Skip to main content
SpringerLink
Log in

Monodromy conjecture for log generic polynomials

  • Published:
Mathematische Zeitschrift Aims and scope Submit manuscript

Abstract

A log generic hypersurface in \(\mathbb {P}^n\) with respect to a birational modification of \(\mathbb {P}^n\) is by definition the image of a generic element of a high power of an ample linear series on the modification. A log very-generic hypersurface is defined similarly but restricting to line bundles satisfying a non-resonance condition. Fixing a log resolution of a product \(f=f_1\ldots f_p\) of polynomials, we show that the monodromy conjecture, relating the motivic zeta function with the complex monodromy, holds for the tuple \((f_1,\ldots ,f_p,g)\) and for the product fg, if g is log generic. We also show that the stronger version of the monodromy conjecture, relating the motivic zeta function with the Bernstein–Sato ideal, holds for the tuple \((f_1,\ldots ,f_p,g)\) and for the product fg, if g is log very-generic. Even the case \(f=1\) is intricate, the proof depending on nontrivial properties of Bernstein–Sato ideals, and it singles out the class of log (very-) generic hypersurfaces as an interesting class of singularities on its own.

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

We’re sorry, something doesn't seem to be working properly.

Please try refreshing the page. If that doesn't work, please contact support so we can address the problem.

References

  1. Aluffi, P.: Euler characteristics of general linear sections and polynomial Chern classes. Rend. Circ. Mat. Palermo 62(1), 3–26 (2013)

    Article  MathSciNet  Google Scholar 

  2. Bath, D.: Bernstein-Sato varieties and annihilation of powers. Trans. Amer. Math. Soc. 373, 8543–8582 (2020)

    Article  MathSciNet  Google Scholar 

  3. Blanco, G., Budur, N., van der Veer, R.: Monodromy conjecture for semi-quasihomogeneous hypersurfaces.arXiv: 2106.11015. To appear in Math. Nachr

  4. Budur, N.: Unitary local systems, multiplier ideals, and polynomial periodicity of Hodge numbers. Adv. Math. 221, 217–250 (2009)

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  7. Budur, N., Mustaţă, M., Teitler, Z.: The monodromy conjecture for hyperplane arrangements. Geom. Dedicata 153, 131–137 (2011)

    Article  MathSciNet  Google Scholar 

  8. Budur, N., van der Veer, R., Wu, L., Zhou, P.: Zero loci of Bernstein-Sato ideals. Invent. Math. 225, 45–72 (2021)

    Article  MathSciNet  Google Scholar 

  9. Denef, J.: Poles of \(p\)-adic complex powers and Newton polyhedra. Groupe de travail d’analyse ultramètrique 12 (1984/85), 1-3

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

    MathSciNet  MATH  Google Scholar 

  11. Denef, J., Loeser, F.: Motivic Igusa zeta functions. J. Algebraic Geom. 7, 505–537 (1998)

    MathSciNet  MATH  Google Scholar 

  12. Esnault, H., Viehweg, E.: Logarithmic de Rham complexes and vanishing theorems. Invent. Math. 86, 161–194 (1986)

    Article  MathSciNet  Google Scholar 

  13. Esnault, H., Viehweg, E.: A remark on a nonvanishing theorem of P. Deligne and G. D. Mostow. J. Reine Angew. Math. 381, 211–213 (1987)

    MathSciNet  MATH  Google Scholar 

  14. Esterov, A., Lemahieu, A., Takeuchi, K.:On the monodromy conjecture for non-degenerate hypersurfaces. arXiv:1309.0630

  15. Guibert, G.: Espaces d’arcs et invariants d’Alexander. Comment. Math. Helv. 77, 783–820 (2002)

    Article  MathSciNet  Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

  17. Hamm, H.:Remarks on asymptotic integrals, the polynomial of I. N. Bernstein and the Picard-Lefschetz monodromy. Several complex variables (Proc. Sympos. Pure Math., Vol. XXX, Part 1, Williams Coll., Williamstown, Mass., 1975), pp. 31-35. Amer. Math. Soc., Providence, R.I., 1977

