finite generation of the canonical ringjmckerna/talks/mori.pdffinite generation of the canonical...

Finite generation of the canonical

ring

James McKernan

MIT

Finite generation of the canonical ring – p. 1

Classification Problem

We would like to give an abstract classification ofcomplex algebraic varieties.




A variety is traditionally given as the zeroes of acollection of polynomials.





Abstract refers to the fact that we would like toclassify varieties intrinsically, without reference tothe embedding into space.






Compare this with the classification of abstractgroups.






Compare this with the classification of abstractgroups.

Groups were originally defined as subsets ofpermutation groups closed under multiplication andtaking inverses.


Calculus on varieties

Calculus is a very powerful tool.




Suppose that we want to integrate on a variety ofdimension n.





Then we should think of our variety as a smoothcomplex manifold and we want to write down adifferential n-form ω.





Then we should think of our variety as a smoothcomplex manifold and we want to write down adifferential n-form ω.

It turns out that the most important invariant of theform ω is its zero locus and polar locus, which is adivisor.


Divisors

A divisor D is a formal linear combination ofcodimension one subvarieties:

D =∑

niDi,

where ni may be positive or negative.


Divisors

A divisor D is a formal linear combination ofcodimension one subvarieties:

D =∑

niDi,

where ni may be positive or negative.

Given a differential n-form ω, locally it is of theform f(z)dz1 ∧ dz2 ∧ dz3 · · · ∧ dzn, and we take thezeroes minus the poles of f

(f)0 − (f)∞,

to get a divisor KX , which we call the canonicaldivisor.


Canonical divisor

There are two crucial properties of the canonicaldivisor, which together make it a very usefulinvariant.


Canonical divisor


Varieties whose canonical divisors are similar havevery similar properties.


Canonical divisor



It is straightforward to compute in concreteexamples.


Canonical divisor




We illustrate these principles in the case of smoothprojective curves.


Canonical divisor




We illustrate these principles in the case of smoothprojective curves.

Smooth projective curves are the same as compactRiemann surfaces.


Riemann sphere

The simplest compact Riemann surface is the

Riemann sphere, P1 = C ∪ {∞}.


Riemann sphere



If we start with dz/z on one coordinate patch thenwe get

d(1/z)/(1/z) = −dz/z,

on the other coordinate patch.


Riemann sphere




d(1/z)/(1/z) = −dz/z,


This differential form has two poles, one at zero(call this p) and one at∞ (call this q).


Riemann sphere




d(1/z)/(1/z) = −dz/z,



Thus the canonical divisor KP1 = −p − q.


Riemann sphere




d(1/z)/(1/z) = −dz/z,



Thus the canonical divisor KP1 = −p − q.

A smooth conic in P2 is isomorphic to P

1

(stereographic projection).


Elliptic curves

The next simplest curves are smooth cubics C in P2.


Elliptic curves


As Riemann surfaces they are isomorphic to C/Λ,where Λ ≃ Z

2 is a lattice.


Elliptic curves



2 is a lattice.

The differential form dz on C is translation invariantand so it descends to a differential form on C withno zeroes or poles.


Elliptic curves



2 is a lattice.


Hence the canonical divisor is trivial.


Elliptic curves



2 is a lattice.


Hence the canonical divisor is trivial.

In fact the lattice is a normal subgroup so that C is acomplex Lie group.


Riemann-Hurwitz and higher genus

More generally, if the genus of C is g, then thecanonical divisor has degree 2g − 2.




If f : C −→ B is a non-constant holomorphic mapof degree d of Riemann surfaces then

2g − 2 = d(2h − 2) + b,

where b ≥ 0 counts the number of branch points.





2g − 2 = d(2h − 2) + b,


C is determined by B and the branch points.





2g − 2 = d(2h − 2) + b,



The general curve of genus g ≥ 2 has finitely many

covers of P1 of degree (g + 2)/2. The number of

branch points is then 2g − 2 + (g + 2) = 3g.





2g − 2 = d(2h − 2) + b,



The general curve of genus g ≥ 2 has finitely many

covers of P1 of degree (g + 2)/2. The number of

branch points is then 2g − 2 + (g + 2) = 3g.

So there is a 3g − 3 dimensional family of curves.Finite generation of the canonical ring – p. 8

Behaviour of the canonical divisor

KC g Topology:Fund gp

Geometry:Aut gp

Arithmetic:# rtl pts

-ve




Geometry:Aut gp


-ve 0




Geometry:Aut gp


-ve 0 simplyconnected




Geometry:Aut gp



PGL(2)




Geometry:Aut gp



PGL(2) Infinite




Geometry:Aut gp



PGL(2) Infinite

0




Geometry:Aut gp



PGL(2) Infinite

0 1




Geometry:Aut gp



PGL(2) Infinite

0 1 abelian




Geometry:Aut gp



PGL(2) Infinite

0 1 abelian almostabelian




Geometry:Aut gp



PGL(2) Infinite


fg abeliangp




Geometry:Aut gp



PGL(2) Infinite


fg abeliangp

+ve




Geometry:Aut gp



PGL(2) Infinite


fg abeliangp

+ve ≥ 2




Geometry:Aut gp



PGL(2) Infinite


fg abeliangp

+ve ≥ 2 large




Geometry:Aut gp



PGL(2) Infinite


fg abeliangp

+ve ≥ 2 large Finite




Geometry:Aut gp



PGL(2) Infinite


fg abeliangp

+ve ≥ 2 large Finite Finite


Adjunction

In higher dimensions, one way to compute thecanonical divisor is by adjunction.


