Characterizations of projective spaces and hyperquadrics for varieties with Picard number one

2011

Characterizations of projective spaces and hyperquadrics for varieties with Picard number one

早稲田大学大学院基幹理工学研究科 数学応用数理専攻

5110A026-5

Abstract. The purpose of this paper is to give a proof of Kov´acs conjecture for varieties with Picard number one by applying a method of slope-stabilities of sheaves.

1 Introduction

Characterizations of projective spaces and hyperquadrics have been studied in several ways. The purpose of this paper is to give a different characterization of them. We will prove the following theorem:

Theorem 1.1. Let X be a smooth complex projective variety of dimension n with Picard number one, and assume that there exist an ample vector bundle E of rank r on X, an integer 1 ≤ p ≤ r and an inclusion ∧

E , → ∧

T

. Then either X ∼ = P

or X ∼ = Q

, where Q

denotes a smooth hyperquadric in P

.

The first important result on characteriztions of them is S. Mori’s proof of the Hartshorne conjecture by using a method of rational curves on varieties in 1979:

Theorem 1.2 ([Mor79]). Let X be a smooth projective variety of dimension n over an algebraic closed field, and assume that the tangent bundle T

is ample.

Then X ∼ = P

.

Generalizing Mori’s result, it was conjectured that X ∼ = P

if the tangent bundle T

contains an ample vector bundle E for a smooth complex projective variety X . In this direction, J. Wahl obtained the case rk(E ) = 1 in 1983 ([Wah83]), and F. Campana and T. Peternell obtained the case rk(E ) ≥ n − 2 in 1998 ([CP98]). Finally M. Andreatta and J. Wi´sniewski proved the case of arbitrary rank in 2001:

Theorem 1.3 ([AW01]). Let X be a smooth complex projective variety of dimension n, and assume that T

contains an ample vector bundle E . Then (X, E ) ∼ = (P

, O

(1)

) or (X, E ) ∼ = (P

, T

).

In 2006, C. Araujo gave a different proof of Theorem 1.3 using a method of the variety of minimal rational tangents. Moreover a generalization of Theorem 1.3 was conjectured by S.J. Kov´acs:

Conjecture 1.4 (Kov´acs). Let X be a smooth complex projective variety of dimension n, and assume that there exist an ample vector bundle E of rank r on X, an integer 1 ≤ p ≤ r and an inclusion ∧

E , → ∧

T

. Then either X ∼ = P

or X ∼ = Q

.

On this conjecture, Araujo, S. Druel and Kov´acs proved the case E = L

for some line bundle L in 2008:

Theorem 1.5 ([ADK08]). Let X be a smooth complex projective variety of

dimension n, and assume that there exist an ample line bundle L on X and

an integer p ≥ 1 such that H

(X, ∧

T

⊗ L

) 6= 0. Then either (X, L ) ∼ =

(P

, O

(1)) or (X, L ) ∼ = (Q

, O

(1)).

Note that our Theorem 1.1 is the case where the Picard number ρ(X ) of X is one in Conjecture 1.4. Now one can easily see that the assumption of Theorem 1.1 implies that X has a minimal dominating family H and degf

E for every [f ] ∈ H is equal to r or to r + 1 (see Section 3). Here I should mention that K.

Ross also claimed the same statement as Theorem 1.1 in [Ros10]. But it seems that the existence of a morphism E → T

is implicitly assumed in the second case of the proof there. Then we give a different proof of Theorem 1.1. We use a method of slopes, semistabilities and Harder-Narasimhan filtrations of sheaves, which was given by [ADK08].

2 Preliminaries

First we recall about rational curves on varieties.

Definition 2.1. Let X be a smooth complex projective variety. An irreducible component H ⊂ RatCurves

(X ) is called a minimal dominating family if H- curves dominate X and the subfamily of H -curves passing through a general point x ∈ X is proper, where an H -curve means a curve parametrized by H . Lemma 2.2 ([Kol96, IV.2.4, 2.9 and 2.10]). If X is uniruled, then X has a minimal dominating family. If H is a minimal dominating family and [f ] ∈ H is a general member, then f

T

∼ = O(2) ⊕ O (1)

⊕ O

, where d =

−degf

(K

) − 2.

Definition 2.3. Let H be a fixed minimal dominating family. Two points x, y ∈ X are H-rationally equivalent if they can be connected by a chain of H- curves. By [Kol96, IV.4.16] there exist an open dense subset X

⊆ X , a normal variety Y

and a proper morphism π

: X

→ Y

whose fibers are H -rationally equivalent class. This π

is called the H-rationally connected quotient. X is said to be H -rationally connected if Y

is a point.

Lemma 2.4 ([KMM90]). Let H be a minimal dominating family. Then X is H-rationally connected if ρ(X ) = 1.

The following result is the key lemma to prove that varieties are isomorphic to projective space.

