Skip to main content

Zero loci of Bernstein-Sato ideals-II


We have recently proved a precise relation between Bernstein-Sato ideals of collections of polynomials and monodromy of generalized nearby cycles. In this article we extend this result to other ideals of Bernstein-Sato type.

This is a preview of subscription content, access via your institution.


  1. 1.

    Björk, J.E.: Rings of differential operators. North-Holland Mathematical Library, 21, 1979, xvii+374 pp

  2. 2.

    Björk, J.-E.: \({\mathscr {D}}\)-modules and applications. Mathematics and its Applications, 247, 1993. xiv+581 pp

  3. 3.

    Briançon, J., Maisonobe, P., Merle, M.: Constructibilité de l’idéal de Bernstein. Singularities-Sapporo 1998, 79-95, Adv. Stud. Pure Math., 29, Kinokuniya, Tokyo (2000)

  4. 4.

    Briançon, J., Maisonobe, P., Merle, M.: Equations fonctionnelles associées à des fonctions analytiques. Proc. Steklov Inst. Math. 238, 77–87 (2002)

    MATH  Google Scholar 

  5. 5.

    Budur, N.: Bernstein-Sato ideals and local systems. Ann. Inst. Fourier 65(2), 549–603 (2015)

    MathSciNet  Article  Google Scholar 

  6. 6.

    Budur, N., Liu, Y., Saumell, L., Wang, B.: Cohomology support loci of local systems. Michigan Math. J. 66(2), 295–307 (2017)

    MathSciNet  Article  Google Scholar 

  7. 7.

    Budur, N., van der Veer, R., Wu, L., Zhou, P.: Zero loci of Bernstein-Sato ideals. arXiv:1907.04010. To appear in Invent. Math

  8. 8.

    Budur, N., Wang, B.: Absolute sets and the decomposition theorem. Ann. Sci. École Norm. Sup. 53, 469–536 (2020)

    MathSciNet  Article  Google Scholar 

  9. 9.

    de Cataldo, M.A., Migliorini, L.: The decomposition theorem, perverse sheaves and the topology of algebraic maps. Bull. Amer. Math. Soc. (N.S.) 46(4), 535–633 (2009)

    MathSciNet  Article  Google Scholar 

  10. 10.

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

    MathSciNet  Article  Google Scholar 

  11. 11.

    Gyoja, A.: Bernstein-Sato’s polynomial for several analytic functions. J. Math. Kyoto Univ. 33, 399–411 (1993)

    MathSciNet  MATH  Google Scholar 

  12. 12.

    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)

    MathSciNet  Article  Google Scholar 

  13. 13.

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

  14. 14.

    Maisonobe, P.: Filtration Relative, l’Idéal de Bernstein et ses pentes. arXiv:1610.03354

  15. 15.

    Sabbah, C.: Proximité évanescente. I. Compositio Math. 62 (1987), no. 3, 283-328. Proximité évanescente. II. ibid. 64 (1987), no. 2, 213-241

  16. 16.

    Wu, L., Zhou, P.: Log \({\mathscr {D}}\)-modules and index theorem. Forum Math. Sigma 9, e3 (2021)

    MathSciNet  Article  Google Scholar 

Download references


The first author was partly supported by the grants STRT/13/005 and Methusalem METH/15/026 from KU Leuven, G097819N and G0F4216N from the Research Foundation—Flanders. The second author is supported by a PhD Fellowship of the Research Foundation—Flanders. The fourth author is supported by the Simons Postdoctoral Fellowship as part of the Simons Collaboration on HMS.

Author information



Corresponding author

Correspondence to Nero Budur.

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

Verify currency and authenticity via CrossMark

Cite this article

Budur, N., Veer, R.v.d., Wu, L. et al. Zero loci of Bernstein-Sato ideals-II. Sel. Math. New Ser. 27, 32 (2021).

Download citation


  • Bernstein-Sato ideal
  • b-function
  • Monodromy
  • Local system
  • \({\mathscr {D}}\)-module

Mathematics Subject Classification

  • 14F10
  • 13N10
  • 32C38
  • 32S40
  • 32S55