Limits of log canonical thresholds

Tommaso de Fernex; Mircea Mustață

Annales scientifiques de l'École Normale Supérieure (2009)

  • Volume: 42, Issue: 3, page 491-515
  • ISSN: 0012-9593

Abstract

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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.

How to cite

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de Fernex, Tommaso, and Mustață, Mircea. "Limits of log canonical thresholds." Annales scientifiques de l'École Normale Supérieure 42.3 (2009): 491-515. <http://eudml.org/doc/272178>.

@article{deFernex2009,
abstract = {Let $\mathcal \{T\}_n$ 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 $\mathcal \{T\}_n$ lies in $\mathcal \{T\}_\{n-1\}$, proving in this setting a conjecture of Kollár. We also show that $\mathcal \{T\}_n$ is closed in $\mathbf \{R\}$; 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 $\mathcal \{T\}_n$, it is enough to show that $1$ is not a point of accumulation from below of any $\mathcal \{T\}_n$. In a different direction, we interpret the ACC Conjecture as a semi-continuity property for log canonical thresholds of formal power series.},
author = {de Fernex, Tommaso, Mustață, Mircea},
journal = {Annales scientifiques de l'École Normale Supérieure},
keywords = {log canonical threshold; multiplier ideals; ultrafilter; resolution of singularities},
language = {eng},
number = {3},
pages = {491-515},
publisher = {Société mathématique de France},
title = {Limits of log canonical thresholds},
url = {http://eudml.org/doc/272178},
volume = {42},
year = {2009},
}

TY - JOUR
AU - de Fernex, Tommaso
AU - Mustață, Mircea
TI - Limits of log canonical thresholds
JO - Annales scientifiques de l'École Normale Supérieure
PY - 2009
PB - Société mathématique de France
VL - 42
IS - 3
SP - 491
EP - 515
AB - Let $\mathcal {T}_n$ 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 $\mathcal {T}_n$ lies in $\mathcal {T}_{n-1}$, proving in this setting a conjecture of Kollár. We also show that $\mathcal {T}_n$ is closed in $\mathbf {R}$; 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 $\mathcal {T}_n$, it is enough to show that $1$ is not a point of accumulation from below of any $\mathcal {T}_n$. In a different direction, we interpret the ACC Conjecture as a semi-continuity property for log canonical thresholds of formal power series.
LA - eng
KW - log canonical threshold; multiplier ideals; ultrafilter; resolution of singularities
UR - http://eudml.org/doc/272178
ER -

References

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