Adjunction


If S is a smooth divisor in a projective variety X ,

adjunction says KS = (KX + S)|S .


Adjunction




KPn = −(n + 1)H , where H is a hyperplane.


Adjunction





So if S is a hypersurface of degree d then

KS = (d − n − 1)H|S.


Adjunction






KS = (d − n − 1)H|S.

KS is negative if d < n + 1, zero if d = n + 1 andpositive if d > n + 1.


Adjunction






KS = (d − n − 1)H|S.

KS is negative if d < n + 1, zero if d = n + 1 andpositive if d > n + 1.

This checks with the case of conics and cubics in P2.


General picture

By now we start to see a general picture emerge.There are three cases:


General picture


• If the canonical divisor is negative then the variety isreasonably well-behaved.


General picture



• If the canonical divisor is zero then the geometry ismore complicated.


General picture




• If the canonical divisor is positive then the geometrycan be very complicated and we can only expect tomake general statements.


General picture





Aim: A classification of higher dimensionalvarieties similar to the case of curves.


General picture





Aim: A classification of higher dimensionalvarieties similar to the case of curves.

The problem is that birational geometry gets in theway. Finite generation of the canonical ring – p. 11

Rational maps

Consider the map C299K C given by

(x, y) −→ y/x. Geometrically this sends the point

(x, y) to the slope of the line through (x, y) and theorigin.


Rational maps




This map is not defined along the y-axis. If wereplace C by C ∪ {∞} = P

1, then we get a new map

C299K P

1.


Rational maps




This map is not defined along the y-axis. If wereplace C by C ∪ {∞} = P

1, then we get a new map

C299K P

1.

However this map is still not defined at origin(geometrically what is the slope of the lineconnecting the origin to the origin?).


The graph

Consider the closure of the graph of this rational

map Γ ⊂ C2 × P

1. This admits two morphisms

p : Γ −→ C2 and q : Γ −→ P

1.


The graph


map Γ ⊂ C2 × P


p : Γ −→ C2 and q : Γ −→ P

1.

The equations for Γ are xT = yS, where (x, y) arecoordinates on C

2 and [S : T ] are coordinates on P1.


The graph


map Γ ⊂ C2 × P


p : Γ −→ C2 and q : Γ −→ P

1.



Focus on p. p is an isomorphism outside the origin.

But over the origin, we get a whole copy of P1. We

have replaced the point p by E = P1.


The graph


map Γ ⊂ C2 × P


p : Γ −→ C2 and q : Γ −→ P

1.






p is called the blow up of C2 at the origin.


The graph


map Γ ⊂ C2 × P


p : Γ −→ C2 and q : Γ −→ P

1.






p is called the blow up of C2 at the origin.

Using local coordinates, we can take any smoothsurface and blow up any point.


The blow up

For example, consider an abelian surface T . As a

complex manifold, T = C2/Λ, where Λ ≃ Z

4 is alattice.


The blow up



4 is alattice.

T is a complex Lie group, since C2 is a group and Λ

is a normal subgroup.


The blow up



4 is alattice.



Let π : S −→ T blow up the origin. Then S is not analgebraic group. Indeed the exceptional divisor E is

the only copy of P1 in S and so S is not even

homogeneous.


The blow up



4 is alattice.



Let π : S −→ T blow up the origin. Then S is not analgebraic group. Indeed the exceptional divisor E is

the only copy of P1 in S and so S is not even

homogeneous.

It is the aim of Mori theory to reverse this process.Start with S and replace it with T .


Back to basics

So, given an arbitrary smooth surface S, how doesone identify the exceptional curves?


Back to basics


First observe that E2 = E · E = −1. On the otherhand KS · E = −1. Thus we are looking for curvesC such that KS · C < 0.


Back to basics



We say that KS is nef if KS · C ≥ 0 for every curveC.


Back to basics



We say that KS is nef if KS · C ≥ 0 for every curveC.

The Minimal Model Program is an algorithm toeliminate all KX-negative curves.


The MMP: surfaces

Start with a smooth surface S.


The MMP: surfaces


Is KS nef? If yes then STOP.


The MMP: surfaces



If not, then there is a curve C and a morphismπ : S −→ Z, with connected fibres, which sends Cto a point, such that one of three things happens:


The MMP: surfaces




• Z is a point. In this case S ≃ P2. STOP.


The MMP: surfaces





• Z is a curve. The fibres are copies of P1. STOP.


The MMP: surfaces





• Z is a curve. The fibres are copies of P1. STOP.

