Revista Matemática Complutense

, Volume 29, Issue 2, pp 455–483

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)
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)
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)
4. 4.
Bittner, F.: The universal Euler characteristic for varieties of characteristic zero. Compos. Math. 140(4), 1011–1032 (2004)
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)
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)
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)
18. 18.
Loeser, F.: Fonctions d’Igusa $$p$$-adiques et polynômes de Bernstein. Am. J. Math. 110(1), 1–21 (1988)
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)
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)
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)
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)
28. 28.
Veys, W.: Zeta functions and “Kontsevich invariants” on singular varieties. Can. J. Math. 53(4), 834–865 (2001)
29. 29.
Veys, W.: Monodromy eigenvalues and zeta functions with differential forms. Adv. Math. 213(1), 341–357 (2007)
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)
32. 32.
Włodarczyk, J.: Toroidal varieties and the weak factorization theorem. Invent. Math. 154(2), 223–331 (2003)