Multi-circled singularities, Lelong numbers, and 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.