Open Access
2014 A sharp lower bound for the log canonical threshold
Jean-Pierre Demailly, Hoàng Hiệp Phạm
Author Affiliations +
Acta Math. 212(1): 1-9 (2014). DOI: 10.1007/s11511-014-0107-4


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.


Download Citation

Jean-Pierre Demailly. Hoàng Hiệp Phạm. "A sharp lower bound for the log canonical threshold." Acta Math. 212 (1) 1 - 9, 2014.


Received: 20 January 2012; Published: 2014
First available in Project Euclid: 30 January 2017

zbMATH: 1298.14006
MathSciNet: MR3179606
Digital Object Identifier: 10.1007/s11511-014-0107-4

Primary: 14B05
Secondary: 32S05 , 32S10 , 32U25

Keywords: Lelong number , log canonical threshold , Monge–Ampère operator

Rights: 2014 © Institut Mittag-Leffler

Vol.212 • No. 1 • 2014
Back to Top