Multi-circled singularities, Lelong numbers, and integrability index

TL;DR

This paper develops formulas for higher Lelong numbers and integrability index of multi-circled plurisubharmonic singularities, extending existing results and providing new characterizations and inequalities for these complex singularities.

Contribution

It extends Howald's multiplier ideal results to multi-circled singularities and provides elementary proofs of relations between Lelong numbers and integrability index.

Findings

Formulas for higher Lelong numbers and integrability index.

Extension of multiplier ideal results to multi-circled singularities.

Characterization of singularities with minimal integrability index.

Abstract

By comparing Green functions of multi-circled plurisubharmonic singularities in the n-domensional complex space to their indicators, we obtain formulas for the higher Lelong numbers and integrability index for such singularities and extend Howald's result on multiplier ideals for monomial ideals to multi-circled singularities. This also leads to an elementary proof of the relations between the k-th Lelong numbers and the integrability index. For k=1 and arbitrary plurisubharmonic functions the inequality is due to Skoda, and for k=n and any plurisubharmonic function with isolated singularity the relation is due to Demailly. We also describe multi-circled functions for which the inequalities are equalities. By a reduction to Demailly's inequality we prove these inequalities in the general case of plurisubharmonic functions as well. In addition, we get a description of all plurisubharmonic singularities whose integrability index is given by the lower bound in Skoda's inequality.