 0.

We also probably need

Conjecture 2.2. Let (X, Δ) be a projective kawamata log terminal pair.

If K_{X} + Δ is pseudo-effective and

h^{0}(X, O_{X}(⌞m(K_{X} + Δ)⌟ + H)),

is not a bounded function of m, for some ample divisor H, then κ(X, K_{X}+ Δ) ≥ 1.

In fact, using the methods of this paper, together with some results of Kawamata (cf. [14] and [15]), (2.1) and (2.2) would seem to imply one of the main outstanding conjectures of higher dimensional geometry:

Conjecture 2.3 (Abundance). Let (X, Δ) be a projective kawamata log terminal pair.

If K_{X} + Δ is nef then it is semiample.

We remark that the following seemingly innocuous generalisation of (1.2) (in dimension n + 1) would seem to imply (2.3) (in dimension n).

Conjecture 2.4. Let (X, Δ) be a projective log canonical pair of dimension n.

If K_{X} + Δ is big, then (X, Δ) has a log canonical model.

It also seems worth pointing out that the other remaining conjecture is:

Conjecture 2.5 (Borisov-Alexeev-Borisov). Fix a positive integer n and a positive real number ϵ > 0.

Then the set of varieties X such that K_{X} + Δ has log discrepancy at least ϵ and −(K_{X} + Δ) is ample, forms a bounded family.

3. Preliminary results

In this section we collect together some definitions and results.

3.1. Notation and conventions. We work over the field of complex numbers ℂ. 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-divisor D is ℚ-Cartier if some integral multiple is Cartier. We say that X is ℚ-factorial if every Weil divisor is ℚ-Cartier. We say that X is analytically ℚ-factorial if every analytic Weil divisor (that is an analytic subset of codimension one) is analytically ℚ-Cartier (i.e. some multiple is locally defined by a single analytic function). We recall some definitions involving divisors with real coefficients.

Definition 3.1.1. Let π: X ⟶ U be a proper morphism of normal algebraic spaces.

(1) An ℝ-Weil divisor (frequently abbreviated to ℝ-divisor) D on X is an ℝ-linear combination of prime divisors.

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

(3) Two ℝ-divisors D and D′ are ℝ-linearly equivalent over U, denoted D ∼_{ℝ,U} D′, if their difference is an ℝ-linear combination of principal divisors and an ℝ-Cartier divisor pulled back from U.

(4) Two ℝ-divisors D and D′ are numerically equivalent over U, denoted D ≡_{U} D′, if their difference is an ℝ-Cartier divisor such that (D − D′) · C = 0 for any curve C contained in a fibre of π.

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

(6) An ℝ-Cartier divisor D on X is nef over U (or π-nef) if D · C ≥ 0 for any curve C ⊂ X, contracted by π.

(7) An ℝ-divisor D is big over U (or π-big) if

lim sup \frac{h^{0}(F, O_{F} (⌞mD⌟))}{m^{dim F}} > 0,

for the fibre F over any generic point of U. Equivalently D is big over U if D ∼_{ℝ,U} A+B where A is ample over U and B ≥ 0 (cf. [28, II 3.16]).

(8) An ℝ-Cartier divisor D is semiample over U (or π-semiample) if there is a morphism f: X ⟶ Y over U such that D is ℝlinearly equivalent to the pullback of an ample ℝ-divisor over U.

(9) An ℝ-divisor D is π-pseudo-effective if the restriction of D to the generic fibre is the limit of divisors D_{i} ≥ 0.

Note that the group of Weil divisors with rational coefficients WDiv_{ℚ}(X), or with real coefficients WDiv_{ℝ}(X), 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. If A = ∑ a_{i}C_{i} and B = ∑ b_{i}C_{i} are two ℝ-divisors,

A ∧ B = ∑ min(a_{i}, b_{i})C_{i}.

Given an ℝ-divisor D and a subvariety Z which is not contained in the singular locus of X, mult_{Z} D denotes the multiplicity of D at the generic point of Z. If Z = E is a prime divisor this is the coefficient of E in D.

A log pair (X, Δ) (sometimes abbreviated by K_{X} + Δ) is a normal variety X and an ℝ-divisor Δ ≥ 0 such that K_{X} + Δ is ℝ-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 birational morphism g: Y ⟶ X is a log resolution of the pair (X, Δ) if g is projective, Y is smooth, the exceptional locus is a divisor and g^{−1}(Δ) union the exceptional set of g is a divisor with global normal crossings support. By Hironaka’s Theorem we may, and often will, assume that the exceptional locus supports an ample divisor over X. If we write

K_{Y} + Γ = K_{Y} + ∑ b_{i}Γ_{i} = g^{∗}(K_{X} + Δ),

where Γ_{i} are distinct prime divisors then the log discrepancy a(Γ_{i}, X, Δ) of Γ_{i} is 1−b_{i}. The log discrepancy of (X, Δ) is then the infimum of the log discrepancy for every Γ_{i} and for every resolution. The image of any component of Γ of coefficient at least one (equivalently log discrepancy at most zero) is a non kawamata log terminal centre of the pair (X, Δ). 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 b_{i} < 1 for all i. Equivalently, the pair (X, Δ) is kawamata log terminal if there are no non kawamata log terminal centres. The non kawamata log terminal locus of (X, Δ) is the union of the non kawamata log terminal centres. We say that the pair (X, Δ) is purely log terminal if the log discrepancy of any exceptional divisor is greater than zero. We say that the pair (X, Δ = ∑ δ_{i}Δ_{i}), where δ_{i} ∈ (0, 1], is divisorially log terminal if there is a log resolution such that the log discrepancy of every exceptional divisor is greater than zero. By [22, (2.40)] (X, Δ) is divisorially log terminal if and only if there is a closed subset Z ⊂ X such that

• (X\Z, Δ|_{X\Z}) is log smooth, and

• if f: Y ⟶ X is a projective birational morphism and E ⊂ Y is an irreducible divisor with centre contained in Z then a(E, X, Δ) > 0.

We will also often write

K_{Y} + Γ = g^{∗}(K_{X} + Δ) + E,

where Γ ≥ 0 and E ≥ 0 have no common components, g_{∗}Γ = Δ and E is g-exceptional. Note that this decomposition is unique.

We say that a birational map ϕ: X −−→ Y is a birational contraction if ϕ is proper and ϕ^{−1} does not contract any divisors. If in addition ϕ^{−1} is also a birational contraction, we say that ϕ is a small birational map.

3.2. Preliminaries.

Lemma 3.2.1. Let π: X ⟶ U be a projective morphism of normal quasi-projective varieties. Let D be an ℝ-Cartier divisor on X and let D′ be its restriction to the generic fibre of π.

If D′ ∼_{ℝ} B′ ≥ 0 for some ℝ-divisor B′ on the generic fibre of π, then there is a divisor B on X such that D ∼_{ℝ,U} B ≥ 0 whose restriction to the generic fibre of π is B′.

Proof. Taking the closure of the generic points of B′, we may assume that there is an ℝ-divisor B_{1} ≥ 0 on X such that the restriction of B_{1} to the generic fibre is B′. As

D′ − B′ ∼_{ℝ} 0,

it follows that there is an open subset U_{1} of U, such that

(D − B_{1})|_{V_{1}} ∼_{ℝ} 0,

where V_{1} is the inverse image of U_{1}. But then there is a divisor G on X such that

D − B_{1} ∼_{ℝ} G,

where Z = π(Supp G) is a proper closed subset. As U is quasiprojective, there is an ample divisor H ≥ 0 on U which contains Z. Possibly rescaling, we may assume that F = π^{∗}H ≥ −G. But then

D ∼_{ℝ} (B_{1} + F + G) − F,

so that

D ∼_{ℝ,U} (B_{1} + F + G) ≥ 0.  □

3.3. Nakayama-Zariski decomposition. We will need some definitions and results from [28].

Definition-Lemma 3.3.1. Let X be a smooth projective variety, B be a big ℝ-divisor and let C be a prime divisor. Let

σ_{C}(B) = inf{ mult_{C}(B′) | B′ ∼_{ℚ} B, B′ ≥ 0 }.

Then σ_{C} is a continuous function on the cone of big divisors.

Now let D be any pseudo-effective ℝ-divisor and let A be any ample ℚ-divisor. Let

σ_{C}(D) = lim_{ϵ→0} σ_{C}(D + ϵA).

Then σ_{C}(D) exists and is independent of the choice of A.

There are only finitely many prime divisors C such that σ_{C}(D) > 0 and the ℝ-divisor N_{σ}(D) = ∑_{C} σ_{C}(D)C is determined by the numerical equivalence class of D. Moreover D − N_{σ}(D) is pseudo-effective and N_{σ}(D − N_{σ}(D)) = 0.

Proof. See §III.1 of [28]. □

Proposition 3.3.2. Let X be a smooth projective variety and let D be a pseudo-effective ℝ-divisor. Let B be any big ℝ-divisor.

If D is not numerically equivalent to N_{σ}(D), then there is a positive integer k and a positive rational number β such that

h^{0}(X, O_{X}(⌞mD⌟ + ⌞kB⌟)) > βm,  for all  m ≫ 0.

Proof. Let A be any integral divisor. Then we may find a positive integer k such that

h^{0}(X, O_{X}(⌞kB⌟ − A)) > 0.

Thus it suffices to exhibit an ample divisor A and a positive rational number β such that

h^{0}(X, O_{X}(⌞mD⌟ + A)) > βm for all m ≫ 0.

Replacing D by D − N_{σ}(D), we may assume that N_{σ}(D) = 0. Now apply (V.1.11) of [28]. □

3.4. Adjunction. We recall some basic facts about adjunction, see [21, §16, §17] for more details.

Definition-Lemma 3.4.1. Let (X, Δ) be a log canonical pair, and let S be a normal component of ⌞Δ⌟ of coefficient one. Then there is a divisor Θ on S such that

(K_{X} + Δ)|_{S} = K_{S} + Θ.

(1) If (X, Δ) is divisorially log terminal then so is K_{S} + Θ.

(2) If (X, Δ) is purely log terminal then K_{S} + Θ is kawamata log terminal.

(3) If (X, Δ = S) is purely log terminal then the coefficients of Θ have the form (r − 1)/r, where r is the index of S at P the generic point of the corresponding divisor D on S (equivalently r is the index of K_{X} + S at P or r is the order of the cyclic group Weil(O_{X,P})). In particular if B is a Weil divisor on X, then the coefficient of B|_{S} in D is an integer multiple of 1/r.

(4) If (X, Δ) is purely log terminal, f: Y ⟶ X is a projective birational morphism and T is the strict transform of S, then (f|_{T})_{∗}Ψ = Θ, where K_{Y} + Γ = f^{∗}(K_{X} + Δ) and Ψ is defined by adjunction,

(K_{Y} + Γ)|_{T} = K_{T} + Ψ.

3.5. Stable base locus. We need to extend the definition of the stable base locus to the case of a real divisor.

Definition 3.5.1. Let π: X ⟶ U be a projective morphism of normal varieties.

Let D be an ℝ-divisor on X. The real linear system associated to D over U is

|D/U|_{ℝ} = { C ≥ 0 | C ∼_{ℝ,U} D }.

The stable base locus of D over U is the Zariski closed set B(D/U) given by the intersection of the support of the elements of the real linear system |D/U|_{ℝ}. If |D/U|_{ℝ} = ∅, then we let B(D/U) = X. The stable fixed divisor is the divisorial support of the stable base locus. The augmented base locus of D over U is the Zariski closed set

B_{+}(D/U) = B((D − ϵA)/U),

for any ample divisor A over U and any sufficiently small rational number ϵ > 0 (compare [23, Definition 10.3.2])

Remark 3.5.2. The stable base locus, the stable fixed divisor and the augmented base locus are only defined as closed subsets, they do not have any scheme structure.

Lemma 3.5.3. Let π: X ⟶ U be a projective morphism of normal varieties and let D be an integral Weil divisor on X.

Then the stable base locus as defined in (3.5.1) coincides with the usual definition of the stable base locus.

Proof. Let

|D/U|_{ℚ} = { C ≥ 0 | C ∼_{ℚ,U} D }.

Let R be the intersection of the elements of |D/U|_{ℝ} and let Q be the intersection of the elements of |D/U|_{ℚ}. It suffices to prove that Q = R. As |D/U|_{ℚ} ⊂ |D/U|_{ℝ}, it is clear that R ⊂ Q.

Suppose that x ∉ R. We want to show that x ∉ Q. We may find D′ ∈ |D/U|_{ℝ} such that mult_{x} D′ = 0. But then

D′ = D + ∑ r_{i}(f_{i}) + π^{∗}E,

where f_{i} are rational functions on X, E is an ℝ-Cartier divisor on U and r_{i} are real numbers. Let V be the subspace of WDiv_{ℝ}(X) spanned by the components of D, D′, π^{∗}E and (f_{i}). We may write E = ∑ e_{j}E_{j}, where E_{i} are Cartier divisors. Let W be the span of the (f_{i}) and the π^{∗}E_{j}. Then W ⊂ V are defined over the rationals. Set

P = { D′′ ∈ V | D′′ ≥ 0, mult_{x} D′′ = 0, D′′ − D ∈ W } ⊂ |D/U|_{ℝ}.

Then P is a rational polyhedron. As D′ ∈ P, P is non-empty, and so it must contain a rational point D′′. We may write

D′′ = D + ∑ s_{i}(f_{i}) + ∑ f_{j}π^{∗}E_{j},

where s_{i} and f_{j} are real numbers. Since D′′ and D have rational coefficients, it follows that we may find s_{i} and f_{j} which are rational. But then D′′ ∈ |D/U|_{ℚ}, and so x ∉ Q. □

Proposition 3.5.4. Let π: X ⟶ U be a projective morphism of normal varieties and let D ≥ 0 be an ℝ-divisor. Then we may find ℝdivisors M and F such that

(1) M ≥ 0 and F ≥ 0,

(2) D ∼_{ℝ,U} M + F,

(3) every component of F is a component of B(D/U), and

(4) if B is a component of M then some multiple of B is mobile.

We need two basic results.

Lemma 3.5.5. Let X be a normal variety and let D and D′ be two ℝ-divisors such that D ∼_{ℝ} D′.

Then we may find rational functions f_{1}, f_{2}, …, f_{k} and real numbers r_{1}, r_{2}, …, r_{k} which are independent over the rationals such that

D = D′ + ∑_{i} r_{i}(f_{i}).

In particular every component of (f_{i}) is either a component of D or D′.

Proof. By assumption we may find rational functions f_{1}, f_{2}, …, f_{k} and real numbers r_{1}, r_{2}, …, r_{k} such that

D = D′ + ∑_{i=1}^{k} r_{i}(f_{i}).

Pick k minimal with this property. Suppose that the real numbers r_{i} are not independent over ℚ. Then we can find rational numbers d_{i}, not all zero, such that

∑_{i} d_{i}r_{i} = 0.

Possibly re-ordering we may assume that d_{k} ≠ 0. Multiplying through by an integer we may assume that d_{i} ∈ ℤ. Possibly replacing f_{i} by f_{i}^{−1}, we may assume that d_{i} ≥ 0. Let d be the least common multiple of the non-zero d_{i}. If d_{i} ≠ 0, we replace f_{i} by f_{i}^{d/d_{i}} (and hence r_{i} by d_{i}r_{i}/d) so that we may assume that either d_{i} = 0 or 1. For 1 ≤ i < k, set

g_{i} = \begin{cases}f_{i}/f_{k} & if d_{i} = 1 \\ f_{i} & if d_{i} = 0. \end{cases}

Then

D = D′ + ∑_{i=1}^{k−1} r_{i}(g_{i}),

which contradicts our choice of k.

Now suppose that B is a component of (f_{i}). Then

mult_{B}(D) = mult_{B}(D′) + ∑ r_{j}n_{j},

where n_{j} = mult_{B}(f_{j}) is an integer and n_{i} ≠ 0. But then mult_{B}(D) − mult_{B}(D′) ≠ 0, so that one of mult_{B}(D) and mult_{B}(D′) must be nonzero. □

Lemma 3.5.6. Let π: X ⟶ U be a projective morphism of normal varieties and let

D′ ∼_{ℝ,U} D,  D ≥ 0,  D′ ≥ 0,

be two ℝ-divisors on X with no common components.

Then we may find D′′ ∈ |D/U|_{ℝ} such that a multiple of every component of D′′ is mobile.

Proof. Pick ample ℝ-divisors on U, H and H′ such that D + π^{∗}H ∼_{ℝ} D′ + π^{∗}H′ and D + π^{∗}H and D′ + π^{∗}H′ have no common components. Replacing D by D + π^{∗}H and D′ by D′ + π^{∗}H′, we may assume that D′ ∼_{ℝ} D.

We may write

D′ = D + ∑ r_{i}(f_{i}) = D + R,

where r_{i} ∈ ℝ and f_{i} are rational functions on X. By (3.5.5) we may assume that every component of R is a component of D + D′.

We proceed by induction on the number of components of D + D′. If q_{1}, q_{2}, …, q_{k} are any rational numbers then we may always write

C′ = C + Q = C + ∑ q_{i}(f_{i}),

where C ≥ 0 and C′ ≥ 0 have no common components. But now if we suppose that q_{i} is sufficiently close to r_{i} then C is supported on D and C′ is supported on D′. We have that mC ∼ mC′ for some integer m > 0. By Bertini we may find C′′ ∼_{ℚ} C such that every component of C′′ has a multiple which is mobile. Pick λ > 0 maximal such that D_{1} = D − λC ≥ 0 and D′_{1} = D′ − λC′ ≥ 0. Note that

D_{1} ∼_{ℝ} D′_{1},  D_{1} ≥ 0,  D′_{1} ≥ 0,

are two ℝ-divisors on X with no common components, and that D_{1}+D′_{1} has fewer components than D + D′. By induction we may then find

D′′_{1} ∈ |D_{1}|_{ℝ},

such that a multiple of every component of D′′_{1} is mobile. But then

D′′ = λC′′ + D′′_{1} ∈ |D|_{ℝ},

and every component of D′′ has a multiple which is mobile. □

Proof of (3.5.4). We may write D = M + F, where every component of F is contained in B(D/U) and no component of M is contained in B(D/U). A prime divisor is bad if none of its multiples are mobile.

We proceed by induction on the number of bad components of M. We may assume that M has at least one bad component B. As B is a component of M, we may find D_{1} ∈ |D/U|_{ℝ} such that B is not a component of D_{1}. If E = D ∧ D_{1} then D′ = D − E ≥ 0 and D′_{1} = D_{1} − E ≥ 0, D′ and D′_{1} have no common components and D′ ∼_{ℝ,U} D′_{1}. By (3.5.6) there is a divisor D′′ ∈ |D′/U|_{ℝ} with no bad components. But then D′′ +E ∈ |D|_{ℝ}, B is not a component of D′′ +E and the only bad components of D′′ + E are components of E, which are also components of D. Therefore D′′+E has fewer bad components than D and we are done by induction. □

3.6. Types of model.

Definition 3.6.1. Let ϕ: X −−→ Y be a proper birational contraction of normal quasi-projective varieties and let D be an ℝ-Cartier divisor on X such that D′ = ϕ_{∗}D is also ℝ-Cartier. We say that ϕ is D-non-positive (respectively D-negative) if for some common resolution p: W ⟶ X and q: W ⟶ Y, we may write

p^{∗}D = q^{∗}D′ + E,

where E ≥ 0 is q-exceptional (respectively E ≥ 0 is q-exceptional and the support of E contains the strict transform of the ϕ-exceptional divisors).

We will often use the following well-known lemma.

Lemma 3.6.2 (Negativity of contraction). Let π: Y ⟶ X be a projective birational morphism of normal quasi-projective varieties.

(1) If E > 0 is an exceptional ℝ-Cartier divisor then there is a component F of E which is covered by curves Σ such that E·Σ < 0.

(2) If π^{∗}L ≡ M+G+E, where L is an ℝ-Cartier divisor on X, M is a π-nef ℝ-Cartier divisor on Y, G ≥ 0, E is π-exceptional, and G and E have no common components, then E ≥ 0. Further if F_{1} is an exceptional divisor such that there is an exceptional divisor F_{2} with the same centre on X as F_{1}, with the restriction of M to F_{2} not numerically π-trivial, then F_{1} is a component of G + E.

(3) If X is ℚ-factorial then there is a π-exceptional divisor E ≥ 0 such that −E is ample over X. In particular the exceptional locus of π is a divisor.

Proof. Cutting by hyperplanes in X, we reduce to the case when X is a surface, in which case (1) reduces to the Hodge Index Theorem.

