Existence of minimal models for varieties of log general type

Abstract

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

1. Introduction

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

Theorem 1.1.

Let $(X,\Delta )$ be a projective Kawamata log terminal pair.

If $\Delta$ is big and $K_X+\Delta$ is pseudo-effective, then $K_X+\Delta$ has a log terminal model.

In particular, it follows that if $K_X+\Delta$ 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

$$R(X,K_X)=\bigoplus _{m\in \mathbb{N}}H^0(X,\mathcal{O}_{X}(mK_X)),$$

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, $\kappa (X,K_X)\geq 0$, then there is a unique smooth surface $Y$ birational to $X$ such that the canonical class $K_Y$ is nef (that is $K_Y\cdot C\geq 0$ for any curve $C\subset Y$). $Y$ is obtained from $X$ simply by contracting all $-1$-curves, that is, all smooth rational curves $E$ with $K_X\cdot E=-1$. If, on the other hand, $\kappa (X,K_X)=-\infty$, then $X$ is birational to either $\mathbb{P}^{2}$ 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 $\kappa (X,K_X)\geq 0$, then there exists a minimal model, that is, a variety $Y$ birational to $X$ such that $K_Y$ is nef.

•

If $\kappa (X,K_X)=-\infty$, then there is a variety $Y$ birational to $X$ which admits a Fano fibration, that is, a morphism $Y \longrightarrow Z$ 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 $K_X\cdot C$ for any curve $C\subset X$. So, we insist that $K_X$ is $\mathbb{Q}$-Cartier (or sometimes we require the stronger property that $X$ is $\mathbb{Q}$-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 $\phi \colon X \mathrel{\rightdasharrow }Y$ 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 a pseudo-effective $K_X$, then one can construct Mori fibre spaces whenever $K_X$ is not pseudo-effective). 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

$$R(X,K_X)=\bigoplus _{m\in \mathbb{N}}H^0(X,\mathcal{O}_{X}(mK_X))$$

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 $\phi \colon X \mathrel{\rightdasharrow }Y$ is $K_X$-negative.

The other natural way to ensure that $\phi$ is $K_X$-negative is to factor $\phi$ 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 $C\subset X$ such that $K_X\cdot C<0$ and a morphism $f\colon X \longrightarrow Z$ which is surjective, with connected fibres, onto a normal projective variety and which contracts an irreducible curve $D$ if and only if $[D]\in \mathbb{R}^+[C] \subset N_1(X)$. Note that $\rho (X/Z)=1$ and $-K_X$ is $f$-ample. We have the following possibilities:

•

If $\operatorname {dim}Z<\operatorname {dim}X$, this is the required Fano fibration.

•

If $\operatorname {dim}Z=\operatorname {dim}X$ and $f$ contracts a divisor, then we say that $f$ is a divisorial contraction and we replace $X$ by $Z$.

•

If $\operatorname {dim}Z=\operatorname {dim}X$ and $f$ does not contract a divisor, then we say that $f$ is a small contraction. In this case $K_Z$ is not $\mathbb{Q}$-Cartier, so that we cannot replace $X$ by $Z$. Instead, we would like to replace $f\colon X \longrightarrow Z$ by its flip $f^+\colon X^+ \longrightarrow Z$, 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 $\rho (X)$ 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 1980s 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^cKernan 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 $\geq 4$ 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 $-(K_X+\Delta )$ is non-negative and the existence of minimal models of Kawamata log terminal $4$-folds when either $\Delta$ or $K_X+\Delta$ 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 $\kappa (X,K_X+\Delta )\geq 0$, 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 $(X,\Delta )$ be a Kawamata log terminal pair, where $K_X+\Delta$ is $\mathbb{R}$-Cartier. Let $\pi \colon X \longrightarrow U$ be a projective morphism of quasi-projective varieties.

If either $\Delta$ is $\pi$-big and $K_X+\Delta$ is $\pi$-pseudo-effective or $K_X+\Delta$ is $\pi$-big, then

(1)

$K_X+\Delta$ has a log terminal model over $U$,

(2)

if $K_X+\Delta$ is $\pi$-big then $K_X+\Delta$ has a log canonical model over $U$, and

(3)

if $K_X+\Delta$ is $\mathbb{Q}$-Cartier, then the $\mathcal{O}_{U}$-algebra$$\mathfrak{R}(\pi , K_X+\Delta )=\bigoplus _{m\in \mathbb{N}} \pi _*\mathcal{O}_{X}(\llcorner m(K_X+\Delta )\lrcorner ),$$

is finitely generated.

We now present some consequences of Theorem 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 Theorem 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$$R(X,K_X)=\bigoplus _{m\in \mathbb{N}} H^0(X,\mathcal{O}_{X}(mK_X))$$

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 $(X,\Delta )$ be a projective Kawamata log terminal pair, where $K_X+\Delta$ is $\mathbb{Q}$-Cartier.

Then the ring

$$R(X,K_X+\Delta )=\bigoplus _{m\in \mathbb{N}} H^0(X,\mathcal{O}_{X}(\llcorner m(K_X+\Delta )\lrcorner ))$$

is finitely generated.

Let us emphasize that in Corollary 1.1.2 we make no assumption about $K_X+\Delta$ or $\Delta$ being big. Indeed Fujino and Mori, 6, proved that Corollary 1.1.2 follows from the case when $K_X+\Delta$ 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 $\pi \colon X \longrightarrow U$ be a projective morphism of normal quasi-projective varieties. Suppose that $K_X+\Delta$ is Kawamata log terminal and $\Delta$ is big over $U$. Let $\phi _i\colon X \mathrel{\rightdasharrow }Y_i$, $i=1$ and $2$, be two log terminal models of $(X,\Delta )$ over $U$. Let $\Gamma _i=\phi _{i*}\Delta$.

Then the birational map $Y_1 \mathrel{\rightdasharrow }Y_2$ is the composition of a sequence of $(K_{Y_1}+\Gamma _1)$-flops over $U$.

Note that Corollary 1.1.3 has been generalised recently to the case when $\Delta$ 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 $\Delta$ is well-behaved. To explain this, we need some definitions:

Definition 1.1.4.

Let $\pi \colon X \longrightarrow U$ be a projective morphism of normal quasi-projective varieties, and let $V$ be a finite dimensional affine subspace of the real vector space $\operatorname {WDiv}_{\mathbb{R}}(X)$ of Weil divisors on $X$. Fix an $\mathbb{R}$-divisor $A\geq 0$ and define

$$\begin{align*} V_A &=\{\,\Delta \,|\, \Delta =A+B, B\in V \,\}, \\ \mathcal{L}_A(V)&=\{\,\Delta =A+B\in V_A \,|\, \text{$K_X+\Delta $ is log canonical and $B\geq 0$} \,\}, \\ \mathcal{E}_{A,\pi }(V) &=\{\,\Delta \in \mathcal{L}_A(V) \,|\, \text{$K_X+\Delta $ is pseudo-effective over $U$} \,\}, \\ \mathcal{N}_{A,\pi }(V)&=\{\,\Delta \in \mathcal{L}_A(V) \,|\, \text{$K_X+\Delta $ is nef over $U$} \,\}. \end{align*}$$

Given a birational contraction $\phi \colon X \mathrel{\rightdasharrow }Y$ over $U$, define

$$\mathcal{W}_{\phi ,A,\pi }(V)=\{\, \Delta \in \mathcal{E}_{A,\pi }(V) \,|\, \text{$\phi $ is a weak log canonical model for $(X,\Delta )$ over $U$} \,\},$$

and given a rational map $\psi \colon X \mathrel{\rightdasharrow }Z$ over $U$, define

$$\mathcal{A}_{\psi ,A,\pi }(V)=\{\, \Delta \in \mathcal{E}_{A,\pi }(V) \,|\, \text{$\psi $ is the ample model for $(X,\Delta )$ over $U$} \,\},$$

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

We will adopt the convention that $\mathcal{L}(V)=\mathcal{L}_0(V)$. If the support of $A$ has no components in common with any element of $V$, then the condition that $B\geq 0$ is vacuous. In many applications, $A$ will be an ample $\mathbb{Q}$-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 $A'\sim _{U} kA$ is very general. With this choice of $A$, we have

$$\mathcal{N}_{A,\pi }(V)\subset \mathcal{E}_{A,\pi }(V)\subset \mathcal{L}_A(V)=\mathcal{L}(V)+A \subset V_A=V+A,$$

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 $\pi \colon X \longrightarrow U$ be a projective morphism of normal quasi-projective varieties. Let $V$ be a finite dimensional affine subspace of $\operatorname {WDiv}_{\mathbb{R}}(X)$ which is defined over the rationals. Suppose there is a divisor $\Delta _0\in V$ such that $K_X+\Delta _0$ is Kawamata log terminal. Let $A$ be a general ample $\mathbb{Q}$-divisor over $U$, which has no components in common with any element of $V$.

(1)

There are finitely many birational contractions $\phi _i\colon X \mathrel{\rightdasharrow }Y_i$ over $U$, $1\leq i\leq p$ such that$$\mathcal{E}_{A,\pi }(V)=\bigcup _{i=1}^p\mathcal{W}_i,$$

where each $\mathcal{W}_i=\mathcal{W}_{\phi _i,A,\pi }(V)$ is a rational polytope. Moreover, if $\phi \colon X \mathrel{\rightdasharrow }Y$ is a log terminal model of $(X,\Delta )$ over $U$, for some $\Delta \in \mathcal{E}_{A,\pi }(V)$, then $\phi =\phi _i$, for some $1\leq i\leq p$.

(2)

There are finitely many rational maps $\psi _j\colon X \mathrel{\rightdasharrow }Z_j$ over $U$, $1\leq j\leq q$ which partition $\mathcal{E}_{A,\pi }(V)$ into the subsets $\mathcal{A}_j=\mathcal{A}_{\psi _j,A,\pi }(V)$.

(3)

For every $1\leq i\leq p$ there is a $1\leq j\leq q$ and a morphism $f_{i,j}\colon Y_i \longrightarrow Z_j$ such that $\mathcal{W}_i\subset \bar{\mathcal{A}}_j$.

In particular $\mathcal{E}_{A,\pi }(V)$ is a rational polytope and each $\bar{\mathcal{A}}_j$ is a finite union of rational polytopes.

Definition 1.1.6.

Let $(X,\Delta )$ be a Kawamata log terminal pair and let $D$ be a big divisor. Suppose that $K_X+\Delta$ is not pseudo-effective. The effective log threshold is

$$\sigma (X,\Delta ,D)=\sup \{\, t\in \mathbb{R} \,|\, \text{$D+t(K_X+\Delta )$ is pseudo-effective} \,\}.$$

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 $(X,\Delta )$ be a projective Kawamata log terminal pair and let $D$ be an ample divisor. Suppose that $K_X+\Delta$ is not pseudo-effective.

If both $K_X+\Delta$ and $D$ are $\mathbb{Q}$-Cartier, then the effective log threshold and the Kodaira energy are rational.

Definition 1.1.8.

Let $\pi \colon X \longrightarrow U$ be a projective morphism of normal quasi-projective varieties. Let $D^{\bullet }=({D}_1,{D}_2,\dots ,{D}_{k})$ be a sequence of $\mathbb{Q}$-divisors on $X$. The sheaf of $\mathcal{O}_{U}$-algebras,

$$\mathfrak{R}(\pi ,D^{\bullet })=\bigoplus _{m\in \mathbb{N}^k} \pi _*\mathcal{O}_{X}(\llcorner \sum m_iD_i\lrcorner ),$$

is called the Cox ring associated to $D^{\bullet }$.

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

Corollary 1.1.9.

Let $\pi \colon X \longrightarrow U$ be a projective morphism of normal quasi-projective varieties. Fix $A\geq 0$ to be an ample $\mathbb{Q}$-divisor over $U$. Let $\Delta _i=A+B_i$, for some $\mathbb{Q}$-divisors ${B}_1,{B}_2,\dots ,{B}_{k}\geq 0$. Assume that $D_i=K_X+\Delta _i$ is divisorially log terminal and $\mathbb{Q}$-Cartier. Then the Cox ring,

$$\mathfrak{R}(\pi ,D^{\bullet })=\bigoplus _{m\in \mathbb{N}^k} \pi _*\mathcal{O}_{X}(\llcorner \sum m_iD_i\lrcorner ),$$

is a finitely generated $\mathcal{O}_{U}$-algebras.

1.2. Moduli spaces

At first sight Corollary 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 $X=\overline{M}_{g,n}$ be the moduli space of stable curves of genus $g$ with $n$ marked points and let $\Delta _i$, $1\leq i\leq k$ denote the boundary divisors.

Let $\Delta =\sum _i a_i\Delta _i$ be a boundary. Then $K_X+\Delta$ is log canonical and if $K_X+\Delta$ is big, then there is a log canonical model $X \mathrel{\rightdasharrow }Y$. Moreover if we fix a positive rational number $\delta$ and require that the coefficient $a_i$ of $\Delta _i$ is at least $\delta$ 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 $K_X+\Delta$, where $\Delta$ is big and $K_X+\Delta$ is Kawamata log terminal.

