American Mathematical Society

Existence of minimal models for varieties of log general type

By Caucher Birkar, Paolo Cascini, Christopher D. Hacon, James McKernan

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 left-parenthesis upper X comma normal upper Delta right-parenthesis be a projective Kawamata log terminal pair.

If normal upper Delta is big and upper K Subscript upper X Baseline plus normal upper Delta is pseudo-effective, then upper K Subscript upper X Baseline plus normal upper Delta has a log terminal model.

In particular, it follows that if upper K Subscript upper X Baseline plus normal upper Delta is big, then it has a log canonical model and the canonical ring is finitely generated. It also follows that if upper X is a smooth projective variety, then the ring

upper R left-parenthesis upper X comma upper K Subscript upper X Baseline right-parenthesis equals circled-plus Underscript m element-of double-struck upper N Endscripts upper H Superscript 0 Baseline left-parenthesis upper X comma script upper O Subscript upper X Baseline left-parenthesis m upper K Subscript upper X Baseline right-parenthesis right-parenthesis comma

is finitely generated.

The birational classification of complex projective surfaces was understood by the Italian algebraic geometers in the early 20th century: If upper X is a smooth complex projective surface of non-negative Kodaira dimension, that is, kappa left-parenthesis upper X comma upper K Subscript upper X Baseline right-parenthesis greater-than-or-equal-to 0 , then there is a unique smooth surface upper Y birational to upper X such that the canonical class upper K Subscript upper Y is nef (that is upper K Subscript upper Y Baseline dot upper C greater-than-or-equal-to 0 for any curve upper C subset-of upper Y ). upper Y is obtained from upper X simply by contracting all negative 1 -curves, that is, all smooth rational curves upper E with upper K Subscript upper X Baseline dot upper E equals negative 1 . If, on the other hand, kappa left-parenthesis upper X comma upper K Subscript upper X Baseline right-parenthesis equals negative normal infinity , then upper X is birational to either double-struck upper P squared or a ruled surface over a curve of genus g greater-than 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 upper X , we have:

If kappa left-parenthesis upper X comma upper K Subscript upper X Baseline right-parenthesis greater-than-or-equal-to 0 , then there exists a minimal model, that is, a variety upper Y birational to upper X such that upper K Subscript upper Y is nef.

If kappa left-parenthesis upper X comma upper K Subscript upper X Baseline right-parenthesis equals negative normal infinity , then there is a variety upper Y birational to upper X which admits a Fano fibration, that is, a morphism upper Y long right-arrow upper Z whose general fibres upper F have ample anticanonical class minus upper K Subscript upper F .

It is possible to exhibit 3-folds which have no smooth minimal model, see for example (16.17) of Reference37, and so one must allow varieties upper X with singularities. However, these singularities cannot be arbitrary. At the very minimum, we must still be able to compute upper K Subscript upper X Baseline dot upper C for any curve upper C subset-of upper X . So, we insist that upper K Subscript upper X is double-struck upper Q -Cartier (or sometimes we require the stronger property that upper X is double-struck upper Q -factorial). We also require that upper X and the minimal model upper Y have the same pluricanonical forms. This condition is essentially equivalent to requiring that the induced birational map phi colon upper X right dasheD arrow upper Y is upper K Subscript upper 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 upper K Subscript upper X , then one can construct Mori fibre spaces whenever upper K Subscript upper 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

upper R left-parenthesis upper X comma upper K Subscript upper X Baseline right-parenthesis equals circled-plus Underscript m element-of double-struck upper N Endscripts upper H Superscript 0 Baseline left-parenthesis upper X comma script upper O Subscript upper X Baseline left-parenthesis m upper K Subscript upper X Baseline right-parenthesis right-parenthesis

is finitely generated and upper K Subscript upper X is big, then the canonical model upper Y is nothing more than the Proj of upper R left-parenthesis upper X comma upper K Subscript upper X Baseline right-parenthesis . It is then automatic that the induced rational map phi colon upper X right dasheD arrow upper Y is upper K Subscript upper X -negative.

The other natural way to ensure that phi is upper K Subscript upper X -negative is to factor phi into a sequence of elementary steps all of which are upper K Subscript upper X -negative. We now explain one way to achieve this factorisation.

If upper K Subscript upper X is not nef, then, by the cone theorem, there is a rational curve upper C subset-of upper X such that upper K Subscript upper X Baseline dot upper C less-than 0 and a morphism f colon upper X long right-arrow upper Z which is surjective, with connected fibres, onto a normal projective variety and which contracts an irreducible curve upper D if and only if left-bracket upper D right-bracket element-of double-struck upper R Superscript plus Baseline left-bracket upper C right-bracket subset-of upper N 1 left-parenthesis upper X right-parenthesis . Note that rho left-parenthesis upper X slash upper Z right-parenthesis equals 1 and minus upper K Subscript upper X is f -ample. We have the following possibilities:

If dimension upper Z less-than dimension upper X , this is the required Fano fibration.

If dimension upper Z equals dimension upper X and f contracts a divisor, then we say that f is a divisorial contraction and we replace upper X by upper Z .

If dimension upper Z equals dimension upper X and f does not contract a divisor, then we say that f is a small contraction. In this case upper K Subscript upper Z is not double-struck upper Q -Cartier, so that we cannot replace upper X by upper Z . Instead, we would like to replace f colon upper X long right-arrow upper Z by its flip f Superscript plus Baseline colon upper X Superscript plus Baseline long right-arrow upper Z , where upper X Superscript plus is isomorphic to upper X in codimension 1 and upper K Subscript upper X Sub Superscript plus is f Superscript plus -ample. In other words, we wish to replace some upper K Subscript upper X -negative curves by upper K Subscript upper X Sub Superscript plus -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 left-parenthesis upper X right-parenthesis 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 Reference26.

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 Reference34 and Hacon and McKernan Reference8 have shown that assuming the minimal model program in dimension n minus 1 (or even better simply finiteness of minimal models in dimension n minus 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 greater-than-or-equal-to 4 only some partial answers are available. Kawamata, Matsuda and Matsuki proved Reference18 the termination of terminal 4 -fold flips, Matsuki has shown Reference25 the termination of terminal 4 -fold flops and Fujino has shown Reference5 the termination of canonical 4 -fold (log) flips. Alexeev, Hacon and Kawamata Reference1 have shown the termination of Kawamata log terminal 4 -fold flips when the Kodaira dimension of minus left-parenthesis upper K Subscript upper X Baseline plus normal upper Delta right-parenthesis is non-negative and the existence of minimal models of Kawamata log terminal 4 -folds when either normal upper Delta or upper K Subscript upper X Baseline plus normal upper 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. Reference35). Moreover, if kappa left-parenthesis upper X comma upper K Subscript upper X Baseline plus normal upper Delta right-parenthesis greater-than-or-equal-to 0 , Birkar has shown Reference2 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 left-parenthesis upper X comma normal upper Delta right-parenthesis be a Kawamata log terminal pair, where upper K Subscript upper X Baseline plus normal upper Delta is double-struck upper R -Cartier. Let pi colon upper X long right-arrow upper U be a projective morphism of quasi-projective varieties.

If either normal upper Delta is pi -big and upper K Subscript upper X Baseline plus normal upper Delta is pi -pseudo-effective or upper K Subscript upper X Baseline plus normal upper Delta is pi -big, then

(1)

upper K Subscript upper X Baseline plus normal upper Delta has a log terminal model over upper U ,

(2)

if upper K Subscript upper X Baseline plus normal upper Delta is pi -big then upper K Subscript upper X Baseline plus normal upper Delta has a log canonical model over upper U , and

(3)

if upper K Subscript upper X Baseline plus normal upper Delta is double-struck upper Q -Cartier, then the script upper O Subscript upper U -algebra German upper R left-parenthesis pi comma upper K Subscript upper X Baseline plus normal upper Delta right-parenthesis equals circled-plus Underscript m element-of double-struck upper N Endscripts pi Subscript asterisk Baseline script upper O Subscript upper X Baseline left-parenthesis bottom left corner m left-parenthesis upper K Subscript upper X Baseline plus normal upper Delta right-parenthesis bottom right corner right-parenthesis comma

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 upper X be a smooth projective variety of general type.

Then

(1)

upper X has a minimal model,

(2)

upper X has a canonical model,

(3)

the ringupper R left-parenthesis upper X comma upper K Subscript upper X Baseline right-parenthesis equals circled-plus Underscript m element-of double-struck upper N Endscripts upper H Superscript 0 Baseline left-parenthesis upper X comma script upper O Subscript upper X Baseline left-parenthesis m upper K Subscript upper X Baseline right-parenthesis right-parenthesis

is finitely generated, and

(4)

upper X has a model with a Kähler-Einstein metric.

Note that (4) follows from (2) and Theorem D of Reference4. Note that Siu has announced a proof of finite generation for varieties of general type using analytic methods; see Reference36.

