We prove that the canonical ring of a smooth projective variety is finitely generated.
The purpose of this paper is to prove the following result in birational algebraic geometry:
Let be a projective Kawamata log terminal pair.
If is big and is pseudo-effective, then has a log terminal model.
In particular, it follows that if is big, then it has a log canonical model and the canonical ring is finitely generated. It also follows that if is a smooth projective variety, then the ring
is finitely generated.
The birational classification of complex projective surfaces was understood by the Italian algebraic geometers in the early 20th century: If is a smooth complex projective surface of non-negative Kodaira dimension, that is, then there is a unique smooth surface , birational to such that the canonical class is nef (that is for any curve ). is obtained from simply by contracting all that is, all smooth rational curves -curves, with If, on the other hand, . then , is birational to either or a ruled surface over a curve of genus .
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 smooth complex projective variety -dimensional we have: ,
If then there exists a minimal model, that is, a variety , birational to such that is nef.
If then there is a variety , birational to which admits a Fano fibration, that is, a morphism whose general fibres have ample anticanonical class .
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 with singularities. However, these singularities cannot be arbitrary. At the very minimum, we must still be able to compute for any curve So, we insist that . is (or sometimes we require the stronger property that -Cartier is We also require that -factorial). and the minimal model have the same pluricanonical forms. This condition is essentially equivalent to requiring that the induced birational map is -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 then one can construct Mori fibre spaces whenever , 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
is finitely generated and is big, then the canonical model is nothing more than the Proj of It is then automatic that the induced rational map . is -negative.
The other natural way to ensure that is is to factor -negative into a sequence of elementary steps all of which are We now explain one way to achieve this factorisation. -negative.
If is not nef, then, by the cone theorem, there is a rational curve such that and a morphism which is surjective, with connected fibres, onto a normal projective variety and which contracts an irreducible curve if and only if Note that . and is We have the following possibilities: -ample.
If this is the required Fano fibration. ,
If and contracts a divisor, then we say that is a divisorial contraction and we replace by .
If and does not contract a divisor, then we say that is a small contraction. In this case is not so that we cannot replace -Cartier, by Instead, we would like to replace . by its flip where , is isomorphic to in codimension and is In other words, we wish to replace some -ample. curves by -negative curves. -positive
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 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 in the 1980s by the work of Kawamata, Kollár, Mori, Reid, Shokurov and others. In particular, the existence of -folds flips was proved by Mori in -foldReference26.
Naturally, one would hope to extend these results to dimension and higher by induction on the dimension.
Recently, Shokurov has shown the existence of flips in dimension Reference34 and Hacon and McKernan Reference8 have shown that assuming the minimal model program in dimension (or even better simply finiteness of minimal models in dimension then flips exist in dimension ), 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 only some partial answers are available. Kawamata, Matsuda and Matsuki proved Reference18 the termination of terminal flips, Matsuki has shown -foldReference25 the termination of terminal flops and Fujino has shown -foldReference5 the termination of canonical (log) flips. Alexeev, Hacon and Kawamata -foldReference1 have shown the termination of Kawamata log terminal flips when the Kodaira dimension of -fold is non-negative and the existence of minimal models of Kawamata log terminal when either -folds or 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 pairs (namely the ascending chain condition for minimal log discrepancies and semicontinuity of log discrepancies; cf. -dimensionalReference35). Moreover, if 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:
Let be a Kawamata log terminal pair, where is Let -Cartier. be a projective morphism of quasi-projective varieties.
If either is and -big is or -pseudo-effective is then -big,
has a log terminal model over ,
if is then -big has a log canonical model over and ,
if is then the -Cartier,-algebra
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.
An immediate consequence of Theorem 1.2 is:
Let be a smooth projective variety of general type.
has a minimal model,
has a canonical model,
is finitely generated, and
has a model with a Kähler-Einstein metric.
Let be a projective Kawamata log terminal pair, where is -Cartier.
Then the ring
is finitely generated.
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:
Let be a projective morphism of normal quasi-projective varieties. Suppose that is Kawamata log terminal and is big over Let . , and be two log terminal models of , over Let ..
Then the birational map is the composition of a sequence of over -flops.