(2) follows easily from (1), see for example (2.19) of [21]. Let H be a general ample ℚ-divisor over X. If X is ℚ-factorial then

E = π^{∗}π_{∗}H − H ≥ 0,

is π-exceptional and −E is ample over X. This is (3). □

Lemma 3.6.3. Let X ⟶ U and Y ⟶ U be two projective morphisms of normal quasi-projective varieties. Let ϕ: X −−→ Y be a birational contraction over U and let D and D′ be ℝ-Cartier divisors such that D′ = ϕ_{∗}D is nef over U.

Then ϕ is D-non-positive (respectively D-negative) if given a common resolution p: W ⟶ X and q: W ⟶ Y, we may write

p^{∗}D = q^{∗}D′ + E,

where p_{∗}E ≥ 0 (respectively p_{∗}E ≥ 0 and the support of p_{∗}E contains the union of all ϕ-exceptional divisors).

Further if D = K_{X} + Δ and D′ = K_{Y} + ϕ_{∗}Δ then this is equivalent to requiring

a(F, X, Δ) ≤ a(F, Y, ϕ_{∗}Δ) (respectively a(F, X, Δ) < a(F, Y, ϕ_{∗}Δ)),

for all ϕ-exceptional divisors F ⊂ X.

Proof. Easy consequence of (3.6.2). □

Lemma 3.6.4. Let X ⟶ U and Y ⟶ U be two projective morphisms of normal quasi-projective varieties. Let ϕ: X −−→ Y be a birational contraction over U and let D and D′ be ℝ-Cartier divisors such ϕ_{∗}D and ϕ_{∗}D′ are ℝ-Cartier. Let p: W ⟶ X and q: W ⟶ Y be common resolutions.

If D and D′ are numerically equivalent over U then

p^{∗}D − q^{∗}ϕ_{∗}D = p^{∗}D′ − q^{∗}ϕ_{∗}D′.

In particular ϕ is D-non-positive (respectively D-negative) if and only if ϕ is D′-non-positive (respectively D′-negative).

Proof. Since

p^{∗}(D − D′) − q^{∗}ϕ_{∗}(D − D′),

is q-exceptional and numerically trivial over Y this follows easily from (3.6.2). □

Definition 3.6.5. Let π: X ⟶ U be a projective morphism of normal quasi-projective varieties and let D be an ℝ-Cartier divisor on X.

We say that a birational contraction f: X −−→ Y over U is a semiample model of D over U, if f is D-non-positive, Y is normal and projective over U and H = f_{∗}D is semiample over U.

We say that g: X −−→ Z is the ample model of D over U, if g is a rational map over U, Z is normal and projective over U and there is an ample divisor H over U on Z such that if p: W ⟶ X and q: W ⟶ Z resolve g then q is a contraction morphism and we may write p^{∗}D ∼_{ℝ,U} q^{∗}H + E, where E ≥ 0 and for every B ∈ |p^{∗}D/U|_{ℝ} then B ≥ E.

Lemma 3.6.6. Let π: X ⟶ U be a projective morphism of normal quasi-projective varieties and let D be an ℝ-Cartier divisor on X.

(1) If g_{i}: X −−→ X_{i}, i = 1, 2 are two ample models of D over U then there is an isomorphism χ: X_{1} ⟶ X_{2} such that g_{2} = χ ∘ g_{1}.

(2) Suppose that g: X −−→ Z is the ample model of D over U and let H be the corresponding ample divisor on Z. If p: W ⟶ X and q: W ⟶ Z resolve g then we may write

p^{∗}D ∼_{ℝ,U} q^{∗}H + E,

where E ≥ 0 and if F is any p-exceptional divisor whose centre lies in the indeterminancy locus of g then F is contained in the support of E.

(3) If f: X −−→ Y is a semiample model of D over U then the ample model g: X −−→ Z of D over U exists and g = h ∘ f, where h: Y ⟶ Z is a contraction morphism and f_{∗}D ∼_{ℝ,U} h^{∗}H. If B is a prime divisor contained in the stable fixed divisor of D over U then B is contracted by f.

(4) If f: X −−→ Y is a birational map over U then f is the ample model of D over U if and only if f is a semiample model of D over U and f_{∗}D is ample over U.

Proof. Let g: Y ⟶ X resolve the indeterminacy of g_{i} and let f_{i} = g_{i}∘g: Y ⟶ X_{i} be the induced contraction morphisms. By assumption g^{∗}D ∼_{ℝ,U} f_{i}^{∗}H_{i}+E_{i}, for some divisor H_{i} on X_{i} ample over U. Since the stable fixed divisor of f_{1}^{∗}H_{1} over U is empty, E_{1} ≥ E_{2}. By symmetry E_{1} = E_{2} and so f_{1}^{∗}H_{1} ∼_{ℝ,U} f_{2}^{∗}H_{2}. But then f_{1} and f_{2} contract the same curves. This is (1).

Suppose that g: X −−→ Z is the ample model of D over U. By assumption this means that we may write

p^{∗}D ∼_{ℝ,U} q^{∗}H + E,

where E ≥ 0. We may write E = E_{1} + E_{2}, where every component of E_{2} is exceptional for p but no component of E_{1} is p-exceptional. Let V = p(F). Possibly blowing up more we may assume that p^{−1}(V) is a divisor. Since V is contained in the indeterminancy locus of g, there is an exceptional divisor F′ with centre V such that dim q(F′) > 0. But then q^{∗}H is not numerically trivial on F′ and we may apply (3.6.2). This is (2).

Now suppose that f: X −−→ Y is a semiample model of D over U. As f_{∗}D is semiample over U, there is a contraction morphism h: Y ⟶ Z over U and an ample divisor H over U on Z such that f_{∗}D ∼_{ℝ,U} h^{∗}H. If p: W ⟶ X and q: W ⟶ Y resolve the indeterminacy of f then p^{∗}D ∼_{ℝ,U} r^{∗}H +E, where E ≥ 0 is q-exceptional and r = h∘q: W ⟶ Z. If B ∈ |p^{∗}D/U|_{ℝ} then B ≥ E. But then g = h ∘ f: X −−→ Z is the ample model of D over U. This is (3).

Now suppose that f: X −−→ Y is birational over U. If f is a semiample model of D over U then (3) implies that the ample model g: X −−→ Z of D over U exists and there is a contraction morphism h: Y ⟶ Z, such that f_{∗}D ∼_{ℝ,U} h^{∗}H where H on Z is ample over U. If f_{∗}D is ample over U then h must be the identity.

Conversely suppose that f is the ample model. Suppose that p: W ⟶ X and q: W ⟶ Y are projective birational morphisms which resolve f. By assumption we may write p^{∗}D ∼_{ℝ,U} q^{∗}H + E, where H is ample over U. We may assume that there is a q-exceptional ℚ-divisor F ≥ 0 such that q^{∗}H − F is ample over U. Then there is a constant δ > 0 such that q^{∗}H − F + δE is ample over U. Suppose B is a component of E. As B does not belong to the stable base locus of q^{∗}H −F +δE over U, B must be a component of F. It follows that E is q-exceptional. If C is a curve contracted by p then

0 = C · p^{∗}D = C · q^{∗}H + C · E,

and so C is contained in the support of E. Thus if G is a divisor contracted by p it is a component of E and G is contracted by q. Therefore f is a birational contraction and f is a semiample model. Further f_{∗}D = H is ample over U. This is (4). □ Definition 3.6.7. Let π: X ⟶ U be a projective morphism of normal quasi-projective varieties. Suppose that K_{X} + Δ is log canonical and let ϕ: X −−→ Y be a birational contraction of normal quasi-projective varieties over U, where Y is projective over U. Set Γ = ϕ_{∗}Δ.

• Y is a weak log canonical model for K_{X} + Δ over U if ϕ is (K_{X} + Δ)-non-positive and K_{Y} + Γ is nef over U.

• Y is the log canonical model for K_{X} + Δ over U if ϕ is the ample model of K_{X} + Δ over U.

• Y is a log terminal model for K_{X}+Δ over U if ϕ is (K_{X}+Δ)- negative, K_{Y} +Γ is divisorially log terminal and nef over U, and Y is ℚ-factorial.

Remark 3.6.8. Note that there is no consensus on the definitions given in (3.6.7). Lemma 3.6.9. Let π: X ⟶ U be a projective morphism of normal quasi-projective varieties. Let ϕ: X −−→ Y be a birational contraction over U. Let (X, Δ) and (X, Δ′) be two log pairs and set Γ = ϕ_{∗}Δ, Γ′ = ϕ_{∗}Δ′. Let μ > 0 be a positive real number.

• If both K_{X} +Δ and K_{X} +Δ′ are log canonical and K_{X} +Δ′ ∼_{ℝ,U} μ(K_{X} + Δ) then ϕ is a weak log canonical model for K_{X} + Δ over U if and only if ϕ is a weak log canonical model for K_{X} +Δ′ over U.

• If both K_{X} + Δ and K_{X} + Δ′ are kawamata log terminal and K_{X} + Δ ≡_{U} μ(K_{X} + Δ′) then ϕ is a log terminal model for K_{X} + Δ over U if and only if ϕ is a log terminal model for K_{X} + Δ′ over U.

Proof. Note first that either K_{Y} + Γ′ ∼_{ℝ,U} μ(K_{Y} + Γ) or Y is ℚfactorial. In particular K_{Y} + Γ is ℝ-Cartier if and only if K_{Y} + Γ′ is ℝ-Cartier. Therefore (3.6.4) implies that ϕ is (K_{X} + Δ)-non-positive (respectively (K_{X} + Δ)-negative) if and only if ϕ is (K_{X} + Δ′)-non-positive (respectively (K_{X} + Δ′)-negative).

Since K_{X} + Δ′ ≡_{U} μ(K_{X} + Δ) and μE − E′ ≡_{Y} 0, it follows that K_{Y} + Γ′ ≡_{U} μ(K_{Y} + Γ), so that K_{Y} + Γ is nef over U if and only if K_{Y} + Γ′ is nef over U. □

Lemma 3.6.10. Let π: X ⟶ U be a projective morphism of normal quasi-projective varieties. Let ϕ: X −−→ Y be a birational contraction over U, where Y is projective over U. Suppose that K_{X} + Δ and K_{Y} + ϕ_{∗}Δ are divisorially log terminal and a(F, X, Δ) < a(F, Y, ϕ_{∗}Δ) for all ϕ-exceptional divisors F ⊂ X.

If φ: Y −−→ Z is a log terminal model of (Y, ϕ_{∗}Δ) over U, then η = φ ∘ ϕ: X −−→ Z is a log terminal model of K_{X} + Δ over U.

Proof. Clearly η is a birational contraction, Z is ℚ-factorial and K_{Z} + η_{∗}Δ is divisorially log terminal and nef over U.

Let p: W ⟶ X, q: W ⟶ Y and r: W ⟶ Z be a common resolution. As φ is a log terminal model of (Y, ϕ_{∗}Δ) we have that q^{∗}(K_{Y} + ϕ_{∗}Δ) − r^{∗}(K_{Z} + η_{∗}Δ) = E ≥ 0 and the support of E contains the exceptional divisors of φ. By assumption K_{X} +Δ−p_{∗}q^{∗}(K_{Y} +ϕ_{∗}Δ) is an effective divisor whose support is the set of all ϕ-exceptional divisors. But then

(K_{X} + Δ) − p_{∗}r^{∗}(K_{Z} + η_{∗}Δ) = K_{X} + Δ − p_{∗}q^{∗}(K_{Y} + ϕ_{∗}Δ) + p_{∗}E

contains all the η-exceptional divisors and (3.6.3) implies that η is a log terminal model of K_{X} + Δ over U. □

Lemma 3.6.11. Let π: X ⟶ U be a projective morphism of normal quasi-projective varieties. Let (X, Δ) be a kawamata log terminal pair, where Δ is big over U. Let f: Z ⟶ X be any log resolution of (X, Δ) and suppose that we write

K_{Z} + Φ_{0} = f^{∗}(K_{X} + Δ) + E,

where Φ_{0} ≥ 0 and E ≥ 0 have no common components, f_{∗}Φ_{0} = Δ and E is exceptional. Let F ≥ 0 be any divisor whose support is equal to the exceptional locus of f.

If η > 0 is sufficiently small and Φ = Φ_{0} + ηF then K_{Z} + Φ is kawamata log terminal and Φ is big over U. Moreover if ϕ: Z −−→ W is a log terminal model of K_{Z} + Φ over U then the induced birational map ψ: X −−→ W is in fact a log terminal model of K_{X} + Δ over U.

Proof. Everything is clear but the last statement. Set Ψ = ϕ_{∗}Φ. By (3.6.10), possibly blowing up more, we may assume that ϕ is a morphism. By assumption if we write

K_{Z} + Φ = ϕ^{∗}(K_{W} + Ψ) + G,

then G ≥ 0 and the support of G is the union of all the ϕ-exceptional divisors. Thus

f^{∗}(K_{X} + Δ) + E + ηF = ϕ^{∗}(K_{W} + Ψ) + G.

By negativity of contraction, (3.6.2), applied to f, G − E − ηF ≥ 0. In particular ϕ must contract every f-exceptional divisor and so ψ is a birational contraction. But then ψ is a log terminal model over U by (3.6.3). □

Lemma 3.6.12. Let π: X ⟶ U be a projective morphism of normal quasi-projective varieties, where X is ℚ-factorial kawamata log terminal. Let ϕ: X −−→ Z be a birational contraction over U and let S be a sum of prime divisors. Suppose that there is a ℚ-factorial quasi-projective variety Y together with a small birational projective morphism f: Y ⟶ Z.

If V is any finite dimensional affine subspace of WDiv_{ℝ}(X) such that L_{S}(V) spans WDiv_{ℝ}(X) modulo numerical equivalence over U and W_{ϕ,S,π}(V) intersects the interior of L_{S}(V) then

W_{ϕ,S,π}(V) = Ā_{ϕ,S,π}(V).

Proof. By (4) of (3.6.6) W_{ϕ,S,π}(V) ⊃ A_{ϕ,S,π}(V). Since W_{ϕ,S,π}(V) is closed, it follows that

W_{ϕ,S,π}(V) ⊃ Ā_{ϕ,S,π}(V).

To prove the reverse inclusion, it suffices to prove that a dense subset of W_{ϕ,S,π}(V) is contained in A_{ϕ,S,π}(V).

Pick Δ belonging to the interior of W_{ϕ,S,π}(V). If ψ: X −−→ Y is the induced birational contraction then ψ is a ℚ-factorial weak log canonical model of K_{X} + Δ over U and

K_{Y} + Γ = f^{∗}(K_{Z} + ϕ_{∗}Δ),

where Γ = ψ_{∗}Δ. As L_{S}(V) spans WDiv_{ℝ}(X) modulo numerical equivalence over U, we may find Δ_{0} ∈ L_{S}(V) such that Δ_{0} −Δ is numerically equivalent over U to μΔ for some μ > 0. Let

Δ′ = Δ + ϵ((Δ_{0} − Δ) − μΔ) = (1 − ϵ)νΔ + ϵΔ_{0},

where

ν = \frac{1 − ϵ − ϵμ}{1 − ϵ} < 1.

Then Δ′ is numerically equivalent to Δ over U and if ϵ > 0 is sufficiently small then Δ′ ≥ 0. As (X, νΔ) is kawamata log terminal it follows that (X, Δ′) is kawamata log terminal. In particular (3.6.9) implies that ψ is a ℚ-factorial weak log canonical model of K_{X} + Δ′ over U and so K_{Y} + Γ′ is kawamata log terminal, where Γ′ = ψ_{∗}Δ′. As Γ and Γ′ are numerically equivalent over U, it follows that K_{Y} + Γ′ is numerically equivalent to zero over Z.

Let H be a general ample ℚ-divisor over U on Z. Let p: W ⟶ X and q: W ⟶ Z resolve the indeterminacy locus of ϕ and let H′ = p_{∗}q^{∗}H. It follows that ϕ is H′-non-positive. Pick Δ_{1} ∈ L_{S}(V) such that B = Δ_{1} − Δ is numerically equivalent over U to ηH′ for some η > 0. Replacing H by ηH we may assume that η = 1. If C = ψ_{∗}B then C is numerically equivalent to f^{∗}H over U. Then both C and C −(K_{Y} +Γ′) are numerically trivial over Z, so that C − (K_{Y} + Γ′) is nef and big over Z and (3.9.1) implies that ϕ_{∗}B = f_{∗}C is ℝ-Cartier. (3.6.4) implies that ϕ is (K_{X} + Δ + λB)-non-positive and ϕ_{∗}(K_{X} + Δ + λB) is ample over U, for any λ > 0. On the other hand, note that

Δ + λB = Δ + λ(Δ_{1} − Δ) ∈ L_{S}(V),

for any λ ∈ [0, 1]. Therefore ϕ is the ample model of K_{X} + Δ + λB over U for any λ ∈ (0, 1]. □

3.7. Convex geometry and Diophantine approximation.

Definition 3.7.1. Let V be a finite dimensional real affine space. If C is a convex subset of V and F is a convex subset of C, then we say that F is a face of C if whenever ∑ r_{i}v_{i} ∈ F, where r_{1}, r_{2}, …, r_{k} are real numbers such that ∑ r_{i} = 1, r_{i} ≥ 0 and v_{1}, v_{2}, …, v_{k} belong to C, then v_{i} ∈ F for some i. We say that v ∈ C is an extreme point if F = {v} is a face of C.

A polyhedron P in V is the intersection of finitely many half spaces. The interior P^{∘} of P is the complement of the proper faces. A polytope P in V is a compact polyhedron.

We say that a real vector space V_{0} is defined over the rationals, if V_{0} = V′ ⊗_{ℚ} ℝ, where V′ is a rational vector space. We say that an affine subspace V of a real vector space V_{0}, which is defined over the rationals, is defined over the rationals, if V is spanned by a set of rational vectors of V_{0}. We say that a polyhedron P is rational if it is defined by rational half spaces.

Note that a polytope is the convex hull of a finite set of points and the polytope is rational if those points can be chosen to be rational.

Lemma 3.7.2. Let X be a normal quasi-projective variety, and let V be a finite dimensional affine subspace of WDiv_{ℝ}(X), which is defined over the rationals.

Then L(V) (cf. (1.1.4) for the definition) is a rational polytope.

Proof. Note that the set of divisors Δ such that K_{X} + Δ is ℝ-Cartier forms an affine subspace W of V, which is defined over the rationals, so that, replacing V by W, we may assume that K_{X} + Δ is ℝ-Cartier for every Δ ∈ V.

Let π: Y ⟶ X be a resolution of X, which is a log resolution of the support of any element of V. Given any divisor Δ ∈ V, if we write

K_{Y} + Γ = π^{∗}(K_{X} + Δ),

then the coefficients of Γ are rational affine linear functions of the coefficients of Δ. On the other hand the condition that K_{X} + Δ is log canonical is equivalent to the condition that the coefficient of every component of Γ is at most one and the coefficient of every component of Δ is at least zero. □

Lemma 3.7.3. Let π: X ⟶ U be a projective morphism of normal quasi-projective varieties. Let V be a finite dimensional affine subspace of WDiv_{ℝ}(X) and let A ≥ 0 be a big ℝ-divisor over U. Let C ⊂ L_{A}(V) be a polytope.

If B_{+}(A/U) does not contain any non kawamata log terminal centres of (X, Δ), for every Δ ∈ C, then we may find a general ample ℚ-divisor A′ over U, a finite dimensional affine subspace V′ of WDiv_{ℝ}(X) and a translation

L: WDiv_{ℝ}(X) ⟶ WDiv_{ℝ}(X),

by an ℝ-divisor T ℝ-linearly equivalent to zero over U such that L(C) ⊂ L_{A′}(V′) and (X, Δ − A) and (X, L(Δ)) have the same non kawamata log terminal centres. Further, if A is a ℚ-divisor then we may choose T ℚ-linearly equivalent to zero over U.

Proof. Let Δ_{1}, Δ_{2}, …, Δ_{k} be the vertices of the polytope C. Let ℨ be the set of non kawamata log terminal centres of (X, Δ_{i}) for 1 ≤ i ≤ k. Note that if Δ ∈ C then any non kawamata log terminal centre of (X, Δ) is an element of ℨ.

By assumption, we may write A ∼_{ℝ,U} C + D, where C is a general ample ℚ-divisor over U and D ≥ 0 does not contain any element of ℨ. Further (3.5.3) implies that if A is a ℚ-divisor then we may assume that A ∼_{ℚ,U} C + D.