Corollary 1.1.2

Let left-parenthesis upper X comma normal upper Delta right-parenthesis be a projective Kawamata log terminal pair, where upper K Subscript upper X Baseline plus normal upper Delta is double-struck upper Q -Cartier.

Then the ring

upper R left-parenthesis upper X comma upper K Subscript upper X Baseline plus normal upper Delta right-parenthesis equals circled-plus Underscript m element-of double-struck upper N Endscripts upper H Superscript 0 Baseline left-parenthesis upper X comma script upper O Subscript upper X Baseline left-parenthesis bottom left corner m left-parenthesis upper K Subscript upper X Baseline plus normal upper Delta right-parenthesis bottom right corner right-parenthesis right-parenthesis

is finitely generated.

Let us emphasize that in Corollary 1.1.2 we make no assumption about upper K Subscript upper X Baseline plus normal upper Delta or normal upper Delta being big. Indeed Fujino and Mori, Reference6, proved that Corollary 1.1.2 follows from the case when upper K Subscript upper X Baseline plus normal upper 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 upper X long right-arrow upper U be a projective morphism of normal quasi-projective varieties. Suppose that upper K Subscript upper X Baseline plus normal upper Delta is Kawamata log terminal and normal upper Delta is big over upper U . Let phi Subscript i Baseline colon upper X right dasheD arrow upper Y Subscript i Baseline , i equals 1 and 2 , be two log terminal models of left-parenthesis upper X comma normal upper Delta right-parenthesis over upper U . Let normal upper Gamma Subscript i Baseline equals phi Subscript i asterisk Baseline normal upper Delta .

Then the birational map upper Y 1 right dasheD arrow upper Y 2 is the composition of a sequence of left-parenthesis upper K Subscript upper Y 1 Baseline plus normal upper Gamma 1 right-parenthesis -flops over upper U .

Note that Corollary 1.1.3 has been generalised recently to the case when normal upper Delta is not assumed big, Reference17. 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 normal upper Delta is well-behaved. To explain this, we need some definitions:

Definition 1.1.4

Let pi colon upper X long right-arrow upper U be a projective morphism of normal quasi-projective varieties, and let upper V be a finite dimensional affine subspace of the real vector space upper W upper D i v Subscript double-struck upper R Baseline left-parenthesis upper X right-parenthesis of Weil divisors on upper X . Fix an double-struck upper R -divisor upper A greater-than-or-equal-to 0 and define

StartLayout 1st Row 1st Column upper V Subscript upper A 2nd Column equals StartSet normal upper Delta vertical-bar normal upper Delta equals upper A plus upper B comma upper B element-of upper V EndSet comma 2nd Row 1st Column script upper L Subscript upper A Baseline left-parenthesis upper V right-parenthesis 2nd Column equals StartSet normal upper Delta equals upper A plus upper B element-of upper V Subscript upper A Baseline vertical-bar upper K Subscript upper X Baseline plus normal upper Delta is log canonical and upper B greater-than-or-equal-to 0 EndSet comma 3rd Row 1st Column script upper E Subscript upper A comma pi Baseline left-parenthesis upper V right-parenthesis 2nd Column equals StartSet normal upper Delta element-of script upper L Subscript upper A Baseline left-parenthesis upper V right-parenthesis vertical-bar upper K Subscript upper X Baseline plus normal upper Delta is pseudo hyphen effective over upper U EndSet comma 4th Row 1st Column script upper N Subscript upper A comma pi Baseline left-parenthesis upper V right-parenthesis 2nd Column equals StartSet normal upper Delta element-of script upper L Subscript upper A Baseline left-parenthesis upper V right-parenthesis vertical-bar upper K Subscript upper X Baseline plus normal upper Delta is nef over upper U EndSet period EndLayout

Given a birational contraction phi colon upper X right dasheD arrow upper Y over upper U , define

script upper W Subscript phi comma upper A comma pi Baseline left-parenthesis upper V right-parenthesis equals StartSet normal upper Delta element-of script upper E Subscript upper A comma pi Baseline left-parenthesis upper V right-parenthesis vertical-bar phi is a weak log canonical model for left-parenthesis upper X comma normal upper Delta right-parenthesis over upper U EndSet comma

and given a rational map psi colon upper X right dasheD arrow upper Z over upper U , define

script upper A Subscript psi comma upper A comma pi Baseline left-parenthesis upper V right-parenthesis equals StartSet normal upper Delta element-of script upper E Subscript upper A comma pi Baseline left-parenthesis upper V right-parenthesis vertical-bar psi is the ample model for left-parenthesis upper X comma normal upper Delta right-parenthesis over upper U EndSet comma

(cf. Definitions 3.6.7 and 3.6.5 for the definitions of weak log canonical model and ample model for left-parenthesis upper X comma normal upper Delta right-parenthesis over upper U ).

We will adopt the convention that script upper L left-parenthesis upper V right-parenthesis equals script upper L 0 left-parenthesis upper V right-parenthesis . If the support of upper A has no components in common with any element of upper V , then the condition that upper B greater-than-or-equal-to 0 is vacuous. In many applications, upper A will be an ample double-struck upper Q -divisor over upper U . In this case, we often assume that upper A is general in the sense that we fix a positive integer such that k upper A is very ample over upper U , and we assume that upper A equals StartFraction 1 Over k EndFraction upper A prime , where upper A prime tilde Subscript upper U Baseline k upper A is very general. With this choice of upper A , we have

script upper N Subscript upper A comma pi Baseline left-parenthesis upper V right-parenthesis subset-of script upper E Subscript upper A comma pi Baseline left-parenthesis upper V right-parenthesis subset-of script upper L Subscript upper A Baseline left-parenthesis upper V right-parenthesis equals script upper L left-parenthesis upper V right-parenthesis plus upper A subset-of upper V Subscript upper A Baseline equals upper V plus upper A comma

and the condition that the support of upper A has no common components with any element of upper V is then automatic. The following result was first proved by Shokurov Reference32 assuming the existence and termination of flips:

Corollary 1.1.5

Let pi colon upper X long right-arrow upper U be a projective morphism of normal quasi-projective varieties. Let upper V be a finite dimensional affine subspace of upper W upper D i v Subscript double-struck upper R Baseline left-parenthesis upper X right-parenthesis which is defined over the rationals. Suppose there is a divisor normal upper Delta 0 element-of upper V such that upper K Subscript upper X Baseline plus normal upper Delta 0 is Kawamata log terminal. Let upper A be a general ample double-struck upper Q -divisor over upper U , which has no components in common with any element of upper V .

(1)

There are finitely many birational contractions phi Subscript i Baseline colon upper X right dasheD arrow upper Y Subscript i Baseline over upper U , 1 less-than-or-equal-to i less-than-or-equal-to p such that script upper E Subscript upper A comma pi Baseline left-parenthesis upper V right-parenthesis equals union Underscript i equals 1 Overscript p Endscripts script upper W Subscript i Baseline comma

where each script upper W Subscript i Baseline equals script upper W Subscript phi Sub Subscript i Subscript comma upper A comma pi Baseline left-parenthesis upper V right-parenthesis is a rational polytope. Moreover, if phi colon upper X right dasheD arrow upper Y is a log terminal model of left-parenthesis upper X comma normal upper Delta right-parenthesis over upper U , for some normal upper Delta element-of script upper E Subscript upper A comma pi Baseline left-parenthesis upper V right-parenthesis , then phi equals phi Subscript i , for some 1 less-than-or-equal-to i less-than-or-equal-to p .

(2)

There are finitely many rational maps psi Subscript j Baseline colon upper X right dasheD arrow upper Z Subscript j Baseline over upper U , 1 less-than-or-equal-to j less-than-or-equal-to q which partition script upper E Subscript upper A comma pi Baseline left-parenthesis upper V right-parenthesis into the subsets script upper A Subscript j Baseline equals script upper A Subscript psi Sub Subscript j Subscript comma upper A comma pi Baseline left-parenthesis upper V right-parenthesis .

(3)

For every 1 less-than-or-equal-to i less-than-or-equal-to p there is a 1 less-than-or-equal-to j less-than-or-equal-to q and a morphism f Subscript i comma j Baseline colon upper Y Subscript i Baseline long right-arrow upper Z Subscript j Baseline such that script upper W Subscript i Baseline subset-of script upper A overbar Subscript j .

In particular script upper E Subscript upper A comma pi Baseline left-parenthesis upper V right-parenthesis is a rational polytope and each script upper A overbar Subscript j is a finite union of rational polytopes.

Definition 1.1.6

Let left-parenthesis upper X comma normal upper Delta right-parenthesis be a Kawamata log terminal pair and let upper D be a big divisor. Suppose that upper K Subscript upper X Baseline plus normal upper Delta is not pseudo-effective. The effective log threshold is

sigma left-parenthesis upper X comma normal upper Delta comma upper D right-parenthesis equals sup left-brace t element-of double-struck upper R vertical-bar upper D plus t left-parenthesis upper K Subscript upper X Baseline plus normal upper Delta right-parenthesis is pseudo hyphen effective right-brace period

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 left-parenthesis upper X comma normal upper Delta right-parenthesis be a projective Kawamata log terminal pair and let upper D be an ample divisor. Suppose that upper K Subscript upper X Baseline plus normal upper Delta is not pseudo-effective.

