Today’s plan
New framework of vanishing theorems
Log canonical pairs
Semi-log canonical pairs

The foundations of the minimal model
programOsamu Fujino

Kyoto University
October 23, 2015

Osamu Fujino

The foundations of the minimal model programToday’s plan
New framework of vanishing theorems
Log canonical pairs
Semi-log canonical pairs

1

Today’s plan2

New framework of vanishing theorems3

Log canonical pairs

4Semi-log canonical pairs

Osamu Fujino

The foundations of the minimal model programToday’s plan
New framework of vanishing theorems
Log canonical pairs
Semi-log canonical pairs

Today’s plan

Part I: Main Results (English, this slide, unfortunately,
somewhat technical)

Osamu Fujino

The foundations of the minimal model programToday’s plan
New framework of vanishing theorems
Log canonical pairs
Semi-log canonical pairs

Today’s plan

Part I: Main Results (English, this slide, unfortunately,
somewhat technical)
Part II: Ideas, Backgroud, History, and so on (Japanese, noslides, fortunately, not technical)

Osamu Fujino

The foundations of the minimal model programToday’s plan
New framework of vanishing theorems
Log canonical pairs
Semi-log canonical pairs

SNC pairs
M : smooth variety /C
X : SNC divisor on M
B: R-divisor on M such that Supp B: SNC divisor
B and X have no common components, Supp(B + X): SNC
divisor
D = B|X

Osamu Fujino

The foundations of the minimal model programToday’s plan
New framework of vanishing theorems
Log canonical pairs
Semi-log canonical pairs

SNC pairs
M : smooth variety /C
X : SNC divisor on M
B: R-divisor on M such that Supp B: SNC divisor
B and X have no common components, Supp(B + X): SNC
divisor
D = B|X
Definition 2.1 (GESNC pair)(X, D): globally embedded simple normal crossing (GESNC) pair

Osamu Fujino

The foundations of the minimal model programToday’s plan
New framework of vanishing theorems
Log canonical pairs
Semi-log canonical pairs

SNC pairs
M : smooth variety /C
X : SNC divisor on M
B: R-divisor on M such that Supp B: SNC divisor
B and X have no common components, Supp(B + X): SNC
divisor
D = B|X
Definition 2.1 (GESNC pair)(X, D): globally embedded simple normal crossing (GESNC) pairDefinition 2.2 (SNC pair)(Y, ∆): simple normal crossing (SNC) pairdef⇐⇒ (Y, ∆): Zariski locally isomorphic to a GESNC pair
Osamu Fujino

The foundations of the minimal model programToday’s plan
New framework of vanishing theorems
Log canonical pairs
Semi-log canonical pairs

Stratum of SNC pair

(X, D): SNC pairD ∈ [0, 1]
ν : X ν → X : normalization
KX ν + Θ = ν∗ (KX + D)

Osamu Fujino

The foundations of the minimal model programToday’s plan
New framework of vanishing theorems
Log canonical pairs
Semi-log canonical pairs

Stratum of SNC pair

(X, D): SNC pairD ∈ [0, 1]
ν : X ν → X : normalization
KX ν + Θ = ν∗ (KX + D)
Definition 2.3 (Stratum)

W : closed subvariety of X
W : stratum of (X, D)
def⇐⇒ W = ν(C), where C is a log canonical center of (X ν , Θ), or W
is an irreducible component of X

Osamu Fujino

The foundations of the minimal model programToday’s plan
New framework of vanishing theorems
Log canonical pairs
Semi-log canonical pairs

Hodge theoretic injectivity theorem

Theorem 2.4 (Relative Hodge theoretic injectivity theorem)(X, ∆): SNC pair, ∆ ∈ [0, 1], π : X → S : proper
L: Cartier divisor on X
D: effective Weil divisor on X
Supp D ⊂ Supp ∆L ∼R,π KX + ∆Osamu Fujino

The foundations of the minimal model programToday’s plan
New framework of vanishing theorems
Log canonical pairs
Semi-log canonical pairs

Hodge theoretic injectivity theorem

Theorem 2.4 (Relative Hodge theoretic injectivity theorem)(X, ∆): SNC pair, ∆ ∈ [0, 1], π : X → S : proper
L: Cartier divisor on X
D: effective Weil divisor on X
Supp D ⊂ Supp ∆L ∼R,π KX + ∆Then

Rq π∗ OX (L) → Rq π∗ OX (L + D)
is injective for every q.

