aa r X i v : . [ m a t h . AG ] A p r Finite generation of a canonical ring
Yujiro KawamataFebruary 21, 2013
Abstract
The purpose of this note is to review an algebraic proof of the finitegeneration theorem due to Birkar-Cascini-Hacon-McKernan [5] whosemethod is based on the Minimal Model Program (MMP). An analyticproof by Siu [57] will be reviewed by Mihai Paun.
The finite generation of canonical rings was a problem considered by Zariski[64], and the proof in the case of dimension 2 due to Mumford [41] in theappendix of [64] is one of the motivations towards the minimal model theoryof higher dimensional algebraic varieties.Let X be a smooth projective variety defined over a field k , D a divisor on X , and O X ( D ) the associated invertible sheaf. Many problems in algebraicgeometry are translated into questions on the vector space of holomorphicsections H ( X, D ) = H ( X, O X ( D )). The Riemann-Roch problem is to de-termine this vector space. For example, the Riemann-Roch theorem tellsus that the alternating sum P np =0 ( − p dim H p ( X, O X ( D )) is expressed interms of topological invariants.Instead of considering a single vector space, we look at the graded ring R ( X, D ) = ∞ M m =0 H ( X, mD )called the section ring for the pair (
X, D ), where we use the additive notationfor divisors instead of multiplicative one for sheaves; we have O X ( mD ) =1 X ( D ) ⊗ m . There are obvious multiplication homomorphisms H ( X, m D ) ⊗ H ( X, m D ) → H ( X, ( m + m ) D )and the section ring becomes a graded algebra over the base field k = H ( X, O X ). The following question arises naturally: Question 1.1.
Is the section ring R ( X, D ) finitely generated as a gradedalgebra over k ?If D is ample, then the answer is yes. On the other hand, if dim X = 2,then there exists an example where R ( X, D ) is not finitely generated ([64]).But the canonical divisor K X , a divisor corresponding to the sheaf of holo-morphic n -forms Ω nX for n = dim X , has a special status in algebraic geom-etry, and the canonical ring R ( X, K X ) is finitely generated: Theorem 1.2 (Finite Generation Theorem) . Let X be a smooth projectivevariety defined over a field k of characteristic . Then the canonical ring R ( X, K X ) is always finitely generated as a graded algebra over k . The canonical divisor, or more precisely its linear equivalence class, isspecial in many senses: • It is naturally attached to any smooth variety as the determinant bun-dle of the cotangent bundle. • It is related to the Serre-Grothendieck duality. • The canonical ring is a birational invariant , i.e., if f : X → Y isa birational morphism between smooth projective varieties, then thenatural homomorphism f ∗ : R ( X, K X ) → R ( Y, K Y ) is an isomorphism.There are two proofs, an algebraic one using the MMP by Birkar-Cascini-Hacon-McKernan [5] and an analytic one using complex analysis by Siu [57].We note that the algebraic proof yields the finite generation for varietieswhich are not necessarily of general type because the proof is naturally “log-arithmic” as we shall explain later.A geometric implication of the finite generation theorem is the existenceof the canonical model Proj R ( X, K X ). It is a traditional basic tool in theinvestigation of algebraic surfaces of general type.2he anti-canonical ring R ( X, − K X ) is not necessarily finitely generated.For example, if X is a ruled surface over a curve of genus greater than 1,then its anti-canonical ring is not finitely generated in general ([49]). It isnot a birational invariant either.It is well known that canonical rings of algebraic curves are generated byelements of degree at most 3, and those of algebraic surfaces are expected tobe generated by elements of degree at most 5. But examples for varieties ofdimension 3 show that there is no bound of degrees of generators for higherdimensional varieties even in fixed dimensions. A correct generalization forthe effective statement is a birationality question. Hacon-McKernan [13] andTakayama [58] proved that there exists a number m ( n ) depending only on thedimension n such that the pluricanonical system | m ( n ) K X | gives a birationalmap for arbitrary n -dimensional variety X of general type. Chen-Chen [6]found an explicit and more realistic bound in the case of dimension three.See also [63] and [46] for development in this direction.The results of the four authors [5] include the complementary case ofnegative Kodaira dimension where the canonical rings tell very little. Thestructure theorem as below, the existence of Mori fiber spaces, was anothermotivation for the MMP: Theorem 1.3.
Let X be a smooth projective variety defined over a fieldof characteristic . Assume that the canonical divisor K X is not pseudo-effective, i.e., not numerically equivalent to a limit of effective Q -divisors.Then there is a birational model of X which has a Mori fiber space structure. A minimal model exists uniquely given a fixed birational class of surfacesof non-negative Kodaira dimension. But the uniqueness of minimal modelsfails in higher dimensions even if the Kodaira dimension is non-negative. It isproved that arbitrary birationally equivalent minimal models are connectedby a sequence of operations called flops . Birationally equivalent Mori fiberspaces are connected by
Sarkisov links which are generalizations of elemen-tary transformations of ruled surfaces.The third and more distant motivation for the MMP is a complete classi-fication of algebraic varieties up to birational equivalence. For example, if theKodaira dimension is zero, then the canonical ring tells very little about thestructure of the variety. A minimal model of such a variety is still important3nd serves as a starting point of more detailed structure theory. We notethat the existence of minimal models in general is still an open problem.We always assume that the charactersitic of the base field is zero in thispaper. We expect that the theorem is still true in positive characteristic, butthe known proofs depend heavily on characteristic zero methods. The re-striction on the characteristic should be eventually removed and the minimalmodel program (MMP) should work in arbitrary characteristic situation. Butthis restriction is necessary for the proof presented here by the two reasons: • The resolution theorem of singularities by Hironaka is used extensively.It would be possible to extend the resolution theorem over arbitrarycharacteristic base or even mixed characteristic base. • The Kodaira vanishing theorem holds true only in characteristic 0.There are counterexamples in positive characteristic. The vanishingtheorem is one of our main technical tools in every corner of the minimalmodel theory. In order to extend our theory to arbitrary characteristic,totally different technique should be developed.We shall explain the following characteristic features of the MMP (cf.[29]): • inductive . The basic idea is to use the induction on the dimension.Then in more finer terms, the final product of the MMP, a minimalmodel or a Mori fiber space, is obtained by a step by step constructionscalled divisorial contractions and flips. • logarithmic . We consider pairs consisting of varieties and boundarydivisors instead of varieties alone. • relativistic . The proof requires formulation in relative situation, i.e.,we consider projective morphisms over base spaces.We shall explain these points in the course of the algebraic proof. It isimportant to note that the MMP is not yet completed because the followingconjectures are still open: • The termination conjecture of flips . This is the conjecture which is stillmissing in order to prove the existence of a minimal model for arbitraryalgebraic variety whose canonical divisor is pseudo-effective.4
The Abundance Conjecture . We expect that some pluricanonical sys-tems are base point free for arbitrary minimal models which are notnecessarily of general type. More precisely, if f : ( X, B ) → T is aprojective morphism from a DLT pair such that K X + B is nef over T ,then K X + B should be semi-ample over T , i.e., numerically equivalentto a pull-back of an ample R -divisor over T .In particular, the following conjecture is still open: “an algebraic varietywith negative Kodaira dimension is uniruled”.The organization of this paper is as follows. In order to motivate the useof the minimal models, we review a classical proof of the finite generationtheorem in the case of dimension two in §
2. We explain our point of viewtowrard minimal models, the numerical geometry, in § §
4, andtry to justify these cumbersome definitions. The algorithm of the MinimalModel Program (MMP) is explained in §
5. The main results are presentedin § § § §
9. The last section §
10 is concerned on thenon-uniqueness problem of the birational models.