Definition 1.3.1.

Let $\pi \colon X \longrightarrow U$ be a projective morphism of normal varieties, where $U$ is affine.

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

Corollary 1.3.2.

Let $\pi \colon X \longrightarrow U$ be a projective morphism of normal varieties, where $U$ is affine. Suppose that $X$ is $\mathbb{Q}$-factorial, $K_X+\Delta$ is divisorially log terminal and $-(K_X+\Delta )$ 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.

Corollary 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 Corollary 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 Theorem 1.2. We show that if $K_X+\Delta$ is not pseudo-effective, then we can run the MMP with scaling to get a Mori fibre space:

Corollary 1.3.3.

Let $(X,\Delta )$ be a $\mathbb{Q}$-factorial Kawamata log terminal pair. Let $\pi \colon X \longrightarrow U$ be a projective morphism of normal quasi-projective varieties. Suppose that $K_X+\Delta$ is not $\pi$-pseudo-effective.

Then we may run $f\colon X \mathrel{\rightdasharrow }Y$ a $(K_X+\Delta )$-MMP over $U$ and end with a Mori fibre space $g\colon Y \longrightarrow W$ over $U$.

Note that we do not claim in Corollary 1.3.3 that however we run the $(K_X+\Delta )$-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 $\Sigma$ is called nef if $B\cdot \Sigma \geq 0$ for all Cartier divisors $B\geq 0$. $\operatorname {NF}(X)$ denotes the cone of nef curves sitting inside $H_2(X,\mathbb{R})$ and $\overline{\operatorname {NF}}(X)$ denotes its closure.

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

Corollary 1.3.5.

Let $(X,\Delta )$ be a projective $\mathbb{Q}$-factorial Kawamata log terminal pair such that $-(K_X+\Delta )$ is ample.

Then $\overline{\operatorname {NF}}(X)$ is a rational polyhedron. If $F=F_i$ is a $(K_X+\Delta )$-co-extremal ray, then there exists an $\mathbb{R}$-divisor $\Theta$ such that the pair $(X,\Theta )$ is Kawamata log terminal and the $(K_X+\Theta )$-MMP $\pi \colon X \mathrel{\rightdasharrow }Y$ ends with a Mori fibre space $f\colon Y \longrightarrow Z$ such that $F$ is spanned by the pullback to $X$ of the class of any curve $\Sigma$ which is contracted by $f$.

1.4. Birational geometry

Another immediate consequence of Theorem 1.2 is the existence of flips:

Corollary 1.4.1.

Let $(X,\Delta )$ be a Kawamata log terminal pair and let $\pi \colon X \longrightarrow Z$ be a small $(K_X+\Delta )$-extremal contraction.

Then the flip of $\pi$ exists.

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

Corollary 1.4.2.

Let $\pi \colon X \longrightarrow U$ be a projective morphism of normal quasi-projective varieties. Let $(X,\Delta )$ be a $\mathbb{Q}$-factorial Kawamata log terminal pair, where $K_X+\Delta$ is $\mathbb{R}$-Cartier and $\Delta$ is $\pi$-big. Let $C\geq 0$ be an $\mathbb{R}$-divisor.

If $K_X+\Delta +C$ is Kawamata log terminal and $\pi$-nef, then we may run the $(K_X+\Delta )$-MMP over $U$ with scaling of $C$.

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

Corollary 1.4.3.

Let $(X,\Delta )$ be a log canonical pair and let $f\colon W \longrightarrow X$ be a log resolution. Suppose that there is a divisor $\Delta _0$ such that $K_X+\Delta _0$ is Kawamata log terminal. Let $\mathfrak{E}$ be any set of valuations of $f$-exceptional divisors which satisfies the following two properties:

(1)

$\mathfrak{E}$ contains only valuations of log discrepancy at most one, and

(2)

the centre of every valuation of log discrepancy one in $\mathfrak{E}$ does not contain any non-Kawamata log terminal centres.

Then we may find a birational morphism $\pi \colon Y \longrightarrow X$, such that $Y$ is $\mathbb{Q}$-factorial and the exceptional divisors of $\pi$ correspond to the elements of $\mathfrak{E}$.

For example, if we assume that $(X,\Delta )$ is Kawamata log terminal and we let $\mathfrak{E}$ be the set of all exceptional divisors with log discrepancy at most one, then the birational morphism $\pi \colon Y \longrightarrow X$ defined in Corollary 1.4.3 above is a terminal model of $(X,\Delta )$. In particular there is an $\mathbb{R}$-divisor $\Gamma \geq 0$ on $Y$ such that $K_Y+\Gamma =\pi ^*(K_X+\Delta )$ and the pair $(Y,\Gamma )$ is terminal.

If instead we assume that $(X,\Delta )$ is Kawamata log terminal but $\mathfrak{E}$ is empty, then the birational morphism $\pi \colon Y \longrightarrow X$ defined in Corollary 1.4.3 above is a log terminal model. In particular $\pi$ is small, $Y$ is $\mathbb{Q}$-factorial and there is an $\mathbb{R}$-divisor $\Gamma \geq 0$ on $Y$ such that $K_Y+\Gamma =\pi ^*(K_X+\Delta )$.

We are able to prove that every log pair admits a birational model with $\mathbb{Q}$-factorial singularities such that the non-Kawamata log terminal locus is a divisor:

Corollary 1.4.4.

Let $(X,\Delta )$ be a log pair.

Then there is a birational morphism $\pi \colon Y \longrightarrow X$, where $Y$ is $\mathbb{Q}$-factorial, such that if we write

$$K_Y+\Gamma =K_Y+\Gamma _1+\Gamma _2=\pi ^*(K_X+\Delta ),$$

where every component of $\Gamma _1$ has coefficient less than one and every component of $\Gamma _2$ has coefficient at least one, then $K_Y+\Gamma _1$ is Kawamata log terminal and nef over $X$ and no component of $\Gamma _1$ is exceptional.

Even though the result in Corollary 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 $(X,\Delta )$ be a log pair and let $\nu \colon S \longrightarrow S'$ be the normalisation of a component $S'$ of $\Delta$ of coefficient one.

If we define $\Theta$ by adjunction,

$$\nu ^*(K_X+\Delta )=K_S+\Theta ,$$

then the log discrepancy of $K_S+\Theta$ is equal to the minimum of the log discrepancy with respect to $K_X+\Delta$ 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 $\mathbb{C}$, 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 $\pi \colon X \longrightarrow U$ be a proper map of normal algebraic spaces, where $X$ is analytically $\mathbb{Q}$-factorial.

If $K_X+\Delta$ is divisorially log terminal and $\pi$ does not contract any rational curves, then $\pi$ is a log terminal model. In particular $\pi$ is projective and $K_X+\Delta$ is $\pi$-nef.

2. Description of the proof

Theorem A (Existence of pl-flips).

Let $f\colon X \longrightarrow Z$ be a pl-flipping contraction for an $n$-dimensional purely log terminal pair $(X,\Delta )$.

Then the flip $f^+\colon X^+ \longrightarrow Z$ of $f$ exists.

Theorem B (Special finiteness).

Let $\pi \colon X \longrightarrow U$ be a projective morphism of normal quasi-projective varieties, where $X$ is $\mathbb{Q}$-factorial of dimension $n$. Let $V$ be a finite dimensional affine subspace of $\operatorname {WDiv}_{\mathbb{R}}(X)$, which is defined over the rationals, let $S$ be the sum of finitely many prime divisors and let $A$ be a general ample $\mathbb{Q}$-divisor over $U$. Let $(X,\Delta _0)$ be a divisorially log terminal pair such that $S\leq \Delta _0$. Fix a finite set $\mathfrak{E}$ of prime divisors on $X$.

Then there are finitely $1\leq i\leq k$ many birational maps $\phi _i\colon X \mathrel{\rightdasharrow }Y_i$ over $U$ such that if $\phi \colon X \mathrel{\rightdasharrow }Y$ is any $\mathbb{Q}$-factorial weak log canonical model over $U$ of $K_{X}+\Delta$, where $\Delta \in \mathcal{L}_{S+A}(V)$, which only contracts elements of $\mathfrak{E}$ and which does not contract every component of $S$, then there is an index $1\leq i\leq k$ such that the induced birational map $\xi \colon Y_i \mathrel{\rightdasharrow }Y$ is an isomorphism in a neighbourhood of the strict transforms of $S$.

Theorem C (Existence of log terminal models).

Let $\pi \colon X \longrightarrow U$ be a projective morphism of normal quasi-projective varieties, where $X$ has dimension $n$. Suppose that $K_X+\Delta$ is Kawamata log terminal, where $\Delta$ is big over $U$.

If there exists an $\mathbb{R}$-divisor $D$ such that $K_X+\Delta \sim _{\mathbb{R},U} D\geq 0$, then $K_X+\Delta$ has a log terminal model over $U$.

Theorem D (Non-vanishing theorem).

Let $\pi \colon X \longrightarrow U$ be a projective morphism of normal quasi-projective varieties, where $X$ has dimension $n$. Suppose that $K_X+\Delta$ is Kawamata log terminal, where $\Delta$ is big over $U$.

If $K_X+\Delta$ is $\pi$-pseudo-effective, then there exists an $\mathbb{R}$-divisor $D$ such that $K_X+\Delta \sim _{\mathbb{R},U} D\geq 0$.

Theorem E (Finiteness of models).

Let $\pi \colon X \longrightarrow U$ be a projective morphism of normal quasi-projective varieties, where $X$ has dimension $n$. Fix a general ample $\mathbb{Q}$-divisor $A\geq 0$ over $U$. Let $V$ be a finite dimensional affine subspace of $\operatorname {WDiv}_{\mathbb{R}}(X)$ which is defined over the rationals. Suppose that there is a Kawamata log terminal pair $(X,\Delta _0)$.

Then there are finitely many birational maps $\psi _j\colon X \mathrel{\rightdasharrow }Z_j$ over $U$, $1\leq j\leq l$ such that if $\psi \colon X \mathrel{\rightdasharrow }Z$ is a weak log canonical model of $K_X+\Delta$ over $U$, for some $\Delta \in \mathcal{L}_A(V)$, then there is an index $1\leq j\leq l$ and an isomorphism $\xi \colon Z_j \longrightarrow Z$ such that $\psi =\xi \circ \psi _j$.

Theorem F (Finite generation).

Let $\pi \colon X \longrightarrow Z$ be a projective morphism to a normal affine variety. Let $(X,\Delta =A+B)$ be a Kawamata log terminal pair of dimension $n$, where $A\geq 0$ is an ample $\mathbb{Q}$-divisor and $B\geq 0$. If $K_X+\Delta$ is pseudo-effective, then

(1)

The pair $(X,\Delta )$ has a log terminal model $\mu \colon X \mathrel{\rightdasharrow }Y$. In particular if $K_X+\Delta$ is $\mathbb{Q}$-Cartier, then the log canonical ring$$R(X,K_X+\Delta )=\bigoplus _{m\in \mathbb{N}}H^0(X,\mathcal{O}_{X}(\llcorner m(K_X+\Delta )\lrcorner ))$$

is finitely generated.

(2)

Let $V\subset \operatorname {WDiv}_{\mathbb{R}}(X)$ be the vector space spanned by the components of $\Delta$. Then there is a constant $\delta >0$ such that if $G$ is a prime divisor contained in the stable base locus of $K_X+\Delta$ and $\Xi \in \mathcal{L} _{A}(V)$ such that $\|\Xi -\Delta \|<\delta$, then $G$ is contained in the stable base locus of $K_X+\Xi$.

(3)

Let $W\subset V$ be the smallest affine subspace of $\operatorname {WDiv}_{\mathbb{R}}(X)$ containing $\Delta$, which is defined over the rationals. Then there is a constant $\eta >0$ and a positive integer $r>0$ such that if $\Xi \in W$ is any divisor and $k$ is any positive integer such that $\|\Xi -\Delta \|<\eta$ and $k(K_X+\Xi )/r$ is Cartier, then every component of $\operatorname {Fix}(k(K_X+\Xi ))$ is a component of the stable base locus of $K_X+\Delta$.