Osamu Fujino

The foundations of the minimal model programToday’s plan
New framework of vanishing theorems
Log canonical pairs
Semi-log canonical pairs

Injectivity theorem for SNC pair
Theorem 2.5 (Injectivity for SNC pair)(X, ∆): SNC pair, ∆ ∈ [0, 1], π : X → S : proper, as before
L: Cartier divisor on X
D: effective Cartier, permissible with respect to (X, ∆)Osamu Fujino

The foundations of the minimal model programToday’s plan
New framework of vanishing theorems
Log canonical pairs
Semi-log canonical pairs

Injectivity theorem for SNC pair
Theorem 2.5 (Injectivity for SNC pair)(X, ∆): SNC pair, ∆ ∈ [0, 1], π : X → S : proper, as before
L: Cartier divisor on X
D: effective Cartier, permissible with respect to (X, ∆)We further assume:(i) L ∼R,π KX + ∆ + H(ii) H : π-semi-ample R-divisor(iii) tH ∼R,π D + D′ , t ∈ R>0 ,D′ : effective R-Cartier R-divisor, permissible with respect to(X, ∆)

Osamu Fujino

The foundations of the minimal model programToday’s plan
New framework of vanishing theorems
Log canonical pairs
Semi-log canonical pairs

Injectivity theorem for SNC pair
Theorem 2.5 (Injectivity for SNC pair)(X, ∆): SNC pair, ∆ ∈ [0, 1], π : X → S : proper, as before
L: Cartier divisor on X
D: effective Cartier, permissible with respect to (X, ∆)We further assume:(i) L ∼R,π KX + ∆ + H(ii) H : π-semi-ample R-divisor(iii) tH ∼R,π D + D′ , t ∈ R>0 ,D′ : effective R-Cartier R-divisor, permissible with respect to(X, ∆)

Then Rq π∗ OX (L) → Rq π∗ OX (L + D) is injective for every q.
Osamu Fujino

The foundations of the minimal model programToday’s plan
New framework of vanishing theorems
Log canonical pairs
Semi-log canonical pairs

Torsion-freeness and Vanishing for SNC pair

Theorem 2.6 (Torsion-freeness and Vanishing thereom)(Y, ∆): SNC pair, ∆ ∈ [0, 1], f : Y → X : proper
L: Cartier divisor on Y such that L − (KY + ∆): f -semi-ample

Osamu Fujino

The foundations of the minimal model programToday’s plan
New framework of vanishing theorems
Log canonical pairs
Semi-log canonical pairs

Torsion-freeness and Vanishing for SNC pair

Theorem 2.6 (Torsion-freeness and Vanishing thereom)(Y, ∆): SNC pair, ∆ ∈ [0, 1], f : Y → X : proper
L: Cartier divisor on Y such that L − (KY + ∆): f -semi-ample
Then we have:(i) Every associated prime of Rq f∗ OY (L) is the generic point of
the f -image of some stratum of (Y, ∆).
(ii)
π : X → V : projective
L − (KY + ∆) ∼R f ∗ H , H : π-ample R-divisor on X=⇒ R p π∗ Rq f∗ OY (L) = 0 for every p > 0 and q ≥ 0.

Osamu Fujino

The foundations of the minimal model programToday’s plan
New framework of vanishing theorems
Log canonical pairs
Semi-log canonical pairs

Vanishing theorem of Reid–Fukuda type
We can generalize Theorem 2.6 (ii) as follows.

Osamu Fujino

The foundations of the minimal model programToday’s plan
New framework of vanishing theorems
Log canonical pairs
Semi-log canonical pairs

Vanishing theorem of Reid–Fukuda type
We can generalize Theorem 2.6 (ii) as follows.
Theorem 2.7 (Vanishing theorem of Reid–Fukuda type)
Use the same notation as in Theorem 2.6(Y, ∆): GESNC, or Y : quasi-projective (extra assumption!)
H : nef and log big over V with respect to f : (Y, ∆) → X , thatis, H : nef over V and H| f (W) : big over V for every stratum W of(Y, ∆)

Osamu Fujino

The foundations of the minimal model programToday’s plan
New framework of vanishing theorems
Log canonical pairs
Semi-log canonical pairs

Vanishing theorem of Reid–Fukuda type
We can generalize Theorem 2.6 (ii) as follows.
Theorem 2.7 (Vanishing theorem of Reid–Fukuda type)
Use the same notation as in Theorem 2.6(Y, ∆): GESNC, or Y : quasi-projective (extra assumption!)
H : nef and log big over V with respect to f : (Y, ∆) → X , thatis, H : nef over V and H| f (W) : big over V for every stratum W of(Y, ∆)
Then R p π∗ Rq f∗ OY (L) = 0 for every p > 0 and q ≥ 0.