If both upper K Subscript upper X Baseline plus normal upper Delta and upper D are double-struck upper Q -Cartier, then the effective log threshold and the Kodaira energy are rational.

Definition 1.1.8

Let pi colon upper X long right-arrow upper U be a projective morphism of normal quasi-projective varieties. Let upper D Superscript bullet Baseline equals left-parenthesis upper D 1 comma upper D 2 comma ellipsis comma upper D Subscript k Baseline right-parenthesis be a sequence of double-struck upper Q -divisors on upper X . The sheaf of script upper O Subscript upper U -algebras,

German upper R left-parenthesis pi comma upper D Superscript bullet Baseline right-parenthesis equals circled-plus Underscript m element-of double-struck upper N Superscript k Baseline Endscripts pi Subscript asterisk Baseline script upper O Subscript upper X Baseline left-parenthesis bottom left corner sigma-summation m Subscript i Baseline upper D Subscript i Baseline bottom right corner right-parenthesis comma

is called the Cox ring associated to upper D Superscript bullet .

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

Corollary 1.1.9

Let pi colon upper X long right-arrow upper U be a projective morphism of normal quasi-projective varieties. Fix upper A greater-than-or-equal-to 0 to be an ample double-struck upper Q -divisor over upper U . Let normal upper Delta Subscript i Baseline equals upper A plus upper B Subscript i , for some double-struck upper Q -divisors upper B 1 comma upper B 2 comma ellipsis comma upper B Subscript k Baseline greater-than-or-equal-to 0 . Assume that upper D Subscript i Baseline equals upper K Subscript upper X Baseline plus normal upper Delta Subscript i is divisorially log terminal and double-struck upper Q -Cartier. Then the Cox ring,

German upper R left-parenthesis pi comma upper D Superscript bullet Baseline right-parenthesis equals circled-plus Underscript m element-of double-struck upper N Superscript k Baseline Endscripts pi Subscript asterisk Baseline script upper O Subscript upper X Baseline left-parenthesis bottom left corner sigma-summation m Subscript i Baseline upper D Subscript i Baseline bottom right corner right-parenthesis comma

is a finitely generated script upper O Subscript upper 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 upper M overbar Subscript g comma 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 Reference7, Reference24 and Reference11).

Corollary 1.2.1

Let upper X equals upper M overbar Subscript g comma n be the moduli space of stable curves of genus g with n marked points and let normal upper Delta Subscript i , 1 less-than-or-equal-to i less-than-or-equal-to k denote the boundary divisors.

Let normal upper Delta equals sigma-summation Underscript i Endscripts a Subscript i Baseline normal upper Delta Subscript i be a boundary. Then upper K Subscript upper X Baseline plus normal upper Delta is log canonical and if upper K Subscript upper X Baseline plus normal upper Delta is big, then there is a log canonical model upper X right dasheD arrow upper Y . Moreover if we fix a positive rational number delta and require that the coefficient a Subscript i of normal upper Delta Subscript 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 upper D , a small multiple of upper D is linearly equivalent to a divisor of the form upper K Subscript upper X Baseline plus normal upper Delta , where normal upper Delta is big and upper K Subscript upper X Baseline plus normal upper Delta is Kawamata log terminal.

Definition 1.3.1

Let pi colon upper X long right-arrow upper U be a projective morphism of normal varieties, where upper U is affine.

We say that upper X is a Mori dream space if h Superscript 1 Baseline left-parenthesis upper X comma script upper O Subscript upper X Baseline right-parenthesis equals 0 and the Cox ring is finitely generated over the coordinate ring of upper U .

Corollary 1.3.2

Let pi colon upper X long right-arrow upper U be a projective morphism of normal varieties, where upper U is affine. Suppose that upper X is double-struck upper Q -factorial, upper K Subscript upper X Baseline plus normal upper Delta is divisorially log terminal and minus left-parenthesis upper K Subscript upper X Baseline plus normal upper Delta right-parenthesis is ample over upper U .

Then upper 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 upper D , one can run the upper D -MMP, and this ends with either a nef model, or a fibration, for which negative upper D is relatively ample, and in fact any sequence of upper D -flips terminates.

Corollary 1.3.2 was conjectured in Reference12 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 Reference12 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 upper K Subscript upper X Baseline plus normal upper Delta is not pseudo-effective, then we can run the MMP with scaling to get a Mori fibre space:

Corollary 1.3.3

Let left-parenthesis upper X comma normal upper Delta right-parenthesis be a double-struck upper Q -factorial Kawamata log terminal pair. Let pi colon upper X long right-arrow upper U be a projective morphism of normal quasi-projective varieties. Suppose that upper K Subscript upper X Baseline plus normal upper Delta is not pi -pseudo-effective.

Then we may run f colon upper X right dasheD arrow upper Y a left-parenthesis upper K Subscript upper X Baseline plus normal upper Delta right-parenthesis -MMP over upper U and end with a Mori fibre space g colon upper Y long right-arrow upper W over upper U .

Note that we do not claim in Corollary 1.3.3 that however we run the left-parenthesis upper K Subscript upper X Baseline plus normal upper Delta right-parenthesis -MMP over upper 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 upper X be a projective variety. A curve normal upper Sigma is called nef if upper B dot normal upper Sigma greater-than-or-equal-to 0 for all Cartier divisors upper B greater-than-or-equal-to 0 . upper N upper F left-parenthesis upper X right-parenthesis denotes the cone of nef curves sitting inside upper H 2 left-parenthesis upper X comma double-struck upper R right-parenthesis and ModifyingAbove upper N upper F With bar left-parenthesis upper X right-parenthesis denotes its closure.

Now suppose that left-parenthesis upper X comma normal upper Delta right-parenthesis is a log pair. A left-parenthesis upper K Subscript upper X Baseline plus normal upper Delta right-parenthesis -co-extremal ray is an extremal ray upper F of the closed cone of nef curves ModifyingAbove upper N upper F With bar left-parenthesis upper X right-parenthesis on which upper K Subscript upper X Baseline plus normal upper Delta is negative.

Corollary 1.3.5

Let left-parenthesis upper X comma normal upper Delta right-parenthesis be a projective double-struck upper Q -factorial Kawamata log terminal pair such that minus left-parenthesis upper K Subscript upper X Baseline plus normal upper Delta right-parenthesis is ample.

Then ModifyingAbove upper N upper F With bar left-parenthesis upper X right-parenthesis is a rational polyhedron. If upper F equals upper F Subscript i is a left-parenthesis upper K Subscript upper X Baseline plus normal upper Delta right-parenthesis -co-extremal ray, then there exists an double-struck upper R -divisor normal upper Theta such that the pair left-parenthesis upper X comma normal upper Theta right-parenthesis is Kawamata log terminal and the left-parenthesis upper K Subscript upper X Baseline plus normal upper Theta right-parenthesis -MMP pi colon upper X right dasheD arrow upper Y ends with a Mori fibre space f colon upper Y long right-arrow upper Z such that upper F is spanned by the pullback to upper X of the class of any curve normal upper 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 left-parenthesis upper X comma normal upper Delta right-parenthesis be a Kawamata log terminal pair and let pi colon upper X long right-arrow upper Z be a small left-parenthesis upper K Subscript upper X Baseline plus normal upper Delta right-parenthesis -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 upper X long right-arrow upper U be a projective morphism of normal quasi-projective varieties. Let left-parenthesis upper X comma normal upper Delta right-parenthesis be a double-struck upper Q -factorial Kawamata log terminal pair, where upper K Subscript upper X Baseline plus normal upper Delta is double-struck upper R -Cartier and normal upper Delta is pi -big. Let upper C greater-than-or-equal-to 0 be an double-struck upper R -divisor.

If upper K Subscript upper X Baseline plus normal upper Delta plus upper C is Kawamata log terminal and pi -nef, then we may run the left-parenthesis upper K Subscript upper X Baseline plus normal upper Delta right-parenthesis -MMP over upper U with scaling of upper C .

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

Corollary 1.4.3

Let left-parenthesis upper X comma normal upper Delta right-parenthesis be a log canonical pair and let f colon upper W long right-arrow upper X be a log resolution. Suppose that there is a divisor normal upper Delta 0 such that upper K Subscript upper X Baseline plus normal upper Delta 0 is Kawamata log terminal. Let German upper E be any set of valuations of f -exceptional divisors which satisfies the following two properties:

(1)

German upper E contains only valuations of log discrepancy at most one, and

(2)

the centre of every valuation of log discrepancy one in German upper E does not contain any non-Kawamata log terminal centres.

Then we may find a birational morphism pi colon upper Y long right-arrow upper X , such that upper Y is double-struck upper Q -factorial and the exceptional divisors of pi correspond to the elements of German upper E .