We start with the low dimensional cases in order to explain the idea howto use the minimal models. We note that the proofs of the finite generationtheorem in dimensions one and two are valid in arbitrary characteristic.Let us consider the case dim X = 1 and let g be the genus of X . Thereare three cases: Case 1 : g ≥
2. Then K X is ample, hence the canonical ring is finitelygenerated. 5 ase 2 : g = 1. Then K X = 0, hence R ( X, K X ) ∼ = k [ x ]. Case 3 : g = 0. Then R ( X, K X ) ∼ = k .Therefore the canonical ring is always finitely generated in dimension one.The above division into cases is generalized to higher dimensions by abirational invariant called the Kodaira dimension of the pair (
X, D ) definedby κ ( X, D ) = trans.deg R ( X, D ) − R ( X, D ) = k , and κ ( X, D ) = −∞ otherwise. The latter convention comesfrom the estimate that dim H ( X, mD ) ∼ m κ for large m . κ ( X, D ) cantake values among −∞ , , , . . . , dim X . In particular, we define κ ( X ) = κ ( X, K X ). If κ ( X, D ) = dim X , then D is called big . X is said to be of general type if K X is big.We recall a proof by Mumford [41] (appendix to [64]) of the finite gener-ation theorem when dim X = 2 and X is of general type. Step 1 : A curve C on X is said to be a ( − -curve if C ∼ = P and( C ) = −
1, where ( C ) denotes the self intersection number. In other words,the normal bundle N C/X has degree −
1. Castelnuovo’s contraction theoremtells us that there exists a birational morphism X → X to another smoothprojective surface which contracts C to a point. By using Castelnuovo’scontraction theorem repeatedly, we obtain a birational morphism X → X ′ toa minimal model on which there is no ( − X ′ is still smooth. It is known that there exists unique minimal model X ′ though the order of contractions of ( − Step 2 : A curve C on X is said to be a ( − -curve if C ∼ = P and( C ) = −
2. Artin’s contraction theorem ([3]) states that all the ( − X ′ → X ′′ to the canonical model . The canonical model X ′′ is a projective surface with isolatedrational singularities called Du Val singularities or canonical singularities .Though X ′′ is singular, we have still define a canonical divisor K X ′′ as a Cartier divisor, hence the canonical ring R ( X ′′ , K X ′′ ). Step 3 : The canonical ring is invariant under the contractions: R ( X, K X ) ∼ = R ( X ′ , K X ′ ) ∼ = R ( X ′′ , K X ′′ ) .K X ′′ is ample, hence R ( X, K X ) is finitely generated.6tep 1 will be generalized to the MMP, and Step 2 to the Base Point FreeTheorem .A proof when dim X = 2 and κ = 1 uses Kodaira’s theory of ellipticsurfaces ([32]). Step 1 is the same as above. Let X ′ be a minimal model of X . It isunique again. Step 2 : By the Enriques classification, there is an elliptic surface structure f : X ′ → Y ; Y is a smooth projective curve and general fibers of f are ellipticcurves. There may be degenerate fibers called singular fibers . Step 3 : We define a Q -divisor , a divisor with rational coefficients, B Y = P i b i B i on Y which measures the degeneration of the morphism f . Thecoefficients b i belong to Z , corresponding to the fact that there exists anautomorphic form of degree 12. The values of the b i are determined by Ko-daira’s classification of singular fibers ([32]). If a singular fiber of f has finitelocal monodromy, then the corresponding coefficient belong to the interval(0 , B Y to another Q -divisor B ′ Y by using the automorphicform so that all the coefficients are less than 1. Thus we obtain the following canonical bundle formula of Kodaira: K X ′ = f ∗ ( K Y + B ′ Y ) . Therefore the canonical ring of X is isomorphic to the log canonical ring fora lower dimensional pair ( Y, B ′ Y ): R ( X, K X ) ∼ = R ( X ′ , K X ′ ) ∼ = R ( Y, K Y + B ′ Y ) . We note that the concept of the section ring is naturally extended to the casewhere the coefficients are no more integers as follows: R ( X, D ) = ∞ M m =0 H ( X, x mD y )where x mD y denotes the round down of mD obtained by taking the rounddowns of the coefficients. Since K Y + B ′ Y is ample, we have the finite gener-atedness of the canonical ring again.This is one of the motivations of the “log” theory which will be explainedlater. 7he case dim X = 2 and κ = 0 is easy; we have R ( X, K X ) ∼ = k [ x ].The classification of surfaces tells us that a minimal model of such a va-riety, which is unique again, is isomorphic to either an abelian surface, a K3surface, or their quotient. We note that the higher dimensional generaliza-tions of this class include interesting varieties such as Calabi-Yau manifoldsand hyperK¨ahler manifolds.In order to explain the second motivation for the MMP, we consider thecase where κ ( X ) = −∞ . In this case, the canonical ring is trivial and tellsvery little. Instead there is an explicit structure theorem when dim X = 2and κ ( X ) = −∞ . Step 1 is the same. Let X ′ be a minimal model of X . Step 2 : There are two cases according to the Enriques classification. Thefirst case is a ruled surface f : X ′ → Y ; Y is a smooth projective curve andall fibers are isomorphic to P . In the second case, we have X ′ ∼ = P .Step 2 will be generalized to a Mori fiber space in the MMP explainedlater. This is another motivation for the MMP.A minimal model X ′ of an algebraic surface X with κ ( X ) ≥ X and is independent of the contractionprocess of ( − − κ ( X ) = −∞ . It is knownthat birationally equivalent minimal models are connected by a sequence of elementary transformations .There is another approach toward the finite generation theorem using the Zariski decomposition ([64]). In the case of a surface X , there is a birationalmorphism f : X → X ′ to its minimal model. We can write K X = f ∗ K X + E ,where E is an effective divisor, a divisor with non-negative coefficients, whosesupport coincides with the exceptional locus, the locus where f is not anisomorphism. The point is that this decomposition can be constructed on X without using the minimal model provided that κ ( X ) ≥ D is an effective divisor on a smooth projective surface X , then there exist uniquely determined Q -divisors P and E , divisors withcoefficients in Q , which satisfy the following conditions:1. P is nef , i.e., the intersection numbers ( P · C ) are non-negative for allcurves C on X . 8. E is effective.3. If { E , . . . , E t } is the set of irreducible components of E , then ( P · E i ) =0 for all i and the matrix [( E i · E j )] is negative definite.For example, K X = f ∗ K X + E is the Zariski decomposition of the canon-ical divisor which is regarded as an effective Q -divisor when mK X is effectivefor a positive integer m . Therefore a minimal model is virtually obtained asthe nef part of the canonical divisor.We obtained a minimal model of a log surface by using the Zariski decom-position in the paper [18] as a log generalization of the minimal model theoryin dimension two. Let X be a smooth projective variety of dimension twoand B a reduced normal crossing divisor. Then the Zariski decomposition K X + B = P + E gives a minimal model of the pair ( X, B ); the supportof E can be contracted to points by a birational morphism f : X → X ′ to a normal surface and we have P = f ∗ K X ′ . We note that the minimalmodel X ′ may have singularities, called log terminal singularities , and thatthe coefficients of P are not necessarily integers even if those of B are equalto one. Moreover we can prove that a linear system | mP | is base point freefor a positive integer m .There is a higher dimensional generalization of the concept of the Zariskidecomposition. The technique of the base point free theorem explained laterimplies that the finite generation theorem holds if there exists a Zariski de-composition of a (log) canonical divisor ([21], [40]). But there is a counterex-ample for the existence of the Zariski decomposition if we consider arbitraryeffective divisors ([44]). On the other hand, the Zariski decomposition of alog canonical divisor should exist because the log canonical divisor of a mini-mal model will give the Zariski decomposition. The non-uniqueness of mini-mal models and the uniqueness of the Zariski decomposition are compatible,because birationally equivalent minimal models have equivalent canonicaldivisors.We note that the Zariski decomposition can be achieved only on a blown-up variety of the original model if it ever exists; this is a phenomenon indimension three or higher due to the existence of flips. The point is to re-solve simultaneously the base loci of all the pluricanonical systems on a fixedvariety. Such simultaneous resolution can be considered on the Zariski space,the inverse limit of blowing up sequences. The b -divisors by Shokurov [53]is a formal concept to consider such situations. The polytope decomposition9heorem explained later will solve this infinity problem and lead to the fliptheorem.There is also a complex analytic version of the Zariski decomposition byTsuji ([60], [9]). It’s existence is rather easily proved though it is not asstrong as the algebraic counterpart. We would like to generalize the minimal model theory to higher dimensionalvarieties. The first task is to generalize contraction theorems of Casteln-uovo and Artin. It is a highly non-linear problem to identify the locus tobe contracted and to construct a contraction morphism. A linearization ofthis problem is achieved by using the intersection numbers of divisors withcurves. The method is called numerical geometry . Please distinguish it fromenumerative geometry. The idea to use cones in real vector spaces in orderfor the investigation of the birational geometry goes back to Hironaka’s thesis[16].We consider a relative situation; let X be an algebraic variety which isprojective over a base space T . If the base space T is a point (absolute case),then X is just a projective variety. We shall need to consider more generalbase space T in order to use inductive arguments. Although the descriptionsof the relative situations are slightly longer, the same methods of proofs workas in the absolute case.We assume that X is normal. Let D = P i d i D i be an R -divisor , a formallinear combination of prime divisors D i , reduced irreducible subvarieties ofcodimension one, with coefficients d i in R . It is said to be an R -Cartierdivisor if it can be expressed as a linear combination of Cartier divisorswith coefficients in R . As a counterpart, let C = P i c i C i be an R - -cycle ,a formal linear combination of curves C i , reduced irreducible subvarieties ofdimension one, with coefficients c i in R . It is said to be relative for f if the C i are mapped to points of the base space. Then we can define a bilinear pairingcalled the intersection number ( D · C ) with values in R if D is R -Cartier and C is relative for f . Two R -Cartier divisors or two relative R -1-cycles aresaid to be numerically equivalent if they give the same intersection numberswhen paired to arbitrary counterparts. We note that we cannot define theintersection number of an arbitrary R -divisor which is not R -Cartier with acurve. 10e consider real vector spaces N ( X/T ) = { R -Cartier divisor } / numerical equivalence N ( X/T ) = { relative R -1-cycle } / numerical equivalencewhich are dual to each other and known to be finite dimensional. The di-mension ρ ( X/T ) = dim N ( X/T ) is called the
Picard number of X over T .It is an important numerical invariant next to the dimension dim X .Let N E ( X/T ) be the closed convex cone in N ( X/T ) generated by thenumerical classes of relative curves, and let Nef(
X/T ) be the nef cone definedas the dual closed convex cone in N ( X/T ):Nef(
X/T ) = { v ∈ N ( X/T ) | ( v · C ) ≥ ∀ C } where the C are arbitrary curves relative for f . An R -divisor is called rel-atively nef , nef over T , or f -nef , if its numerical class belongs to Nef( X/T ).The following Kleiman’s criterion is fundamental ([30]):
Theorem 3.1.
