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.

[1] V. Alexeev, Two two-dimensional terminations, Duke Math. J. 69 (1993), 527-545. | MR | Zbl

[2] V. V. Batyrev, Stringy Hodge numbers of varieties with Gorenstein canonical singularities, in Integrable systems and algebraic geometry (Kobe/Kyoto, 1997), World Sci. Publ., River Edge, NJ, 1998, 1-32. | MR | Zbl

[3] C. Birkar, Ascending chain condition for log canonical thresholds and termination of log flips, Duke Math. J. 136 (2007), 173-180. | MR | Zbl

[4] C. Birkar, P. Cascini, C. D. Hacon & J. Mckernan, Existence of minimal models for varieties of general type, preprint arXiv:math.AG/0610203. | Zbl

[5] M. Blickle, M. Mustață & K. E. Smith, $F$-thresholds of hypersurfaces, to appear in Trans. Amer. Math. Soc., available at arXiv:math/0705.1210. | MR | Zbl

[6] L. Van Den Dries & K. Schmidt, Bounds in the theory of polynomial rings over fields. A nonstandard approach, Invent. Math. 76 (1984), 77-91. | EuDML | MR | Zbl

[7] L. Ein, R. Lazarsfeld & M. Mustațǎ, Contact loci in arc spaces, Compos. Math. 140 (2004), 1229-1244. | MR | Zbl

[8] L. Ein, R. Lazarsfeld, K. E. Smith & D. Varolin, Jumping coefficients of multiplier ideals, Duke Math. J. 123 (2004), 469-506. | MR | Zbl

[9] L. Ein & M. Mustaţă, Jet schemes and singularities, to appear in Proceedings of the 2005 AMS Summer Research Institute in Algebraic Geometry, available at arXiv:math.AG/0612862. | MR | Zbl

[10] R. Goldblatt, Lectures on the hyperreals. An introduction to nonstandard analysis, Graduate Texts in Math. 188, Springer, 1998. | MR | Zbl

[11] J. Kollár, Singularities of pairs, in Algebraic geometry - Santa Cruz 1995, Proc. Sympos. Pure Math. 62, Amer. Math. Soc., 1997, 221-287. | Zbl

[12] J. Kollár et al., Flips and abundance for algebraic threefolds. A summer seminar at the University of Utah (Salt Lake City, 1991), Astérisque 211 (1992), 1-258. | Numdam | Zbl

[13] R. Lazarsfeld, Positivity in algebraic geometry. II, Ergebnisse Math. Grenzg. 49, Springer, 2004. | Zbl

[14] H. Matsumura, Commutative ring theory, second éd., Cambridge Studies in Advanced Mathematics 8, Cambridge University Press, 1989. | Zbl

[15] J. Mckernan & Y. Prokhorov, Threefold thresholds, Manuscripta Math. 114 (2004), 281-304. | Zbl

[16] M. Mustaţǎ, Singularities of pairs via jet schemes, J. Amer. Math. Soc. 15 (2002), 599-615. | Zbl

[17] H. Schoutens, Bounds in cohomology, Israel J. Math. 116 (2000), 125-169. | Zbl

[18] V. V. Shokurov, Three-dimensional log perestroikas, Izv. Ross. Akad. Nauk Ser. Mat. 56 (1992), 105-203.

[19] M. Temkin, Desingularization of quasi-excellent schemes in characteristic zero, Adv. Math. 219 (2008), 488-522. | Zbl