The proofs of Theorem A, Theorem B, Theorem C, Theorem D, Theorem E and Theorem F proceed 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 Theorem 1.2, we review a natural approach to proving that the canonical ring

$$R(X,K_X)=\bigoplus _{m\in \mathbb{N}} H^0(X,\mathcal{O}_{X}(mK_X))$$

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 $D\in |kK_X|$, whose existence is guaranteed as we are assuming that $K_X$ is big, and then to restrict to $D$. One obtains an exact sequence

$$0\longrightarrow H^0(X,\mathcal{O}_{X}((l-k)K_X)) \longrightarrow H^0(X,\mathcal{O}_{X}(lK_X)) \longrightarrow H^0(D,\mathcal{O}_{D}(mK_D)),$$

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

$$H^0(X,\mathcal{O}_{X}(m(1+k)K_X)) \longrightarrow H^0(D,\mathcal{O}_{D}(mK_D)),$$

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 $\omega _D$). 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

$$H^0(X,\mathcal{O}_{X}(m(1+k)K_X)) \longrightarrow H^0(S,\mathcal{O}_{S}(mK_S))$$

no longer has any obvious connection with

$$H^0(X,\mathcal{O}_{X}((m(1+k)-k)K_X)),$$

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

$$\bigoplus _{m\in \mathbb{N}}H^0(S,\mathcal{O}_{S}(mK_S)),$$

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 non-empty for any $k>0$. Finally, even though this aspect of the problem may not be apparent from the description above, in practice 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 $\mathbb{P}^{1}$-bundle over $V$. In the best case, the projection $\pi \colon C \longrightarrow V$ will be a $\mathbb{P}^{1}$-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 $\mathbb{N}^2$, rather than just $\mathbb{N}$. Let us consider then the simplest possible algebras over $\mathbb{C}$ which are graded by $\mathbb{N}^2$. If we are given a submonoid $M\subset \mathbb{N}^2$ (that is, a subset of $\mathbb{N}^2$ which contains the origin and is closed under addition), then we get a subalgebra $R\subset \mathbb{C}^2[x,y]$ spanned by the monomials

$$\{\, x^iy^j \,|\, (i,j)\in M \,\}.$$

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,

$$M=\{\, (i,j)\in \mathbb{N}^2 \,|\, j>0 \,\}\cup \{(0,0)\} \quad \text{and} \quad \{\,(i,j)\in \mathbb{N}^2 \,|\, i\geq \sqrt {2}j \,\}.$$

In fact, if $\mathcal{C}\subset \mathbb{R}^2$ is the convex hull of the set $M$, then $M$ is finitely generated iff $\mathcal{C}$ is a rational polytope. In the general case, we will be given a convex subset $\mathcal{C}$ of a finite dimensional vector space of Weil divisors on $X$ and a key part of the proof is to show that the set $\mathcal{C}$ is in fact a rational cone. 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 Theorem 1.2. As is clear from the plan of the proof given in the previous subsection, the proof of Theorem 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 $D\in |kK_X|$, 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 $\pi \colon X \longrightarrow Z$ be the corresponding contraction. Then $S\cdot R<0$, so that every curve $\Sigma$ contracted by $\pi$ must be contained in $S$. In particular $\pi$ cannot be a divisorial contraction, as $S$ is not uniruled. Hence $\pi$ is a pl-flip and by Theorem A$_n$, we can construct the flip of $\pi$, $\phi \colon X \mathrel{\rightdasharrow }Y$. Consider the restriction $\psi \colon S \mathrel{\rightdasharrow }T$ of $\phi$ to $S$, where $T$ is the strict transform of $S$. Since log discrepancies increase under flips and $S$ is irreducible, $\psi$ is a birational contraction. After finitely many flips, we may therefore assume that $\psi$ 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 $\psi \colon S \mathrel{\rightdasharrow }T$ 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 $(K_X+S+tH)\cdot R=0$. If $t=0$, then $K_X+S$ is nef and we are done. Otherwise $(K_X+S)\cdot R<0$, 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 $G'=\psi _*G$. 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 $\psi$ might contract divisors on which $K_S+tG$ is trivial and more importantly because $T$ need not be $\mathbb{Q}$-factorial). In this case we are then done, by finiteness of weak log canonical models for $(S,tG)$, where $t\in [0,1]$ (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 $\phi \colon X \mathrel{\rightdasharrow }Y$ 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 $\mathbb{R}$-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 $K_X+\Delta =K_X+S+A+B$, 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 $D\geq 0$ such that $K_X+\Delta \sim _{\mathbb{R}} D$. 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 that $K_X+\Delta$ 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 $(X,\Delta )$ is log smooth and if $B=\sum b_iB_i$, then the coefficients $({b}_1,{b}_2,\dots ,{b}_{k})$ of $B$ lie in $[0,1]^k$. Let $\Theta =(\Delta -S)|_S$ so that $(K_X+\Delta )|_S=K_S+\Theta$.

Suppose that $f\colon X \mathrel{\rightdasharrow }Y$ is a log terminal model of $(X,\Delta )$. There are three problems that arise, two of which are quite closely related. Suppose that $g\colon S \mathrel{\rightdasharrow }T$ 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 $\Sigma$ intersecting $S$, which is not contained in $S$. Then $S\cdot \Sigma >0$ so that $T\cdot E<0$, where $E$ is the flipped curve. In this case $E\subset T$ so that the induced birational map $S \mathrel{\rightdasharrow }T$ extracts the curve $E$. The basic observation is that $E$ must have log discrepancy less than one with respect to $(S,\Theta )$. Since the pair $(X,\Delta )$ is purely log terminal if we replace $(X,\Delta )$ by a fixed model which is high enough, then we can ensure that the pair $(S,\Theta )$ 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 $E\cap S$ is not contracted by $g$. For this reason, $g$ is not necessarily a weak log canonical model of $(S,\Theta )$. However we can construct a divisor $0\leq \Xi \leq \Theta$ such that $g$ is a weak log canonical model for $(S,\Xi )$. Suppose that we start with a smooth threefold $Y$ and a smooth surface $T\subset Y$ which contains a $-2$-curve $\Sigma$, such that $K_Y+T$ is nef. Let $f\colon X \longrightarrow Y$ be the blowup of $Y$ along $\Sigma$ 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 $S$, $g\colon S \longrightarrow T$ is the identity, but $g$ is not a log terminal model for $K_S+e\Sigma$, since $K_S+e\Sigma$ is negative along $\Sigma$. It is a weak log canonical model for $K_T$, so that in this case $\Xi =0$. The details of the construction of $\Xi$ are contained in Lemma 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,

$$(K_Y+T)|_T=K_T+\Phi .$$

The other parts of the third problem only occur in dimension three or more. For example, suppose that $Z$ is the cone over a smooth quadric in $\mathbb{P}^{3}$ and $p\colon X \longrightarrow Z$ and $q\colon Y \longrightarrow Z$ are the two small resolutions, so that the induced birational map $f\colon X \mathrel{\rightdasharrow }Y$ is the standard flop. Let $\pi \colon W \longrightarrow Z$ blow up the maximal ideal, so that the exceptional divisor $E$ is a copy of $\mathbb{P}^{1}\times \mathbb{P}^{1}$. Pick a surface $R$ which intersects $E$ along a diagonal curve $\Sigma$. If $S$ and $T$ are the strict transforms of $R$ in $X$ and $Y$, then the induced birational map $g\colon S \longrightarrow T$ is an isomorphism (both $S$ and $T$ are isomorphic to $R$). To get around this problem, one can perturb $\Delta$ 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 $(S,\Xi )$. Finally it is not hard to write down examples of flops which fix $\Xi$, but switch the individual components of $\Xi$. In this case one needs to keep track not only of $\Xi$ but the individual pieces $(g_*B_i)|_T$, $1\leq i\leq k$. We prove that an ample model $f$ is determined in a neighbourhood of $T$ by $g$, the different $\Phi$ and $(f_*B_i)|_T$; see Lemma 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 $\Phi$ 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 $\Delta +D$ has normal crossings.

Suppose that there is a divisor $C$ such that

$$\begin{equation*} K_X+\Delta \sim _{\mathbb{R},U} D+\alpha C, \tag{$\ast $} \end{equation*}$$

where $K_X+\Delta +C$ is divisorially log terminal and nef and the support of $D$ is contained in $S$. If $R$ is an extremal ray which is $(K_X+\Delta )$-negative, then $D\cdot R<0$, so that $S_i\cdot R<0$ 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 $(K_X+\Delta )$-MMP is not an isomorphism in a neighbourhood of $S$. Therefore the $(K_X+\Delta )$-MMP with scaling of $C$ must terminate with a log terminal model for $K_X+\Delta$. To summarise, whenever the conditions above hold, we can always construct a log terminal model of $K_X+\Delta$.

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 $K_X+\Delta$ has a log terminal model. If $D_2\neq 0$, 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 $(X,\Theta =\Delta +\lambda D_2)$, where $\lambda$ is the largest real number so that the coefficients of $\Theta$ are at most one. Then more components of $\llcorner \Theta \lrcorner$ are components of $D$. By induction $(X,\Theta )$ has a neutral model, $f\colon X \mathrel{\rightdasharrow }Y$. It is then easy to check that the conditions in the paragraph above apply, and we can construct a log terminal model $g\colon Y \mathrel{\rightdasharrow }Z$ for $K_Y+g_*\Delta$. It is then automatic that the composition $h=g\circ f\colon X \mathrel{\rightdasharrow }Z$ is a neutral model of $K_X+\Delta$ (since $f$ is not $(K_X+\Delta )$-negative, it is not true in general that $h$ is a log terminal model of $K_X+\Delta$). However $g$ is automatically a log terminal model provided we only contract components of the stable base locus of $K_X+\Delta$. 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 to prove that if $K_X+\Delta =K_X+A+B$ is pseudo-effective, then $K_X+\Delta \sim _{\mathbb{R}}D\geq 0$. 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 $H$,

$$h^0(X,\mathcal{O}_{X}(\llcorner m(K_X+\Delta )\lrcorner +H))$$

is a bounded function of $m$. In this case it follows that $K_X+\Delta$ is numerically equivalent to the divisor $N_{\sigma }(K_X+\Delta )\geq 0$. It is then not hard to prove that Theorem C$_n$ implies that $K_X+\Delta$ 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

$$m(K_X+\Delta )+H,$$

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

$$K_X+\Delta =K_X+S+A+B,$$

where $S$ is irreducible and $(K_X+\Delta )|_S$ is pseudo-effective, and the support of $\Delta$ has global normal crossings. Suppose first that $K_X+\Delta$ is $\mathbb{Q}$-Cartier. We may write

$$(K_X+S+A+B)|_S=K_S+C+D,$$

where $C$ is ample and $D\geq 0$. By induction we know that there is a positive integer $m$ such that $h^0(S,\mathcal{O}_{S}(m(K_S+C+D)))>0$. To lift sections, we need to know that $h^1(X,\mathcal{O}_{X}(m(K_X+S+A+B)-S))=0$. Now

$$\begin{align*} m(K_X+\Delta )-(K_X+B)-S&=(m-1)(K_X+\Delta )+A \\ &=(m-1)(K_X+\Delta +\frac{1}{m-1}A). \end{align*}$$

As $K_X+\Delta +A/(m-1)$ is big, we can construct a log terminal model $\phi \colon X \mathrel{\rightdasharrow }Y$ for $K_X+\Delta +A/(m-1)$, 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 $\mathbb{R}$-divisor. The argument is now a little more delicate as $h^0(S,\mathcal{O}_{S}(m(K_S+C+D)))$ 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 practice 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 $K_X+\Delta =K_X+A+B$, where the coefficients of $B$ are variable. For ease of exposition, we assume that the supports of $A$ and $B$ have global normal crossings, so that $K_X+\Delta =K_X+A+\sum b_iB_i$ is log canonical if and only if $0\leq b_i\leq 1$ 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 $\mathbb{R}^k$. 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 is equal to one or zero and so we may even assume that $K_X+\Delta$ 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 $b\in [0,1]^k$. There are two cases. If $K_X+\Delta$ is not pseudo-effective, 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 $K_X+\Delta$ is pseudo-effective. By induction we know that $K_X+\Delta \sim _{\mathbb{R}} D\geq 0$. Then we know that there is a log terminal model $\phi \colon X \mathrel{\rightdasharrow }Y$. Replacing $(X,\Delta )$ by $(Y,\Gamma =\phi _*\Delta )$, we may assume that $K_X+\Delta$ is nef. By the base point free theorem, it is semiample. Let $X \longrightarrow Z$ be the corresponding morphism. The key observation is that locally about $\Delta$, any log terminal model over $Z$ is an absolute log terminal model. Working over $Z$, we may assume that $K_X+\Delta$ is numerically trivial. In this case the problem of finding a log terminal model for $K_X+\Delta '$ only depends on the line segment spanned by $\Delta$ and $\Delta '$. Working in a small box about $\Delta$, 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 practice, 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 $(X,\Delta )$, where $\Delta$ is big. The first observation is that since the Kawamata log terminal condition is open, it is straightforward to show that $\Delta$ is $\mathbb{Q}$-linearly equivalent to $A+B$, where $A$ is an ample $\mathbb{Q}$-divisor, $B\geq 0$ and $K_X+A+B$ is Kawamata log terminal. 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 $(X,\Delta )$ is divisorially log terminal and $f\colon X \longrightarrow Y$ is a $(K_X+\Delta )$-trivial contraction, then $K_Y+\Gamma =K_Y+f_*\Delta$ is not necessarily divisorially log terminal, only log canonical. For example, suppose that $Y$ is a surface with a simple elliptic singularity and $f\colon X \longrightarrow Y$ is the blowup 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 $\Delta =A+B$, where $A$ is ample, then $K_Y+\Gamma$ is always divisorially log terminal.

2.2. Standard conjectures of the MMP

Having sketched the proof of Theorem 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 $\kappa (X,K_X)\geq 0$. In other words we need:

Conjecture 2.1.