A Cartier divisor D is relatively ample over T if and onlyif its numerical class belongs to the interior Amp ( X/T ) of the nef coneNef ( X/T ) called the ample cone . An R -Cartier divisor D is said to be ample over T , or f -ample , if itsnumerical class belongs to the ample cone Amp( X/T ), i.e., if it is a linearcombination of ample Cartier divisors with positive linear coefficients. Wenote that an ample R -Cartier divisor does not in general become a Cartierdivisor by multiplying a positive number and it is not directly related to aprojective embedding.An R -Cartier divisor D is said to be big over T , or f -big , if one can write D = A + E for an f -ample R -Cartier divisor A and an effective R -divisor E , an R -divisor with non-negative coefficients. In other words, an R -divisoris big if it is bigger than an ample divisor. The set Big( X/T ) of numericalclasses of all the f -big R -divisors is an open cone in N ( X/T ) called the bigcone . D is said to be pseudo-effective over T if its numerical class belongs tothe the closure Psef( X/T ) of the big cone callled a pseudo-effective cone . Inother words, D is pseudo-effective if its numerical class is a limit of effective Q -divisors. We have the following commutative diagram of inclusions:Amp( X/S ) −−−→ Nef(
X/S ) y y Big(
X/S ) −−−→ Psef(
X/S )11here the cones on the left are open and on the right their closures.Let f : X → Y be an arbitrary surjective morphism to another normalvariety which is projective over T such that all the geometric fibers of f areconnected. Then the natural homomorphism f ∗ : N ( Y /T ) → N ( X/T ) isinjective, and the image of the nef cone satisfies f ∗ Nef(
Y /T ) = Nef(
X/T ) ∩ f ∗ N ( Y /T ) . If f is not an isomorphism, then F = f ∗ Nef(
Y /T ) is a face of Nef(
X/T ) inthe sense that F ⊂ ∂ Nef(
X/T ), because the pull-back of an ample divisor on Y is never ample on X . There is a dual face F ∗ of N E ( X/T ) defined by F ∗ = { w ∈ N E ( X/T ) | f ∗ w = 0 } . Indeed we have F ∗ = { w ∈ N E ( X/T ) | ( D · w ) = 0 ∀ D } where the D are arbitrary ample divisors on Y . By the Zariski main theorem,the morphism f is uniquely determined by the face F or F ∗ as long as Y isnormal.The converse is not true; there are faces which do not correspond to anymorphisms. The point of the MMP is that there always exists a morphism ifthe face F ∗ on the cone of curves lies in the negative side for the log canonicaldivisor, i.e., if the intersection numbers with the log canonical divisor arenegative for all non-zero vectors in F ∗ .In the case where X is smooth and T is a point, Mori proved in hisfamous paper [38] that, if the canonical divisor K X is not nef, then therealways exists an extremal ray , a face F ∗ with dim F ∗ = 1 on which K X isnegative. Moreover he proved that there exists a corresponding contractionmorphism from X in the case dim X = 3. These results were generalized tothe cone and contraction theorems explained later.The key point in [38] is to prove that there exists a rational curve whichgenerates an extremal ray F ∗ . In order to prove this, he employed the defor-mation theory over a base field of positive characteristic, a purely algebraicmethod. It is remarkable that this is still the only method to prove theexistence of rational curves; there is no purely characteristic zero proof norcomplex analytic proof. The result is generalized to arbitrary pairs with logterminal singularities in [22] using [37]; an extremal ray is always generatedby a rational curve. 12n this paper we shall only deal with cones in N ( X/T ) and not considerthose in N ( X/T ). The reason will be clear in the polytope decompositiontheorem.
In order to extend the concept of the minimality to higher dimensional va-rieties, we have to change the point of view. We look at canonical divisorsinstead of varieties themselves. Moreover, we have to deal with logarithmicpairs instead of varieties. Singularities appear naturally in this way.We explain how to compare canonical divisors of birationally equivalentvarieties as a way to compare different birational models. Let α : X X ′ be a birational map between smooth projective varieties. Then there exists athird smooth projective variety X ′′ with birational morphisms f : X ′′ → X and f ′ : X ′′ → X ′ such that α = f ′ ◦ f − . If we define canonical divisors K X and K X ′ by using the same rational differential n -form, then the difference f ∗ K X − ( f ′ ) ∗ K X ′ is a well defined divisor which is supported above the locuswhere α is not an isomorphism. Moreover, it is independent of the choice ofthe rational differential n -form which was used to define K X and K X ′ . Wedefine that an equality K X ≥ K X ′ holds if and only if their pull-backs satisfy f ∗ K X ≥ ( f ′ ) ∗ K X ′ .Let us consider some examples before stating a formal definition of log ter-minal singularities. If f : X → X ′ is a contraction morphism of a ( − C , then K X − C = f ∗ K X ′ . An important observation is that the canoicaldivisor becomes smaller when the variety becomes smaller. Therefore, wecome to the following (temporary) defeinition: A variety X is said to be a minimal model if K X is minimal among all birationally equivalent varieties .We note that, if f : X → X ′ is a contraction morphism of ( − K X = f ∗ K X ′ and X has canonical singularities .Let us consider an example in higher dimension. Let X be a smoothvariety of dimension n and E a prime divisor which is isomorphic to P n − and such that the normal bundle N E/X is isomorphic to O P n − ( − d ) for apositive integer d . Then there exists a birational morphism f : X → X ′ which contracts E to a point. If d ≥
2, then X ′ has an isolated quotientsingularity, a typical example of a log terminal singularity . We can still13efine a canonical divisor K X ′ as a Q -Cartier divisor , namely dK X ′ is aCartier divisor. We have an equation K X + (1 − nd ) E = f ∗ K X ′ . Thus K X > K X ′ if n > d . In this case, the singularity of X ′ is said to be terminal . For example, if n = 3 and d = 2, then K X > K X ′ . This terminalsingularity was first discovered by Mori [38]. d is the smallest positive integer for which dK X ′ becomes a Cartier divisor.Such an integer is called a Cartier index of the singularity. Therefore thereexists an isolated quotient singularity in dimension three which is terminaland has arbitrarily large Cartier index. This is the reason why there is nobound of degrees for the generators of canonical rings in dimension three.Indeed let X be a projective variety having only terminal singularities suchthat there is a point at which the Cartier index is d and that the canonicaldivisor K X is ample. Then the canonical ring of X is not generated byelements whose degrees are less than d .We have to consider the log version of a canonical divisor, i.e., the logcanonical divisor K X + B of a log pair ( X, B ) in the MMP. There are manyreasons which lead us to consider the log version: • The canonical divisor of a fiber space is described by using a log canon-ical divisor as explained already in the case of elliptic surfaces. Thecanonical divisor of a finite covering is similarly described. • The adjunction formula for log canonical divisors is the basic tool forthe inductive argument on dimensions. • The natural range for the power of vanishing theorems is the categoryof log varieties.An objact of the log theory is a pair (
X, B ) consisting of a normal variety X and an R -divisor B called a boundary . In many cases, we assume that B iseffective, i.e., the coefficients are non-negative. If B is not effective, we oftencall B a subboundary in order to distinguish from the effective case. Thecoefficients of B are not necessarily integers, but rational or even real num-bers. The rational coefficients naturally appear when we consider positivemultiples of divisors, while the real coefficients when taking the limits. The14anonical divisor K X is still defined as a Weil divisor as long as the variety X is normal. The log canonical divisor is the sum K X + B . For example, if X is smooth and B is a reduced divisor, i.e., all the coefficients are equal to1, with normal crossing support, then K X + B is the divisor correspondingto the determinant line bundle of logarithmic differential forms Ω X (log B ).This is the origin of the name “log”.We note that our log structures are different from the log structures ofFontaine-Illusie-Kato [17]. Both concepts are derived as generalizations ofnormal crossing divisors on smooth varieties. There is no log theory whichincludes both at the moment.We have to introduce singularities in higher dimensions by two resasons.The first is that the canoical divisor can become smaller when the varietyacquires singularities in the case dim X ≥ Definition 4.1.
