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 that 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 that can be used to test various K-stability notions of Fano varieties.
"On positivity of the CM line bundle on K-moduli spaces." Ann. of Math. (2) 192 (3) 1005 - 1068, November 2020. https://doi.org/10.4007/annals.2020.192.3.7