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 φ with an isolated singularity at 0 in an open subset of Cn. 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) to the intermediate multiplicity numbers ej(φ), defined as the Lelong numbers of (ddcφ)j at 0 (so that in particular e0(φ)=1). Our main result is that c(φ)⩾∑j=0n-1ej(φ)/ej+1(φ). This inequality is shown to be sharp; it simultaneously improves the classical result c(φ)⩾1/e1(φ) due to Skoda, as well as the lower estimate c(φ)⩾n/en(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.