Let X be a normal variety. It is said to be Q -factorial if anarbitrary prime divisor on X is a Q -Cartier divisor , a Q -linear combinationof Cartier divisors.Let ( X, B ) be a pair of a normal variety X and an R -divisor B = P i b i B i ,and let µ : Y → X be a projective birational morphism from another variety.The exceptional locus of µ is the smallest closed subset of Y such that µ is an isomorphism when restricted on its complement. µ is said to be a logresolution (in a strict sense) of the pair ( X, B ), if Y is smooth, the exceptionallocus is a normal crossing divisor, and moreover the union of the exceptionallocus and the strict transforms µ − ∗ B i of the prime divisors B i is a normalcrossing divisor.The pair ( X, B ) is said to be divisorially log terminal (DLT) or to haveonly divisorially log terminal singularities if there exists a log resolution suchthat the following conditions are satisfied:1. The coefficients b i belong to the interval (0 , K X + B is an R -Cartier divisor.3. We can write µ ∗ ( K X + B ) = K Y + µ − ∗ B + X d j D j d j < j , where µ − ∗ B = P b i µ − ∗ B i is the strict transform of B and the D j are prime divisors which are contained in the exceptionallocus of µ .If 0 < b i < i instead of the condition 1, then the pair is called KLT . In other words, we can write µ ∗ ( K X + B ) = K Y + B Y where the coefficients of B Y belong the interval (0 , log canonial (LC) if all the coefficients of B Y belong to the interval (0 ,
1] inthe above formula.If d j ≤ d j < j instead of the condition 3, then the pair iscalled canonical or terminal , respectively. We usually assume that B = 0 forcanonical or terminal singularities, but those with B = 0 are also useful.For example, if X is smooth and the support of B is a normal crossingdivisor, then the pair ( X, B ) is DLT (resp. KLT) if and only if the coefficientsare in (0 ,
1] (resp. (0 , K X + B is strictly smaller than K Y + µ − ∗ B + P D j . In other words, the pair is locally minimal among allbirational models in the log sense. DLT and KLT pairs are respectively called weal log terminal and log terminal in [29].The log canonical singularites do not behave so well as log terminal sin-gularities. For example, the underlying variety X may have non Cohen-Macaulay singularities. We can compare this with the fact that the under-lying variety of a DLT pair has always rational singularities in characteristiczero. But we have often a situation where there exists another boundary B ′ on an LC pair ( X, B ) such that (
X, B ′ ) is KLT. In this case there is noproblem because we can easily extend arguments in the MMP by usnig theperturbation to a new boundary (1 − ǫ ) B + ǫB ′ .To be KLT is an open condition under varying coefficients of the bound-ary, LC is closed, and DLT is intermediate. Each of them has advantage anddisadvantage. We can use a compactness argument for the LC pairs.The KLT condition is equivalent to the L condition in complex analysis.This stronger version of the log terminality is easier to handle and morenatural in some cases. For example, KLT condition is independent of a logresolution. Moreover, some statements are only true for KLT pairs. On the16ther hand, it is necessary to consider general DLT pairs for the inductionargument on the dimension. Indeed the adjunction formula holds only alonga boundary component with coefficient one.The pairs having Q -factorial DLT singularities can be characterized asthose which may appear in the process of the MMP if we start with pairs ofsmooth varieties and reduced normal crossing divisors. This is the categoryof pairs we work in the MMP. We explain the algorithm of the MMP in this section. There are some stan-dard references including [29], [33], [35], [36] and [8].The cone and the contraction theorems are fundamental for the MMP.As we already explained, some of the faces of the nef cone correspond tomorphisms while some are not. Mori’s discovery in [38] is that, if the dualface of the cone of curves is contained in the half space on which the canonicaldivisor is negative, then there should always exist a corresponding morphism.In particular we define an extremal ray to be such a one dimensional face ofthe cone of curves.Now we state the cone and contraction theorems for the log pairs ([29]).We state the theorem in the dual terminology on the cone of divisors:
Theorem 5.1 (Cone Theorem) . Let f : ( X, B ) → T be a projective mor-phism from a DLT pair, H an ample divisor for f , and ǫ a positive num-ber. Then the part of the nef cone Nef ( X/T ) which is visible from the point [ K X + B + ǫH ] ∈ N ( X.T ) is generated by finitely many points whose coor-dinates are rational numbers. If K X + B is nef, then the assertion is empty. If K X + B is not nef, then K X + B + ǫH is not nef either if ǫ is sufficiently small. The Come Theoremsays that the boundary of the nef cone facing the point [ K X + B + ǫH ] is likea boundary of a polyhedral cone, and consists of finitely many rational faces ,intersections of Nef( X/T ) with linear subspaces of N ( X/T ). When thepositive number ǫ approaches to zero, the number of visible faces increases,so that there may be eventually infinitely many faces. Theorem 5.2 (Contraction Theorem) . Let f : ( X, B ) → T be a projectivemorphism from a DLT pair, and let F ⊂ Nef ( X/T ) be a face which is visible rom the point [ K X + B + ǫH ] as above. Then there exists a surjective mor-phism φ : X → Y to a normal variety which is projective over the base space T such that the geometric fibers of φ are connected and that F = φ ∗ Nef ( Y /T ) . In particular, if the codimension of the face F is equal to one, then wecall φ a primitive contraction .The Contraction Theorem is equivalent to the Base Point Free Theorem: Theorem 5.3 (Base Point Free Theorem) . Let f : ( X, B ) → T be a pro-jective morphism from a DLT pair, and let D be a Cartier divisor on X .Assume that D is f -nef and that D − ǫ ( K X + B ) is f -ample for a smallpositive number ǫ . Then there exists a positive integer m such that mD isrelatively base point free for arbitrary m ≥ m . The conclusion is saying that the natural homomorphism f ∗ f ∗ O X ( mD ) → O X ( mD )is surjective. The Base Point Free Theorem implies in particular the Non-Vanishing Theorem which states that f ∗ O X ( mD ) = 0.We explain steps of the MMP formulated by Reid ([48]) modified to thelog and relative situation. The coefficients of the boundaries are real numbers,so that the limiting arguments are made possible.We consider the category of Q -factorial DLT pairs ( X, B ) consisting ofvarieties X which are projective over a base space T together with R -divisors B . Let f : X → T be the structure morphism. This category includes pairsconsisting of smooth varieties and normal crossing divisors with coefficientsbelonging to in the interval (0 , Case 1 : If K X + B is relatively nef over the base (i.e., f -nef), then ( X, B )is minimal . We stop here in this case.
Case 2 : If K X + B is not relatively nef, then there exists a primitivecontraction morphism φ : X → Y by the Cone and Contraction Theorems.It is important to note that we need to choose one of the codimension onefaces of the nef cone.There are three cases for φ . 18 If dim
X > dim Y , then φ is called a Mori fiber space . We stop here inthis case. • If φ is a birational morphism and contracts a divisor, i.e, if φ is a divisorial contraction , then the exceptional locus of φ is proved to bea prime divisor thanks to the Q -factoriality condition. The new pair( X ′ , B ′ ) = ( Y, φ ∗ B ) is again Q -factorial and DLT. We go back to Case1 or 2, and continue the program. • If φ is small , i.e., if φ changes the variety only in codimension two orhigher, then there exists another small projective birational morphism φ ′ : X ′ → Y for which K X ′ + B ′ is positive where B ′ is the stricttransform of B . We note that the set of prime divisors stay unchangedunder the birational map ( φ ′ ) − ◦ φ which is called a flip . The existenceof a flip is now a theorem, called the Flip Theorem , by Hacon andMcKernan [14]. The new pair ( X ′ , B ′ ) is again Q -factorial and DLT.We can continue the program.We should prove that there does not exist an infinite chain of flips inorder to obtain the final product, a minimal model or a Mori fiber space.This Termination of Flips is still a conjecture in the general case.It is important to note that an inequality K X + B > K X ′ + B ′ alwaysholds after a divisorial contraction or a flip.The Picard number drops by one after a divisorial contraction: ρ ( Y /T ) = ρ ( X/T ) −
1. Therefore the number of possible divisorial contractions isbounded by the Picard number.But the Picard number stays the same after a flip. Indeed there is anatural isomorphism ( φ ′∗ ) − ◦ φ ∗ : N ( X/T ) → N ( X ′ /T ). Under the iden-tification of real vector spaces by this isomorphism, the nef cones Nef( X/T )and Nef( X ′ /T ) are adjacent and touch each other along the face F , the pull-back of the nef cone Nef( Y /T ). Therefore we can investigate the changesof birational models by looking at a partial cone decomposition of the realvector space.
Example 5.4.
The following is an example of a flip called
Francia’s flip [10].This example was first considered as a counterexample to the existence of aminimal model in higher dimension. Now it is incorporated into the MMP.19e have dim X = 3 and B = 0 in this example. Let X ′ be a smoothprojective variety containing a smooth rational curve C ′ , i.e., a curve isomor-phic to P , whose normal bundle is isomorphic to O P ( − ⊕ O P ( − X with an isolated quotient singularity P and a smooth rational curve C on X passing through P such that X ′ \ C ′ is isomorphic to X \ C .There is a small contraction morphism φ : X → Y which contracts C to a point Q . K X is negative for φ ; ( K X · C ) = − <
0. There is a smallbiratiotnal morphism φ ′ : X ′ → Y on the other side such that φ ′ ( C ′ ) = Q . K X ′ is positive for φ ′ ; ( K X ′ · C ′ ) = 1.The MMP is a way to obtain a minimal model. We note that this is notthe only way. We give a formal definition of a minimal model and a canonicalmodel: Definition 5.5.
