Acta Mathematica

A sharp lower bound for the log canonical threshold

Jean-Pierre Demailly and Hoàng Hiệp Phạm

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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.

Article information

Source
Acta Math., Volume 212, Number 1 (2014), 1-9.

Dates
Received: 20 January 2012
First available in Project Euclid: 30 January 2017

Permanent link to this document
https://projecteuclid.org/euclid.acta/1485801724

Digital Object Identifier
doi:10.1007/s11511-014-0107-4

Mathematical Reviews number (MathSciNet)
MR3179606

Zentralblatt MATH identifier
1298.14006

Subjects
Primary: 14B05: Singularities [See also 14E15, 14H20, 14J17, 32Sxx, 58Kxx]
Secondary: 32S05: Local singularities [See also 14J17] 32S10: Invariants of analytic local rings 32U25: Lelong numbers

Keywords
Lelong number Monge–Ampère operator log canonical threshold

Rights
2014 © Institut Mittag-Leffler

Citation

Demailly, Jean-Pierre; Phạm, Hoàng Hiệp. A sharp lower bound for the log canonical threshold. Acta Math. 212 (2014), no. 1, 1--9. doi:10.1007/s11511-014-0107-4. https://projecteuclid.org/euclid.acta/1485801724


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