Rational curves on holomorphic symplectic fourfolds

One main problem in the theory of irreducible holomorphic symplectic manifolds is the description of the ample cone in the Picard group. The goal of this paper is to formulate explicit Hodge-theoretic criteria for the ampleness of line bundles on certain irreducible holomorphic symplectic manifolds. It is well known that for K3 surfaces the ample cone is governed by (−2)-curves. More generally, we expect a framework relating the existence of distinguished two-dimensional classes in the cohomology to the existence of certain explicit families of rational curves. Our philosophy is that these are precisely the rational curves bounding the ample cone. Let F be an irreducible holomorphic symplectic fourfold deformation equivalent to the punctual Hilbert scheme S [2] for some K3 surface S. Given its Hodge structure H(F,Z), we describe explicitly (but conjecturally) the corresponding ample cone. It is known that H(F,Z) carries a natural quadratic form, the Beauville form. As in the case of K3 surfaces, each divisor class of square (−2) induces a reflection preserving the Hodge structure. The ‘birational ample cone’ is conjectured to be the interior of a fundamental domain for this reflection group. However, the ample cone may be strictly smaller than the birational ample cone, owing to the existence of elementary transformations along P’s in F . The corresponding classes have square −10 with respect to the Beauville form.