University of Utah January, 2007
Outline of the talk 1 Introduction
Outline of the talk 1 Introduction 2
Outline of the talk 1 Introduction 2 3
The main Theorem Curves Surfaces Outline of the talk 1 Introduction The main Theorem Curves Surfaces 2 3
The main Theorem Curves Surfaces Notation I am interested in the birational classication of complex projective varieties. X P N C is dened by homogeneous polynomials P 1 ; ; P t 2 C[x 0 ; ; x N ]. X ; X 0 are birational if they have isomorphic open subsets U = U 0. If X is smooth of dimension n, then T X denotes the tangent bundle of X and! X = n T _ X is the canonical line bundle of X. K X denotes (a choice of) the canonical divisor i.e. a formal linear combination of codimension 1 subvarieties of X given by the zeroes of a section s 6= 0 of! X. We would like to use! X to understand the geometry of X.
The main Theorem Curves Surfaces Birational invariants If X and X 0 are birational, then 8m 0 H 0 (X ;! m ) X = H 0 (X 0 ;! m ): X 0 The numbers P m (X ) := h 0 (! m ) are the plurigenera of X. X The pluricanonical ring of X is given by R(K X ) = M m2n The Kodaira dimension of X is H 0 (X ;! m X ): (X ) := tr:deg: C R(K X ) 1: If (X ) = dimx, we say that X is of general type. One would like to understand the structure of this ring and to use its features to classify complex projective varieties.
The main Theorem Curves Surfaces Finite generation Today I would like to discuss a recent result of Birkar-Cascini-Hacon-M c Kernan and Siu: Theorem Let X be a smooth projective variety, then R(K X ) is nitely generated. The two proofs are independent. Our proof is algebraic and uses the ideas of the MMP (minimal model program). Siu's proof is analytic and requires X to be of general type.
The main Theorem Curves Surfaces Curves Any two birational curves are isomorphic. Curves are topologically determined by their genus. g = 0: P 1 and R(K ) = C g = 1: There is a 1-dimensional family of elliptic curves; K = 0 so R(K ) = C[t]. g 2: Curves of general type. These form a 3g 3-dimensional family. K is ample and j3k j determines an embedding. R(K ) is then nitely generated (by Serre-Vanishing).
The main Theorem Curves Surfaces Surfaces of general type If S is a surface of general type and K S is nef (i.e. K S C 0 for all curves C S), then we say that S is a minimal surface. In this case K S is semiample. Therefore, there is a birational morphism : S! S P N to a surface (with rational double points) so that! m = O X S (1). It is known that R(K S ) is nitely generated if and only if R(K S ) (m) = M i2n H 0 (O S (imk S )) is nitely generated (for any m > 0). By Serre vanishing R(K S ) (m) = R(O S (1)) is nitely generated. Note that we may take m = 5, contracts rational curves E with E K S = 0 and S = ProjR(K S ).
The main Theorem Curves Surfaces Surfaces of general type II If S is of general type, but K S is not nef, then there is a rational curve E such that E 2 = E K S = 1. We may contract E, i.e. there is a morphism of smooth surfaces 1 : S! S 1 which is an isomorphism on S E and such that E maps to a point. Since K S = 1 K S 1 + E, we have R(K S ) = R(K S1 ). If K S1 is not nef, we may repeat the procedure and we obtain a sequence of morphisms S! S 1! S 2!. It is easy to see that 0 < (S i+1 ) = (S i ) 1 and so this sequence of morphism is nite, i.e. there is an integer t > 0 such that K St is nef i.e. S t is a minimal model of S. Then R(K St ) is nitely generated and so is R(K S ).
The main Theorem Curves Surfaces Surfaces with 2 If S is not of general type, then (S) 2 f 1; 0; 1g. If (S) = 1, then R(K S ) = C. If (S) = 0, then R(K S ) (m) = C[t] for some m > 0. (For surfaces m = 12.) If (S) = 1, then we rst contract all 1 curves to obtain a morphism : S! S 0 such that K S 0 is nef (i.e. S 0 is minimal). Then K S 0 is semiample and it gives a morphism f : S 0! C whose general ber is an elliptic curve. We have that mk S 0 = f H where m > 0 is an integer and H is an ample divisor on C. Therefore, R(K S ) (m) = R(K S 0) (m) = R(H) is nitely generated.