• Z is a surface. φ blows down a −1-curve. Replace Swith Z = T and go back to (2).


MMP: general case

Start with a smooth projective variety X .


MMP: general case


If KX is nef, then STOP.


MMP: general case



If not, then there is a morphism π : X −→ Z and:


MMP: general case




• (Mori fibre space) dim Z < dim X . The fibres Fhave −KF ample (KF negative). STOP.


MMP: general case





• dim Z = dim X , exceptional locus is a divisor.Replace X by Z and go back to (2).


MMP: general case





• dim Z = dim X , exceptional locus is a divisor.Replace X by Z and go back to (2).

• dim Z = dim X , exceptional locus has codimensiongreater than one. We cannot replace X by Z, sinceZ is too singular.


Flips

Instead of contracting C, we try to replace X by another

birational model X+, X 99K X+, such thatπ+ : X+ −→ Z is KX+-ample.

Xφ

- X+

Z.

�

π+π

-


Existence and termination

This operation is called a flip.




Even supposing we can perform a flip, how do weknow that this process terminates?




Even supposing we can perform a flip, how do weknow that this process terminates?

It is clear that we cannot keep contracting divisors,but why could there not be an infinite sequence offlips?



Theorem. [Hacon,-] Flips exist.




Theorem. [Birkar,Cascini,Hacon,-] Let X be a smoothprojective variety, such that KX is big.Then the MMP with scaling terminates. In particular Xhas a minimal model.




Theorem. [Birkar,Cascini,Hacon,-] Let X be a smoothprojective variety, such that KX is big.Then the MMP with scaling terminates. In particular Xhas a minimal model.

Theorem. [Birkar,Cascini,Hacon,-] Let X be a smoothprojective variety. If KX · C < 0 for some coveringfamily of curves, then X is birational to a Mori fibrespace π : Y −→ Z.


The canonical ring

Let H0(X,OX(mKX)) be the vector space of weight mdifferential n-forms, that transform as the Jacobianraised to the mth power.


The canonical ring


Theorem. [Birkar,Cascini,Hacon,-; Siu] Let X be asmooth projective variety. Then the canonical ring

⊕

m∈N

H0(X,OX(mKX)),

is finitely generated, as an algebra over C.


The canonical ring


Theorem. [Birkar,Cascini,Hacon,-; Siu] Let X be asmooth projective variety. Then the canonical ring

⊕

m∈N

H0(X,OX(mKX)),

is finitely generated, as an algebra over C.

There are some new proofs of this result, due to Lazicand Paun.


Existence of flips

Suppose that we are given π : X −→ Z a smallcontraction, −KX is ample over Z. Our aim is to

construct the flip π+ : X+ −→ Z of π.


Existence of flips



The existence of the flip is local, so that we mayassume that Z = Spec A is affine.


Existence of flips



The existence of the flip is local, so that we mayassume that Z = Spec A is affine.

In fact it suffices to prove that the canonical ring

R(X, KX) =⊕

m∈N

H0(X,OX(mKX)),

is a finitely generated A-algebra.


Reduction to pl-flips

Shokurov proved that to prove the existence ofKX-flips, it suffices to prove the existence of flipsfor KX + S + B, where S has coefficient one.




The key point is that

(KX + S + B)|S = KS + C,

so that we have the start of an induction.





(KX + S + B)|S = KS + C,


In fact there is a natural restriction map,

H0(X,OX(m(KX+S+B)) −→ H0(X,OS(m(KS+C)).





(KX + S + B)|S = KS + C,


In fact there is a natural restriction map,

H0(X,OX(m(KX+S+B)) −→ H0(X,OS(m(KS+C)).

Let RS be the restricted algebra in R(S, KS + C),the direct sum of all the images.


Lifting sections

It is easy to see that R(X, KX + S + B) is finitelygenerated iff RS is finitely generated.


Lifting sections


If the natural restriction maps are surjective, then

RS = R(S, KS + C) and we would be done byinduction.


Lifting sections


If the natural restriction maps are surjective, then

RS = R(S, KS + C) and we would be done byinduction.

In fact it is a very natural question to ask whichsections can one lift. Fortunately Siu, using ideasfrom PDE’s (multiplier ideal sheaves) and thenKawamata gave some important partial answers tothis question.


Limiting algebras

In fact we show that in every degree, one can find adivisor Θm on some fixed higher model of S, suchthat

RS =⊕

m∈N

H0(S,OS(mk(KS + Θm)).


Limiting algebras


RS =⊕

m∈N


Θ• forms a convex sequence. Let Θ = lim Θm.


Limiting algebras


RS =⊕

m∈N



Then we can prove, using ideas of Shokurov(saturation and Diophantine approximation) thatΘ = Θm is constant.


Limiting algebras


RS =⊕

m∈N



Then we can prove, using ideas of Shokurov(saturation and Diophantine approximation) thatΘ = Θm is constant.

But then RS =⊕

m∈NH0(S,OS(mk(KS + Θ)) and

we are done.


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