A valuative criterion for uniform K-stability of ℚ-Fano varieties

Fujita, Kento

doi:10.1515/crelle-2016-0055

Cited by 168 publications

(242 citation statements)

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“…Remark 3.10. The invariants S(v) and T (v) for v divisorial have been explored by K. Fujita [Fuj16b], C. Li [Li15b], and Y. Liu [Liu16]. In the notation of [Fuj16b],…”

Section: Global Invariantsmentioning

confidence: 99%

Thresholds, valuations, and K-stability

Blum

Jönsson

2020

Advances in Mathematics

118

181

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Let X be a normal complex projective variety with at worst klt singularities, and L a big line bundle on X. We use valuations to study the log canonical threshold of L, as well as another invariant, the stability threshold. The latter generalizes a notion by Fujita and Odaka, and can be used to characterize when a Q-Fano variety is K-semistable or uniformly K-stable. It can also be used to generalize volume bounds due to Fujita and Liu. The two thresholds can be written as infima of certain functionals on the space of valuations on X. When L is ample, we prove that these infima are attained. In the toric case, toric valuations acheive these infima, and we obtain simple expressions for the two thresholds in terms of the moment polytope of L.arXiv:1706.04548v1 [math.AG] 14 Jun 2017for all positive integers p, q. This, together with the Noetherianity of X, implies that {J (X, (c/p) · a p )} p∈N has a unique maximal element that is called the c-th asymptotic multiplier ideal and denoted by J (X, c · a • ). Note that J (X, c · a • ) = J (X, (c/p) · a p ) for all p divisible enough.Asymptotic multiplier ideals also satisfy a subadditivity property. See [Laz04, Theorem 11.2.3] for the case when X is smooth.Corollary 5.6. Let a • be a graded sequence of ideals on X. If m ∈ N * and c ∈ Q * + , thenNext we give a containment relation for the multiplier ideal associated to the graded sequence of valuation ideals. The result appears in [ELS03] in the case when v is divisorial.Proof. It is an immediate consequence of the valuative criterion for membership in the multiplier ideal [BdFFU15, Theorem1.2] that J (X, c · a • (v)) ⊂ a cv(a•(v))−A(v) (v).Since v(a • (v)) = 1 (see [Blu16b, Lemma 3.5]), the proof is complete.

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“…Remark 3.10. The invariants S(v) and T (v) for v divisorial have been explored by K. Fujita [Fuj16b], C. Li [Li15b], and Y. Liu [Liu16]. In the notation of [Fuj16b],…”

Section: Global Invariantsmentioning

confidence: 99%

Thresholds, valuations, and K-stability

Blum

Jönsson

2020

Advances in Mathematics

118

181

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“…Definition 2.4 ( [Fuj16]). Let X be a Q-Fano variety of dimension n, and F a prime divisor over X corresponding to a projective birational morphism σ : Y → X.…”

Section: K-stabilitymentioning

confidence: 99%

Higher codimensional alpha invariants and characterization of projective spaces

Zhu

2019

Int. J. Math.

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We generalize the definition of alpha invariant to arbitrary codimension. We also give a lower bound of these alpha invariants for K-semistable Q-Fano varieties and show that we can characterize projective spaces among all K-semistable Fano manifolds in terms of higher codimensional alpha invariants. Our results demonstrate the relation between alpha invariants of any codimension and volumes of Fano manifolds in the characterization of projective spaces.

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“…Definition 2.3 ( [Fuj16b]). Let X be a Q-Fano variety of dimension n. Take any projective birational morphism σ : Y → X with Y normal and any prime divisor F on Y , that is, F is a prime divisor over X.…”

Section: Preliminariesmentioning

confidence: 99%