Osamu Fujino

The foundations of the minimal model programToday’s plan
New framework of vanishing theorems
Log canonical pairs
Semi-log canonical pairs

Vanishing theorem of Reid–Fukuda type
We can generalize Theorem 2.6 (ii) as follows.
Theorem 2.7 (Vanishing theorem of Reid–Fukuda type)
Use the same notation as in Theorem 2.6(Y, ∆): GESNC, or Y : quasi-projective (extra assumption!)
H : nef and log big over V with respect to f : (Y, ∆) → X , thatis, H : nef over V and H| f (W) : big over V for every stratum W of(Y, ∆)
Then R p π∗ Rq f∗ OY (L) = 0 for every p > 0 and q ≥ 0.
We can see that these results contain various classical results as
special cases.

Osamu Fujino

The foundations of the minimal model programToday’s plan
New framework of vanishing theorems
Log canonical pairs
Semi-log canonical pairs

Kawamata–Viehweg
Theorem 2.8 (Kawamata–Viehweg)

X : smooth projective variety
D: nef and big Q-divisor
Supp{D}: SNC divisor
Then H q (X, OX (KX + ⌈D⌉)) = 0 for every q > 0.

Osamu Fujino

The foundations of the minimal model programToday’s plan
New framework of vanishing theorems
Log canonical pairs
Semi-log canonical pairs

Kawamata–Viehweg
Theorem 2.8 (Kawamata–Viehweg)

X : smooth projective variety
D: nef and big Q-divisor
Supp{D}: SNC divisor
Then H q (X, OX (KX + ⌈D⌉)) = 0 for every q > 0.
Kawamata–Viehweg is a generalization of Kodaira.
Theorem 2.9 (Kodaira)X : smooth projective variety
D: ample Cartier divisor
Then H q (X, OX (KX + D)) = 0 for every q > 0.
Osamu Fujino

The foundations of the minimal model programToday’s plan
New framework of vanishing theorems
Log canonical pairs
Semi-log canonical pairs

Nadel

Theorem 2.10 ((algebraic version of) Nadel)

X : smooth projective variety
L: Cartier divisor
D: effective Q-divisor
L − D: nef and big
Then H q (X, OX (KX + L) ⊗ J(X, D)) = 0 for every q > 0, where
J(X, D): multiplier ideal sheaf of (X, D).

Osamu Fujino

The foundations of the minimal model programToday’s plan
New framework of vanishing theorems
Log canonical pairs
Semi-log canonical pairs

Kollár

Theorem 2.11 (Kollár)

X : smooth projective variety
Y : projective variety
f : X → Y : surjective morphism

Osamu Fujino

The foundations of the minimal model programToday’s plan
New framework of vanishing theorems
Log canonical pairs
Semi-log canonical pairs

Kollár

Theorem 2.11 (Kollár)

X : smooth projective variety
Y : projective variety
f : X → Y : surjective morphism
Then we have:(i) Rq f∗ OX (KX ): torsion-free(ii) H p (Y, Rq f∗ OX (KX ) ⊗ OY (H)) = 0 for every p > 0 and q ≥ 0,
where H : ample Cartier divisor on Y .

Osamu Fujino

The foundations of the minimal model programToday’s plan
New framework of vanishing theorems
Log canonical pairs
Semi-log canonical pairs

Main statement

Our result for SNC pairs contains Kodaira, Kawamata–Viehweg,
Nadel, Kollár, and many other powerful and useful vanishing
results as very special cases.

Osamu Fujino

The foundations of the minimal model programToday’s plan
New framework of vanishing theorems
Log canonical pairs
Semi-log canonical pairs

MHS for cohomology with compact support
Almost all the classical vanishing theorems (Kawamata–Viehweg,
Kollár, etc.) can be proved by the E1 -degeneration ofpqp

E1 = H q (X, ΩX ) ⇒ H p+q (X, C).

Osamu Fujino

The foundations of the minimal model programToday’s plan
New framework of vanishing theorems
Log canonical pairs
Semi-log canonical pairs

MHS for cohomology with compact support
Almost all the classical vanishing theorems (Kawamata–Viehweg,
Kollár, etc.) can be proved by the E1 -degeneration ofpqp

E1 = H q (X, ΩX ) ⇒ H p+q (X, C).
My idea is to use the E1 -degeneration of
pq

p+q

p

E1 = H q (X, ΩX (log D) ⊗ OX (−D)) ⇒ Hc

(X \ D, C),

where X : smooth projective variety, D: SNC divisor.