Let ( X , B ) be an LC pair which is projective over a base T . We assume that there exists another boundary B ′ such that ( X , B ′ ) isKLT. A new Q -factorial LC pair ( X, B ) projective over a base T togetherwith a birational map α : X X over T is said to be a minimal model of( X , B ) if the following conditions are satisfied:1. α is surjective in codimension one, i.e., an arbitrary codimension onescheme theoretic point of X is in the image of a morphism which rep-resents α .2. B = α ∗ B is the strict transform.3. If µ : Y → X and µ : Y → X are common resolutions, then thedifference µ ∗ ( K X + B ) − µ ∗ ( K X + B ) is effective. Moreover arbitrarycodimension one point of Y which remains codimension one on X butnot on X is contained in the support of the difference µ ∗ ( K X + B ) − µ ∗ ( K X + B ).A surjective morphism f : X → Z with connected geometric fibers froma minimal model to a normal variety projective over T is said to be the canonical model if K X + B is numerically equivalent to the pull-back f ∗ H for a relatively ample R -divisor H on Z .The triple ( X, B, α ) is more precisely called a marked minimal model .The last condition for the minimal model means that a prime divisor is20ontracted by α only if there is a reason to be contracted. The ellipticfibration considered in § B of a minimalmodel ( X, B ) is big over T , then a canonical model always exists. Theexistence of a canonical model in general is a conjecture called the AbundanceConjecture .A minimal model of a given pair may not be unique if it ever exists. Butthey are always equivalent; if (
X, B ) and ( X ′ , B ′ ) are two minimal modelsof a pair ( X , B ) over T , and if µ : Y → X and µ ′ : Y → X ′ are commonresolutions, then we have µ ∗ ( K X + B ) = ( µ ′ ) ∗ ( K X ′ + B ′ ). It follows that acanonical model is unique if it exists.We shall need a modified version of the MMP, called the MMP with scaling or the directed MMP . It is much easier version than the general MMP.The process of the MMP is not unique because we should choose a facein each step. The scaled version of the MMP has smaller ambiguity of thechoice. The termination conjecture for arbitrary sequence of flips is in factnot necessary in order to prove the existence of a minimal model or a Morifiber space. What we have to prove is that some sequence of flips terminates.It turned out that the termination conjecture is easier for the sequence of flipswhich is directed by an additional divisor. We can expect that a sequence offlips has more tendency to terminate when the boundary moves along a linesegment in the space of divisors.We take an additional effective R -Cartier divisor H besides the pair( X, B ) such that (
X, B + H ) is still DLT. We assume that K X + B + H is f -nef. Let t = min { t ∈ R ≥ | K X + B + tH is f -nef } ∈ [0 , . If K X + B is not f -nef, then we choose a face F such that [ K X + B + t H ] ∈ F .The existence of such a face is clear in the case where B is big. In the generalcase, the existence is proved by Birkar [4] using the boundedness of extremalrays [22].We perform a divisorial contraction or a flip to the pair ( X, B ). Let usdenote by (
X, B ) and H the new pair and the strict transform of H by abuseof language. Then the threshold t for the new pair decreases or stays thesame. In other words, the MMP is directed by the scaling H . If it reachesto 0, then K X + B becomes f -nef, so we are done.21he point is that the log pair ( X, B + t H ) is already minimal, since K X + B + t H is f -nef, though it is an intermediate model of the MMP for( X, B ). It will be proved, in the course of inductive argument, that the set ofunderlying varieties of minimal models is finite when we move the coefficientsof the boundary on the line segment joining B and B + H as above. This isthe Polytope Decomposition Theorem. We shall apply this theorem to theMMP with scaling for the termination argument. The paper [5] proved that minimal models exist for a Q -factorial DLT pairs( X, B ) in the case where the boundary divisor B is big with respect to thepair ( X, B ) over the base in the following sense:
Definition 6.1.
Let (
X, B ) be an LC pair. A subvariety Z of X is said to bean LC center of (
X, B ) if there exists a log resolution µ : Y → ( X, B ) suchthat, when one writes µ ∗ ( K X + B ) = K Y + B Y , there exists an irreduciblecomponent E of B Y whose coefficient is equal to one and such that µ ( E ) = Z .If ( X, B ) is a DLT pair, then it is easy to see that Z is an LC center ifand only if it is an irreducible component of the intersection of some of theirreducible components of B whose coefficients are equal to one. Definition 6.2.
Let f : ( X, B ) → T be a projective morphism from a DLTpair. An R -Cartier divisor D on X is said to be big with respect to ( X, B ) ifone can write D = A + E for an ample R -Cartier divisor A and an effective R -Cartier divisor E whose support does not contain any LC center of ( X, B ). D is called pseudo-effective with respect to ( X, B ) if D + A is big withrespect to ( X, B ) for any ample R -Cartier divisor A .There are many advantages to consider big boundaries. If B is big withrespect to ( X, B ), then the following hold: • The log canonical divisor K X + B is semi-ample, i.e., it is numericallyequivalent to a pull-back of an ample R -Cartier divisor by a morphism.In other words, the Abundance Conjecture holds in this case. In par-ticular, if B is a Q -divisor, then there exists a positive integer m suchthat m ( K X + B ) is relatively base point free over the base. This is aconsequence of the Base Point Free Theorem.22 The number of faces on the nef cone which was considered in the MMPis finite even if ǫ goes to zero. This is a consequence of the ConeTheorem. • The number of marked minimal models for a fixed pair is finite, wherea marked minimal model is a minimal model together with a birationalmap from the original pair. This is one of the results proved in [5].We note that there are examples where items 2 and 3 are false when B is not big.Hacon and McKernan [14] proved the following existence theorem of theflip (when combined with results in [5]): Theorem 6.3 (Flip Theorem) . Let ( X, B ) be a Q -factorial DLT pair whichis projective over a base, and φ : X → Y a small contraction morphism inthe MMP for K X + B . Then there exists a flip φ ′ : X ′ → Y . In order to prove the Flip Theorem, it is sufficient to consider the casewhere the pair (
X, B ) is KLT and B is a Q -divisor by perturbing the coef-ficients of B because the ampleness is an open condition. In this case, theexistence of a flip is equivalent to the finite generatedness of the sheaf ofgraded O Y -algebras ∞ M m =0 φ ∗ O X ( x m ( K X + B ) y )over O Y , which is a a special case of the Finite Generation Theorem. Inother words, the inductive procedure of the MMP decomposes a difficultglobal finite generation problem into easier local finite generation problems.The termination conjecture of flips has not yet been fully proved. But [5]proved a partial termination theorem for directed flips with scaling: Theorem 6.4 (Scaled Termination Theorem) . Let f : ( X, B ) → T be a pro-jective morphism from a Q -factorial DLT pair to a quasi-projective variety.Assume that B is big over T with respect to ( X, B ) . Then the MMP for ( X, B ) with scaling always terminates after a finite number of steps. By the MMP, we have the following consequence:
Theorem 6.5 (Existence of Models) . Let f : ( X, B ) → T be a projectivemorphism from a Q -factorial DLT pair to a quasi-projective variety. Assume hat B is big over T with respect to ( X, B ) . Then there exists is a birationalmap α : ( X, B ) ( X ′ , B ′ ) over T to a Q -factorial DLT pair with a projec-tive morphism f ′ : X ′ → T which satisfies the following conditions:1. α is surjective in codimension one, and B ′ = α ∗ B .2. If µ : Y → X and µ ′ : Y → X ′ are common resolutions, then µ ∗ ( K X + B ) − ( µ ′ ) ∗ ( K X ′ + B ′ ) is an effective R -divisor whose support containsstrict transforms of all the exceptional prime divisors of α .3. If [ K X + B ] ∈ Psef ( X/T ) , then ( X ′ , B ′ ) is a minimal model of ( X, B ) over T , i.e., K X ′ + B ′ is f ′ -nef. On the other hand, if [ K X + B ] Psef ( X/T ) , then ( X ′ , B ′ ) has a Mori fiber space structure φ : X ′ → Y over T , where dim Y < dim X ′ and − ( K X ′ + B ′ ) is φ -ample. Now we state the Finite Generation Theorem. We note that we do notneed to assume the bigness of the boundary:
Theorem 6.6 (Finite Generation Theorem) . Let f : ( X, B ) → T be a pro-jective morphism from a KLT pair to a quasi-projective variety. Assume inaddition that the boundary B is a Q -divisor. Then the relative canonical ring R ( X/T, K X + B ) = ∞ M m =0 f ∗ O X ( x m ( K X + B ) y ) is finitely generated as a graded O T -algebra. We note that the assumption of B being a Q -divisor is indispensable.Indeed the statement is clearly false for a non-rational R -divisor. On theother hand, the limiting argument in the proof of the existence of minimalmodels makes it necessary to consider R -divisors in the proof.The proof of the Finite Generation Theorem is as follows. If B is bigwith respect to the pair, then the theorem follows from the Base Point Freetheorem together with the existence theorem of a minimal model. For thegeneral case, we use the following reduction theorem ([12]): Theorem 6.7.
