On projective varieties with group actions

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Kiwamu Watanabe

楫研究室所属

February

Kiwamu Watanabe (楫研究室所属) On projective varieties with group actions February 1 / 11

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1

Classification of polarized manifolds admitting homogeneous varieties

as ample divisors, preprint (May 2007), submitted.

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1

Classification of polarized manifolds admitting homogeneous varieties as ample divisors, preprint (May 2007), submitted.

2

Actions of linear algebraic groups of exceptional type on projective varieties, preprint (December 2007), submitted.

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1

Classification of polarized manifolds admitting homogeneous varieties as ample divisors, preprint (May 2007), submitted.

2

Actions of linear algebraic groups of exceptional type on projective varieties, preprint (December 2007), submitted.

3

Homogeneous varieties with structures of algebraic fiber spaces,

preprint (December 2007).

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1

Classification of polarized manifolds admitting homogeneous varieties as ample divisors, preprint (May 2007), submitted.

2

Actions of linear algebraic groups of exceptional type on projective varieties, preprint (December 2007), submitted.

3

Homogeneous varieties with structures of algebraic fiber spaces, preprint (December 2007).

4

A study of homogeneous varieties in the viewpoint of classification theories of polarized varieties, preparation.

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Classify smooth polarized varieties (X , L) such that the linear system |L|

has a homogeneous member A.

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Classify smooth polarized varieties (X , L) such that the linear system |L|

has a homogeneous member A.

We say that a projective variety X is homogeneous if there exists a group variety which acts transitively on X .

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Example

∗ P : projective space,

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Example

∗ P : projective space,

∗ Q: smooth quadric hypersurface,

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Example

∗ P : projective space,

∗ Q: smooth quadric hypersurface,

∗ G (r , C

ⁿ⁺¹

): Grassmann variety,

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Example

∗ P : projective space,

∗ Q: smooth quadric hypersurface,

∗ G (r , C

ⁿ⁺¹

): Grassmann variety,

∗ Abelian variety.

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Example

∗ P : projective space,

∗ Q: smooth quadric hypersurface,

∗ G (r , C

ⁿ⁺¹

): Grassmann variety,

∗ Abelian variety.

Dynkin diagram and a subset of its nodes = ⇒ rational homogeneous

variety.

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Example

∗ P : projective space, • ω

1

ω ◦

2

· · · ◦ ω

n−1

ω ◦

n

: A

n

(ω

₁

)

∗ Q: smooth quadric hypersurface,

∗ G (r , C

ⁿ⁺¹

): Grassmann variety,

∗ Abelian variety.

Dynkin diagram and a subset of its nodes = ⇒ rational homogeneous variety.

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Example

∗ P : projective space, • ω

1

ω ◦

2

· · · ◦ ω

n−1

ω ◦

n

: A

n

(ω

₁

)

∗ Q: smooth quadric hypersurface, ω •

₁

◦

ω

₂

· · · ◦ ω

n−1

ω ◦

n

ω •

₁

◦

ω

₂

· · · ◦ ω

n−2

ω

n−

◦

1

ω ◦

n

∗ G (r , C

ⁿ⁺¹

): Grassmann variety,

∗ Abelian variety.

Dynkin diagram and a subset of its nodes = ⇒ rational homogeneous

variety.

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Example

∗ P : projective space, •

ω

₁

◦

ω

₂

· · · ◦ ω

n−1

ω ◦

n

: A

n

(ω

1

)

∗ Q: smooth quadric hypersurface, ω •

₁

◦

ω

₂

· · · ◦ ω

n−1

ω ◦

n

ω •

1

ω ◦

2

· · · ◦ ω

n−2

ω

n−

◦

1

ω ◦

n

∗ G (r , C

ⁿ⁺¹

): Grassmann variety, ω ◦

1

ω ◦

2

· · · • ω

r

· · · ◦ ω

n−1

ω ◦

n

: A

n

(ω

r

)

∗ Abelian variety.

Dynkin diagram and a subset of its nodes = ⇒ rational homogeneous

variety.

