Analytical approximations and Monge-Amp\`{e}re masses of plurisubharmonic singularities

TL;DR

This paper constructs examples of plurisubharmonic functions with isolated singularities where the residual Monge-Ampère masses cannot be approximated by analytic methods, disproving a conjecture by Demailly.

Contribution

It provides counterexamples showing the limitations of analytic approximation of Monge-Ampère masses for certain singularities.

Findings

Residual Monge-Ampère masses defy approximation by multiplier ideal-based analytic methods.

Counterexamples challenge existing conjectures in pluripotential theory.

Highlights the complexity of singularity behavior in plurisubharmonic functions.

Abstract

We construct examples of plurisubharmonic functions with isolated singularities at $0\in \mathbb{C}^n$, whose residual Monge-Amp\`{e}re masses at the origin can not be approximated by masses of analytic approximations obtained via multiplier ideals. This answers negatively a conjecture of Demailly.