The struture of the proof
The direted MMP
Existene of models
Finiteness of models
Non-vanishing
Finite generation of anonial rings II
Christopher Haon
University of Utah
Deember, 2006
Christopher Haon
Finite generation of anonial rings II
The struture of the proof
The direted MMP
Existene of models
Finiteness of models
Non-vanishing
Outline of the talk
1
The struture of the proof
Christopher Haon
Finite generation of anonial rings II
The struture of the proof
The direted MMP
Existene of models
Finiteness of models
Non-vanishing
Outline of the talk
1
The struture of the proof
2
The direted MMP
Christopher Haon
Finite generation of anonial rings II
The struture of the proof
The direted MMP
Existene of models
Finiteness of models
Non-vanishing
Outline of the talk
1
The struture of the proof
2
The direted MMP
3
Existene of models
Christopher Haon
Finite generation of anonial rings II
The struture of the proof
The direted MMP
Existene of models
Finiteness of models
Non-vanishing
Outline of the talk
1
The struture of the proof
2
The direted MMP
3
Existene of models
4
Finiteness of models
Christopher Haon
Finite generation of anonial rings II
The struture of the proof
The direted MMP
Existene of models
Finiteness of models
Non-vanishing
Outline of the talk
1
The struture of the proof
2
The direted MMP
3
Existene of models
4
Finiteness of models
5
Non-vanishing
Christopher Haon
Finite generation of anonial rings II
The struture of the proof
The direted MMP
Existene of models
The struture of the proof
Finiteness of models
Non-vanishing
Outline of the talk
1
The struture of the proof
The struture of the proof
2
The direted MMP
3
Existene of models
4
Finiteness of models
5
Non-vanishing
Christopher Haon
Finite generation of anonial rings II
The struture of the proof
The direted MMP
Existene of models
The struture of the proof
Finiteness of models
Non-vanishing
The rough struture of the proof
Reall that there are 3 main theorems we have to prove:
Christopher Haon
Finite generation of anonial rings II
The struture of the proof
The direted MMP
Existene of models
The struture of the proof
Finiteness of models
Non-vanishing
The rough struture of the proof
Reall that there are 3 main theorems we have to prove:
Thm A. Existene of log terminal models
Christopher Haon
Finite generation of anonial rings II
The struture of the proof
The direted MMP
Existene of models
The struture of the proof
Finiteness of models
Non-vanishing
The rough struture of the proof
Reall that there are 3 main theorems we have to prove:
Thm A. Existene of log terminal models
Thm B. Nonvanishing
Christopher Haon
Finite generation of anonial rings II
The struture of the proof
The direted MMP
Existene of models
The struture of the proof
Finiteness of models
Non-vanishing
The rough struture of the proof
Reall that there are 3 main theorems we have to prove:
Thm A. Existene of log terminal models
Thm B. Nonvanishing
Thm C. Finiteness of models
Christopher Haon
Finite generation of anonial rings II
The struture of the proof
The direted MMP
Existene of models
The struture of the proof
Finiteness of models
Non-vanishing
The rough struture of the proof
Reall that there are 3 main theorems we have to prove:
Thm A. Existene of log terminal models
Thm B. Nonvanishing
Thm C. Finiteness of models
Roughly speaking we would like to show that:
An 1 +Bn 1 +Cn 1 imply An ;
Bn 1 +Cn 1 +An imply Bn ;
Cn 1 +An +Bn imply Cn .
Christopher Haon
Finite generation of anonial rings II
The struture of the proof
The direted MMP
Existene of models
The struture of the proof
Finiteness of models
Non-vanishing
The rough struture of the proof
Reall that there are 3 main theorems we have to prove:
Thm A. Existene of log terminal models
Thm B. Nonvanishing
Thm C. Finiteness of models
Roughly speaking we would like to show that:
An 1 +Bn 1 +Cn 1 imply An ;
Bn 1 +Cn 1 +An imply Bn ;
Cn 1 +An +Bn imply Cn .
We will use the fat that by a result of Haon-M Kernan,
An 1 +Bn 1 +Cn 1 imply that PL-ips exist in dimension n.
Christopher Haon
Finite generation of anonial rings II
The struture of the proof
The direted MMP
Existene of models
The struture of the proof
Finiteness of models
Non-vanishing
The struture of the proof I
We now reall the preise statements of Thm's A, B, C:
Christopher Haon
Finite generation of anonial rings II
The struture of the proof
The direted MMP
Existene of models
The struture of the proof
Finiteness of models
Non-vanishing
The struture of the proof I
We now reall the preise statements of Thm's A, B, C:
Let : X ! U be a projetive morphism of normal
quasi-projetive varieties, where dim(X ) = n.
Christopher Haon
Finite generation of anonial rings II
The struture of the proof
The direted MMP
Existene of models
The struture of the proof
Finiteness of models
Non-vanishing
The struture of the proof I
We now reall the preise statements of Thm's A, B, C:
Let : X ! U be a projetive morphism of normal
quasi-projetive varieties, where dim(X ) = n.
Thm A. (Existene of log terminal models.) Suppose that
KX + is KLT and is big over U.
If KX + R;U D 0, then KX + has a log terminal
model over U.
Christopher Haon
Finite generation of anonial rings II
The struture of the proof
The direted MMP
Existene of models
The struture of the proof
Finiteness of models
Non-vanishing
The struture of the proof I
We now reall the preise statements of Thm's A, B, C:
Let : X ! U be a projetive morphism of normal
quasi-projetive varieties, where dim(X ) = n.
Thm A. (Existene of log terminal models.) Suppose that
KX + is KLT and is big over U.
If KX + R;U D 0, then KX + has a log terminal
model over U.
Thm B. (Non-vanishing.) Suppose that KX + is KLT and
is big over U.
If KX + is -pseudo-eetive over U, then there is an
eetive R-divisor D suh that KX + R;U D.