Note that Corollary 1.1.3 has been generalised recently to the case when 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 is well-behaved. To explain this, we need some definitions:
Let be a projective morphism of normal quasi-projective varieties, and let be a finite dimensional affine subspace of the real vector space of Weil divisors on Fix an . -divisor and define
Given a birational contraction over define ,
and given a rational map over define ,
We will adopt the convention that If the support of . has no components in common with any element of then the condition that , is vacuous. In many applications, will be an ample over -divisor In this case, we often assume that . is general in the sense that we fix a positive integer such that is very ample over and we assume that , where , is very general. With this choice of we have ,
and the condition that the support of has no common components with any element of is then automatic. The following result was first proved by Shokurov Reference32 assuming the existence and termination of flips:
Let be a projective morphism of normal quasi-projective varieties. Let be a finite dimensional affine subspace of which is defined over the rationals. Suppose there is a divisor such that is Kawamata log terminal. Let be a general ample over -divisor which has no components in common with any element of ,.
There are finitely many birational contractions over , such that
where each is a rational polytope. Moreover, if is a log terminal model of over for some , then , for some ,.
There are finitely many rational maps over , which partition into the subsets .
For every there is a and a morphism such that .
In particular is a rational polytope and each is a finite union of rational polytopes.
Let be a Kawamata log terminal pair and let be a big divisor. Suppose that is not pseudo-effective. The effective log threshold is
The Kodaira energy is the reciprocal of the effective log threshold.
Following ideas of Batyrev, one can easily show that:
Let be a projective Kawamata log terminal pair and let be an ample divisor. Suppose that is not pseudo-effective.
If both and are then the effective log threshold and the Kodaira energy are rational. -Cartier,
Let be a projective morphism of normal quasi-projective varieties. Let be a sequence of on -divisors The sheaf of .-algebras,
is called the Cox ring associated to .
Using Corollary 1.1.5 one can show that adjoint Cox rings are finitely generated:
Let be a projective morphism of normal quasi-projective varieties. Fix to be an ample over -divisor Let . for some , -divisors Assume that . is divisorially log terminal and Then the Cox ring, -Cartier.
is a finitely generated -algebras.
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 of stable curves of genus -pointed 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).
Let be the moduli space of stable curves of genus with marked points and let , denote the boundary divisors.
Let be a boundary. Then is log canonical and if is big, then there is a log canonical model Moreover if we fix a positive rational number . and require that the coefficient of is at least for each then the set of all log canonical models obtained this way is finite. ,
The next set of applications is to Fano varieties. The key observation is that given any divisor a small multiple of , is linearly equivalent to a divisor of the form where , is big and is Kawamata log terminal.
Let be a projective morphism of normal varieties, where is affine.
We say that is a Mori dream space if and the Cox ring is finitely generated over the coordinate ring of .
Let be a projective morphism of normal varieties, where is affine. Suppose that is -factorial, is divisorially log terminal and is ample over .
Then 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 one can run the , and this ends with either a nef model, or a fibration, for which -MMP, is relatively ample, and in fact any sequence of terminates. -flips
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 is not pseudo-effective, then we can run the MMP with scaling to get a Mori fibre space:
Let be a Kawamata log terminal pair. Let -factorial be a projective morphism of normal quasi-projective varieties. Suppose that is not -pseudo-effective.
Then we may run a over -MMP and end with a Mori fibre space over .
Note that we do not claim in Corollary 1.3.3 that however we run the over -MMP 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.
Let be a projective variety. A curve is called nef if for all Cartier divisors . denotes the cone of nef curves sitting inside and denotes its closure.
Now suppose that is a log pair. A co-extremal ray is an extremal ray - of the closed cone of nef curves on which is negative.
Let be a projective Kawamata log terminal pair such that -factorial is ample.
Then is a rational polyhedron. If is a ray, then there exists an -co-extremal -divisor such that the pair is Kawamata log terminal and the -MMP ends with a Mori fibre space such that is spanned by the pullback to of the class of any curve which is contracted by .
Another immediate consequence of Theorem 1.2 is the existence of flips:
Let be a Kawamata log terminal pair and let be a small contraction. -extremal
Then the flip of 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:
Let be a projective morphism of normal quasi-projective varieties. Let be a Kawamata log terminal pair, where -factorial is and -Cartier is Let -big. be an -divisor.
If is Kawamata log terminal and then we may run the -nef, over -MMP with scaling of .
