On the symplectic eightfold associated to a Pfaffian cubic fourfold

TL;DR

This paper proves that a special eight-dimensional symplectic manifold related to certain cubic fourfolds is deformation-equivalent to a Hilbert scheme of four points on a K3 surface, using birational maps and moduli space interpretations.

Contribution

It establishes the deformation equivalence of the symplectic eightfold to a Hilbert scheme of points on a K3 surface for Pfaffian cubic fourfolds, via explicit birational constructions.

Findings

Z is deformation-equivalent to Hilb^4(X)

Constructs a birational map for generic Pfaffian cubic Y

Interprets Z as a moduli space of complexes on X

Abstract

We show that the irreducible holomorphic symplectic eightfold Z associated to a cubic fourfold Y not containing a plane is deformation-equivalent to the Hilbert scheme of four points on a K3 surface. We do this by constructing for a generic Pfaffian cubic Y a birational map Z ---> Hilb^4(X), where X is the K3 surface associated to Y by Beauville and Donagi. We interpret Z as a moduli space of complexes on X and observe that at some point of Z, hence on a Zariski open subset, the complex is just the ideal sheaf of four points.