Acta Mathematica

, Volume 212, Issue 1, pp 1–9 | Cite as

A sharp lower bound for the log canonical threshold

  • Jean-Pierre DemaillyEmail author
  • Hoàng Hiệp Phạm


In this note, we prove a sharp lower bound for the log canonical threshold of a plurisubharmonic function \({\varphi}\) with an isolated singularity at 0 in an open subset of \({\mathbb{C}^n}\). This threshold is defined as the supremum of constants c > 0 such that \({e^{-2c\varphi}}\) is integrable on a neighborhood of 0. We relate \({c(\varphi)}\) to the intermediate multiplicity numbers \({e_j(\varphi)}\), defined as the Lelong numbers of \({(dd^c\varphi)^j}\) at 0 (so that in particular \({e_0(\varphi)=1}\)). Our main result is that \({c(\varphi)\geqslant\sum_{j=0}^{n-1} e_j(\varphi)/e_{j+1}(\varphi)}\). This inequality is shown to be sharp; it simultaneously improves the classical result \({c(\varphi)\geqslant 1/e_1(\varphi)}\) due to Skoda, as well as the lower estimate \({c(\varphi)\geqslant n/e_n(\varphi)^{1/n}}\) which has received crucial applications to birational geometry in recent years. The proof consists in a reduction to the toric case, i.e. singularities arising from monomial ideals.

2000 Math. Subject Classification

14B05 32S05 32S10 32U25 


Lelong number Monge–Ampère operator log canonical threshold 


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Copyright information

© Institut Mittag-Leffler 2014

Authors and Affiliations

  1. 1.Université de Grenoble I, Département de Mathématiques, Institut FourierSaint-Martin d’HèresFrance
  2. 2.Department of MathematicsHanoi National University of EducationCau Giay, HanoiVietnam

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