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Example

∗ P : projective space, •

ω

₁

◦

ω

₂

· · · ◦ ω

n−1

ω ◦

n

: A

n

(ω

1

)

∗ Q: smooth quadric hypersurface, ω •

₁

◦

ω

₂

· · · ◦ ω

n−1

ω ◦

n

ω •

1

ω ◦

2

· · · ◦ ω

n−2

ω

n−

◦

1

ω ◦

n

∗ G (r , C

ⁿ⁺¹

): Grassmann variety, ω ◦

1

ω ◦

2

· · · • ω

r

· · · ◦ ω

n−1

ω ◦

n

: A

n

(ω

r

)

∗ Abelian variety.

∗ • ω

₁

• ω

₂

: G

₂

(ω

₁

+ ω

₂

)

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( 2 ) If A ∼ = P

ⁿ

with n ≥ 2, then (X , L) ∼ = ( P

ⁿ⁺¹

, O (1)).

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( 2 ) If A ∼ = P

ⁿ

with n ≥ 2, then (X , L) ∼ = ( P

ⁿ⁺¹

, O (1)).

( 3 ) If A ∼ = Q

ⁿ

with n ≥ 3, then (X , L) ∼ = ( P

ⁿ⁺¹

, O (2)) or (Q

ⁿ⁺¹

, O (1)).

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( 2 ) If A ∼ = P

ⁿ

with n ≥ 2, then (X , L) ∼ = ( P

ⁿ⁺¹

, O (1)).

( 3 ) If A ∼ = Q

ⁿ

with n ≥ 3, then (X , L) ∼ = ( P

ⁿ⁺¹

, O (2)) or (Q

ⁿ⁺¹

, O (1)).

( 4 ) ¬∃(X , L) with A: ab. var. of dim A ≥ 2 (A. J. Sommese, ’76).

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( 2 ) If A ∼ = P

ⁿ

with n ≥ 2, then (X , L) ∼ = ( P

ⁿ⁺¹

, O (1)).

( 3 ) If A ∼ = Q

ⁿ

with n ≥ 3, then (X , L) ∼ = ( P

ⁿ⁺¹

, O (2)) or (Q

ⁿ⁺¹

, O (1)).

( 4 ) ¬∃(X , L) with A: ab. var. of dim A ≥ 2 (A. J. Sommese, ’76).

( 5 ) ¬∃(X , L) with A ∼ = G (r, C

ⁿ

) unless r = 1, r = n − 1 or (n, r ) = (4, 2) (T. Fujita, ’81, ’82).

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( 2 ) If A ∼ = P

ⁿ

with n ≥ 2, then (X , L) ∼ = ( P

ⁿ⁺¹

, O (1)).

( 3 ) If A ∼ = Q

ⁿ

with n ≥ 3, then (X , L) ∼ = ( P

ⁿ⁺¹

, O (2)) or (Q

ⁿ⁺¹

, O (1)).

( 4 ) ¬∃(X , L) with A: ab. var. of dim A ≥ 2 (A. J. Sommese, ’76).

( 5 ) ¬∃(X , L) with A ∼ = G (r, C

ⁿ

) unless r = 1, r = n − 1 or (n, r ) = (4, 2) (T. Fujita, ’81, ’82).

( 6 ) If A and X are rational homogeneous varieties with ρ(A) = ρ(X ) = 1,

then a classification of such (X , L) is obtained by K.Konno (’88).

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Theorem (W1)

Let (X , L) be as in Problem 1. Assume that dim A ≥ 2. Then (X , L) is one of the following:

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Theorem (W1)

Let (X , L) be as in Problem 1. Assume that dim A ≥ 2. Then (X , L) is one of the following:

( 1 ) ( P

ⁿ⁺¹

, O

Pⁿ⁺¹

(i)), i = 1, 2,

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Theorem (W1)

Let (X , L) be as in Problem 1. Assume that dim A ≥ 2. Then (X , L) is one of the following:

( 1 ) ( P

ⁿ⁺¹

, O

Pⁿ⁺¹

(i)), i = 1, 2, ( 2 ) (Q

ⁿ⁺¹

, O

_Q_n+1

(1)),

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Theorem (W1)

