Rational curves on holomorphic symplectic fourfolds

TL;DR

This paper explores the structure of effective and ample cones on certain holomorphic symplectic fourfolds, proposing conjectures analogous to K3 surfaces and verifying them in specific cases, with implications for cubic fourfolds.

Contribution

It formulates precise conjectures for the cones of effective curves and ample divisors on polarized irreducible holomorphic symplectic fourfolds and proves these in an open subset of the moduli space.

Findings

Conjectures are verified in an open subset of the moduli space.

The structure of cones is described in terms of the Beauville form and Picard group.

Implications for the geometry of cubic fourfolds are established.

Abstract

Let F be a polarized irreducible holomorphic symplectic fourfold, deformation equivalent to the Hilbert scheme parametrizing length-two zero-dimensional subschemes of a K3 surface. The homology group H^2(F,Z) is equipped with an integral symmetric nondegenerate form, the Beauville form. We give precise conjectures for the structure of the cone of effective curves - and by duality - the cone of ample divisors. Formally they are completely analogous to the results known for K3 surfaces, and they are expressed entirely in terms of the integers represented by the Beauville form restricted to Pic(F). We prove that these conjectures are true in an open subset of the moduli space using deformation theory. The Fano variety of lines contained in a cubic fourfold is an example of a holomorphic symplectic fourfold. Our conjectures imply many concrete geometric statements about cubic fourfolds which may be verified using projective geometry.