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

Abstract

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 K\"ahler--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, K\"ahler--Einstein edge metrics exist when either $m=0$ or $m\geq3$ for the first family, and the cases $m=1,2$ have been studied by Fujita--Liu--S\"uss--Zhang--Zhuang and Fujita. The final remaining family, denoted $\mathrm{(II.6A.n.m)}$ 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 $\mathrm{(II.6A.0.0)}$ being exactly the football times $\mathbb{P}^1$. The pairs $\mathrm{(II.6A.n.0)}$ 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 $\mathrm{(II.6A.n.m)}$ for $m\geq1$. 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 K\"ahler--Einstein edge metrics, generalizing the Rubinstein--Zhang condition. Thus, we resolve the Cheltsov--Rubinstein problem for strongly asymptotically log del Pezzo surfaces.