Outline of the talk 1 Introduction 2 The MMP with scaling 3
Log pairs One would like to generalize the above approach to higher dimensional varieties. In order to achieve this one has to allow varieties with mild singularities and the minimal models will not be unique. We would also like to have more exibility and so we work with log-pairs. The proof will in fact proceed by induction on the dimension. We will need to relate K X to it's restriction to a divisor S (which typically belongs to S 2 jmk X j). This is achieved by the adjunction formula which has the form (K X + S)j S = K S + S for some Q-divisor S.
KLT Log Pairs We consider log pairs (X ; ) where X is a normal Q-factorial variety and is a divisor with positive real coecients. Then K X + is R-Cartier and so (K X + ) C and f (K X + ) still make sense. Consider a map f : X 0! X such that X 0 is smooth and f + Exc(f ) has simple normal crossings support. Write K X 0 + 0 = f (K X + ): (X ; ) is KLT if each coecient of 0 is less than 1.
A Theorem of Fujino-Mori Mori and Fujino have shown that: Theorem Let (X ; ) be any KLT Q-factorial pair with 2 Div Q (X ). Then there exists a big KLT Q-factorial pair (Y ; ) with 2 Div Q (Y ) such that R(K X + ) (m) = R(K Y + ) (m) for any suciently divisible integer m > 0. Recall that K Y + is big if tr:deg: C R(K Y + ) (m) = dim Y or equivalently if K Y + Q A + B with A ample and B 0. It follows that to show the main theorem (i.e. that R(K X ) is nitely generated), it suces to prove the corresponding statement for big KLT log pairs.
The MMP with scaling From now on we will consider Q-factorial KLT pairs (X ; ) such that is big. We allow 2 Div R (X ). Note that if K X + is big, then there is an eective divisor D R K X + and so for all 0 < 1 the pair (X ; + D) is KLT, + D is big and K X + + D R (1 + )(K X + ). We would like to nd a nite sequence of KLT log pairs (X i ; i ) with i big and birational maps X 99K X 1 99K X 2 such that R(K Xi + i ) = R(K Xi+1 + i+1 ); and K Xt + t is nef. Then by the base point free theorem, K Xt + t is semiample so that R(K Xt + t ) is nitely generated (if 2 Div Q ).
The MMP with scaling II Assume that A is ample and K X + + A is nef. Let = infft 0jK X + + tag is nef. If = 0 we STOP (K X + is nef). Otherwise, by the cone theorem, there is an extremal ray R such that (K X + + A) R = 0 and (K X + + ta) R < 0 for 0 t <. Let cont R : X! Z be the corresponding morphism which contracts a curve C i [C ] = R. If cont R is not birational, we STOP (we have a Mori-Fano ber space and R(K X + ) = C).
The MMP with scaling III If cont R is birational and divisorial (i.e. it contracts a divisor), then we replace X by the image of the corresponding contraction and we may continue this procedure. If cont R is birational and small, (i.e. Exc(cont R ) has codimension 2) then the image of the corresponding contraction is not KLT. Therefore, we can NOT replace X by the corresponding contraction! Instead we replace (X ; ) by the corresponding ip : X 99K X + (assuming it exists). Let + =. The ip is a codimension 2 surgery which replaces K X + negative curves by K X + + + positive curves.
Flips More precisely, f = cont R : X! Z is a ipping contraction i.e. a small birational morphism with (X =Z ) = 1 and (K X + ) ample over Z. The ip f + : X +! Z (if it exists) is a small birational morphism such that (X + =Z ) = 1 and K X + + + is ample over Z. The ip f + : X +! Z (if it exists) is given by M X + = Proj Z f O X ([m(k X + )]): m2n Note that after peturbing, we may assume that 2 Div Q (X ).
