We describe a compactification by KSBA stable pairs of the five-dimensional moduli space of K3 surfaces with a 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 pairs compactification is isomorphic to Kirwan's partial desingularization of the GIT quotient $(\mathbb {P}^{1})^{8}{/\!/}\mathrm {SL}_2$ with the symmetric linearization.

中文翻译：

---

具有四阶纯非辛自同构的 K3 曲面模空间的 KSBA 紧化

我们描述了 K3 表面的五维模空间的 KSBA 稳定对的紧化，具有四阶纯非辛自同构和$U(2)\oplus D_4^{\oplus 2}$晶格极化。这些K3表面可以实现为双覆盖的最小分辨率$\mathbb {P}^{1}\times \mathbb {P}^{1}$沿着特定的分支$(4,\,4)$曲线。我们表明，在有限群作用下，这种稳定的对紧化与 Kirwan 对 GIT 商的部分去奇异化同构$(\mathbb {P}^{1})^{8}{/\!/}\mathrm {SL}_2$与对称线性化。