For example, if we assume that left-parenthesis upper X comma normal upper Delta right-parenthesis is Kawamata log terminal and we let German upper E be the set of all exceptional divisors with log discrepancy at most one, then the birational morphism pi colon upper Y long right-arrow upper X defined in Corollary 1.4.3 above is a terminal model of left-parenthesis upper X comma normal upper Delta right-parenthesis . In particular there is an double-struck upper R -divisor normal upper Gamma greater-than-or-equal-to 0 on upper Y such that upper K Subscript upper Y Baseline plus normal upper Gamma equals pi Superscript asterisk Baseline left-parenthesis upper K Subscript upper X Baseline plus normal upper Delta right-parenthesis and the pair left-parenthesis upper Y comma normal upper Gamma right-parenthesis is terminal.

If instead we assume that left-parenthesis upper X comma normal upper Delta right-parenthesis is Kawamata log terminal but German upper E is empty, then the birational morphism pi colon upper Y long right-arrow upper X defined in Corollary 1.4.3 above is a log terminal model. In particular pi is small, upper Y is double-struck upper Q -factorial and there is an double-struck upper R -divisor normal upper Gamma greater-than-or-equal-to 0 on upper Y such that upper K Subscript upper Y Baseline plus normal upper Gamma equals pi Superscript asterisk Baseline left-parenthesis upper K Subscript upper X Baseline plus normal upper Delta right-parenthesis .

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

Corollary 1.4.4

Let left-parenthesis upper X comma normal upper Delta right-parenthesis be a log pair.

Then there is a birational morphism pi colon upper Y long right-arrow upper X , where upper Y is double-struck upper Q -factorial, such that if we write

upper K Subscript upper Y Baseline plus normal upper Gamma equals upper K Subscript upper Y Baseline plus normal upper Gamma 1 plus normal upper Gamma 2 equals pi Superscript asterisk Baseline left-parenthesis upper K Subscript upper X Baseline plus normal upper Delta right-parenthesis comma

where every component of normal upper Gamma 1 has coefficient less than one and every component of normal upper Gamma 2 has coefficient at least one, then upper K Subscript upper Y Baseline plus normal upper Gamma 1 is Kawamata log terminal and nef over upper X and no component of normal upper 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. Reference31, Reference21, Reference13):

Corollary 1.4.5 (Inversion of adjunction).

Let left-parenthesis upper X comma normal upper Delta right-parenthesis be a log pair and let nu colon upper S long right-arrow upper S prime be the normalisation of a component upper S prime of normal upper Delta of coefficient one.

If we define normal upper Theta by adjunction,

nu Superscript asterisk Baseline left-parenthesis upper K Subscript upper X Baseline plus normal upper Delta right-parenthesis equals upper K Subscript upper S Baseline plus normal upper Theta comma

then the log discrepancy of upper K Subscript upper S Baseline plus normal upper Theta is equal to the minimum of the log discrepancy with respect to upper K Subscript upper X Baseline plus normal upper Delta of any valuation whose centre on upper X is of codimension at least two and intersects upper 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 double-struck upper C , which are not projective. In fact the appendix to Reference10 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 upper X is a terminal threefold with Picard number one; see Reference19. In a slightly different but related direction, it is conjectured that if a complex Kähler manifold upper M does not contain any rational curves, then upper K Subscript upper M is nef (see for example Reference30), 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. Reference33, gives an affirmative answer to the first conjecture and at the same time connects the two conjectures:

Corollary 1.4.6

Let pi colon upper X long right-arrow upper U be a proper map of normal algebraic spaces, where upper X is analytically double-struck upper Q -factorial.

If upper K Subscript upper X Baseline plus normal upper 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 upper K Subscript upper X Baseline plus normal upper Delta is pi -nef.

2. Description of the proof

Theorem A (Existence of pl-flips).

Let f colon upper X long right-arrow upper Z be a pl-flipping contraction for an n -dimensional purely log terminal pair left-parenthesis upper X comma normal upper Delta right-parenthesis .

Then the flip f Superscript plus Baseline colon upper X Superscript plus Baseline long right-arrow upper Z of f exists.

Theorem B (Special finiteness).

Let pi colon upper X long right-arrow upper U be a projective morphism of normal quasi-projective varieties, where upper X is double-struck upper Q -factorial of dimension n . Let upper V be a finite dimensional affine subspace of upper W upper D i v Subscript double-struck upper R Baseline left-parenthesis upper X right-parenthesis , which is defined over the rationals, let upper S be the sum of finitely many prime divisors and let upper A be a general ample double-struck upper Q -divisor over upper U . Let left-parenthesis upper X comma normal upper Delta 0 right-parenthesis be a divisorially log terminal pair such that upper S less-than-or-equal-to normal upper Delta 0 . Fix a finite set German upper E of prime divisors on upper X .

Then there are finitely 1 less-than-or-equal-to i less-than-or-equal-to k many birational maps phi Subscript i Baseline colon upper X right dasheD arrow upper Y Subscript i Baseline over upper U such that if phi colon upper X right dasheD arrow upper Y is any double-struck upper Q -factorial weak log canonical model over upper U of upper K Subscript upper X Baseline plus normal upper Delta , where normal upper Delta element-of script upper L Subscript upper S plus upper A Baseline left-parenthesis upper V right-parenthesis , which only contracts elements of German upper E and which does not contract every component of upper S , then there is an index 1 less-than-or-equal-to i less-than-or-equal-to k such that the induced birational map xi colon upper Y Subscript i Baseline right dasheD arrow upper Y is an isomorphism in a neighbourhood of the strict transforms of upper S .

Theorem C (Existence of log terminal models).

Let pi colon upper X long right-arrow upper U be a projective morphism of normal quasi-projective varieties, where upper X has dimension n . Suppose that upper K Subscript upper X Baseline plus normal upper Delta is Kawamata log terminal, where normal upper Delta is big over upper U .

If there exists an double-struck upper R -divisor upper D such that upper K Subscript upper X Baseline plus normal upper Delta tilde Subscript double-struck upper R comma upper U Baseline upper D greater-than-or-equal-to 0 , then upper K Subscript upper X Baseline plus normal upper Delta has a log terminal model over upper U .

Theorem D (Non-vanishing theorem).

Let pi colon upper X long right-arrow upper U be a projective morphism of normal quasi-projective varieties, where upper X has dimension n . Suppose that upper K Subscript upper X Baseline plus normal upper Delta is Kawamata log terminal, where normal upper Delta is big over upper U .

If upper K Subscript upper X Baseline plus normal upper Delta is pi -pseudo-effective, then there exists an double-struck upper R -divisor upper D such that upper K Subscript upper X Baseline plus normal upper Delta tilde Subscript double-struck upper R comma upper U Baseline upper D greater-than-or-equal-to 0 .

Theorem E (Finiteness of models).

Let pi colon upper X long right-arrow upper U be a projective morphism of normal quasi-projective varieties, where upper X has dimension n . Fix a general ample double-struck upper Q -divisor upper A greater-than-or-equal-to 0 over upper U . Let upper V be a finite dimensional affine subspace of upper W upper D i v Subscript double-struck upper R Baseline left-parenthesis upper X right-parenthesis which is defined over the rationals. Suppose that there is a Kawamata log terminal pair left-parenthesis upper X comma normal upper Delta 0 right-parenthesis .

Then there are finitely many birational maps psi Subscript j Baseline colon upper X right dasheD arrow upper Z Subscript j Baseline over upper U , 1 less-than-or-equal-to j less-than-or-equal-to l such that if psi colon upper X right dasheD arrow upper Z is a weak log canonical model of upper K Subscript upper X Baseline plus normal upper Delta over upper U , for some normal upper Delta element-of script upper L Subscript upper A Baseline left-parenthesis upper V right-parenthesis , then there is an index 1 less-than-or-equal-to j less-than-or-equal-to l and an isomorphism xi colon upper Z Subscript j Baseline long right-arrow upper Z such that psi equals xi ring psi Subscript j .

Theorem F (Finite generation).

Let pi colon upper X long right-arrow upper Z be a projective morphism to a normal affine variety. Let left-parenthesis upper X comma normal upper Delta equals upper A plus upper B right-parenthesis be a Kawamata log terminal pair of dimension n , where upper A greater-than-or-equal-to 0 is an ample double-struck upper Q -divisor and upper B greater-than-or-equal-to 0 . If upper K Subscript upper X Baseline plus normal upper Delta is pseudo-effective, then

(1)

The pair left-parenthesis upper X comma normal upper Delta right-parenthesis has a log terminal model mu colon upper X right dasheD arrow upper Y . In particular if upper K Subscript upper X Baseline plus normal upper Delta is double-struck upper Q -Cartier, then the log canonical ringupper R left-parenthesis upper X comma upper K Subscript upper X Baseline plus normal upper Delta right-parenthesis equals circled-plus Underscript m element-of double-struck upper N Endscripts upper H Superscript 0 Baseline left-parenthesis upper X comma script upper O Subscript upper X Baseline left-parenthesis bottom left corner m left-parenthesis upper K Subscript upper X Baseline plus normal upper Delta right-parenthesis bottom right corner right-parenthesis right-parenthesis

is finitely generated.