Given any rational number δ > 0, let

L: WDiv_{ℝ}(X) ⟶ WDiv_{ℝ}(X) given by L(Δ) = Δ+δ(C +D −A),

be the translation by the divisor T = δ(C + D − A) ∼_{ℝ,U} 0. Note that T ∼_{ℚ,U} 0 if A is a ℚ-divisor. As C + D does not contain any element of ℨ, if δ is sufficiently small then

K_{X} + L(Δ_{i}) = K_{X} + Δ_{i} + δ(C + D − A) = K_{X} + δC + (Δ_{i} − δA + δD),

is log canonical for every 1 ≤ i ≤ k and has the same non kawamata log terminal centres as (X, Δ_{i} − A). But then L(C) ⊂ L_{A′}(V′), where A′ = δC and V′ = V_{(1−δ)A+δD} and (X, Δ − A) and (X, L(Δ)) have the same non kawamata log terminal centres. □

Lemma 3.7.4. Let π: X ⟶ U be a projective morphism of normal quasi-projective varieties. Let V be a finite dimensional affine subspace of WDiv_{ℝ}(X), which is defined over the rationals, and let A be a general ample ℚ-divisor over U. Let S be a sum of prime divisors. Suppose that there is a divisorially log terminal pair (X, Δ_{0}), where S = ⌞Δ_{0}⌟, and let G ≥ 0 be any divisor whose support does not contain any non kawamata log terminal centres of (X, Δ_{0}).

Then we may find a general ample ℚ-divisor A′ over U, an affine subspace V′ of WDiv_{ℝ}(X), which is defined over the rationals, and a rational affine linear isomorphism

L: V_{S+A} ⟶ V′_{S+A′},

such that • L preserves ℚ-linear equivalence over U,

• L(L_{S+A}(V)) is contained in the interior of L_{S+A′}(V′),

• for any Δ ∈ L(L_{S+A}(V)), K_{X} + Δ is divisorially log terminal and ⌞Δ⌟ = S, and

• for any Δ ∈ L(L_{S+A}(V)), the support of Δ contains the support of G.

Proof. Let W be the vector space spanned by the components of Δ_{0}. Then Δ_{0} ∈ L_{S}(W) and (3.7.2) implies that L_{S}(W) is a non-empty rational polytope. But then L_{S}(W) contains a rational point and so, possibly replacing Δ_{0}, we may assume that K_{X} + Δ_{0} is ℚ-Cartier.

We first prove the result in the case that K_{X}+Δ is ℝ-Cartier for every Δ ∈ V_{S+A}. By compactness, we may pick ℚ-divisors Δ_{1}, Δ_{2}, …, Δ_{l} ∈ V_{S+A} such that L_{S+A}(V) is contained in the simplex spanned by Δ_{1}, Δ_{2}, …, Δ_{l} (we do not assume that Δ_{i} ≥ 0). Pick a rational number ϵ ∈ (0, 1/4] such that

ϵ(Δ_{i} − Δ_{0}) + (1 − 2ϵ)A,

is an ample ℚ-divisor over U, for 1 ≤ i ≤ l. Pick

A_{i} ∼_{ℚ,U} ϵ(Δ_{i} − Δ_{0}) + (1 − 2ϵ)A,

general ample ℚ-divisors over U. Pick A′ ∼_{ℚ,U} ϵA a general ample ℚ-divisor over U. If we define L: V_{S+A} ⟶ WDiv_{ℝ}(X) by

L(Δ_{i}) = (1 − ϵ)Δ_{i} + A_{i} + ϵΔ_{0} + A′ − (1 − ϵ)A ∼_{ℚ,U} Δ_{i},

and extend to the whole of V_{S+A} by linearity, then L is an injective rational linear map which preserves ℚ-linear equivalence over U. We let V′ be the rational affine subspace of WDiv_{ℝ}(X) defined by V′_{S+A′} = L(V_{S+A}). Note also that L is the composition L_{2} ∘ L_{1} of

L_{1}(Δ_{i}) = Δ_{i}+A_{i}/(1−ϵ)+A′−A and L_{2}(Δ) = (1−ϵ)Δ+ϵ(A′+Δ_{0}).

If Δ ∈ L_{S+A}(V) then K_{X} + Δ + A′ − A is log canonical, and as A_{i} is a general ample ℚ-divisor over U it follows that K_{X}+Δ+4/3A_{i}+A′−A is log canonical as well. As 1/(1−ϵ) ≤ 4/3, it follows that if Δ ∈ L_{S+A}(V) then K_{X} + L_{1}(Δ) is log canonical. Therefore, if Δ ∈ L_{S+A}(V) then K_{X} + L(Δ) is divisorially log terminal and ⌞L(Δ)⌟ = S.

Pick a divisor G′ such that S + A′ + G′ belongs to the interior of L_{S+A′}(V′). As G + G′ contains no log canonical centres of (X, Δ_{0}) and X is smooth at the generic point of every log canonical centre of (X, Δ_{0}), we may pick a ℚ-Cartier divisor H ≥ G + G′ which contains no log canonical centres of (X, Δ_{0}). Pick a rational number η > 0 such that A′ − ηH is ample over U. Pick A′′ ∼_{ℚ,U} A′ − ηH a general ample ℚ-divisor over U. Let δ > 0 by any rational number and let

T: WDiv_{ℝ}(X) ⟶ WDiv_{ℝ}(X),

be translation by δ(ηH+A′′−A′) ∼_{ℚ,U} 0. If V′′ is the span of V′, A′ and H and δ > 0 is sufficiently small then T (L(L_{S+A}(V))) is contained in the interior of L_{δA′′+S}(V′′), K_{X} + T (Δ) is divisorially log terminal and the support of T (Δ) contains the support of G, for all Δ ∈ L(L_{S+A}(V)). If we replace L by T ∘ L, V′_{S+A′} by T (L(V_{S+A})) and A′ by δA′′ then this finishes the case when K_{X} + Δ is ℝ-Cartier for every Δ ∈ V_{S+A}.

We now turn to the general case. If

W_{0} = { B ∈ V | K_{X} + S + A + B is ℝ-Cartier },

then W_{0} ⊂ V is an affine subspace of V, which is defined over the rationals. Note that L_{S+A}(V) = L_{S+A}(W_{0}). By what we have already proved, there is a rational affine linear isomorphism L_{0} : W_{0} ⟶ W′_{0}, which preserves ℚ-linear equivalence over U, a general ample ℚ-divisor A′ over U, such that L_{0}(L_{S+A}(W_{0})) is contained in the interior of L_{S+A′}(W′_{0}), and for every divisor Δ ∈ L_{0}(L_{S+A}(W_{0})), K_{X} + Δ is divisorially log terminal and the support of Δ contains the support of G.

Let W_{1} be any vector subspace of WDiv_{ℝ}(X), which is defined over the rationals, such that V = W_{0} + W_{1} and W_{0} ∩ W_{1} ⊂ {0}. Let V′ = W′_{0} + W_{1}. Since L_{0} preserves ℚ-linear equivalence over U, W′_{0} ∩ W_{1} = W_{0} ∩ W_{1} and L_{S+A′}(V′) = L_{S+A′}(W′_{0}). If we define L: V_{A} ⟶ V′_{A′}, by sending A + B_{0} + B_{1} to L_{0}(A + B_{0}) + B_{1}, where B_{i} ∈ W_{i}, then L is a rational affine linear isomorphism, which preserves ℚ-linear equivalence over U and L(L_{S+A}(V)) is contained in the interior of L_{S+A′}(V′). □

Lemma 3.7.5. Let π: X ⟶ U be a projective morphism of normal quasi-projective varieties. Let (X, Δ = A + B) be a log canonical pair, where A ≥ 0 and B ≥ 0.

If A is π-big and B_{+}(A/U) does not contain any non kawamata log terminal centres of (X, Δ) and there is a kawamata log terminal pair (X, Δ_{0}) then we may find a kawamata log terminal pair (X, Δ′ = A′ + B′), where A′ ≥ 0 is a general ample ℚ-divisor over U, B′ ≥ 0 and K_{X} + Δ′ ∼_{ℝ,U} K_{X} + Δ. If in addition A is a ℚ-divisor then K_{X} + Δ′ ∼_{ℚ,U} K_{X} + Δ.

Proof. By (3.7.3) we may assume that A is a general ample ℚ-divisor over U. If V is the vector space spanned by the components of Δ then Δ ∈ L_{A}(V) and the result follows by (3.7.4). □

Lemma 3.7.6. Let V be a finite dimensional real vector space, which is defined over the rationals. Let Λ ⊂ V be a lattice spanned by rational vectors. Suppose that v ∈ V is a vector which is not contained in any proper affine subspace W ⊂ V which is defined over the rationals.

Then the set

X = { mv + λ | m ∈ ℕ, λ ∈ Λ },

is dense in V.

Proof. Let

q: V ⟶ V/Λ,

be the quotient map and let G be the closure of the image of X. As G is infinite and V/Λ is compact, G has an accumulation point. It then follows that zero is also an accumulation point and that G is a closed subgroup.

The connected component G_{0} of G containing the identity is a Lie subgroup of V/Λ and so by Theorem 15.1 of [3], G_{0} is a torus. Thus G_{0} = W/Λ_{0}, where

W = H_{1}(G_{0}, ℝ) = Λ_{0}⊗_{ℤ}ℝ = H_{1}(G_{0}, ℤ)⊗_{ℤ}ℝ ⊂ H_{1}(G, ℤ)⊗_{ℤ}ℝ = H_{1}(G, ℝ),

is a subspace of V which is defined over the rationals. On the other hand, G/G_{0} is finite as it is discrete and compact. Thus a translate of v by a rational vector is contained in W and so W = V. □

Lemma 3.7.7. Let C be a rational polytope contained in a real vector space V of dimension n, which is defined over the rationals. Fix a positive integer k and a positive real number α.

If v ∈ C then we may find vectors v_{1}, v_{2}, …, v_{p} ∈ C and positive integers m_{1}, m_{2}, …, m_{p}, which are divisible by k, such that v is a convex linear combination of the vectors v_{1}, v_{2}, …, v_{p} and

∥v_{i} − v∥ < \frac{α}{m_{i}}  where  \frac{m_{i}v_{i}}{k} is integral.

Proof. Rescaling by k, we may assume that k = 1. We may assume that v is not contained in any proper affine linear subspace which is defined over the rationals. In particular v is contained in the interior of C since the faces of C are rational.

After translating by a rational vector, we may assume that 0 ∈ C. After fixing a suitable basis for V and possibly shrinking C, we may assume that C = [0, 1]^{n} ⊆ ℝ^{n} and v = (x_{1}, x_{2}, …, x_{n}) ∈ (0, 1)^{n}. By (3.7.6), for each subset I ⊆ {1, 2, …, n}, we may find

v_{I} = (s_{1}, s_{2}, …, s_{n}) ∈ (0, 1)^{n} ∩ ℚ^{n},

and an integer m_{I} such that m_{I}v_{I} is integral, such that

∥v − v_{I}∥ < \frac{α}{m_{I}} and s_{j} < x_{j} if and only if j ∈ I.

In particular v is contained inside the rational polytope B ⊆ C generated by the v_{I}. Thus v is a convex linear combination of a subset v_{1}, v_{2}, …, v_{p} of the extreme points of B. □

3.8. Rational curves of low degree. We will need the following generalisation of a result of Kawamata, see Theorem 1 of [16], which is proved by Shokurov in the appendix to [29].

Theorem 3.8.1. Let π: X ⟶ U be a projective morphism of normal quasi-projective varieties. Suppose that (X, Δ) is a log canonical pair of dimension n, where K_{X} + Δ is ℝ-Cartier. Suppose that there is a divisor Δ_{0} such that K_{X} + Δ_{0} is kawamata log terminal.

If R is an extremal ray of \overline{NE}(X/U) that is (K_{X} + Δ)-negative, then there is a rational curve Σ spanning R, such that

0 < −(K_{X} + Δ) · Σ ≤ 2n.

Proof. Passing to an open subset of U, we may assume that U is affine. Let V be the vector pace spanned by the components of Δ + Δ_{0}. By (3.7.2) the space L(V) of log canonical divisors is a rational polytope. Since Δ_{0} ∈ L(V), we may find ℚ-divisors Δ_{i} ∈ V with limit Δ, such that K_{X} + Δ_{i} is kawamata log terminal. In particular we may assume that (K_{X} + Δ_{0}) · R < 0. Replacing π by the contraction defined by the extremal ray R, we may assume that −(K_{X} + Δ) is π-ample.

Theorem 1 of [16] implies that we can find a rational curve Σ_{i} contracted by π such that

−(K_{X} + Δ_{i}) · Σ_{i} ≤ 2n.

Pick a π-ample ℚ-divisor A such that −(K_{X} + Δ + A) is also π-ample. In particular −(K_{X} + Δ_{i} + A) is π-ample for i ≫ 0. Now

A · Σ_{i} = (K_{X} + Δ_{i} + A) · Σ_{i} − (K_{X} + Δ_{i}) · Σ_{i} < 2n.

It follows that the curves Σ_{i} belong to a bounded family. Thus, possibly passing to a subsequence, we may assume that Σ = Σ_{i} is constant. In this case

−(K_{X} + Δ) · Σ = lim_{i} −(K_{X} + Δ_{i}) · Σ ≤ 2n.  □

Corollary 3.8.2. Let π: X ⟶ U be a projective morphism of normal quasi-projective varieties. Suppose the pair (X, Δ = A + B) has log canonical singularities, where A ≥ 0 is an ample ℝ-divisor over U and B ≥ 0. Suppose that there is a divisor Δ_{0} such that K_{X} + Δ_{0} is kawamata log terminal.

Then there are only finitely many (K_{X} + Δ)-negative extremal rays R_{1}, R_{2}, …, R_{k} of \overline{NE}(X/U).

Proof. We may assume that A is a ℚ-divisor. Let R be a (K_{X} + Δ)- negative extremal ray of \overline{NE}(X/U). Then

−(K_{X} + B) · R = −(K_{X} + Δ) · R + A · R > 0.

By (3.8.1) R is spanned by a curve Σ such that

−(K_{X} + B) · Σ ≤ 2n.

But then

A · Σ = −(K_{X} + B) · Σ + (K_{X} + Δ) · Σ ≤ 2n.  □

Therefore the curve Σ belongs to a bounded family.

3.9. Effective base point free theorem.

Theorem 3.9.1 (Effective Base Point Free Theorem). Fix a positive integer n. Then there is a positive integer m > 0 with the following property:

Let f: X ⟶ U be a projective morphism of normal quasi-projective varieties, and let D be a nef ℝ-divisor over U, such that aD−(K_{X} +Δ) is nef and big over U, for some positive real number a, where (X, Δ) is kawamata log terminal and X has dimension n.

Then D is semiample over U and if aD is Cartier then maD is globally generated over U.

Proof. Replacing D by aD we may assume that a = 1. As the property that D is either semiample or globally generated over U is local over U, we may assume that U is affine.

By assumption we may write D − (K_{X} + Δ) ∼_{ℝ,U} A + B, where A is an ample ℚ-divisor and B ≥ 0. Pick ϵ ∈ (0, 1) such that (X, Δ + ϵB) is kawamata log terminal. Then

D − (K_{X} + Δ + ϵB) = (1 − ϵ)(D − (K_{X} + Δ)) + ϵ(D − (K_{X} + Δ + B)),

is ample. Replacing (X, Δ) by (X, Δ + ϵB) we may therefore assume that D − (K_{X} + Δ) is ample. Let V be the subspace of WDiv_{ℝ}(X) spanned by the components of Δ. As L(V) is a rational polytope, cf. (3.7.2), which contains Δ, we may find Δ′ such that K_{X} + Δ′ is ℚ-Cartier and kawamata log terminal, sufficiently close to Δ so that D−(K_{X} +Δ′) is ample. Replacing (X, Δ) by (X, Δ′) we may therefore assume that K_{X} + Δ is ℚ-Cartier.

The existence of the integer m is Kollár’s effective version of the base point free theorem [20].

Pick a general ample ℚ-divisor A such that D − (K_{X} + Δ + A) is ample. Replacing Δ by Δ + A, we may assume that Δ = A + B where A is ample and B ≥ 0. By (3.8.2) there are finitely many (K_{X} + Δ)- negative extremal rays R_{1}, R_{2}, …, R_{k} of \overline{NE}(X). Let

F = \{ α ∈ \overline{NE}(X) | D · α = 0 \}.

Then F is a face of \overline{NE}(X) and if α ∈ F then (K_{X} + Δ) · α < 0, and so F is spanned by a subset of the extremal rays R_{1}, R_{2}, …, R_{k}. Let V be the smallest affine subspace of WDiv_{ℝ}(X), which is defined over the rationals and contains D. Then

C = { C ∈ V | C · α = 0, ∀α ∈ F },

is a rational polyhedron. It follows that we may find positive real numbers r_{1}, r_{2}, …, r_{q} and nef ℚ-Cartier divisors D_{1}, D_{2}, …, D_{q} such that D = ∑ r_{p}D_{p}. Possibly re-choosing D_{1}, D_{2}, …, D_{q} we may assume that D_{p} − (K_{X} + Δ) is ample. By the usual base point free theorem, D_{p} is semiample and so D is semiample. □

Corollary 3.9.2. Fix a positive integer n. Then there is a constant m > 0 with the following property:

Let f: X ⟶ U be a projective morphism of normal quasi-projective varieties such that K_{X} + Δ is nef over U and Δ is big over U, where (X, Δ) is kawamata log terminal and X has dimension n.

Then K_{X} + Δ is semiample over U and if r is a positive constant such that r(K_{X} +Δ) is Cartier, then mr(K_{X} +Δ) is globally generated over U.

Proof. By (3.7.5) we may find a kawamata log terminal pair

K_{X} + A + B ∼_{ℝ,U} K_{X} + Δ,

where A ≥ 0 is a general ample ℚ-divisor over U and B ≥ 0. As

(K_{X} + Δ) − (K_{X} + B) ∼_{ℝ,U} A,

is ample over U and K_{X} +B is kawamata log terminal, the result follows by (3.9.1). □

Lemma 3.9.3. Let π: X ⟶ U be a projective morphism of quasiprojective varieties Suppose that (X, Δ) is a kawamata log terminal pair, where Δ is big over U.

If ϕ: X −−→ Y is a weak log canonical model of K_{X} + Δ over U then (1) ϕ is a semiample model over U,

(2) the ample model ψ: X −−→ Z of K_{X} + Δ over U exists, and (3) there is a contraction morphism h: Y ⟶ Z such that K_{Y} +

Γ ∼_{ℝ,U} h^{∗}H, for some ample ℝ-divisor H over U, where Γ = ϕ_{∗}Δ.

Proof. K_{Y} + Γ is semiample over U by (3.9.2). (3) of (3.6.6) implies (2) and (3). □

3.10. The MMP with scaling. In order to run a minimal model program, two kinds of operations, known as flips and divisorial contractions are required. We begin by recalling their definitions.

Definition 3.10.1. Let (X, Δ) be a log canonical pair and f: X ⟶ Z be a projective morphism of normal varieties. Then f is a flipping contraction if

(1) X is ℚ-factorial and Δ is an ℝ-divisor,

(2) f is a small birational morphism of relative Picard number ρ(X/Z) = 1, and (3) −(K_{X} + Δ) is f-ample. The flip f^{+} : X^{+} ⟶ Z of a flipping contraction f: X ⟶ Z is a

small birational projective morphism of normal varieties f^{+} : X^{+} ⟶ Z such that K_{X^{+}} + Δ^{+} is f^{+}-ample, where Δ^{+} is the strict transform of Δ.

Lemma 3.10.2. Let (X, Δ) be a kawamata log terminal pair, let f: X ⟶ Z be a flipping contraction and let f^{+} : X^{+} ⟶ Z be the flip of f. Let ϕ: X −−→ X^{+} be the induced birational map.

Then

(1) ϕ is the log canonical model of (X, Δ) over Z,

(2) (X^{+}, Δ^{+} = ϕ_{∗}Δ) is log canonical,

(3) X^{+} is ℚ-factorial, and

(4) ρ(X^{+}/Z) = 1.

Proof. As ϕ is small, (3.6.2) implies that ϕ is (K_{X} + Δ)-negative and this implies (1) and (2).

Let B be a divisor on X^{+} and let C ⊂ X be the strict transform of B. Since ρ(X/Z) = 1, we may find λ ∈ ℝ such C + λ(K_{X} + Δ) is numerically trivial over U. But then (3.9.1) implies that C + λ(K_{X} + Δ) = f^{∗}D, for some ℝ-Cartier divisor D on Z. Therefore B +λ(K_{X^{+}} + Δ^{+}) = f^{+∗}D, and this implies (3) and (4). □

Remark 3.10.3. Results of Ambro and Fujino imply that the results of (3.10.2) hold in the case when (X, Δ) is log canonical.

In terms of our induction, we will need to work with a more restrictive notion of flipping contraction.

