A Special Class of Hyper-kähler Manifolds

We consider hyper-Kähler manifolds of complex dimension 4 which are fibrations. It is known that the fibers are abelian varieties and the base is P. We assume that the general fiber is isomorphic to a product of two elliptic curves. We prove that such a hyper-Kähler manifold is deformation equivalent to a Hilbert scheme of two points on a K3 surface. 1. Preliminaries First we define our main objects of study, irreducible symplectic manifolds or hyperKähler manifolds. Definition 1.1. A compact complex Kähler manifold X is called irreducible symplectic if it is simply connected and if H(X,ΩX) is spanned by an everywhere non-degenerate 2-form ω. Any holomorphic two-form σ induces a homomorphism TX → ΩX . The two-form is everywhere non-degenerate if and only if TX → ΩX is bijective. The last condition in the definition implies that h(X) = h(X) = 1 and KX ∼= OX , i.e., c1(X) = 0. Definition 1.2. A compact connected 4n-dimensional Riemannian manifold (M, g) is called irreducible hyper-Kähler if its holonomy is Sp(n). As Hybrechts notes [7], irreducible symplectic manifolds with a fixed Kähler class and compact irreducible hyper-Kähler manifolds are the same object. In the rest of the paper, we are going to refer to irreducible hyper-Kähler manifolds just as hyperKähler manifolds for simplicity. Definition 1.3. An abelian fibration on a 2n-dimensional hyper-Kähler manifold X is the structure of a fibration over P whose generic fibre is a smooth abelian variety of dimension n. This is a higher dimensional analogue of elliptic fibrations on K3 surfaces. Any fibration structure of a hyper-Kahler manifold looks like an abelian fibration due to the following theorem by Matsushita [8]: Theorem 1.1. For projective symplectic manifold X, let f : X → B be a proper surjective morphism such that the generic fibre F is connected. Assume that B is smooth and 0 < dimB < dimX. Then Research partially supported by NSF Grants DMS-0111298 and DMS-0805594 1