On the duality theorem on an analytic variety

TL;DR

This paper investigates the conditions under which the duality theorem for Coleff-Herrera products extends from complex manifolds to analytic varieties, focusing on intersection properties with singularity subvarieties.

Contribution

It provides necessary and sufficient conditions related to zero set intersections for the duality theorem to hold on analytic varieties.

Findings

Duality theorem holds under specific intersection conditions.

Necessary and sufficient criteria are identified.

Results extend understanding of Coleff-Herrera products on singular spaces.

Abstract

The duality theorem for Coleff-Herrera products on a complex manifold says that if $f = (f_1,\dots,f_p)$ defines a complete intersection, then the annihilator of the Coleff-Herrera product $\mu^f$ equals (locally) the ideal generated by $f$. This does not hold unrestrictedly on an analytic variety $Z$. We give necessary, and in many cases sufficient conditions for when the duality theorem holds. These conditions are related to how the zero set of $f$ intersects certain singularity subvarieties of the sheaf $\mathcal{O}_Z$.