History and Moriwaki’s early works
On canonical rings
On log canonical rings of projective varieties

On canonical rings
Osamu Fujino

Kyoto University
November 29, 2021

Osamu Fujino

On canonical rings

History and Moriwaki’s early works
On canonical rings
On log canonical rings of projective varieties

1

History and Moriwaki’s early works

2

On canonical rings

3

On log canonical rings of projective varieties

Osamu Fujino

On canonical rings

History and Moriwaki’s early works
On canonical rings
On log canonical rings of projective varieties

History and Moriwaki’s earlyworks

Osamu Fujino

On canonical rings

History and Moriwaki’s early works
On canonical rings
On log canonical rings of projective varieties

History

In 1996:
Moriwaki’s seminar
Text Book: D. Mumford: Algebraic Geometry I. Complex
Projective Varieties
Students: O. Fujino, K. Yamaki, and 4 others (Ono, Tane,
Higuchi, Yuyama)Moriwaki’s lecture
Topic: elementary proof of Modell–Faltings
The lecture note was published in 2017 by A. Moriwaki,
S. Kawaguchi, and H. Ikoma

Osamu Fujino

On canonical rings

History and Moriwaki’s early works
On canonical rings
On log canonical rings of projective varieties

Moriwaki’s early works

Zariski decomposition (1986, 1988)
There are many variants.
Some generalization of the Zariski decomposition still plays acrucial role in MMP.
Torsion-freeness (1987)
His result was already generalized completely.
Openness of nefness (1992)
This is related to my recent work on MMP.
This problem is still widely open.

Osamu Fujino

On canonical rings

History and Moriwaki’s early works
On canonical rings
On log canonical rings of projective varieties

Zariski decomposition, 1
Definition 1.1 (Zariski decomposition in C-K-M’s sense)
Let f : X → Y be a proper surjective morphism of normal varieties.
An expression D = P + N of R-Cartier divisors D, P and N is called
the Zariski decomposition in C-K-M’s sense if the following
conditions are satisfied:(1) P is f -nef,(2) N is effective, and(3) the natural homomorphisms f∗ OX (⌊mP⌋) → f∗ OX (⌊mD⌋) arebijective for all m ∈ N.
In his master thesis, Moriwaki proved that P is semiample in somespecial case. Zariski decomposition is closely related to the finite
generation of canonical rings.
Osamu Fujino

On canonical rings

History and Moriwaki’s early works
On canonical rings
On log canonical rings of projective varieties

Zariski decomposition, 2

Zariski decomposition still plays an important role in MMP.
Theorem 1.2 (V. Lazić and N. Tsakanikas)
Assume that any smooth projective variety V such that KV ispseudo-effective has an NQC weak Zariski decomposition. Thenevery projective log canonical pair (X, ∆) such that KX + ∆ ispseudo-effective has a minimal model.
The existence problem of minimal models for log canonical pairs
can be reduced to the existence problem of NQC weak Zariski
decompositions for smooth projective varieties.

Osamu Fujino

On canonical rings

History and Moriwaki’s early works
On canonical rings
On log canonical rings of projective varieties

Torsion-freeness, 1
Theorem 1.3 (J. Kollár, 1986)
Let f : X → Y be a surjective morphism between complex
projective varieties such that X is smooth. Then Ri f∗ ωX is
torsion-free for every i.
This theorem can be proved easily by using Hodge theory.
Theorem 1.4 (A. Moriwaki, 1987)
Let f : X → Y be a projective surjective morphism between
complex analytic varieties such that X is smooth. Then Ri f∗ ωX is
torsion-free for every i.
Here, X is not necessarily compact. Hence we can not directly use
the Hodge structure on X .
Osamu Fujino

On canonical rings

History and Moriwaki’s early works
On canonical rings
On log canonical rings of projective varieties

Torsion-freeness, 2
Takegoshi completely generalized Kollár’s torsion-freeness.
Theorem 1.5 (K. Takegoshi, 1995)
Let f : X → Y be a proper surjective morphism between complex
analytic varieties such that X is a Kähler manifold. Then Ri f∗ ωX is
torsion-free for every i.
We skip my contributions on this topic. Then we finally get:
Theorem 1.6 (S. Matsumura)Let f : X → Y be a proper surjective morphism between complex
analytic varieties such that X is a Kähler manifold. Let (L, h) be a
pseudo-effective line bundle on X . Then Ri f∗ (ωX ⊗ L ⊗ J(h)) is
torsion-free for every i.
Osamu Fujino

