Residue currents with prescribed annihilator ideals on singular varieties

TL;DR

This paper extends the construction of residue currents with prescribed annihilator ideals from smooth complex manifolds to singular varieties, broadening the applicability of duality theorems in complex analysis.

Contribution

It introduces a method to generalize residue current constructions to singular varieties, enabling the analysis of ideals in more complex geometric settings.

Findings

Constructed residue currents with prescribed annihilators on singular varieties.

Generalized duality theorem for Coleff-Herrera products to singular spaces.

Provided a framework for analyzing ideals on singular complex varieties.

Abstract

Given an ideal $\mathcal{J}$ on a complex manifold, Andersson and Wulcan constructed a current $R^\mathcal{J}$ such that the annihilator of $R^\mathcal{J}$ is $\mathcal{J}$, generalizing the duality theorem for Coleff-Herrera products. We describe a way to generalize this construction to ideals on singular varieties.