K-stability of birationally superrigid Fano varieties

TL;DR

This paper establishes conditions under which birationally superrigid Fano varieties are K-stable or K-semistable based on their alpha invariants, and explores their moduli space properties.

Contribution

It proves new stability criteria for birationally superrigid Fano varieties using alpha invariants and analyzes their moduli space separation.

Findings

Alpha invariant of such varieties is at least 1/(n+1).

Varieties with alpha > 1/2 are K-stable.

Moduli space of these varieties is separated.

Abstract

We prove that every birationally superrigid Fano variety whose alpha invariant is greater than (resp. no smaller than) $\frac{1}{2}$ is K-stable (resp. K-semistable). We also prove that the alpha invariant of a birationally superrigid Fano variety of dimension $n$ is at least $\frac{1}{n+1}$ (under mild assumptions) and that the moduli space (if exists) of birationally superrigid Fano varieties is separated.