(2)

Let upper V subset-of upper W upper D i v Subscript double-struck upper R Baseline left-parenthesis upper X right-parenthesis be the vector space spanned by the components of normal upper Delta . Then there is a constant delta greater-than 0 such that if upper G is a prime divisor contained in the stable base locus of upper K Subscript upper X Baseline plus normal upper Delta and normal upper Xi element-of script upper L Subscript upper A Baseline left-parenthesis upper V right-parenthesis such that double-vertical-bar normal upper Xi minus normal upper Delta double-vertical-bar less-than delta , then upper G is contained in the stable base locus of upper K Subscript upper X Baseline plus normal upper Xi .

(3)

Let upper W subset-of upper V be the smallest affine subspace of upper W upper D i v Subscript double-struck upper R Baseline left-parenthesis upper X right-parenthesis containing normal upper Delta , which is defined over the rationals. Then there is a constant eta greater-than 0 and a positive integer r greater-than 0 such that if normal upper Xi element-of upper W is any divisor and k is any positive integer such that double-vertical-bar normal upper Xi minus normal upper Delta double-vertical-bar less-than eta and k left-parenthesis upper K Subscript upper X Baseline plus normal upper Xi right-parenthesis slash r is Cartier, then every component of upper F i x left-parenthesis k left-parenthesis upper K Subscript upper X Baseline plus normal upper Xi right-parenthesis right-parenthesis is a component of the stable base locus of upper K Subscript upper X Baseline plus normal upper Delta .

The proofs of Theorem A, Theorem B, Theorem C, Theorem D, Theorem E and Theorem F proceed by induction:

Theorem F Subscript n minus 1 implies Theorem A Subscript n ; see the main result of Reference9.

Theorem E Subscript n minus 1 implies Theorem B Subscript n ; cf. 4.4.

Theorem A Subscript n and Theorem B Subscript n imply Theorem C Subscript n ; cf. 5.6.

Theorem D Subscript n minus 1 , Theorem B Subscript n and Theorem C Subscript n imply Theorem D Subscript n ; cf. 6.6.

Theorem C Subscript n and Theorem D Subscript n imply Theorem E Subscript n ; cf. 7.3.

Theorem C Subscript n and Theorem D Subscript n imply Theorem F Subscript 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

upper R left-parenthesis upper X comma upper K Subscript upper X Baseline right-parenthesis equals circled-plus Underscript m element-of double-struck upper N Endscripts upper H Superscript 0 Baseline left-parenthesis upper X comma script upper O Subscript upper X Baseline left-parenthesis m upper K Subscript upper X Baseline right-parenthesis right-parenthesis

of a smooth projective variety upper 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 upper D element-of StartAbsoluteValue k upper K Subscript upper X Baseline EndAbsoluteValue , whose existence is guaranteed as we are assuming that upper K Subscript upper X is big, and then to restrict to upper D . One obtains an exact sequence

0 long right-arrow upper H Superscript 0 Baseline left-parenthesis upper X comma script upper O Subscript upper X Baseline left-parenthesis left-parenthesis l minus k right-parenthesis upper K Subscript upper X Baseline right-parenthesis right-parenthesis long right-arrow upper H Superscript 0 Baseline left-parenthesis upper X comma script upper O Subscript upper X Baseline left-parenthesis l upper K Subscript upper X Baseline right-parenthesis right-parenthesis long right-arrow upper H Superscript 0 Baseline left-parenthesis upper D comma script upper O Subscript upper D Baseline left-parenthesis m upper K Subscript upper D Baseline right-parenthesis right-parenthesis comma

where l equals m left-parenthesis 1 plus k right-parenthesis is divisible by k plus 1 , and it is easy to see that it suffices to prove that the restricted algebra, given by the image of the maps

upper H Superscript 0 Baseline left-parenthesis upper X comma script upper O Subscript upper X Baseline left-parenthesis m left-parenthesis 1 plus k right-parenthesis upper K Subscript upper X Baseline right-parenthesis right-parenthesis long right-arrow upper H Superscript 0 Baseline left-parenthesis upper D comma script upper O Subscript upper D Baseline left-parenthesis m upper K Subscript upper D Baseline right-parenthesis right-parenthesis comma

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

upper H Superscript 0 Baseline left-parenthesis upper X comma script upper O Subscript upper X Baseline left-parenthesis m left-parenthesis 1 plus k right-parenthesis upper K Subscript upper X Baseline right-parenthesis right-parenthesis long right-arrow upper H Superscript 0 Baseline left-parenthesis upper S comma script upper O Subscript upper S Baseline left-parenthesis m upper K Subscript upper S Baseline right-parenthesis right-parenthesis

no longer has any obvious connection with

upper H Superscript 0 Baseline left-parenthesis upper X comma script upper O Subscript upper X Baseline left-parenthesis left-parenthesis m left-parenthesis 1 plus k right-parenthesis minus k right-parenthesis upper K Subscript upper X Baseline right-parenthesis right-parenthesis comma

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

circled-plus Underscript m element-of double-struck upper N Endscripts upper H Superscript 0 Baseline left-parenthesis upper S comma script upper O Subscript upper S Baseline left-parenthesis m upper K Subscript upper S Baseline right-parenthesis right-parenthesis comma

since it is only the latter that we can handle by induction. Yet another problem is that if upper C is a component of upper S , it is no longer the case that upper 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 upper K Subscript upper C is pseudo-effective, it is not clear that the linear system StartAbsoluteValue k upper K Subscript upper C Baseline EndAbsoluteValue is non-empty for any k greater-than 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 upper D equals upper D Subscript k , which entails working with infinitely many different birational models of upper X (since for every different value of k , one needs to resolve the singularities of upper 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 upper D we need to blow up a subvariety upper V . The corresponding divisor upper C will typically fibre over upper V and if upper V has codimension two, then upper C will be close to a double-struck upper P Superscript 1 -bundle over upper V . In the best case, the projection pi colon upper C long right-arrow upper V will be a double-struck upper P Superscript 1 -bundle with two disjoint sections (this is the toroidal case) and sections of tensor powers of a line bundle on upper C will give sections of an algebra on upper V which is graded by double-struck upper N squared , rather than just double-struck upper N . Let us consider then the simplest possible algebras over double-struck upper C which are graded by double-struck upper N squared . If we are given a submonoid upper M subset-of double-struck upper N squared (that is, a subset of double-struck upper N squared which contains the origin and is closed under addition), then we get a subalgebra upper R subset-of double-struck upper C squared left-bracket x comma y right-bracket spanned by the monomials

StartSet x Superscript i Baseline y Superscript j Baseline vertical-bar left-parenthesis i comma j right-parenthesis element-of upper M EndSet period

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

upper M equals StartSet left-parenthesis i comma j right-parenthesis element-of double-struck upper N squared vertical-bar j greater-than 0 EndSet union StartSet left-parenthesis 0 comma 0 right-parenthesis EndSet and StartSet left-parenthesis i comma j right-parenthesis element-of double-struck upper N squared vertical-bar i greater-than-or-equal-to StartRoot 2 EndRoot j EndSet period

In fact, if script upper C subset-of double-struck upper R squared is the convex hull of the set upper M , then upper M is finitely generated iff script upper C is a rational polytope. In the general case, we will be given a convex subset script upper C of a finite dimensional vector space of Weil divisors on upper X and a key part of the proof is to show that the set script upper 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 upper X . The first part is to prove the existence of pl-flips. This is proved by induction in Reference8, 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 upper D element-of StartAbsoluteValue k upper K Subscript upper X Baseline EndAbsoluteValue , and to pass to a log resolution of the support upper S of upper D . By way of induction we want to work with upper K Subscript upper X Baseline plus upper S rather than upper K Subscript upper X . As before this is tricky since a log terminal model for upper K Subscript upper X Baseline plus upper S is not the same as a log terminal model for upper K Subscript upper X . In other words, having added upper S , we really want to subtract it as well. The trick however is to first add upper S , construct a log terminal model for upper K Subscript upper X Baseline plus upper S and then subtract upper S (almost literally component by component). This is one of the key steps to show that Theorem A Subscript n and Theorem B Subscript n imply Theorem C Subscript 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 upper K Subscript upper X ; see §5.