  18. Igusa, J.-i.: An introduction to the theory of local zeta functions. Amer. Math. Soc., Providence, RI; International Press, Cambridge, MA, 2000. xii+232 pp

  19. Kouchnirenko, A.G.: Polyèdres de Newton et nombres de Milnor. Invent. Math. 32, 1–31 (1976)

    Article  MathSciNet  Google Scholar 

  20. Lazarsfeld, R.: Positivity in algebraic geometry I. Springer-Verlag, Berlin, II (2004)

    Book  Google Scholar 

  21. Lemahieu, A., Van Proeyen, L.: Monodromy conjecture for nondegenerate surface singularities. Trans. Amer. Math. Soc. 363, 4801–4829 (2011)

    Article  MathSciNet  Google Scholar 

  22. Lichtin, B.: Poles of \(|f(z, w)|^{2s}\) and roots of the \(b\)-function. Ark. Mat. 27, 283–304 (1989)

    Article  MathSciNet  Google Scholar 

  23. Loeser, F.: Fonctions d’Igusa \(p\)-adiques et polynômes de Bernstein. Amer. J. Math. 110, 1–21 (1988)

    Article  MathSciNet  Google Scholar 

  24. Loeser, F.: Fonctions zêta locales d’Igusa à plusieurs variables, intégration dans les fibres, et discriminants. Ann. Sci. École Norm. Sup. 22, 435–471 (1989)

    Article  MathSciNet  Google Scholar 

  25. Loeser, F.: Fonctions d’Igusa \(p\)-adiques, polynômes de Bernstein, et polyèdres de Newton. J. reine angew. Math. 412, 75–96 (1990)

    MathSciNet  MATH  Google Scholar 

  26. Malgrange, B.: Sur les polynômes de I. N. Bernstein. Séminaire Goulaouic-Schwartz 1973-1974: Équations aux dérivées partielles et analyse fonctionnelle, Exp. No. 20, 10 pp. Centre de Math., École Polytech., Paris, 1974

  27. Meuser, D.: A survey of Igusa’s local zeta function. Amer. J. Math. 138, 149–179 (2016)

    Article  MathSciNet  Google Scholar 

  28. Nicaise, J.: Zeta functions and Alexander modules. math.AG/0404212

  29. Sabbah, C.: Modules d’Alexander et \(D\)-modules. Duke Math. J. 60, 729–814 (1990)

    Article  MathSciNet  Google Scholar 

  30. Varchenko, A.N.: Zeta-function of monodromy and Newton’s diagram. Invent. Math. 37, 253–262 (1976)

    Article  MathSciNet  Google Scholar 

  31. Walther, U.: The Jacobian module, the Milnor fiber, and the D-module generated by \(f^s\). Invent. Math. 207, 1239–1287 (2017)

    Article  MathSciNet  Google Scholar 

  32. Wu, L.: Bernstein-Sato ideals and hyperplane arrangements. arXiv:2005.13502

Download references

Acknowledgements

We thank M. Mustaţă, L. Wu, P. Zhao, and the referee for useful comments. We are especially grateful to the referee for formulating Theorem 1.10. The first author would like to thank MPI Bonn for hospitality during the writing of this article. The first author was partly supported by the grants STRT/13/005 and Methusalem METH/15/026 from KU Leuven, G097819N and G0F4216N from FWO (Research Foundation - Flanders). The second author is supported by a PhD Fellowship from FWO.

Author information

Authors and Affiliations

  1. KU Leuven, Celestijnenlaan 200B, 3001, Leuven, Belgium

    Nero Budur & Robin van der Veer

  2. BCAM, Mazarredo 14, 48009, Bilbao, Spain

    Nero Budur

Authors
  1. Nero Budur
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Robin van der Veer
    View author publications

    You can also search for this author in PubMed Google Scholar

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

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Budur, N., van der Veer, R. Monodromy conjecture for log generic polynomials. Math. Z. 301, 713–737 (2022). https://doi.org/10.1007/s00209-021-02914-4

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00209-021-02914-4

Keywords

Mathematics Subject Classification

Search

Navigation