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Enhanced perversities. (English) Zbl 1423.32015
The Riemann-Hilbert correspondence of the authors [Publ. Math., Inst. Hautes Étud. Sci. 123, 69–197 (2016; Zbl 1351.32017)] concerns with two categories:
(1) the triangulated category \(\mathscr{D}\) of complexes of \(\mathcal D\)-modules with holonomic (but not necessarily regular) cohomology sheaves on a complex manifold \(M\), and
(2) the triangulated category \(\mathscr{E}\) of \(\mathbb{R}\)-constructible enhanced ind-sheaves on \(M\).
The correspondence [loc. cit] says that there is an enhanced de Rham functor \[ \mathcal{DR}: \mathscr{D} \to \mathscr{E} \] which is full and faithful.
The category \(\mathscr{D}\) has its standard t-structure, and its heart is the category of holonomic \(\mathcal D\)-modules. However, the category \(\mathscr{E}\) does not have a perverse t-structure that corresponds to the one in the classical Riemann-Hilbert correspondence.
The notion of a “slicing” [T. Bridgeland, Ann. Math. (2) 166, No. 2, 317–345 (2007; Zbl 1137.18008)] manifestly generalizes the notion of t-structures by allowing a non-integral index set. In the paper under review, for each real-valued perversity function the authors put a generalized t-structure on the category \(\mathscr{E}\). In the generalized situation, the middle perverse t-structure always exists and its index set is \(\frac{1}{2}\mathbb{Z}\). It is proved in the paper under review, that if the middle perversity is used, then the functor \(\mathcal{DR}\) will be “exact”, in the sense that for any real number \(c\), \(\mathcal{DR}(\mathscr{D}^{\leq c}) \subset \mathscr{E}^{\leq c} \) and \( \mathcal{DR}(\mathscr{D}^{\geq c}) \subset \mathscr{E}^{\geq c}\).

MSC:
32C38 Sheaves of differential operators and their modules, \(D\)-modules
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