Definition 3.10.4. Let (X, Δ) be a purely log terminal pair and let f: X ⟶ Z be a projective morphism of normal varieties. Then f is a pl-flipping contraction if

(1) X is ℚ-factorial and Δ is an ℝ-divisor,

(2) f is a small birational morphism of relative Picard number ρ(X/Z) = 1,

(3) −(K_{X} + Δ) is f-ample, and

(4) S = ⌞Δ⌟ is irreducible and −S is f-ample.

A pl-flip is the flip of a pl-flipping contraction.

Definition 3.10.5. Let (X, Δ) be a log canonical pair and f: X ⟶ Z be a projective morphism of normal varieties. Then f is a divisorial contraction if

(1) X is ℚ-factorial and Δ is an ℝ-divisor, (2) f is a birational morphism of relative Picard number ρ(X/Z) =

1 with exceptional locus a divisor, and

(3) −(K_{X} + Δ) is f-ample.

Remark 3.10.6. If f: X ⟶ Z is a divisorial contraction, then an argument similar to (3.10.2) shows that (Z, f_{∗}Δ) is log canonical and Z is ℚ-factorial.

Definition 3.10.7. Let (X, Δ) be a log canonical pair and f: X ⟶ Z be a projective morphism of normal varieties. Then f is a Mori fibre space if

(1) X is ℚ-factorial and Δ is an ℝ-divisor,

(2) f is a contraction morphism, ρ(X/Z) = 1 and dim Z < dim X, and

(3) −(K_{X} + Δ) is f-ample.

The objective of the MMP is to produce either a log terminal model or a Mori fibre space. Note that if K_{X} + Δ has a log terminal model then K_{X} + Δ is pseudo-effective and if K_{X} + Δ has a Mori fibre space then K_{X} + Δ is not pseudo-effective, so these two cases are mutually exclusive.

There are several versions of the MMP, depending on the singularities that are allowed (typically, one restricts to kawamata log terminal singularities or divisorially log terminal singularities or terminal singularities with Δ = 0) and depending on the choices of negative extremal rays that are allowed (traditionally any choice of an extremal ray is acceptable).

In this paper, we will run the MMP with scaling for divisorially log terminal pairs satisfying certain technical assumptions. We will need the following key result.

Lemma 3.10.8. Let π: X ⟶ U be a projective morphism of normal quasi-projective varieties. Suppose that the pair (X, Δ = A + B) has kawamata log terminal singularities, where A ≥ 0 is big over U, B ≥ 0, D is a nef ℝ-Cartier divisor over U, but K_{X} + Δ is not nef over U. Set

λ = sup { μ | D + μ(K_{X} + Δ) is nef over U }.

Then there is a (K_{X} + Δ)-negative extremal ray R over U, such that

(D + λ(K_{X} + Δ)) · R = 0.

Proof. By (3.7.5) we may assume that A is ample over U.

By (3.8.2) there are only finitely many (K_{X} + Δ)-negative extremal rays R_{1}, R_{2}, …, R_{k} over U. For each (K_{X} + Δ)-negative extremal ray R_{i}, pick a curve Σ_{i} which generates R_{i}. Let

μ = min_{i} \frac{D · Σ_{i}}{−(K_{X} + Δ) · Σ_{i}} .

Then D + μ(K_{X} + Δ) is nef over U, since it is non-negative on each R_{i}, but it is zero on one of the extremal rays R = R_{i}. Thus λ = μ. □

Lemma 3.10.9. Let π: X ⟶ U be a projective morphism of normal quasi-projective varieties and let (X, Δ_{0}) be a kawamata log terminal pair. Suppose that (X, Δ = A+B) is a log canonical pair, where A ≥ 0 is big over U, B ≥ 0, B_{+}(A/U) contains no non kawamata log terminal centres of (X, Δ) and C is an ℝ-Cartier divisor such that K_{X} + Δ is not nef over U, whilst K_{X} + Δ + C is nef over U.

Then there is a (K_{X} +Δ)-negative extremal ray R and a real number 0 < λ ≤ 1 such that K_{X} + Δ + λC is nef over U but trivial on R.

Proof. By (3.7.5) we may assume K_{X} + Δ is kawamata log terminal and that A is ample over U. Apply (3.10.8) to D = K_{X} + Δ + C. □

Remark 3.10.10. Assuming existence and termination of the relevant flips, we may use (3.10.9) to define a special minimal model program, which we will refer to as the (K_{X} + Δ)-MMP with scaling of C.

Let π: X ⟶ U be a projective morphism of normal quasi-projective varieties, where X is ℚ-factorial, (X, Δ + C = S + A + B + C) is a divisorially log terminal pair, such that ⌞Δ⌟ = S, A ≥ 0 is big over U, B_{+}(A/U) does not contain any non kawamata log terminal centres of (X, Δ), and B ≥ 0, C ≥ 0. We pick λ ≥ 0 and a (K_{X} + Δ)-negative extremal ray R over U as in (3.10.9) above. If K_{X} + Δ is nef over U we stop. Otherwise λ > 0 and we let f: X ⟶ Z be the extremal contraction over U defined by R. If f is not birational, we have a Mori fibre space over U and we stop. If f is birational then either f is divisorial and we replace X by Z or f is small and assuming the existence of the flip f^{+} : X^{+} ⟶ Z, we replace X by X^{+}. In either case K_{X} +Δ+λC is nef over U and K_{X} +Δ is divisorially log terminal and so we may repeat the process.

In this way, we obtain a sequence ϕ_{i}: X_{i} −−→ X_{i+1} of K_{X} + Δ flips and divisorial contractions over U and real numbers 1 ≥ λ_{1} ≥ λ_{2} ≥ ⋯ such that K_{X_{i}} + Δ_{i} + λ_{i}C_{i} is nef over U where Δ_{i} = (ϕ_{i−1})_{∗}Δ_{i−1} and C_{i} = (ϕ_{i−1})_{∗}C_{i−1}.

Note that by (3.10.11), each step of this MMP preserves the condition that (X, Δ) is divisorially log terminal and B_{+}(A/U) does not contain any non kawamata log terminal centres of (X, Δ) so that we may apply (3.10.9). By (3.10.2) and (3.10.6), (X, Δ + λC) is log canonical. However the condition that B_{+}(A/U) does not contain any non kawamata log terminal centres of (X, Δ + λC) is not necessarily preserved.

Lemma 3.10.11. Let π: X ⟶ U be a projective morphism of normal quasi-projective varieties. Suppose that K_{X} + Δ is divisorially log terminal, X is ℚ-factorial and let ϕ: X −−→ Y be a sequence of steps of the (K_{X} + Δ)-MMP over U.

If Γ = ϕ_{∗}Δ then

(1) ϕ is an isomorphism at the generic point of every non kawamata log terminal centre of K_{Y} +Γ. In particular (Y, Γ) is divisorially log terminal.

(2) If Δ = S + A + B, where S = ⌞Δ⌟, A ≥ 0 is big over U, B_{+}(A/U) does not contain any non kawamata log terminal centres of K_{X} + Δ, and B ≥ 0 then ϕ_{∗}S = ⌞Γ⌟, ϕ_{∗}A is big over U and B_{+}(ϕ_{∗}A/U) does not contain any non kawamata log terminal centres of K_{Y} + Γ.

In particular Γ ∼_{ℝ,U} Γ′ where (Y, Γ′) is kawamata log terminal and Γ′ is big over U.

Proof. We may assume that ϕ is either a flip or a divisorial contraction over U. We first prove (1). Let p: W ⟶ X and q: W ⟶ Y be common log resolutions, which resolve the indeterminacy of ϕ. We may write

p^{∗}(K_{X} + Δ) = q^{∗}(K_{Y} + Γ) + E,

where E is exceptional and contains every exceptional divisor over the locus where ϕ^{−1} is not an isomorphism. In particular the log discrepancy of every valuation with centre on Y contained in the locus where ϕ^{−1} is not an isomorphism with respect to K_{Y} + Γ is strictly greater than the log discrepancy with respect to K_{X} + Δ. Hence (1).

Now suppose that Δ = S + A + B, A is big over U and no non kawamata log terminal centre of (X, Δ) is contained in B_{+}(A/U). Pick a divisor C on Y which is a general ample ℚ-divisor over U. Possibly replacing C by a smaller multiple, we may assume that B((A − ϕ_{∗}^{−1}C)/U) does not contain any log canonical centres of (X, Δ). Thus A − ϕ_{∗}^{−1}C ∼_{ℝ,U} D ≥ 0, where D does not contain any non kawamata log terminal centres of (X, Δ). If ϵ > 0 is sufficiently small, then (X, Δ′ = Δ + ϵD) is divisorially log terminal, (X, Δ) and (X, Δ′) have the same non kawamata log terminal centres and ϕ is a K_{X} +Δ′ flip or a divisorial contraction over U. (1) implies that (Y, Γ′ = ϕ_{∗}Δ′) is divisorially log terminal and hence ϕ_{∗}D does not contain any non kawamata log terminal centres of (Y, Γ). (2) follows as ϕ_{∗}A−C ∼_{ℝ,U} ϕ_{∗}D ≥ 0. □

Lemma 3.10.12. Let π: X ⟶ U be a projective morphism of quasiprojective varieties. Suppose that (X, Δ) is a divisorially log terminal pair. Let S be a prime divisor.

Let f_{i}: X_{i} −−→ X_{i+1} a sequence of flips and divisorial contractions over U, starting with X_{1} := X, for the (K_{X} + Δ)-MMP, which does not contract S. If f_{i} is not an isomorphism in a neighbourhood of the strict transform S_{i} of S, then neither is the induced birational map X_{i} −−→ X_{j}, j > i.

Proof. Since the map f_{i} is (K_{X_{i}} + Δ_{i})-negative and X_{i+1} −−→ X_{j} is (K_{X_{i+1}} + Δ_{i+1})-negative, there is some valuation ν whose centre intersects S_{i}, such that

a(ν, X_{i}, Δ_{i}) < a(ν, X_{i+1}, Δ_{i+1}) and a(ν, X_{i+1}, Δ_{i+1}) ≤ a(ν, X_{j}, Δ_{j}).

But then a(ν, X_{i}, Δ_{i}) < a(ν, X_{j}, Δ_{j}), so that X_{i} −−→ X_{j} is not an isomorphism in a neighbourhood of S_{i}. □

3.11. Shokurov’s polytopes. We will need some results from [32]. First some notation. Given a ray R ⊂ \overline{NE}(X), let

R^{⊥} = { Δ ∈ L(V) | (K_{X} + Δ) · R = 0 }.

Theorem 3.11.1. Let π: X ⟶ U be a projective morphism of normal quasi-projective varieties. Let V be a finite dimensional affine subspace of WDiv_{ℝ}(X), which is defined over the rationals. Fix an ample ℚdivisor A over U. Suppose that there is a kawamata log terminal pair (X, Δ_{0}).

Then the set of hyperplanes R^{⊥} is finite in L_{A}(V), as R ranges over the set of extremal rays of \overline{NE}(X/U). In particular, N_{A,π}(V) is a rational polytope.

Corollary 3.11.2. Let π: X ⟶ U be a projective morphism of normal quasi-projective varieties. Let V be a finite dimensional affine subspace of WDiv_{ℝ}(X), which is defined over the rationals. Fix a general ample ℚ-divisor A over U. Suppose that there is a kawamata log terminal pair (X, Δ_{0}). Let ϕ: X −−→ Y be any birational contraction over U.

Then W_{ϕ,A,π}(V) is a rational polytope. Moreover there are finitely many morphisms f_{i}: Y ⟶ Z_{i} over U, 1 ≤ i ≤ k, such that if f: Y ⟶ Z is any contraction morphism over U and there is an ℝ-divisor D on Z, which is ample over U, such that K_{Y} + Γ = ϕ_{∗}(K_{X} + Δ) ∼_{ℝ,U} f^{∗}D for some Δ ∈ W_{ϕ,A,π}(V), then there is an index 1 ≤ i ≤ k and an isomorphism η: Z_{i} ⟶ Z such that f = η ∘ f_{i}.

Corollary 3.11.3. Let π: X ⟶ U be a projective morphism of normal quasi-projective varieties. Let V be a finite dimensional affine subspace of WDiv_{ℝ}(X), which is defined over the rationals. Fix a general ample ℚ-divisor A over U. Let (X, Δ_{0}) be a kawamata log terminal pair, let f: X ⟶ Z be a morphism over U such that Δ_{0} ∈ L_{A}(V) and K_{X} + Δ_{0} ∼_{ℝ,U} f^{∗}H, where H is an ample divisor over U. Let ϕ: X −−→ Y be a birational contraction over Z.

Then there is a neighbourhood P_{0} of Δ_{0} in L_{A}(V) such that for all Δ ∈ P_{0}, ϕ is a log terminal model for K_{X} + Δ over Z if and only if ϕ is a log terminal model for K_{X} + Δ over U.

Proof of (3.11.1). Since L_{A}(V) is compact it suffices to prove this locally about any point Δ ∈ L_{A}(V). By (3.7.4) we may assume that K_{X} + Δ is kawamata log terminal. Fix ϵ > 0 such that if Δ′ ∈ L_{A}(V) and ∥Δ′ − Δ∥ < ϵ, then Δ′ − Δ + A/2 is ample over U. Let R be an extremal ray over U such that (K_{X} + Δ′) · R = 0, where Δ′ ∈ L_{A}(V) and ∥Δ′ − Δ∥ < ϵ. We have

(K_{X} + Δ − A/2) · R = (K_{X} + Δ′) · R − (Δ′ − Δ + A/2) · R < 0.

Finiteness then follows from (3.8.2).

N_{A,π}(V) is surely a closed subset of L_{A}(V). If K_{X} + Δ is not nef over U then (3.8.1) implies that K_{X} +Δ is negative on a rational curve Σ which generates an extremal ray R of \overline{NE}(X/U). Thus N_{A,π}(V) is the intersection of L_{A}(V) with the half-spaces determined by finitely many of the extremal rays of \overline{NE}(X/U). □

Proof of (3.11.2). Since L_{A}(V) is a rational polytope its span is an affine subspace of V_{A}, which is defined over the rationals. Possibly replacing V, we may therefore assume that L_{A}(V) spans V_{A}. By compactness, to prove that W_{ϕ,A,π}(V) is a rational polytope, we may work locally about a divisor Δ ∈ W_{ϕ,A,π}(V). By (3.7.4) we may assume that K_{X} + Δ is kawamata log terminal in which case K_{Y} + Γ is kawamata log terminal as well. Let W ⊂ WDiv_{ℝ}(Y) be the image of V. If C = ϕ_{∗}A then C is big over U and by (3.7.3) and (3.7.4), we may find a rational affine linear isomorphism L: W ⟶ W′ and an ample ℚ-divisor C′ over U such that L(Γ) belongs to the interior of L_{C′}(W′) and L(Ψ) ∼_{ℚ,U} Ψ for any Ψ ∈ W. (3.11.1) implies that N_{C′,ψ}(W′) is a rational polytope, where ψ: Y ⟶ U is the structure morphism, and so N_{C,ψ}(W) is a rational polytope locally about Γ.

Let p: Z ⟶ X be a log resolution of (X, Δ) which resolves the indeterminacy locus of ϕ, via a birational map q: Z ⟶ Y. We may write

K_{Z} + Ψ = p^{∗}(K_{X} + Δ)
K_{Z} + Φ = q^{∗}(K_{Y} + Γ).

Note that Δ ∈ W_{ϕ,A,π}(V) if and only if Γ = ϕ_{∗}Δ ∈ N_{C,ψ}(W) and Ψ − Φ ≥ 0. Since the map L: V ⟶ W given by Δ ⟶ Γ = ϕ_{∗}Δ is rational linear, the first statement is clear.

Note that if f: Y ⟶ Z and f′: Y ⟶ Z′ are two contraction morphisms over U then there is an isomorphism η: Z ⟶ Z′ such that f′ = η ∘ f if and only if the curves contracted by f and f′ coincide.

Let f: Y ⟶ Z be a contraction morphism over U, such that

K_{Y} + Γ = K_{Y} + ϕ_{∗}Δ ∼_{ℝ,U} f^{∗}D,

where Δ ∈ W_{ϕ,A,π}(V) and D is an ample over U ℝ-divisor on Z. Γ belongs to the interior of a unique face G of N_{C,ψ}(W) and the curves contracted by f are determined by G. Now Δ belongs to the interior of a unique face F of W_{ϕ,A,π}(V) and G is determined by F. But as W_{ϕ,A,π}(V) is a rational polytope it has only finitely many faces F. □

Proof of (3.11.3). By (3.11.1) we may find finitely many extremal rays R_{1}, R_{2}, …, R_{k} of Y over U such that if K_{Y} + Γ = K_{Y} + ϕ_{∗}Δ is not nef over U, then it is negative on one of these rays. If Γ_{0} = ϕ_{∗}Δ_{0} then we may write

K_{Y} + Γ = K_{Y} + Γ_{0} + (Γ − Γ_{0}) ∼_{ℝ,U} g^{∗}H + ϕ_{∗}(Δ − Δ_{0}),

where g: Y ⟶ Z is the structure morphism. Therefore there is a neighbourhood P_{0} of Δ_{0} in L_{A}(V) such that if K_{Y} + Γ is not nef over U then it is negative on an extremal ray R_{i}, which is extremal over Z. In particular if ϕ is a log terminal model of K_{X} + Δ over Z then it is a log terminal model over U. The other direction is clear. □

4. Special finiteness

Lemma 4.1. Let π: X ⟶ U be a projective morphism of quasiprojective varieties. Let (X, Δ = S + A + B) be a log smooth pair, where S = ⌞Δ⌟ is a prime divisor, A is a general ample ℚ-divisor over U and B ≥ 0. Let C = A|_{S} and Θ = (Δ − S)|_{S}. Let ϕ: X −−→ Y be a birational map over U which does not contract S, let T be the strict transform of S and let τ: S −−→ T be the induced birational map. Let Γ = ϕ_{∗}Δ and define Ψ on T by adjunction,

(K_{Y} + Γ)|_{T} = K_{T} + Ψ.

If (S, Θ) is terminal and ϕ is a weak log canonical model of K_{X} + Δ over U then there is a divisor C ≤ Ξ ≤ Θ such that τ is a weak log canonical model of K_{S} + Ξ over U, where τ_{∗}Ξ = Ψ.

Proof. Let p: W ⟶ X and q: W ⟶ Y be log resolutions of (X, Δ) and (Y, Γ), which resolve the indeterminacy of ϕ, where Γ = ϕ_{∗}Δ. Then we may write

K_{W} + Δ′ = p^{∗}(K_{X} + Δ) + E and K_{W} + Γ′ = q^{∗}(K_{Y} + Γ) + F,

where Δ′ ≥ 0 and E ≥ 0 have no common components, Γ′ ≥ 0 and F ≥ 0 have no common components, p_{∗}Δ′ = Δ, q_{∗}Γ′ = Γ, p_{∗}E = 0 and q_{∗}F = 0. Since ϕ is (K_{X} +Δ)-non-positive, we have F −E +Δ′−Γ′ ≥ 0 and so

F ≥ E and Γ′ ≤ Δ′.

If R is the strict transform of S on W then there are two birational morphisms f = p|_{R}: R ⟶ S and g = q|_{R}: R ⟶ T. Since (X, Δ) is purely log terminal, (Y, Γ) is purely log terminal. In particular (T, Ψ) is kawamata log terminal. It follows that

K_{R} + Θ′ = f^{∗}(K_{S} + Θ) + E′ and K_{R} + Ψ′ = g^{∗}(K_{T} + Ψ) + F′,

where Θ′ = (Δ′ − R)|_{R}, E′ = E|_{R}, Ψ′ = (Γ′ − R)|_{R} and F′ = F |_{R}. Moreover, every component of F′ is g-exceptional, by (4) of (3.4.1). As p and q are log resolutions

F′ ≥ E′ and Ψ′ ≤ Θ′.

Suppose that B is a prime divisor on R which is f-exceptional but not g-exceptional. Then B is not a component of F′, and so it is not a component of E′. But then the log discrepancy of B with respect to (S, Θ) is at most one, which contradicts the fact that (S, Θ) is terminal. Thus τ is a birational contraction.

As g_{∗}E′ = 0,

τ_{∗}Θ = g_{∗}Θ′ ≥ g_{∗}Ψ′ = Ψ.

Let Ξ ≤ Θ be the biggest divisor on S such that τ_{∗}Ξ = Ψ. In other words, if a component of Θ is not τ-exceptional, we replace its coefficient by the corresponding coefficient of Ψ and if a component of Θ is τ-exceptional, then we do not change its coefficient. As A is a general ample ℚ-divisor over U, C is not contained in the locus where ϕ is not an isomorphism. It follows that C ≤ Ξ. We have

f^{∗}(K_{S} + Ξ) = g^{∗}(K_{T} + Ψ) + L,

where g_{∗}L = 0. Contrast this with

f^{∗}(K_{S} + Θ) = g^{∗}(K_{T} + Ψ) + M.

