# A note on the singularities of residue currents of integrally closed ideals

## TL;DR

This paper investigates the singularities of residue currents associated with ideals of holomorphic functions, establishing a connection between the size of these singularities and the integrally closed property of Artinian monomial ideals.

## Contribution

It characterizes when the singularities of residue currents are small, specifically linking this to the integrally closed condition for Artinian monomial ideals.

## Key findings

- Residue currents coincide with Coleff-Herrera products for complete intersections.
- Small singularities of residue currents occur if and only if the ideal is integrally closed.
- The result applies specifically to Artinian monomial ideals.

## Abstract

Given a free resolution of an ideal $\mathfrak a$ of holomorpic functions there is an associated residue current $R$ that coincides with the classical Coleff-Herrera product if $\mathfrak a$ is a complete intersection ideal and whose annihilator ideal equals $\mathfrak a$. In the case when $\mathfrak a$ is an Artinian monomial ideal, we show that the singularities of $R$ are small in a certain sense if and only if $\mathfrak a$ is integrally closed.