To gain intuition for how this part of the proof works, let us first consider a simplified case. Suppose that upper D equals upper S is irreducible. In this case it is clear that upper S is of general type and upper K Subscript upper X is nef if and only if upper K Subscript upper X Baseline plus upper S is nef and in fact a log terminal model for upper K Subscript upper X is the same as a log terminal model for upper K Subscript upper X Baseline plus upper S . Consider running the left-parenthesis upper K Subscript upper X Baseline plus upper S right-parenthesis -MMP. Then every step of this MMP is a step of the upper K Subscript upper X -MMP and vice versa. Suppose that we have a left-parenthesis upper K Subscript upper X Baseline plus upper S right-parenthesis -extremal ray upper R . Let pi colon upper X long right-arrow upper Z be the corresponding contraction. Then upper S dot upper R less-than 0 , so that every curve normal upper Sigma contracted by pi must be contained in upper S . In particular pi cannot be a divisorial contraction, as upper S is not uniruled. Hence pi is a pl-flip and by Theorem A Subscript n , we can construct the flip of pi , phi colon upper X right dasheD arrow upper Y . Consider the restriction psi colon upper S right dasheD arrow upper T of phi to upper S , where upper T is the strict transform of upper S . Since log discrepancies increase under flips and upper 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 upper S cannot keep dropping. Consider what happens if we restrict to upper S . By adjunction, we have

left-parenthesis upper K Subscript upper X Baseline plus upper S right-parenthesis vertical-bar Subscript upper S Baseline equals upper K Subscript upper S Baseline period

Thus psi colon upper S right dasheD arrow upper T is upper K Subscript upper S -negative. We have to show that this cannot happen infinitely often. If we knew that every sequence of flips on upper 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 upper S , so that we have to do something slightly different. Instead we throw in an auxiliary ample divisor upper H on upper X , and consider upper K Subscript upper X Baseline plus upper S plus t upper H , where t is a positive real number. If t is large enough, then upper K Subscript upper X Baseline plus upper S plus t upper H is ample. Decreasing t , we may assume that there is an extremal ray upper R such that left-parenthesis upper K Subscript upper X Baseline plus upper S plus t upper H right-parenthesis dot upper R equals 0 . If t equals 0 , then upper K Subscript upper X Baseline plus upper S is nef and we are done. Otherwise left-parenthesis upper K Subscript upper X Baseline plus upper S right-parenthesis dot upper R less-than 0 , so that we are still running a left-parenthesis upper K Subscript upper X Baseline plus upper S right-parenthesis -MMP, but with the additional restriction that upper K Subscript upper X Baseline plus upper S plus t upper H is nef and trivial on any ray we contract. This is the left-parenthesis upper K Subscript upper X Baseline plus upper S right-parenthesis -MMP with scaling of upper H . Let upper G equals upper H vertical-bar Subscript upper S Baseline . Then upper K Subscript upper S Baseline plus t upper G is nef and so is upper K Subscript upper T Baseline plus t upper G prime , where upper G prime equals psi Subscript asterisk Baseline upper G . In this case upper K Subscript upper T Baseline plus t upper G prime is a weak log canonical model for upper K Subscript upper S Baseline plus t upper G (it is not a log terminal model, both because psi might contract divisors on which upper K Subscript upper S Baseline plus t upper G is trivial and more importantly because upper T need not be double-struck upper Q -factorial). In this case we are then done, by finiteness of weak log canonical models for left-parenthesis upper S comma t upper G right-parenthesis , where t element-of left-bracket 0 comma 1 right-bracket (cf. Theorem E Subscript n minus 1 ).

We now turn to the general case. The idea is similar. First we want to use finiteness of log terminal models on upper S to conclude that there are only finitely many log terminal models in a neighbourhood of upper 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 upper X right dasheD arrow upper Y is upper K Subscript upper X -negative, then upper K Subscript upper X is bigger than upper K Subscript upper Y (the difference is an effective divisor on a common resolution) so that we can never return to the same neighbourhood of upper S . As already pointed out, in the general case we need to work with double-struck upper 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 upper K Subscript upper X Baseline plus normal upper Delta equals upper K Subscript upper X Baseline plus upper S plus upper A plus upper B , where upper S is a sum of prime divisors, upper A is an ample divisor (with rational coefficients) and the coefficients of upper B are real numbers between zero and one. We are also given a divisor upper D greater-than-or-equal-to 0 such that upper K Subscript upper X Baseline plus normal upper Delta tilde Subscript double-struck upper R Baseline upper 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 upper S is a prime divisor and that upper K Subscript upper X Baseline plus normal upper Delta is purely log terminal. We fix upper S and upper A but we allow upper B to vary and we want to show that finiteness of log terminal models for upper S implies finiteness of log terminal models in a neighbourhood of upper S . We are free to pass to a log resolution, so we may assume that left-parenthesis upper X comma normal upper Delta right-parenthesis is log smooth and if upper B equals sigma-summation b Subscript i Baseline upper B Subscript i , then the coefficients left-parenthesis b 1 comma b 2 comma ellipsis comma b Subscript k Baseline right-parenthesis of upper B lie in left-bracket 0 comma 1 right-bracket Superscript k . Let normal upper Theta equals left-parenthesis normal upper Delta minus upper S right-parenthesis vertical-bar Subscript upper S Baseline so that left-parenthesis upper K Subscript upper X Baseline plus normal upper Delta right-parenthesis vertical-bar Subscript upper S Baseline equals upper K Subscript upper S Baseline plus normal upper Theta .

Suppose that f colon upper X right dasheD arrow upper Y is a log terminal model of left-parenthesis upper X comma normal upper Delta right-parenthesis . There are three problems that arise, two of which are quite closely related. Suppose that g colon upper S right dasheD arrow upper T is the restriction of f to upper S , where upper T is the strict transform of upper S . The first problem is that g need not be a birational contraction. For example, suppose that upper X is a threefold and f flips a curve normal upper Sigma intersecting upper S , which is not contained in upper S . Then upper S dot normal upper Sigma greater-than 0 so that upper T dot upper E less-than 0 , where upper E is the flipped curve. In this case upper E subset-of upper T so that the induced birational map upper S right dasheD arrow upper T extracts the curve upper E . The basic observation is that upper E must have log discrepancy less than one with respect to left-parenthesis upper S comma normal upper Theta right-parenthesis . Since the pair left-parenthesis upper X comma normal upper Delta right-parenthesis is purely log terminal if we replace left-parenthesis upper X comma normal upper Delta right-parenthesis by a fixed model which is high enough, then we can ensure that the pair left-parenthesis upper S comma normal upper Theta right-parenthesis is terminal, so that there are no such divisors upper E , and g is then always a birational contraction. The second problem is that if upper E is a divisor intersecting upper S which is contracted to a divisor lying in upper T , then upper E intersection upper S is not contracted by g . For this reason, g is not necessarily a weak log canonical model of left-parenthesis upper S comma normal upper Theta right-parenthesis . However we can construct a divisor 0 less-than-or-equal-to normal upper Xi less-than-or-equal-to normal upper Theta such that g is a weak log canonical model for left-parenthesis upper S comma normal upper Xi right-parenthesis . Suppose that we start with a smooth threefold upper Y and a smooth surface upper T subset-of upper Y which contains a negative 2 -curve normal upper Sigma , such that upper K Subscript upper Y Baseline plus upper T is nef. Let f colon upper X long right-arrow upper Y be the blowup of upper Y along normal upper Sigma with exceptional divisor upper E and let upper S be the strict transform of upper T . Then f is a step of the left-parenthesis upper K Subscript upper X Baseline plus upper S plus e upper E right-parenthesis -MMP for any e greater-than 0 and f is a log terminal model of upper K Subscript upper X Baseline plus upper S plus e upper E . The restriction of f to upper S , g colon upper S long right-arrow upper T is the identity, but g is not a log terminal model for upper K Subscript upper S Baseline plus e normal upper Sigma , since upper K Subscript upper S Baseline plus e normal upper Sigma is negative along normal upper Sigma . It is a weak log canonical model for upper K Subscript upper T , so that in this case normal upper Xi equals 0 . The details of the construction of normal upper 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 upper X is a surface and upper 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,

left-parenthesis upper K Subscript upper Y Baseline plus upper T right-parenthesis vertical-bar Subscript upper T Baseline equals upper K Subscript upper T Baseline plus normal upper Phi period

The other parts of the third problem only occur in dimension three or more. For example, suppose that upper Z is the cone over a smooth quadric in double-struck upper P cubed and p colon upper X long right-arrow upper Z and q colon upper Y long right-arrow upper Z are the two small resolutions, so that the induced birational map f colon upper X right dasheD arrow upper Y is the standard flop. Let pi colon upper W long right-arrow upper Z blow up the maximal ideal, so that the exceptional divisor upper E is a copy of double-struck upper P Superscript 1 Baseline times double-struck upper P Superscript 1 . Pick a surface upper R which intersects upper E along a diagonal curve normal upper Sigma . If upper S and upper T are the strict transforms of upper R in upper X and upper Y , then the induced birational map g colon upper S long right-arrow upper T is an isomorphism (both upper S and upper T are isomorphic to upper R ). To get around this problem, one can perturb normal upper Delta so that f is the ample model, and one can distinguish between upper X and upper Y by using the fact that g is the ample model of left-parenthesis upper S comma normal upper Xi right-parenthesis . Finally it is not hard to write down examples of flops which fix normal upper Xi , but switch the individual components of normal upper Xi . In this case one needs to keep track not only of normal upper Xi but the individual pieces left-parenthesis g Subscript asterisk Baseline upper B Subscript i Baseline right-parenthesis vertical-bar Subscript upper T Baseline , 1 less-than-or-equal-to i less-than-or-equal-to k . We prove that an ample model f is determined in a neighbourhood of upper T by g , the different normal upper Phi and left-parenthesis f Subscript asterisk Baseline upper B Subscript i Baseline right-parenthesis vertical-bar Subscript upper T Baseline ; 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 normal upper Phi and the divisors left-parenthesis f Subscript asterisk Baseline upper B Subscript i Baseline right-parenthesis vertical-bar Subscript upper T Baseline , and this shows that there are only finitely many possibilities for f . This explains the implication Theorem E Subscript n minus 1 implies Theorem B Subscript n . The details are contained in §4.

