Disclaimer: I am far from expert in the topics under discussion.
Let $A \subseteq V$ be a closed affine cone in a vector space, defined by polynomial equations.
In classical "projective duality", one takes the conormal bundle to $A_{reg}$ (the smooth points) inside $T^* V \cong V\times V^*$, takes the closure, and projects to $V^*$ to get the "projectively dual cone" $A' \subseteq V^*$. The same construction applied to $A'$ gives $A$ back.
Alternatively, one can start with the constant sheaf on $A_{reg}$, take its perverse extension on $V$, then the Fourier transform of that on $V^*$ (again an IC sheaf), and finally take the support of that.
Must the support of that Fourier transform be $A'$?
For the answer to be "no", the following must happen, as I understand it. The characteristic cycle of the perverse extension may be reducible (which is sad but at least to some extent understood; in particular in work of Tom Braden). One component $CA$ of it will be the closure of the conormal bundle, but there may be others, say $B$. But then we project to $V^*$, and find out that $\pi(B) \supset \pi(CA)$.
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