We have already seen that M = F′ − E′ + Θ′ − Ψ′ ≥ 0. By definition of Ξ, f_{∗}L and f_{∗}M agree on the τ-exceptional divisors. Since f_{∗}L is τ-exceptional, it follows that f_{∗}L ≥ 0. But then (3.6.3) implies that L ≥ 0 and so τ is a weak log canonical model of K_{S} + Ξ over U. □

Lemma 4.2. Let X ⟶ U and Y ⟶ U be two projective morphisms of normal quasi-projective varieties, where X and Y are ℚ-factorial. Let f: X −−→ Y be a small birational map over U. Let S be a prime divisor on X, let T be its strict transform on Y, let g: S −−→ T be the induced birational map and let H be an ample ℝ-divisor on Y over U. Let D be the strict transform of H in X.

If g is an isomorphism and D|_{S} = g^{∗}(H|_{T}) then f is an isomorphism in a neighbourhood of S and T.

Proof. If p: W ⟶ X is the normalisation of the graph of f and q: W ⟶ Y is the induced birational morphism then we may write

p^{∗}D = q^{∗}H + E,

where E is p-exceptional. Since f is small the exceptional divisors of p and q are the same. Since X and Y are ℚ-factorial the exceptional locus of p and q are divisors by (3) of (3.6.2). Notice also that if F is a p-exceptional divisor then it is covered by a family of p-exceptional curves which are not contracted by q. Thus the image of F is contained in the indeterminancy locus of f and also in the support of E by (2) of (3.6.2). Putting all of this together, the support of E is equal to the inverse image of the indeterminancy locus of f and of f^{−1}.

Let R be the strict transform of S in W and let p_{1} = p|_{R}: R ⟶ S and q_{1} = q|_{R}: R ⟶ T. Since

p_{1}^{∗}(D|_{S}) = (p^{∗}D)|_{R} = (q^{∗}H + E)|_{R} = q_{1}^{∗}(H|_{T}) + E|_{R},

we have E|_{R} = 0. Thus S does not intersect the indeterminancy locus of f and T does not intersect the indeterminancy locus of f^{−1} and so f is an isomorphism in neighbourhood of S and T. □

Lemma 4.3. Let π: X ⟶ U be a projective morphism of normal quasi-projective varieties. Let (X, Δ_{i}) be two i = 1, 2, purely log terminal pairs, where S = ⌞Δ_{i}⌟ is a prime divisor, which is independent of i. Let ϕ_{i}: X −−→ Y_{i} be ℚ-factorial log canonical models of K_{X} + Δ_{i} over U, where the ϕ_{i} are birational and do not contract S for i = 1, 2. Let T_{i} be the strict transform of S and let τ_{i}: S −−→ T_{i} be the induced birational maps. Let Φ_{i} be the different

(K_{Y_{i}} + T_{i})|_{T_{i}} = K_{T_{i}} + Φ_{i}.

If (1) the induced birational map χ: Y_{1} −−→ Y_{2} is small,

(2) the induced birational map σ: T_{1} −−→ T_{2} is an isomorphism,

(3) σ^{∗}Φ_{2} = Φ_{1}, and

(4) for every component B of the support of (Δ_{2} − S), we have

(ϕ_{1∗}B)|_{T_{1}} = σ^{∗}((ϕ_{2∗}B)|_{T_{2}}),

then χ is an isomorphism in a neighbourhood of T_{1} and T_{2}.

Proof. Let Γ_{i} = ϕ_{i∗}Δ_{i} and define Ψ_{i} by adjunction,

(K_{Y_{i}} + Γ_{i})|_{T_{i}} = K_{T_{i}} + Ψ_{i}.

We have

K_{T_{i}} + Ψ_{i} = (K_{Y_{i}} + T_{i} + Γ_{i} − T_{i})|_{T_{i}}
= K_{T_{i}} + Φ_{i} + (Γ_{i} − T_{i})|_{T_{i}},

so that Ψ_{i} = Φ_{i} + (ϕ_{i∗}(Δ_{i} − S))|_{T_{i}}. Conditions (3) and (4) then imply that

σ^{∗}Ψ_{2} = σ^{∗}(Φ_{2} + (ϕ_{2∗}(Δ_{2} − S))|_{T_{2}}) = Φ_{1} + (ϕ_{1∗}(Δ_{2} − S))|_{T_{1}}.

It follows that

(χ^{∗}(K_{Y_{2}} + Γ_{2}))|_{T_{1}} = (χ^{∗}ϕ_{2∗}(K_{X} + Δ_{2}))|_{T_{1}}
= (ϕ_{1∗}(K_{X} + Δ_{2}))|_{T_{1}}
= (K_{Y_{1}} + T_{1} + ϕ_{1∗}(Δ_{2} − S))|_{T_{1}}
= K_{T_{1}} + Φ_{1} + (ϕ_{1∗}(Δ_{2} − S))|_{T_{1}}
= σ^{∗}(K_{T_{2}} + Ψ_{2}).

Thus (4.2) implies that χ is an isomorphism in a neighbourhood of T_{1} and T_{2}. □

Lemma 4.4. Theorem E_{n−1} implies Theorem B_{n}.

Proof. Suppose not. Then there would be an infinite sequence of ℚfactorial weak log canonical models over U, ϕ_{i}: X −−→ Y_{i} for K_{X} + Δ_{i} where Δ_{i} ∈ L_{S+A}(V), which only contract components of 𝔈 and which do not contract every component of S, such that if the induced birational map f_{ij}: Y_{i} −−→ Y_{j} is an isomorphism in a neighbourhood of the strict transforms S_{i} and S_{j} of S then i = j. Since 𝔈 is finite, possibly passing to a subsequence, we may assume that f_{ij} is small. Possibly passing to a further subsequence, we may also assume that there is a fixed component T of S such that ϕ_{i} does not contract T and f_{ij} is not an isomorphism in a neighbourhood of the strict transforms of T. Replacing S by T, we may therefore assume that S is irreducible.

Pick H_{1}, H_{2}, …, H_{k} general ample over U ℚ-divisors which span WDiv_{ℝ}(X) modulo numerical equivalence over U and let H = H_{1} + H_{2} + ⋯ + H_{k} be their sum. We may replace V by the subspace of WDiv_{ℝ}(X) generated by V and H_{1}, H_{2}, …, H_{k}. Passing to a subsequence, we may assume that lim Δ_{i} = Δ_{∞} ∈ L_{S+A}(V). By (3.7.4) we may assume that K_{X} + Δ_{∞} is purely log terminal and Δ_{∞} is in the interior of L_{S+A}(V). Possibly passing to a subsequence we may therefore assume that Δ_{i} and Δ_{∞} have the same support. We may therefore assume that K_{X} + Δ_{i} is purely log terminal and Δ_{i} contains the support of H. By (3.6.12) we may therefore assume that ϕ_{i} is the ample model of K_{X} + Δ_{i} over U. In particular Δ_{i} = Δ_{j} implies i = j.

Pick Δ ∈ L_{S+A}(V) such that K_{X} + Δ is purely log terminal and Δ_{i} ≤ Δ for all i > 0. Let f: Y ⟶ X be a log resolution of (X, Δ). Then we may write

K_{Y} + Γ = f^{∗}(K_{X} + Δ) + E,

where Γ ≥ 0 and E ≥ 0 have no common components, f_{∗}Γ = Δ and f_{∗}E = 0. Let T be the strict transform of S. Possibly blowing up more, we may assume that (T, Θ) is terminal, where Θ = (Γ − T)|_{T}. We may find F ≥ 0 an exceptional ℚ-divisor so that f^{∗}A − F is ample over U, (Y, Γ + F) is purely log terminal and (T, Θ + F |_{T}) is terminal. Let A′ ∼_{ℚ,U} f^{∗}A − F be a general ample ℚ-divisor over U. For every i, we may write

K_{Y} + Γ_{i} = f^{∗}(K_{X} + Δ_{i}) + E_{i},

where Γ_{i} ≥ 0 and E_{i} ≥ 0 have no common components, f_{∗}Γ_{i} = Δ_{i} and f_{∗}E_{i} = 0. Let

Γ′_{i} = Γ_{i} − f^{∗}A + F + A′ ∼_{ℚ,U} Γ_{i}.

Then K_{Y} + Γ′_{i} is purely log terminal and (3.6.9) implies that f ∘ ϕ_{i} is both a weak log canonical model, and the ample model, of K_{Y} + Γ_{i} over U.

Replacing X by Y we may therefore assume that (X, Δ) is log smooth and (S, Θ) is terminal, where Θ = (Δ − S)|_{S}. Let

C = A|_{S} ≤ Ξ_{i} ≤ (Δ_{i} − S)|_{S} ≤ Θ,

be the divisors whose existence is guaranteed by (4.1) so that τ_{i} = ϕ_{i}|_{S}: S −−→ S_{i} is a weak log canonical model (and the ample model) of K_{S} + Ξ_{i} over U, where S_{i} is the strict transform of S. As we are assuming Theorem E_{n−1}, possibly passing to a subsequence we may assume that the restriction g_{ij} = f_{ij}|_{S_{i}}: S_{i} −−→ S_{j} is an isomorphism.

Since Y_{i} is ℚ-factorial, we may define two ℝ-divisors on S_{i} by adjunction,

(K_{Y_{i}} + Γ_{i})|_{S_{i}} = K_{S_{i}} + Ψ_{i} and (K_{Y_{i}} + S_{i})|_{S_{i}} = K_{S_{i}} + Φ_{i},

where Γ_{i} = ϕ_{i∗}Δ_{i}. Then

0 ≤ Φ_{i} ≤ Ψ_{i} = τ_{i∗}Ξ_{i} ≤ τ_{i∗}Θ.

As the coefficients of Φ_{i} have the form \frac{(r−1)}{r} for some integer r > 0, by (3) of (3.4.1), and the coefficients of Θ (and hence also of τ_{i∗}Θ) are all less than one, it follows that there are only finitely many possibilities for Φ_{i}. Let B be a component of the support of Δ_{i} − S and let G = (ϕ_{i∗}(B))|_{S_{i}}. By (3) of (3.4.1), the coefficients of G are integer multiples of 1/r. Pick a real number β > 0 such that the coefficients of B in Δ_{i} − S are at least β. As

βG ≤ ϕ_{i∗}(Δ_{i} − S)|_{S_{i}} ≤ τ_{i∗}Θ,

there are only finitely many possibilities for G. Possibly passing to a subsequence, we may assume Φ_{i} = f_{ij}^{∗}Φ_{j} and that if B is a component of the support of Δ_{j} − S then (ϕ_{i∗}(B))|_{S_{i}} = g_{ij}^{∗}((ϕ_{j∗}B)|_{S_{j}}). (4.3) implies that f_{ij} is an isomorphism in a neighbourhood of S_{i} and S_{j}, a contradiction. □

5. Log terminal models

Lemma 5.1. Assume Theorem B_{n}.

Let π: X ⟶ U be a projective morphism of normal quasi-projective varieties, where X is ℚ-factorial of dimension n. Suppose that

K_{X} + Δ + C = K_{X} + S + A + B + C,

is divisorially log terminal and nef over U, where S is a sum of prime divisors, B_{+}(A/U) does not contain any non kawamata log terminal centres of (X, Δ + C) and B ≥ 0, C ≥ 0.

Then any sequence of flips and divisorial contractions for the (K_{X} + Δ)-MMP over U with scaling of C, which does not contract S, is eventually disjoint from S.

Proof. We may assume that S is irreducible and by (3.7.5) we may assume that A is a general ample ℚ-divisor over U. Let f_{i}: X_{i} −−→ X_{i+1} be a sequence of flips and divisorial contractions over U, starting with X_{1} := X, for the (K_{X} + Δ)-MMP with scaling of C.

Let 𝔈 be the set of prime divisors in X which are contracted by any of the induced birational maps ϕ_{i}: X −−→ X_{i}. Then the cardinality of 𝔈 is less than the relative Picard number of X over U. In particular 𝔈 is finite.

By assumption there is a non-increasing sequence of real numbers λ_{1}, λ_{2}, … ∈ [0, 1], such that K_{X_{i}} + Δ_{i} + λ_{i}C_{i} is nef over U, where Δ_{i} is the strict transform of Δ and C_{i} is the strict transform of C. Further the birational map f_{i} is (K_{X_{i}} + Δ_{i})-negative and (K_{X_{i}} + Δ_{i} + λ_{i}C_{i})- trivial, so that f_{i} is (K_{X_{i}} +Δ_{i} +λC_{i})-non-positive if and only if λ ≤ λ_{i}. By induction, the birational map ϕ_{i} is therefore (K_{X} + Δ + λ_{i}C)-non-positive. In particular ϕ_{i} is a ℚ-factorial weak log canonical model over U of (X, Δ + λ_{i}C).

Let V be the smallest affine subspace of WDiv_{ℝ}(X) containing Δ − S − A and C, which is defined over the rationals. As Δ + λ_{i}C ∈ L_{S+A}(V), Theorem B_{n} implies that there is an index k and infinitely many indices l such that the induced birational map X_{k} −−→ X_{l} is an isomorphism in a neighbourhood of S_{k} and S_{l}. But then (3.10.12) implies that f_{i} is an isomorphism in a neighbourhood of S_{i} for all i ≥ k. □

We use (5.1) to run a special MMP:

Lemma 5.2. Assume Theorem A_{n} and Theorem B_{n}.

Let π: X ⟶ U be a projective morphism of normal quasi-projective varieties, where X is ℚ-factorial of dimension n. Suppose that (X, Δ+ C = S + A + B + C) is a divisorially log terminal pair, such that ⌞Δ⌟ = S, A ≥ 0 is big over U, B_{+}(A/U) does not contain any non kawamata log terminal centres of (X, Δ + C), and B ≥ 0, C ≥ 0. Suppose that there is an ℝ-divisor D ≥ 0 whose support is contained in S and a real number α ≥ 0, such that

(∗)  K_{X} + Δ ∼_{ℝ,U} D + αC.

If K_{X}+Δ+C is nef over U, then there is a log terminal model ϕ: X −−→ Y for K_{X} + Δ over U, where B_{+}(ϕ_{∗}A/U) does not contain any non kawamata log terminal centres of (Y, Γ = ϕ_{∗}Δ).

Proof. By (3.10.9) and (3.10.11) we may run the (K_{X} + Δ)-MMP with scaling of C over U, and this will preserve the condition that B_{+}(A/U) does not contain any non kawamata log terminal centres of (X, Δ). Pick t ∈ [0, 1] minimal such that K_{X} + Δ + tC is nef over U. If t = 0 we are done. Otherwise we may find a (K_{X} +Δ)-negative extremal ray R over U, such that (K_{X} + Δ + tC) · R = 0. Let f: X ⟶ Z be the associated contraction over U. As t > 0, C · R > 0 and so D · R < 0. In particular f is always birational.

If f is divisorial, then we can replace X, S, A, B, C and D by their images in Z. Note that (∗) continues to hold.

Otherwise f is small. As D · R < 0, R is spanned by a curve Σ which is contained in a component T of S, where T · Σ < 0. Note that K_{X} +S +A+B −ϵ(S −T) is purely log terminal for any ϵ ∈ (0, 1), and so f is a pl-flip. As we are assuming Theorem A_{n}, the flip f′: X′ ⟶ Z of f: X ⟶ Z exists. Again, if we replace X, S, A, B, C and D by their images in X′, then (∗) continues to hold.

On the other hand this MMP is certainly not an isomorphism in a neighbourhood of S and so the MMP terminates by (5.1). □

Definition 5.3. Let π: X ⟶ U be a projective morphism of normal quasi-projective varieties. Let (X, Δ = A + B) be a ℚ-factorial divisorially log terminal pair and let D be an ℝ-divisor, where A ≥ 0, B ≥ 0 and D ≥ 0. A neutral model over U for (X, Δ), with respect to A and D, is any birational map f: X −−→ Y over U, such that

• f is a birational contraction,

• the only divisors contracted by f are components of D,

• Y is ℚ-factorial and projective over U,

• B_{+}(f_{∗}A/U) does not contain any non kawamata log terminal centres of (Y, Γ = f_{∗}Δ), and

• K_{Y} + Γ is divisorially log terminal and nef over U.

Lemma 5.4. Assume Theorem A_{n} and Theorem B_{n}.

Let π: X ⟶ U be a projective morphism of normal quasi-projective varieties, where X has dimension n. Let (X, Δ = A + B) be a divisorially log terminal log pair and let D be an ℝ-divisor, where A ≥ 0 is big over U, B ≥ 0 and D ≥ 0 and D and A have no common components.

If

(i) K_{X} + Δ ∼_{ℝ,U} D, (ii) (X, G) is log smooth, where G is the support of Δ + D, and (iii) B_{+}(A/U) does not contain any non kawamata log terminal cen-

tres of (X, G)

then (X, Δ) has a neutral model over U, with respect to A and D.

Proof. We may write D = D_{1} + D_{2}, where every component of D_{1} is a component of ⌞Δ⌟ and no component of D_{2} is a component of ⌞Δ⌟. We proceed by induction on the number of components of D_{2}.

Suppose D_{2} = 0. If H is any general ample ℚ-divisor over U, which is sufficiently ample then K_{X} + Δ + H is divisorially log terminal and ample over U. As the support of D is contained in ⌞Δ⌟, (5.2) implies that (X, Δ) has a neutral model f: X −−→ Y over U, with respect to A and D.

Now suppose that D_{2} ≠ 0. Let

λ = sup{ t ≥ 0 | (X, Δ + tD_{2}) is log canonical },

be the log canonical threshold of D_{2}. Then λ > 0 and (X, Θ = Δ+λD_{2}) is divisorially log terminal and log smooth, K_{X} + Θ ∼_{ℝ,U} D + λD_{2} and the number of components of D +λD_{2} that are not components of ⌞Θ⌟ is smaller than the number of components of D_{2}. By induction there is a neutral model f: X −−→ Y over U for (X, Θ), with respect to A and D.

Now

K_{Y} + f_{∗}Δ ∼_{ℝ,U} f_{∗}D_{1} + f_{∗}D_{2},
K_{Y} + f_{∗}Θ = K_{Y} + f_{∗}Δ + λf_{∗}D_{2},

where K_{Y} + f_{∗}Θ is divisorially log terminal and nef over U, and the support of f_{∗}D_{1} is contained in ⌞f_{∗}Δ⌟. Since B_{+}(f_{∗}A/U) does not contain any non kawamata log terminal centres of (Y, f_{∗}Θ), (5.2) implies that (Y, f_{∗}Δ) has a neutral model g: Y −−→ Z over U, with respect to f_{∗}A and f_{∗}D. The composition g ∘ f: X −−→ Z is then a neutral model over U for (X, Δ), with respect to A and D. □

Lemma 5.5. Let π: X ⟶ U be a projective morphism of normal quasi-projective varieties. Let (X, Δ = A + B) be a ℚ-factorial divisorially log terminal log pair and let D be an ℝ-divisor, where A ≥ 0 is big over U, B ≥ 0 and D ≥ 0.

If every component of D is either semiample over U or a component of B((K_{X} + Δ)/U) and f: X −−→ Y is a neutral model over U for (X, Δ), with respect to A and D, then f is a log terminal model for (X, Δ) over U.

Proof. By hypothesis the only divisors contracted by f are components of B((K_{X} + Δ)/U). Since the question is local over U, we may assume that U is affine. Since B_{+}(f_{∗}A/U) does not contain any non kawamata log terminal centres of (Y, Γ = f_{∗}Δ), (3.7.3) implies that we may find K_{Y} + Γ′ ∼_{ℝ,U} K_{Y} + Γ where K_{Y} + Γ′ is kawamata log terminal and Γ′ is big over U. (3.9.2) implies that K_{Y} + Γ is semiample over U.

If p: W ⟶ X and q: W ⟶ Y resolve the indeterminacy of f then we may write

p^{∗}(K_{X} + Δ) + E = q^{∗}(K_{Y} + Γ) + F,

where E ≥ 0 and F ≥ 0 have no common components, and both E and F are exceptional for q.

As K_{Y} + Γ is semiample over U, B((q^{∗}(K_{Y} + Γ) + F)/U) and F have the same support. On the other hand, every component of E is a component of B((p^{∗}(K_{X} + Δ) + E)/U). Thus E = 0 and any divisor contracted by f is contained in the support of F, and so f is a log terminal model of (X, Δ) over U. □

Lemma 5.6. Theorem A_{n} and Theorem B_{n} imply Theorem C_{n}.

Proof. By (3.7.5) we may assume that Δ = A+B, where A is a general ample ℚ-divisor over U and B ≥ 0. By (3.5.4) we may assume that D = M + F, where every component of F is a component of B(D/U) and there is a positive integer m such that if L is a component of M then mL is mobile.

