We prove that K-polystable log Fano pairs have reductive automorphism groups. In fact, we deduce this statement by establishing more general results concerning the S-completeness and $\Theta$-reductivity of the moduli of K-semistable log Fano pairs. Assuming the conjecture that K-semistability is an open condition, we prove that the Artin stack parametrizing K-semistable Fano varieties admits a separated good moduli space.

中文翻译：

---

K-polystable Fano变种自同构群的还原性

我们证明了 K-polystable log Fano 对具有还原自同构群。事实上，我们通过建立关于 K-半稳定对数 Fano 对的模的 S-完备性和 $\Theta$-还原性的更一般结果来推导出这个陈述。假设 K-semistability 是一个开放条件的猜想，我们证明了参数化 K-semistable Fano 变体的 Artin stack 允许分离的良好模空间。