On the K-stability of Fano varieties and anticanonical divisors

Fujita, Kento; Odaka, Yuji

doi:10.2748/tmj/1546570823

Cited by 152 publications

(188 citation statements)

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“…Next, is the δ-invariant of (X, ∆), which, as defined in [FO18], measures log canonical thresholds of a certain classes of anti-log canonical divisors of (X, ∆). It is shown in [BJ17] that…”

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confidence: 99%

Uniqueness of K-polystable degenerations of Fano varieties

Blum

2019

Ann. of Math. (2)

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We prove that K-polystable degenerations of Q-Fano varieties are unique. Furthermore, we show that the moduli stack of K-stable Q-Fano varieties is separated. Together with recently proven boundedness and openness statements, the latter result yields a separated Deligne-Mumford stack parametrizing all uniformly K-stable Q-Fano varieties of fixed dimension and volume. The result also implies that the automorphism group of a K-stable Q-Fano variety is finite.

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confidence: 99%

Uniqueness of K-polystable degenerations of Fano varieties

Blum

2019

Ann. of Math. (2)

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“…• Fujita and Odaka [43] defined a numerical invariant δ(X) of a Fano manifold and conjectured that uniform K-stability is equivalent to δ(X) > 1, which was later confirmed by Blum and Jonsson [5]. This invariant δ is obtained from the theory of the log canonical threshold (lct) (which in this context is a numerical invariant measuring how singular a divisor is).…”

Section: Fano Manifolds and Kähler-einstein Metricsmentioning

confidence: 94%

Some Recent Developments in Kähler Geometry and Exceptional Holonomy

Donaldson

2019

Proceedings of the International Congress of Mathematicians (ICM 2018)

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“…of KE metrics. Recently, this has been remedied by Fujita-Odaka [38] that introduced a new invariant, that we refer to as the basis log canonical threshold (blct) reminiscent of the global log canonical threshold (the blct has also been referred to as the delta invariant and the stability threshold in the literature, see Definition 2.5 below for detailed references). The advantage of estimating this modified threshold is that it provides a necessary and sufficient condition for K-stability, which in turn is equivalent to the existence of KE metrics [27,62].…”

Section: Estimating Basis Log Canonical Thresholdsmentioning

confidence: 99%

Basis log canonical thresholds, local intersection estimates, and asymptotically log del Pezzo surfaces

Cheltsov

Rubinstein

Zhang

2019

Sel. Math. New Ser.

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The purpose of this article is to develop techniques for estimating basis log canonical thresholds on logarithmic surfaces. To that end, we develop new local intersection estimates that imply log canonicity. Our main motivation and application is to show the existence of Kähler-Einstein edge metrics on all but finitely many families of asymptotically log del Pezzo surfaces, partially confirming a conjecture of two of us. In an appendix we show that the basis log canonical threshold of Fujita-Odaka coincides with the greatest lower Ricci bound invariant of Tian.is also the necessary and sufficient condition for (b) if µ ≤ 0 [40, Theorem 2]; moreover, a classification (i.e., part (a)) is essentially impossible when µ ≤ 0 [32], [54, §8], and so we will restrict our attention in (a)-(b) exclusively to the case µ > 0, that we have previously called the asymptotically log Fano regime [21, Definition 1.1].Our previous work accomplished (a) in dimension 2, providing a complete classification [21, Theorem 2.1]. Furthermore, we also obtained the "necessary" portion of (b) [21,22], and this was extended to higher dimensions by Fujita [34]. The purpose of this article is to complete the "sufficient" portion of (b) in dimension 2 in all but finitely many (in fact, all but 6) of the (infinite list of) cases classified in [21]. The Calabi problem for asymptotically log Fano varietiesA special class of asymptotically log Fano varieties is as follows. This is a special case of [21, Definition 1.1].Definition 1.1. We say that a pair (X, D) consisting of a smooth projective variety X and a smooth irreducible divisor D on X is asymptotically log Fano if the divisor −K X − (1 − β)D is ample for sufficiently small β ∈ (0, 1].This definition contains the class of smooth Fano varieties (D = 0) as well as the classical notion of a smooth log Fano pair due to Maeda (β = 0) [48].One can show using a result of Kawamata-Shokurov that if (X, D) is asymptotically log Fano then |k(K X + D)| (for some k ∈ N) is free from base points and gives a morphism [21, §1] η : X → Z.The following conjecture, posed in our earlier work, gives a rather complete picture concerning (b) when D is smooth. Conjecture 1.2. [21, Conjecture 1.11] Suppose that (X, D) is an asymptotically log Fano manifold with D smooth and irreducible. There exist KEE metrics with angle 2πβ along D for all sufficiently small β if and only if η is not birational.This conjecture stipulates that the existence problem for KEE metrics in the small angle regime boils down to a simple birationality criterion. In fact, this amounts to computing a single intersection number, i.e., checking whether (K X + D) n = 0.This would be a rather far-reaching simplification as compared to checking the much harder condition of log K-stability that involves, in theory, computing the Futaki invariant of an infinite number of log test configurations, or else estimating the blct which involves, in theory, estimation of singularities of pairs that may occur after an unbounded number of blow-ups.3

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On the K-stability of Fano varieties and anticanonical divisors

Cited by 152 publications

References 40 publications

Uniqueness of K-polystable degenerations of Fano varieties

Uniqueness of K-polystable degenerations of Fano varieties

Some Recent Developments in Kähler Geometry and Exceptional Holonomy

Basis log canonical thresholds, local intersection estimates, and asymptotically log del Pezzo surfaces

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