Let $(X,\Delta )$ be a projective Kawamata log terminal pair.

If $K_X+\Delta$ is pseudo-effective, then $\kappa (X,K_X+\Delta )\geq 0$.

We also probably need

Conjecture 2.2.

Let $(X,\Delta )$ be a projective Kawamata log terminal pair.

If $K_X+\Delta$ is pseudo-effective and

$$h^0(X,\mathcal{O}_{X}(\llcorner m(K_X+\Delta )\lrcorner +H))$$

is not a bounded function of $m$, for some ample divisor $H$, then $\kappa (X,K_X+\Delta )\geq 1$.

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

Conjecture 2.3 (Abundance).

Let $(X,\Delta )$ be a projective Kawamata log terminal pair.

If $K_X+\Delta$ is nef, then it is semiample.

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

Conjecture 2.4.

Let $(X,\Delta )$ be a projective log canonical pair of dimension $n$.

If $K_X+\Delta$ is big, then $(X,\Delta )$ 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 $\epsilon >0$.

Then the set of varieties $X$ such that $K_X+\Delta$ has log discrepancy at least $\epsilon$ and $-(K_X+\Delta )$ 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 $\mathbb{C}$. We say that two $\mathbb{Q}$-divisors $D_1$, $D_2$ are $\mathbb{Q}$-linearly equivalent ($D_1\sim _{\mathbb{Q}} D_2$) if there exists an integer $m>0$ such that $mD_i$ are linearly equivalent. We say that a $\mathbb{Q}$-divisor $D$ is $\mathbb{Q}$-Cartier if some integral multiple is Cartier. We say that $X$ is $\mathbb{Q}$-factorial if every Weil divisor is $\mathbb{Q}$-Cartier. We say that $X$ is analytically $\mathbb{Q}$-factorial if every analytic Weil divisor (that is, an analytic subset of codimension one) is analytically $\mathbb{Q}$-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 $\pi \colon X \longrightarrow U$ be a proper morphism of normal algebraic spaces.

(1)

An $\mathbb{R}$-Weil divisor (frequently abbreviated to $\mathbb{R}$-divisor) $D$ on $X$ is an $\mathbb{R}$-linear combination of prime divisors.

(2)

An $\mathbb{R}$-Cartier divisor $D$ is an $\mathbb{R}$-linear combination of Cartier divisors.

(3)

Two $\mathbb{R}$-divisors $D$ and $D'$ are $\mathbb{R}$-linearly equivalent over $U$, denoted $D\sim _{\mathbb{R},U} D'$, if their difference is an $\mathbb{R}$-linear combination of principal divisors and an $\mathbb{R}$-Cartier divisor pulled back from $U$.

(4)

Two $\mathbb{R}$-divisors $D$ and $D'$ are numerically equivalent over $U$, denoted $D\equiv _{U} D'$, if their difference is an $\mathbb{R}$-Cartier divisor such that $(D-D')\cdot C=0$ for any curve $C$ contained in a fibre of $\pi$.

(5)

An $\mathbb{R}$-Cartier divisor $D$ is ample over $U$ (or $\pi$-ample) if it is $\mathbb{R}$-linearly equivalent to a positive linear combination of ample (in the usual sense) Cartier divisors over $U$.

(6)

An $\mathbb{R}$-Cartier divisor $D$ on $X$ is nef over $U$ (or $\pi$-nef) if $D\cdot C\geq 0$ for any curve $C\subset X$, contracted by $\pi$.

(7)

An $\mathbb{R}$-divisor $D$ is big over $U$ (or $\pi$-big) if$$\limsup \frac{h^0(F,\mathcal{O}_{F}(\llcorner mD\lrcorner ))}{m^{\operatorname {dim}F}}>0,$$

for the fibre $F$ over any generic point of $U$. Equivalently $D$ is big over $U$ if $D\sim _{\mathbb{R},U}A+B$, where $A$ is ample over $U$ and $B\geq 0$ (cf. 28, II 3.16).

(8)

An $\mathbb{R}$-Cartier divisor $D$ is semiample over $U$ (or $\pi$-semiample) if there is a morphism $f\colon X \longrightarrow Y$ over $U$ such that $D$ is $\mathbb{R}$-linearly equivalent to the pullback of an ample $\mathbb{R}$-divisor over $U$.

(9)

An $\mathbb{R}$-divisor $D$ is $\pi$-pseudo-effective if the restriction of $D$ to the fibre over each generic point of every component of $U$ is the limit of divisors $D_i\geq 0$.

Note that the group of Weil divisors with rational coefficients $\operatorname {WDiv}_{\mathbb{Q}}(X)$, or with real coefficients $\operatorname {WDiv}_{\mathbb{R}}(X)$, forms a vector space, with a canonical basis given by the prime divisors. Given an $\mathbb{R}$-divisor, $\|D\|$ denotes the sup norm with respect to this basis. If $A=\sum a_iC_i$ and $B=\sum b_iC_i$ are two $\mathbb{R}$-divisors, then

$$A \wedge B= \sum \min (a_i,b_i) C_i.$$

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

A log pair $(X,\Delta )$ (sometimes abbreviated by $K_X+\Delta$) is a normal variety $X$ and an $\mathbb{R}$-divisor $\Delta \geq 0$ such that $K_X+\Delta$ is $\mathbb{R}$-Cartier. We say that a log pair $(X,\Delta )$ is log smooth if $X$ is smooth and the support of $\Delta$ is a divisor with global normal crossings. A birational morphism $g\colon Y \longrightarrow X$ is a log resolution of the pair $(X,\Delta )$ if $g$ is projective, $Y$ is smooth, the exceptional locus is a divisor and $g^{-1}(\Delta )$ 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

$$K_Y+\Gamma =K_Y+\sum b_i\Gamma _i=g^*(K_X+\Delta ),$$

where $\Gamma _i$ are distinct prime divisors, then the log discrepancy $a(\Gamma _i,X,\Delta )$ of $\Gamma _i$ is $1-b_i$. The log discrepancy of $(X,\Delta )$ is then the infimum of the log discrepancy for every $\Gamma _i$ and for every resolution. The image of any component of $\Gamma$ of coefficient at least one (equivalently log discrepancy at most zero) is a non-Kawamata log terminal centre of the pair $(X,\Delta )$. The pair $(X,\Delta )$ is Kawamata log terminal if for every (equivalently for one) log resolution $g\colon Y \longrightarrow X$ as above, the coefficients of $\Gamma$ are strictly less than one, that is, $b_i<1$ for all $i$. Equivalently, the pair $(X,\Delta )$ is Kawamata log terminal if there are no non-Kawamata log terminal centres. The non-Kawamata log terminal locus of $(X,\Delta )$ is the union of the non-Kawamata log terminal centres. We say that the pair $(X,\Delta )$ is purely log terminal if the log discrepancy of any exceptional divisor is greater than zero. We say that the pair $(X,\Delta =\sum \delta _i\Delta _i)$, where $\delta _i\in (0,1]$, 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), $(X,\Delta )$ is divisorially log terminal if and only if there is a closed subset $Z\subset X$ such that

•

$(X\backslash Z,\Delta |_{X\backslash Z})$ is log smooth, and

•

if $f\colon Y \longrightarrow X$ is a projective birational morphism and $E\subset Y$ is an irreducible divisor with centre contained in $Z$, then $a(E,X,\Delta )>0$.

We will also often write

$$K_Y+\Gamma =g^*(K_X+\Delta )+E,$$

where $\Gamma \geq 0$ and $E\geq 0$ have no common components, $g_*\Gamma =\Delta$ and $E$ is $g$-exceptional. Note that this decomposition is unique.

We say that a birational map $\phi \colon X \mathrel{\rightdasharrow }Y$ is a birational contraction if $\phi$ is proper and $\phi ^{-1}$ does not contract any divisors. If in addition $\phi ^{-1}$ is also a birational contraction, we say that $\phi$ is a small birational map.

3.2. Preliminaries

Lemma 3.2.1.

Let $\pi \colon X \longrightarrow U$ be a projective morphism of normal quasi-projective varieties. Let $D$ be an $\mathbb{R}$-Cartier divisor on $X$ and let $D'$ be its restriction to the generic fibre of $\pi$.

If $D'\sim _{\mathbb{R}}B'\geq 0$ for some $\mathbb{R}$-divisor $B'$ on the generic fibre of $\pi$, then there is a divisor $B$ on $X$ such that $D\sim _{\mathbb{R},U} B\geq 0$ whose restriction to the generic fibre of $\pi$ is $B'$.

Proof.

Taking the closure of the generic points of $B'$, we may assume that there is an $\mathbb{R}$-divisor $B_1\geq 0$ on $X$ such that the restriction of $B_1$ to the generic fibre is $B'$. As

$$D'-B'\sim _{\mathbb{R}} 0,$$

it follows that there is an open subset $U_1$ of $U$, such that

$$(D-B_1)|_{V_1}\sim _{\mathbb{R}} 0,$$

where $V_1$ is the inverse image of $U_1$. But then there is a divisor $G$ on $X$ such that

$$D-B_1\sim _{\mathbb{R}} G,$$

where $Z=\pi (\operatorname {Supp}G)$ is a proper closed subset. As $U$ is quasi-projective, there is an ample divisor $H\geq 0$ on $U$ which contains $Z$. Possibly rescaling, we may assume that $F=\pi ^* H\geq -G$. But then

$$D \sim _{\mathbb{R}} (B_1+F+G)-F,$$

so that

$$\begin{equation*} D \sim _{\mathbb{R},U} (B_1+F+G) \geq 0. \end{equation*}$$■

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 $\mathbb{R}$-divisor and let $C$ be a prime divisor. Let

$$\sigma _C(B)=\inf \{\, \operatorname {mult}_C(B') \,|\, B'\sim _\mathbb{Q}B, B'\geq 0\,\}.$$

Then $\sigma _C$ is a continuous function on the cone of big divisors.

Now let $D$ be any pseudo-effective $\mathbb{R}$-divisor and let $A$ be any ample $\mathbb{Q}$-divisor. Let

$$\sigma _C(D)=\lim _{\epsilon \to 0} \sigma _C(D+\epsilon A).$$

Then $\sigma _C(D)$ exists and is independent of the choice of $A$.

There are only finitely many prime divisors $C$ such that $\sigma _{C}(D)>0$ and the $\mathbb{R}$-divisor $N_\sigma (D)=\sum _C\sigma _C(D)C$ is determined by the numerical equivalence class of $D$. Moreover $D-N_\sigma (D)$ is pseudo-effective and $N_\sigma (D-N_\sigma (D))=0$.

Proof.

See §III.1 of 28.

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Proposition 3.3.2.

Let $X$ be a smooth projective variety and let $D$ be a pseudo-effective $\mathbb{R}$-divisor. Let $B$ be any big $\mathbb{R}$-divisor.

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

$$h^0(X,\mathcal{O}_{X}(\llcorner mD\lrcorner +\llcorner kB\lrcorner ))> \beta m, \qquad \text{for all} \ m\gg 0.$$

Proof.

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

$$h^0(X,\mathcal{O}_{X} (\llcorner kB\lrcorner -A))> 0.$$

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

$$h^0(X,\mathcal{O}_{X}(\llcorner mD\lrcorner +A)) > \beta m \qquad \text{for all} \ m\gg 0.$$

Replacing $D$ by $D-N_{\sigma }(D)$, we may assume that $N_{\sigma }(D)=0$. Now apply (V.1.11) of 28.

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3.4. Adjunction

We recall some basic facts about adjunction; see 21, §16, §17 for more details.

Definition-Lemma 3.4.1.

Let $(X,\Delta )$ be a log canonical pair, and let $S$ be a normal component of $\llcorner \Delta \lrcorner$ of coefficient one. Then there is a divisor $\Theta$ on $S$ such that

$$(K_X+\Delta )|_S=K_S+\Theta .$$

(1)

If $(X,\Delta )$ is divisorially log terminal, then so is $K_S+\Theta$.

(2)

If $(X,\Delta )$ is purely log terminal, then $K_S+\Theta$ is Kawamata log terminal.

(3)

If $(X,\Delta =S)$ is purely log terminal, then the coefficients of $\Theta$ 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 $\text{Weil}(\mathcal{O}_{X,P})$). 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 $(X,\Delta )$ is purely log terminal, $f\colon Y \longrightarrow X$ is a projective birational morphism and $T$ is the strict transform of $S$, then $(f|_T)_*\Psi =\Theta$, where $K_Y+\Gamma =f^*(K_X+\Delta )$ and $\Psi$ is defined by adjunction,$$(K_Y+\Gamma )|_T=K_T+\Psi .$$

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 $\pi \colon X \longrightarrow U$ be a projective morphism of normal varieties.

