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

Abstract

Motivated by the problem for 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, recently, Fujita, Liu, Süß, Zhang and Zhuang 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 delta-invariant 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.

arXiv comment41 pages

References in corpus (9)

• K-stability and Kähler-Einstein metrics
• Big rational surfaces
• Basis log canonical thresholds, local intersection estimates, and asymptotically log del Pezzo surfaces
• Asymptotically log Fano varieties
• K-stability of log Fano hyperplane arrangements
• K-stability of constant scalar curvature Kähler manifolds
• On the Yau-Tian-Donaldson conjecture for singular Fano varieties
• Uniqueness of K-polystable degenerations of Fano varieties
• On the existence of conic Kahler-Einstein metrics

Cited by in corpus (1)

• Unstable singular del Pezzo hypersurfaces with lower index