zbMATH — the first resource for mathematics

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 \( \mathcal{DR}(\mathscr{D}^{\geq c}) \subset \mathscr{E}^{\geq c}\).

32C38 Sheaves of differential operators and their modules, \(D\)-modules
Full Text: DOI
[1] A. A. Beilinson, J. Bernstein and P. Deligne, Faisceaux pervers, Analysis and topology on singular spaces, I (Luminy 1981), Astérisque 100, Société Mathématique de France, Paris (1982), 5-171. · Zbl 0536.14011
[2] E. Bierstone and P.-D. Milman, Semi-analytic sets and subanalytic sets, Publ. Math. Inst. Hautes Études Sci. 67 (1988), 5-42. · Zbl 0674.32002
[3] T. Bridgeland, Stability conditions on triangulated categories, Ann. of Math. (2) 166 (2007), no. 2, 317-345. · Zbl 1137.18008
[4] A. D’Agnolo and M. Kashiwara, Riemann-Hilbert correspondence for holonomic D-modules, Publ. Math. Inst. Hautes Études Sci. 123 (2016), no. 1, 69-197. · Zbl 1351.32017
[5] S. Guillermou and P. Schapira, Microlocal theory of sheaves and Tamarkin’s non-displaceability theorem, Homological mirror symmetry and tropical geometry, Lect. Notes Unione Mat. Ital. 15, Springer, Berlin (2014), 43-85. · Zbl 1319.32006
[6] H. Hironaka, Subanalytic sets, Number theory, algebraic geometry and commutative algebra, Kinokuniya, Tokyo (1973), 453-493. · Zbl 0297.32008
[7] M. Kashiwara, The Riemann-Hilbert problem for holonomic systems, Publ. Res. Inst. Math. Sci. 20 (1984), no. 2, 319-365. · Zbl 0566.32023
[8] M. Kashiwara, \mathcalD-modules and microlocal calculus, Transl. Math. Monogr. 217, American Mathematical Society, Providence 2003.
[9] M. Kashiwara, Equivariant derived category and representation of real semisimple Lie groups, Representation theory and complex analysis, Lecture Notes in Math. 1931, Springer, Berlin (2008), 137-234. · Zbl 1173.22010
[10] M. Kashiwara, Self-dual t-structure, preprint (2015), .
[11] M. Kashiwara and P. Schapira, Sheaves on manifolds, Grundlehren Math. Wiss. 292, Springer, Berlin 1990.
[12] M. Kashiwara and P. Schapira, Moderate and formal cohomology associated with constructible sheaves, Mém. Soc. Math. Fr. (N.S.) 64 (1996), 1-76. · Zbl 0881.58060
[13] M. Kashiwara and P. Schapira, Ind-sheaves, Astérisque 271, Société Mathématique de France, Paris 2001.
[14] M. Kashiwara and P. Schapira, Categories and sheaves, Grundlehren Math. Wiss. 332, Springer, Berlin 2006.
[15] M. Kashiwara and P. Schapira, Irregular holonomic Kernels and Laplace transform, Selecta Math. (N.S.) 22 (2016), no. 1, 55-109. · Zbl 1337.32020
[16] M. Kashiwara and P. Schapira, Regular and irregular holonomic D-modules, London Math. Soc. Lecture Note Ser. 433, Cambridge University Press, Cambridge 2016.
[17] D. Lazard, Autour de la platitude, Bull. Soc. Math. France 97 (1969), 81-128. · Zbl 0174.33301
[18] J.-P. Schneiders, Quasi-abelian categories and sheaves, Mém. Soc. Math. Fr. (N.S.) 76 (1999), 1-134.
[19] D. Tamarkin, Microlocal condition for non-displaceability, preprint (2008), . · Zbl 1416.35019
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.