Limits of log canonical thresholds

de Fernex, Tommaso; Mustață, Mircea

Annales scientifiques de l'École Normale Supérieure, Série 4, Tome 42 (2009) no. 3, pp. 491-515.

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.

Classification : 14B05, 03H05, 14E30
Keywords: log canonical threshold, multiplier ideals, ultrafilter, resolution of singularities
Mot clés : seuils log canoniques, idéaux multiples, ultra-filtres, résolution de singularités

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