In this paper, we consider the CM line bundle on the K-moduli space, i.e., the moduli space parametrizing K-polystable Fano varieties. We prove it is ample on any proper subspace parametrizing reduced uniformly K-stable Fano varieties which conjecturally should be the entire moduli space. As a corollary, we prove that the moduli space parametrizing smoothable K-polystable Fano varieties is projective. During the course of proof, we develop a new invariant for filtrations which can be used to test various K-stability notions of Fano varieties.

中文翻译：

---

K-模空间上CM线丛的正性

在本文中，我们考虑 K-模空间上的 CM 线丛，即参数化 K-polystable Fano 簇的模空间。我们证明了它在任何适当的子空间参数化上都是充分的，即归约的一致 K 稳定 Fano 变体，推测应该是整个模空间。作为推论，我们证明了模空间参数化平滑 K-polystable Fano 变体是射影的。在证明过程中，我们开发了一个新的过滤不变量，可用于测试 Fano 变体的各种 K 稳定性概念。