  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.
