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

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 $\frac{n-1}{n}$.

arXiv comment34 pages. To appear in Nagoya Math. J

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Cited by in corpus (4)

• K-stability of Fano varieties: an algebro-geometric approach
• Equivariant K-stability under finite group action
• On K-stability of Fano weighted hypersurfaces
• Delta invariants of projective bundles and projective cones of Fano type