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