On canonical rings

History and Moriwaki’s early works
On canonical rings
On log canonical rings of projective varieties

Openness of nefness, 1
f : X → S : projective morphism of algebraic varieties
L: line bundle on X
We put:A := {s ∈ S | L s is ample on X s }N := {s ∈ S | L s is nef on X s }Theorem 1.7
A is Zariski open in S
This theorem is well known. It is natural to ask:
Question 1.8
Is N Zariski open?
Osamu Fujino

On canonical rings

History and Moriwaki’s early works
On canonical rings
On log canonical rings of projective varieties

Openness of nefness, 2Theorem 1.9 (Moriwaki?)

N is not Zariski open
Moriwaki constructed an explicit example in positive characteristic.
He used Frobenius pull-backs of semistable rank 2 vector bundles
on curves.
Question 1.10
Is N Zariski open in characteristic zero?
I think that everyone believes that N is not open. However, no
example has been founded yet. When L is an R-Cartier divisor,
Lesieutre constructed an example such that N is not open.
Osamu Fujino

On canonical rings

History and Moriwaki’s early works
On canonical rings
On log canonical rings of projective varieties

On canonical rings

Osamu Fujino

On canonical rings

History and Moriwaki’s early works
On canonical rings
On log canonical rings of projective varieties

On canonical rings, 1

X : compact complex manifold∑∆ = i ai ∆i is a Q-divisor on X with 0 ≤ ai ≤ 1 for every i,
Supp ∆: simple normal crossing divisor
We putR(X, ∆) :=⊕H 0 (X, OX (⌊m(KX + ∆)⌋))

m≥0

and call it the log canonical ring of (X, ∆). When ∆ = 0, we write

R(X) :=⊕H 0 (X, OX (mKX ))m≥0and call it the canonical ring of X .
Osamu Fujino

On canonical rings

History and Moriwaki’s early works
On canonical rings
On log canonical rings of projective varieties

On canonical rings, 2
We are mainly interested in:Conjecture 2.1 (Finite generation of R(X, ∆))R(X, ∆) is a finitely generated C-algebra.
If dim X = 1, then X is a compact Riemann surface. In thiscase, the above conjecture is almost obvious.
If dim X = 2, then we can check that R(X, ∆) is always finitely
generated.
When dim X = 2, KX + ∆ is big, and ai = 1 for some i, theproof is much more difficult than we expected. This was
carried out by Takao Fujita.
Hence we may assume that dim X ≥ 3.
Osamu Fujino

On canonical rings

History and Moriwaki’s early works
On canonical rings
On log canonical rings of projective varieties

On canonical rings, 3

In dimension 3, Moriwaki established:
Theorem 2.2 (A. Moriwaki, 1988)
In dimension ≤ 3, R(X) is always a finitely generated C-algebra.
In Theorem 2.2, it is sufficient to treat the case where
dim X = 3, κ(X) = 2, and X is not algebraic.
Although I have not checked the details, I believe that we canprove that R(X, ∆) is always finitely generated in dimension≤ 3.

Osamu Fujino

On canonical rings

History and Moriwaki’s early works
On canonical rings
On log canonical rings of projective varieties

On canonical rings, 4

We need the condition that X is Kähler in dimension ≥ 4.
Theorem 2.3 (P. M. H. Wilson, 1981)
There exists a compact non-Kähler manifold in dimension 4 suchthat R(X) is not finitely generated.
Hence the correct conjecture is:
Conjecture 2.4 (Finite generation of R(X, ∆))We further assume that X is Kähler. Then R(X, ∆) is a finitely
generated C-algebra.

Osamu Fujino

On canonical rings

History and Moriwaki’s early works
On canonical rings
On log canonical rings of projective varieties

On canonical rings, 5When X is projective, we have:
Theorem 2.5 (Birkar–Cascini–Hacon–Mc Kernan, 2010)
We assume that X is projective and that ai < 1 holds for every i.
Then R(X, ∆) is a finitely generated C-algebra. In particular, R(X) isa finitely generated C-algebra.
BCHM partially established MMP for kawamata log terminal
pairs and obtained the above finite generation. It is a great
work on MMP.
Idea of the proof: By Fujino–Mori’s canonical bundle formula, we
can reduce the problem to the case where KX + ∆ is big. In thiscase, by BCHM, we can prove the existence of good minimalmodel. Then R(X, ∆) is a finitely generated C-algebra.
Osamu Fujino

