A sharp lower bound for the log canonical threshold

By J. Demailly, Hoàng Hiệp Phạm. 2012

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In this note, we prove a sharp lower bound for the log canonical threshold of a plurisubharmonic function $${\varphi}$$φ with an isolated singularity at 0 in an open subset of $${\mathbb{C}^n}$$Cn. This threshold is defined as the supremum of constants c > 0 such that $${e^{-2c\varphi}}$$e-2cφ is integrable on a neighborhood of 0. We relate $${c(\varphi)}$$c(φ) to the intermediate multiplicity numbers $${e_j(\varphi)}$$ej(φ), defined as the Lelong numbers of $${(dd^c\varphi)^j}$$(ddcφ)j at 0 (so that in particular $${e_0(\varphi)=1}$$e0(φ)=1). Our main result is that $${c(\varphi)\geqslant\sum_{j=0}^{n-1} e_j(\varphi)/e_{j+1}(\varphi)}$$c(φ)⩾∑j=0n-1ej(φ)/ej+1(φ). This inequality is shown to be sharp; it simultaneously improves the classical result $${c(\varphi)\geqslant 1/e_1(\varphi)}$$c(φ)⩾1/e1(φ) due to Skoda, as well as the lower estimate $${c(\varphi)\geqslant n/e_n(\varphi)^{1/n}}$$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.

Subfields: Applied Mathematics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/aab2a67643188f77afc80557bdb34aaefd7a2050