Christopher Haon
Finite generation of anonial rings II
The struture of the proof
The direted MMP
Existene of models
The struture of the proof
Finiteness of models
Non-vanishing
The struture of the proof II
Thm C. (resp. C ) (Finiteness of models.) Fix A an ample
divisor over U. Suppose that there exists a divisor 0 suh
that KX + 0 is KLT.
Let C fjKX + + A is log anonial g be a rational
polytope (resp. a rational polytope s.t. C + KX + A is
ontained in the big one).
Then the set of isomorphism lasses
fY jY is a WLCM for (X; + A) over U ;
+ A 2 Cg
is nite.
Christopher Haon
Finite generation of anonial rings II
The struture of the proof
The direted MMP
Existene of models
The struture of the proof
Finiteness of models
Non-vanishing
The struture of the proof II
Thm C. (resp. C ) (Finiteness of models.) Fix A an ample
divisor over U. Suppose that there exists a divisor 0 suh
that KX + 0 is KLT.
Let C fjKX + + A is log anonial g be a rational
polytope (resp. a rational polytope s.t. C + KX + A is
ontained in the big one).
Then the set of isomorphism lasses
fY jY is a WLCM for (X; + A) over U ;
is nite.
Claim 1.
+ A 2 Cg
Theorems An 1 , Bn 1 and Cn 1 imply Theorem An
Christopher Haon
Finite generation of anonial rings II
The struture of the proof
The direted MMP
Existene of models
The struture of the proof
Finiteness of models
Non-vanishing
The struture of the proof II
Thm C. (resp. C ) (Finiteness of models.) Fix A an ample
divisor over U. Suppose that there exists a divisor 0 suh
that KX + 0 is KLT.
Let C fjKX + + A is log anonial g be a rational
polytope (resp. a rational polytope s.t. C + KX + A is
ontained in the big one).
Then the set of isomorphism lasses
fY jY is a WLCM for (X; + A) over U ;
is nite.
Claim 1.
Claim 2.
+ A 2 Cg
Theorems An 1 , Bn 1 and Cn 1 imply Theorem An
Theorem An implies Theorem Cn ,
Christopher Haon
Finite generation of anonial rings II
The struture of the proof
The direted MMP
Existene of models
The struture of the proof
Finiteness of models
Non-vanishing
The struture of the proof II
Thm C. (resp. C ) (Finiteness of models.) Fix A an ample
divisor over U. Suppose that there exists a divisor 0 suh
that KX + 0 is KLT.
Let C fjKX + + A is log anonial g be a rational
polytope (resp. a rational polytope s.t. C + KX + A is
ontained in the big one).
Then the set of isomorphism lasses
fY jY is a WLCM for (X; + A) over U ;
is nite.
Claim 1.
Claim 2.
Claim 3.
+ A 2 Cg
Theorems An 1 , Bn 1 and Cn 1 imply Theorem An
Theorem An implies Theorem Cn ,
Theorem An and Theorem Cn imply Theorem Bn ,
Christopher Haon
Finite generation of anonial rings II
The struture of the proof
The direted MMP
Existene of models
The struture of the proof
Finiteness of models
Non-vanishing
The struture of the proof II
Thm C. (resp. C ) (Finiteness of models.) Fix A an ample
divisor over U. Suppose that there exists a divisor 0 suh
that KX + 0 is KLT.
Let C fjKX + + A is log anonial g be a rational
polytope (resp. a rational polytope s.t. C + KX + A is
ontained in the big one).
Then the set of isomorphism lasses
fY jY is a WLCM for (X; + A) over U ;
is nite.
Claim
Claim
Claim
Claim
1.
2.
3.
4.
+ A 2 Cg
Theorems An 1 , Bn 1 and Cn 1 imply Theorem An
Theorem An implies Theorem Cn ,
Theorem An and Theorem Cn imply Theorem Bn ,
Theorem An and Theorem Bn imply Theorem Cn .
Christopher Haon
Finite generation of anonial rings II
The struture of the proof
The direted MMP
Existene of models
The struture of the proof
Finiteness of models
Non-vanishing
Remarks
This is joint work with C. Birkar, P. Casini and J. M Kernan.
Christopher Haon
Finite generation of anonial rings II
The struture of the proof
The direted MMP
Existene of models
The struture of the proof
Finiteness of models
Non-vanishing
Remarks
This is joint work with C. Birkar, P. Casini and J. M Kernan.
We work over the eld of omplex numbers C .
Christopher Haon
Finite generation of anonial rings II
The struture of the proof
The direted MMP
Existene of models
The struture of the proof
Finiteness of models
Non-vanishing
Remarks
This is joint work with C. Birkar, P. Casini and J. M Kernan.
We work over the eld of omplex numbers C .
Throughout the proof I will usually assume for simpliity that
U = SpeC .
Christopher Haon
Finite generation of anonial rings II
The struture of the proof
The direted MMP
Existene of models
Finiteness of models
Non-vanishing
Outline of the talk
1
The struture of the proof
2
The direted MMP
3
Existene of models
4
Finiteness of models
5
Non-vanishing
Christopher Haon
Finite generation of anonial rings II
The struture of the proof
The direted MMP
Existene of models
Finiteness of models
Non-vanishing
The direted MMP
The main idea of the proof of Claim 1 is to show that any
sequene of ips with saling terminate.
Christopher Haon
Finite generation of anonial rings II
The struture of the proof
The direted MMP
Existene of models
Finiteness of models
Non-vanishing
The direted MMP
The main idea of the proof of Claim 1 is to show that any
sequene of ips with saling terminate.
To run the MMP with saling, assume that X is Q -fatorial,
(X ; ) is KLT and C 0 is an R-divisor suh that
KX + + C is nef and KLT.
