Alpha invariants of birationally bi-rigid Fano 3-folds I
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Alpha invariants of birationally bi-rigid Fano 3-folds I In-Kyun Kim1 · Takuzo Okada2 · Joonyeong Won3
Received: 17 April 2019 / Revised: 4 January 2020 / Accepted: 13 October 2020 © Springer Nature Switzerland AG 2020
Abstract We compute global log canonical thresholds of certain birationally bi-rigid Fano 3-folds embedded in weighted projective spaces as complete intersections of codimension 2 and prove that they admit an orbifold Kähler–Einstein metric and are K-stable. As an application, we give examples of super-rigid affine Fano 4-folds. Keywords K-stability · Log del Pezzo surface · Delta invariant Mathematics Subject Classification 14J45 · 53C25
1 Introduction Throughout the article, the ground field is assumed to be the field of complex numbers. The alpha invariant, which is also known as the global log canonical threshold, of a Fano variety is an invariant which measures singularities of pluri-anticanonical divi-
The first author was supported by the National Research Foundation of Korea (NRF-2020R1A2C4002510). The second author is partially supported by JSPS KAKENHI Grant Number JP18K03216. The third author was supported by the National Research Foundation of Korea (NRF-2020R1A2C1A01008018) and a KIAS Individual Grant (SP037003) via the Center for Mathematical Challenges at Korea Institute for Advanced Study.
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Joonyeong Won [email protected] In-Kyun Kim [email protected] Takuzo Okada [email protected]
1
Department of mathematics, Yonsei University, Seoul 03722, Republic of Korea
2
Department of Mathematics, Faculty of Science and Engineering, Saga University, Saga 840-8502, Japan
3
Center for Mathematical Challenges, Korea Institute for Advanced Study, Seoul 02455, Republic of Korea
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sors. The most important application of this invariant is as follows: if a Fano variety X with only quotient singularities satisfies lct(X ) > dim X /(dim X + 1), where lct(X ) denotes the global log canonical threshold of X (see Definition 2.1), then X admits an orbifold Kähler–Einstein metric (see [11,18,30]) and is K-stable (see [20]). We recall known results for Fano 3-folds which are complete intersections in weighted projective spaces. In this paper, a weighted hypersurface in P(a0 , . . . , a4 ) defined by an equation of d that is quasi-smooth, well formed, has only terdegree 4 ai − d = 1 is simply called a Fano 3-fold weighted minal singularities, and i=0 hypersurface of index 1. It is known that they form 95 families (see [7,13]) and their global log canonical thresholds have been computed in [1–3,6]. Theorem 1.1 ([2, Corollary 1.45]) Let X be a general Fano 3-fold weighted hypersurface of index 1 (i.e., a general member of one of the 95 families). Then X admits an orbifold Kähler–Einsten metric and is K-stable. We note that it is proved in [5,8] that a Fano 3-fold weighted hypersurface of index 1 is birationally rigid, that is, it has a unique (up to isomorphism) structure of a Mori fiber space in its birational equivalence class. A codimension 2 weighted complete interse
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