Let (X , L) be as in Problem 1. Assume that dim A ≥ 2. Then (X , L) is one of the following:

( 1 ) ( P

ⁿ⁺¹

, O

Pⁿ⁺¹

(i)), i = 1, 2, ( 2 ) (Q

ⁿ⁺¹

, O

_Q_n+1

(1)),

( 3 ) ( P (E ), H(E)), E is an ample vector bundle on a smooth curve C with g (C ) = 0 or 1 and L an ample line bundle on C with an exact sequence:

0 → O

_C

→ E → L

^⊕n

→ 0,

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Theorem (W1)

Let (X , L) be as in Problem 1. Assume that dim A ≥ 2. Then (X , L) is one of the following:

( 1 ) ( P

ⁿ⁺¹

, O

Pⁿ⁺¹

(i)), i = 1, 2, ( 2 ) (Q

ⁿ⁺¹

, O

_Q_n+1

(1)),

( 3 ) ( P (E ), H(E)), E is an ample vector bundle on a smooth curve C with g (C ) = 0 or 1 and L an ample line bundle on C with an exact sequence:

0 → O

_C

→ E → L

^⊕n

→ 0,

( 4 ) ( P

^m

× P

^m

, O

_Pm×P^m

(1, 1)),

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Theorem (W1)

Let (X , L) be as in Problem 1. Assume that dim A ≥ 2. Then (X , L) is one of the following:

( 1 ) ( P

ⁿ⁺¹

, O

Pⁿ⁺¹

(i)), i = 1, 2, ( 2 ) (Q

ⁿ⁺¹

, O

_Q_n+1

(1)),

( 3 ) ( P (E ), H(E)), E is an ample vector bundle on a smooth curve C with g (C ) = 0 or 1 and L an ample line bundle on C with an exact sequence:

0 → O

_C

→ E → L

^⊕n

→ 0,

( 4 ) ( P

^m

× P

^m

, O

_Pm×P^m

(1, 1)),

( 5 ) (G (2, C

^2m

), O

_Pl¨_ucker

(1)),

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Theorem (W1)

Let (X , L) be as in Problem 1. Assume that dim A ≥ 2. Then (X , L) is one of the following:

( 1 ) ( P

ⁿ⁺¹

, O

Pⁿ⁺¹

(i)), i = 1, 2, ( 2 ) (Q

ⁿ⁺¹

, O

_Q_n+1

(1)),

( 3 ) ( P (E ), H(E)), E is an ample vector bundle on a smooth curve C with g (C ) = 0 or 1 and L an ample line bundle on C with an exact sequence:

0 → O

_C

→ E → L

^⊕n

→ 0,

( 4 ) ( P

^m

× P

^m

, O

_Pm×P^m

(1, 1)), ( 5 ) (G (2, C

^2m

), O

_Pl¨_ucker

(1)), ( 6 ) (E

₆

(ω

₁

), O

_E

6(ω₁)

(1)).

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Theorem (M. Andreatta, ’01)

X : a smooth projective variety of dimension n

G : a simple, simply connected linear algebraic group acting regularly and non-trivially on X .

Then ( 1 ) n ≥ r

G

,

( 2 ) if moreover n = r

G

, then X is homogeneous.

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G : a simple, simply connected linear algebraic group of classical type acting regularly and non-trivially on X .

Assume that n = r

G

+ 1.

Then X is one of the following:

( 1 ) P

ⁿ

, ( 2 ) Q

ⁿ

,

( 3 ) Y × C , where Y is P

ⁿ⁻¹

or Q

ⁿ⁻¹

,

( 4 ) P ( O

_Y

⊕ O

_Y

(m)), where Y is as in ( 3 ) and m > 0, ( 5 ) P (T

_P2

),

( 6 ) C

₂

(ω

1

+ ω

2

).

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Theorem (W2)

X : a smooth projective variety of dimension n.

G : a simple, simply connected linear algebraic group of exceptional type acting regularly and non-trivially on X .