Let f : ( X, B ) → T be a projective morphism from a KLTpair to a quasi-projective variety. Assume that the boundary B is a Q -divisor,and that κ ( X η , ( K X + B ) | X η ) ≥ for the generic fiber X η of f . Then thereexists a projective birational morphism µ : Y → X from a smooth variety, surjective morphism f : Y → Z to a smooth variety projective over T , a Q -divisor C = P c j C j on Z such that P C j is a normal crossing divisorand c i ∈ (0 , , and a Q -divisor L on Z which is nef over T such that thefollowing are satisfied:1. K Z + C + L is big over T .2. f ◦ µ − induces an isomorphism of graded O T -algebras: R ( X/T, K X + B ) ∼ = R ( Z/T, K Z + C + L ) . The outline of the proof of this theorem is as follows. We consider arational map, called the
Iitaka fibration , defined by some positive multiple of K X + B whose image has relative dimension equal to κ ( X η , ( K X + B ) | X η ) over T , and modify it by changing birational models and using covering tricks.Then we apply the semi-positivity theorem of the Hodge bundles in [19].The rest of the proof of the Finite Generation Theorem is as follows. Wewrite K Z + C + L = A + E for an ample Q -divisor A and an effective Q -divisor E . Then we apply the big case to a new pair (1 + ǫ )( K Z + C + L ) = K Z + ( C + ǫE ) + ( L + ǫA ) for a small positive rational number ǫ .The following Polytope Decomposition Theorem for varying boundarieswas proved in [5] after Shokurov [52]. This theorem is interesting in its ownright besides its importance for the termination argument. We formulate thetheorem in a more exact form and in the case where the boundary is notnecessarily big. The first part is a decomposition according to the canonicalmodels: Theorem 6.8 (Polytope Decomposition Theorem 1) . Let ( X, ¯ B ) be a Q -factorial KLT pair with a projective morphism f : X → T to a base space, B , . . . , B r effective Q -Cartier divisors, and ¯ V a polytope contained in theset { B = P i b i B i | b i ∈ R } ∼ = R r such that the pairs ( X, B ) are LC for all B ∈ ¯ V . Consider a closed convex subset V = { B ∈ ¯ V | [ K X + B ] ∈ Psef ( X/T ) } . Assume that for each B ∈ V , there exist a minimal model α : ( X, B ) ( Y, C ) and a canonical model g : Y → Z for f : ( X, B ) → T . Moreover as-sume that there exists a real number ǫ > for each B such that the morphism : ( Y, α ∗ B ′ ) → Z for B ′ ∈ ¯ V has minimal and canonical models whenever B ′ ∈ Psef ( Y /Z ) and k B ′ − B k ≤ ǫ , where kk denotes the maximum norm ofthe coefficients. Then there exists a finite decomposition to disjoint subsets V = s a j =1 V j with rational maps β j : X Z j which satisfies the following conditions:1. B ∈ V j if and only if β j gives the canonical model for f : ( X, B ) → T .2. The closures ¯ V j , hence V , are polytopes for all j . Moreover, if ¯ V is arational polytope, then so are the ¯ V j and V . The second part is a finer decomposition according to the minimal models:
Theorem 6.9 (Polytope Decomposition Theorem 2) . Under the same as-sumptions as the above theorem, each V j are further decomposed into a finitedisjoint union V j = t a k =1 W j,k which satisfies the following conditions: let α : X Y be a birational mapsuch that W = { B ∈ V | α is a minimal model for ( X, B ) } is non-empty. Then1. There exists an index j such that W ⊂ ¯ V j .2. If W ∩ V j is non-empty for some j , then W coincides with one of the W j,k .3. The closure ¯ W j,k is a polytope for any j and k . Moreover, if ¯ V is arational polytope, then so are the ¯ W j,k . The existence of ¯ B is assumed only to ensure that the MMP works well.In the case where the B ∈ V are big with respect to the pairs, the existenceof minimal and canonical models is already proved, and the conditions in theabove theorems are always satisfied. 26e note that the conclusions of the above theorems do not contradictwith examples in [25], where there are infinitely many chambers, even if theboundaries are not big. The reason is that our finite decomposition theoremis a statement of local nature in a sense.The termination of the flips for the scaled MMP is an easy consequenceof the Polytope Decomposition Theorem. Indeed the models before andafter a flip correspond to different chambers. There are only finitely manychambers on the line segment on which the scaled MMP is played, hence thetermination. Now we start to explain some ideas of proofs of the theorems. The vanishingtheorems of Kodaira type have surprizingly diverse applications in the bira-tional geometry. We need to assume that the characteristic of the base fieldis zero.We start with the original Kodaira vanishing theorem ([31]):
Theorem 7.1 (Kodaira Vanishing Theorem) . Let X be a smooth projectivevariety, and D a divisor. If D − K X is ample, then H p ( X, D ) = 0 for p > . By applying a covering trick to the Kodaira vanishing theorem, we deduceits log and relative version ([20], [62], [29]):
Theorem 7.2 (Kawamata-Viehweg Vanishing Theorem) . Let X be a normalvariety, f : X → T a projective morphism to a base, B an effective R -divisor,and D a Cartier divisor. If ( X, B ) is KLT and D − ( K X + B ) is f -ample,then R p f ∗ O X ( D ) = 0 for p > . This generalization is used as in the following way. Let D be a divisoron a smooth projective variety X . Suppose that D − K X is not ample, butclose to ample. We can sometimes find a small effective R -divisor B by a perturbation such that ( X, B ) is KLT and D − ( K X + B ) is ample. Then wehave still the vanishing H p ( X, D ) = 0 for p > x -method , which is acohomological technique for proving base point freeness using the vanishing27heorem. The name came from the surprizingly similar applications of thevanishing theorem toward apparently different first two problems in Reid’slist ([48]). The same method was later applied toward Fujita’s conjecture onthe base point freeness of adjoint systems.The idea of the proof of the base point free theorem is as follows. Thepoint is to extend a holomorphic section from a codimension one subvarietyto the whole space, and use the induction on the dimension. Suppose thatwe want to prove the base point freeness of a complete linear system | mD | ona smooth projective variety X for a large integer m . First we take a positiveinteger m such that | m D | is non-empty, where the existence of such an m is guaranteed by the Non-Vanishing Theorem which is also proved by the x -method. We want to make the base locus smaller by taking a larger integer m so that the base locus disappears eventually. We construct a projectivebirational morphism µ : Y → X from a smooth variety with an effectivedivisor E and a smooth prime divisor Z such that:1. The support of E is contracted by µ .2. The image of Z by µ is contained in the base locus of | m D | .The argument for finding such a situation looked tricky when it was found,but we now know that the log canonical threshold is the concept hidden inthis argument. We consider an exact sequence0 → O Y ( m µ ∗ D + E − Z ) → O Y ( m µ ∗ D + E ) → O Z (( m µ ∗ D + E ) | Z ) → m of m . If m is suitably large, then we prove that H ( Z, ( m µ ∗ D + E ) | Z ) = 0by using the Non-Vanishing Theorem, and H ( Y, m µ ∗ D + E − Z ) = 0by using the Vanishing Theorem. By condition 1, the natural homomorphism µ ∗ : H ( Y, m µ ∗ D + E ) → H ( X, m D )is bijective. It follows that µ ( Z ) is not contained in the base locus of | m D | ,hence it is strictly smaller that that of | m D | .28he proof of the Non-Vanishing Theorem is similar, but we take an ar-tificial non-complete linear system compared to the natural one | mD | . Theproof of the Cone Theorem is also similar.We worked “upstairs” in the above proof, i.e., on a resolution which liesabove the original variety, and extend holomorphic sections from a divisor tothe whole variety by using the Vanishing Theorem for line bundles. If we usethe vanishing theorem for multiplier ideal sheaves explained below, we canwork “downstairs”, i.e., on the original variety, for the extension argument.Then we can obtain more powerful extension theorems because it becomespossible to consider an infinite series of linear systems simultaneously.We extend the vanishing theoren to the case where a pair ( X, B ) consist-ing of a normal variety and an effective R -divisor such that K X + B is an R -Cartier divisor but the pair is not necessarily KLT. Definition 7.3.
Let (
X, B ) be a pair consisting of a normal variety and aneffective R -divisor such that K X + B is an R -Cartier divisor. Let µ : Y → X be a log resolution and we write µ ∗ ( K X + B ) = K Y + P e j E j , where P E j isa normal crossing divisor. The multiplier ideal sheaf I ( X, B ) ⊂ O X is definedby the following formula I ( X, B ) = µ ∗ O Y ( − X x e j y E j ) . It is independent of the choice of the log resolution. We have I ( X, B ) = O X if and only if e j < j , i.e., if ( X, B ) is KLT.Let f : X → T be a projective morphism and D a Cartier divisor on X .If D − ( K X + B ) is f -ample, then Theorem 7.2 implies that R p µ ∗ O X ( µ ∗ D − X x e j y E j ) = 0and R p ( f ◦ µ ) ∗ O X ( µ ∗ D − X x e j y E j ) = 0for p >
0. Hence we have a vanishing theorem for non-KLT pairs:
Theorem 7.4 (Nadel Vanishing Theorem) . Let f : X → T be a projectivemorphism from a normal variety, B an effective R -divisor such that K X + B is an R -Cartier divisor, and D a Cartier divisor. If D − ( K X + B ) is f -ample,then R p f ∗ ( I ( X, B ) O X ( D )) = 0 for p > . L with a singular hermitianmetric on a complex manifold X . A singular hermitian metric h is a degener-ate hermitian metric which can be written locally as h = h e − φ , where h is a C ∞ metric and φ is a locally integrable weight function. The multiplier idealsheaf I ( L, h ) is defined as the largest ideal sheaf such that all local sectionsof I ( L, h ) L satisfy locally the L condition with respect to the metric. It isproved to be a coherent sheaf on X .For example, assume that X is an open subset of C n and B = P b i B i is aneffective R -divisor. If the prime divisors B i have local equations g i = 0, thenwe can define an algebraically defined singular hermitian metric on a trivialbundle L by using the weight function φ = P i b i log | g i | . In this case, thealgebraic and analytic multiplier ideal sheaves coincide: I ( X, B ) = I ( L, h ).The point is that there are a lot more metrics which are essentially dif-ferent from algebraic metrics. One can use a limit of a sequence of algebraicmetrics to produce a non-algebraic one provided that we can prove certainconvergence. For example, the analytic Zariski decomposition ([60], [9]) is asingular hermitian metric which is defined naturally for an arbitrary pseudo-effective line bundle. The metric on a Hodge bundle is another example ofanalytic metrics.A new extension technique using the multiplier sheaves was developed bySiu [55] when he proved the deformation invariance of plurigenera:
Theorem 7.5.