Pick a log resolution f: Y ⟶ X of the support of D and Δ, which resolves the base locus of each linear system |mL|, for every component L of M. If Φ is the divisor defined in (3.6.11) then every component of the exceptional locus belongs to B((K_{Y} +Φ)/U) and replacing Φ by an ℝ-linearly equivalent divisor, we may assume that Φ contains an ample divisor over U. In particular, replacing mπ^{∗}L by a general element of the linear system |mπ^{∗}L|, we may assume that K_{Y} + Φ ∼_{ℝ,U} N + G, where every component of N is semiample, every component of G is a component of B((K_{Y} + Φ)/U), and (Y, Φ + N + G) is log smooth. By (3.6.11), we may replace X by Y and the result follows by (5.4) and (5.5). □

6. Non-vanishing

We follow the general lines of the proof of the non-vanishing theorem, see for example Chapter 3, §5 of [22]. In particular there are two cases:

Lemma 6.1. Assume Theorem C_{n}. Let (X, Δ) be a projective, log smooth pair of dimension n, where ⌞Δ⌟ = 0, such that K_{X} + Δ is pseudo-effective and Δ − A ≥ 0 for an ample ℚ-divisor A. Suppose that for every positive integer k such that kA is integral,

h^{0}(X, O_{X}(⌞mk(K_{X} + Δ)⌟ + kA)),

is a bounded function of m.

Then there is an ℝ-divisor D such that K_{X} + Δ ∼_{ℝ} D ≥ 0.

Proof. By (3.3.2) it follows that K_{X} + Δ is numerically equivalent to N_{σ}(K_{X} + Δ). Since N_{σ}(K_{X} + Δ) − (K_{X} + Δ) is numerically trivial and ampleness is a numerical condition, it follows that

A′′ = A + N_{σ}(K_{X} + Δ) − (K_{X} + Δ),

is ample and numerically equivalent to A. Thus, since A′′ is ℝ-linearly equivalent to a positive linear combination of ample ℚ-divisors, there exists 0 ≤ A′ ∼_{ℝ} A′′ such that

K_{X} + Δ′ = K_{X} + A′ + (Δ − A),

is kawamata log terminal and numerically equivalent to K_{X} + Δ, and

K_{X} + Δ′ ∼_{ℝ} N_{σ}(K_{X} + Δ) ≥ 0.

Thus by Theorem C_{n}, K_{X} + Δ′ has a log terminal model ϕ: X −−→ Y, which, by (3.6.9), is also a log terminal model for K_{X} + Δ. Replacing (X, Δ) by (Y, Γ) we may therefore assume that K_{X} + Δ is nef and the result follows by the base point free theorem, cf. (3.9.2). □

Lemma 6.2. Let (X, Δ = A+B) be a projective, log smooth pair where A is a general ample ℚ-divisor and ⌞B⌟ = 0. Suppose that there is a positive integer k such that kA is integral and

h^{0}(X, O_{X}(⌞mk(K_{X} + Δ)⌟ + kA)),

is an unbounded function of m.

Then we may find a projective, log smooth pair (Y, Γ) and a general ample ℚ-divisor C on Y, where

• Y is birational to X,

• Γ − C ≥ 0,

• T = ⌞Γ⌟ is an irreducible divisor, and

• Γ and N_{σ}(K_{Y} + Γ) have no common components. Moreover the pair (Y, Γ) has the property that K_{X} + Δ ∼_{ℝ} D ≥ 0 for

some ℝ-divisor D if and only if K_{Y} + Γ ∼_{ℝ} G ≥ 0 for some ℝ-divisor G.

Proof. Pick m large enough so that

h^{0}(X, O_{X}(⌞mk(K_{X} + Δ)⌟ + kA)) > \binom{kn + n}{n},

where n is the dimension of X. By standard arguments, given any point x ∈ X, we may find a divisor H′ ≥ 0 which is ℝ-linearly equivalent to

⌞mk(K_{X} + Δ)⌟ + kA,

of multiplicity greater than kn at x. In particular, we may find an ℝ-Cartier divisor

0 ≤ H ∼_{ℝ} m(K_{X} + Δ) + A,

of multiplicity greater than n at x. Given t ∈ [0, m], consider

(t + 1)(K_{X} + Δ) = K_{X} + \frac{m − t}{m}A + B + t(K_{X} + Δ + \frac{1}{m}A)

∼_{ℝ} K_{X} + \frac{m − t}{m}A + B + \frac{t}{m}H

= K_{X} + Δ_{t}.

Fix 0 < ϵ ≪ 1, let A′ = (ϵ/m)A and u = m − ϵ. We have:

(1) K_{X} + Δ_{0} is kawamata log terminal,

(2) Δ_{t} ≥ A′, for any t ∈ [0, u] and

(3) the non kawamata log terminal locus of (X, Δ_{u}) contains a very general point x of X.

Let π: Y ⟶ X be a log resolution of (X, Δ + H). We may write

K_{Y} + Ψ_{t} = π^{∗}(K_{X} + Δ_{t}) + E_{t},

where E_{t} ≥ 0 and Ψ_{t} ≥ 0 have no common components, π_{∗}Ψ_{t} = Δ_{t} and E_{t} is exceptional. Pick an exceptional divisor F ≥ 0 such that π^{∗}A′−F is ample and let C ∼_{ℚ} π^{∗}A′ − F be a general ample ℚ-divisor. For any t ∈ [0, u], let

Φ_{t} = Ψ_{t} −π^{∗}A′ +C +F ∼_{ℝ} Ψ_{t} and Γ_{t} = Φ_{t} −Φ_{t} ∧N_{σ}(K_{Y} +Φ_{t}).

Then properties (1-3) above become (1) K_{Y} + Γ_{0} is kawamata log terminal,

(2) Γ_{t} ≥ C, for any t ∈ [0, u], and

(3) (Y, Γ_{u}) is not kawamata log terminal. Moreover

(4) (Y, Γ_{t}) is log smooth, for any t ∈ [0, u], and

(5) Γ_{t} and N_{σ}(K_{Y} + Γ_{t}) do not have any common components. Let

s = sup{ t ∈ [0, u] | K_{Y} + Γ_{t} is log canonical }.

Note that

N_{σ}(K_{Y} + Φ_{t}) = N_{σ}(K_{Y} + Ψ_{t})
= N_{σ}(π^{∗}(K_{X} + Δ_{t})) + E_{t}
= N_{σ}((t + 1)π^{∗}(K_{X} + Δ)) + E_{t}
= (t + 1)N_{σ}(π^{∗}(K_{X} + Δ)) + E_{t}.

Thus K_{Y} + Γ_{t} is a continuous piecewise affine linear function of t. Setting Γ = Γ_{s}, we may write

Γ = T + C + B′,

where T = ⌞Γ⌟ ≠ 0, C is ample and B′ ≥ 0. Possibly perturbing Γ, we may assume that T is irreducible, so that K_{Y} + Γ is purely log terminal. □

We will need the following consequence of Kawamata-Viehweg vanishing:

Lemma 6.3. Let (X, Δ = S +A+B) be a ℚ-factorial projective purely log terminal pair and let m > 1 be an integer. Suppose that

(1) S = ⌞Δ⌟ is irreducible, (2) m(K_{X} + Δ) is integral,

(3) m(K_{X} + Δ) is Cartier in a neighbourhood of S,

(4) h^{0}(S, O_{S}(m(K_{X} + Δ))) > 0,

(5) K_{X} + G + B is kawamata log terminal, where G ≥ 0,

(6) A ∼_{ℚ} (m − 1)tH + G for some tH, and

(7) K_{X} + Δ + tH is big and nef.

Then h^{0}(X, O_{X}(m(K_{X} + Δ))) > 0.

Proof. Considering the long exact sequence associated to the restriction exact sequence,

0 ⟶ O_{X}(m(K_{X}+Δ)−S) ⟶ O_{X}(m(K_{X}+Δ)) ⟶ O_{S}(m(K_{X}+Δ)) ⟶ 0,

it suffices to observe that

H^{1}(X, O_{X}(m(K_{X} + Δ) − S)) = 0,

by Kawamata-Viehweg vanishing, since

m(K_{X} + Δ) − S = (m − 1)(K_{X} + Δ) + K_{X} + A + B
∼_{ℚ} K_{X} + G + B + (m − 1)(K_{X} + Δ + tH),

and K_{X} + Δ + tH is big and nef. □

Lemma 6.4. Let X be a normal projective variety, let S be a prime divisor and let D_{i} be three, i = 1, 2, 3, ℝ-Cartier divisors on X. Suppose that f_{i}: X −−→ Z_{i} are ample models of D_{i}, i = 1, 2, 3, where f_{i} is birational and f_{i} does not contract S. Suppose that Z_{i} is ℚ-factorial, i = 1, 2, and the induced birational map Z_{1} −−→ Z_{2} is an isomorphism in a neighbourhood of the strict transforms of S.

If D_{3} is a positive linear combination of D_{1} and D_{2} then the induced birational map Z_{i} −−→ Z_{3} is an isomorphism in a neighbourhood of the strict transforms of S.

Proof. By assumption D_{3} = λ_{1}D_{1}+λ_{2}D_{2}, where λ_{i} > 0. If g_{i}: Z ⟶ Z_{i} is the normalisation of the graph of Z_{1} −−→ Z_{2} then g_{i} is by assumption an isomorphism in a neighbourhood of the strict transforms of S, for i = 1, 2. Let f: X −−→ Z be the induced birational map. Let g: W ⟶ X resolve the indeterminacy of f and f_{3}. Replacing X by W and D_{i} by g^{∗}D_{i}, we may assume that f and f_{3} are morphisms.

Let H_{i} = f_{i∗}D_{i}, i = 1, 2, 3. As the ample model is a semiample model, by (4) of (3.6.6), E_{i} = D_{i} − f_{i}^{∗}H_{i} ≥ 0 is f_{i}-exceptional. As H_{i} is ample, i = 1 and 2, H = λ_{1}g_{1}^{∗}H_{1} + λ_{2}g_{2}^{∗}H_{2} is semiample. Let Σ ⊂ Z be a curve. As Σ is not contracted by both g_{1} and g_{2}, g_{i}^{∗}H_{i} · Σ ≥ 0, with equality for at most one i. Thus H is ample. Now

f_{3}^{∗}H_{3} + E_{3} = D_{3}
= λ_{1}D_{1} + λ_{2}D_{2}
= λ_{1}f_{1}^{∗}H_{1} + λ_{2}f_{2}^{∗}H_{2} + λ_{1}E_{1} + λ_{2}E_{2}
= f^{∗}(λ_{1}g_{1}^{∗}H_{1} + λ_{2}g_{2}^{∗}H_{2}) + λ_{1}E_{1} + λ_{2}E_{2}
= f^{∗}H + λ_{1}E_{1} + λ_{2}E_{2}.

Note that E_{3} ≤ λ_{1}E_{1} + λ_{2}E_{2}, as f^{∗}H has no stable base locus. Let T = f(S). We may write

f^{∗}H = f_{3}^{∗}H_{3} + E_{3} − (λ_{1}E_{1} + λ_{2}E_{2}).

Since λ_{1}E_{1} +λ_{2}E_{2} is f-exceptional in a neighbourhood of f^{−1}(T), E_{3} = λ_{1}E_{1} + λ_{2}E_{2} in a neighbourhood of f^{−1}(T), by (3.6.2). But then f and f_{3} contract precisely the same curves in the same neighbourhood and so Z is isomorphic to Z_{3} in a neighbourhood of the strict transforms of S. □

Lemma 6.5. Assume Theorem B_{n} and Theorem C_{n}.

Let (X, Δ_{0} = S+A+B_{0}) be a log smooth projective pair of dimension n, where A ≥ 0 is a general ample ℚ-divisor, ⌞Δ_{0}⌟ = S is a prime divisor and B_{0} ≥ 0. Suppose that K_{X} + Δ_{0} is pseudo-effective and S is not a component of N_{σ}(K_{X} + Δ_{0}). Let V_{0} be a finite dimensional affine subspace of WDiv_{ℝ}(X) containing B_{0}, which is defined over the rationals.

Then we may find a general ample ℚ-divisor H ≥ 0, a log terminal model ϕ: X −−→ Y for K_{X} + Δ_{0} + H and a positive constant α, such that if B ∈ V_{0} and

∥B − B_{0}∥ < αt,

for some t ∈ (0, 1], then there is a log terminal model ψ: X −−→ Z of K_{X} + Δ + tH = K_{X} + S + A + B + tH, which does not contract S, such that the induced birational map χ: Y −−→ Z is an isomorphism in a neighbourhood of the strict transforms of S.

Proof. Pick general ample ℚ-Cartier divisors H_{1}, H_{2}, …, H_{k}, which span WDiv_{ℝ}(X) modulo numerical equivalence, and let V be the affine subspace of WDiv_{ℝ}(X) spanned by V_{0} and H_{1}, H_{2}, …, H_{k}. Let C ⊂ L_{S+A}(V) be a convex subset spanning V, which does not contain Δ_{0}, but whose closure contains Δ_{0}, such that if Δ ∈ C then K_{X} + Δ is purely log terminal, Δ − Δ_{0} is ample and the support of Δ − Δ_{0} contains the support of the sum H_{1} +H_{2} +⋯+H_{k}. In particular if Δ ∈ C then N_{σ}(K_{X} + Δ) ≤ N_{σ}(K_{X} + Δ_{0}). As the coefficients σ_{C} of N_{σ} are continuous on the big cone, (3.3.1), possibly replacing C by a subset, we may assume that if Δ ∈ C then N_{σ}(K_{X} + Δ) and N_{σ}(K_{X} + Δ_{0}) share the same support. Moreover, if Δ ∈ C then K_{X} + Δ is big and so, as we are assuming Theorem C_{n}, K_{X} + Δ has a log terminal model ϕ: X −−→ Y, whose exceptional divisors are given by the support of N_{σ}(K_{X} + Δ_{0}). In particular ϕ does not contract S.

Given ϕ: X −−→ Y, define a subset S_{ϕ} ⊂ C as follows: Δ ∈ S_{ϕ} if and only if there is a log terminal model ψ: X −−→ Z of K_{X} + Δ such that

• the induced rational map χ: Y −−→ Z is an isomorphism in a

neighbourhood of the strict transforms of S. Define a subset S′_{ϕ} ⊂ S_{ϕ} by requiring in addition that

• ϕ is isomorphic to the ample model of K_{X} + Δ in a neighbour-

hood of the strict transforms of S. As we are assuming Theorem B_{n}, there are finitely many 1 ≤ j ≤ l

birational maps ϕ_{j}: X −−→ Y_{i} such that

C = ⋃_{j=1}^{l} S_{ϕ_{j}}.

On the other hand, (3.6.12) implies that

⋃_{j=1}^{l} S′_{ϕ_{j}},

is a dense open subset of C.

We will now show that the sets S′_{ϕ} are convex. Suppose that Δ_{i} ∈ S′_{ϕ}, i = 1, 2 and Δ is a convex linear combination of Δ_{1} and Δ_{2}. Then Δ ∈ C and so K_{X} + Δ has a log terminal model ψ: X −−→ Z. By (3) of (3.6.6) there is a birational morphism h: Z ⟶ Z′ to the ample model of K_{X} + Δ. Let ψ_{i}: X −−→ Z_{i} be the ample model of K_{X} + Δ_{i}. (6.4) implies that the induced birational map Z_{i} −−→ Z′ is an isomorphism in a neighbourhood of the strict transforms of S. As Z and Z_{i} are isomorphic in codimension 1, Z ⟶ Z′ is small in a neighbourhood of the strict transforms of S. As Z and Z′ are ℚfactorial in a neighbourhood of the strict transforms of S, the morphism Z ⟶ Z′ is also an isomorphism in a neighbourhood of the strict transforms of S. Thus Δ ∈ S′_{ϕ} and so S′_{ϕ} is convex.

Shrinking C, we may therefore assume that C = S′_{ϕ_{j}} for some 1 ≤ j ≤ l.

Pick Δ = S + A + B ∈ C and pick H ∼_{ℚ} Δ − Δ_{0} a general ample ℚ-divisor. Pick a positive constant α such that if ∥B − B_{0}∥ < αt for some B ∈ V_{0} and t ∈ (0, 1] then S +A+B +t(Δ−Δ_{0}) ∈ C. By (3.6.9), K_{X} + S + A + B + t(Δ − Δ_{0}) and K_{X} + S + A + B + tH have the same log terminal models. It follows that α and H have the required properties. □

Lemma 6.6. Theorem D_{n−1}, Theorem B_{n} and Theorem C_{n} imply Theorem D_{n}.

Proof. By (3.2.1), it suffices to prove this result for the generic fibre of U. Thus we may assume that U is a point, so that X is a projective variety.

Let f: Y ⟶ X be a log resolution of (X, Δ). We may write

K_{Y} + Γ = f^{∗}(K_{X} + Δ) + E,

where Γ ≥ 0 and E ≥ 0 have no common components, f_{∗}Γ = Δ and f_{∗}E = 0. If F ≥ 0 is an ℝ-divisor whose support equals the union of all f-exceptional divisors then Γ + F is big. Pick F so that (X, Γ + F) is kawamata log terminal. Replacing (X, Δ) by (Y, Γ + F) we may therefore assume that (X, Δ) is log smooth. By (3.7.5) we may assume that Δ = A + B, where A is a general ample ℚ-divisor and B ≥ 0. By (6.1) and (6.2), we may therefore assume that Δ = S + A + B, where (X, Δ) is log smooth, A is a general ample ℚ-divisor, and ⌞Δ⌟ = S is a prime divisor, which is not a component of N_{σ}(K_{X} + Δ).

Let V be the subspace of WDiv_{ℝ}(X) spanned by the components of B. By (6.5) we may find a constant α > 0, a general ample ℚ-divisor H on X, and a log terminal model ϕ: X −−→ Y of K_{X} + Δ + H such that if B′ ∈ V, t ∈ (0, 1] and ∥B −B′∥ < αt then there is a log terminal model ϕ′: X −−→ Y′ of K_{X} + S + A + B′ + tH such that the induced rational map χ: Y −−→ Y′ is an isomorphism in a neighbourhood of the strict transforms of S. Pick ϵ > 0 such that A − ϵH is ample.

Let T be the strict transform of S on Y and define Φ_{0} on T by adjunction

(K_{Y} + T)|_{T} = K_{T} + Φ_{0}.

Let W be the subspace of WDiv_{ℝ}(T) spanned by the components of (ϕ_{∗}B)|_{T} + Φ_{0} and let L: V ⟶ W be the rational affine linear map L(B′) = Φ_{0} + ϕ_{∗}B′|_{T}. Let Γ = ϕ_{∗}Δ and let C = ϕ_{∗}A|_{T}. If we define Ψ on T by adjunction

(K_{Y} + Γ)|_{T} = K_{T} + Ψ,

then Ψ = C + L(B). As K_{T} + C + L(B) + tϕ_{∗}H|_{T} is nef for any t > 0, it follows that K_{T} + Ψ is nef. Since C is big and K_{T} + Ψ is kawamata log terminal, (3.7.3) implies that there is a rational affine linear isomorphism L′: WDiv_{ℝ}(T) ⟶ WDiv_{ℝ}(T) which preserves ℚlinear equivalence, an ample ℚ-divisor G on T and a rational affine linear subspace W′ of WDiv_{ℝ}(T) such that L′(U) ⊂ L_{G}(W′), where U ⊂ L_{C}(W) is a neighbourhood of Ψ. N_{G}(W′) is a rational polytope, by (3.11.1). In particular we may find a rational polytope C ⊂ V containing B such that if B′ ∈ C then K_{X} + Δ′ = K_{X} + S + A + B′ is purely log terminal and

(K_{Y} + Γ′)|_{T} = K_{T} + Ψ′,

is nef, where Γ′ = ϕ_{∗}Δ′ and Ψ′ = C + L(B′). Pick a positive integer k such that if r(K_{Y} + Γ′) is integral then rk(K_{Y} + Γ′) is Cartier in a neighbourhood of T. By (3.9.2) there is then a constant m > 0 such that mr(K_{Y} + Γ′) is Cartier in a neighbourhood of T and mr(K_{T} + Ψ′) is base point free.

(3.7.7) implies that there are real numbers r_{i} > 0 with ∑ r_{i} = 1, positive integers p_{i} > 0 and ℚ-divisors B_{i} ∈ C such that

p_{i}(K_{X} + Δ_{i}),

is integral, where Δ_{i} = S + A + B_{i},

K_{X} + Δ = ∑ r_{i}(K_{X} + Δ_{i}),

and

∥B_{i} − B∥ < \frac{αϵ}{m_{i}} ,

where m_{i} = mp_{i} . Let Ψ_{i} = C+L(B_{i}). By our choice of p_{i}, m_{i}(K_{T} +Ψ_{i}) is base point free and so

h^{0}(T, O_{T} (m_{i}(K_{T} + Ψ_{i}))) > 0.

