Analytic approximation of plurisubharmonic singularities

TL;DR

This paper investigates classes of isolated singularities of plurisubharmonic functions, providing analytic approximations with controlled Monge-Ampère masses, and links these classes to Green functions, valuations, and ideal families.

Contribution

It characterizes singularity classes via Green functions, valuations, and ideal families, advancing understanding of plurisubharmonic singularities and their analytic approximations.

Findings

Characterization of singularity classes using Green functions and valuations

Establishment of approximation methods with controlled residual Monge-Ampère masses

Connection between singularity classes and graded families of ideals

Abstract

We study several classes of isolated singularities of plurisubharmonic functions that can be approximated by analytic singularities with control over their residual Monge--Amp\`ere masses. They are characterized in terms of Green functions for Demailly's approximations, relative types, and valuations. Furthermore, the classes are shown to appear when studying graded families of ideals of analytic functions and the corresponding asymptotic multiplier ideals.