On the uniform K-stability for some asymptotically log del Pezzo surfaces

Abstract

Motivated by the problem for the existence of K\"ahler-Einstein edge metrics, Cheltsov and Rubinstein conjectured the K-polystability of asymptotically log Fano varieties with small cone angles when the anti-log-canonical divisors are not big. Cheltsov, Rubinstein and Zhang proved it affirmatively in dimension $2$ with irreducible boundaries except for the type $(\operatorname{I.9B.}n)$ with $1\leq n\leq 6$. Unfortunately, recently, Fujita, Liu, S\"u\ss, Zhang and Zhuang showed the non-K-polystability for some members of type $(\operatorname{I.9B.}1)$ and for some members of type $(\operatorname{I.9B.}2)$. In this article, we show that Cheltsov--Rubinstein's problem is true for all of the remaining cases. More precisely, we explicitly compute the delta-invariant for asymptotically log del Pezzo surfaces of type $(\operatorname{I.9B.}n)$ for all $n\geq 1$ with small cone angles. As a consequence, we finish Cheltsov--Rubinstein's problem in dimension $2$ with irreducible boundaries.