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A sharp lower bound for the log canonical threshold


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.

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Correspondence to Jean-Pierre Demailly.

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Demailly, JP., Phạm, H.H. A sharp lower bound for the log canonical threshold. Acta Math 212, 1–9 (2014).

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2000 Math. Subject Classification

  • 14B05
  • 32S05
  • 32S10
  • 32U25


  • Lelong number
  • Monge–Ampère operator
  • log canonical threshold