Let f : X → T be a smooth projective morphism. Then theplurigenus dim H ( X t , mK X t ) of a fiber X t = f − ( t ) is independent of t ∈ T for any positive integer m . Nakayama [43] proved that positive solutions for conjectures in the MMPincluding the Abundance Conjecture imply the invariance of plurigenera. ButSiu proved the theorem in one step without using an inductive approach ofthe MMP. The theorem is proved in the case where a fiber X t is of generaltype in [55], and in general in [56] (see also [59] and [47]). The vanishingtheorem used in these proofs is the Ohsawa-Takegoshi type extension theorem([45], [55]): Theorem 7.6 (Ohsawa-Takegoshi Extension Theorem) . Let X be a Steinmanifold, and Y a smooth hypersurface defined by a bounded holomorphic unction t . Let ( L, h ) be a line bundle on X with a singular hermitian metric,and s ∈ H ( Y, K Y + L | Y ) a holomorphic section. Assume that the curvaturecurrent − dd c log h is semipositive as a real current of type (1 , , and that R Y | s | h < ∞ . Then there exists an extension ˜ s ∈ H ( X, K X + L ) such that ˜ s = s ∧ dt on Y and Z X | ˜ s | h ≤ C sup | t | Z Y | s | h for a universal constant C . The advantage of Theorem 7.6 is that we do not need to assume thestrict positivity of the metric, which corresponds roughly to the amplenessor bigness. But the semipositivity for the metric is strictly stronger thecorresponding algebraic concept of the nefness. This is the reason why thereis still no algebraic proof for the invariance theorem of plurigenera.The idea of the proof of the invariance of plurigenera in the case wherethe fibers are of general type is to use a construction downstairs which issimilar to the Zariski decomposition. The spaces of sections of pluricanonicalsystems on the central fiber define a series of metrics on the canonical linebundle, hence a sequence of multiplier ideal sheaves. Similar constructions onthe total space of the deformations define different metrics, hence a differentsequence of multiplier ideal sheaves. A suitable vanishing theorem relatesthese ideals, and the extension of sections is proved.This proof allowed an algebraic analogue in [26] (see also [44] and [27])because the canonical line bundle is big in this case. The algebraic versionof the extension theorem were generalized to the logarithmic situation in[13] and [58] (see also [61]). The logarithmic extension theorem was usedin the proof of the PL flip theorem [14] explained later. We note that thealgebraization of the proof in the general case, i.e., non general type case, isstill an open problem.Now we state the logarithmic extension theorem:
Theorem 7.7.
Let f : X → T be a projective morphism from a smoothvariety to an affine variety, and B = P b i B i a Q -divisor whose support isa normal crossing divisor and such that only one coefficient b is equal to and other coefficients satisfy b i ∈ (0 , for i = 0 . Set Y = B . Let r be a positive integer such that r ( K X + B ) has integral coefficients. Set ( K X + B ) | Y = K Y + B Y . Assume the following conditions: . K X + B is pseudo-effective with respect to ( X, p B q ) .2. B − Y is big with respect to ( X, Y ) .Then natural homomorphisms H ( X, mr ( K X + B )) → H ( Y, mr ( K Y + B Y )) are surjective for all positive integers m . This is a correct log generalization of the extension theorem, while [27]Example 4.3 showed that a naive extension is false.
We start to explain how the remaining two conjectures on the flips, theexistence and the termination, are proved for the scaled MMP under theadditional assumption that the boundary is big.The basic idea is to use the induction on the dimensions using the ad-junction formula which relates canonical divisors in different dimensions: if Y is a smooth divisor on a smooth variety X , then we have( K X + Y ) | Y = K Y . This is a very different approach from the proof in dimension three in [39].We need two theorems, the Special Termination Theorem and the reduc-tion to PL flips, due to Shokurov [53] (see also [11]) preceding to the proof ofthe flip theorem. The first one is on the termination of flips along boundarycomponents with coefficient 1:
Theorem 8.1 (Special Termination) . Let ( X, B ) be a Q -factorial DLT pairof dimension n which is projective over a base. Assume that the MMP holds,i.e., the existence and termination conjectures hold, in dimension less than n . Let ( X, B ) = ( X , B ) ( X , B ) · · · be an infinite sequence of flips. Then there exists a positive integer m suchthat the flip ( X m , B m ) ( X m +1 , B m +1 ) is an isomorphism in a neighbor-hood of x B m y for every m ≥ m . imilarly, if the MMP holds in dimension less than n under the additionalcondition that the MMP is scaled or the boundary is big, then the specialtermination holds under the additional assumption that the MMP is scaledor the boundary B is big with respect to ( X, B ) . The idea of the proof is to use the adjunction formula to reduced boundarycomponents. It turned out that the Special Termination Theorem, not thegeneral termination, is sufficient to prove the existence of minimal modelsthanks to the Non-Vanishing Thoerem explained in the next section.A small contraction morphism of a Q -factorial DLT pair φ : ( X, B ) → Y in the MMP is said to be a PL (prelimiting) contraction, if there is anirreducible component S of x B y such that − S is φ -ample. A flip for a PLcontraction is called a PL flip . The second one is the reduction to PL flips:
Theorem 8.2.
Assume that the special termination theorem for the scaledMMP with big boundaries and the existence of the PL flip hold in dimension n . Then the existence of the flip in general holds in the same dimension. The idea of the proof is to introduce additional artificial boundary as ascaling, and then reduce the boundary back to the original state gradually byusing the scaled MMP. This is the first appearance of the MMP with scaling.Hacon and McKernan [14] proved the existence of a PL flip:
Theorem 8.3.
Assume that a minimal models exists for any KLT pair ofdimension n − when the boundary is big. Then a flip exists for arbitraryPL small contraction in dimension n . The idea of proof is as follows. Let φ : ( X, B ) → Y be a PL contractionmorphism. We may assume that B is a Q -divisor. Let S be an irreduciblecomponent of x B y such that − S is φ -ample. We use the adjunction formula:( K X + B ) | S = K S + B S , where B S is usually larger than the restriction B | S because of the singularities of X , namely the subadjunction . In order to provethe finite generation of the relative canonical ring R ( X/Y, K X + B ) = ∞ M m =0 φ ∗ O X ( x m ( K X + B ) y )it is sufficient to prove the finite generation of the image of the restrictionhomomorphism R ( X/Y, K X + B ) → R ( S/Y, K S + B S ) . not surjective.The point of the proof of the PL flip theorem is to identify the image of thishomomorphism.By restricting m -canonical systems on X to S and cancelling fixed compo-nents from the boundary B S , we obtain a series of boundaries B S ′ ,m depend-ing on m on a fixed birational model S ′ of S . By using the extension theorem(Theorem 7.7), we prove that all the pluricanonical forms with respect theboundaries B S ′ ,m on S ′ are in the images of the restriction homomorphisms.By the Polytope Decomposition Theorem in dimension n −
1, we find a fixedbirational model S ′′ of S which dominates all the minimal models correspond-ing to this series of boundary divisors. Then we prove that certain stabilityof the movable parts of m -canonical systems holds when m goes to infinityby using the vanishing theorem and the effective version of the Base PointFree Theorem by Koll´ar [34].We note that arbitrary divisor is big for a birational map such as φ . Thedifficulty concerning the infinity arising from the fact that we should considerall the m -canonical systems at the same time is solved by the finitenessstatement of the Polytope Decomposition Theorem. An old approach to the termination conjecture is to use invariants of singu-larities initiated by Shokurov [50]. Let us consider the simplest case; let X bea three dimensional variety with only terminal singulsrities, and µ : Y → X aresolution of singularities. We write µ ∗ K X = K Y + P e j E j . By the assump-tion, we have e j < j . The difficulty of the variety X is the numberof j such that − < e j . One can prove that the difficulty is a well definednon-negative integer, and it decreases strictly after a flip. The terminationfollows immediately.If the singularity is worse or if the dimension is higher, then similar prop-erties, the well-definedness and the monotoneness, fail. But we can modifythe definition of the difficulty and obtain some termination theorems. Thefollowing is the known results up to now obtained by using the concept ofdifficulty: 34 heorem 9.1. (1) ([23], [52]) The termination conjecture holds for arbitraryDLT pairs in dimension three.(2) ([1]) The termination conjecture holds for a KLT pair ( X, B ) of di-mension four if − ( K X + B ) is numerically equivalent to an effective R -divisorover the base.(3) ([1]) If ( X, P b i B i ) is a KLT pair of dimension four such that c K X + P c i B i is relatively big over the base for some numbers c i ∈ R . Then thereexists a process of the MMP with scaling which terminates. In particular, the existence of a minimal model for arbitrary DLT pair(
X, B ) over a base is proved only in the case dim X = 3 at the moment.Birkar, Cascini, Hacon and McKernan [5] took a very different approachto the termination conjecture. They did not prove the termination as an iso-lated statement, but rather included it into a chain of statements concerningthe MMP. By using the induction on dimension, the termination is reducedto the special termination as in the following way.Let ( X, B ) be a DLT pair which is projective over a base T . Assume thatthe log canonical divisor K X + B is numerically equivalent to an effective R -divisor M over T . Then the minimality question for ( X, B ) is equivalentto that for (
X, B + tM ) for t > X, B + tM ) is DLT for some t > x B + tM y coincides with that of M . By applying the Special TerminationTheorem, we conclude that the flips terminates near x B + tM y . Since M isnumerically equivalent to K X + B , it follows that K X + B becomes nef in thisprocess. In the general case, we need more careful argument on the supportof M during the process of the MMP with scaling.There is a modified version of the above termination argument in Birkar[4]. In particular it is proved that, if K X + B is numerically equivalent to aneffective R -divisor, then a minimal model exists if dim X ≤ R -divisor M . This is a generalization of the Non-Vanishing Theorem: Theorem 9.2.