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

Suppose that there is a divisor upper C such that

StartLayout 1st Row with Label left-parenthesis asterisk right-parenthesis EndLabel upper K Subscript upper X Baseline plus normal upper Delta tilde Subscript double-struck upper R comma upper U Baseline upper D plus alpha upper C comma EndLayout

where upper K Subscript upper X Baseline plus normal upper Delta plus upper C is divisorially log terminal and nef and the support of upper D is contained in upper S . If upper R is an extremal ray which is left-parenthesis upper K Subscript upper X Baseline plus normal upper Delta right-parenthesis -negative, then upper D dot upper R less-than 0 , so that upper S Subscript i Baseline dot upper R less-than 0 for some component upper S Subscript i of upper S . As before this guarantees the existence of flips. It is easy to see that the corresponding step of the left-parenthesis upper K Subscript upper X Baseline plus normal upper Delta right-parenthesis -MMP is not an isomorphism in a neighbourhood of upper S . Therefore the left-parenthesis upper K Subscript upper X Baseline plus normal upper Delta right-parenthesis -MMP with scaling of upper C must terminate with a log terminal model for upper K Subscript upper X Baseline plus normal upper Delta . To summarise, whenever the conditions above hold, we can always construct a log terminal model of upper K Subscript upper X Baseline plus normal upper Delta .

We now explain how to construct log terminal models in the general case. We may write upper D equals upper D 1 plus upper D 2 , where every component of upper D 1 is a component of upper S and no component of upper D 2 is a component of upper S . If upper D 2 is empty, that is, every component of upper D is a component of upper S , then we take upper C to be a sufficiently ample divisor, and the argument in the previous paragraph implies that upper K Subscript upper X Baseline plus normal upper Delta has a log terminal model. If upper D 2 not-equals 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 left-parenthesis upper X comma normal upper Theta equals normal upper Delta plus lamda upper D 2 right-parenthesis , where lamda is the largest real number so that the coefficients of normal upper Theta are at most one. Then more components of bottom left corner normal upper Theta bottom right corner are components of upper D . By induction left-parenthesis upper X comma normal upper Theta right-parenthesis has a neutral model, f colon upper X right dasheD arrow upper 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 upper Y right dasheD arrow upper Z for upper K Subscript upper Y Baseline plus g Subscript asterisk Baseline normal upper Delta . It is then automatic that the composition h equals g ring f colon upper X right dasheD arrow upper Z is a neutral model of upper K Subscript upper X Baseline plus normal upper Delta (since f is not left-parenthesis upper K Subscript upper X Baseline plus normal upper Delta right-parenthesis -negative, it is not true in general that h is a log terminal model of upper K Subscript upper X Baseline plus normal upper Delta ). However g is automatically a log terminal model provided we only contract components of the stable base locus of upper K Subscript upper X Baseline plus normal upper Delta . For this reason, we pick upper D so that we may write upper D equals upper M plus upper F , where every component of upper M is semiample and every component of upper F is a component of the stable base locus. This explains the implication Theorem A Subscript n and Theorem B Subscript n imply Theorem C Subscript n . The details are contained in §5.

Now we explain how to prove that if upper K Subscript upper X Baseline plus normal upper Delta equals upper K Subscript upper X Baseline plus upper A plus upper B is pseudo-effective, then upper K Subscript upper X Baseline plus normal upper Delta tilde Subscript double-struck upper R Baseline upper D greater-than-or-equal-to 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 upper H ,

h Superscript 0 Baseline left-parenthesis upper X comma script upper O Subscript upper X Baseline left-parenthesis bottom left corner m left-parenthesis upper K Subscript upper X Baseline plus normal upper Delta right-parenthesis bottom right corner plus upper H right-parenthesis right-parenthesis

is a bounded function of m . In this case it follows that upper K Subscript upper X Baseline plus normal upper Delta is numerically equivalent to the divisor upper N Subscript sigma Baseline left-parenthesis upper K Subscript upper X Baseline plus normal upper Delta right-parenthesis greater-than-or-equal-to 0 . It is then not hard to prove that Theorem C Subscript n implies that upper K Subscript upper X Baseline plus normal upper 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 left-parenthesis upper K Subscript upper X Baseline plus normal upper Delta right-parenthesis plus upper H comma

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

upper K Subscript upper X Baseline plus normal upper Delta equals upper K Subscript upper X Baseline plus upper S plus upper A plus upper B comma

where upper S is irreducible and left-parenthesis upper K Subscript upper X Baseline plus normal upper Delta right-parenthesis vertical-bar Subscript upper S Baseline is pseudo-effective, and the support of normal upper Delta has global normal crossings. Suppose first that upper K Subscript upper X Baseline plus normal upper Delta is double-struck upper Q -Cartier. We may write

left-parenthesis upper K Subscript upper X Baseline plus upper S plus upper A plus upper B right-parenthesis vertical-bar Subscript upper S Baseline equals upper K Subscript upper S Baseline plus upper C plus upper D comma

where upper C is ample and upper D greater-than-or-equal-to 0 . By induction we know that there is a positive integer m such that h Superscript 0 Baseline left-parenthesis upper S comma script upper O Subscript upper S Baseline left-parenthesis m left-parenthesis upper K Subscript upper S Baseline plus upper C plus upper D right-parenthesis right-parenthesis right-parenthesis greater-than 0 . To lift sections, we need to know that h Superscript 1 Baseline left-parenthesis upper X comma script upper O Subscript upper X Baseline left-parenthesis m left-parenthesis upper K Subscript upper X Baseline plus upper S plus upper A plus upper B right-parenthesis minus upper S right-parenthesis right-parenthesis equals 0 . Now

StartLayout 1st Row 1st Column m left-parenthesis upper K Subscript upper X Baseline plus normal upper Delta right-parenthesis minus left-parenthesis upper K Subscript upper X Baseline plus upper B right-parenthesis minus upper S 2nd Column equals left-parenthesis m minus 1 right-parenthesis left-parenthesis upper K Subscript upper X Baseline plus normal upper Delta right-parenthesis plus upper A 2nd Row 1st Column Blank 2nd Column equals left-parenthesis m minus 1 right-parenthesis left-parenthesis upper K Subscript upper X Baseline plus normal upper Delta plus StartFraction 1 Over m minus 1 EndFraction upper A right-parenthesis period EndLayout

As upper K Subscript upper X Baseline plus normal upper Delta plus upper A slash left-parenthesis m minus 1 right-parenthesis is big, we can construct a log terminal model phi colon upper X right dasheD arrow upper Y for upper K Subscript upper X Baseline plus normal upper Delta plus upper A slash left-parenthesis m minus 1 right-parenthesis , and running this argument on upper Y , the required vanishing holds by Kawamata-Viehweg vanishing. In the general case, upper K Subscript upper X Baseline plus upper S plus upper A plus upper B is an double-struck upper R -divisor. The argument is now a little more delicate as h Superscript 0 Baseline left-parenthesis upper S comma script upper O Subscript upper S Baseline left-parenthesis m left-parenthesis upper K Subscript upper S Baseline plus upper C plus upper D right-parenthesis right-parenthesis right-parenthesis does not make sense. We need to approximate upper K Subscript upper S Baseline plus upper C plus upper 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 upper Y constructed above does not depend on m , at least locally in a neighbourhood of upper T , the strict transform of upper S , and then the result follows by Diophantine approximation. This explains the implication Theorem D Subscript n minus 1 , Theorem B Subscript n and Theorem C Subscript n imply Theorem D Subscript 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 upper A and work with divisors of the form upper K Subscript upper X Baseline plus normal upper Delta equals upper K Subscript upper X Baseline plus upper A plus upper B , where the coefficients of upper B are variable. For ease of exposition, we assume that the supports of upper A and upper B have global normal crossings, so that upper K Subscript upper X Baseline plus normal upper Delta equals upper K Subscript upper X Baseline plus upper A plus sigma-summation b Subscript i Baseline upper B Subscript i is log canonical if and only if 0 less-than-or-equal-to b Subscript i Baseline less-than-or-equal-to 1 for all i . The key point is that we allow the coefficients of upper B to be real numbers, so that the set of all possible choices of coefficients left-bracket 0 comma 1 right-bracket Superscript k is a compact subset of double-struck upper R Superscript k . Thus we may check finiteness locally. In fact since upper A is ample, we can always perturb the coefficients of upper B so that none of the coefficients is equal to one or zero and so we may even assume that upper K Subscript upper X Baseline plus normal upper Delta is Kawamata log terminal.