Let $D$ be an $\mathbb{R}$-divisor on $X$. The real linear system associated to $D$ over $U$ is

$$|D/U|_{\mathbb{R}}=\{\, C\geq 0\,|\, C\sim _{\mathbb{R},U} D \,\}.$$

The stable base locus of $D$ over $U$ is the Zariski closed set $\mathbf{B}(D/U)$ given by the intersection of the support of the elements of the real linear system $|D/U|_{\mathbb{R}}$. If $|D/U|_{\mathbb{R}}=\emptyset$, then we let $\mathbf{B}(D/U)=X$. 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

$$\mathbf{B}_{+}(D/U)=\mathbf{B}((D-\epsilon A)/U),$$

for any ample divisor $A$ over $U$ and any sufficiently small rational number $\epsilon >0$ (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 $\pi \colon X \longrightarrow U$ 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 Definition 3.5.1 coincides with the usual definition of the stable base locus.

Proof.

Let

$$|D/U|_{\mathbb{Q}}=\{\, C\geq 0\,|\, C\sim _{\mathbb{Q},U} D \,\}.$$

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

Suppose that $x\notin R$. We want to show that $x\notin Q$. We may find $D'\in |D/U|_{\mathbb{R}}$ such that $\operatorname {mult}_xD'=0$. But then

$$D'=D+\sum r_i(f_i)+\pi ^*E,$$

where $f_i$ are rational functions on $X$, $E$ is an $\mathbb{R}$-Cartier divisor on $U$, and $r_i$ are real numbers. Let $V$ be the subspace of $\operatorname {WDiv}_{\mathbb{R}}(X)$ spanned by the components of $D$, $D'$, $\pi ^*E$ and $(f_i)$. We may write $E=\sum e_jE_j$, where $E_i$ are Cartier divisors. Let $W$ be the span of the $(f_i)$ and the $\pi ^*E_j$. Then $W\subset V$ are defined over the rationals. Set

$$\mathcal{P}=\{\, D''\in V \,|\, D''\geq 0,\ \operatorname {mult}_xD''=0, \ D''-D\in W\,\}\subset |D/U|_\mathbb{R}.$$

Then $\mathcal{P}$ is a rational polyhedron. As $D'\in \mathcal{P}$, $\mathcal{P}$ is non-empty, and so it must contain a rational point $D''$. We may write

$$D''=D+\sum s_i(f_i)+\sum f_j\pi ^*E_j,$$

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 $D''\in |D/U|_{\mathbb{Q}}$, and so $x\notin Q$.

■

Proposition 3.5.4.

Let $\pi \colon X \longrightarrow U$ be a projective morphism of normal varieties and let $D\geq 0$ be an $\mathbb{R}$-divisor. Then we may find $\mathbb{R}$-divisors $M$ and $F$ such that

(1)

$M\geq 0$ and $F\geq 0$,

(2)

$D\sim _{\mathbb{R},U}M+F$,

(3)

every component of $F$ is a component of $\mathbf{B}(D/U)$, 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 $\mathbb{R}$-divisors such that $D\sim _{\mathbb{R}} D'$.

Then we may find rational functions ${f}_1,{f}_2,\dots ,{f}_{k}$ and real numbers ${r}_1,{r}_2,\dots ,{r}_{k}$ which are independent over the rationals such that

$$D=D'+\sum _i r_i(f_i).$$

In particular every component of $(f_i)$ is either a component of $D$ or $D'$.

Proof.

By assumption we may find rational functions ${f}_1,{f}_2,\dots ,{f}_{k}$ and real numbers ${r}_1,{r}_2,\dots ,{r}_{k}$ such that

$$D=D'+\sum _{i=1}^k r_i(f_i).$$

Pick $k$ minimal with this property. Suppose that the real numbers $r_i$ are not independent over $\mathbb{Q}$. Then we can find rational numbers $d_i$, not all zero, such that

$$\sum _i d_ir_i=0.$$

Possibly reordering we may assume that $d_k\neq 0$. Multiplying through by an integer we may assume that $d_i\in \mathbb{Z}$. Possibly replacing $f_i$ by $f_i^{-1}$, we may assume that $d_i\geq 0$. Let $d$ be the least common multiple of the non-zero $d_i$. If $d_{i}\neq 0$, 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 $1\leq i < k$, set

$$g_i=\begin{cases} f_i/f_k & \text{if $d_i=1$}, \\ f_i & \text{if $d_i=0$.} \end{cases}$$

Then

$$D=D'+\sum _{i=1}^{k-1} r_i(g_i),$$

which contradicts our choice of $k$.

Now suppose that $B$ is a component of $(f_i)$. Then

$$\operatorname {mult}_B(D)=\operatorname {mult}_B(D')+\sum r_j n_j,$$

where $n_j=\operatorname {mult}_B(f_j)$ is an integer and $n_i\neq 0$. But then $\operatorname {mult}_B(D)-\operatorname {mult}_B(D')\neq 0$, so that one of $\operatorname {mult}_B(D)$ and $\operatorname {mult}_B(D')$ must be non-zero.

■

Lemma 3.5.6.

Let $\pi \colon X \longrightarrow U$ be a projective morphism of normal varieties and let

$$D' \sim _{\mathbb{R},U} D, \qquad D\geq 0, \qquad D'\geq 0,$$

be two $\mathbb{R}$-divisors on $X$ with no common components.

Then we may find $D''\in |D/U|_{\mathbb{R}}$ such that a multiple of every component of $D''$ is mobile.

Proof.

Pick ample $\mathbb{R}$-divisors on $U$, $H$ and $H'$ such that $D+\pi ^*H\sim _{\mathbb{R}} D'+\pi ^*H'$ and $D+\pi ^*H$ and $D'+\pi ^*H'$ have no common components. Replacing $D$ by $D+\pi ^*H$ and $D'$ by $D'+\pi ^*H'$, we may assume that $D' \sim _{\mathbb{R}} D$.

We may write

$$D'=D+\sum r_i(f_i)=D+R,$$

where $r_i\in \mathbb{R}$ and $f_i$ are rational functions on $X$. By Lemma 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 ${q}_1,{q}_2,\dots ,{q}_{k}$ are any rational numbers, then we may always write

$$C'=C+Q=C+\sum q_i(f_i),$$

where $C\geq 0$ and $C'\geq 0$ 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 $mC\sim mC'$ for some integer $m>0$. By Bertini we may find $C''\sim _{\mathbb{Q}}C$ such that every component of $C''$ has a multiple which is mobile. Pick $\lambda >0$ maximal such that $D_1=D-\lambda C\geq 0$ and $D_1'=D'-\lambda C'\geq 0$. Note that

$$D_1\sim _{\mathbb{R}} D_1', \qquad D_1\geq 0, \qquad D_1'\geq 0,$$

are two $\mathbb{R}$-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

$$D_1''\in |D_1|_{\mathbb{R}},$$

such that a multiple of every component of $D_1''$ is mobile. But then

$$D''=\lambda C''+D_1''\in |D|_{\mathbb{R}},$$

and every component of $D''$ has a multiple which is mobile.

■

Proof of Proposition 3.5.4.

We may write $D=M+F$, where every component of $F$ is contained in $\mathbf{B}(D/U)$ and no component of $M$ is contained in $\mathbf{B}(D/U)$. A prime divisor is bad if none of its multiples is 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 $D_1\in |D/U|_{\mathbb{R}}$ such that $B$ is not a component of $D_1$. If $E=D\wedge D_1$, then $D'=D-E\geq 0$ and $D'_1=D_1-E\geq 0$, $D'$ and $D'_1$ have no common components and $D' \sim _{\mathbb{R},U} D'_1$. By Lemma 3.5.6 there is a divisor $D''\in |D'/U|_{\mathbb{R}}$ with no bad components. But then $D''+E\in |D|_{\mathbb{R}}$, $B$ 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.

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3.6. Types of models

Definition 3.6.1.

Let $\phi \colon X \mathrel{\rightdasharrow }Y$ be a proper birational contraction of normal quasi-projective varieties and let $D$ be an $\mathbb{R}$-Cartier divisor on $X$ such that $D'=\phi _*D$ is also $\mathbb{R}$-Cartier. We say that $\phi$ is $D$-non-positive (respectively $D$-negative) if for some common resolution $p\colon W \longrightarrow X$ and $q\colon W \longrightarrow Y$, we may write

$$p^*D=q^*D'+E,$$

where $E\geq 0$ is $q$-exceptional (respectively $E\geq 0$ is $q$-exceptional and the support of $E$ contains the strict transform of the $\phi$-exceptional divisors).

We will often use the following well-known lemma.

Lemma 3.6.2 (Negativity of contraction).

Let $\pi \colon Y \longrightarrow X$ be a projective birational morphism of normal quasi-projective varieties.

(1)

If $E>0$ is an exceptional $\mathbb{R}$-Cartier divisor, then there is a component $F$ of $E$ which is covered by curves $\Sigma$ such that $E\cdot \Sigma <0$.

(2)

If $\pi ^*L\equiv M+G+E$, where $L$ is an $\mathbb{R}$-Cartier divisor on $X$, $M$ is a $\pi$-nef $\mathbb{R}$-Cartier divisor on $Y$, $G\geq 0$, $E$ is $\pi$-exceptional, and $G$ and $E$ have no common components, then $E\geq 0$. 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 $\pi$-trivial, then $F_1$ is a component of $G+E$.

(3)

If $X$ is $\mathbb{Q}$-factorial, then there is a $\pi$-exceptional divisor $E\geq 0$ such that $-E$ is ample over $X$. In particular the exceptional locus of $\pi$ 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 $\mathbb{Q}$-divisor over $X$. If $X$ is $\mathbb{Q}$-factorial, then

$$E=\pi ^*\pi _*H-H\geq 0$$

is $\pi$-exceptional and $-E$ is ample over $X$. This is (3).

■

Lemma 3.6.3.

Let $X \longrightarrow U$ and $Y \longrightarrow U$ be two projective morphisms of normal quasi-projective varieties. Let $\phi \colon X \mathrel{\rightdasharrow }Y$ be a birational contraction over $U$ and let $D$ and $D'$ be $\mathbb{R}$-Cartier divisors such that $D'=\phi _*D$ is nef over $U$.

Then $\phi$ is $D$-non-positive (respectively $D$-negative) if given a common resolution $p\colon W \longrightarrow X$ and $q\colon W \longrightarrow Y$, we may write

$$p^*D=q^*D'+E,$$

where $p_*E\geq 0$ (respectively $p_*E\geq 0$ and the support of $p_*E$ contains the union of all $\phi$-exceptional divisors).

Further if $D=K_X+\Delta$ and $D'=K_Y+\phi _*\Delta$, then this is equivalent to requiring

$$\begin{equation*} a(F,X,\Delta )\leq a(F,Y,\phi _*\Delta ) \qquad \text{(respectively $a(F,X,\Delta )< a(F,Y,\phi _*\Delta )$),} \end{equation*}$$

for all $\phi$-exceptional divisors $F\subset X$.

Proof.

This is an easy consequence of Lemma 3.6.2.

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Lemma 3.6.4.

Let $X \longrightarrow U$ and $Y \longrightarrow U$ be two projective morphisms of normal quasi-projective varieties. Let $\phi \colon X \mathrel{\rightdasharrow }Y$ be a birational contraction over $U$ and let $D$ and $D'$ be $\mathbb{R}$-Cartier divisors such that $\phi _*D$ and $\phi _*D'$ are $\mathbb{R}$-Cartier. Let $p\colon W \longrightarrow X$ and $q\colon W \longrightarrow Y$ be common resolutions.

If $D$ and $D'$ are numerically equivalent over $U$, then

$$p^*D-q^*\phi _*D=p^*D'-q^*\phi _*D'.$$

In particular $\phi$ is $D$-non-positive (respectively $D$-negative) if and only if $\phi$ is $D'$-non-positive (respectively $D'$-negative).

Proof.

Since

$$p^*(D-D')-q^*\phi _*(D-D')$$

is $q$-exceptional and numerically trivial over $Y$, this follows easily from Lemma 3.6.2.

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Definition 3.6.5.

Let $\pi \colon X \longrightarrow U$ be a projective morphism of normal quasi-projective varieties and let $D$ be an $\mathbb{R}$-Cartier divisor on $X$.

We say that a birational contraction $f\colon X \mathrel{\rightdasharrow }Y$ 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 $g\colon X \mathrel{\rightdasharrow }Z$ 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 $p\colon W \longrightarrow X$ and $q\colon W \longrightarrow Z$ resolve $g$, then $q$ is a contraction morphism and we may write $p^*D \sim _{\mathbb{R},U} q^*H+E$, where $E\geq 0$ and for every $B\in |p^*D/U|_{\mathbb{R}}$, then $B\geq E$.

Lemma 3.6.6.

Let $\pi \colon X \longrightarrow U$ be a projective morphism of normal quasi-projective varieties and let $D$ be an $\mathbb{R}$-Cartier divisor on $X$.

(1)

If $g_i\colon X \mathrel{\rightdasharrow }X_i$, $i=1$, $2$, are two ample models of $D$ over $U$, then there is an isomorphism $\chi \colon X_1 \longrightarrow X_2$ such that $g_2=\chi \circ g_1$.

