KSBA compactification of the moduli space of K3 surfaces with purely non-symplectic automorphism of order four

TL;DR

This paper constructs a KSBA stable pair compactification of a specific moduli space of K3 surfaces with non-symplectic automorphisms of order four, linking it to GIT quotients and lattice polarization.

Contribution

It provides a new compactification of the moduli space using KSBA stable pairs and relates it to Kirwan's desingularization of a GIT quotient.

Findings

The moduli space is realized as minimal resolutions of double covers of  with specific branch curves.

The compactification is isomorphic to Kirwan's partial desingularization of a GIT quotient.

The construction involves lattice polarization and automorphism considerations.

Abstract

We describe a compactification by KSBA stable pairs of the five-dimensional moduli space of K3 surfaces with purely non-symplectic automorphism of order four and $U(2)\oplus D_4^{\oplus2}$ lattice polarization. These K3 surfaces can be realized as the minimal resolution of the double cover of $\mathbb{P}^1\times\mathbb{P}^1$ branched along a specific $(4,4)$ curve. We show that, up to a finite group action, this stable pair compactification is isomorphic to Kirwan's partial desingularization of the GIT quotient $(\mathbb{P}^1)^8//\mathrm{SL}_2$ with the symmetric linearization.