K-polystability of two smooth Fano threefolds

Abstract

We give new proofs of the K-polystability of two smooth Fano threefolds. One of them is a~smooth divisor in ${\mathbb{P}}^1\times {\mathbb{P}}^1\times {\mathbb{P}}^2$ of degree $(1,1,1)$, which is unique up to isomorphism. Another one is the~blow up of the complete intersection $$\{x_0{x}_3+x_1{x}_4+x_2{x}_5=x_0^2+\omega x_1^2+{\omega }^2{x}_2^2+(x_3^2+\omega x_4^2+{\omega }^2{x}_5^2)+(x_0{x}_3+\omega x_1{x}_4+{\omega }^2{x}_2{x}_5\left)\right\}\subset {\mathbb{P}}^5$$ in the conic cut out by $x_0=x_1=x_2=0$, where $\omega$ is a~primitive cube root of unity.

22 pages