Limits of log canonical thresholds

de Fernex, Tommaso; Mustață, Mircea

Limits of log canonical thresholds
[Limites de seuils log canoniques]

Annales scientifiques de l'École Normale Supérieure, Tome 42 (2009), p. 491-515 / Harvested from Numdam

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Résumé
Abstract

Dans cet article, nous analysons les ensembles $𝒯_{n}$ de seuils log canoniques de paires $(X, Y)$ , où $X$ est une variété lisse de dimension $n$ , et $Y$ est un sous-schéma fermé non-vide de $X$ . En employant des méthodes non-standard, nous montrons que chaque limite d’une suite strictement décroissante de $𝒯_{n}$ appartient à l’ensemble $𝒯_{n - 1}$ (ce résultat a été conjecturé par J. Kollár dans ses travaux sur le sujet). Nous montrons également que l’ensemble $𝒯_{n}$ est fermé dans $𝐑$ , et en déduisons que les valeurs adhérentes de l’ensemble des seuils log canoniques des pairs $(X, Y)$ sont rationnelles, si la dimension de $X$ est majorée. Une autre conséquence de nos résultats concerne la conjecture ACC de Shokurov pour les $𝒯_{n}$ . En effet, nous montrons qu’elle est une conséquence de l’énoncé suivant : pour tout $n$ , la valeur $1$ ne peut pas être obtenue comme limite d’une suite strictement croissante de nombres contenus dans $𝒯_{n}$ . Dans une autre perspective, nous interprétons la conjecture ACC comme une propriété de semi-continuité de seuils log canoniqes des séries formelles.

Let $𝒯_{n}$ denote the set of log canonical thresholds of pairs $(X, Y)$ , with $X$ a nonsingular variety of dimension $n$ , and $Y$ a nonempty closed subscheme of $X$ . Using non-standard methods, we show that every limit of a decreasing sequence in $𝒯_{n}$ lies in $𝒯_{n - 1}$ , proving in this setting a conjecture of Kollár. We also show that $𝒯_{n}$ is closed in $𝐑$ ; in particular, every limit of log canonical thresholds on smooth varieties of fixed dimension is a rational number. As a consequence of this property, we see that in order to check Shokurov’s ACC Conjecture for all $𝒯_{n}$ , it is enough to show that $1$ is not a point of accumulation from below of any $𝒯_{n}$ . In a different direction, we interpret the ACC Conjecture as a semi-continuity property for log canonical thresholds of formal power series.

Publié le : 2009-01-01

MR 2543330 | Zbl 1186.14007 | 1 citation dans Numdam

DOI : https://doi.org/10.24033/asens.2100
Classification: 14B05, 03H05, 14E30
Mots clés: seuils log canoniques, idéaux multiples, ultra-filtres, résolution de singularités

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     author = {de Fernex, Tommaso and Musta\c t\u a, Mircea},
     title = {Limits of log canonical thresholds},
     journal = {Annales scientifiques de l'\'Ecole Normale Sup\'erieure},
     volume = {42},
     year = {2009},
     pages = {491-515},
     doi = {10.24033/asens.2100},
     mrnumber = {2543330},
     zbl = {1186.14007},
     language = {en},
     url = {http://dml.mathdoc.fr/item/ASENS_2009_4_42_3_491_0}
}

de Fernex, Tommaso; Mustață, Mircea. Limits of log canonical thresholds. Annales scientifiques de l'École Normale Supérieure, Tome 42 (2009) pp. 491-515. doi : 10.24033/asens.2100. http://gdmltest.u-ga.fr/item/ASENS_2009_4_42_3_491_0/

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