Osamu Fujino

The foundations of the minimal model programToday’s plan
New framework of vanishing theorems
Log canonical pairs
Semi-log canonical pairs

MHS for cohomology with compact support
Almost all the classical vanishing theorems (Kawamata–Viehweg,
Kollár, etc.) can be proved by the E1 -degeneration ofpqp

E1 = H q (X, ΩX ) ⇒ H p+q (X, C).
My idea is to use the E1 -degeneration of
pq

p+q

p

E1 = H q (X, ΩX (log D) ⊗ OX (−D)) ⇒ Hc

(X \ D, C),

where X : smooth projective variety, D: SNC divisor.
In my framework,OX (KX + D) ≃ Hom(Ω0X (log D) ⊗ OX (−D), ωX ).
∧We do not see OX (KX + D) as dim X Ω1X (log D).
Osamu Fujino

The foundations of the minimal model programToday’s plan
New framework of vanishing theorems
Log canonical pairs
Semi-log canonical pairs

Some remarks
Precisely speaking:(X, D): SNC pair, or finite cyclic cover of SNC pair
We have to consider MHS on

Hck (X \ ⌊D⌋, C).

Osamu Fujino

The foundations of the minimal model programToday’s plan
New framework of vanishing theorems
Log canonical pairs
Semi-log canonical pairs

Some remarks
Precisely speaking:(X, D): SNC pair, or finite cyclic cover of SNC pair
We have to consider MHS on

Hck (X \ ⌊D⌋, C).
By considering VMHS, we have various semipositivity theorems(Fujino–Fujisawa, Fujino–Fujisawa–Saito).
(X, D): projective SNC pair, D: reduced
f : X → Y : surjective, Y : smooth projective variety
Under some suitable assumptions, we obtain thatRq f∗ OX (KX/Y + D)
is a semipositive locally free sheaf for every q.
Osamu Fujino

The foundations of the minimal model programToday’s plan
New framework of vanishing theorems
Log canonical pairs
Semi-log canonical pairs

Cone Theorem
Theorem 3.1 (Cone and contraction theorem)(X, ∆): projective log canonical pairThen

NE(X) = NE(X)KX +∆≥0 +

∑

R j.

j

Osamu Fujino

The foundations of the minimal model programToday’s plan
New framework of vanishing theorems
Log canonical pairs
Semi-log canonical pairs

Cone Theorem
Theorem 3.1 (Cone and contraction theorem)(X, ∆): projective log canonical pairThen

NE(X) = NE(X)KX +∆≥0 +

∑

R j.

j

R j : (KX + ∆)-negative extremal ray
Then there is a contraction morphism

φR j : X → Y
associated to R j .

Osamu Fujino

The foundations of the minimal model programToday’s plan
New framework of vanishing theorems
Log canonical pairs
Semi-log canonical pairs

Cone Theorem
Theorem 3.1 (Cone and contraction theorem)(X, ∆): projective log canonical pairThen

NE(X) = NE(X)KX +∆≥0 +

∑

R j.

j

R j : (KX + ∆)-negative extremal ray
Then there is a contraction morphism

φR j : X → Y
associated to R j .
It was classically well-know for “log terminal” pairs.
Osamu Fujino

The foundations of the minimal model programToday’s plan
New framework of vanishing theorems
Log canonical pairs
Semi-log canonical pairs

MMP for log canonical pairs(X, ∆): projective log canonical pair
Then we can run the minimal model program (MMP) (with scaling).
Thus we obtain a sequence of flips and divisorial contractions.

(X, ∆) = (X0 , ∆0 ) d (X1 , ∆1 ) d · · · d (Xk , ∆k ) d

Osamu Fujino

The foundations of the minimal model programToday’s plan
New framework of vanishing theorems
Log canonical pairs
Semi-log canonical pairs

MMP for log canonical pairs(X, ∆): projective log canonical pair
Then we can run the minimal model program (MMP) (with scaling).
Thus we obtain a sequence of flips and divisorial contractions.

(X, ∆) = (X0 , ∆0 ) d (X1 , ∆1 ) d · · · d (Xk , ∆k ) dConjecture 3.2 (Flip Conjecture II)
There are no infinite sequences of flips.
This conjecture is widely open. It is well-known that it is sufficient
to prove that Conjecture 3.2 holds for kawamata log terminal pairs.
Osamu Fujino

The foundations of the minimal model programToday’s plan
New framework of vanishing theorems
Log canonical pairs
Semi-log canonical pairs

Open problem for log canonical pairs
Conjecture 3.3 (Finite generation of log canonical ring)X : smooth projective variety∆: Q-divisor, Supp ∆: SNC divisor, ∆ ∈ [0, 1]
Then

R(X, ∆) =⊕H 0 (X, OX (m(KX + ∆)))

m≥0

is a finitely generated C-algebra.