Christopher Haon
Finite generation of anonial rings II
The struture of the proof
The direted MMP
Existene of models
Finiteness of models
Non-vanishing
The direted MMP
The main idea of the proof of Claim 1 is to show that any
sequene of ips with saling terminate.
To run the MMP with saling, assume that X is Q -fatorial,
(X ; ) is KLT and C 0 is an R-divisor suh that
KX + + C is nef and KLT.
Let = inf ft 0jKX + + tC g is nef
Christopher Haon
Finite generation of anonial rings II
The struture of the proof
The direted MMP
Existene of models
Finiteness of models
Non-vanishing
The direted MMP
The main idea of the proof of Claim 1 is to show that any
sequene of ips with saling terminate.
To run the MMP with saling, assume that X is Q -fatorial,
(X ; ) is KLT and C 0 is an R-divisor suh that
KX + + C is nef and KLT.
Let = inf ft 0jKX + + tC g is nef
If = 0 we STOP. Otherwise there is an extremal ray R suh
that (KX + + C ) R = 0 and (KX + + tC ) R < 0 for
all 0 t < .
Christopher Haon
Finite generation of anonial rings II
The struture of the proof
The direted MMP
Existene of models
Finiteness of models
Non-vanishing
The direted MMP
The main idea of the proof of Claim 1 is to show that any
sequene of ips with saling terminate.
To run the MMP with saling, assume that X is Q -fatorial,
(X ; ) is KLT and C 0 is an R-divisor suh that
KX + + C is nef and KLT.
Let = inf ft 0jKX + + tC g is nef
If = 0 we STOP. Otherwise there is an extremal ray R suh
that (KX + + C ) R = 0 and (KX + + tC ) R < 0 for
all 0 t < .
If the ontration indued by R is not birational, we have a
Mori ber spae and we STOP.
Christopher Haon
Finite generation of anonial rings II
The struture of the proof
The direted MMP
Existene of models
Finiteness of models
Non-vanishing
The direted MMP
The main idea of the proof of Claim 1 is to show that any
sequene of ips with saling terminate.
To run the MMP with saling, assume that X is Q -fatorial,
(X ; ) is KLT and C 0 is an R-divisor suh that
KX + + C is nef and KLT.
Let = inf ft 0jKX + + tC g is nef
If = 0 we STOP. Otherwise there is an extremal ray R suh
that (KX + + C ) R = 0 and (KX + + tC ) R < 0 for
all 0 t < .
If the ontration indued by R is not birational, we have a
Mori ber spae and we STOP.
If the ontration indued by R is birational, we replae X by
the orresponding ip/divisorial ontration and repeat the
proedure.
Christopher Haon
Finite generation of anonial rings II
The struture of the proof
The direted MMP
Existene of models
Finiteness of models
Non-vanishing
The direted MMP
The advantage is that for eah ip : X 9 9 K X + there is a suh that KX + + C is nef and hene X is a log terminal
model.
Christopher Haon
Finite generation of anonial rings II
The struture of the proof
The direted MMP
Existene of models
Finiteness of models
Non-vanishing
The direted MMP
The advantage is that for eah ip : X 9 9 K X + there is a suh that KX + + C is nef and hene X is a log terminal
model.
Thm Cn says that there are only nitely many suh models
(under appropriate hypothesis) and so we expet that these
ips should terminate.
Christopher Haon
Finite generation of anonial rings II
The struture of the proof
The direted MMP
Existene of models
Finiteness of models
Non-vanishing
The direted MMP
The advantage is that for eah ip : X 9 9 K X + there is a suh that KX + + C is nef and hene X is a log terminal
model.
Thm Cn says that there are only nitely many suh models
(under appropriate hypothesis) and so we expet that these
ips should terminate.
In terms of our indution, we are only allowed to assume Thm
Cn 1 and this makes the proof more tehnial. (An analog of
speial termination will be required.)
Christopher Haon
Finite generation of anonial rings II
The struture of the proof
The direted MMP
Existene of models
Finiteness of models
Non-vanishing
Outline of the talk
1
The struture of the proof
2
The direted MMP
3
Existene of models
4
Finiteness of models
5
Non-vanishing
Christopher Haon
Finite generation of anonial rings II
The struture of the proof
The direted MMP
Existene of models
Finiteness of models
Non-vanishing
Proof of Claim 1, Speial termination
Throughout the remainder of the talk, we will assume that
Thms An 1 , Bn 1 , Cn 1 hold so that in partiular PL-ips
exist in dimension n.
Christopher Haon
Finite generation of anonial rings II
The struture of the proof
The direted MMP
Existene of models
Finiteness of models
Non-vanishing
Proof of Claim 1, Speial termination
Throughout the remainder of the talk, we will assume that
Thms An 1 , Bn 1 , Cn 1 hold so that in partiular PL-ips
exist in dimension n.
Lemma. Suppose that
KX + = KX + S + A + B
is DLT where A is ample and B 0.
Then any sequene of ips X = X1 9 9 K X2 9 9 K X3 for the
KX + -MMP with saling is eventually disjoint from S = [℄.
Christopher Haon
Finite generation of anonial rings II
The struture of the proof
The direted MMP
Existene of models
Finiteness of models
Non-vanishing
Proof of Claim 1, Speial termination
Throughout the remainder of the talk, we will assume that
Thms An 1 , Bn 1 , Cn 1 hold so that in partiular PL-ips
exist in dimension n.
Lemma. Suppose that
KX + = KX + S + A + B
is DLT where A is ample and B 0.
Then any sequene of ips X = X1 9 9 K X2 9 9 K X3 for the
KX + -MMP with saling is eventually disjoint from S = [℄.
Proof. We may assume that S is irreduible and that the
indued maps Si 9 9 K Si +1 are isomorphisms in odim. 1.