Flips II The problem of existence of ips is local over Z and hence we may assume that Z is ane. It suces then to show that R(K X + ) is nitely generated. So this is a local version of the original problem. To run this MMP with scaling, we must therefore show that such ips exist and terminate (i.e. there is no innite sequence of such ips). Note that after each divisorial contraction, the Picard number drops by 1 and so there are only nitely many divisorial contractions.
Outline of the talk Recent results The structure of the proof Existence of Flips Existence of minimal models Finiteness of models Non-vanishing Open problems 1 Introduction 2 3 Recent results The structure of the proof Existence of Flips Existence of minimal models Finiteness of models Non-vanishing Open problems
Existence and termination of ips Recent results The structure of the proof Existence of Flips Existence of minimal models Finiteness of models Non-vanishing Open problems Theorem (Birkar-Cascini-Hacon-M c Kernan) Let (X ; ) be a KLT log pair and f : X! Z a ipping contraction, then the ip f + : X +! Z of f exists. If moreover is big, then any sequence of ips for the K X + MMP with scaling must terminate. Corollary Let (X ; ) be a KLT pair with K X + -pseudo-eective and big, then K X + has a minimal model. (In particular ips exist.) K X + -pseudo-eective means that given any ample Q-divisor H, K X + + H is big (i.e. (K X + + H) = dim X ). This is a necessary condition.
Finite Generation Recent results The structure of the proof Existence of Flips Existence of minimal models Finiteness of models Non-vanishing Open problems Corollary (Birkar-Cascini-Hacon-M c Kernan, Siu) Let X be a smooth projective variety, then R(K X ) is nitely generated. Proof: Assume that X is of general type. We may nd K X with 0 < 1. A minimal model for K X + is then a minimal model for K X. The result then follows from the base point free theorem. As mentioned above, the general case is a consequence of a result of Fujino and Mori. Siu's proof relies on analytic techniques.
The structure of the proof Recent results The structure of the proof Existence of Flips Existence of minimal models Finiteness of models Non-vanishing Open problems We proceed by induction on n = dim X. We show that for an n-dimensional KLT pair (X ; ): Claim 1. Flips exist. Claim 2. If K X + R D 0 and is big, then (X ; ) has a minimal model. Claim 3. The set of minimal models for KLT pairs with A 0 where A is ample and (X ; 0 ) is KLT, is nite. Claim 4. If is big and K X + is pseudo-eective, then there is a Q-divisor D such that K X + R D 0. All statements should be relative over a base.
The structure of the proof II Recent results The structure of the proof Existence of Flips Existence of minimal models Finiteness of models Non-vanishing Open problems Roughly speaking, the inductive structure of the proof is: (1 4) n 1 ) 1 n ) 2 n ) 3 n ) 4 n : Note that that (1 4) n 1 ) 1 n was obtained by Hacon M c Kernan in 2005 using ideas of Shokurov and an extension result base on work of Siu, Kawamata and Tsuji. The remaining implications heavily use the ideas of the MMP established by Kawamata, Kollar, Mori, Reid, Shokurov and others.
Reduction to PL Flips Recent results The structure of the proof Existence of Flips Existence of minimal models Finiteness of models Non-vanishing Open problems Shokurov has shown that to construct ips it suces to construct PL-ips. That is, we assume that has a component S of multiplicity 1, S is relatively ample and K X + is \almost KLT". By adjunction (K X + )j S = K S + S and (S; S ) is KLT. As we have remarked above, the existence of ips is equivalent to showing that R = R(K X + ) is nitely generated. (Here we assume Z is ane.) This is equivalent to showing that the algebra R S given by the image of the restriction map M m2n H 0 (O X (m(k X + )))! M m2n is nitely generated. H 0 (O S (m(k S + S )))
Existence of ips Recent results The structure of the proof Existence of Flips Existence of minimal models Finiteness of models Non-vanishing Open problems If R S = m2n H 0 (O S (m(k S + S ))) we would be done by induction on the dimension. This is too much to hope for. Instead, we show that R S has very special properties. Roughly speaking we show that there exists a divisor 0 S on S such that R S = M m2n H 0 (O S (m(k S + ))): This is proved using some extension theorems that were inspired by work of Siu, Kawamata and Tsuji. These extension theorems determine/dene.
