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 (Formula presented.) with an isolated singularity at 0 in an open subset of (Formula presented.). This threshold is defined as the supremum of constants c > 0 such that (Formula presented.) is integrable on a neighborhood of 0. We relate (Formula presented.) to the intermediate multiplicity numbers (Formula presented.), defined as the Lelong numbers of (Formula presented.) at 0 (so that in particular (Formula presented.). Our main result is that (Formula presented.). This inequality is shown to be sharp; it simultaneously improves the classical result (Formula presented.) due to Skoda, as well as the lower estimate (Formula presented.) 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. © 2014 Institut Mittag-Leffler.