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 with an isolated singularity at 0 in an open subset of . This threshold is defined as the supremum of constants c > 0 such that is integrable on a neighborhood of 0. We relate to the intermediate multiplicity numbers , defined as the Lelong numbers of at 0 (so that in particular ). Our main result is that . This inequality is shown to be sharp; it simultaneously improves the classical result due to Skoda, as well as the lower estimate 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.