Let t_{i} = ϵ/m_{i}. By (6.5) there is a log terminal model ϕ_{i}: X −−→ Y_{i} of K_{X} + Δ_{i} + t_{i}H, such that the induced birational map χ: Y −−→ Y_{i} is an isomorphism in a neighbourhood of T and the strict transform T_{i} of S. In particular if Γ_{i} = T_{i} + ϕ_{i∗}A + ϕ_{i∗}B_{i} and τ: T ⟶ T_{i} is the induced isomorphism then

τ^{∗}(K_{Y_{i}} + Γ_{i})|_{T_{i}} = K_{T} + Ψ_{i},

and so the pair (Y_{i}, Γ_{i}) clearly satisfies conditions (1), (2), (4) and (7) of (6.3). As the induced birational map Y_{i} −−→ Y is an isomorphism in a neighbourhood of T_{i}, (3) of (6.3) also holds. As

(m_{i} − 1)t_{i} = (\frac{m_{i} − 1}{m_{i}}) ϵ < ϵ,

A−(m_{i} −1)t_{i}H is ample and so we may pick a general ample ℚ-divisor L_{i} such that L_{i} ∼_{ℚ} A − (m_{i} − 1)t_{i}H. Then K_{X} + S + L_{i} + B_{i} is purely log terminal and as ϕ_{i} is (K_{X} + Δ_{i} + t_{i}H)-negative and H is ample, ϕ_{i} is (K_{X} +S +L_{i} +B_{i})-negative. It follows that K_{Y_{i}} +T_{i} +ϕ_{i∗}(L_{i} +B_{i}) is purely log terminal so that K_{Y_{i}} +ϕ_{i∗}(L_{i} +B_{i}) is kawamata log terminal, and conditions (5) and (6) of (6.3) hold. Therefore (6.3) implies that

h^{0}(Y_{i}, O_{Y_{i}}(m_{i}(K_{Y_{i}} + Γ_{i}))) > 0.

As ϕ_{i} is (K_{X} + Δ_{i} + t_{i}H)-negative and H is ample, it follows that ϕ_{i} is (K_{X} + Δ_{i})-negative. But then

h^{0}(X, O_{X}(m_{i}(K_{X} + Δ_{i}))) = h^{0}(Y, O_{Y} (m_{i}(K_{Y} + Γ_{i}))) > 0.

In particular there is an ℝ-divisor D such that

K_{X} + Δ = ∑ r_{i}(K_{X} + Δ_{i}) ∼_{ℝ} D ≥ 0.  □

7. Finiteness of models

Lemma 7.1. Assume Theorem C_{n} and Theorem D_{n}.

Let π: X ⟶ U be a projective morphism of normal quasi-projective varieties, where X has dimension n. Let V be a finite dimensional affine subspace of WDiv_{ℝ}(X), which is defined over the rationals. Fix a general ample ℚ-divisor A over U. Let C ⊂ L_{A}(V) be a rational polytope such that if Δ ∈ C then K_{X} + Δ is kawamata log terminal.

Then there are finitely many rational maps ϕ_{i}: X −−→ Y_{i} over U, 1 ≤ i ≤ k, with the property that if Δ ∈ C ∩ E_{A,π}(V) then there is an index 1 ≤ i ≤ k such that ϕ_{i} is a log terminal model of K_{X} + Δ over U.

Proof. Possibly replacing V_{A} by the span of C, we may assume that C spans V_{A}. We proceed by induction on the dimension of C.

Suppose that Δ_{0} ∈ C ∩ E_{A,π}(V). As we are assuming Theorem D_{n} there is an ℝ-divisor D_{0} ≥ 0 such that K_{X} + Δ_{0} ∼_{ℝ,U} D_{0} and so, as we are assuming Theorem C_{n}, there is a log terminal model ϕ: X −−→ Y over U for K_{X} + Δ_{0}. In particular we may assume that dim C > 0.

First suppose that there is a divisor Δ_{0} ∈ C such that K_{X} + Δ_{0} ∼_{ℝ,U} 0. Pick Θ ∈ C, Θ ≠ Δ_{0}. Then there is a divisor Δ on the boundary of C such that

Θ − Δ_{0} = λ(Δ − Δ_{0}),

for some 0 < λ ≤ 1. Now

K_{X} + Θ =λ(K_{X} + Δ) + (1 − λ)(K_{X} + Δ_{0})
∼_{ℝ,U}λ(K_{X} + Δ).

In particular Δ ∈ E_{A,π}(V) if and only if Θ ∈ E_{A,π}(V) and (3.6.9) implies that K_{X} +Δ and K_{X} +Θ have the same log terminal models over U. On the other hand the boundary of C is contained in finitely many affine hyperplanes defined over the rationals, and we are done by induction on the dimension of C.

We now prove the general case. By (3.7.4), we may assume that C is contained in the interior of L_{A}(V). Since L_{A}(V) is compact and C ∩ E_{A,π}(V) is closed, it suffices to prove this result locally about any divisor Δ_{0} ∈ C ∩ E_{A,π}(V). Let ϕ: X −−→ Y be a log terminal model over U for K_{X} + Δ_{0}. Let Γ_{0} = ϕ_{∗}Δ_{0}.

Pick a neighbourhood C_{0} ⊂ L_{A}(V) of Δ_{0}, which is a rational polytope. As ϕ is (K_{X} + Δ_{0})-negative we may pick C_{0} such that for any Δ ∈ C_{0}, a(F, K_{X} + Δ) < a(F, K_{Y} + Γ) for all ϕ-exceptional divisors F ⊂ X, where Γ = ϕ_{∗}Δ. Since K_{Y} + Γ_{0} is kawamata log terminal and Y is ℚ-factorial, possibly shrinking C_{0}, we may assume that K_{Y} + Γ is kawamata log terminal for all Δ ∈ C_{0}. In particular, replacing C by C_{0}, we may assume that the rational polytope C′ = ϕ_{∗}(C) is contained in L_{ϕ_{∗}A}(W), where W = ϕ_{∗}(V). By (3.7.3), there is a rational affine linear isomorphism L: W ⟶ V′ and a general ample ℚ-divisor A′ over U such that L(C′) ⊂ L_{A′}(V′), L(Γ) ∼_{ℚ,U} Γ for all Γ ∈ C′ and K_{Y} + Γ is kawamata log terminal for any Γ ∈ L(C′).

Note that dim V′ ≤ dim V. By (3.6.9) and (3.6.10), any log terminal model of (Y, L(Γ)) over U is a log terminal model of (X, Δ) over U for any Δ ∈ C. Replacing X by Y and C by L(C′), we may therefore assume that K_{X} + Δ_{0} is π-nef.

By (3.9.2) K_{X} + Δ_{0} has an ample model ψ: X ⟶ Z over U. In particular K_{X} + Δ_{0} ∼_{ℝ,Z} 0. By what we have already proved there are finitely many birational maps ϕ_{i}: X −−→ Y_{i} over Z, 1 ≤ i ≤ k, such that for any Δ ∈ C ∩ E_{A,ψ}(V), there is an index i such that ϕ_{i} is a log terminal model of K_{X} + Δ over Z. Since there are only finitely many indices 1 ≤ i ≤ k, possibly shrinking C, (3.11.3) implies that if Δ ∈ C then ϕ_{i} is a log terminal model for K_{X} + Δ over Z if and only if it is log terminal model for K_{X} + Δ over U.

Suppose that Δ ∈ C ∩ E_{A,π}(V). Then Δ ∈ C ∩ E_{A,ψ}(V) and so there is an index 1 ≤ i ≤ k such that ϕ_{i} is a log terminal model for K_{X} + Δ over Z. But then ϕ_{i} is a log terminal model for K_{X} + Δ over U. □

Lemma 7.2. Assume Theorem C_{n} and Theorem D_{n}.

Let π: X ⟶ U be a projective morphism of normal quasi-projective varieties, where X has dimension n. Suppose that there is a kawamata log terminal pair (X, Δ_{0}). Fix A ≥ 0, a general ample ℚ-divisor over U. Let V be a finite dimensional affine subspace of WDiv_{ℝ}(X) which is defined over the rationals. Let C ⊂ L_{A}(V) be a rational polytope.

Then there are finitely many birational maps ψ_{j}: X −−→ Z_{j} over U, 1 ≤ j ≤ l such that if ψ: X −−→ Z is a weak log canonical model of K_{X} + Δ over U, for some Δ ∈ C then there is an index 1 ≤ j ≤ l and an isomorphism ξ: Z_{j} ⟶ Z such that ψ = ξ ∘ ψ_{j}.

Proof. Suppose that Δ ∈ C and K_{X} + Δ′ ∼_{ℝ,U} K_{X} + Δ is kawamata log terminal. (3.6.9) implies that ψ: X −−→ Z is a weak log canonical model of K_{X} + Δ over U if and only if ψ is a weak log canonical model of K_{X} + Δ′ over U. By (3.7.4) we may therefore assume that if Δ ∈ C then K_{X} + Δ is kawamata log terminal.

Let G be any divisor which contains the support of every element of V and let f: Y ⟶ X be a log resolution of (X, G). Given Δ ∈ L_{A}(V) we may write

K_{Y} + Γ = f^{∗}(K_{X} + Δ) + E,

where Γ ≥ 0 and E ≥ 0 have no common components, f_{∗}Γ = Δ and f_{∗}E = 0. If ψ: X −−→ Z is a weak log canonical model of K_{X} + Δ over U then ψ ∘ f: Y −−→ Z is a weak log canonical model of K_{Y} + Γ over U. If C′ denotes the image of C under the map Δ ⟶ Γ then C′ is a rational polytope, cf. (3.7.2), and if Γ ∈ C′ then K_{Y} + Γ is kawamata log terminal. In particular B_{+}(f^{∗}A/U) does not contain any non kawamata log terminal centres of K_{Y} + Γ, for any Γ ∈ C′. Let W be the subspace of WDiv_{ℝ}(Y) spanned by the components of the strict transform of G and the exceptional locus of f. By (3.7.3) and (3.6.9), we may assume that there is a general ample ℚ-divisor A′ on Y over U such that C′ ⊂ L_{A′}(W). Replacing X by Y and C by C′, we may therefore assume that X is smooth.

Pick general ample ℚ-Cartier divisors H_{1}, H_{2}, …, H_{p} over U, which generate WDiv_{ℝ}(X) modulo relative numerical equivalence over U and let H = H_{1} + H_{2} + ⋯ + H_{p} be their sum. By (3.7.4) we may assume that if Δ ∈ C then Δ contains the support of H. Let W be the affine subspace of WDiv_{ℝ}(X) spanned by V and the divisors H_{1}, H_{2}, …, H_{p}. Pick C′ a rational polytope in L_{A}(W) containing C in its interior such that if Δ ∈ C′ then K_{X} + Δ is kawamata log terminal.

By (7.1) there are finitely many 1 ≤ i ≤ k rational maps ϕ_{i}: X −−→ Y_{i} over U, such that given any Δ′ ∈ C′ ∩ E_{A,π}(W), we may find an index 1 ≤ i ≤ k such that ϕ_{i} is a log terminal model of K_{X} + Δ′ over U. By (3.11.2) for each index 1 ≤ i ≤ k there are finitely many contraction morphisms f_{i,m}: Y_{i} ⟶ Z_{i,m} over U such that if Δ′ ∈ W_{ϕ_{i},A,π}(W) and there is a contraction morphism f: Y_{i} ⟶ Z over U, with

K_{Y_{i}} + Γ_{i} = K_{Y_{i}} + ϕ_{i}_{∗}Δ′ ∼_{ℝ,U} f^{∗}D,

for some ample over U ℝ-divisor D on Z, then there is an index (i, m) and an isomorphism ξ: Z_{i,m} ⟶ Z such that f = ξ ∘ f_{i,m}. Let ψ_{j}: X −−→ Z_{j}, 1 ≤ j ≤ l be the finitely many rational maps obtained by composing every ϕ_{i} with every f_{i,j}.

Pick Δ ∈ C and let ψ: X −−→ Z be a weak log canonical model of K_{X} +Δ over U. Then K_{Z} +Θ is kawamata log terminal and nef over U, where Θ = ψ_{∗}Δ. As we are assuming Theorem C_{n}, we may find a log terminal model η: Z −−→ Y′ of K_{Z} + Θ over Z. Then Y′ is ℚ-factorial and the structure morphism ξ: Y′ ⟶ Z is a small birational map, the inverse of η. By (3.6.12), we may find Δ′ ∈ C′ ∩ E_{A,π}(W) such that ψ is an ample model of K_{X} + Δ′ over U. Pick an index 1 ≤ i ≤ k such that ϕ_{i} is a log terminal model of K_{X} + Δ′ over U. By (4) of (3.6.6) there is a contraction morphism f: Y_{i} ⟶ Z such that

K_{Y_{i}} + Γ_{i} = f^{∗}(K_{Z} + Θ′),

where Γ′ = ϕ_{∗}Δ′ and Θ′ = ψ_{∗}Δ′. As K_{Z} +Θ′ is ample over U, it follows that there is an index m and an isomorphism ξ: Z_{i,m} ⟶ Z such that f = ξ ∘ f_{i,m}. But then

ψ = f ∘ ϕ_{i} = ξ ∘ f_{i,m} ∘ ϕ_{i} = ξ ∘ ψ_{j},

for some index 1 ≤ j ≤ l. □

Lemma 7.3. Theorem C_{n} and Theorem D_{n} imply Theorem E_{n}.

Proof. Since L_{A}(V) is itself a rational polytope by (3.7.2), this is immediate from (7.2). □

8. Finite generation

Lemma 8.1. Theorem C_{n} and Theorem D_{n} imply Theorem F_{n}.

Proof. Theorem C_{n} and Theorem D_{n} imply that there is is a log terminal model μ: X −−→ Y of K_{X} + Δ, and K_{Y} + Γ = μ_{∗}(K_{X} + Δ) is semiample by (1) of (3.9.3). (1) follows, as

R(X, K_{X} + Δ) ≃ R(Y, K_{Y} + Γ).

As K_{Y} + Γ is semiample the prime divisors contained in the stable base locus of K_{X} + Δ are precisely the exceptional divisors of μ. But there is a constant δ > 0 such that if Ξ ∈ V and ∥Ξ − Δ∥ < δ then the exceptional divisors of μ are also (K_{X} + Ξ)-negative. Hence (2).

Note that by (3.11.2), there is a constant η > 0 such that if Ξ ∈ W and ∥Ξ − Δ∥ < η then μ is also a log terminal model of K_{X} + Ξ. (3.9.2) implies that there is a constant r > 0 such that if m(K_{Y} + Γ) is integral and nef, then mr(K_{Y} + Γ) is base point free. It follows that if k(K_{X} + Ξ)/r is Cartier then every component of Fix(k(K_{X} + Ξ)) is contracted by μ and so every such component is in the stable base locus of K_{X} + Δ. This is (3). □

9. Proof of Theorems

Proof of Theorems A, B, C, D, E and F. Immediate from the main result of [9], (4.4), (5.6), (6.6), (7.3) and (8.1). □

Proof of (1.2). Suppose K_{X} + Δ is π-big. Then we may write K_{X} + Δ ∼_{ℝ,U} B ≥ 0. If ϵ > 0 is sufficiently small then K_{X} + Δ + ϵB is kawamata log terminal. (3.6.9) implies that K_{X} + Δ and K_{X} + Δ + ϵB have the same log terminal models over U. Replacing Δ by Δ + ϵB we may therefore assume that Δ is big over U. (1) follows by Theorem C and Theorem D.

(2) and (3) follow from (1) and (3.9.2). □

10. Proof of Corollaries

Proof of (1.1.1). (1), (2) and (3) are immediate from (1.2). (4) is Theorem D of [4]. □

Proof of (1.1.2). Immediate by Theorem 5.2 of [6] and (3) of (1.2). □

Proof of (1.1.3). Note that Y_{1} and Y_{2} are isomorphic in codimension one. Replacing U by the common ample model of (X, Δ), we may assume that K_{Y_{i}} + Γ_{i} is numerically trivial over U. Let H_{2} be a divisor on Y_{2}, which is ample over U. Let H_{1} be the strict transform on Y_{1}. Possibly replacing H_{2} by a small multiple, we may assume that K_{Y_{i}} + Γ_{i} + H_{i} is kawamata log terminal.

Suppose that K_{Y_{1}} + Γ_{1} + H_{1} is not nef over U. Then there is a (K_{Y_{1}} + Γ_{1} + H_{1})-flip over U which is automatically a (K_{Y_{1}} + Γ_{1})-flop over U. By finiteness of log terminal models for K_{X} + Δ over U, this (K_{Y_{1}} + Γ_{1} + H_{1})-MMP terminates. Thus we may assume that K_{Y_{1}} +Γ_{1}+H_{1} is nef over U. But then K_{Y_{2}} +Γ_{2}+H_{2} is the corresponding ample model, and so there is a small birational morphism f: Y_{1} ⟶ Y_{2}. As Y_{2} is ℚ-factorial, f is an isomorphism. □

Proof of (1.1.5). We first prove (1) and (2). By Theorem E and (3.11.2), and since ample models are unique by (1) of (3.6.6), it suffices to prove that if Δ ∈ E_{A,π}(V) then K_{X} + Δ has both a log terminal model over U and an ample model over U.

By (3.7.5) and (3.6.9) we may assume that K_{X} + Δ is kawamata log terminal. (1.2) implies the existence of a log terminal model over U and the existence of an ample model then follows from (3.9.3).

(3) follows as in the proof of (3.11.2). □

Proof of (1.1.7). Immediate consequence of (1.1.5). □

Proof of (1.1.9). Let V_{A} be the affine subspace of WDiv_{ℝ}(X) generated by Δ_{1}, Δ_{2}, …, Δ_{k}. (1.1.5) implies that there are finitely many 1 ≤ p ≤ q rational maps ϕ_{p}: X −−→ Y_{p} over U such that if Δ ∈ E_{A,π}(V) then there is an index 1 ≤ p ≤ q such that ϕ_{p} is a log terminal model of K_{X} + Δ over U. Let C ⊂ L_{A}(V) be the polytope spanned by Δ_{1}, Δ_{2}, …, Δ_{k} and let

C_{p} = W_{ϕ_{p},A,π}(V) ∩ C.

Then C_{p} is a rational polytope. Replacing Δ_{1}, Δ_{2}, …, Δ_{k} by the vertices of C_{p}, we may assume that C = C_{p}, and we will drop the index p. Let π′: Y ⟶ U be the induced morphism. Let Γ_{i} = ϕ_{∗}Δ_{i}. If we pick a positive integer m so that both G_{i} = m(K_{Y} +Γ_{i}) and D_{i} = m(K_{X} +Δ_{i}) are Cartier for every 1 ≤ i ≤ k, then

ℜ(π, D^{•}) ≃ ℜ(π′, G^{•}).

Replacing X by Y, we may therefore assume that K_{X} + Δ_{i} is nef over U. By (3.9.2) K_{X} + Δ_{i} is semiample over U and so the Cox ring is finitely generated. Aliter: By (3.7.5), we may assume that each K_{X} + Δ_{i} is kawamata log terminal. Pick a log resolution f: Y ⟶ X of (X, Δ), where Δ is the support of the sum Δ_{1} + Δ_{2} + ⋯ + Δ_{k}. Then we may write

K_{Y} + Γ_{i} = π^{∗}(K_{X} + Δ_{i}) + E_{i},

where Γ_{i} and E_{i} have no common components, f_{∗}Γ_{i} = Δ_{i} and f_{∗}E_{i} = 0. We may assume that there is an exceptional ℚ-divisor F ≥ 0 such that f^{∗}A − F is ample over U and (Y, Γ_{i} + F) is kawamata log terminal. Pick A′ ∼_{ℚ,U} f^{∗}A − F a general ample ℚ-divisor over U. Then

G_{i} = K_{Y} + Γ_{i} + F − f^{∗}A + A′ ∼_{ℚ,U} K_{Y} + Γ_{i},

is kawamata log terminal and ℜ(π, D^{•}) if finitely generated if and only if ℜ(π ∘ f, G^{•}) if finitely generated, since they have isomorphic truncations. Replacing X by Y we may therefore assume that (X, Δ) is log smooth.