Another application of Theorem 1.2 is the existence of log terminal models which extract certain divisors:
Let be a log canonical pair and let be a log resolution. Suppose that there is a divisor such that is Kawamata log terminal. Let be any set of valuations of divisors which satisfies the following two properties: -exceptional
contains only valuations of log discrepancy at most one, and
the centre of every valuation of log discrepancy one in does not contain any non-Kawamata log terminal centres.
Then we may find a birational morphism such that , is and the exceptional divisors of -factorial correspond to the elements of .
For example, if we assume that is Kawamata log terminal and we let be the set of all exceptional divisors with log discrepancy at most one, then the birational morphism defined in Corollary 1.4.3 above is a terminal model of In particular there is an . -divisor on such that and the pair is terminal.
If instead we assume that is Kawamata log terminal but is empty, then the birational morphism defined in Corollary 1.4.3 above is a log terminal model. In particular is small, is and there is an -factorial -divisor on such that .
We are able to prove that every log pair admits a birational model with singularities such that the non-Kawamata log terminal locus is a divisor: -factorial
Let be a log pair.
Then there is a birational morphism where , is such that if we write -factorial,
where every component of has coefficient less than one and every component of has coefficient at least one, then is Kawamata log terminal and nef over and no component of 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 be a log pair and let be the normalisation of a component of of coefficient one.
If we define by adjunction,
then the log discrepancy of is equal to the minimum of the log discrepancy with respect to of any valuation whose centre on is of codimension at least two and intersects .
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 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 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 does not contain any rational curves, then 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:
Let be a proper map of normal algebraic spaces, where is analytically -factorial.
If is divisorially log terminal and does not contract any rational curves, then is a log terminal model. In particular is projective and is -nef.
Theorem A (Existence of pl-flips).
Let be a pl-flipping contraction for an purely log terminal pair -dimensional.
Then the flip of exists.
Theorem B (Special finiteness).
Let be a projective morphism of normal quasi-projective varieties, where is of dimension -factorial Let . be a finite dimensional affine subspace of which is defined over the rationals, let , be the sum of finitely many prime divisors and let be a general ample over -divisor Let . be a divisorially log terminal pair such that Fix a finite set . of prime divisors on .
Then there are finitely many birational maps over such that if is any weak log canonical model over -factorial of where , which only contracts elements of , and which does not contract every component of then there is an index , such that the induced birational map is an isomorphism in a neighbourhood of the strict transforms of .
Theorem C (Existence of log terminal models).
Let be a projective morphism of normal quasi-projective varieties, where has dimension Suppose that . is Kawamata log terminal, where is big over .
If there exists an -divisor such that then , has a log terminal model over .
Theorem D (Non-vanishing theorem).
Let be a projective morphism of normal quasi-projective varieties, where has dimension Suppose that . is Kawamata log terminal, where is big over .
If is then there exists an -pseudo-effective, -divisor such that .
Theorem E (Finiteness of models).
Let be a projective morphism of normal quasi-projective varieties, where has dimension Fix a general ample . -divisor over Let . be a finite dimensional affine subspace of which is defined over the rationals. Suppose that there is a Kawamata log terminal pair .
Then there are finitely many birational maps over , such that if is a weak log canonical model of over for some , then there is an index , and an isomorphism such that .
Theorem F (Finite generation).
Let be a projective morphism to a normal affine variety. Let be a Kawamata log terminal pair of dimension where , is an ample and -divisor If . is pseudo-effective, then
The pair has a log terminal model In particular if . is then the log canonical ring -Cartier,
is finitely generated.
Let be the vector space spanned by the components of Then there is a constant . such that if is a prime divisor contained in the stable base locus of and such that then , is contained in the stable base locus of .
Let be the smallest affine subspace of containing which is defined over the rationals. Then there is a constant , and a positive integer such that if is any divisor and is any positive integer such that and is Cartier, then every component of is a component of the stable base locus of .