Assume that n = r

G

+ 1.

Then X is one of the following and the action of G is unique for each case:

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Theorem (W2)

X : a smooth projective variety of dimension n.

G : a simple, simply connected linear algebraic group of exceptional type acting regularly and non-trivially on X .

Assume that n = r

G

+ 1.

Then X is one of the following and the action of G is unique for each case:

( 1 ) E

₆

(ω

1

),

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Theorem (W2)

X : a smooth projective variety of dimension n.

G : a simple, simply connected linear algebraic group of exceptional type acting regularly and non-trivially on X .

Assume that n = r

G

+ 1.

Then X is one of the following and the action of G is unique for each case:

( 1 ) E

₆

(ω

1

),

( 2 ) G

₂

(ω

₁

+ ω

₂

),

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Theorem (W2)

X : a smooth projective variety of dimension n.

G : a simple, simply connected linear algebraic group of exceptional type acting regularly and non-trivially on X .

Assume that n = r

G

+ 1.

Then X is one of the following and the action of G is unique for each case:

( 1 ) E

₆

(ω

1

), ( 2 ) G

₂

(ω

₁

+ ω

₂

),

( 3 ) Y × Z , where Y is E

₆

(ω

₁

), E

₇

(ω

₁

), E

₈

(ω

₁

), F

₄

(ω

₁

), F

₄

(ω

₄

), G

₂

(ω

₁

) or G

₂

(ω

2

) and Z is a smooth projective curve,

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Theorem (W2)

X : a smooth projective variety of dimension n.

G : a simple, simply connected linear algebraic group of exceptional type acting regularly and non-trivially on X .

Assume that n = r

G

+ 1.

Then X is one of the following and the action of G is unique for each case:

( 1 ) E

₆

(ω

1

), ( 2 ) G

₂

(ω

₁

+ ω

₂

),

( 3 ) Y × Z , where Y is E

₆

(ω

₁

), E

₇

(ω

₁

), E

₈

(ω

₁

), F

₄

(ω

₁

), F

₄

(ω

₄

), G

₂

(ω

₁

) or G

₂

(ω

2

) and Z is a smooth projective curve,

( 4 ) P ( O

_Y

⊕ O

_Y

(m)), where Y is as in ( 3 ) and m > 0.

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Theorem (W2)

X : a smooth projective variety of dimension n.

G : a simple, simply connected linear algebraic group of exceptional type acting regularly and non-trivially on X .

Assume that n = r

G

+ 1.

Then X is one of the following and the action of G is unique for each case:

( 1 ) E

₆

(ω

1

), ( 2 ) G

₂

(ω

₁

+ ω

₂

),

( 3 ) Y × Z , where Y is E

₆

(ω

₁

), E

₇

(ω

₁

), E

₈

(ω

₁

), F

₄

(ω

₁

), F

₄

(ω

₄

), G

₂

(ω

₁

) or G

₂

(ω

2

) and Z is a smooth projective curve,

( 4 ) P ( O

_Y

⊕ O

_Y

(m)), where Y is as in ( 3 ) and m > 0.

A linear algebraic group of Dynkin type F

₄

acts on E

₆

(ω

1

).

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(1) 等質多様体を豊富な因子として含む非特異偏極多様体の分類を行った．

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(1) 等質多様体を豊富な因子として含む非特異偏極多様体の分類を行った．

今まで知られていた結果の大幅な一般化．

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(1) 等質多様体を豊富な因子として含む非特異偏極多様体の分類を行った．

今まで知られていた結果の大幅な一般化．

(2) n = r

G

+ 1 なる例外型単純線型代数群の作用をもつ非特異射影多様体

の分類を行った．

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(1) 等質多様体を豊富な因子として含む非特異偏極多様体の分類を行った．

今まで知られていた結果の大幅な一般化．

(2) n = r

G

+ 1 なる例外型単純線型代数群の作用をもつ非特異射影多様体の分類を行った．

Andreatta ^{の結果と合わせると，} n = r

G

+ 1 なる単純線型代数群の作用を

もつ非特異射影多様体の完全な分類を得る．