On the sharpness of Tian's criterion for K-stability

TL;DR

This paper demonstrates the sharpness of Tian's criterion for K-stability by constructing specific Fano varieties with critical alpha invariants that are not K-stable or are K-unstable, especially in higher dimensions.

Contribution

It provides explicit examples showing the limits of Tian's criterion for K-stability, including non-K-stable and K-unstable Fano varieties at the critical alpha invariants.

Findings

Constructed singular Fano varieties with alpha invariant n/(n+1) that are not K-polystable.

Constructed K-unstable Fano varieties with alpha invariant (n-1)/n.

Confirmed the sharpness of Tian's criterion in high dimensions.

Abstract

Tian's criterion for K-stability states that a Fano variety of dimension $n$ whose alpha invariant is greater than $\frac{n}{n+1}$ is K-stable. We show that this criterion is sharp by constructing singular Fano varieties with alpha invariants $\frac{n}{n+1}$ that are not K-polystable for sufficiently large $n$. We also construct K-unstable Fano varieties with alpha invariants $\frac{n-1}{n}$.