Skip to main content
SpringerLink
Log in

Bernstein–Sato polynomials of hyperplane arrangements

  • Published:
Selecta Mathematica Aims and scope Submit manuscript

Abstract

We show that the Bernstein–Sato polynomial (that is, the b-function) of a hyperplane arrangement with a reduced equation is calculable by combining a generalization of Malgrange’s formula with the theory of Aomoto complexes due to Esnault, Schechtman, Terao, Varchenko, and Viehweg in certain cases. We prove in general that the roots are greater than \(-2\) and the multiplicity of the root \(-1\) is equal to the (effective) dimension of the ambient space. We also give an estimate of the multiplicities of the roots in terms of the multiplicities of the arrangement at the dense edges, and provide a method to calculate the Bernstein–Sato polynomial at least in the case of 3 variables with degree at most 7 and generic multiplicities at most 3. Using our argument, we can terminate the proof of a conjecture of Denef and Loeser on the relation between the topological zeta function and the Bernstein–Sato polynomial of a reduced hyperplane arrangement in the 3 variable case.

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

References

  1. Artal-Bartolo, E.: Combinatorics and topology of line arrangements in the complex projective plane. Proc. Amer. Math. Soc. 121, 385–390 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  2. Barlet, D., Saito, M.: Brieskorn modules and Gauss-Manin systems for non-isolated hypersurface singularities. J. Lond. Math. Soc. (2) 76, 211–224 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  3. Beilinson, A.A., Bernstein, J.N., Deligne, P.: Faisceaux Pervers, Astérisque 100. Soc. Math. France, Paris (1982)

  4. Bernstein, J.N.: The analytic continuation of generalized functions with respect to a parameter. Funct. Anal. Appl. 6, 273–285 (1972)

    Article  MathSciNet  MATH  Google Scholar 

  5. Brieskorn, E.: Die Monodromie der isolierten Singularitäten von Hyperflächen. Manuscr. Math. 2, 103–161 (1970)

    Article  MathSciNet  MATH  Google Scholar 

  6. Brieskorn, E.: Sur les groupes de tresses, Séminaire Bourbaki, 24ème année (1971/1972), Exp. No. 401, Lecture Notes in Mathematics 317, Springer, Berlin, pp. 21–44 (1973)

  7. Budur, N.: On Hodge spectrum and multiplier ideals. Math. Ann. 327, 257–270 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  8. Budur, N., Dimca, A., Saito, M.: First Milnor cohomology of hyperplane arrangements. In: Topology of Algebraic Varieties and Singularities, Contemporary Mathematics, vol. 538, pp. 279–292. American Mathematical Society, Providence (2011)

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

    Article  MathSciNet  MATH  Google Scholar 

  10. Budur, N., Saito, M.: Jumping coefficients and spectrum of a hyperplane arrangement. Math. Ann. 347, 545–579 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  11. Budur, N., Saito, M.: On the local zeta functions and the \(b\)-functions of certain hyperplane arrangements (with an appendix by W. Veys). J. Lond. Math. Soc. 84, 631–648 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  12. Cohen, D.C., Dimca, A., Orlik, P.: Nonresonance conditions for arrangements. Ann. Inst. Fourier 53, 1883–1896 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  13. Cohen, D., Suciu, A.: On Milnor fibrations of arrangements. J. Lond. Math. Soc. 51, 105–119 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  14. Deligne, P.: Equations Différentielles à Points Singuliers Réguliers, Lecture Notes in Mathematics, vol. 163. Springer, Berlin (1970)

  15. Deligne, P.: Théorie de Hodge II. Publ. Math. IHES 40, 5–58 (1971)

    Article  MathSciNet  MATH  Google Scholar 

  16. Deligne, P.: Le formalisme des cycles évanescents. In: SGA7 XIII and XIV, Lecture Notes in Mathematics, vol. 340, pp. 82–115 and 116–164. Springer, Berlin (1973)

  17. Denef, J., Loeser, F.: Caractéristiques d’Euler-Poincaré, fonctions zéta locales et modifications analytiques. J. Amer. Math. Soc. 5, 705–720 (1992)

    MathSciNet  MATH  Google Scholar 

  18. Dimca, A.: Singularities and Topology of Hypersurfaces. Springer, Berlin (1992)

    Book  MATH  Google Scholar 

  19. Dimca, A., Saito, M.: Some consequences of perversity of vanishing cycles. Ann. Inst. Fourier 54, 1769–1792 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  20. Dimca, A., Saito, M.: A generalization of Griffiths’ theorem on rational integrals. Duke Math. J. 135, 303–326 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  21. Dimca, A., Saito, M.: Koszul complexes and spectra of projective hypersurfaces with isolated singularities. arXiv:1212.1081

  22. Dimca, A., Saito, M.: Generalization of theorems of Griffiths and Steenbrink to hypersurfaces with ordinary double points. arXiv:1403.4563

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

    Article  MathSciNet  MATH  Google Scholar 

  24. Esnault, H., Schechtman, V., Viehweg, E.: Cohomology of local systems on the complement of hyperplanes. Inv. Math. 109, 557–561 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  25. Falk, M.: Arrangements and cohomology. Ann. Combin. 1, 135–157 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  26. Falk, M., Yuzvinsky, S.: Multinets, resonance varieties, and pencils of plane curves. Compos. Math. 143, 1069–1088 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  27. Kashiwara, M.: \(B\)-functions and holonomic systems. Inv. Math. 38, 33–53 (1976/77)

