Skip to content
Licensed Unlicensed Requires Authentication Published by De Gruyter November 8, 2016

Enhanced perversities

  • Andrea D’Agnolo EMAIL logo and Masaki Kashiwara

Abstract

On a complex manifold, the Riemann–Hilbert correspondence embeds the triangulated category of (not necessarily regular) holonomic 𝒟-modules into the triangulated category of ℝ-constructible enhanced ind-sheaves. The source category has a standard t-structure. Here, we provide the target category with a middle perversity t-structure, and prove that the embedding is exact.

In the paper, we also discuss general perversities in the framework of â„ť-constructible enhanced ind-sheaves on bordered subanalytic spaces.

Award Identifier / Grant number: 15H03608

Award Identifier / Grant number: CPDA159224

Funding statement: The first author was partially supported by grant CPDA159224, Padova University. The second author was supported by Grant-in-Aid for Scientific Research (B) 15H03608, Japan Society for the Promotion of Science.

Acknowledgements

The first author acknowledges the kind hospitality at RIMS, Kyoto University, during the preparation of this paper.

References

[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. Search in Google Scholar

[2] E. Bierstone and P.-D. Milman, Semi-analytic sets and subanalytic sets, Publ. Math. Inst. Hautes Études Sci. 67 (1988), 5–42. 10.1007/BF02699126Search in Google Scholar

[3] T. Bridgeland, Stability conditions on triangulated categories, Ann. of Math. (2) 166 (2007), no. 2, 317–345. 10.4007/annals.2007.166.317Search in Google Scholar

[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. 10.1007/s10240-015-0076-ySearch in Google Scholar

[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. 10.1007/978-3-319-06514-4_3Search in Google Scholar

[6] H. Hironaka, Subanalytic sets, Number theory, algebraic geometry and commutative algebra, Kinokuniya, Tokyo (1973), 453–493. Search in Google Scholar

[7] M. Kashiwara, The Riemann–Hilbert problem for holonomic systems, Publ. Res. Inst. Math. Sci. 20 (1984), no. 2, 319–365. 10.2977/prims/1195181610Search in Google Scholar

[8] M. Kashiwara, đť’ź-modules and microlocal calculus, Transl. Math. Monogr. 217, American Mathematical Society, Providence 2003. 10.1090/mmono/217Search in Google Scholar

[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. 10.1007/978-3-540-76892-0_3Search in Google Scholar

[10] M. Kashiwara, Self-dual t-structure, preprint (2015), https://arxiv.org/abs/1507.03384. 10.4171/PRIMS/181Search in Google Scholar

[11] M. Kashiwara and P. Schapira, Sheaves on manifolds, Grundlehren Math. Wiss. 292, Springer, Berlin 1990. 10.1007/978-3-662-02661-8Search in Google Scholar

[12] M. Kashiwara and P. Schapira, Moderate and formal cohomology associated with constructible sheaves, Mém. Soc. Math. Fr. (N.S.) 64 (1996), 1–76. 10.24033/msmf.378Search in Google Scholar

[13] M. Kashiwara and P. Schapira, Ind-sheaves, Astérisque 271, Société Mathématique de France, Paris 2001. Search in Google Scholar

[14] M. Kashiwara and P. Schapira, Categories and sheaves, Grundlehren Math. Wiss. 332, Springer, Berlin 2006. 10.1007/3-540-27950-4Search in Google Scholar

[15] M. Kashiwara and P. Schapira, Irregular holonomic Kernels and Laplace transform, Selecta Math. (N.S.) 22 (2016), no. 1, 55–109. 10.1007/s00029-015-0185-ySearch in Google Scholar

[16] M. Kashiwara and P. Schapira, Regular and irregular holonomic D-modules, London Math. Soc. Lecture Note Ser. 433, Cambridge University Press, Cambridge 2016. 10.1017/CBO9781316675625Search in Google Scholar

[17] D. Lazard, Autour de la platitude, Bull. Soc. Math. France 97 (1969), 81–128. 10.24033/bsmf.1675Search in Google Scholar

[18] J.-P. Schneiders, Quasi-abelian categories and sheaves, Mém. Soc. Math. Fr. (N.S.) 76 (1999), 1–134. 10.24033/msmf.389Search in Google Scholar

[19] D. Tamarkin, Microlocal condition for non-displaceability, preprint (2008), https://arxiv.org/abs/0809.1584. 10.1007/978-3-030-01588-6_3Search in Google Scholar

Received: 2015-12-18
Revised: 2016-07-08
Published Online: 2016-11-08
Published in Print: 2019-06-01

© 2019 Walter de Gruyter GmbH, Berlin/Boston

Downloaded on 9.2.2023 from https://www.degruyter.com/document/doi/10.1515/crelle-2016-0062/html
Scroll Up Arrow