2017
DOI: 10.1093/imrn/rnw310
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Alpha Invariants of Birationally Rigid Fano Three-folds

Abstract: Abstract. We compute global log canonical thresholds, or equivalently alpha invariants, of birationally rigid orbifold Fano threefolds embedded in weighted projective spaces as codimension two or three. As an important application, we prove that most of them are weakly exceptional, K-stable and admit Kähler-Einstien metric.

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Cited by 13 publications
(14 citation statements)
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“…In other words, both notions are tied to the singularities of certain anticanonical -divisors and so it is very natural to expect some relation between them. Indeed, the slope stability (a weaker notion of K-stability) of birationally superrigid Fano manifolds has been established by [OO13] under some mild assumptions, and it is conjectured [OO13, KOW18] that birationally rigid Fano varieties are always K-stable.…”
Section: Introductionmentioning
confidence: 99%
“…In other words, both notions are tied to the singularities of certain anticanonical -divisors and so it is very natural to expect some relation between them. Indeed, the slope stability (a weaker notion of K-stability) of birationally superrigid Fano manifolds has been established by [OO13] under some mild assumptions, and it is conjectured [OO13, KOW18] that birationally rigid Fano varieties are always K-stable.…”
Section: Introductionmentioning
confidence: 99%
“…The special importance of those papers is in that they connected some concepts of complex differential geometry with some objects of higher-dimensional birational geometry, which makes it possible to use the results of birational geometry to prove the existence of Kähler-Einstein metrics. That work was started in [1] and continued in [2][3][4][5][6]8,11,16,17]. Every time, a computation or estimate for the global log canonical threshold, obtained by the methods of birational geometry (the connectedness principle, inversion of adjunction, the technique of hypertangent divisors) yielded a proof of existence of Kähler-Einstein metrics for new classes of varieties.…”
Section: Historical Remarksmentioning
confidence: 99%
“…It is stronger than the K-polystability which is equivalent to the existence of Kähler-Einstein metric [6,7,8,32]. Birational superrigidity and K-stability are unexpectedtly related according to Odaka-Okada and Stibitz-Zhuang [23,30], and it is conjectured by Kim-Okada-Won [18] that every birationally superrigid Fano manifold is K-stable. Both the notions are intensively studied in the case of smooth Fano complete intersections of index 1: birational superrigidity by Iskovskih-Manin, Pukhlikov, Cheltsov, de Fernex-Ein-Mustat ¸ȃ, de Fernex, Suzuki, and Zhuang [17,24,25,26,28,29,2,12,9,10,31,34] (see also the note [20] written by Kollár), and K-stability by Fujita and Zhuang [14,34].…”
Section: Introductionmentioning
confidence: 99%