Shokurov's ACC conjecture for log canonical thresholds on smooth varieties

Fernex, Tommaso de; Ein, Lawrence; Mustaţă, Mircea

doi:10.1215/00127094-2010-008

Cited by 59 publications

(68 citation statements)

References 14 publications

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“…The latter, independently proven also by Ishii in [Ish11], extends to the setting considered here the main theorems of [EMY03,EM04,Kaw08,EM09], and has a number of consequences that were previously obtained in [Mus01,EM04,dFEM10] for normal, locally complete intersection varieties. These include the semi-continuity of minimal log J-discrepancies (see Corollaries 4.15 and 4.14, see also [Ish11]) and the fact that the set of all log J-canonical thresholds in any fixed dimension satisfies the ascending chain condition (see Corollary 4.13).…”

Section: Introductionsupporting

confidence: 67%

“…In short, it goes as follows. The same argument of the proof of Proposition 6.3 of [dFEM10] shows that if X is a reduced equidimensional scheme with log J-canonical singularities then dim T p X ≤ 2 dim X for every p ∈ X. Then, using this bound on the embedded dimension in combination with inversion of adjunction (Theorem 4.10) one deduces the above ACC property directly from the analogous property of mixed log canonical thresholds on smooth varieties, which is established in Theorem 6.1 of [dFEM10].…”

Section: Singularitiesmentioning

confidence: 72%

“…The proof of this property is a straightforward extensions of the corresponding proof given in [dFEM10] in the case of normal varieties with locally complete intersection singularities. In short, it goes as follows.…”

Section: Singularitiesmentioning

confidence: 80%

“…The problem arises from a conjecture of Shokurov for log canonical thresholds in the usual setting [Sho92], which has recently being solved for certain classes of singularities in [dFEM10,dFEM11]. In the framework considered in this paper we obtain the following unconditional result.…”

Section: Singularitiesmentioning

confidence: 93%

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Jacobian discrepancies and rational singularities

Fernex¹,

Docampo²

2013

J. Eur. Math. Soc.

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Abstract. Inspired by several works on jet schemes and motivic integration, we consider an extension to singular varieties of the classical definition of discrepancy for morphisms of smooth varieties. The resulting invariant, which we call Jacobian discrepancy, is closely related to the jet schemes and the Nash blow-up of the variety. This notion leads to a framework in which adjunction and inversion of adjunction hold in full generality, and several consequences are drawn from these properties. The main result of the paper is a formula measuring the gap between the dualizing sheaf and the Grauert-Riemenschneider canonical sheaf of a normal variety. As an application, we give characterizations for rational and Du Bois singularities on normal Cohen-Macaulay varieties in terms of Jacobian discrepancies. In the case when the canonical class of the variety is Q-Cartier, our result provides the necessary corrections for the converses to hold in theorems of Elkik, of Kovács, Schwede and Smith, and of Kollár and Kovács on rational and Du Bois singularities.

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Section: Introductionsupporting

confidence: 67%

Section: Singularitiesmentioning

confidence: 72%

Section: Singularitiesmentioning

confidence: 80%

Section: Singularitiesmentioning

confidence: 93%

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Jacobian discrepancies and rational singularities

Fernex¹,

Docampo²

2013

J. Eur. Math. Soc.

Self Cite

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“…Note that the set T n = T n ∩ (0, 1] consists precisely of the set of log canonical thresholds for divisors on smooth n-dimensional varieties. This set is known to satisfy ACC: this was a conjecture of Shokurov, proved in [dFEM10]. Question 6.11.…”

Section: Indeed It Follows From Theoremmentioning

confidence: 92%

Hodge Ideals for -Divisors, -Filtration, and Minimal Exponent

Mustaţă

Popa

2020

Forum of Mathematics, Sigma

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We compute the Hodge ideals of Q-divisors in terms of the V -filtration induced by a local defining equation, inspired by a result of Saito in the reduced case. We deduce basic properties of Hodge ideals in this generality, and relate them to Bernstein-Sato polynomials. As a consequence of our study we establish general properties of the minimal exponent, a refined version of the log canonical threshold, and bound it in terms of discrepancies on log resolutions, addressing a question of Lichtin and Kollár.

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Notes on Divisors Computing MLD’s and LCT’s

Blum

2018

Trends in Mathematics

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We show that if a divisor centered over a point on a smooth surface computes a minimal log discrepancy, then the divisor also computes a log canonical threshold. To prove the result, we study the asymptotic log canonical threshold of the graded sequence of ideals associated to a divisor over a variety. We systematically study this invariant and also prove a result describing which divisors compute asymptotic log canonical thresholds.

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Shokurov's ACC conjecture for log canonical thresholds on smooth varieties

Cited by 59 publications

References 14 publications

Jacobian discrepancies and rational singularities

Jacobian discrepancies and rational singularities

Hodge Ideals for -Divisors, -Filtration, and Minimal Exponent

Notes on Divisors Computing MLD’s and LCT’s

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