  28. Kashiwara, M.: Vanishing cycle sheaves and holonomic systems of differential equations. Lecture Notes in Mathematics, vol. 1016, pp. 134–142. Springer, Berlin (1983)

  29. Kloosterman, R.: On the relation between Alexander polynomials and Mordell-Weil ranks, equianalytic deformations and a variant of Nagata’s conjecture (preprint)

  30. Kollár, J.: Singularities of pairs. Proc. Symp. Pure Math. A.M.S. 62(Part 1), 221–287 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  31. Lazarsfeld, R.: Positivity in Algebraic Geometry II. Springer, Berlin (2004)

    Book  MATH  Google Scholar 

  32. Libgober, A., Yuzvinsky, S.: Cohomology of the Orlik-Solomon algebras and local systems. Compos. Math. 121, 337–361 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  33. Malgrange, B.: Le polynôme de Bernstein d’une singularité isolée, Lecture Notes in Mathematics, vol. 459, pp. 98–119. Springer, Berlin (1975)

  34. Malgrange, B.: Polynôme de Bernstein-Sato et cohomologie évanescente. Analysis and topology on singular spaces, II, III (Luminy, 1981), Astérisque 101–102, 243–267 (1983)

    MathSciNet  MATH  Google Scholar 

  35. Mustaţǎ, M.: Multiplier ideals of hyperplane arrangements. Trans. Amer. Math. Soc. 358, 5015–5023 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  36. Orlik, P., Randell, R.: The Milnor fiber of a generic arrangement. Ark. Mat. 31, 71–81 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  37. Orlik, P., Solomon, L.: Combinatorics and topology of complements of hyperplanes. Inv. Math. 56, 167–189 (1980)

    Article  MathSciNet  MATH  Google Scholar 

  38. Saito, M.: Mixed Hodge modules. Publ. RIMS Kyoto Univ. 26, 221–333 (1990)

    Article  MathSciNet  MATH  Google Scholar 

  39. Saito, M.: On microlocal \(b\)-function. Bull. Soc. Math. France 122, 163–184 (1994)

    MathSciNet  MATH  Google Scholar 

  40. Saito, M.: Multiplier ideals, \(b\)-function, and spectrum of a hypersurface singularity. Compos. Math. 143, 1050–1068 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  41. Saito, M.: On real log canonical thresholds. arXiv:0707.2308

  42. Saito, M.: Hilbert series of graded Milnor algebras and roots of Bernstein-Sato polynomials. arXiv:1509.06288

  43. Sato, M. (ed.), Singularities of Hypersurfaces and \(b\)-function. In: Proceedings of Workshop in 1973, RIMS Kokyuroku vol. 225, (in Japanese) (1975)

  44. Sato, M., Shintani, T.: On zeta functions associated with prehomogeneous vector spaces. Proc. Nat. Acad. Sci. USA 69, 1081–1082 (1972)

    Article  MathSciNet  MATH  Google Scholar 

  45. Schechtman, V., Terao, H., Varchenko, A.: Local systems over complements of hyperplanes and the Kac-Kazhdan conditions for singular vectors. J. Pure Appl. Algebra 100, 93–102 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  46. Scherk, J., Steenbrink, J.H.M.: On the mixed Hodge structure on the cohomology of the Milnor fibre. Math. Ann. 271, 641–665 (1985)

    Article  MathSciNet  MATH  Google Scholar 

  47. Steenbrink, J.H.M.: Intersection form for quasi-homogeneous singularities. Compositio Math. 34, 211–223 (1977)

    MathSciNet  MATH  Google Scholar 

  48. Steenbrink, J.H.M.: Mixed Hodge structure on the vanishing cohomology. In: Real and Complex Singularities (Proceedings of the Nordic Summer School, Oslo, 1976), pp. 525–563. Sijthoff & Noordhoff, Alphen a/d Rijn (1977)

  49. Steenbrink, J.H.M.: The spectrum of hypersurface singularity. Astérisque 179–180, 163–184 (1989)

    MathSciNet  MATH  Google Scholar 

  50. Walther, U.: Bernstein-Sato polynomial versus cohomology of the Milnor fiber for generic hyperplane arrangements. Compos. Math. 141, 121–145 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  51. Walther, U.: The Jacobian module, the Milnor fiber, and the \(D\)-module generated by \(f^s\) (preprint)

Download references

Author information

Authors and Affiliations

  1. RIMS Kyoto University, Kyoto, 606-8502, Japan

    Morihiko Saito

Authors
  1. Morihiko Saito
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Morihiko Saito.

Additional information

Dedicated to Joseph Bernstein.

This work was partially supported by JSPS Kakenhi 17540023 and 24540039.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Saito, M. Bernstein–Sato polynomials of hyperplane arrangements. Sel. Math. New Ser. 22, 2017–2057 (2016). https://doi.org/10.1007/s00029-016-0268-4

Download citation

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00029-016-0268-4

Mathematics Subject Classification

Search

Navigation