On canonical rings

History and Moriwaki’s early works
On canonical rings
On log canonical rings of projective varieties

On canonical rings, 6

For X , which is Kähler but is not projective, we have:
Theorem 2.6 (O. Fujino, 2015)
If X is a compact Kähler manifold and ai < 1 holds for every i, thenR(X, ∆) is a finitely generated C-algebra.
In particular, R(X) is always a finitely generated C-algebra when X
is Kähler.
By this theorem, Moriwak’s theorem, and Wilson’s example,
we completely solved the finite generation conjecture ofcanonical rings for compact complex manifolds

Osamu Fujino

On canonical rings

History and Moriwaki’s early works
On canonical rings
On log canonical rings of projective varieties

On canonical rings, 7

Idea of Proof:
We may assume that κ(X, KX + ∆) ≥ 1.
We take the Iitaka fibration f : X → Y with respect to KX + ∆.
By a generalization of Fujino–Mori’s canonical bundle formula,
we can construct ∆Y such that R(X, ∆) is finitely generated if
and only if R(Y, ∆Y ) is finitely generated.
Here, we need that X is Kähler and ai < 1 for every i.
By construction, KY + ∆Y is big and Y is a smooth projective
variety. Therefore, we can apply the result by BCHM.

Osamu Fujino

On canonical rings

History and Moriwaki’s early works
On canonical rings
On log canonical rings of projective varieties

On log canonical rings ofprojective varieties

Osamu Fujino

On canonical rings

History and Moriwaki’s early works
On canonical rings
On log canonical rings of projective varieties

On log canonical rings, 1

From now on, we always assume that X is a complex projective
variety.

X : smooth projective variety defined over C∑∆ = i ai ∆i is an R-divisor on X with 0 ≤ ai ≤ 1 for every i,
Supp ∆: simple normal crossing divisor∑⌊∆⌋ := ai =1 ∆i , the round-down of ∆Osamu Fujino

On canonical rings

History and Moriwaki’s early works
On canonical rings
On log canonical rings of projective varieties

On log canonical rings, 2

We state our main goal again.
Conjecture 3.1 (Finite generation of R(X, ∆))If we further assume that ∆ is a Q-divisor, then R(X, ∆) is finitely
generated.
Note that we have already known that R(X, ∆) is finitely
generated when ⌊∆⌋ = 0 by BCHM. Hence we are onlyinterested in the case where ⌊∆⌋ , 0.

Osamu Fujino

On canonical rings

History and Moriwaki’s early works
On canonical rings
On log canonical rings of projective varieties

On log canonical rings, 3Conjecture (Conjecture An )If dim X = n and ∆ is a Q-divisor, then R(X, ∆) is a finitely
generated C-algebra.
Conjecture (Conjecture Bn )If we further assume that dim X = n, ∆ is a Q-divisor, ⌊∆⌋ isirreducible, and KX + ∆ is big, then R(X, ∆) is a finitely generated
C-algebra.
Conjecture (Conjecture Cn )If dim X = n and KX + ∆ is pseudo-effective, then (X, ∆) has a goodminimal model.
Osamu Fujino

On canonical rings

History and Moriwaki’s early works
On canonical rings
On log canonical rings of projective varieties

On log canonical rings, 4Conjecture (Conjecture An )If dim X = n and ∆ is a Q-divisor, then R(X, ∆) is a finitely
generated C-algebra.
This conjecture is the main goal when X is a projective variety.
In n ≤ 4, Conjecture An holds true. We note:
Conjecture C≤3 =⇒ Conjecture A4
If ⌊∆⌋ = 0, then R(X, ∆) is finitely generated by BCHM

Osamu Fujino

On canonical rings

History and Moriwaki’s early works
On canonical rings
On log canonical rings of projective varieties

On log canonical rings, 5Conjecture (Conjecture Bn )If we further assume that dim X = n, ∆ is a Q-divisor, ⌊∆⌋ isirreducible, and KX + ∆ is big, then R(X, ∆) is a finitely generated
C-algebra.
This conjecture is a very special case of Conjecture An .
We can run a (KX + ∆)-MMP with scaling. However, we havenot known whether it always terminates or not. If ⌊∆⌋ = 0, thenit terminates at a good minimal model. Then R(X, ∆) is finitely
generated.