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
of a smooth projective variety 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 whose existence is guaranteed as we are assuming that , is big, and then to restrict to One obtains an exact sequence .
where is divisible by and it is easy to see that it suffices to prove that the restricted algebra, given by the image of the maps ,
is finitely generated. Various problems arise at this point. First is neither smooth nor even reduced (which, for example, means that the symbol is only formally defined; strictly speaking we ought to work with the dualising sheaf It is natural then to pass to a log resolution, so that the support ). of has simple normal crossings, and to replace by The second problem is that the kernel of the map .
no longer has any obvious connection with
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
since it is only the latter that we can handle by induction. Yet another problem is that if is a component of it is no longer the case that , 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 is pseudo-effective, it is not clear that the linear system is non-empty for any 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 . and hence which entails working with infinitely many different birational models of , (since for every different value of one needs to resolve the singularities of ,).
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 we need to blow up a subvariety The corresponding divisor . will typically fibre over and if has codimension two, then will be close to a over -bundle In the best case, the projection . will be a with two disjoint sections (this is the toroidal case) and sections of tensor powers of a line bundle on -bundle will give sections of an algebra on which is graded by rather than just , Let us consider then the simplest possible algebras over . which are graded by If we are given a submonoid . (that is, a subset of which contains the origin and is closed under addition), then we get a subalgebra spanned by the monomials
The basic observation is that is finitely generated iff is a finitely generated monoid. There are two obvious cases when is not finitely generated,
In fact, if is the convex hull of the set then , is finitely generated iff is a rational polytope. In the general case, we will be given a convex subset of a finite dimensional vector space of Weil divisors on and a key part of the proof is to show that the set 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 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 and to pass to a log resolution of the support , of By way of induction we want to work with . rather than As before this is tricky since a log terminal model for . is not the same as a log terminal model for In other words, having added . we really want to subtract it as well. The trick however is to first add , construct a log terminal model for , and then subtract (almost literally component by component). This is one of the key steps to show that Theorem A and Theorem B imply Theorem C 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 see § ;5.
To gain intuition for how this part of the proof works, let us first consider a simplified case. Suppose that is irreducible. In this case it is clear that is of general type and is nef if and only if is nef and in fact a log terminal model for is the same as a log terminal model for Consider running the . Then every step of this MMP is a step of the -MMP. and vice versa. Suppose that we have a -MMP ray -extremal Let . be the corresponding contraction. Then so that every curve , contracted by must be contained in In particular . cannot be a divisorial contraction, as is not uniruled. Hence is a pl-flip and by Theorem A we can construct the flip of , , Consider the restriction . of to where , is the strict transform of Since log discrepancies increase under flips and . is irreducible, is a birational contraction. After finitely many flips, we may therefore assume that does not contract any divisors, since the Picard number of cannot keep dropping. Consider what happens if we restrict to By adjunction, we have .
Thus is We have to show that this cannot happen infinitely often. If we knew that every sequence of flips on -negative. 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 so that we have to do something slightly different. Instead we throw in an auxiliary ample divisor , on and consider , where , is a positive real number. If is large enough, then is ample. Decreasing we may assume that there is an extremal ray , such that If . then , is nef and we are done. Otherwise so that we are still running a , but with the additional restriction that -MMP, is nef and trivial on any ray we contract. This is the with scaling of -MMP Let . Then . is nef and so is where , In this case . is a weak log canonical model for (it is not a log terminal model, both because might contract divisors on which is trivial and more importantly because need not be In this case we are then done, by finiteness of weak log canonical models for -factorial). where , (cf. Theorem E).
We now turn to the general case. The idea is similar. First we want to use finiteness of log terminal models on to conclude that there are only finitely many log terminal models in a neighbourhood of 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 . is then -negative, is bigger than (the difference is an effective divisor on a common resolution) so that we can never return to the same neighbourhood of As already pointed out, in the general case we need to work with . 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 -divisors. where , is a sum of prime divisors, is an ample divisor (with rational coefficients) and the coefficients of are real numbers between zero and one. We are also given a divisor such that 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 is a prime divisor and that is purely log terminal. We fix and but we allow to vary and we want to show that finiteness of log terminal models for implies finiteness of log terminal models in a neighbourhood of We are free to pass to a log resolution, so we may assume that . is log smooth and if then the coefficients , of lie in Let . so that .
