On positivity of the CM line bundle on K-moduli spaces

Abstract

In this paper, we consider the CM line bundle on the $\mathrm {K}$-moduli space, i.e., the moduli space parametrizing $\mathrm {K}$-polystable Fano varieties. We prove it is ample on any proper subspace parametrizing reduced uniformly $\mathrm {K}$-stable Fano varieties that conjecturally should be the entire moduli space. As a corollary, we prove that the moduli space parametrizing smoothable $\mathrm {K}$-polystable Fano varieties is projective.

During the course of proof, we develop a new invariant for filtrations that can be used to test various $\mathrm {K}$-stability notions of Fano varieties.