On the proper moduli spaces of smoothable K\"ahler-Einstein Fano varieties

TL;DR

This paper constructs a proper moduli space for smoothable K"ahler-Einstein Fano varieties, proving key properties like openness of K-semistability and uniqueness of Gromov-Hausdorff limits, advancing the understanding of their geometric classification.

Contribution

It introduces a new proper scheme parameterizing S-equivalence classes of K-semistable Fano varieties, establishing it as a good moduli space with desirable geometric properties.

Findings

K-semistability is a Zariski open condition.

Uniqueness of Gromov-Hausdorff limits for families of Fano K"ahler-Einstein manifolds.

Construction of a proper moduli scheme for smoothable K"ahler-Einstein Fano varieties.

Abstract

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