EPW cubes

Abstract

Abstract We construct a new 20-dimensional family of projective six-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-three subvarieties of the Grassmannian <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"><m:mrow><m:mi>G</m:mi><m:mo>⁢</m:mo><m:mrow><m:mo>(</m:mo><m:mn>3</m:mn><m:mo>,</m:mo><m:mn>6</m:mn><m:mo>)</m:mo></m:mrow></m:mrow></m:math> G(3,6) . These codimension-three 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 <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"><m:mrow><m:mi>K</m:mi><m:mo>⁢</m:mo><m:mn>3</m:mn></m:mrow></m:math> K3 -type, Beauville–Bogomolov degree 4 and divisibility 2 is unirational.