On K-Polystability for Log Del Pezzo Pairs of Maeda Type
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On K-Polystability for Log Del Pezzo Pairs of Maeda Type Kento Fujita1 Received: 9 July 2019 / Revised: 3 November 2019 / Accepted: 29 November 2019 / © Institute of Mathematics, Vietnam Academy of Science and Technology (VAST) and Springer Nature Singapore Pte Ltd. 2020
Abstract We give an algebraic proof for which log del Pezzo pairs of Maeda type are K-polystable or not. If the base field is the complex number field, then the result is already known by Li and Sun. Keywords Fano varieties · K-Stability Mathematics Subject Classification (2010) 14J45 · 14L24
1 Introduction We work over an arbitrary algebraically closed field k with the characteristic zero. Let X be a Fano manifold, that is, X is a smooth projective variety over k such that the anti-canonical divisor −KX is ample. We are interested in the problem whether X is K-polystable or not. In fact, if k is equal to the complex number field C, then K-polystability of X is known to be equivalent to the existence of K¨ahler-Einstein metrics on X thanks to the works [3, 8– 10, 13, 38–40] and the references therein. It is natural to consider K-polystability for not only Fano manifolds but also log Fano pairs (X, Δ) (see Definition 1 (4)). However, in general, it is difficult to test K-polystability purely algebraically. Recently, Li, Wang, and Xu in [30, Theorem 1.4] gave a purely algebraic proof for which toric log Fano pairs are K-polystable or not. However, when a log Fano pair is not a toric pair, it is difficult to test K-polystability. See also Remark 6. In this article, we mainly consider K-polystability of log del Pezzo pairs, that is, log Fano pairs of dimension two. The purpose of this article is to give an algebraic proof for Kpolystability of the log del Pezzo pair (P2 , δC), where δ is a non-negative rational number with δ < 3/4 and C ⊂ P2 is a smooth conic, and the log del Pezzo pair (P1 × P1 , δC), where δ is a non-negative rational number with δ < 1/2 and C ⊂ P1 × P1 is the diagonal. Kento Fujita
[email protected] 1
Department of Mathematics, Graduate School of Science, Osaka University, Toyonaka, Osaka 560-0043, Japan
K. Fujita
Theorem 1 (cf. [29, Example 3.12]) (1) Assume that C ⊂ P2 is a smooth conic and let δ ∈ [0, 1) ∩ Q. Then, the log del Pezzo pair (P2 , δC) is K-polystable (resp. K-semistable) if and only if δ < 3/4 (resp. δ ≤ 3/4). (2) Assume that C ⊂ P1 × P1 is the diagonal and let δ ∈ (0, 1) ∩ Q. Then, the log del Pezzo pair (P1 × P1 , δC) is K-polystable (resp. K-semistable) if and only if δ < 1/2 (resp. δ ≤ 1/2). If k = C, then the above result is known by [29, Example 3.12] and [3, Theorem 4.8]. We emphasize that our proof is based on the work [18], purely algebraic, direct, and easy. Moreover, in Theorem 1 (1), we give a very easy and purely algebraic proof for K-polystability of P2 . For the proof of Theorem 1 (2), we use the fact that P1 × P1 is K-semistable. We can prove this fact purely algebraically (see [5, 7, 22, 27]). Remark 1 The referee pointed out to the author that the method in [29] can work for any
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