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By the Riemann-Hilbert correspondence, there is an equivalence between

(1) $\mathcal{D}\operatorname{-mod}(X)$ , the (derived) category of holonomic D-modules on a complex variety X, and

(2) $D^b_c(X)$ , the (derived) category of constructible sheaves on X.

There is a "naive" t-structure we can put on both categories. In $\mathcal{D}\operatorname{-mod}(X)$ , we can look at a t-structure whose heart $\mathcal{D}\operatorname{-mod}^\heartsuit$ is a complex (of D-modules) concentrated in degree 0. In $D^b_c(X)$ , we can look at the naive t-structure whose heart $D^{b \heartsuit}_c$ is a complex (of constructible sheaves) concentrated in degree 0.

It's known that if we transfer the naive t-structure on $\mathcal{D}\operatorname{-mod}(X)$ to $D^b_c(X)$ (using the equivalence above), $\mathcal{D}\operatorname{-mod}^\heartsuit$ is identified with "perverse sheaves" on X.

My question is:

If we map $D^{b\heartsuit}_c$ to the category of D-modules using the Riemann-Hilbert correspondence, what subcategory of $\mathcal{D}\operatorname{-mod}$ do we get? Does this have a well-known name?

More generally, is there some geometric/nice description of what the naive t-structure on $D^b_c$ becomes on $\mathcal{D}{\operatorname{-mod}}$ ?

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There must be a kind of tautological answer along the lines of: consider objects in the derived category of D-modules such that *-restriction to a subvariety lies in such and such degrees and the !-restriction to a subvariety lies in such and such degrees. (By tautological I mean for instance that it would apply to any perversity). Are you looking for something more non-trivial? –  t3suji Oct 6 '10 at 16:18
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I guess the naive t-structure viewed from the vantage of D-modules will be as perverse as the perverse t-structure for sheaves. In other words, I doubt there's a simple answer. You can describe it inductively by applying Theorem 1.4.10 of BBD (the tautological answer in the above sense). –  Donu Arapura Oct 6 '10 at 16:34
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up vote 9 down vote accepted

The t-structure is described in this paper of Kashiwara. It looks essentially like Donu and t3suji suggest in their comments: defined by conditions that look like middle-perversity support conditions.

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