Suppose that is a log terminal model of There are three problems that arise, two of which are quite closely related. Suppose that . is the restriction of to where , is the strict transform of The first problem is that . need not be a birational contraction. For example, suppose that is a threefold and flips a curve intersecting which is not contained in , Then . so that where , is the flipped curve. In this case so that the induced birational map extracts the curve The basic observation is that . must have log discrepancy less than one with respect to Since the pair . is purely log terminal if we replace by a fixed model which is high enough, then we can ensure that the pair is terminal, so that there are no such divisors and , is then always a birational contraction. The second problem is that if is a divisor intersecting which is contracted to a divisor lying in then , is not contracted by For this reason, . is not necessarily a weak log canonical model of However we can construct a divisor . such that is a weak log canonical model for Suppose that we start with a smooth threefold . and a smooth surface which contains a -curve such that , is nef. Let be the blowup of along with exceptional divisor and let be the strict transform of Then . is a step of the for any -MMP and is a log terminal model of The restriction of . to , is the identity, but is not a log terminal model for since , is negative along It is a weak log canonical model for . so that in this case , The details of the construction of . are contained in Lemma 4.1.
The third problem is that the birational contraction does not determine This is most transparent in the case when . is a surface and is a curve, since in this case is always an isomorphism. To remedy this particular part of the third problem we use the different, which is defined by adjunction,
The other parts of the third problem only occur in dimension three or more. For example, suppose that is the cone over a smooth quadric in and and are the two small resolutions, so that the induced birational map is the standard flop. Let blow up the maximal ideal, so that the exceptional divisor is a copy of Pick a surface . which intersects along a diagonal curve If . and are the strict transforms of in and then the induced birational map , is an isomorphism (both and are isomorphic to To get around this problem, one can perturb ). so that is the ample model, and one can distinguish between and by using the fact that is the ample model of Finally it is not hard to write down examples of flops which fix . but switch the individual components of , In this case one needs to keep track not only of . but the individual pieces , We prove that an ample model . is determined in a neighbourhood of by the different , and see Lemma ;4.3. To finish this part, by induction we assume that there are finitely many possibilities for and it is easy to see that there are then only finitely many possibilities for the different and the divisors and this shows that there are only finitely many possibilities for , This explains the implication Theorem .E implies Theorem B The details are contained in § .4.
The second part consists of using finiteness of models in a neighbourhood of to run a sequence of minimal model programs to construct a log terminal model. We may assume that is smooth and the support of has normal crossings.
Suppose that there is a divisor such that
where is divisorially log terminal and nef and the support of is contained in If . is an extremal ray which is then -negative, so that , for some component of As before this guarantees the existence of flips. It is easy to see that the corresponding step of the . is not an isomorphism in a neighbourhood of -MMP Therefore the . with scaling of -MMP must terminate with a log terminal model for To summarise, whenever the conditions above hold, we can always construct a log terminal model of ..
We now explain how to construct log terminal models in the general case. We may write where every component of , is a component of and no component of is a component of If . is empty, that is, every component of is a component of then we take , to be a sufficiently ample divisor, and the argument in the previous paragraph implies that has a log terminal model. If 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 , where , is the largest real number so that the coefficients of are at most one. Then more components of are components of By induction . has a neutral model, It is then easy to check that the conditions in the paragraph above apply, and we can construct a log terminal model . for It is then automatic that the composition . is a neutral model of (since is not it is not true in general that -negative, is a log terminal model of However ). is automatically a log terminal model provided we only contract components of the stable base locus of For this reason, we pick . so that we may write where every component of , is semiample and every component of is a component of the stable base locus. This explains the implication Theorem A and Theorem B imply Theorem C The details are contained in § .5.
Now we explain how to prove that if is pseudo-effective, then 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 .,
is a bounded function of In this case it follows that . is numerically equivalent to the divisor It is then not hard to prove that Theorem .C implies that 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
when is sufficiently large. Passing to a log resolution, and using standard arguments, we are reduced to the case when
where is irreducible and is pseudo-effective, and the support of has global normal crossings. Suppose first that is We may write -Cartier.
where is ample and By induction we know that there is a positive integer . such that To lift sections, we need to know that . Now .
As is big, we can construct a log terminal model for and running this argument on , the required vanishing holds by Kawamata-Viehweg vanishing. In the general case, , is an The argument is now a little more delicate as -divisor. does not make sense. We need to approximate by rational divisors, which we can do by induction. But then it is not so clear how to choose In practice we need to prove that the log terminal model . constructed above does not depend on at least locally in a neighbourhood of , the strict transform of , and then the result follows by Diophantine approximation. This explains the implication Theorem ,D Theorem ,B and Theorem C imply Theorem D 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 and work with divisors of the form where the coefficients of , are variable. For ease of exposition, we assume that the supports of and have global normal crossings, so that is log canonical if and only if for all The key point is that we allow the coefficients of . to be real numbers, so that the set of all possible choices of coefficients is a compact subset of Thus we may check finiteness locally. In fact since . is ample, we can always perturb the coefficients of so that none of the coefficients is equal to one or zero and so we may even assume that is Kawamata log terminal.