Lemma 2.5 ([ADK08, 2.7]). Let X be a smooth complex projective variety, H a minimal dominating family of rational curves on X, and π

: X

→ Y

the H-rationally connected quotient of X . Suppose that T

contains a subsheaf D such that f

D is an ample vector bundle for a general member [f ] ∈ H. Then, after shrinking X

and Y

if necessary, π

becomes a projective space bundle.

Next we recall some facts about slopes of sheaves.

Definition 2.6. Let X be a projective variety of dimension n and H an ample line bundle on X. The slope of a non-zero torsion-free sheaf F with respect to H is defined by

µ

(F ) = c

(F ) · c

(H )

rk(F ) .

A torsion-free sheaf F is µ

-semistable if µ

(E ) ≤ µ

(F ) for any subsheaf E of F .

Lemma 2.7 ([HN75] or [HL97, 1.3.4]). Let X and H be as above. Any torsion- free sheaf F on X has a filtration

F = F

) F

) · · · ) F

= 0,

such that Q

= F

/F

is µ

-semistable for 0 ≤ i ≤ h − 1, and µ

(Q

) <

· · · < µ

(Q

). This is called the Harder-Narasimhan filtration of F . Lemma 2.8 ([HL97, 3.1.4 and 3.2.10]). Let X be a normal projective variety over an algebraic closed field of characteristic zero, H an ample line bundle on X . Suppose that torsion-free sheaves F and G are µ

-semistable. Then F ⊗ G and ∧

F are also µ

-semistable if they are torsion-free.

Lemma 2.9 ([MR82] or [HL97, 7.2.1]). Let X be a smooth projective variety of dimension n ≥ 2, H an ample line bundle on X, and O(1) a very ample line bundle on X . If a torsion-sheaf F on X is µ

-semistable, then F |

is also µ

-semistable for a sufficiently large integer a and a general hypersurface H ∈ |O(a)|.

3 Proof of Theorem 1.1

Proof. The result is clear if n = 1, so we assume n ≥ 2. First it immedi- ately follows from [Miy87] that X is uniruled. Hence X has a minimal dom- inating family H ⊂ RatCurves

(X ) of rational curves (Lemma 2.2). Let [f ] be a general member of H . Then f

T

∼ = O(2) ⊕ O(1)

⊕ O

where d = −degf

(K

)−2, and f

(∧

E ) ⊆ f

(∧

T

). Since E is ample, we have pos- itive integers a

≥ · · · ≥ a

≥ 1 such that f

E ∼ = L

i=1

O (a

). Then it follows that a

+ · · · + a

≤ p+ 1. Therefore f

E ∼ = O(1)

or f

E ∼ = O (2) ⊕O (1)

. Since degf

E depends only on H but not on the choice of [f ], this is true for all [f] ∈ H .

Case 1. f

E ∼ = O(1)

for every [f ] ∈ H.

In this case, the result is proved in [Ros10]. By the induction on r we can reduce to the result of [ADK08] (Theorem 1.5).

Case 2. f

E ∼ = O(2) ⊕ O(1)

for every [f ] ∈ H.

In this case, we prove that X is isomorphic to projective space. First we prove the following claim.

Claim 3.1. Fix a very ample vector bundle H on X . Then T

contains a

subsheaf D (possibly not locally-free) such that µ

(D) ≥ µ

(E ).

Proof. We write µ simply instead of µ

. Now we take the Harder-Narasimhan filtration of T

(Lemma 2.7):

T

= D

) D

) · · · ) D

) D

= 0,

with µ-semistable Q

= D

/D

and µ(Q

) < · · · < µ(Q

). We would like to show that D = D

= Q

satisfies the condition. Let a be a sufficiently large integer, and let C be an intersection curve of n − 1 general hypersurfaces in |H

|. Let E

= E |

, D

= D

|

, Q

= Q

|

, T

= T

|

, and µ

= µ

. By Bertini’s theorem, we may assume that C is smooth, and that D

and Q

are locally-free. Moreover by Lemma 2.9 we may assume that Q

is µ

-semistable.

By the property of exterior products, the short exact sequence 0 → D

→ D

→ Q

→ 0

yields a filtration

∧

D

= F

⊇ F

⊇ · · · ⊇ F

= 0

for any non-negative integer t, such that F

/F

∼ = ∧

Q

⊗ ∧

D

. For any non-negative integers s, t, i, j and vector bundle A on C, again the short exact sequence

0 → A ⊗ F

→ A ⊗ F

→ A ⊗ ∧

Q

⊗ ∧

D

→ 0 yields a filtration

∧

(A ⊗ F

) = G

⊇ G

⊇ · · · ⊇ G

= 0 such that G

/G

∼ = ∧

(A ⊗ ∧

Q

⊗ ∧

D

) ⊗ ∧

(A ⊗ F

).

