On fibre space structures of a projective irreducible symplectic manifold
Abstract
In this note, we investigate fibre space structures of a projective irreducible symplectic manifold. We prove that an 2ndimensional projective irreducible symplectic manifold admits only an ndimensional fibration over a Fano variety which has only factorial logterminal singularities and whose Picard number is one. Moreover we prove that a general fibre is an abelian variety up to finite unramified cover, especially, a general fibre is an abelian surface for 4fold.
1 Introduction
We first define an irreducible symplectic manifold.
Definition 1
A complex manifold is called irreducible symplectic if satisfies the following three conditions:

is compact and Kähler.

is simply connected.

is spanned by an everywhere nondegenerate twofrom .
Such a manifold can be considered as an unit of compact Kähler manifold with due to the following Bogomolov decomposition theorem.
Theorem 1 (Bogomolov decomposition theorem [2])
A compact Kähler manifold with admits a finite unramified covering of which is isomorphic to a product where is a complex torus, are irreducible symplectic manifolds and is a projective manifold with , .
In dimension 2, surfaces are the only irreducible symplectic manifolds, and irreducible symplectic manifolds are considered as higherdimensional analogies of surfaces. In this note, we investigate fibre space structures of a projective irreducible symplectic manifolds.
Definition 2
For an algebraic variety , a fibre space structure of is a proper surjective morphism which satisfies the following two conditions:

and are normal varieties such that

A general fibre of is connected.
Some of surface has a fibre space structure whose general fibre is an elliptic curve. In higher dimensional analogy, we obtain the following results.
Theorem 2
Let be a fibre space structure of a projective irreducible symplectic fold with projective base . Then a general fibre of and satisfy the following three conditions:

is an abelian variety up to finite unramified cover and .

is dimensional and has only factorial logterminal singularities

is ample and Picard number is one.
Especially, if is dimensional, a general fibre of is an abelian surface.
Example. Let be a surface with an elliptic fibration and a pointed Hilbert scheme of . It is known that is an irreducible symplectic fold and there exists a birational morphism where is the symmetric product of (cf. [1]). We can consider dimensional abelian fibration for the symmetric product of . Then the composition morphism gives an example of a fibre space structure of an irreducible symplectic manifold.
Remark. Markushevich obtained some result of theorem 2 in [6, Theorem 1, Proposition 1] under the assumption and is the moment map. In general, a fibre space structure of an irreducible symplectic manifold is not a moment map. Markushevich constructs in [7, Remark 4.2] counterexample.
Acknowledgment. The author express his thanks to Professors Y. Miyaoka, S. Mori and N. Nakayama for their advice and encouragement. He also thanks to Prof. D. Huybrechts [4] for his nice survey articles of irreducible symplectic manifolds.
2 Proof of Theorems
Theorem 3 ([3] Theorem 4.7, Lemma 4.11, Remark 4.12 [1] Thèoréme 5)
Let be an irreducible symplectic fold. Then there exists a nondegenerate quadratic form of signature on which satisfies
where and ’s are constants depending on .
We shall prove theorem 2 in five steps.

and has only logterminal singularities;

A general fibre of is an abelian variety up to unramified finite cover and ;

;

is factorial;

is ample.
Step 1. and has only logterminal singularities.
Lemma 1
Let be an irreducible symplectic projective fold and be a divisor on such that . Then,

If for some ample divisor , .

If for an ample divisor on , then
Proof of lemma. Let . By [3, Lemma 4.13], is negative definite on where . Thus, if and , . Next we prove (2). From Theorem 3, for every integer ,
(1) 
Because ,
Thus the right hand side of the equation (1) has order at most . Comparing the both hand side of the equation (1), we can obtain for . If , comparing the first order term of of both hand of the equation (1) we can obtain . Because coefficients of other terms of left hand side of (1) can be written and , we can obtain for .
Let be a very ample divisor on . Then is a nef divisor such that , for an ample divisor on . Thus . From [8, Theorem 2], has only logterminal singularities.
Step 2. A general fibre of is an abelian variety up to unramified finite cover and .
By adjunction, . Moreover
by Theorem 3. Thus has an étale cover such that is an Abelian variety by [9].
Step 3. .
Lemma 2
Let be a divisor of such that and . Then for some rational number .
Proof of lemma. Considering the following equation
we can obtain where is a constant. Thus . Because for every ample divisor on , we can take a rational number such that Then by lemma 1.
Let be a Cartier divisor on . Then and , thus and .
Step 4. is factorial.
Let be an irreducible and reduced Weil divisor on and , divisors on whose supports are contained in . We construct a divisor , such that . Let be a very ample divisor on , and . Then there exists a surjective morphism . If we choose and general, we may assume that and are smooth and are contained smooth locus of . Because is a Cartier divisor in a neighborhood of , we can define in a neighborhood of . We can express in and let . Note that if , because we choose generally. Compareing the th order term of of the both hand side of the following equotion
we can see that . Since , and . Considering the following equation
we can obtain where is a constant. Since is a fibre of , . Thus . Considering , we can obtain by Lemma 1, and because . Therefore is factorial.
Step 5. is ample.
From Step 3,4, we can write . It is enough to prove . Because and a general fibre of is a minimal model, by [5, Theorem 1.1] and . Assume that . If , we can consider the following diagram:
where is an unramified finite cover and . Because , is the direct sum of . Thus there exists a morphism from to and we may assume that . Then there exists a holomorphic form on coming from . However, if is odd, it is a contradiction because there exist no holomorphic form on . If is even, it is also a contradiction because dose not generated by . Thus and we completed the proof of Theorem 2. Q.E.D.
References
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Resarch Institute of Mathematical Science,
Kyoto University.
KITASHIRAKAWA, OIWAKECHO,
KYOTO, 60601, JAPAN.