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

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