Revista Matemática Complutense

, Volume 29, Issue 2, pp 455–483 | Cite as

Splicing motivic zeta functions

Article

Abstract

The first part of the paper discusses how to lift the splicing formula of Némethi and Veys to the motivic level. After defining the motivic zeta function and the monodromic motivic zeta function with respect to a differential form, we prove a splicing formula for them, which specializes to this formula of Némethi and Veys. We also show that we cannot introduce a monodromic motivic zeta functions in terms of a (splice) diagram since it does not contain all the necessary information. In the last part we discuss the generalized monodromy conjecture of Némethi and Veys. The statement also holds for motivic zeta functions but it turns out that the analogous statement for monodromic motivic zeta functions is not correct. We illustrate this with some examples.

Keywords

Splicing Motivic zeta functions Monodromy conjecture 

Mathematics Subject Classification

Primary 14E18 Secondary 14H50 32S05 

Notes

Acknowledgments

I am very grateful to Wim Veys and Johannes Nicaise for their valuable suggestions. I am also very grateful for the suggestions made by the referees.

References

  1. 1.
    Abramovich, D., Karu, K., Matsuki, K., Włodarczyk, J.: Torification and factorization of birational maps. J. Am. Math. Soc. 15(3), 531–572 (2002). (electronic)MathSciNetCrossRefMATHGoogle Scholar
  2. 2.
    Artal Bartolo, E., Cassou-Noguès, P., Luengo, I., Melle Hernández, A.: Monodromy conjecture for some surface singularities. Ann. Sci. École Norm. Sup. (4) 35(4), 605–640 (2002)MathSciNetMATHGoogle Scholar
  3. 3.
    Artal Bartolo, E., Cassou-Noguès, P., Luengo, I., Melle Hernández, A.: Quasi-ordinary power series and their zeta functions. Mem. Am. Math. Soc 178(841), vi+85 (2005)MathSciNetMATHGoogle Scholar
  4. 4.
    Bittner, F.: The universal Euler characteristic for varieties of characteristic zero. Compos. Math. 140(4), 1011–1032 (2004)MathSciNetCrossRefMATHGoogle Scholar
  5. 5.
    Bories, B.: Zeta functions, Bernstein–Sato polynomials, and the monodromy conjecture. Ph.D. thesis, KU Leuven, Leuven (2013)Google Scholar
  6. 6.
    Cox, D.A., Little, J.B., Schenck, H.K.: Toric varieties. In: Graduate Studies in Mathematics, vol. 124. American Mathematical Society, Providence (2011)Google Scholar
  7. 7.
    de Jong, J., Noot, R.: Jacobians with complex multiplication. In: Arithmetic Algebraic Geometry (Texel, 1989). Progr. Math., vol. 89, pp. 177–192. Birkhäuser, Boston (1991)Google Scholar
  8. 8.
    Denef, J., Loeser, F., d’Euler-Poincaré, C.: fonctions zêta locales et modifications analytiques. J. Am. Math. Soc. 5(4), 705–720 (1992)Google Scholar
  9. 9.
    Denef, J., Loeser, F.: Motivic Igusa zeta functions. J. Algebraic Geom. 7(3), 505–537 (1998)MathSciNetMATHGoogle Scholar
  10. 10.
    Denef, J., Loeser, F.: Geometry on arc spaces of algebraic varieties. In: European Congress of Mathematics, Vol. I (Barcelona, 2000). Progr. Math., vol. 201, pp. 327–348. Birkhäuser, Basel (2001)Google Scholar
  11. 11.
    Denef, J., Loeser, F.: Lefschetz numbers of iterates of the monodromy and truncated arcs. Topology 41(5), 1031–1040 (2002)MathSciNetCrossRefMATHGoogle Scholar
  12. 12.
    Dimca, A.: Sheaves in topology. In: Universitext. Springer, Berlin (2004)Google Scholar
  13. 13.
    Eisenbud, D., Neumann, W.: Three-dimensional link theory and invariants of plane curve singularities. In: Annals of Mathematics Studies, vol. 110. Princeton University Press, Princeton (1985)Google Scholar
  14. 14.
    Ekedahl, T.: The Grothendieck group of algebraic stacks. 21 (2009). arXiv:0903.3143
