On Uniform K-Stability for Some Asymptotically log del Pezzo Surfaces

Abstract

Motivated by the problem of the existence of Kähler–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 (I.9B.$n$) with $1\le n\le 6$. Unfortunately, Fujita, Liu, Süß, Zhang and Zhuang recently showed the non-K-polystability for some members of type (I.9B.1) and for some members of type (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 deltainvariant for asymptotically log del Pezzo surfaces of type (I.9B.$n$) for all $n\ge 1$ with small cone angles. As a consequence, we finish Cheltsov–Rubinstein’s problem in dimension 2 with irreducible boundaries.