EPW Cubes

TL;DR

This paper introduces a new 20-dimensional family of irreducible holomorphic symplectic manifolds called EPW cubes, constructed as double covers of special Lagrangian degeneracy loci in Grassmannians, and proves related moduli space properties.

Contribution

It constructs EPW cubes as a new class of IHS sixfolds and establishes their deformation equivalence to Hilbert schemes of points on K3 surfaces, expanding the understanding of such manifolds.

Findings

Construction of EPW cubes as double covers of Lagrangian degeneracy loci.

Proof that the moduli space of certain polarized IHS sixfolds is unirational.

Identification of EPW cubes as deformation equivalent to Hilbert schemes of three points on K3 surfaces.

Abstract

We construct a new 20-dimensional family of projective 6-dimensional irreducible holomorphic symplectic manifolds. The elements of this family are deformation equivalent with the Hilbert scheme of three points on a K3 surface and are constructed as natural double covers of special codimension 3 subvarieties of the Grassmanian G(3,6). These codimension 3 subvarieties are defined as Lagrangian degeneracy loci and their construction is parallel to that of EPW sextics, we call them the EPW cubes. As a consequence we prove that the moduli space of polarized IHS sixfolds of K3-type, Beauville-Bogomolov degree 4 and divisibility 2 is unirational.