Limits of log canonical thresholds
Annales scientifiques de l'École Normale Supérieure, Serie 4, Volume 42 (2009) no. 3, p. 491-515

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.

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.

DOI : https://doi.org/10.24033/asens.2100
Classification:  14B05,  03H05,  14E30
Keywords: log canonical threshold, multiplier ideals, ultrafilter, resolution of singularities
@article{ASENS_2009_4_42_3_491_0,
     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},
     publisher = {Soci\'et\'e math\'ematique de France},
     volume = {Ser. 4, 42},
     number = {3},
     year = {2009},
     pages = {491-515},
     doi = {10.24033/asens.2100},
     zbl = {1186.14007},
     mrnumber = {2543330},
     language = {en},
     url = {http://www.numdam.org/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, Serie 4, Volume 42 (2009) no. 3, pp. 491-515. doi : 10.24033/asens.2100. http://www.numdam.org/item/ASENS_2009_4_42_3_491_0/

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