Observe that we are certainly free to add components to upper B (formally we add components with coefficient zero and then perturb so that their coefficients are non-zero). In particular we may assume that upper 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 left-bracket 0 comma 1 right-bracket Superscript k by finitely many log terminal models. By compactness, it suffices to do this locally.

So pick b element-of left-bracket 0 comma 1 right-bracket Superscript k . There are two cases. If upper K Subscript upper X Baseline plus normal upper Delta is not pseudo-effective, then upper K Subscript upper X Baseline plus upper A plus upper B prime is not pseudo-effective, for upper B prime in a neighbourhood of upper B , and there are no weak log canonical models at all. Otherwise we may assume that upper K Subscript upper X Baseline plus normal upper Delta is pseudo-effective. By induction we know that upper K Subscript upper X Baseline plus normal upper Delta tilde Subscript double-struck upper R Baseline upper D greater-than-or-equal-to 0 . Then we know that there is a log terminal model phi colon upper X right dasheD arrow upper Y . Replacing left-parenthesis upper X comma normal upper Delta right-parenthesis by left-parenthesis upper Y comma normal upper Gamma equals phi Subscript asterisk Baseline normal upper Delta right-parenthesis , we may assume that upper K Subscript upper X Baseline plus normal upper Delta is nef. By the base point free theorem, it is semiample. Let upper X long right-arrow upper Z be the corresponding morphism. The key observation is that locally about normal upper Delta , any log terminal model over upper Z is an absolute log terminal model. Working over upper Z , we may assume that upper K Subscript upper X Baseline plus normal upper Delta is numerically trivial. In this case the problem of finding a log terminal model for upper K Subscript upper X Baseline plus normal upper Delta prime only depends on the line segment spanned by normal upper Delta and normal upper Delta prime . Working in a small box about normal upper 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 upper 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 upper B (and not just the space spanned by the components of upper B ). This poses no significant problem. This explains the implication Theorem C Subscript n and Theorem D Subscript n imply Theorem E Subscript n . The details are contained in §7.

The implication Theorem C Subscript n , Theorem D Subscript n and Theorem E Subscript n imply Theorem F Subscript 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 left-parenthesis upper X comma normal upper Delta right-parenthesis , where normal upper Delta is big. The first observation is that since the Kawamata log terminal condition is open, it is straightforward to show that normal upper Delta is double-struck upper Q -linearly equivalent to upper A plus upper B , where upper A is an ample double-struck upper Q -divisor, upper B greater-than-or-equal-to 0 and upper K Subscript upper X Baseline plus upper A plus upper B is Kawamata log terminal. The presence of the ample divisor upper 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 upper S is ample, so that if upper B does not contain upper S in its support, then the restriction of upper A plus upper B to upper S is big. This is very useful for induction.

Secondly, as we vary the coefficients of upper 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 upper A to perturb the coefficients of upper B so that they are bounded away from zero and upper K Subscript upper X Baseline plus upper A plus upper B is always Kawamata log terminal.

Finally, if left-parenthesis upper X comma normal upper Delta right-parenthesis is divisorially log terminal and f colon upper X long right-arrow upper Y is a left-parenthesis upper K Subscript upper X Baseline plus normal upper Delta right-parenthesis -trivial contraction, then upper K Subscript upper Y Baseline plus normal upper Gamma equals upper K Subscript upper Y Baseline plus f Subscript asterisk Baseline normal upper Delta is not necessarily divisorially log terminal, only log canonical. For example, suppose that upper Y is a surface with a simple elliptic singularity and f colon upper X long right-arrow upper Y is the blowup with exceptional divisor upper E . Then f is a weak log canonical model of upper K Subscript upper X Baseline plus upper E , but upper Y is not log terminal as it does not have rational singularities. On the other hand, if normal upper Delta equals upper A plus upper B , where upper A is ample, then upper K Subscript upper Y Baseline plus normal upper 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 upper X is not of general type. The main issue seems to be the implication upper K Subscript upper X pseudo-effective implies kappa left-parenthesis upper X comma upper K Subscript upper X Baseline right-parenthesis greater-than-or-equal-to 0 . In other words we need:

Conjecture 2.1

Let left-parenthesis upper X comma normal upper Delta right-parenthesis be a projective Kawamata log terminal pair.

If upper K Subscript upper X Baseline plus normal upper Delta is pseudo-effective, then kappa left-parenthesis upper X comma upper K Subscript upper X Baseline plus normal upper Delta right-parenthesis greater-than-or-equal-to 0 .

We also probably need

Conjecture 2.2

Let left-parenthesis upper X comma normal upper Delta right-parenthesis be a projective Kawamata log terminal pair.

If upper K Subscript upper X Baseline plus normal upper Delta is pseudo-effective and

h Superscript 0 Baseline left-parenthesis upper X comma script upper O Subscript upper X Baseline left-parenthesis bottom left corner m left-parenthesis upper K Subscript upper X Baseline plus normal upper Delta right-parenthesis bottom right corner plus upper H right-parenthesis right-parenthesis

is not a bounded function of m , for some ample divisor upper H , then kappa left-parenthesis upper X comma upper K Subscript upper X Baseline plus normal upper Delta right-parenthesis greater-than-or-equal-to 1 .

In fact, using the methods of this paper, together with some results of Kawamata (cf. Reference14 and Reference15), 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 left-parenthesis upper X comma normal upper Delta right-parenthesis be a projective Kawamata log terminal pair.

If upper K Subscript upper X Baseline plus normal upper Delta is nef, then it is semiample.

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

Conjecture 2.4

Let left-parenthesis upper X comma normal upper Delta right-parenthesis be a projective log canonical pair of dimension n .

If upper K Subscript upper X Baseline plus normal upper Delta is big, then left-parenthesis upper X comma normal upper Delta right-parenthesis 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 greater-than 0 .

Then the set of varieties upper X such that upper K Subscript upper X Baseline plus normal upper Delta has log discrepancy at least epsilon and minus left-parenthesis upper K Subscript upper X Baseline plus normal upper Delta right-parenthesis 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 double-struck upper C . We say that two double-struck upper Q -divisors upper D 1 , upper D 2 are double-struck upper Q -linearly equivalent ( upper D 1 tilde Subscript double-struck upper Q Baseline upper D 2 ) if there exists an integer m greater-than 0 such that m upper D Subscript i are linearly equivalent. We say that a double-struck upper Q -divisor upper D is double-struck upper Q -Cartier if some integral multiple is Cartier. We say that upper X is double-struck upper Q -factorial if every Weil divisor is double-struck upper Q -Cartier. We say that upper X is analytically double-struck upper Q -factorial if every analytic Weil divisor (that is, an analytic subset of codimension one) is analytically double-struck upper 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 upper X long right-arrow upper U be a proper morphism of normal algebraic spaces.

(1)

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

(2)

An double-struck upper R -Cartier divisor upper D is an double-struck upper R -linear combination of Cartier divisors.

(3)

Two double-struck upper R -divisors upper D and upper D prime are double-struck upper R -linearly equivalent over upper U , denoted upper D tilde Subscript double-struck upper R comma upper U Baseline upper D prime , if their difference is an double-struck upper R -linear combination of principal divisors and an double-struck upper R -Cartier divisor pulled back from upper U .

(4)

Two double-struck upper R -divisors upper D and upper D prime are numerically equivalent over upper U , denoted upper D identical-to Subscript upper U Baseline upper D prime , if their difference is an double-struck upper R -Cartier divisor such that left-parenthesis upper D minus upper D prime right-parenthesis dot upper C equals 0 for any curve upper C contained in a fibre of pi .

(5)

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

(6)

An double-struck upper R -Cartier divisor upper D on upper X is nef over upper U (or pi -nef) if upper D dot upper C greater-than-or-equal-to 0 for any curve upper C subset-of upper X , contracted by pi .

(7)

An double-struck upper R -divisor upper D is big over upper U (or pi -big) if limit sup StartFraction h Superscript 0 Baseline left-parenthesis upper F comma script upper O Subscript upper F Baseline left-parenthesis bottom left corner m upper D bottom right corner right-parenthesis right-parenthesis Over m Superscript dimension upper F Baseline EndFraction greater-than 0 comma

for the fibre upper F over any generic point of upper U . Equivalently upper D is big over upper U if upper D tilde Subscript double-struck upper R comma upper U Baseline upper A plus upper B , where upper A is ample over upper U and upper B greater-than-or-equal-to 0