  15. 15.
    Fantechi, B., Göttsche, L., Illusie, L., Kleiman, S.L., Nitsure, N., Vistoli, A.: Fundamental algebraic geometry: Grothendieck’s FGA explained, vol. 123. American Mathematical Society (AMS), Providence (2005). (English)Google Scholar
  16. 16.
    Griffiths, P., Harris, J.: Principles of algebraic geometry. In: Wiley Classics Library. Wiley, New York (1994). (Reprint of the 1978 original)Google Scholar
  17. 17.
    Hartshorne, R.: Algebraic Geometry, Graduate Texts in Mathematics, vol. 52. Springer, New York (1977)CrossRefGoogle Scholar
  18. 18.
    Loeser, F.: Fonctions d’Igusa \(p\)-adiques et polynômes de Bernstein. Am. J. Math. 110(1), 1–21 (1988)MathSciNetCrossRefMATHGoogle Scholar
  19. 19.
    Loeser, F.: Seattle lectures on motivic integration. In: Algebraic Geometry–Seattle 2005. Part 2, Proc. Sympos. Pure Math., vol. 80, pp. 745–784. Am. Math. Soc., Providence (2009)Google Scholar
  20. 20.
    Loeser, F.: Microlocal geometry and valued fields. Publ. Res. Inst. Math. Sci. 47(2), 613–627 (2011)MathSciNetCrossRefMATHGoogle Scholar
  21. 21.
    Mumford, D.: Abelian varieties. In: Tata Institute of Fundamental Research Studies in Mathematics, vol. 5. Published for the Tata Institute of Fundamental Research, Bombay; by Hindustan Book Agency, New Delhi (2008). [With appendices by C. P. Ramanujam and Yuri Manin, corrected reprint of the second (1974) edition]Google Scholar
  22. 22.
    Némethi, A., Veys, W.: Generalized monodromy conjecture in dimension two. Geom. Topol. 16(1), 155–217 (2012)MathSciNetCrossRefMATHGoogle Scholar
  23. 23.
    Nicaise, J.: An introduction to \(p\)-adic and motivic zeta functions and the monodromy conjecture. In: Algebraic and Analytic Aspects of Zeta Functions and \(L\)-Functions, MSJ Mem., vol. 21, pp. 141–166. Math. Soc. Japan (2010)Google Scholar
  24. 24.
    Rodrigues, B., Veys, W.: Poles of zeta functions on normal surfaces. Proc. Lond. Math. Soc. (3) 87(1), 164–196 (2003)MathSciNetCrossRefMATHGoogle Scholar
  25. 25.
    Schrauwen, R.: Series of singularities and their topology. Ph.D. thesis, Rijksuniversiteit Utrecht, Utrecht (1990)Google Scholar
  26. 26.
    Schrauwen, R., Steenbrink, J., Stevens, J.: Spectral pairs and the topology of curve singularities. In: Complex Geometry and Lie Theory (Sundance, UT, 1989). Proc. Sympos. Pure Math., vol. 53, pp. 305–328. Am. Math. Soc., Providence (1991)Google Scholar
  27. 27.
    Veys, W.: Zeta functions for curves and log canonical models. Proc. Lond. Math. Soc. (3) 74(2), 360–378 (1997)MathSciNetCrossRefMATHGoogle Scholar
  28. 28.
    Veys, W.: Zeta functions and “Kontsevich invariants” on singular varieties. Can. J. Math. 53(4), 834–865 (2001)MathSciNetCrossRefMATHGoogle Scholar
  29. 29.
    Veys, W.: Monodromy eigenvalues and zeta functions with differential forms. Adv. Math. 213(1), 341–357 (2007)MathSciNetCrossRefMATHGoogle Scholar
  30. 30.
    Voisin, C.: Hodge theory and complex algebraic geometry. I, English ed. In: Cambridge Studies in Advanced Mathematics, vol. 76. Cambridge University Press, Cambridge (2007). (Translated from the French by Leila Schneps)Google Scholar
  31. 31.
    Weil, A.: On Picard varieties. Am. J. Math. 74, 865–894 (1952)MathSciNetCrossRefMATHGoogle Scholar
  32. 32.
    Włodarczyk, J.: Toroidal varieties and the weak factorization theorem. Invent. Math. 154(2), 223–331 (2003)MathSciNetCrossRefMATHGoogle Scholar

Copyright information

© Universidad Complutense de Madrid 2016

Authors and Affiliations

  1. 1.Dept. WiskundeKU LeuvenLouvainBelgium

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