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 $\phi$ 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 $e^{-2c\phi }$ is integrable on a neighborhood of $0$. We relate $c(\phi )$ with the intermediate multiplicity numbers $e_j(\phi )$, defined as the Lelong numbers of $(d{d}^c\phi {)}^j$ at $0$ (so that in particular $e_0(\phi )=1$). Our main result is that $c(\phi )\ge \sum e_j(\phi )\mathrm{/}{e}_{j+1}(\phi )$, $0\le j\le n-1$. This inequality is shown to be sharp; it simultaneously improves the classical result $c(\phi )\ge 1\mathrm{/}{e}_1(\phi )$ due to Skoda, as well as the lower estimate $c(\phi )\ge n\mathrm{/}{e}_n(\phi {)}^{1\mathrm{/}n}$ 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.