  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 $\mathcal{O}_Z$.