(2)

Suppose that $g\colon X \mathrel{\rightdasharrow }Z$ is the ample model of $D$ over $U$ and let $H$ be the corresponding ample divisor on $Z$. If $p\colon W \longrightarrow X$ and $q\colon W \longrightarrow Z$ resolve $g$, then we may write$$p^*D \sim _{\mathbb{R},U} q^*H+E,$$

where $E\geq 0$ 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 $f\colon X \mathrel{\rightdasharrow }Y$ is a semiample model of $D$ over $U$, then the ample model $g\colon X \mathrel{\rightdasharrow }Z$ of $D$ over $U$ exists and $g=h\circ f$, where $h\colon Y \longrightarrow Z$ is a contraction morphism and $f_*D \sim _{\mathbb{R},U} h^*H$. If $B$ is a prime divisor contained in the stable fixed divisor of $D$ over $U$, then $B$ is contracted by $f$.

(4)

If $f\colon X \mathrel{\rightdasharrow }Y$ 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 $g\colon Y \longrightarrow X$ resolve the indeterminacy of $g_i$ and let $f_i=g_i\circ g\colon Y \longrightarrow X_i$ be the induced contraction morphisms. By assumption $g^*D \sim _{\mathbb{R},U} f_i^*H_i+E_i$, 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, $E_1\geq E_2$. By symmetry $E_1=E_2$ and so $f_1^*H_1 \sim _{\mathbb{R},U} f_2^*H_2$. But then $f_1$ and $f_2$ contract the same curves. This is (1).

Suppose that $g\colon X \mathrel{\rightdasharrow }Z$ is the ample model of $D$ over $U$. By assumption this means that we may write

$$p^*D \sim _{\mathbb{R},U} q^*H+E,$$

where $E\geq 0$. 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 $\operatorname {dim}q(F')>0$. But then $q^*H$ is not numerically trivial on $F'$ and we may apply Lemma 3.6.2. This is (2).

Now suppose that $f\colon X \mathrel{\rightdasharrow }Y$ is a semiample model of $D$ over $U$. As $f_*D$ is semiample over $U$, there is a contraction morphism $h\colon Y \longrightarrow Z$ over $U$ and an ample divisor $H$ over $U$ on $Z$ such that $f_*D \sim _{\mathbb{R},U} h^*H$. If $p\colon W \longrightarrow X$ and $q\colon W \longrightarrow Y$ resolve the indeterminacy of $f$, then $p^*D \sim _{\mathbb{R},U} r^*H+E$, where $E\geq 0$ is $q$-exceptional and $r=h\circ q\colon W \longrightarrow Z$. If $B\in |p^*D/U|_{\mathbb{R}}$, then $B\geq E$. But then $g=h\circ f\colon X \mathrel{\rightdasharrow }Z$ is the ample model of $D$ over $U$. This is (3).

Now suppose that $f\colon X \mathrel{\rightdasharrow }Y$ is birational over $U$. If $f$ is a semiample model of $D$ over $U$, then (3) implies that the ample model $g\colon X \mathrel{\rightdasharrow }Z$ of $D$ over $U$ exists and there is a contraction morphism $h\colon Y \longrightarrow Z$, such that $f_*D \sim _{\mathbb{R},U} h^*H$, 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 $p\colon W \longrightarrow X$ and $q\colon W \longrightarrow Y$ are projective birational morphisms which resolve $f$. By assumption we may write $p^*D \sim _{\mathbb{R},U} q^*H+E$, where $H$ is ample over $U$. We may assume that there is a $q$-exceptional $\mathbb{Q}$-divisor $F\geq 0$ such that $q^*H-F$ is ample over $U$. Then there is a constant $\delta >0$ such that $q^*H-F+\delta E$ is ample over $U$. Suppose $B$ is a component of $E$. As $B$ does not belong to the stable base locus of $q^*H-F+\delta E$ 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

$$0=C\cdot p^*D=C\cdot q^{*}H+ C\cdot E,$$

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).

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Definition 3.6.7.

Let $\pi \colon X \longrightarrow U$ be a projective morphism of normal quasi-projective varieties. Suppose that $K_X+\Delta$ is log canonical and let $\phi \colon X \mathrel{\rightdasharrow }Y$ be a birational contraction of normal quasi-projective varieties over $U$, where $Y$ is projective over $U$. Set $\Gamma =\phi _*\Delta$.

•

$Y$ is a weak log canonical model for $K_X+\Delta$ over $U$ if $\phi$ is $(K_X+\Delta )$-non-positive and $K_Y+\Gamma$ is nef over $U$.

•

$Y$ is the log canonical model for $K_X+\Delta$ over $U$ if $\phi$ is the ample model of $K_X+\Delta$ over $U$.

•

$Y$ is a log terminal model for $K_X+\Delta$ over $U$ if $\phi$ is $(K_X+\Delta )$-negative, $K_Y+\Gamma$ is divisorially log terminal and nef over $U$, and $Y$ is $\mathbb{Q}$-factorial.

Remark 3.6.8.

Note that there is no consensus on the definitions given in Definition 3.6.7.

Lemma 3.6.9.

Let $\pi \colon X \longrightarrow U$ be a projective morphism of normal quasi-projective varieties. Let $\phi \colon X \mathrel{\rightdasharrow }Y$ be a birational contraction over $U$. Let $(X,\Delta )$ and $(X,\Delta ')$ be two log pairs and set $\Gamma =\phi _*\Delta$, $\Gamma '=\phi _*\Delta '$. Let $\mu >0$ be a positive real number.

•

If both $K_X+\Delta$ and $K_X+\Delta '$ are log canonical and $K_X+\Delta ' \sim _{\mathbb{R} ,U} \mu (K_X+\Delta )$, then $\phi$ is a weak log canonical model for $K_X+\Delta$ over $U$ if and only if $\phi$ is a weak log canonical model for $K_X+\Delta '$ over $U$.

•

If both $K_X+\Delta$ and $K_X+\Delta '$ are Kawamata log terminal and $K_X+\Delta \equiv _U \mu (K_X+\Delta ')$, then $\phi$ is a log terminal model for $K_X+\Delta$ over $U$ if and only if $\phi$ is a log terminal model for $K_X+\Delta '$ over $U$.

Proof.

Note first that either $K_Y+\Gamma ' \sim _{\mathbb{R},U} \mu (K_Y+\Gamma )$ or $Y$ is $\mathbb{Q}$-factorial. In particular $K_Y+\Gamma$ is $\mathbb{R}$-Cartier if and only if $K_Y+\Gamma '$ is $\mathbb{R}$-Cartier. Therefore Lemma 3.6.4 implies that $\phi$ is $(K_X+\Delta )$-non-positive (respectively $(K_X+\Delta )$-negative) if and only if $\phi$ is $(K_X+\Delta ')$-non-positive (respectively $(K_X+\Delta ')$-negative).

Since $K_X+\Delta '\equiv _U \mu (K_X+\Delta )$ and $\mu E-E'\equiv _Y 0$, it follows that $K_Y+\Gamma '\equiv _U \mu (K_Y+\Gamma )$, so that $K_Y+\Gamma$ is nef over $U$ if and only if $K_Y+\Gamma '$ is nef over $U$.

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Lemma 3.6.10.

Let $\pi \colon X \longrightarrow U$ be a projective morphism of normal quasi-projective varieties. Let $\phi \colon X \mathrel{\rightdasharrow }Y$ be a birational contraction over $U$, where $Y$ is projective over $U$. Suppose that $K_X+\Delta$ and $K_Y+\phi _*\Delta$ are divisorially log terminal and $a(F,X,\Delta )<a(F,Y,\phi _*\Delta )$ for all $\phi$-exceptional divisors $F\subset X$.

If $\varphi \colon Y \mathrel{\rightdasharrow }Z$ is a log terminal model of $(Y,\phi _*\Delta )$ over $U$, then $\eta =\varphi \circ \phi \colon X \mathrel{\rightdasharrow }Z$ is a log terminal model of $K_X+\Delta$ over $U$.

Proof.

Clearly $\eta$ is a birational contraction, $Z$ is $\mathbb{Q}$-factorial and $K_Z+\eta _*\Delta$ is divisorially log terminal and nef over $U$.

Let $p\colon W \longrightarrow X$, $q\colon W \longrightarrow Y$ and $r\colon W \longrightarrow Z$ be a common resolution. As $\varphi$ is a log terminal model of $(Y,\phi _*\Delta )$ we have that $q^*(K_Y+\phi _*\Delta )-r^*(K_Z+\eta _*\Delta )=E\geq 0$ and the support of $E$ contains the exceptional divisors of $\varphi$. By assumption $K_X+\Delta -p_*q^*(K_Y+\phi _*\Delta )$ is an effective divisor whose support is the set of all $\phi$-exceptional divisors. But then

$$(K_X+\Delta )-p_*r^*(K_Z+\eta _*\Delta )=K_X+\Delta -p_*q^*(K_Y+\phi _*\Delta )+p_*E$$

contains all the $\eta$-exceptional divisors and Lemma 3.6.3 implies that $\eta$ is a log terminal model of $K_X+\Delta$ over $U$.

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Lemma 3.6.11.

Let $\pi \colon X \longrightarrow U$ be a projective morphism of normal quasi-projective varieties. Let $(X,\Delta )$ be a Kawamata log terminal pair, where $\Delta$ is big over $U$. Let $f\colon Z \longrightarrow X$ be any log resolution of $(X,\Delta )$ and suppose that we write

$$K_Z+\Phi _0=f^*(K_X+\Delta )+E,$$

where $\Phi _0\geq 0$ and $E\geq 0$ have no common components, $f_*\Phi _0=\Delta$ and $E$ is exceptional. Let $F\geq 0$ be any divisor whose support is equal to the exceptional locus of $f$.

If $\eta >0$ is sufficiently small and $\Phi =\Phi _0+\eta F$, then $K_Z+\Phi$ is Kawamata log terminal and $\Phi$ is big over $U$. Moreover if $\phi \colon Z \mathrel{\rightdasharrow }W$ is a log terminal model of $K_Z+\Phi$ over $U$, then the induced birational map $\psi \colon X \mathrel{\rightdasharrow }W$ is in fact a log terminal model of $K_X+\Delta$ over $U$.

Proof.

Everything is clear but the last statement. Set $\Psi =\phi _*\Phi$. By Lemma 3.6.10, possibly blowing up more, we may assume that $\phi$ is a morphism. By assumption if we write

$$K_Z+\Phi =\phi ^*(K_W+\Psi )+G,$$

then $G\geq 0$ and the support of $G$ is the union of all the $\phi$-exceptional divisors. Thus

$$f^*(K_X+\Delta )+E+\eta F=\phi ^*(K_W+\Psi )+G.$$

By negativity of contraction, Lemma 3.6.2, applied to $f$, $G-E-\eta F\geq 0$. In particular $\phi$ must contract every $f$-exceptional divisor and so $\psi$ is a birational contraction. But then $\psi$ is a log terminal model over $U$ by Lemma 3.6.3.

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Lemma 3.6.12.

Let $\pi \colon X \longrightarrow U$ be a projective morphism of normal quasi-projective varieties, where $X$ is $\mathbb{Q}$-factorial Kawamata log terminal. Let $\phi \colon X \mathrel{\rightdasharrow }Z$ be a birational contraction over $U$ and let $S$ be a sum of prime divisors. Suppose that there is a $\mathbb{Q}$-factorial quasi-projective variety $Y$ together with a small birational projective morphism $f\colon Y \longrightarrow Z$.

If $V$ is any finite dimensional affine subspace of $\operatorname {WDiv}_{\mathbb{R}}(X)$ such that $\mathcal{L}_S(V)$ spans $\operatorname {WDiv}_{\mathbb{R}}(X)$ modulo numerical equivalence over $U$ and $\mathcal{W}_{\phi ,S,\pi }(V)$ intersects the interior of $\mathcal{L}_S(V)$, then

$$\mathcal{W}_{\phi ,S,\pi }(V)=\bar{\mathcal{A}}_{\phi ,S,\pi }(V).$$

Proof.

By (4) of Lemma 3.6.6 $\mathcal{W}_{\phi ,S,\pi }(V)\supset \mathcal{A}_{\phi ,S,\pi }(V)$. Since $\mathcal{W}_{\phi ,S,\pi }(V)$ is closed, it follows that

$$\mathcal{W}_{\phi ,S,\pi }(V)\supset \bar{\mathcal{A}}_{\phi ,S,\pi }(V).$$

To prove the reverse inclusion, it suffices to prove that a dense subset of $\mathcal{W}_{\phi ,S,\pi }(V)$ is contained in $\mathcal{A}_{\phi ,S,\pi }(V)$.

