A sharp lower bound for the log canonical threshold

Jean-Pierre Demailly Université de Grenoble I, Département de Mathématiques, Institut Fourier, Saint-Martin d’Hères, France Hoàng Hiệp Phạm Department of Mathematics, Hanoi National University of Education

Analysis of PDEs mathscidoc:1701.03011

Acta Mathematica, 212, (1), 1-9, 2012.1
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.
Lelong number; Monge–Ampère operator; log canonical threshold
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@inproceedings{jean-pierre2012a,
  title={A sharp lower bound for the log canonical threshold},
  author={Jean-Pierre Demailly, and Hoàng Hiệp Phạm},
  url={http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20170108203402193544783},
  booktitle={Acta Mathematica},
  volume={212},
  number={1},
  pages={1-9},
  year={2012},
}
Jean-Pierre Demailly, and Hoàng Hiệp Phạm. A sharp lower bound for the log canonical threshold. 2012. Vol. 212. In Acta Mathematica. pp.1-9. http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20170108203402193544783.
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