Christopher Haon
Finite generation of anonial rings II
The struture of the proof
The direted MMP
Existene of models
Finiteness of models
Non-vanishing
Proof of Claim 1, Speial termination
Throughout the remainder of the talk, we will assume that
Thms An 1 , Bn 1 , Cn 1 hold so that in partiular PL-ips
exist in dimension n.
Lemma. Suppose that
KX + = KX + S + A + B
is DLT where A is ample and B 0.
Then any sequene of ips X = X1 9 9 K X2 9 9 K X3 for the
KX + -MMP with saling is eventually disjoint from S = [℄.
Proof. We may assume that S is irreduible and that the
indued maps Si 9 9 K Si +1 are isomorphisms in odim. 1.
We write (KX + i )jS = KS + i .
i
i
Christopher Haon
i
Finite generation of anonial rings II
The struture of the proof
The direted MMP
Existene of models
Finiteness of models
Non-vanishing
Proof of Claim 1, Speial termination II
Note that as we may assume that all Si are isomorphi in
odimension 1, then i is just the strit transform of 1 .
Christopher Haon
Finite generation of anonial rings II
The struture of the proof
The direted MMP
Existene of models
Finiteness of models
Non-vanishing
Proof of Claim 1, Speial termination II
Note that as we may assume that all Si are isomorphi in
odimension 1, then i is just the strit transform of 1 .
The KS + i + i C jS are weak log anonial models of
KS1 + 1 + i C jS1 where 0 1.
i
i
Christopher Haon
Finite generation of anonial rings II
The struture of the proof
The direted MMP
Existene of models
Finiteness of models
Non-vanishing
Proof of Claim 1, Speial termination II
Note that as we may assume that all Si are isomorphi in
odimension 1, then i is just the strit transform of 1 .
The KS + i + i C jS are weak log anonial models of
KS1 + 1 + i C jS1 where 0 1.
By Thm Cn 1 there are only nitely many distint pairs
(Si ; i ).
i
i
Christopher Haon
Finite generation of anonial rings II
The struture of the proof
The direted MMP
Existene of models
Finiteness of models
Non-vanishing
Proof of Claim 1, Speial termination II
Note that as we may assume that all Si are isomorphi in
odimension 1, then i is just the strit transform of 1 .
The KS + i + i C jS are weak log anonial models of
KS1 + 1 + i C jS1 where 0 1.
By Thm Cn 1 there are only nitely many distint pairs
(Si ; i ).
If i is a ipping urve for Xi 9 9 K Xi +1 that intersets Si ,
then either i Si or i Si > 0 in whih ase the ipped
urves are ontained in Si +1 . This means that there is a
enter suh that a(; Si ; i ) < a(; Si +1 ; i +1 ).
i
i
Christopher Haon
Finite generation of anonial rings II
The struture of the proof
The direted MMP
Existene of models
Finiteness of models
Non-vanishing
Proof of Claim 1, Speial termination II
Note that as we may assume that all Si are isomorphi in
odimension 1, then i is just the strit transform of 1 .
The KS + i + i C jS are weak log anonial models of
KS1 + 1 + i C jS1 where 0 1.
By Thm Cn 1 there are only nitely many distint pairs
(Si ; i ).
If i is a ipping urve for Xi 9 9 K Xi +1 that intersets Si ,
then either i Si or i Si > 0 in whih ase the ipped
urves are ontained in Si +1 . This means that there is a
enter suh that a(; Si ; i ) < a(; Si +1 ; i +1 ).
Sine a(; Si +1 ; i +1 ) a(; Sj ; j ) then Sj 6
= Si for all j > i .
But there are only nitely many models Sj and so the ips are
eventually disjoint from Sj .
i
i
Christopher Haon
Finite generation of anonial rings II
The struture of the proof
The direted MMP
Existene of models
Finiteness of models
Non-vanishing
Proof of Claim 1, Cont.
Proof of Claim 1.
Reall that KX + R D 0 and write D = M + F where
Supp(F ) SBs(KX + ) and M is mobile.
Christopher Haon
Finite generation of anonial rings II
The struture of the proof
The direted MMP
Existene of models
Finiteness of models
Non-vanishing
Proof of Claim 1, Cont.
Proof of Claim 1.
Reall that KX + R D 0 and write D = M + F where
Supp(F ) SBs(KX + ) and M is mobile.
We may assume that X is smooth, + D has simple normal
rossings, = A + B where A is ample and B 0 and
^ M = 0. (I.e. and M have no ommon omponents.)
Christopher Haon
Finite generation of anonial rings II
The struture of the proof
The direted MMP
Existene of models
Finiteness of models
Non-vanishing
Proof of Claim 1, Cont.
Proof of Claim 1.
Reall that KX + R D 0 and write D = M + F where
Supp(F ) SBs(KX + ) and M is mobile.
We may assume that X is smooth, + D has simple normal
rossings, = A + B where A is ample and B 0 and
^ M = 0. (I.e. and M have no ommon omponents.)
For simpliity assume that M is irreduible (this avoids some
linear algebra).
Christopher Haon
Finite generation of anonial rings II
The struture of the proof
The direted MMP
Existene of models
Finiteness of models
Non-vanishing
Proof of Claim 1, Cont.
Proof of Claim 1.
Reall that KX + R D 0 and write D = M + F where
Supp(F ) SBs(KX + ) and M is mobile.
We may assume that X is smooth, + D has simple normal
rossings, = A + B where A is ample and B 0 and
^ M = 0. (I.e. and M have no ommon omponents.)
For simpliity assume that M is irreduible (this avoids some
linear algebra).
Set = Supp(M + F ) _ = S + A + B 0 where 0 B 0 B .
(If G = gi Gi , G 0 = gi0 Gi then
G _ G 0 := maxfgi ; gi0 gGi .)
