On weak Fano varieties with log canonical singularities

On weak Fano varieties with log canonical singularities

権業善範

^∗

東京大学大学院数理科学研究科

1 準備

この報告集では

, nef

big

[KoM]

[KaMM]

.

定義

1.1.

X

,

Q -

Weil

∆

, Q -

Weil

K _X + ∆

Q -Cartier

, (X, ∆)

.

(X, ∆)

Fano

, − (K _X + ∆)

nef

big

.

∆ = 0

,

X

Fano

.

定義

1.2.

(X, ∆)

,

ϕ : Y → X

.

.

K _Y = ϕ

(K _X + ∆) + ∑ a _i E _i ,

ここで

E i

Y

.

,

(1)

,

i

a i > − 1

, (X, ∆)

, (2)

,

i

a i ≥ − 1

, (X, ∆)

.

2 問題

対数的標準弱対数的

Fano

(X, ∆)

(cf. [S, 2.6. Remark-Corollary], [P, 11.1]):

(i) − (K X + ∆)

semi-ample

.

(ii) Q -complements

,

, K X + ∆ + D ∼

0

(X, ∆ + D)

Q -

D

.

(iii) Kleiman-

N E(X)

.

(i)

(ii)

.

(X, ∆)

,

-Shokurov

(cf. [KoM]).

Shokurov

(cf. [S, 2.5. Proposition]).

.

∗

[email protected]

3 主結果 ( 反例 )

しかし

,

,

Fano

,

,

.

, plt

Fano

semi-ample

(

,

[Kar1]

[Kar2]

).

.

例

3.1.

S

− K S

nef

semi-ample

.

P

9

.

X

S ⊂ P ^N

,

X

X

.

E

.

.

X

φ

~~||| ||| || π

ÃÃ @

@ @

@ @

@ @

@

X

_ _ _ _ _ _ _ // S.

このとき

(X, E )

plt

Fano

− (K X + E)

semi-ample

.

S

0

2

P

-

, (X, ∆)

Q -complements

.

, S

P

9

, X

Kleiman-

.

4 主結果 ( 肯定的な結果 )

この例たちを考察することにより

, (X, ∆)

1

lc center

, (i)-(iii)

.

,

.

定理

4.1.

3

Fano

X

, − K X

semi-ample

, N E(X )

.

定理

4.2.

4

Fano

X

,

1

.

− K X

semi-ample

, N E(X)

.

特に三次元の場合に注目すると

,

Fano

(i)-(iii)

,

Fano

.

4.1 Semi-ample

さらに

,

Semi-ample

.

予想

4.3 (

).

(

sdlt

)(S, ∆)

,

K S + ∆

.

K S + ∆

Q -

.

この予想は極小モデル理論における重要な予想の一つである

.

,

(cf. [F1]).

,

,

, Ambro

.

, S

.

sdlt

,

dlt

,

ものである

:

定義

4.4.

d

S

-

X

,

Q -

Weil

∆

, Q -

Weil

K X + ∆

Q -Cartier

.

X

1

.

X = ∪

X i

とする

.

ν : X

:= ∐

X _i

→ X = ∪

X i

.

.

X

Q -

Θ

K X

+ Θ := ν

(K X + ∆)

,

Θ i := Θ | X

.

(X, ∆)

(

, sdlt

)

, X i

(X _i

, Θ i )

dlt

.

予想

4.3

S

,

,

4.3

(cf. [G2]).

,

,

,

(

6

).

この予想を用いて次の結果が得られる

.

定理

4.5.

d − 1

4.3

.

(X, ∆)

d

Fano

.

, (X, ∆)

lc center

1

− (K X + ∆)

semi-ample

.

ここで定理

6.7

.

.

(1) ”

”dlt

, i.e.

lc

とれる

(

[F2]

),

(2)

-Shokurov

− (K X + ∆)

lc center

semi-ample

,

(3)

,

(4) 1

.

実際

,

. [F2, Theorem 10.5]

,

ϕ : Y → X

(

”

”dlt

).

(i) Y

, Q -

,

(ii) Γ

K Y + Γ = ϕ

K X

(Y, Γ)

dlt

,

(iii) Γ

1

,

ϕ-

Γ

,

(iv) { C i }

lc center

,

W = ∪

C i , Γ W

ϕ

W

Γ

.

(ϕ |

)

O

= O W

.

klt

C

,

C = ∪

C _i

.

,

.