Osamu Fujino

On canonical rings

History and Moriwaki’s early works
On canonical rings
On log canonical rings of projective varieties

On log canonical rings, 6

Conjecture (Conjecture Cn )If dim X = n and KX + ∆ is pseudo-effective, then (X, ∆) has a goodminimal model.
This conjecture says that after finitely many flips and divisorial
contractions we can always obtain a birational model (Y, ∆Y )such that KY + ∆Y is semiample.
This conjecture is the main goal of the theory of minimalmodels.

Osamu Fujino

On canonical rings

History and Moriwaki’s early works
On canonical rings
On log canonical rings of projective varieties

On log canonical rings, 7

The main result is:
Theorem 3.2 (O. Fujino and Y. Gongyo, 2017)
Conjectures An , Bn , and C≤n−1 are all equivalent.
This theorem shows that Conjecture An is a much moredifficult conjecture than we expected. If Conjecture An holds
true for every n, then we can prove that almost all conjectureson the theory of minimal models hold true.

Osamu Fujino

On canonical rings

History and Moriwaki’s early works
On canonical rings
On log canonical rings of projective varieties

On log canonical rings, 8

Idea of Proof:
Conjecture An =⇒ Conjecture Bn
Conjecture Bn looks much easier to treat than Conjecture An ...
This step is obvious

Osamu Fujino

On canonical rings

History and Moriwaki’s early works
On canonical rings
On log canonical rings of projective varieties

On log canonical rings, 9Conjecture Bn =⇒ Conjecture C≤n−1
We consider (X, ∆) with dim X = n − 1
Y : a cone over X
Z → Y : blow-up at the vertex
We apply Conjecture Bn to (Z, ∆Z ) for some ∆Z . We can prove
Conjecture C≤n−1 .
Key point: we can prove the nonvanishing conjecture for smoothprojective varieties in dimension n − 1 by Conjecture Bn . Then we
can freely run the minimal model program with ample scaling in
dimension n − 1 by Hashizume’s result. The main problem is thesemiampleness of K + ∆.

Osamu Fujino

On canonical rings

History and Moriwaki’s early works
On canonical rings
On log canonical rings of projective varieties

On log canonical rings, 10

Conjecture C≤n−1 =⇒ Conjecture An
We consider (X, ∆) with dim X = n and κ(X, KX + ∆) ≥ 1. Then
we can prove that (X, ∆) has a good minimal model.
More precisely, by running the minimal model program withscaling, we can prove that (X, ∆) has a minimal model. The
difficult part is to prove the semiampleness of K + ∆.
Fortunately, this part was already carried out by
Fujino–Gongyo.

Osamu Fujino

On canonical rings

History and Moriwaki’s early works
On canonical rings
On log canonical rings of projective varieties

On log canonical rings, 11

One of the most difficult conjectures in MMP is:
Conjecture 3.3 (Abundance conjecture)
Let (X, ∆) be a projective log canonical pair such that KX + ∆ is nef.
Then KX + ∆ is semiample.

dim X ≤ 3, the abundance conjecture holds true in fullgenerality.
dim X ≥ 4, it is still widely open.

Osamu Fujino

On canonical rings

History and Moriwaki’s early works
On canonical rings
On log canonical rings of projective varieties

On log canonical rings, 12

Key result is:
Theorem 3.4 (O. Fujino and Y. Gongyo, 2014)
Let (X, ∆) be a projective semi-log canonical pair such that ∆ is a
Q-divisor and let ν : X ν → X be the normalization. We put
KX ν + Θ = ν∗ (KX + ∆). Suppose that KX ν + Θ is semiample. Then
KX + ∆ is also semiample.
This theorem plays a crucial role in various inductive
arguments.

Osamu Fujino

On canonical rings

History and Moriwaki’s early works
On canonical rings
On log canonical rings of projective varieties

On Kähler manifolds
The following conjecture is still widely open.
Conjecture 3.5
When X is a nonprojective compact Kähler manifold and ∆ is a
Q-divisor, R(X, ∆) is a finitely generated C-algebra.
This conjecture holds true when ⌊∆⌋ = 0. Therefore, we may
assume that ⌊∆⌋ , 0.
When ⌊∆⌋ , 0, we can not use canonical bundle formula toreduce the problem to the case where X is projective.
Since X is not projective, we can not use the framework of
MMP.

Osamu Fujino

On canonical rings

History and Moriwaki’s early works
On canonical rings
On log canonical rings of projective varieties

Thank youThank you very much!

Osamu Fujino

On canonical rings