P
P
P
Christopher Haon
Finite generation of anonial rings II
The struture of the proof
The direted MMP
Existene of models
Finiteness of models
Non-vanishing
Proof of Claim 1, Cont.
we will peform a sequene of ips/ontrations (for various
divisors i ) suh that:
1) the indued rational map : X 9 9 K Y only ontrats
divisors in SBs(KX + ), and
2) KY + is nef for some divisor = ( + ) where
Supp() SBs(KX + ).
Christopher Haon
Finite generation of anonial rings II
The struture of the proof
The direted MMP
Existene of models
Finiteness of models
Non-vanishing
Proof of Claim 1, Cont.
we will peform a sequene of ips/ontrations (for various
divisors i ) suh that:
1) the indued rational map : X 9 9 K Y only ontrats
divisors in SBs(KX + ), and
2) KY + is nef for some divisor = ( + ) where
Supp() SBs(KX + ).
It is then not hard to show that Y is a log terminal model for
(X ; ).
Christopher Haon
Finite generation of anonial rings II
The struture of the proof
The direted MMP
Existene of models
Finiteness of models
Non-vanishing
Proof of Claim 1, Step 1
Pik H a suÆiently ample divisor and run the KX + MMP
with saling of H .
Christopher Haon
Finite generation of anonial rings II
The struture of the proof
The direted MMP
Existene of models
Finiteness of models
Non-vanishing
Proof of Claim 1, Step 1
Pik H a suÆiently ample divisor and run the KX + MMP
with saling of H .
Reall that = S + A + B 0 where S = Supp(M + F ) and
0 B 0 B . Moreover, = + with
0 Supp(M + F ) = S .
Christopher Haon
Finite generation of anonial rings II
The struture of the proof
The direted MMP
Existene of models
Finiteness of models
Non-vanishing
Proof of Claim 1, Step 1
Pik H a suÆiently ample divisor and run the KX + MMP
with saling of H .
Reall that = S + A + B 0 where S = Supp(M + F ) and
0 B 0 B . Moreover, = + with
0 Supp(M + F ) = S .
Let C be a ipping urve. We have
0 > (KX + ) C = (D + ) C
and so the ipping lous is ontained in S .
Christopher Haon
Finite generation of anonial rings II
The struture of the proof
The direted MMP
Existene of models
Finiteness of models
Non-vanishing
Proof of Claim 1, Step 1
Pik H a suÆiently ample divisor and run the KX + MMP
with saling of H .
Reall that = S + A + B 0 where S = Supp(M + F ) and
0 B 0 B . Moreover, = + with
0 Supp(M + F ) = S .
Let C be a ipping urve. We have
0 > (KX + ) C = (D + ) C
and so the ipping lous is ontained in S .
Therefore, this is a sequene of PL-ips that must terminate
and we may assume that KX + is nef.
Christopher Haon
Finite generation of anonial rings II
The struture of the proof
The direted MMP
Existene of models
Finiteness of models
Non-vanishing
Proof of Claim 1, Step 2
We wish to remove M , so we write
KX + S + A + B 0 = KX + Fred + A + B 0 + Mred
and we run a MMP with saling of Mred .
Christopher Haon
Finite generation of anonial rings II
The struture of the proof
The direted MMP
Existene of models
Finiteness of models
Non-vanishing
Proof of Claim 1, Step 2
We wish to remove M , so we write
KX + S + A + B 0 = KX + Fred + A + B 0 + Mred
and we run a MMP with saling of Mred .
By a similar argument to the one above, the ipping lous is
always ontaned in Fred so that this is a sequene of PL-ips
that must terminate.
Christopher Haon
Finite generation of anonial rings II
The struture of the proof
The direted MMP
Existene of models
Finiteness of models
Non-vanishing
Proof of Claim 1, Step 2
We wish to remove M , so we write
KX + S + A + B 0 = KX + Fred + A + B 0 + Mred
and we run a MMP with saling of Mred .
By a similar argument to the one above, the ipping lous is
always ontaned in Fred so that this is a sequene of PL-ips
that must terminate.
Therefore, we have a model where
KX + S + A + B 0 = KX + + is nef and SBs(KX + ).
As remarked above, this is a log terminal model of (X ; ).
Christopher Haon
Finite generation of anonial rings II
The struture of the proof
The direted MMP
Existene of models
Finiteness of models
Non-vanishing
Outline of the talk
1
The struture of the proof
2
The direted MMP
3
Existene of models
4
Finiteness of models
5
Non-vanishing
Christopher Haon
Finite generation of anonial rings II
The struture of the proof
The direted MMP
Existene of models
Finiteness of models
Non-vanishing
Proof of Claims 2 and 4
In proving Thm C (resp. C-), we need to know that for any
pseudo-eetive lass KX + that we onsider, there is an
eetive divisor D 0 suh that KX + R D .
Christopher Haon
Finite generation of anonial rings II
The struture of the proof
The direted MMP
Existene of models
Finiteness of models
Non-vanishing
Proof of Claims 2 and 4
In proving Thm C (resp. C-), we need to know that for any
pseudo-eetive lass KX + that we onsider, there is an
eetive divisor D 0 suh that KX + R D .
This will hold as we are assuming Thm B (resp. we are
assuming that KX + is big).
Christopher Haon
Finite generation of anonial rings II
The struture of the proof
The direted MMP
Existene of models
Finiteness of models
Non-vanishing
Proof of Claims 2 and 4
In proving Thm C (resp. C-), we need to know that for any
pseudo-eetive lass KX + that we onsider, there is an
eetive divisor D 0 suh that KX + R D .
This will hold as we are assuming Thm B (resp. we are
assuming that KX + is big).
We will just show that there exists an integer k > 0 and
rational maps i : X 9 9 K Yi suh that if 2 C is
pseudo-eetive then there is an integer i with 1 i k suh
that i is a log terminal model for (X ; ).
Christopher Haon
Finite generation of anonial rings II
The struture of the proof
The direted MMP
Existene of models
Finiteness of models
Non-vanishing
Proof of Claims 2 and 4 ont.
