Delta Invariants of Singular del Pezzo Surfaces

Abstract

We use the methods introduced by Cheltsov-Rubinstein-Zhang in [CRZ18] to estimate δ-invariants of the seven singular del Pezzo surfaces with quotient singularities studied by Cheltsov-Park-Shramov in [CPS10] that have α-invariants less than 2 3 . As a result, we verify that each of these surfaces admits an orbifold Kähler-Einstein metric.All varieties are assumed to be complex, projective and normal unless otherwise stated.2010 Mathematics Subject Classification. 14J17, 14J45, 32Q20. 1 holds, where α(S d ) is the α-invariant of the surface S d . Indeed, Johnson and Kollár verified (1.3) in the case when I = 1, the surface S d is singular, and the quintuple (a 0 , a 1 , a 2 , a 3 , d) is not one of the four exceptions (1.2). Two of the four remaining cases (1.2) have been treated in [A02] by Araujo, who proved the following result:Theorem 1.4 ([A02, Theorem 4.1]). In the following two cases:• (a 0 , a 1 , a 2 , a 3 , d) = (1, 2, 3, 5, 10),• (a 0 , a 1 , a 2 , a 3 , d) = (1, 3, 5, 7, 15) and the equation of S d contains yzt, the inequality α(S d ) > 2 3 holds. In particular, S d admits an orbifold Kähler-Einstein metric. The remaining two cases of (1.2) have been dealt with in the paper [CPS10], which was published in Journal of Geometric Analysis in 2010. In this paper, we succeeded in estimating their α-invariants from below by large enough numbers for the criterion (1.3). To be precise, we proved the following result:Theorem 1.5 ([CPS10, Theorem 1.10]). Suppose that (a 0 , a 1 , a 2 , a 3 , d) = (1, 3, 5, 8, 16) or (2, 3, 5, 9, 18). Then α(S d ) > 2 3 . In particular, S d admits an orbifold Kähler-Einstein metric. In particular, if I = 1, then S d admits an orbifold Kähler-Einstein metric except possibly the case when (a 0 , a 1 , a 2 , a 3 , d) = (1, 3, 5, 7, 15) and the defining equation of the surface S d does not contain yzt. Note that in the latter case one has α(S d ) = 8 15 < 2 3 by [CPS10, Theorem 1.10], so that the criterion by the α-invariant could not be applied.Meanwhile, since 2010 we have witnessed dramatic developments in the study of the Yau-Tian-Donaldson conjecture concerning the existence of Kähler-Einstein metrics on Fano manifolds and stability. The challenge to the conjecture has been heightened by Chen, Donaldson, Sun and Tian who have completed the proof for the case of Fano manifolds with anticanonical polarisations [CDS15, T15]. Following this celebrated achievement, useful technologies have been introduced to determine whether given Fano varieties are Kähler-Einstein or not, via the theorem of Chen-Donaldson-Sun and Tian.Recently Fujita and Odaka introduced a new invariant of Fano varieties, which they called δ-invariant (for the definition, see [FO18, Definition 1.2]), that serves as a strong criterion for uniform K-stability (see [FO18]). Theorem 1.6 ([FO18, BJ17]). Let X be a Fano variety with at most Kawamata log terminal singularities. Then X is uniformly K-stable if and only if δ(X) > 1. This powerful tool has been practiced for del Pezzo surfaces in [PW18, CRZ18, CZ18]. Around...

Unstable singular del Pezzo hypersurfaces with lower index

“…Moreover the paper [7] expect that the similar phenomenon happens in surfaces with lower index. The present article is motivated by Conjecture 1.1 and answer it by following theorem.…”

“…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.…”

“…The remaining two cases have been dealt with in the paper [6] who shows the alpha invariant is large enough to admit orbifold Kähler-Einstein metrics except for possibly the case when (a 0 , a 1 , a 2 , a 3 , d) = (1,3,5,7,15) and the defining equation of the surface S d does not contain a monomial yzt. Finally, the paper [7] proves K-stability of remaining one case by estimating the delta invariant so that it has orbifold Kähler-Einstein metrics. This ends the existence of the orbifold Kähler-Einstein metric on the log del Pezzo hypersurfaces with index 1.…”

K-stability of log del Pezzo hypersurfaces with index 2

“…The only remaining case was when pa 0 , a 1 , a 2 , a 3 , dq " p1, 3, 5, 7, 15q and the equation of S d does not contain yzt. But this was solved in [9]. In this case, since αpS d q ă 2 3 , the α-invariant criterion could not be used.…”

“…Note that this approach works under the assumption that I ă 3a 0 {2 and pa 0 , a 1 , a 2 , a 3 , dq ‰ pI ´k, I `k, a, a `k, 2a `k `Iq, for any non-negative integer k ă I and any positive integer a ě I `k, because if I ě 3a 0 {2 or if I ă 3a 0 {2 and pa 0 , a 1 , a 2 , a 3 , dq " pI ´k, I `k, a, a `k, 2a `k `Iq, then αpS d q ď 2{3 ( [11,Corollary 1.15]) where αpS d q is the α-invariant of S d and hence cannot be used to prove the existence of Kähler-Einstein metrics. Recently, the existence of Kähler-Einstein metrics on few of the remaining cases has been shown using the δ-invariant in [9]. As a consequence, it was conjectured in [9,Conjecture 1.10], that for I " 2, all S d admit an orbifold Kähler-Einstein metric.…”