The Cheltsov--Rubinstein problem for strongly asymptotically log del Pezzo surfaces

The notion of (strongly) asymptotically log Fano varieties was introduced in 2013 by Cheltsov–Rubinstein, who posed the problem of classifying all strongly asymptotically log del Pezzo surfaces with smooth boundary that admit Kähler–Einstein edge metrics. Thanks to the Cheltsov–Rubinstein classification, this amounts to considering 10 families. In 8 families the problem has been solved by work of Cheltsov–Rubinstein, Fujita and Mazzeo–Rubinstein. The remaining 2 families are rational surfaces parameterized by the self-intersection of the 0-section $n$ and the number of blow-ups $m$. By Cheltsov–Rubinstein, Cheltsov–Rubinstein–Zhang and Fujita, Kähler–Einstein edge metrics exist when either $m=0$ or $m\ge 3$ for the first family, and the cases $m=1,2$ have been studied by Fujita–Liu–Süss–Zhang–Zhuang and Fujita. The final remaining family, denoted $(\mathrm{I}\mathrm{I}.6\mathrm{A}\mathrm{.}\mathrm{n}\mathrm{.}\mathrm{m})$ in the Cheltsov–Rubinstein classification, is more difficult as the boundary consists of two components, unlike any of the other 9 families. It is the generalization of the football to complex surfaces with the pair $(\mathrm{I}\mathrm{I}.6\mathrm{A}\mathrm{.0.0})$ being exactly the football times ${\mathbb{P}}^1$. The pairs $(\mathrm{I}\mathrm{I}.6\mathrm{A}\mathrm{.}\mathrm{n}.0)$ have been completely understood by the work of Rubinstein–Zhang using the ${\mathbb{P}}^1$-bundle structure of Hirzebruch surfaces. This article studies the family $(\mathrm{I}\mathrm{I}.6\mathrm{A}\mathrm{.}\mathrm{n}\mathrm{.}\mathrm{m})$ for $m\ge 1$. These pairs no longer have a ${\mathbb{P}}^1$-bundle structure and are therefore more difficult to tackle. The main result is a necessary and sufficient condition on the angles for the existence of Kähler–Einstein edge metrics, generalizing the Rubinstein–Zhang condition. Thus, we resolve the Cheltsov–Rubinstein problem for strongly asymptotically log del Pezzo surfaces.