Pick $\Delta$ belonging to the interior of $\mathcal{W}_{\phi ,S,\pi }(V)$. If $\psi \colon X \mathrel{\rightdasharrow }Y$ is the induced birational contraction, then $\psi$ is a $\mathbb{Q}$-factorial weak log canonical model of $K_X+\Delta$ over $U$ and

$$K_Y+\Gamma =f^*(K_Z+\phi _*\Delta ),$$

where $\Gamma =\psi _*\Delta$. As $\mathcal{L}_S(V)$ spans $\operatorname {WDiv}_{\mathbb{R}}(X)$ modulo numerical equivalence over $U$, we may find $\Delta _0\in \mathcal{L}_S(V)$ such that $\Delta _0-\Delta$ is numerically equivalent over $U$ to $\mu \Delta$ for some $\mu >0$. Let

$$\Delta '=\Delta +\epsilon ((\Delta _0-\Delta )-\mu \Delta )=(1-\epsilon )\nu \Delta +\epsilon \Delta _0,$$

where

$$\nu =\frac{1-\epsilon -\epsilon \mu }{1-\epsilon }<1.$$

Then $\Delta '$ is numerically equivalent to $\Delta$ over $U$ and if $\epsilon >0$ is sufficiently small, then $\Delta '\geq 0$. As $(X,\nu \Delta )$ is Kawamata log terminal it follows that $(X,\Delta ')$ is Kawamata log terminal. In particular Lemma 3.6.9 implies that $\psi$ is a $\mathbb{Q}$-factorial weak log canonical model of $K_X+\Delta '$ over $U$ and so $K_Y+\Gamma '$ is Kawamata log terminal, where $\Gamma '=\psi _*\Delta '$. As $\Gamma$ and $\Gamma '$ are numerically equivalent over $U$, it follows that $K_Y+\Gamma '$ is numerically equivalent to zero over $Z$.

Let $H$ be a general ample $\mathbb{Q}$-divisor over $U$ on $Z$. Let $p\colon W \longrightarrow X$ and $q\colon W \longrightarrow Z$ resolve the indeterminacy locus of $\phi$ and let $H'=p_*q^*H$. It follows that $\phi$ is $H'$-non-positive. Pick $\Delta _1\in \mathcal{L}_S(V)$ such that $B=\Delta _1-\Delta$ is numerically equivalent over $U$ to $\eta H'$ for some $\eta >0$. Replacing $H$ by $\eta H$ we may assume that $\eta =1$. If $C=\psi _*B$, then $C$ is numerically equivalent to $f^*H$ over $U$. Then both $C$ and $C-(K_Y+\Gamma ')$ are numerically trivial over $Z$, so that $C-(K_Y+\Gamma ')$ is nef and big over $Z$ and Theorem 3.9.1 implies that $\phi _*B=f_*C$ is $\mathbb{R}$-Cartier. Lemma 3.6.4 implies that $\phi$ is $(K_X+\Delta +\lambda B)$-non-positive and $\phi _*(K_X+\Delta +\lambda B)$ is ample over $U$, for any $\lambda >0$. On the other hand, note that

$$\Delta +\lambda B=\Delta +\lambda (\Delta _1-\Delta ) \in \mathcal{L}_S(V),$$

for any $\lambda \in [0,1]$. Therefore $\phi$ is the ample model of $K_X+\Delta +\lambda B$ over $U$ for any $\lambda \in (0,1]$.

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3.7. Convex geometry and Diophantine approximation

Definition 3.7.1.

Let $V$ be a finite dimensional real affine space. If $\mathcal{C}$ is a convex subset of $V$ and $F$ is a convex subset of $\mathcal{C}$, then we say that $F$ is a face of $\mathcal{C}$ if whenever $\sum r_i v_i \in F$, where ${r}_1,{r}_2,\dots ,{r}_{k}$ are real numbers such that $\sum r_i=1$, $r_i\geq 0$ and ${v}_1,{v}_2,\dots ,{v}_{k}$ belong to $\mathcal{C}$, then $v_i\in F$ for some $i$. We say that $v\in \mathcal{C}$ is an extreme point if $F=\{v\}$ is a face of $\mathcal{C}$.

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

We say that a real vector space $V_0$ is defined over the rationals, if $V_0=V'\underset{\mathbb{Q}}{\otimes } \mathbb{R}$, 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 $\mathcal{P}$ 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 $\operatorname {WDiv}_{\mathbb{R}}(X)$, which is defined over the rationals.

Then $\mathcal{L}(V)$ (cf. Definition 1.1.4 for the definition) is a rational polytope.

Proof.

Note that the set of divisors $\Delta$ such that $K_X+\Delta$ is $\mathbb{R}$-Cartier forms an affine subspace $W$ of $V$, which is defined over the rationals, so that, replacing $V$ by $W$, we may assume that $K_X+\Delta$ is $\mathbb{R}$-Cartier for every $\Delta \in V$.

Let $\pi \colon Y \longrightarrow X$ be a resolution of $X$, which is a log resolution of the support of any element of $V$. Given any divisor $\Delta \in V$, if we write

$$K_Y+\Gamma =\pi ^*(K_X+\Delta ),$$

then the coefficients of $\Gamma$ are rational affine linear functions of the coefficients of $\Delta$. On the other hand, the condition that $K_X+\Delta$ is log canonical is equivalent to the condition that the coefficient of every component of $\Gamma$ is at most one and the coefficient of every component of $\Delta$ is at least zero.

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Lemma 3.7.3.

Let $\pi \colon X \longrightarrow U$ be a projective morphism of normal quasi-projective varieties. Let $V$ be a finite dimensional affine subspace of $\operatorname {WDiv}_{\mathbb{R}}(X)$ and let $A\geq 0$ be a big $\mathbb{R}$-divisor over $U$. Let $\mathcal{C}\subset \mathcal{L}_A(V)$ be a polytope.

If $\mathbf{B}_+(A/U)$ does not contain any non-Kawamata log terminal centres of $(X,\Delta )$, for every $\Delta \in \mathcal{C}$, then we may find a general ample $\mathbb{Q}$-divisor $A'$ over $U$, a finite dimensional affine subspace $V'$ of $\operatorname {WDiv}_{\mathbb{R}}(X)$ and a translation

$$L\colon \operatorname {WDiv}_{\mathbb{R}}(X) \longrightarrow \operatorname {WDiv}_{\mathbb{R}}(X),$$

by an $\mathbb{R}$-divisor $T$ $\mathbb{R}$-linearly equivalent to zero over $U$ such that $L(\mathcal{C})\subset \mathcal{L}_{A'}(V')$ and $(X,\Delta -A)$ and $(X,L(\Delta ))$ have the same non-Kawamata log terminal centres. Further, if $A$ is a $\mathbb{Q}$-divisor, then we may choose $T$ $\mathbb{Q}$-linearly equivalent to zero over $U$.

Proof.

Let ${\Delta }_1,{\Delta }_2,\dots ,{\Delta }_{k}$ be the vertices of the polytope $\mathcal{C}$. Let $\mathfrak{Z}$ be the set of non-Kawamata log terminal centres of $(X,\Delta _i)$ for $1\leq i \leq k$. Note that if $\Delta \in \mathcal{C}$, then any non-Kawamata log terminal centre of $(X,\Delta )$ is an element of $\mathfrak{Z}$.

By assumption, we may write $A \sim _{\mathbb{R},U}C+D$, where $C$ is a general ample $\mathbb{Q}$-divisor over $U$ and $D\geq 0$ does not contain any element of $\mathfrak{Z}$. Further Lemma 3.5.3 implies that if $A$ is a $\mathbb{Q}$-divisor, then we may assume that $A \sim _{\mathbb{Q},U} C+D$.

Given any rational number $\delta >0$, let

$$L\colon \operatorname {WDiv}_{\mathbb{R}}(X) \longrightarrow \operatorname {WDiv}_{\mathbb{R}}(X) \quad \text{given by} \quad L(\Delta )=\Delta +\delta (C+D-A)$$

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

$$K_X+L(\Delta _i)=K_X+\Delta _i+\delta (C+D-A)=K_X+\delta C+(\Delta _i-\delta A+\delta D)$$

is log canonical for every $1\leq i\leq k$ and has the same non-Kawamata log terminal centres as $(X,\Delta _i-A)$. But then $L(\mathcal{C})\subset \mathcal{L}_{A'}(V')$, where $A'=\delta C$ and $V'=V_{(1-\delta )A+\delta D}$ and $(X,\Delta -A)$ and $(X,L(\Delta ))$ have the same non-Kawamata log terminal centres.

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Lemma 3.7.4.

Let $\pi \colon X \longrightarrow U$ be a projective morphism of normal quasi-projective varieties. Let $V$ be a finite dimensional affine subspace of $\operatorname {WDiv}_{\mathbb{R}}(X)$, which is defined over the rationals, and let $A$ be a general ample $\mathbb{Q}$-divisor over $U$. Let $S$ be a sum of prime divisors. Suppose that there is a divisorially log terminal pair $(X,\Delta _0)$, where $S=\llcorner \Delta _0\lrcorner$, and let $G\geq 0$ be any divisor whose support does not contain any non-Kawamata log terminal centres of $(X,\Delta _0)$.

Then we may find a general ample $\mathbb{Q}$-divisor $A'$ over $U$, an affine subspace $V'$ of $\operatorname {WDiv}_{\mathbb{R}}(X)$, which is defined over the rationals, and a rational affine linear isomorphism

$$L\colon V_{S+A} \longrightarrow V'_{S+A'},$$

such that

•

$L$ preserves $\mathbb{Q}$-linear equivalence over $U$,

•

$L(\mathcal{L}_{S+A}(V))$ is contained in the interior of $\mathcal{L}_{S+A'}(V')$,

•

for any $\Delta \in L(\mathcal{L}_{S+A}(V))$, $K_X+\Delta$ is divisorially log terminal and $\llcorner \Delta \lrcorner =S$, and

•

for any $\Delta \in L(\mathcal{L}_{S+A}(V))$, the support of $\Delta$ contains the support of $G$.

Proof.

Let $W$ be the vector space spanned by the components of $\Delta _0$. Then $\Delta _0\in \mathcal{L}_S(W)$ and Lemma 3.7.2 implies that $\mathcal{L}_S(W)$ is a non-empty rational polytope. But then $\mathcal{L}_S(W)$ contains a rational point and so, possibly replacing $\Delta _0$, we may assume that $K_X+\Delta _0$ is $\mathbb{Q}$-Cartier.

We first prove the result in the case that $K_X+\Delta$ is $\mathbb{R}$-Cartier for every $\Delta \in V_{S+A}$. By compactness, we may pick $\mathbb{Q}$-divisors ${\Delta }_1,{\Delta }_2,\dots ,{\Delta }_{l}\in V_{S+A}$ such that $\mathcal{L}_{S+A}(V)$ is contained in the simplex spanned by ${\Delta }_1,{\Delta }_2,\dots ,{\Delta }_{l}$ (we do not assume that $\Delta _i\geq 0$). Pick a rational number $\epsilon \in (0,1/4]$ such that

$$\epsilon (\Delta _i-\Delta _0)+(1-2\epsilon )A$$

is an ample $\mathbb{Q}$-divisor over $U$, for $1\leq i\leq l$. Pick

$$A_i \sim _{\mathbb{Q},U} \epsilon (\Delta _i-\Delta _0)+(1-2\epsilon )A,$$

general ample $\mathbb{Q}$-divisors over $U$. Pick $A' \sim _{\mathbb{Q},U} \epsilon A$ a general ample $\mathbb{Q}$-divisor over $U$. If we define $L\colon V_{S+A} \longrightarrow \operatorname {WDiv}_{\mathbb{R}}(X)$ by

$$L(\Delta _i)=(1-\epsilon )\Delta _i+A_i+\epsilon \Delta _0+A'-(1-\epsilon ) A \sim _{\mathbb{Q},U} \Delta _i,$$

and extend to the whole of $V_{S+A}$ by linearity, then $L$ is an injective rational linear map which preserves $\mathbb{Q}$-linear equivalence over $U$. We let $V'$ be the rational affine subspace of $\operatorname {WDiv}_{\mathbb{R}}(X)$ defined by $V'_{S+A'}=L(V_{S+A})$. Note also that $L$ is the composition $L_2\circ L_1$ of

$$L_1(\Delta _i)=\Delta _i+A_i/(1-\epsilon )+A'-A \quad \text{and} \quad L_2(\Delta )=(1-\epsilon )\Delta +\epsilon (A'+\Delta _0).$$

If $\Delta \in \mathcal{L}_{S+A}(V)$, then $K_X+\Delta +A'-A$ is log canonical, and as $A_i$ is a general ample $\mathbb{Q}$-divisor over $U$ it follows that $K_X+\Delta +4/3A_i+A'-A$ is log canonical as well. As $1/(1-\epsilon )\leq 4/3$, it follows that if $\Delta \in \mathcal{L}_{S+A}(V)$, then $K_X+L_1(\Delta )$ is log canonical. Therefore, if $\Delta \in \mathcal{L}_{S+A}(V)$, then $K_X+L(\Delta )$ is divisorially log terminal and $\llcorner L(\Delta )\lrcorner =S$.

