On the proper moduli spaces of smoothable Kähler-Einstein Fano varieties

In this paper we investigate the geometry of the orbit space of the closure of the subscheme parameterizing smooth Kähler-Einstein Fano manifolds inside an appropriate Hilbert scheme. In particular, we prove that being K-semistable is a Zariski-open condition, and we establish the uniqueness of the Gromov-Hausdorff limit for a punctured flat family of Kähler-Einstein Fano manifolds. Based on these, we construct a proper scheme parameterizing the S-equivalent classes of Q-Gorenstein smoothable, K-semistable Q-Fano varieties, and we verify various necessary properties to guarantee that it is a good moduli space.