Positivity of CM line bundle on the K-moduli space
-- Ziquan Zhuang, February 21, 2020

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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.