Osamu Fujino

The foundations of the minimal model programToday’s plan
New framework of vanishing theorems
Log canonical pairs
Semi-log canonical pairs

Open problem for log canonical pairs
Conjecture 3.3 (Finite generation of log canonical ring)X : smooth projective variety∆: Q-divisor, Supp ∆: SNC divisor, ∆ ∈ [0, 1]
Then

R(X, ∆) =⊕H 0 (X, OX (m(KX + ∆)))

m≥0

is a finitely generated C-algebra.
Conjecture 3.3 was completely solved for kawamata log
terminal (KLT) pairs by BCHM

Osamu Fujino

The foundations of the minimal model programToday’s plan
New framework of vanishing theorems
Log canonical pairs
Semi-log canonical pairs

Open problem for log canonical pairs
Conjecture 3.3 (Finite generation of log canonical ring)X : smooth projective variety∆: Q-divisor, Supp ∆: SNC divisor, ∆ ∈ [0, 1]
Then

R(X, ∆) =⊕H 0 (X, OX (m(KX + ∆)))

m≥0

is a finitely generated C-algebra.
Conjecture 3.3 was completely solved for kawamata log
terminal (KLT) pairs by BCHM
It also holds for KLT pairs in Fujiki’s class C (Fujino).

Osamu Fujino

The foundations of the minimal model programToday’s plan
New framework of vanishing theorems
Log canonical pairs
Semi-log canonical pairs

Open problem for log canonical pairs
Conjecture 3.3 (Finite generation of log canonical ring)X : smooth projective variety∆: Q-divisor, Supp ∆: SNC divisor, ∆ ∈ [0, 1]
Then

R(X, ∆) =⊕H 0 (X, OX (m(KX + ∆)))

m≥0

is a finitely generated C-algebra.
Conjecture 3.3 was completely solved for kawamata log
terminal (KLT) pairs by BCHM
It also holds for KLT pairs in Fujiki’s class C (Fujino).
It implies the existence of good minimal models, abundance
conjecture, and so on (Fujino–Gongyo).
Osamu Fujino

The foundations of the minimal model programToday’s plan
New framework of vanishing theorems
Log canonical pairs
Semi-log canonical pairs

SLC pairs

X : equidimensional variety, Serre’s S 2 condition, normal
crossing in codimension one∆: effective R-divisor on X , no components of ∆ are containedin Sing X .
KX + ∆: R-Cartierν : X ν → X : normalization, KX ν + Θ = ν∗ (KX + ∆)Definition 4.1 (SLC pair)def(X, ∆): semi-log canonical (SLC) pair ⇐⇒ (X ν , Θ): log canonicalpair

Osamu Fujino

The foundations of the minimal model programToday’s plan
New framework of vanishing theorems
Log canonical pairs
Semi-log canonical pairs

Why SLC?

Example 4.2
nodal pointed curve is SLC

Osamu Fujino

The foundations of the minimal model programToday’s plan
New framework of vanishing theorems
Log canonical pairs
Semi-log canonical pairs

Why SLC?

Example 4.2
nodal pointed curve is SLC
We need the notion of SLC pairs in order to compactify somemoduli spaces (Kollár–Shepherd-Barron, Alexeev, ...)

Osamu Fujino

The foundations of the minimal model programToday’s plan
New framework of vanishing theorems
Log canonical pairs
Semi-log canonical pairs

Cone Theorem for SLC pairs
Theorem 4.3 (Cone and contraction theorem)(X, ∆): projective SLC pairThen

NE(X) = NE(X)KX +∆≥0 +

∑

R j.

j

R j : (KX + ∆)-negative extremal ray
Then there is a contraction morphism

φR j : X → Y
associated to R j .
Osamu Fujino

The foundations of the minimal model programToday’s plan
New framework of vanishing theorems
Log canonical pairs
Semi-log canonical pairs

Remarks on SLC pairsquasi-projective SLC pair has a natural quasi-log structure(Ambro, Fujino)
Kodaira-type vanishing theorems hold for SLC pairs !
We can generalize many results for kawamata log terminal
pairs to SLC pairs !!
Unfortunately, we can not run the minimal model program for
SLC pairs (Kollár, Fujino)

Osamu Fujino

The foundations of the minimal model programToday’s plan
New framework of vanishing theorems
Log canonical pairs
Semi-log canonical pairs

Thank youThank you very much!

Osamu Fujino

The foundations of the minimal model program