Observe that we are certainly free to add components to (formally we add components with coefficient zero and then perturb so that their coefficients are non-zero). In particular we may assume that 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 by finitely many log terminal models. By compactness, it suffices to do this locally.
So pick There are two cases. If . is not pseudo-effective, then is not pseudo-effective, for in a neighbourhood of and there are no weak log canonical models at all. Otherwise we may assume that , is pseudo-effective. By induction we know that Then we know that there is a log terminal model . Replacing . by we may assume that , is nef. By the base point free theorem, it is semiample. Let be the corresponding morphism. The key observation is that locally about any log terminal model over , is an absolute log terminal model. Working over we may assume that , is numerically trivial. In this case the problem of finding a log terminal model for only depends on the line segment spanned by and Working in a small box about . 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 , 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 . (and not just the space spanned by the components of This poses no significant problem. This explains the implication Theorem ).C and Theorem D imply Theorem E The details are contained in § .7.
Let us end the sketch of the proof by pointing out some of the technical advantages with working with Kawamata log terminal pairs where , is big. The first observation is that since the Kawamata log terminal condition is open, it is straightforward to show that is equivalent to -linearly where , is an ample -divisor, and is Kawamata log terminal. The presence of the ample divisor 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 is ample, so that if does not contain in its support, then the restriction of to is big. This is very useful for induction.
Secondly, as we vary the coefficients of 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 , to perturb the coefficients of so that they are bounded away from zero and is always Kawamata log terminal.
Finally, if is divisorially log terminal and is a contraction, then -trivial is not necessarily divisorially log terminal, only log canonical. For example, suppose that is a surface with a simple elliptic singularity and is the blowup with exceptional divisor Then . is a weak log canonical model of but , is not log terminal as it does not have rational singularities. On the other hand, if where , is ample, then is always divisorially log terminal.
Having sketched the proof of Theorem 1.2, we should point out the main obstruction to extending these ideas to the case when is not of general type. The main issue seems to be the implication pseudo-effective implies In other words we need: .
Let be a projective Kawamata log terminal pair.
If is pseudo-effective, then .
We also probably need
Let be a projective Kawamata log terminal pair.
If is pseudo-effective and
is not a bounded function of for some ample divisor , then ,.
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 be a projective Kawamata log terminal pair.
If is nef, then it is semiample.
Let be a projective log canonical pair of dimension .
If is big, then 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 and a positive real number .
Then the set of varieties such that has log discrepancy at least and is ample forms a bounded family.
In this section we collect together some definitions and results.
We work over the field of complex numbers We say that two . -divisors , are equivalent -linearly( if there exists an integer ) such that are linearly equivalent. We say that a -divisor is if some integral multiple is Cartier. We say that -Cartier is if every Weil divisor is -factorial We say that -Cartier. is analytically if every analytic Weil divisor (that is, an analytic subset of codimension one) is analytically -factorial (i.e., some multiple is locally defined by a single analytic function). We recall some definitions involving divisors with real coefficients. -Cartier
Let be a proper morphism of normal algebraic spaces.
An divisor (frequently abbreviated to -Weil -divisor) on is an combination of prime divisors. -linear
An divisor -Cartier is an combination of Cartier divisors. -linear
Two -divisors and are equivalent over -linearly, denoted if their difference is an , combination of principal divisors and an -linear divisor pulled back from -Cartier.
Two -divisors and are numerically equivalent over , denoted if their difference is an , divisor such that -Cartier for any curve contained in a fibre of .
An divisor -Cartier is ample over (or if it is -ample) equivalent to a positive linear combination of ample (in the usual sense) Cartier divisors over -linearly.
An divisor -Cartier on is nef over (or if -nef) for any curve contracted by ,.
An -divisor is big over (or if -big)
for the fibre over any generic point of Equivalently . is big over if where , is ample over and