Σ := { i | ( − K X ) | C

≡ 0 } , C

:= ∪

i

C i ,

C

:= ∪

i

C i .

ここでわかることだが

, − (K _X ) | C

ample

.

, S

ϕ

C

Γ

, (K _Y + Γ) | S

= K _S

+ Γ _S

= ϕ |

S

(D | C

) ≡ 0

.

4.3

(

!, cf. [AFKM])

, K S

+ Γ S

∼

0

.

, ”

”dlt

D | C

∼

0

.

D | C

ample

C

∩ C

O (mD | C

)

C

.

, D | C

semi-ample

.

,

-Shokurov

, − K X

semi-ample

.

4.2

次に

, Kleiman-

. (iii)

,

Ambro

([A, Theorem 5.10], [F2, Theorem 16.5])

.

定理

4.6.

(X, ∆)

,

.

N E(X) = N E(X ) K

+ N E(X )

+ ∑ R j ,

ここで

R j

(K X + ∆)-

, { R j }

.

定理

5.1

Nlc(X, ∆)

, N E(X )

1-

.

[F2]

.

.

定理

4.7.

(X, ∆)

d

Fano

.

, (X, ∆)

lc center

1

Kleiman-

N E(X)

.

実際

,

4.1

4.2

.

5 アバンダンス予想について

折角なので

, [G2]

.

.

予想

5.1 (

).

(X, ∆)

.

ν (K _X + ∆) = κ(K _X + ∆)

.

,

K _X + ∆

nef

,

semi-ample

.

数値的対数的小平次元

ν(K X + ∆)

κ(K X + ∆)

, [N]

.

,

.

.

,

6.1

(cf. [B]).

今回扱うのは

,

6.1

(X, ∆)

ν (K X + ∆) = 0

(i.e. K X + ∆ ≡ 0

)

.

.

定理

5.2.

(X, ∆)

.

K X + ∆ ≡ 0

.

K X + ∆ ∼

0

.

先に述べたように

,

[A], [N]

.

,

[BCHM]

.

,

,

6.2

[CKP]

Simpson

[Sim]

(cf. [Ka]).

Simpson

.

次に半対数的標準対に対する定理

6.2

,

.

.

定義

5.3.

d

S

-

X

,

Q -

Weil

∆

, Q -

Weil

K X + ∆

Q -Cartier

.

X

1

.

X = ∪

X i

とする

.

ν : X

:= ∐

X _i

→ X = ∪

X i

.

.

X

Q -

Θ

K _X

+ Θ := ν

(K _X + ∆)

,

Θ _i := Θ | X

.

(X, ∆)

(

, slc

)

, (X _i

, Θ _i )

lc

をいう

.

sdlt

slc

.

定理

5.4.

(X, ∆)

.

K X + ∆ ≡ 0

.

K X + ∆ ∼

0

.

この証明は基本的には

[F1]

[BCHM]

.

.

定理

5.5.

(X, ∆)

.

K X + ∆ ∼

0

.

m

ρ _m (Bir(X, ∆))

.

ここで

ρ _m (Bir(X, ∆))

B -pluricanonical

,

.

ϕ : X 99K X

B-

α, β : W → X

ϕ ◦ α = β

α

(K _X + ∆) = β

(K _X + ∆)

.

,

ρ m : Bir(X, ∆) := { ϕ : X 99K X | ϕ

B-

} → GL

(H

(X, m(K X + ∆)))

は引き戻しで定義される自然な群準同型写像である

.

–

, Deligne

Moishezon

pluricanonical

(cf. [NU])

, (X, ∆)

K X + ∆

nef

,

(cf. [F1]).

[S]

n-

.

応用として

,

6.7

.

系

5.6.

(X, ∆)

Fano

.

, (X, ∆)

lc center

1

− (K _X + ∆)

semi-ample

.

またセクション

4.1

.

系

5.7.

(X, ∆)

.

K X + ∆

nef

big

.

, (X, ∆)

lc center

1

(K X + ∆)

semi-ample

.

6 位相的性質

最後に

,

6.7

semi-ample

Fano

に触れておく

.

6.7

[HM, Corollary 1.4]

.

系

6.1.

(X, ∆)

Fano

.

, (X, ∆)

lc center

1

,

π

(Nklt(X, ∆)) → π

(X)

は全射

.

Nklt(X, ∆)

lc center

.

Nklt(X, ∆)

X

.

7 最後に

今回

,

,

.

参考文献

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