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 ${n}{/(n+1)}$ is K-stable. We show that this criterion is sharp by constructing n-dimensional singular Fano varieties with alpha invariants ${n}{/(n+1)}$ that are not K-polystable for sufficiently large n. We also construct K-unstable Fano varieties with alpha invariants ${(n-1)}{/n}$ .

K-stability and birational models of moduli of quartic K3 surfaces

“…(5.1) By[68, Theorem 1.4], we know that the analogous statement of [71, Proposition 5.3] for K-polystability is true (see also[74, Proposition 2.11]). Hence(Y, bV Y +cD 0 ) is the K-polystable since (V, cC) is K-polystable.…”