Sine C is ompat and the pseudo-eetive one is losed, we
may work loally in a neighborhood
0 2 U
Christopher Haon
C \ PSEF :
Finite generation of anonial rings II
The struture of the proof
The direted MMP
Existene of models
Finiteness of models
Non-vanishing
Proof of Claims 2 and 4 ont.
Sine C is ompat and the pseudo-eetive one is losed, we
may work loally in a neighborhood
0 2 U
Let
:X
99 K
C \ PSEF :
Y be a log terminal model of KX + 0 .
Christopher Haon
Finite generation of anonial rings II
The struture of the proof
The direted MMP
Existene of models
Finiteness of models
Non-vanishing
Proof of Claims 2 and 4 ont.
Sine C is ompat and the pseudo-eetive one is losed, we
may work loally in a neighborhood
0 2 U
C \ PSEF :
Let : X 9 9 K Y be a log terminal model of KX + 0 .
is KX + 0 negative in the sense that for any exeptional
divisor E X , a(E ; X ; 0 ) < a(E ; Y ; 0 ).
Christopher Haon
Finite generation of anonial rings II
The struture of the proof
The direted MMP
Existene of models
Finiteness of models
Non-vanishing
Proof of Claims 2 and 4 ont.
Sine C is ompat and the pseudo-eetive one is losed, we
may work loally in a neighborhood
0 2 U
C \ PSEF :
Let : X 9 9 K Y be a log terminal model of KX + 0 .
is KX + 0 negative in the sense that for any exeptional
divisor E X , a(E ; X ; 0 ) < a(E ; Y ; 0 ).
So after shrinking U , we have that is KX + negative for
all 2 U .
Christopher Haon
Finite generation of anonial rings II
The struture of the proof
The direted MMP
Existene of models
Finiteness of models
Non-vanishing
Proof of Claims 2 and 4 ont.
Sine C is ompat and the pseudo-eetive one is losed, we
may work loally in a neighborhood
0 2 U
C \ PSEF :
Let : X 9 9 K Y be a log terminal model of KX + 0 .
is KX + 0 negative in the sense that for any exeptional
divisor E X , a(E ; X ; 0 ) < a(E ; Y ; 0 ).
So after shrinking U , we have that is KX + negative for
all 2 U .
So we may replae X by Y and assume that KX + 0 is nef.
Christopher Haon
Finite generation of anonial rings II
The struture of the proof
The direted MMP
Existene of models
Finiteness of models
Non-vanishing
Proof of Claims 2 and 4 ont.
Sine C is ompat and the pseudo-eetive one is losed, we
may work loally in a neighborhood
0 2 U
C \ PSEF :
Let : X 9 9 K Y be a log terminal model of KX + 0 .
is KX + 0 negative in the sense that for any exeptional
divisor E X , a(E ; X ; 0 ) < a(E ; Y ; 0 ).
So after shrinking U , we have that is KX + negative for
all 2 U .
So we may replae X by Y and assume that KX + 0 is nef.
In partiular, as 0 is big, by the base point free theorem,
KX + 0 = f H for some morphism f : X ! Z and some
ample R -divisor H on Z .
Christopher Haon
Finite generation of anonial rings II
The struture of the proof
The direted MMP
Existene of models
Finiteness of models
Non-vanishing
Proof of Claims 2 and 4 ont.
Sine C is a rational polytope, we may assume that U is a
rational polytope and so by indution on the dimension (of
the aÆne spae spanned by the polytope) there exist nitely
many rational maps i : X 9 9 K Yi over Z suh that for all
2 C \ PSEF(X =Z ) then i is a log terminal model of
(X ; ) for some 1 i k .
Christopher Haon
Finite generation of anonial rings II
The struture of the proof
The direted MMP
Existene of models
Finiteness of models
Non-vanishing
Proof of Claims 2 and 4 ont.
We have seen that KX + 0 = f H and KX + 0 R;Z 0
Christopher Haon
Finite generation of anonial rings II
The struture of the proof
The direted MMP
Existene of models
Finiteness of models
Non-vanishing
Proof of Claims 2 and 4 ont.
We have seen that KX + 0 = f H and KX + 0 R;Z 0
Further shrinking U , we may assume that for all 2 U
KX + is nef if and only if KX + is nef/Z.
Christopher Haon
Finite generation of anonial rings II
The struture of the proof
The direted MMP
Existene of models
Finiteness of models
Non-vanishing
Proof of Claims 2 and 4 ont.
We have seen that KX + 0 = f H and KX + 0 R;Z 0
Further shrinking U , we may assume that for all 2 U
KX + is nef if and only if KX + is nef/Z.
Let 0 2 U . For any 2 [0 ; ℄, we have
0 = (KX + )
(KX + 0 ) R;Z (KX + )
so that 0 is PSEF/Z i KX + is PSEF/Z i KX + is PSEF/Z.
Christopher Haon
Finite generation of anonial rings II
The struture of the proof
The direted MMP
Existene of models
Finiteness of models
Non-vanishing
Proof of Claims 2 and 4 ont.
We have seen that KX + 0 = f H and KX + 0 R;Z 0
Further shrinking U , we may assume that for all 2 U
KX + is nef if and only if KX + is nef/Z.
Let 0 2 U . For any 2 [0 ; ℄, we have
0 = (KX + )
(KX + 0 ) R;Z (KX + )
so that 0 is PSEF/Z i KX + is PSEF/Z i KX + is PSEF/Z.
If KX + is PSEF then it is PSEF/Z and so KX + is
PSEF/Z. A log terminal model/Z of KX + is a log terminal
model/Z of KX + and hene it is a log terminal model of
KX + .