Now let q = rk(∧

E ) = ¡

p

¢ . The assumption of Theorem 1.1 implies ∧

∧

E ⊆ ∧

∧

T

. By the generality of C we may assume that ∧

∧

E

⊆ ∧

∧

T

. Note that ∧

∧

T

= G

. First from the filtration G

, we have an integer k

such that ∧

∧

E

⊆ G

and ∧

∧

E

6⊆ G

. Then an inclusion

∧

∧

E

, → ∧

(∧

Q

⊗ ∧

D

) ⊗ ∧

F

is induced. Next from the filtration ∧

(∧

Q

⊗ ∧

D

) ⊗ G

, we have an integer k

such that ∧

∧

E

⊆ ∧

(∧

Q

⊗ ∧

D

) ⊗ G

and ∧

∧

E

6⊆ ∧

(∧

Q

⊗ ∧

D

) ⊗ G

. Then an inclusion

∧

∧

E

, → ∧

(∧

Q

⊗ ∧

D

) ⊗ ∧

(∧

Q

⊗ ∧

D

) ⊗ ∧

F

is induced. After a similar procedure of p steps, we have non-negative integers b

, . . . , b

such that b

+ · · · + b

= q and an inclusion

∧

∧

E

, → ∧

(∧

Q

⊗∧

D

)⊗∧

(∧

Q

⊗∧

D

)⊗· · ·⊗∧

(∧

Q

⊗∧

D

).

Next for 0 ≤ m ≤ p, let B

= N

t6=m

∧

(∧

Q

⊗ ∧

D

). Then

∧

∧

E

, → B

⊗ ∧

(∧

Q

⊗ ∧

D

).

Note that ∧

(∧

Q

⊗ ∧

D

) = G

. From the filtration B

⊗ G

, we have an integer l

and an inclusion

∧

∧

E

, → B

⊗ ∧

(∧

Q

⊗ ∧

Q

⊗ ∧

D

) ⊗ ∧

(∧

Q

⊗F

).

Next from the filtration B

⊗∧

(∧

Q

⊗∧

Q

⊗∧

D

)⊗G

, we have an integer l

and an inclusion

∧

∧

E

, → B

⊗ ∧

(∧

Q

⊗ ∧

Q

⊗ ∧

D

)

⊗ ∧

(∧

Q

⊗ ∧

Q

⊗ ∧

D

)

⊗ ∧

(∧

Q

⊗ F

).

After a similar procedure of m steps, we have non-negative integers a

, . . . , a

such that a

+ · · · + a

= b

and an inclusion

∧

∧

E

, → B

⊗ ∧

(∧

Q

⊗ ∧

Q

⊗ ∧

D

)

⊗ ∧

(∧

Q

⊗ ∧

Q

⊗ ∧

D

)

· · ·

⊗ ∧

(∧

Q

⊗ ∧

Q

⊗ ∧

D

).

Repeating a similar procedure for m = 0, 1, . . . , p, we have integers {a

}

such that P

a

= q and an inclusion

∧

∧

E

, → O

α,β,γ

∧

(∧

Q

⊗ ∧

Q

⊗ ∧

D

).

After a similar procedure of h−1 steps, finally we have integers {a

}

such that P

a

= q and an inclusion

∧

∧

E

, → O

α0,...,α_h−1

∧

(∧

Q

⊗ ∧

Q

⊗ · · · ⊗ ∧

Q

) =: Q

. Now since each Q

is µ

-semistable, so is Q

(Lemma 2.8). Therefore µ

(∧

∧

E

) ≤ µ

(Q

). On the other hand,

µ

(∧

∧

E

) = qpa

µ(E ), µ

(Q

) = X

α0,...,α_h−1

a

{α

µ

(Q

) + · · · + α

µ

(Q

)}

≤ X

α0,...,αh−1

a

pµ

(Q

)

= qpa

µ(Q

).

Thus µ(E ) ≤ µ(Q

).

Now let [f ] ∈ H be a general member. Then we have integers a

≥ . . . ≥ a

such that f

D ∼ = L

i=1

O(a

), and f

D ⊆ f

T

∼ = O(2) ⊕ O (1)

⊕ O

implies that a

≤ 2 and a

≤ 1. On the other hand, µ

(D) ≥ µ

(E ) implies

that P

a

s = degf

D

s ≥ degf

E

r = r + 1 r > 1.

Thus it follows that f

D ∼ = O(2) ⊕ O(1)

. Hence we have a H -rationally connected quotient which is projective space bundle by Lemma 2.5. But then, the assumption ρ(X ) = 1 implies that X is H -rationally connected (Lemma 2.4). Therefore X ∼ = P

as desired.

Acknowledgements. I would like to express my gratitude to my supervisor Professor Hajime Kaji for beneficial discussions and continued encouragement.

His advice and suggestions were greatly valuable for my study. I would also like to thank Professor Yasunari Nagai for helpful adivice and useful comments.

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