Pick a divisor $G'$ such that $S+A'+G'$ belongs to the interior of $\mathcal{L}_{S+A'}(V')$. As $G+G'$ contains no log canonical centres of $(X,\Delta _0)$ and $X$ is smooth at the generic point of every log canonical centre of $(X,\Delta _0)$, we may pick a $\mathbb{Q}$-Cartier divisor $H\geq G+G'$ which contains no log canonical centres of $(X,\Delta _0)$. Pick a rational number $\eta >0$ such that $A'-\eta H$ is ample over $U$. Pick $A''\sim _{\mathbb{Q},U} A'-\eta H$ a general ample $\mathbb{Q}$-divisor over $U$. Let $\delta >0$ by any rational number and let

$$T\colon \operatorname {WDiv}_{\mathbb{R}}(X) \longrightarrow \operatorname {WDiv}_{\mathbb{R}}(X)$$

be translation by $\delta (\eta H+A''-A') \sim _{\mathbb{Q},U} 0$. If $V''$ is the span of $V'$, $A'$ and $H$ and $\delta >0$ is sufficiently small, then $T(L(\mathcal{L}_{S+A}(V)))$ is contained in the interior of $\mathcal{L}_{\delta A''+S}(V'')$, $K_X+T(\Delta )$ is divisorially log terminal and the support of $T(\Delta )$ contains the support of $G$, for all $\Delta \in L(\mathcal{L}_{S+A}(V))$. If we replace $L$ by $T\circ L$, $V'_{S+A'}$ by $T(L(V_{S+A}))$ and $A'$ by $\delta A''$, then this finishes the case when $K_X+\Delta$ is $\mathbb{R}$-Cartier for every $\Delta \in V_{S+A}$.

We now turn to the general case. If

$$W_0=\{\, B\in V \,|\, \text{$K_X+S+A+B$ is $\mathbb{R}$-Cartier} \,\},$$

then $W_0\subset V$ is an affine subspace of $V$, which is defined over the rationals. Note that $\mathcal{L}_{S+A}(V)=\mathcal{L}_{S+A}(W_0)$. By what we have already proved, there is a rational affine linear isomorphism $L_0\colon W_0 \longrightarrow W'_0$, which preserves $\mathbb{Q}$-linear equivalence over $U$, a general ample $\mathbb{Q}$-divisor $A'$ over $U$, such that $L_0(\mathcal{L}_{S+A}(W_0))$ is contained in the interior of $\mathcal{L}_{S+A'}(W'_0)$, and for every divisor $\Delta \in L_0(\mathcal{L}_{S+A}(W_0))$, $K_X+\Delta$ is divisorially log terminal and the support of $\Delta$ contains the support of $G$.

Let $W_1$ be any vector subspace of $\operatorname {WDiv}_{\mathbb{R}}(X)$, which is defined over the rationals, such that $V=W_0+W_1$ and $W_0\cap W_1\subset \{0\}$. Let $V'=W'_0+W_1$. Since $L_0$ preserves $\mathbb{Q}$-linear equivalence over $U$, $W'_0\cap W_1=W_0\cap W_1$ and $\mathcal{L}_{S+A'}(V')=\mathcal{L}_{S+A'}(W'_0)$. If we define $L\colon V_A \longrightarrow V'_{A'}$, by sending $A+B_0+B_1$ to $L_0(A+B_0)+B_1$, where $B_i\in W_i$, then $L$ is a rational affine linear isomorphism, which preserves $\mathbb{Q}$-linear equivalence over $U$ and $L(\mathcal{L}_{S+A}(V))$ is contained in the interior of $\mathcal{L}_{S+A'}(V')$.

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Lemma 3.7.5.

Let $\pi \colon X \longrightarrow U$ be a projective morphism of normal quasi-projective varieties. Let $(X,\Delta =A+B)$ be a log canonical pair, where $A\geq 0$ and $B\geq 0$.

If $A$ is $\pi$-big and $\mathbf{B}_+(A/U)$ does not contain any non-Kawamata log terminal centres of $(X,\Delta )$ and there is a Kawamata log terminal pair $(X,\Delta _0)$, then we may find a Kawamata log terminal pair $(X,\Delta '=A'+B')$, where $A'\geq 0$ is a general ample $\mathbb{Q}$-divisor over $U$, $B'\geq 0$ and $K_X+\Delta ' \sim _{\mathbb{R},U} K_X+\Delta$. If in addition $A$ is a $\mathbb{Q}$-divisor, then $K_X+\Delta ' \sim _{\mathbb{Q},U} K_X+\Delta$.

Proof.

By Lemma 3.7.3 we may assume that $A$ is a general ample $\mathbb{Q}$-divisor over $U$. If $V$ is the vector space spanned by the components of $\Delta$, then $\Delta \in \mathcal{L}_A(V)$ and the result follows by Lemma 3.7.4.

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Lemma 3.7.6.

Let $V$ be a finite dimensional real vector space, which is defined over the rationals. Let $\Lambda \subset V$ be a lattice spanned by rational vectors. Suppose that $v\in V$ is a vector which is not contained in any proper affine subspace $W\subset V$ which is defined over the rationals.

Then the set

$$X=\{\, mv+\lambda \,|\, m\in \mathbb{N}, \lambda \in \Lambda \,\}$$

is dense in $V$.

Proof.

Let

$$q\colon V \longrightarrow V/\Lambda$$

be the quotient map and let $G$ be the closure of the image of $X$. As $G$ is infinite and $V/\Lambda$ 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 $V/\Lambda$ and so by Theorem 15.1 of 3, $G_0$ is a torus. Thus $G_0=W/\Lambda _0$, where

$$W=H_1(G_0,\mathbb{R})=\Lambda _0\underset{\mathbb{Z}}{\otimes } \mathbb{R}=H_1(G_0,\mathbb{Z})\underset{\mathbb{Z}}{\otimes } \mathbb{R}\subset H_1(G,\mathbb{Z})\underset{\mathbb{Z}}{\otimes } \mathbb{R}=H_1(G,\mathbb{R})$$

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 $W=V$.

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Lemma 3.7.7.

Let $\mathcal{C}$ 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 $\alpha$.

If $v\in \mathcal{C}$, then we may find vectors ${v}_1,{v}_2,\dots ,{v}_{p}\in \mathcal{C}$ and positive integers ${m}_1,{m}_2,\dots ,{m}_{p}$, which are divisible by $k$, such that $v$ is a convex linear combination of the vectors ${v}_1,{v}_2,\dots ,{v}_{p}$ and

$$\|v_i-v\|< \frac{\alpha }{m_i}, \qquad \text{where} \qquad \text{$\frac{m_i v_i}{k}$ is integral.}$$

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 $\mathcal{C}$ since the faces of $\mathcal{C}$ are rational.

After translating by a rational vector, we may assume that $0\in \mathcal{C}$. After fixing a suitable basis for $V$ and possibly shrinking $\mathcal{C}$, we may assume that $\mathcal{C}=[0,1]^n\subseteq \mathbb{R}^n$ and $v=({x}_1,{x}_2,\dots ,{x}_{n})\in (0,1)^n$. By Lemma 3.7.6, for each subset $I\subseteq \{1,2,\dots ,n\}$, we may find

$$v_I=({s}_1,{s}_2,\dots ,{s}_{n})\in (0,1)^n\cap \mathbb{Q}^n,$$

and an integer $m_I$ such that $m_I v_I$ is integral, such that

$$\qquad \|v-v_I\|< \frac{\alpha }{m_I} \qquad \text{and } s_j <x_j \quad \text{if and only if }j\in I.$$

In particular $v$ is contained inside the rational polytope $\mathcal{B}\subseteq \mathcal{C}$ generated by the $v_I$. Thus $v$ is a convex linear combination of a subset ${v}_1,{v}_2,\dots ,{v}_{p}$ of the extreme points of $\mathcal{B}$.

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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 $\pi \colon X \longrightarrow U$ be a projective morphism of normal quasi-projective varieties. Suppose that $(X,\Delta )$ is a log canonical pair of dimension $n$, where $K_X+\Delta$ is $\mathbb{R}$-Cartier. Suppose that there is a divisor $\Delta _0$ such that $K_X+\Delta _0$ is Kawamata log terminal.

If $R$ is an extremal ray of $\overline{\operatorname {NE}}(X/U)$ that is $(K_X+\Delta )$-negative, then there is a rational curve $\Sigma$ spanning $R$, such that

$$0<-(K_X+\Delta )\cdot \Sigma \leq 2n.$$

Proof.

Passing to an open subset of $U$, we may assume that $U$ is affine. Let $V$ be the vector space spanned by the components of $\Delta +\Delta _0$. By Lemma 3.7.2 the space $\mathcal{L}(V)$ of log canonical divisors is a rational polytope. Since $\Delta _0\in \mathcal{L}(V)$, we may find $\mathbb{Q}$-divisors $\Delta _i\in V$ with limit $\Delta$, such that $K_X+\Delta _i$ is Kawamata log terminal. In particular we may assume that $(K_X+\Delta _0)\cdot R<0$. Replacing $\pi$ by the contraction defined by the extremal ray $R$, we may assume that $-(K_X+\Delta )$ is $\pi$-ample.

Theorem 1 of 16 implies that we can find a rational curve $\Sigma _i$ contracted by $\pi$ such that

$$-(K_X+\Delta _i)\cdot \Sigma _i\leq 2n.$$

Pick a $\pi$-ample $\mathbb{Q}$-divisor $A$ such that $-(K_X+\Delta +A)$ is also $\pi$-ample. In particular $-(K_X+\Delta _i+A)$ is $\pi$-ample for $i\gg 0$. Now

$$A\cdot \Sigma _i=(K_X+\Delta _{i}+A)\cdot \Sigma _i-(K_X+\Delta _i)\cdot \Sigma _i<2n.$$

It follows that the curves $\Sigma _i$ belong to a bounded family. Thus, possibly passing to a subsequence, we may assume that $\Sigma =\Sigma _i$ is constant. In this case

$$\begin{equation*} -(K_X+\Delta )\cdot \Sigma =\lim _i -(K_X+\Delta _i)\cdot \Sigma \leq 2n. \end{equation*}$$■

Corollary 3.8.2.

Let $\pi \colon X \longrightarrow U$ be a projective morphism of normal quasi-projective varieties. Suppose the pair $(X,\Delta =A+B)$ has log canonical singularities, where $A\geq 0$ is an ample $\mathbb{R}$-divisor over $U$ and $B\geq 0$. Suppose that there is a divisor $\Delta _0$ such that $K_X+\Delta _0$ is Kawamata log terminal.

Then there are only finitely many $(K_X+\Delta )$-negative extremal rays ${R}_1,{R}_2,\dots ,{R}_{k}$ of $\overline{\operatorname {NE}}(X/U)$.

Proof.

We may assume that $A$ is a $\mathbb{Q}$-divisor. Let $R$ be a $(K_X+\Delta )$-negative extremal ray of $\overline{\operatorname {NE}}(X/U)$. Then

$$-(K_X+B)\cdot R=-(K_X+\Delta )\cdot R+A\cdot R>0.$$

By Theorem 3.8.1 $R$ is spanned by a curve $\Sigma$ such that

$$-(K_X+B)\cdot \Sigma \leq 2n.$$

But then

$$\begin{equation*} A\cdot \Sigma =-(K_X+B)\cdot \Sigma +(K_X+\Delta )\cdot \Sigma \leq 2n. \end{equation*}$$

Therefore the curve $\Sigma$ belongs to a bounded family.

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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 $f\colon X \longrightarrow U$ be a projective morphism of normal quasi-projective varieties, and let $D$ be a nef $\mathbb{R}$-divisor over $U$, such that $aD-(K_X+\Delta )$ is nef and big over $U$, for some positive real number $a$, where $(X,\Delta )$ 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 $D-(K_X+\Delta ) \sim _{\mathbb{R},U} A+B$, where $A$ is an ample $\mathbb{Q}$-divisor and $B\geq 0$. Pick $\epsilon \in (0,1)$ such that $(X,\Delta +\epsilon B)$ is Kawamata log terminal. Then

$$D-(K_X+\Delta +\epsilon B)=(1-\epsilon )(D-(K_X+\Delta ))+\epsilon (D-(K_X+\Delta +B))$$

is ample. Replacing $(X,\Delta )$ by $(X,\Delta +\epsilon B)$ we may therefore assume that $D-(K_X+\Delta )$ is ample. Let $V$ be the subspace of $\operatorname {WDiv}_{\mathbb{R}}(X)$ spanned by the components of $\Delta$. As $\mathcal{L}(V)$ is a rational polytope, cf. Lemma 3.7.2, which contains $\Delta$, we may find $\Delta '$ such that $K_X+\Delta '$ is $\mathbb{Q}$-Cartier and Kawamata log terminal, sufficiently close to $\Delta$ so that $D-(K_X+\Delta ')$ is ample. Replacing $(X,\Delta )$ by $(X,\Delta ')$ we may therefore assume that $K_X+\Delta$ is $\mathbb{Q}$-Cartier.

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

Pick a general ample $\mathbb{Q}$-divisor $A$ such that $D-(K_X+\Delta +A)$ is ample. Replacing $\Delta$ by $\Delta +A$, we may assume that $\Delta =A+B$, where $A$ is ample and $B\geq 0$. By Corollary 3.8.2 there are finitely many $(K_X+\Delta )$-negative extremal rays ${R}_1,{R}_2,\dots ,{R}_{k}$ of $\overline{\operatorname {NE}}(X)$. Let