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 $φ$ 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 $e^{-2cφ}$ is integrable on a neighborhood of $0$. We relate $c(φ)$ with the intermediate multiplicity numbers $e_j(φ)$, defined as the Lelong numbers of $(dd^cφ)^j$ at $0$ (so that in particular $e_0(φ)=1$). Our main result is that $c(φ)\ge\sum e_j(φ)/e_{j+1}(φ)$, $0\le j\le n-1$. This inequality is shown to be sharp; it simultaneously improves the classical result $c(φ)\ge 1/e_1(φ)$ due to Skoda, as well as the lower estimate $c(φ)\ge n/e_n(φ)^{1/n}$ 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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