recently there has been a lot of work on the construction of the k-moduli space, i.e. a good moduli space parametrizing k-polystable fano varieties. it is conjectured that this moduli space is projective and the polarization is given by a natural line bundle, the chow-mumford (cm) line bundle. in this talk, i will present a recent joint work with chenyang xu where we show the cm line bundle is ample on any proper subspace parametrizing reduced uniformly k-stable fano varieties, which conjecturally should be the entire k-moduli. as an application, we prove that the moduli space parametrizing smoothable k-polystable fano varieties is projective.