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Let $j:U\to S$ be an (open) immesrion; let $P_U$ be a perverse (\'etale, though my question makes sense in the topological setting also) sheaf on $U$. Then I would like to say that the intermediate extension $j_{!*}P$ is 'based at' $U$. Note here that for any closed immersion $i:Z\to S$, $Z$ is disjoint from $U$, the zeroth perverse cohomology of both $Ri^*P_U$ and $Ri^!P_U$ is $0$.

In particular, if $P$ is a constant sheaf on a (connected smooth) variety $U$, I would like to say that $P$ is supported on any dense subvariety of $U$ (or at the generic point of $U$). Did anybody introduce something like this notion of 'base' for a perverse sheaf? The discussion Is there a 'classical' definition for the support of a perverse sheaves. made me think that the word 'support' is not appropriate here.

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This seems to me a strange convention, trying to call U some sort of support. As YBL pointed out on your previous comment, you may decide that rather than taking the usual notion of closed support to assign its generic point (when the support is irreducible) --- which is just another (nonstandard) way to say the same thing, but I don't see any way in which you can pick out the open U itself. e.g. suppose the closure of U happens to be smooth. Then the intermediate extension of the constant sheaf is just the constant sheaf on the closure, which certainly doesn't care which dense open you started from (so doesn't recover U).

If the closure is interestingly singular, you can detect a meaningful open using characteristic varieties --- ie you want a maximal open on which the restriction of your perverse sheaf is actually a local system.

This of course brings to mind the important point that a perverse sheaf (in the topological setting) has a more refined and interesting invariant than its support, namely its microsupport (or characteristic variety), which is the conical Lagrangian subset of the cotangent bundle comprising all codirections in which your perverse sheaf fails to be locally constant. This again allows you to pick our your open in interesting cases --- you look at the complement in the support of the closed subset over which there are nonzero characteristic directions...

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Dear David, actually I don't want to recover $U$; I want to say that $P$ is supported at $U$ if it could be presented as $j_{!*}P_U$. In the 'constant-smooth' case I would prefer to say that $P$ is 'based' at the generic point of $U$. –  Mikhail Bondarko Feb 4 '11 at 19:57
perhaps a better terminology would be "attached to" or "associated to U"? eg when classifying simple perverse sheaves one thinks of them as attached to irreducible local systems on various strata via intermediate extension (see eg Beilinson-Bernstein "Proof of Jantzen Conjectures" in the representation theoretic context, their "Langlands classification" of representations).. –  David Ben-Zvi Feb 4 '11 at 20:18
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