Skip to main content
SpringerLink
Log in

Characteristic cycles associated to holonomic \({\mathscr {D}}\)-modules

  • Published:
Mathematische Zeitschrift Aims and scope Submit manuscript

Abstract

We study relative and logarithmic characteristic cycles associated to holonomic \({\mathscr {D}}\)-modules. As applications, we obtain: (1) an alternative proof of Ginsburg’s log characteristic cycle formula for lattices of regular holonomic \({\mathscr {D}}\)-modules following ideas of Sabbah and Briancon–Maisonobe–Merle, and (2) the constructibility of the log de Rham complexes for lattices of holonomic \({\mathscr {D}}\)-modules, which is a natural generalization of Kashiwara’s constructibility theorem.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

Notes

  1. The V-filtration in loc. cit. is the \({\mathbb {Q}}\)-indexed one. One can refine the \({\mathbb {Z}}\)-index to the \({\mathbb {Q}}\)-index by a standard procedure using b-functions.

  2. This relative holonomicity is slightly more general than the ones in [6, 7, 12, 34], where the latter requires additional geometric conditions on the relative Lagrangian subvarieties.

  3. In the literature, some authors define the Kashiwara–Malgrange filtration on \({\mathscr {D}}_X\) as the decreasing filtration, that is, \(V_Y^k{\mathscr {D}}_X:=V^Y_{-k}{\mathscr {D}}_X\).

References

  1. Björk, J.-E.: Analytic \({\cal{D}}\)-modules and applications, Mathematics and its Applications, vol. 247. Kluwer Academic Publishers Group, Dordrecht (1993)

    Book  Google Scholar 

  2. Briançon, J., Maisonobe, P., Merle, M.: Équations fonctionnelles associées à des fonctions analytiques. Tr. Mat. Inst. Steklova 238, 86–96 (2002)

    MATH  Google Scholar 

  3. Budur, N., Mustaţǎ, M., Saito, M.: Bernstein–Sato polynomials of arbitrary varieties. Compos. Math. 142(3), 779–797 (2006)

    Article  MathSciNet  Google Scholar 

  4. Budur, N., Saito, M.: Multiplier ideals, V-filtration, and spectrum. J. Algebr. Geom. 14, 269–282 (2005)

    Article  MathSciNet  Google Scholar 

  5. Budur, N.: Bernstein-sato ideals and local systems. Annales de l’Institut Fourier 65(2), 549–603 (2015)

    Article  MathSciNet  Google Scholar 

  6. Budur, N., van der Veer, R., Wu, L., Zhou, P.: Zero loci of Bernstein–Sato ideals, Invent. Math. 1–28 (2021)

  7. Budur, N., van der Veer, R., Lei, W., Zhou, P.: Zero loci of Bernstein–Sato ideals II. Sel. Math. New Ser. 27, 32 (2021)

    Article  MathSciNet  Google Scholar 

  8. Calderón-Moreno, F.J.: Logarithmic differential operators and logarithmic de Rham complexes relative to a free divisor. Ann. Sci. l’E.N.S. 32(5), 701–714 (1999)

  9. Calderón-Moreno, F., Narváez-Macarro, L.: The module \({\cal{D}}f^s\) for locally quasi-homogeneous free divisors. Compos. Math. 134(1), 59–74 (2002)

    Article  MathSciNet  Google Scholar 

  10. D’Agnolo, A., Kashiwara, M.: Riemann–Hilbert correspondence for holonomic D-modules. Publ. Math. l’IHÉS 123(1), 69–197 (2016)

    Article  MathSciNet  Google Scholar 

  11. Ein, L., Lazarsfeld, R., Smith, K.E., Varolin, D., et al.: Jumping coefficients of multiplier ideals. Duke Math. J. 123(3), 469–506 (2004)

    Article  MathSciNet  Google Scholar 

  12. Fernandes, T.M., Sabbah, C.: Riemann–Hilbert correspondence for mixed twistor D-modules. arXiv:1609.04192 (2016) (preprint)

  13. Fiorot, L.: Teresa Monteiro Fernandes, and Claude Sabbah, Relative regular Riemann–Hilbert correspondence. Proceedings of the London Mathematical Society (2020)

  14. Ginsburg, V.: Characteristic varieties and vanishing cycles. Invent. Math. 84(2), 327–402 (1986)

    Article  MathSciNet  Google Scholar 

  15. Ginsburg, V.: Admissible modules on a symmetric space. Astérisque (173–174), 9–10, 199–255 (1989). Orbites unipotentes et représentations, III

  16. Ginsburg, V.: Vanishing cycle sheaves and holonomic systems of differential equations, Algebraic geometry (Tokyo/Kyoto, 1982), Lecture Notes in Math., vol. 1016. Springer, Berlin, pp. 134–142 (1983)

  17. Ginsburg, V.: \(D\)-modules and microlocal calculus, Translations of Mathematical Monographs, vol. 217, American Mathematical Society, Providence (2003). Translated from the 2000 Japanese original by Mutsumi Saito, Iwanami Series in Modern Mathematics

  18. Ginsburg, V.: \(B\)-functions and holonomic systems. Rationality of roots of \(b\)-functions. Invent. Math. 38(1), 33–53 (1976/1977)

  19. Ginsburg, V.: L’Idéal de Bernstein d’un arrangement libre d’hyperplans linéaires. arXiv:1610.03356 (2016) (preprint)

  20. Ginsburg, V.: Hodge ideals for Q-divisors, V-filtration, and minimal exponent. Forum Math. Sigma 8 (2020)

  21. Ginsburg, V.: Hodge ideals for Q-divisors: birational approach. Journal de l’École polytechnique 6, 283–328 (2019)

    Article  MathSciNet  Google Scholar 

  22. Ginsburg, V.: Proximité évanescente. II. Équations fonctionnelles pour plusieurs fonctions analytiques. Compos. Math. 64(2), 213–241 (1987)

