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

Abstract

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 $\mathbb{Q}$-Gorenstein smoothable, K-semistable $\mathbb{Q}$-Fano varieties, and we verify various necessary properties to guarantee that it is a good moduli space.