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 ≤ n ≤ 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 ≥ 1 with small cone angles. As a consequence, we finish Cheltsov-Rubinstein's problem in dimension 2 with irreducible boundaries. Contents 1. Introduction 1 2. Preliminaries 3 3. On the del Pezzo surface of degree seven 9 4. Prime divisors centered at special points, I 25 5. Prime divisors centered at special points, II 32 6. Proof of Theorem 1.3 39 References 40

On the body of ample angles of asymptotically log Fano varieties

“…Asymptotically log Fano pairs, introduced by Cheltsov-Rubinstein [5], generalizing work of [20], have received attention in the last decade within the theory of K-stability and the Calabi problem [6,7,10,11]. We believe asymptotically log Fanos to be interesting objects in their own right.…”

Unstable singular del Pezzo hypersurfaces with lower index

“…Then S d is a log del Pezzo surface with at most quotient singularities. For I = 1 Johnson and Kollár [10] found all possibilities for quintuple (a 0 , a 1 , a 2 , a 3 , d) and then computed the alpha invariant to show the existence of the orbifold Kähler-Einstein metric in the case when the quintuple (a 0 , a 1 , a 2 , a 3 , d) is not of the following four quintuples : (1,2,3,5,10), (1,3,5,7,15), (1,3,5,8,16), (2,3,5,9,18). Later, Araujo [1] shows for the two of these four cases.…”