A sharp lower bound for the log canonical threshold

Abstract

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 $c(\varphi)$ with 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 $c(\varphi)\ge\sum e_j(\varphi)/e_{j+1}(\varphi)$, $0\le j\le n-1$. This inequality is shown to be sharp; it simultaneously improves the classical result $c(\varphi)\ge 1/e_1(\varphi)$ due to Skoda, as well as the lower estimate $c(\varphi)\ge n/e_n(\varphi)^{1/n}$ 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.