A sharp lower bound for the log canonical threshold

TL;DR

This paper establishes a precise lower bound for the log canonical threshold of plurisubharmonic functions with isolated singularities, linking it to intermediate multiplicity numbers, and improves upon classical bounds with sharp inequalities.

Contribution

It introduces a sharp lower bound for the log canonical threshold in terms of multiplicity numbers, refining previous estimates and applying reduction to the toric case for proof.

Findings

The bound c(φ) ≥ ∑ e_j(φ)/e_{j+1}(φ) is sharp.

Improves classical bounds c(φ) ≥ 1/e_1(φ) and c(φ) ≥ n/e_n(φ)^{1/n}.

Reduction to the toric case simplifies the proof.

Abstract

In this note, we prove a sharp lower bound for the log canonical threshold of a plurisubharmonic function $\varphi$ 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 $e^{-2c\varphi}$ is integrable on a neighborhood of $0$. We relate $c(\varphi)$ with the intermediate multiplicity numbers $e_j(\varphi)$, defined as the Lelong numbers of $(dd^c\varphi)^j$ at $0$ (so that in particular $e_0(\varphi)=1$). Our main result is that $c(\varphi)\ge\sum e_j(\varphi)/e_{j+1}(\varphi)$, $0\le j\le n-1$. This inequality is shown to be sharp; it simultaneously improves the classical result $c(\varphi)\ge 1/e_1(\varphi)$ due to Skoda, as well as the lower estimate $c(\varphi)\ge n/e_n(\varphi)^{1/n}$ 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.