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

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 $U(2)\oplus D_4^{\oplus 2}$ 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.

27 pages, 6 figures. Final version. To appear in Proceedings of the Edinburgh Mathematical Society