Cubic threefolds and hyperk\"ahler manifolds uniformized by the 10-dimensional complex ball

TL;DR

This paper establishes a deep connection between the moduli spaces of smooth cubic threefolds and certain hyperk"ahler fourfolds, showing they are both uniformized by the same complex ball and exploring automorphisms and degenerations.

Contribution

It proves an isomorphism between the moduli spaces of cubic threefolds and hyperk"ahler fourfolds with specific automorphisms, and constructs explicit examples of automorphisms.

Findings

Both moduli spaces are uniformized by the same 10-dimensional complex ball.

Degenerations of automorphisms correspond to specific loci related to hyperk"ahler fourfolds.

Explicit construction of a non-natural automorphism on the Hilbert square of a K3 surface.

Abstract

We first prove an isomorphism between the moduli space of smooth cubic threefolds and the moduli space of hyperkaehler fourfolds of K3^{[2]}-type with a non-symplectic automorphism of order three, whose invariant lattice has rank one and is generated by a class of square 6, both these spaces are uniformized by the same 10-dimensional arithmetic complex ball quotient. We then study the degeneration of the automorphism along the loci of nodal or chordal degenerations of the cubic threefold, showing the birationality of these loci with some moduli spaces of hyperkaehler fourfolds of K3^{[2]}-type with non-symplectic automorphism of order three belonging to different families. Finally, we construct a cyclic Pfaffian cubic fourfold to give an explicit construction of a non-natural automorphism of order three on the Hilbert square of a K3 surface.