Multi-circled Singularities, Lelong Numbers, and Integrability Index

Abstract

By comparing Green functions of multi-circled plurisubharmonic singularities u in ℂ n to their indicators, we prove formulas for higher Lelong numbers L k (u) and integrability index λ u (the latter one being due to Kiselman) 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 λ u ≤k -1 L k (u)1/k , 1≤k≤l, for the multi-circled singularities, where l is the codimension of the set u -1(-∞). For k=1 and arbitrary plurisubharmonic function u the inequality is due to Skoda, and for k=n and any plurisubharmonic u with isolated singularity the relation is due to Demailly. We also describe all multi-circled functions for which the inequalities are equalities. We also prove these inequalities, by a reduction to Demailly’s result, in the general case of (not necessarily multi-circled) plurisubharmonic functions. In addition, we get a description of all plurisubharmonic singularities u whose integrability index is given by the lower bound in Skoda’s inequality, i.e., λ u =n -1 L 1(u).