    Google Scholar 

  23. Grothendieck, A.: On the de Rham cohomology of algebraic varieties. Publications Mathématiques de l’Institut des Hautes Études Scientifiques 29(1), 95–103 (1966)

    Article  MathSciNet  Google Scholar 

  24. Hotta, R., Takeuchi, K., Tanisaki, T.: \(D\)-modules, perverse sheaves, and representation theory, Progress in Mathematics, vol. 236. Birkhäuser Boston, Inc., Boston (2008). Translated from the 1995 Japanese edition by Takeuchi

  25. Kashiwara, M., Kawai, T.: On holonomic systems for \(\prod _{l=1}^n(f_l+\sqrt{-1}{O})^{\lambda _l}\). Publ. Res. Inst. Math. Sci. 15(2), 551–575 (1979)

    Article  MathSciNet  Google Scholar 

  26. Kashiwara, M.: On the maximally overdetermined system of linear differential equations, I. Publ. Res. Inst. Math. Sci. 10(2), 563–579 (1975)

    Article  MathSciNet  Google Scholar 

  27. Kashiwara, M., Schapira, P.: Sheaves on manifolds: With a short history. les débuts de la théorie des faisceaux, by Christian Houzel, vol. 292. Springer Science & Business Media (2013)

  28. Kato, K., Nakayama, C.: Log betti cohomology, log étale cohomology, and log de Rham cohomology of log schemes over C. Kodai Math. J. 22(2), 161–186 (1999)

    Article  MathSciNet  Google Scholar 

  29. Kollár, J.: Singularities of pairs, Proceedings of Symposia in Pure Mathematics, vol. 62. American Mathematical Society, pp. 221–288 (1997)

  30. Koppensteiner, C.: The de Rham functor for logarithmic D-modules. Select. Math. 26(3), 1–44 (2020)

    Article  MathSciNet  Google Scholar 

  31. Koppensteiner, C., Talpo, M.: Holonomic and perverse logarithmic D-modules. Adv. Math. 346, 510–545 (2019)

    Article  MathSciNet  Google Scholar 

  32. Laumon, G.: Sur la catégorie dérivée des \({\cal{D}}\)-modules filtrés, Algebraic geometry (Tokyo/Kyoto, 1982), Lecture Notes in Math., vol. 1016. Springer, Berlin, pp. 151–237 (1983)

  33. Lazarsfeld, R.: Positivity in algebraic geometry. II, vol. 49. Springer 18(385), 8 (2004)

  34. Maisonobe, P.: Filtration Relative, l’Idéal de Bernstein et ses pentes. arXiv:1610.03354 (2016) (preprint)

  35. Malgrange, B.: Polynômes de Bernstein-Sato et cohomologie évanescente, Analysis and topology on singular spaces, II, III (Luminy, 1981), Astérisque, vol. 101, Soc. Math. France, Paris, pp. 243–267 (1983)

  36. Mustaţă, M., Popa, M.: Hodge ideals. Mem. Am. Math. Soc. 262(1268) (2019)

  37. Nadler, D., Zaslow, E.: Constructible sheaves and the Fukaya category. J. Am. Math. Soc. 22 (2009)

  38. Ogus, A.: On the logarithmic Riemann–Hilbert correspondence. Doc. Math. 655 (2003)

  39. Sabbah, C.: Proximité évanescente. I. La structure polaire d’un \({\cal{D}}\)-module. Compos. Math. 62(3), 283–328 (1987)

    MATH  Google Scholar 

  40. Saito, M.: Hodge ideals and microlocal V-filtration. arXiv:1612.08667 (2016) (preprint)

  41. Schapira, P.: Microdifferential Systems in the Complex Domain, vol. 269. Springer Science & Business Media, New York (2012)

    MATH  Google Scholar 

  42. Schapira, P., Schneiders, J.-P.: Elliptic pairs I. Relative finiteness and duality. Astérisque (224), 5–60 (1994)

  43. Stevens, J.: On canonical singularities as total spaces of deformations, Abhandlungen aus dem Mathematischen Seminar der Universität Hamburg, vol. 58. Springer, p. 275 (1988)

  44. Wu, L.: Riemann–Hilbert correspondence for Alexander complexes. arXiv:2104.06941 (2021) (preprint)

  45. Wu, L., Zhou, P.: Log D-modules and index theorems, Forum of Mathematics, Sigma, vol. 9. Cambridge University Press, Cambridge (2021)

    Google Scholar 

Download references

Acknowledgements

The author thanks Peng Zhou and Nero Budur for useful discussions, Claude Sabbah for answering questions and Yajnaseni Dutta and Ruijie Yang for useful comments. He is grateful to an anonymous referee for very useful comments in improving the exposition. The author was supported an FWO postdoctoral fellowship.

Author information

Authors and Affiliations

  1. Department of Mathematics, KU Leuven, Celestijnenlaan 200B, 3001, Leuven, Belgium

    Lei Wu

Authors
  1. Lei Wu
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Lei Wu.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Wu, L. Characteristic cycles associated to holonomic \({\mathscr {D}}\)-modules. Math. Z. 301, 2059–2098 (2022). https://doi.org/10.1007/s00209-022-02974-0

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00209-022-02974-0

Mathematics Subject Classification

Search

Navigation