Termination of MMP with scaling Recent results The structure of the proof Existence of Flips Existence of minimal models Finiteness of models Non-vanishing Open problems Since we are unable to show that an arbitrary sequence of ips must terminate, we run a MMP with scaling. The idea is that if A is suciently ample, then K X + + A is ample and so it is a minimal model. We then consider K X + + ta and let t go to 0. Proceeding this way we get a sequence of pairs (X i ; i + i A i ) such that K Xi + i + i A i is nef and trivial on some K Xi + i extremal ray. We then ip/contract. Each (X i ; i + i A i ) is a minimal model for (X ; + i A).
Termination of MMP with scaling II Recent results The structure of the proof Existence of Flips Existence of minimal models Finiteness of models Non-vanishing Open problems If we could assume Claim 3 n, we would be done. (As is big, we may assume that A 0 + ta for some ample divisor A 0 and we let 0 = + A.) Since we can only assume 3 n 1, we replace (X ; ) by an appropriate DLT pair (X 0 ; 0 ) and we run a MMP with scaling such that all ips/contractions are contained in [ 0 ]. This step is similar to Shokurov's reduction to PL-ips.
Finiteness of models Recent results The structure of the proof Existence of Flips Existence of minimal models Finiteness of models Non-vanishing Open problems Theorem Assume that (X ; 0 ) is KLT, A is an ample Q-divisor and = P r i=1 i i. Let C [0; 1] r be a rational polytope such that for any ( 1 ; ; r ) 2 C, the pair (X ; P r i=1 i i + A) is KLT. Then the set of isomorphism classes is nite. fy jy is a min: model for (X ; + A); ( 1 ; ; r ) 2 Cg The proof is based on ideas of Shokurov (1996). It requires a compactness argument and so it is necessary to work with R-divisors.
Finiteness of models II Recent results The structure of the proof Existence of Flips Existence of minimal models Finiteness of models Non-vanishing Open problems We pick 2 C and work in a suciently small neighborhood U of. After running a MMP with scaling, we may assume that K X + is nef. By the base-point free theorem, there is a morphism f : X! Z such that K X + R f H. Working over Z, we may assume that K X + R 0. Therefore C \ U is a cone over and we are done by induction on the dimension of the subspace of Div R (X ) spanned by the components of.
Non-vanishing Recent results The structure of the proof Existence of Flips Existence of minimal models Finiteness of models Non-vanishing Open problems We must show that if is big and K X + is pseudo-eective, then K X + R D 0. If h 0 (O X (m(k X + ) + A)) is bounded (for A ample), then the result follows from work of Nakayama. If h 0 (O X (m(k X + ) + A)) is un-bounded, then we can produce a log canonical center not contained in the stable base locus of K X + + A for 0 < 1. So we study a log smooth PLT pair (X 0 ; 0 ) where S = [ 0 ] is irreducible. Then K S + ( 0 S)j S is pseudo-eective, ( 0 S)j S is big and so h 0 (O S (m(k S + ( 0 S)j S ))) > 0 for some m > 0. Using Kawamata-Viehweg vanishing, we can lift this section to m(k X + ).
Open problems Recent results The structure of the proof Existence of Flips Existence of minimal models Finiteness of models Non-vanishing Open problems Abundance: If K X + is nef and KLT show that it is semiample. (When is big, this follows from the Base Point Free Theorem. In dim. 2, it follows from the classication. It also holds in dim. 3) Existence of Minimal models for Log Canonical pairs (should be equivalent to Abundance). Termination of ips (would follow from the Borisov-Alexeev-Borisov conjecture). Characteristic p (need resolution of singularities; there is no Kawamata-Viehweg vanishing Thm, but... ).