Christopher Haon
Finite generation of anonial rings II
The struture of the proof
The direted MMP
Existene of models
Finiteness of models
Non-vanishing
Outline of the talk
1
The struture of the proof
2
The direted MMP
3
Existene of models
4
Finiteness of models
5
Non-vanishing
Christopher Haon
Finite generation of anonial rings II
The struture of the proof
The direted MMP
Existene of models
Finiteness of models
Non-vanishing
Proof of Claim 3
We may assume that = A + B has simple normal rossings,
A is ample and B 0.
Christopher Haon
Finite generation of anonial rings II
The struture of the proof
The direted MMP
Existene of models
Finiteness of models
Non-vanishing
Proof of Claim 3
We may assume that = A + B has simple normal rossings,
A is ample and B 0.
If for any xed k 0 suÆiently divisible,
h0 (OX ([mk (KX + )℄ + kA))
is a bounded funtion of m, then by a result of Nakayama, we
have KX + N 0. Reall that N = N (KX + ) is the
limit as ! 0 of divisors in SBs(KX + + A).
Christopher Haon
Finite generation of anonial rings II
The struture of the proof
The direted MMP
Existene of models
Finiteness of models
Non-vanishing
Proof of Claim 3
We may assume that = A + B has simple normal rossings,
A is ample and B 0.
If for any xed k 0 suÆiently divisible,
h0 (OX ([mk (KX + )℄ + kA))
is a bounded funtion of m, then by a result of Nakayama, we
have KX + N 0. Reall that N = N (KX + ) is the
limit as ! 0 of divisors in SBs(KX + + A).
So A0 R A + N (KX + ) is ample and
KX + A0 + B R N 0 and hene KX + A0 + B has a log
terminal model whih is also a log terminal model for
KX + A + B .
Christopher Haon
Finite generation of anonial rings II
The struture of the proof
The direted MMP
Existene of models
Finiteness of models
Non-vanishing
Proof of Claim 3
We may assume that = A + B has simple normal rossings,
A is ample and B 0.
If for any xed k 0 suÆiently divisible,
h0 (OX ([mk (KX + )℄ + kA))
is a bounded funtion of m, then by a result of Nakayama, we
have KX + N 0. Reall that N = N (KX + ) is the
limit as ! 0 of divisors in SBs(KX + + A).
So A0 R A + N (KX + ) is ample and
KX + A0 + B R N 0 and hene KX + A0 + B has a log
terminal model whih is also a log terminal model for
KX + A + B .
By the base point free theorem KX + A + B R D 0.
Christopher Haon
Finite generation of anonial rings II
The struture of the proof
The direted MMP
Existene of models
Finiteness of models
Non-vanishing
Proof of Claim 3 ont.
If h0 (OX ([mk (KX + )℄ + kA)) is not bounded, then
we may assume that = S + A + B where [℄ = S , A is
ample, S 6 N (KX + ).
Claim.
Christopher Haon
Finite generation of anonial rings II
The struture of the proof
The direted MMP
Existene of models
Finiteness of models
Non-vanishing
Proof of Claim 3 ont.
If h0 (OX ([mk (KX + )℄ + kA)) is not bounded, then
we may assume that = S + A + B where [℄ = S , A is
ample, S 6 N (KX + ).
Proof. Pik H R m(KX + ) + A with multx (H ) > n for
some general x 2 X .
Claim.
Christopher Haon
Finite generation of anonial rings II
The struture of the proof
The direted MMP
Existene of models
Finiteness of models
Non-vanishing
Proof of Claim 3 ont.
If h0 (OX ([mk (KX + )℄ + kA)) is not bounded, then
we may assume that = S + A + B where [℄ = S , A is
ample, S 6 N (KX + ).
Proof. Pik H R m(KX + ) + A with multx (H ) > n for
some general x 2 X .
For t 2 [0; m℄ onsider
Claim.
(t + 1)(KX + ) = KX +
KX +
m
m
t
m
m
t
A + B + t (K X + +
A+B +
Christopher Haon
t
H = KX + t :
m
Finite generation of anonial rings II
1
m
A) R
The struture of the proof
The direted MMP
Existene of models
Finiteness of models
Non-vanishing
Proof of Claim 3 ont.
Then for some 0 < 1 we have KX + 0 is KLT;
t A0 = (=m)A for t 2 [0; m ℄, and KX + m
log anonial enter not ontained in SBs(KX + ).
Christopher Haon
Finite generation of anonial rings II
has a
The struture of the proof
The direted MMP
Existene of models
Finiteness of models
Non-vanishing
Proof of Claim 3 ont.
Then for some 0 < 1 we have KX + 0 is KLT;
t A0 = (=m)A for t 2 [0; m ℄, and KX + m
log anonial enter not ontained in SBs(KX + ).
We pass to a log resolution : Y ! X and write
KY +
t
= (KX + t ) + Et
and we then anel ommon omponents of t and
SBs( (KX + t )). The laim now follows easily.
Christopher Haon
Finite generation of anonial rings II
has a
The struture of the proof
The direted MMP
Existene of models
Finiteness of models
Non-vanishing
Proof of Claim 3 ont.
We now run a KX + S + A + B MMP with saling of (some
multiple of) A.
Christopher Haon
Finite generation of anonial rings II
The struture of the proof
The direted MMP
Existene of models
Finiteness of models
Non-vanishing
Proof of Claim 3 ont.
We now run a KX + S + A + B MMP with saling of (some
multiple of) A.
For 0 < t 1 we get log terminal models t : X 9 9 K Yt of
KX + S + A + B + tA and S 9 9 K Tt .
Christopher Haon
Finite generation of anonial rings II
The struture of the proof
The direted MMP
Existene of models
Finiteness of models
Non-vanishing
Proof of Claim 3 ont.
We now run a KX + S + A + B MMP with saling of (some
multiple of) A.
For 0 < t 1 we get log terminal models t : X 9 9 K Yt of
KX + S + A + B + tA and S 9 9 K Tt .