Pick a positive integer m such that D_{i} = m(K_{X} + Δ_{i}) is integral for 1 ≤ i ≤ k. Let

E = ⨁_{i=1}^{k} O_{X}(mΔ_{i}).

and let Y = ℙ_{X}(E) with projection map f: Y ⟶ X. Pick σ_{i} ∈ O_{X}(mΔ_{i} − mA) with zero locus mΔ_{i} and let σ = (σ_{1}, σ_{2}, …, σ_{k}) ∈ H^{0}(X, E(−mA)). Let S be the divisor corresponding to σ in Y. Let T_{1}, T_{2}, …, T_{k} be the divisors on Y, given by the summands of E, let T be their sum, and let Γ = T + f^{∗}A + S/m. Note that O_{Y} (m(K_{Y} + Γ)) is the tautological bundle associated to E(mK_{X}); indeed if k = 1 this is clear and if k > 1 then it has degree one on the fibres and O_{Y} (m(K_{Y} + Γ)) restricts to the tautological line bundle on each T_{i} by adjunction and induction on k. Thus

ℜ(π, D^{•}) ≃ ℜ(π ∘ f, m(K_{Y} + Γ)).

On the other hand (Y, Γ) is log smooth outside T and restricts to a divisorially log terminal pair on each T_{i} by adjunction and induction. Therefore (Y, Γ) is divisorially log terminal by inversion of adjunction. Since T is ample over X, there is a positive rational number ϵ > 0 such that f^{∗}A + ϵT is ample over U. Let A′ ∼_{ℚ,U} f^{∗}A + ϵT be a general ample ℚ-divisor over U. Then

K_{Y} + Γ′ = K_{Y} + Γ − ϵT − f^{∗}A + A′ ∼_{ℚ,U} K_{Y} + Γ,

is kawamata log terminal. Thus ℜ(π∘f, m(K_{Y} +Γ)) is finitely generated over U by (1.1.2). □

We will need a well-known result on the geometry of the moduli spaces of n-pointed curves of genus g:

Lemma 10.1. Let X = \overline{M}_{g,n} and let D be the sum of the boundary divisors.

Then • X is ℚ-factorial,

• K_{X} is kawamata log terminal and

• K_{X} + D is log canonical and ample.

Proof. X is ℚ-factorial, K_{X} is kawamata log terminal and K_{X} + D is log canonical as the pair (X, D) is locally a quotient of a normal crossings pair.

It is proved in [27] that K_{X} + D is ample when n = 0. To prove the general case, consider the natural map

π: \overline{M}_{g,n+1} ⟶ \overline{M}_{g,n},

which drops the last point. Let Y = \overline{M}_{g,n+1} and G be the sum of the boundary divisors. If \overline{M}_{g,n} is the moduli stack of stable curves of genus g, then \overline{M}_{g,n} is the coarse moduli space and so there is a representable morphism

f: \overline{M}_{g,n} ⟶ \overline{M}_{g,n},

which only ramifies over the locus of stable curves with automorphisms. If g = 1 and n ≤ 1 then K_{Y} + G is obviously π-ample. Otherwise the locus of smooth curves with extra automorphisms has codimension at least two and in this case K_{\overline{M}_{g,n}} + Δ = f^{∗}(K_{X} + D), where Δ is the sum of the boundary divisors. On the other hand there is a fibre square

[figure: \overline{M}_{g,n+1}; \overline{M}_{g,n+1}; \overline{M}_{g,n}; ψ; f; \overline{M}_{g,n},; π]

where ψ is the universal morphism. Since the stack is a fine moduli space, ψ is the universal curve. If Γ is the sum of the boundary divisors then K_{\overline{M}_{g,n+1}} + Γ has positive degree on the fibres of ψ, by adjunction and the definition of a stable pair. In particular K_{Y} + G is ample on the fibres of π.

On the other hand, we may write

K_{Y} + G = π^{∗}(K_{X} + D) + ψ,

for some ℚ-divisor ψ. It is proved in [7] that ψ is nef. We may assume that K_{X} + D is ample, by induction on n. If ϵ > 0 is sufficiently small, it follows that ϵ(K_{Y} + G) + (1 − ϵ)π^{∗}(K_{X} + D) is ample. But then

K_{Y} + G = ϵ(K_{Y} + G) + (1 − ϵ)(K_{Y} + G)
= ϵ(K_{Y} + G) + (1 − ϵ)π^{∗}(K_{X} + D) + (1 − ϵ)ψ,

is also ample and so the result follows by induction on n. □

Proof of (1.2.1). By (10.1) K_{X}+D is ample and log canonical, where D is the sum of the boundary divisors. In particular K_{X} +Δ is kawamata log terminal, provided none of the a_{i} are equal to one.

Pick a general ample ℚ-divisor A ∼_{ℚ} δ(K_{X} + D). Note that

(1 + δ)(K_{X} + Δ) = K_{X} + δ(K_{X} + D) + (1 + δ)Δ − δD
∼_{ℚ} K_{X} + A + B.

Now

0 ≤ (Δ − δD) + δΔ = B = Δ + δ(Δ − D) ≤ D.

Thus the result is an immediate consequence of Theorem C, Theorem D and Theorem E. □

Proof of (1.3.2). Immediate by (1.1.9) and the main result of [12] (note that h^{1}(X, O_{X}) = 0 by Kawamata-Viehweg vanishing). □

Proof of (1.3.3). Pick any π-ample divisor A such that K_{X} + Δ + A is π-ample and kawamata log terminal. We may find ϵ > 0 such that K_{X} +Δ+ϵA is not π-pseudo-effective. We run the (K_{X} +Δ+ϵA)-MMP over U with scaling of A. Since every step of this MMP is A-positive, it is automatically a (K_{X} + Δ)-MMP as well. But as K_{X} + Δ + ϵA is not π-pseudo-effective, this MMP must terminate with a Mori fibre space g: Y ⟶ W over U. □

Proof of (1.3.5). As X is a Mori dream space, the cone of pseudoeffective divisors is a rational polyhedron. It follows that \overline{NF}(X) is also a rational polyhedron, as it is the dual of the cone of pseudo-effective divisors.

Now suppose that F is a co-extremal ray. As \overline{NF}(X) is polyhedral, we may pick a pseudo-effective divisor D which supports F. Pick ϵ > 0 such that D − ϵ(K_{X} + Δ) is ample. Pick a general ample ℚ-divisor A ∼_{ℚ} \frac{1}{ϵ}D − (K_{X} + Δ) such that K_{X} + Δ + kA is ample and kawamata log terminal for some k > 1. Then (X, Θ = Δ + A) is kawamata log terminal and K_{X} + Θ ∼_{ℚ} \frac{1}{ϵ}D supports F. As in the proof of (1.3.3), the (K_{X} + Δ)-MMP with scaling of A ends with a (K_{X} + Θ)-trivial Mori fibre space f: Y ⟶ Z, and it is easy to see that the pullback to X of a general curve in the fibre of f generates F. □

Proof of (1.4.1). The flip of π is precisely the log canonical model, so that this result follows from Theorem C. □

Proof of (1.4.2). Immediate from (3.10.12) and Theorem E. □

Proof of (1.4.3). Pick an ample ℚ-divisor A ≥ 0 which contains the centre of every element of 𝔈 of log discrepancy one, but no non kawamata log terminal centres. If ϵ > 0 is sufficiently small, then (X, Δ+ϵA) is log canonical and so replacing Δ by Δ + ϵA, we may assume that 𝔈 contains no valuations of log discrepancy one. Replacing Δ by (1 − η)Δ + ηΔ_{0}, where η > 0 is sufficiently small, we may assume that K_{X} + Δ is kawamata log terminal.

We may write

K_{W} + Ψ = f^{∗}(K_{X} + Δ) + E,

where Ψ ≥ 0 and E ≥ 0 have no common components, f_{∗}Ψ = Δ and E is exceptional. Let F be the sum of all the exceptional divisors which are neither components of E nor correspond to elements of 𝔈.

Pick ϵ > 0 such that K_{W} +Φ = K_{Y} +Ψ+ϵF is kawamata log terminal. As f is birational, Φ is big over X and so by (1.2) we may find a log terminal model g: W −−→ Y for K_{W} + Φ over X. Let π: Y ⟶ X be the induced morphism. If Γ = g_{∗}Φ and E′ = g_{∗}(E + ϵF) then

K_{Y} + Γ = π^{∗}(K_{X} + Δ) + E′,

where K_{Y} + Γ is nef over X. Negativity of contraction implies that E′ = 0 so that

K_{Y} + Γ = π^{∗}(K_{X} + Δ).

But then we must have contracted every exceptional divisor which does correspond to an element of 𝔈. □

Proof of (1.4.4). Pick a log resolution f: Z ⟶ X of (X, Δ). We may write

K_{Z} + Ψ = f^{∗}(K_{X} + Δ) + E_{0},

where Ψ ≥ 0 and E_{0} ≥ 0 have no common components, E_{0} is fexceptional and f_{∗}Ψ = Δ. We may write Ψ = Ψ_{1} + Ψ_{2} where every component of Ψ_{1} has coefficient less than one and every component of Ψ_{2} has coefficient at least one. Let E_{1} be the sum of the components of Ψ_{1} which are exceptional. Pick δ > 0 such that K_{Z} + Ψ_{3} is kawamata log terminal, where Ψ_{3} = Ψ_{1} + δE_{1}. Let g: Z −−→ Y be a log terminal model of K_{Z} + Ψ_{3} over X, whose existence is guaranteed by (1.2). If π: Y ⟶ X is the induced birational morphism then we may write

K_{Y} + Γ = π^{∗}(K_{X} + Δ) + E,

where Γ = g_{∗}(Ψ + δE_{1}) ≥ 0, E = g_{∗}(E_{0} + δE_{1}) ≥ 0 and K_{Y} + Γ_{3} is nef over X, where Γ_{3} = g_{∗}Ψ_{3}. If E ≠ 0 then by (3.6.2) there is a family of curves Σ ⊂ Y contracted by π, which sweeps out a component of E such that E · Σ < 0. Since E and Γ − Γ_{3} have no common components, (Γ−Γ_{3})·Σ ≥ 0, so that (K_{Y} +Γ_{3})·Σ < 0, a contradiction. Thus E = 0, in which case Γ_{3} = Γ_{1} and no component of Γ_{1} is exceptional. □

Lemma 10.2. Let (X, Δ) be a quasi-projective divisorially log terminal pair and let S be a component of the support of Δ.

Then there is a small projective birational morphism π: Y ⟶ X, where Y is ℚ-factorial and −T is nef over X, where T is the strict transform of S.

In particular if Σ is a curve in Y which is contracted by π and Σ intersects T then Σ is contained in T.

Proof. By (2.43) of [22] we may assume that (X, Δ) is kawamata log terminal. Let f: Z ⟶ X be a log terminal model of (X, Δ) over X, let Ψ be the strict transform of Δ and let R be the strict transform of S. Let g: Z −−→ Y be a log terminal model of K_{Z} + Ψ − ϵR over X, for any ϵ > 0 sufficiently small. As Ψ − ϵR is big over X and K_{Z} + Ψ is kawamata log terminal, (3.7.3) and (1.1.5) imply that we may assume that Y is independent of ϵ. In particular −T is π-nef, where π: Y ⟶ X is the induced morphism. □

Proof of (1.4.5). We work locally about S. Let a be the log discrepancy of K_{S} +Θ and let b be the minimum of the log discrepancy with respect to K_{X} + Δ of any valuation whose centre on X is of codimension at least two. It is straightforward to prove that b ≤ a, cf. (17.2) of [21].

If B is a prime divisor on S which is not a component of Θ then the log discrepancy of B with respect to (S, Θ) is one, so that a ≤ 1. In particular we may assume that b < 1.

By the main theorem of [13], (X, Δ) is log canonical near the image of S if and only if (S, Θ) is log canonical and so we may assume that (X, Δ) is log canonical and hence b ≥ 0.

Suppose that (X, Δ) is not purely log terminal. By (1.4.4) we may find a birational projective morphism π: Y ⟶ X which only extracts divisors of log discrepancy zero and if we write K_{Y} + Γ = π^{∗}(K_{X} + Δ), then Y is ℚ-factorial and K_{Y} is kawamata log terminal. By connectedness, see (17.4) of [21], the non kawamata log terminal locus of K_{Y} + Γ contains the strict transform of S and is connected (indeed the fibres are connected and we work locally about S). Thus we are free to replace X by Y and so we may assume that X is ℚ-factorial and K_{X} is kawamata log terminal.

Suppose that (X, Δ) is purely log terminal. By (10.2) there is a birational projective morphism π: Y ⟶ X such that Y is ℚ-factorial and the exceptional locus over S is contained in the strict transform of S. Replacing X by Y, we may assume that X is ℚ-factorial.

We may therefore assume that X is ℚ-factorial and K_{X} is kawamata log terminal. Let ν be any valuation of log discrepancy b. Suppose that the centre of ν is not a divisor. By (1.4.3) there is a birational projective morphism π: Y ⟶ X which extracts a single exceptional divisor E corresponding to ν. Since X is ℚ-factorial, the exceptional locus of π is equal to the support of E and so E intersects the strict transform of S. Let Γ = Δ′ +(1−b)E, where Δ′ is the strict transform of Δ. Then K_{Y} + Γ = π^{∗}(K_{X} + Δ).

Replacing X by Y we may therefore assume there is a prime divisor D ≠ S on X, whose coefficient Δ is 1 − b. By (3.4.1) some component B of Θ has coefficient at least 1 − b, that is log discrepancy b and so a ≤ b. □

Proof of (1.4.6). Since this result is local in the étale topology, we may assume that U is affine. Let f: Y ⟶ X be a log resolution of (X, Δ), so that the composition ψ: Y ⟶ U of f and π is projective. We may write

K_{Y} + Γ′ = f^{∗}(K_{X} + Δ) + E,

where Γ′ ≥ 0 and E ≥ 0 have no common components and E is f-exceptional. If F ≥ 0 is a ℚ-divisor whose support equals the exceptional locus of f, then for any 0 < ϵ ≪ 1, (Y, Γ = Γ′ + ϵF) is kawamata log terminal.

Pick a ψ-ample divisor H such that K_{Y} + Γ + H is ψ-ample. We run the (K_{Y} + Γ)-MMP over U with scaling of H. Since X contains no rational curves contracted by π, this MMP is automatically a MMP over X. Since Γ is big over X and termination is local in the étale topology, this MMP terminates by (1.4.2).

Thus we may assume that K_{Y} + Γ is f-nef, so that E + ϵF is empty, and hence f is small. As X is analytically ℚ-factorial it follows that f is an isomorphism. But then π is a log terminal model. □

References

[1] V. Alexeev, C. Hacon, and Y. Kawamata, Termination of (many) 4dimensional log flips, Invent. Math. 168 (2007), no. 2, 433–448.

[2] C. Birkar, Ascending chain condition for log canonical thresholds and termination of log flips, Duke Math. J. 136 (2007), no. 1.

[3] D. Bump, Lie groups, Graduate Texts in Mathematics, vol. 225, Springer-Verlag, New York, 2004.

[4] P. Eyssidieux, V. Guedj, and A. Zeriahi, Singular Kahler-Einstein metrics, arXiv:math.AG/0603431.

[5] O. Fujino, Termination of 4-fold canonical flips, Publ. Res. Inst. Math. Sci. 40 (2004), no. 1, 231–237.

[6] O. Fujino and S. Mori, A canonical bundle formula, J. Differential Geom. 56 (2000), no. 1, 167–188.

[7] A. Gibney, S. Keel, and I. Morrison, Towards the ample cone of \overline{M}_{g,n}, J. Amer. Math. Soc. 15 (2002), no. 2, 273–294.

[8] C. Hacon and J. M^{c}Kernan, On the existence of flips, math.AG/0507597.

[9] , Extension theorems and the existence of flips, Flips for 3-folds and 4-folds (Alessio Corti, ed.), Oxford University Press, 2007, pp. 79–100.

[10] R. Hartshorne, Algebraic geometry, Springer-Verlag, 1977.

[11] B. Hassett and D. Hyeon, Log canonical models for the moduli space of curves: First divisorial contraction, arXiv:math.AG/0607477.

[12] Y. Hu and S. Keel, Mori dream spaces and GIT, Michigan Math. J. 48 (2000), 331–348, Dedicated to William Fulton on the occasion of his 60th birthday.

[13] M. Kawakita, Inversion of adjunction on log canonicity, Invent. Math. 167 (2007), no. 1, 129–133.

[14] Y. Kawamata, The Zariski decomposition of log-canonical divisors, Algebraic Geometry Bowdoin, Proc. Symp. Pure Math. 46 (1987), 1985, pp. 425–433.

[15] , The crepant blowing up of 3-dimensional canonical singularities and its application to degeneration of surfaces, Ann. of Math. 127 (1988), 93–163.

[16] , On the length of an extremal rational curve, Invent. Math. 105 (1991), 609–611.

[17] , Flops connect minimal models, Publ. Res. Inst. Math. Sci. 44 (2008), no. 2, 419–423.

[18] Y. Kawamata, K. Matsuda, and K. Matsuki, Introduction to the minimal model program, Algebraic Geometry, Sendai (T. Oda, ed.), Kinokuniya-North-Holland, 1987, Adv. Stud. Pure Math, vol 10, pp. 283–360.

[19] J. Kollár, Flips, flops, minimal models, etc, Surveys in differential geometry (Cambridge, MA, 1990), Lehigh Univ., Bethlehem, PA, 1991, pp. 113–199.

[20] , Effective base point freeness, Math. Ann. 296 (1993), no. 4, 595–605.

[21] J. Kollár et al., Flips and abundance for algebraic threefolds, Société Mathématique de France, Paris, 1992, Papers from the Second Summer Seminar on Algebraic Geometry held at the University of Utah, Salt Lake City, Utah, August 1991, Astérisque No. 211 (1992).

[22] J. Kollár and S. Mori, Birational geometry of algebraic varieties, Cambridge tracts in mathematics, vol. 134, Cambridge University Press, 1998.

[23] R. Lazarsfeld, Positivity in algebraic geometry. II, Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. A Series of Modern Surveys in Mathematics [Results in Mathematics and Related Areas. 3rd Series. A Series of Modern Surveys in Mathematics], vol. 49, Springer-Verlag, Berlin, 2004, Positivity for vector bundles, and multiplier ideals.

[24] Y. Manin, Moduli stacks \overline{L}_{g,S}, Mosc. Math. J. 4 (2004), no. 1, 181–198, 311.

[25] K. Matsuki, Termination of flops for 4-folds, Amer. J. Math. 113 (1991), no. 5, 835–859.

[26] S. Mori, Flip theorem and the existence of minimal models for 3-folds, J. Amer. Math. Soc. 1 (1988), no. 1, 117–253.

[27] D. Mumford, Stability of projective varieties, Enseignement Math. (2) 23 (1977), no. 1-2, 39–110.

[28] N. Nakayama, Zariski-decomposition and abundance, MSJ Memoirs, vol. 14, Mathematical Society of Japan, Tokyo, 2004.

[29] V. Nikulin, The diagram method for 3-folds and its application to the Kähler cone and Picard number of Calabi-Yau 3-folds. I, Higher-dimensional complex varieties (Trento, 1994), de Gruyter, Berlin, 1996, With an appendix by V. V. Shokurov, pp. 261–328.

[30] T. Peternell, Towards a Mori theory on compact Kähler threefolds. II, Math. Ann. 311 (1998), no. 4, 729–764.

[31] V. V. Shokurov, Three-dimensional log perestroikas, Izv. Ross. Akad. Nauk Ser. Mat. 56 (1992), no. 1, 105–203.

[32] , 3-fold log models, J. Math. Sci. 81 (1996), no. 3, 2667–2699, Algebraic geometry, 4.

[33] , Letters of a bi-rationalist. I. A projectivity criterion, Birational algebraic geometry (Baltimore, MD, 1996), Contemp. Math., vol. 207, Amer. Math. Soc., Providence, RI, 1997, pp. 143–152.

[34] , Prelimiting flips, Proc. Steklov Inst. of Math. 240 (2003), 82–219.

[35] , Letters of a bi-rationalist. V. Minimal log discrepancies and termination of log flips, Tr. Mat. Inst. Steklova 246 (2004), no. Algebr. Geom. Metody, Svyazi i Prilozh., 328–351.

[36] Y-T. Siu, A General Non-Vanishing Theorem and an Analytic Proof of the Finite Generation of the Canonical Ring, arXiv:math.AG/0610740.

[37] K. Ueno, Bimeromorphic geometry of algebraic and analytic threefolds, Algebraic Threefolds, Lecture Notes in Mathematics 947 (A. Conte, ed.), Springer, 1982, pp. 1–34.

DPMMS, Centre for Mathematical Sciences, University of Cambridge, Wilberforce Road, Cambridge CB3 0WB, UK

E-mail address: c.birkar@dpmms.cam.ac.uk

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

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

Department of Mathematics, University of Utah, 155 South 1400 East, 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

Department of Mathematics, MIT, 77 Massachusetts Avenue, Cambridge, MA 02139, USA

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