Let ( X, B ) be a KLT pair which is projective over a base T .Assume the following conditions:1. K X + B is pseudo-effective over T .2. B is big over T . hen there exists an effective R -divisor M which is numerically equivalentto K X + B over T . The assertion is only a part of the existence theorem of a minimal model,but this is the key point in its proof. The proof in [5] is also a generalization ofthe Non-Vanishing Theorem which is a part of the Base Point Free Theoremwhich is already proved when K X + B is nef, together with the idea similarto the proof of the flip theorem. We use the minimal model theorem whichis already proved under the additional assumption that K X + B is effective,and the Polytope Decomposition Theorem in dimension one less.The proof proceeds roughly as follows. As in the case of the Non-Vanishing Thorem or the reduction theorem to the PL flip, we increase theboundary artificially so that the pair ( X, B ) becomes DLT and x B y = Z isirreducible. We consider the case where B is a Q -divisor for simplicity. If weadd an ample Q -divisor ǫH to the boundary, then K X + B + ǫH becomesbig, and there exists a minimal model for the pair ( X, B + ǫH ). If we takethe limit ǫ →
0, then there may be infinitely many chambers in the space ofdivisors corresponding to the minimal models of the pairs (
X, B + ǫH ). Byusing the Special Termination Theorem, we can prove that certain neighbor-hood of Z becomes stable under this infinite chain of wall crossings. Then wecan fix a minimal model of Z , on which there exists a desired section thanksto the usual Base Point Free Theorem. By using the Vanishing Theorem, weinfer that this section is extended to a birational model of X .The assumption that the boundary B is big is indispensable in the firststep of the above proof where we find Z . Indeed as is shown in [4], the Non-Vanishing Thorem is the most difficult point if we try to extend the minimalmodel theory to the case where the boundary is not necessarily big.
10 Birational maps between birational mod-els
The output of the MMP, a minimal model or a Mori fiber space, is notuniquely determined in general when we start with a fixed pair, because thereare choices of the faces in the process of the MMP. We would like to controlthe non-uniqueness. The answer is given by the Polytope DecompositionTheorem again. 36he non-uniqueness of a minimal model for a fixed variety is a new phe-nomena in dimension three or higher. The following theorem asserts that theexistence of flops is the only reason.A flop for a pair (
X, B ) is a diagram X φ −−−→ Y φ + ←−−− X + which is a flip for another pair ( X, B ′ ), where B ′ is a suitably chosen differentboundary, and such that K X + B is numerically trivial for φ ; [ K X + B ] = 0in N ( X/Y ). Theorem 10.1.
Let f : ( X, B ) → T be a projective morphism from a KLTpair, let g : ( Y, C ) → T and g ′ : ( Y ′ , C ′ ) → T be its minimal models, andlet α : Y Y ′ be the induced birational map over T . Assume that thereexists a canonical model of ( X, B ) . Then α is decomposed into a sequence offlops in the follwoing way: there exists an effective Q -Cartier divisor D on Y such that ( Y, C + D ) is still KLT and such that α becomes a compositionof a sequence of birational maps Y = Y Y · · · Y l = Y ′ such that α k : Y k − → Y k ( ≤ k ≤ l ) is a flop for the pair ( Y k − , C k − ) aswell as a flip for the pair ( Y k − , C k − + D k − ) , where C k − and D k − are thestrict transforms of C and D , respectively. We note that α is an isomorphism in codimension one as we alreadyknow. We remark that the boundary B need not to be big. There is adiffernt version of the factorization theorem in [28], where we do not need toassume the existence of a canonical model, but we have to assume that B isa Q -divisor.A marked minimal model is a pair consisting of a minimal model anda birational map to a fixed reference model. The number of birationallyequivalent marked minimal models is finite if the boundary B is big, butit is not the case in general ([25]). The above theorem claims that thereare still only finitely many marked minimal models which lie between twodifferent minimal models. It is conjectured that the number of birationallyequivalent minimal models is finite up to isomorphisms , i.e., when we forgetthe markings.We use the Polytope Decomposition Theorem to prove the above theoremin the following way. We take a general ample R -Cartier divisor H and H ′ Y and Y ′ , respectively, and we consider a triangle spanned by C , C + H and C + H ′ in the space of divisors on Y , where the strict transform of H ′ on Y is denoted by the same letter. The canonical model coresponds to thechamber { } . If we choose H and H ′ small enough, then the closures ofthe chambers V and V ′ corresponding to the models Y and Y ′ contain 0.Moreover the union of the chambers between V and V ′ which contain 0 inthe closures contains the line segment joining the points C + H and C + H ′ .Then the wall crossing process provides a decomposition of α . By inductionon the relative Picard numbers, we obtain eventually the decomposition toflops.As for the Mori fiber spaces, the non-uniqueness phenomenon appearsalready in dimension two. For example, ruled surfaces P ( O P ⊕ O P ( d ))over P for different integers d are all birationally equivalent. An elementarytransformation of a ruled surface is a combination of a blowing up at apoint in a fiber and the blowing down of the strcit transform of the fiber.If g i : X i → C for i = 1 , C , thenthey are connected each other by a sequence of elementary transformations.There is a different kind of decompositions; any birational map P P isdecomposed into linear and quadratic transformations. The latter is furtherdecomposed into point blowings up, point blowings down, and elementarytansformations.The Sarkisov program is a higher dimensional generalization. We candecompose any birational map between the total spaces of Mori fiber spacesinto elementary links , generalizations of elementary transformations.Hacon and McKernan [15] proved the following:
Theorem 10.2.
Let f : ( X, B ) → T be a projective morphism from a KLTpair, let φ : ( Y, C ) → Z and φ ′ : ( Y ′ , C ′ ) → Z ′ be Mori fiber spaces obtainedby the MMP from a KLT pair ( X, B ) over T , and let α : Y Y ′ be theinduced birational map over T : Y α −−−→ Y ′ φ y y φ ′ Z Z ′ Then there exists a sequence of the following type commutative diagrams alled elementary links for ≤ k ≤ l with some positive integer l : U ( k )1 β ( k ) −−−→ U ( k )2 f ( k )1 y y f ( k )2 V ( k )1 V ( k )2 g ( k )1 y y g ( k )2 W ( k ) = −−−→ W ( k ) where β ( k ) is a composition of a sequence of flips for a suitably chosen bound-ary on U ( k )1 and such that α is decomposed as α = α l ◦ · · · ◦ α where each α k is a birational map described in one of the following cases:1. g ( k )1 and f ( k )2 are Mori fiber spaces, f ( k )1 is a divisorial contraction, g ( k )2 is a morphism with relative Picard number one, and α k = β ( k ) ◦ ( f ( k )1 ) − .2. g ( k )1 and g ( k )2 are Mori fiber spaces, f ( k )1 and f ( k )2 are divisorial contrac-tions, and α k = f ( k )2 ◦ β ( k ) ◦ ( f ( k )1 ) − .3. f ( k )1 and g ( k )2 are Mori fiber spaces, f ( k )2 is a divisorial contraction, g ( k )1 is a morphism with relative Picard number one, and α k = f ( k )2 ◦ β ( k ) .4. f ( k )1 and f ( k )2 are Mori fiber spaces, g ( k )1 and g ( k )2 are morphisms withrelative Picard number one, and α k = β ( k ) . The theorem was proved in dimension three by Corti [7]. The crucialpoint of the proof is to prove that a sequence of elementary links terminates.[15] interpreted the sequence of elementary links as a wall crossing process,and proved the termination by using the Polytope Decomposition Theorem.It is remarkable that the four types of elementary links which are apparentlydifferent have the same interpretation as a wall crossing process in terms ofthe polytope decomposition.The argument of [15] is as follows. We take a general ample R -Cartierdivisor H (resp. H ′ ) on Y (resp. Y ′ ) such that K Y + C + H = φ ∗ L (resp. K Y ′ + C ′ + H ′ = ( φ ′ ) ∗ L ′ ) for some ample R -Cartier divisor L (resp. L ′ ) on39 (resp. Z ′ ), and let D (resp. D ′ ) be the strict transforms of H (resp. H ′ )on X . Then the pair ( Y, C + H ) (resp. ( Y ′ , C ′ + H ′ )) is a minimal model of( X, B + D ) (resp. ( X, B + D ′ )), and Z (resp. Z ′ ) is the canonical model.We consider a triangle ¯ V spanned by B , B + D and B + D ′ in the space ofdivisors on X , and let V be the subset corresponding to the pseudo-effectivelog canonical divisors. The points ( X, B + D ) and ( X, B + D ′ ) are on theboundary of V . We consider a path on the boundary which connects thesepoints, and look at the chambers whose closures intersect this path. Thenthe wall crossing process provides a decomposition of α . References [1] Valery Alexeev, Christopher Hacon and Yujiro Kawamata.
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