S is not ontained in SBs(KX + S + A + B ) and so it is not
ontrated.
Christopher Haon
Finite generation of anonial rings II
The struture of the proof
The direted MMP
Existene of models
Finiteness of models
Non-vanishing
Proof of Claim 3 ont.
We now run a KX + S + A + B MMP with saling of (some
multiple of) A.
For 0 < t 1 we get log terminal models t : X 9 9 K Yt of
KX + S + A + B + tA and S 9 9 K Tt .
S is not ontained in SBs(KX + S + A + B ) and so it is not
ontrated.
We would like to say that for 0 < t 1, we have Yt =Y a
log terminal model of KX + S + A + B . Then by the base
point free theorem, KX + S + A + B R D 0.
Christopher Haon
Finite generation of anonial rings II
The struture of the proof
The direted MMP
Existene of models
Finiteness of models
Non-vanishing
Proof of Claim 3 ont.
We now run a KX + S + A + B MMP with saling of (some
multiple of) A.
For 0 < t 1 we get log terminal models t : X 9 9 K Yt of
KX + S + A + B + tA and S 9 9 K Tt .
S is not ontained in SBs(KX + S + A + B ) and so it is not
ontrated.
We would like to say that for 0 < t 1, we have Yt =Y a
log terminal model of KX + S + A + B . Then by the base
point free theorem, KX + S + A + B R D 0.
We an only onlude that Tt Yt = T Y in a
neighborhood of Tt .
Christopher Haon
Finite generation of anonial rings II
The struture of the proof
The direted MMP
Existene of models
Finiteness of models
Non-vanishing
Proof of Claim 3 ont.
We now run a KX + S + A + B MMP with saling of (some
multiple of) A.
For 0 < t 1 we get log terminal models t : X 9 9 K Yt of
KX + S + A + B + tA and S 9 9 K Tt .
S is not ontained in SBs(KX + S + A + B ) and so it is not
ontrated.
We would like to say that for 0 < t 1, we have Yt =Y a
log terminal model of KX + S + A + B . Then by the base
point free theorem, KX + S + A + B R D 0.
We an only onlude that Tt Yt = T Y in a
neighborhood of Tt .
So KT + = (KY + t )jT is nef, is big and hene KT + is semiample.
t
Christopher Haon
Finite generation of anonial rings II
The struture of the proof
The direted MMP
Existene of models
Finiteness of models
Non-vanishing
Proof of Claim 3 ont.
We want to lift setions of KT + .
Christopher Haon
Finite generation of anonial rings II
The struture of the proof
The direted MMP
Existene of models
Finiteness of models
Non-vanishing
Proof of Claim 3 ont.
We want to lift setions of KT + .
For simpliity assume KX + is Q -Cartier (this avoids a
diophantine approximation argument).
Christopher Haon
Finite generation of anonial rings II
The struture of the proof
The direted MMP
Existene of models
Finiteness of models
Non-vanishing
Proof of Claim 3 ont.
We want to lift setions of KT + .
For simpliity assume KX + is Q -Cartier (this avoids a
diophantine approximation argument).
We may then assume that for some integer m > 0, we have
m(KX + ) is integral Weil, m(KY + ) is Cartier in a
neighborhood of T and h0 (m(KT + )) > 0,
Christopher Haon
Finite generation of anonial rings II
The struture of the proof
The direted MMP
Existene of models
Finiteness of models
Non-vanishing
Proof of Claim 3 ont.
We want to lift setions of KT + .
For simpliity assume KX + is Q -Cartier (this avoids a
diophantine approximation argument).
We may then assume that for some integer m > 0, we have
m(KX + ) is integral Weil, m(KY + ) is Cartier in a
neighborhood of T and h0 (m(KT + )) > 0,
Then we look at the orresponding short exat sequene and
apply Kawamata-Viehweg vanishing:
Christopher Haon
Finite generation of anonial rings II
The struture of the proof
The direted MMP
Existene of models
Finiteness of models
Non-vanishing
Proof of Claim 3 ont.
We have
0 ! OY (m(KY + )
t
t
T ) ! OY (m(KY + ))
t
t
! OT (m(KT + )) ! 0:
Christopher Haon
Finite generation of anonial rings II
The struture of the proof
The direted MMP
Existene of models
Finiteness of models
Non-vanishing
Proof of Claim 3 ont.
We have
0 ! OY (m(KY + )
t
t
T ) ! OY (m(KY + ))
t
t
! OT (m(KT + )) ! 0:
T (m 1)tA is KLT (for 0 < t
Then KY +
(m 1)(KY + + tA) is nef and big.
t
1) and
t
Christopher Haon
Finite generation of anonial rings II
The struture of the proof
The direted MMP
Existene of models
Finiteness of models
Non-vanishing
Proof of Claim 3 ont.
We have
0 ! OY (m(KY + )
t
t
T ) ! OY (m(KY + ))
t
t
! OT (m(KT + )) ! 0:
T (m 1)tA is KLT (for 0 < t 1) and
Then KY +
(m 1)(KY + + tA) is nef and big.
By Kawamata-Viehweg vanishing, we may lift setions to
m(KY + ).
t
t
t
Christopher Haon
Finite generation of anonial rings II
The struture of the proof
The direted MMP
Existene of models
Finiteness of models
Non-vanishing
Proof of Claim 3 ont.
We have
0 ! OY (m(KY + )
t
t
T ) ! OY (m(KY + ))
t
t
! OT (m(KT + )) ! 0:
T (m 1)tA is KLT (for 0 < t 1) and
Then KY +
(m 1)(KY + + tA) is nef and big.
By Kawamata-Viehweg vanishing, we may lift setions to
m(KY + ).
It is easy to see that these setions lift to setions of KX + .
t
t
t
Christopher Haon
Finite generation of anonial rings II