Acta Mathematica Vietnamica

, Volume 43, Issue 1, pp 31–44 | Cite as

Igusa Zeta Functions and the Non-archimedean SYZ Fibration

  • Johannes NicaiseEmail author


We explain the proof, obtained in collaboration with Chenyang Xu, of a 1999 conjecture of Veys about poles of maximal order of Igusa zeta functions. The proof technique is based on the Minimal Model Program in birational geometry, but the proof was heavily inspired by ideas coming from non-archimedean geometry and mirror symmetry; we will outline these relations at the end of the paper. This text is intended to be a low-tech introduction to these topics; we only assume that the reader has a basic knowledge of algebraic geometry.


Igusa zeta functions Non-archimedean geometry Mirror symmetry 

Mathematics Subject Classification (2010)

14E30 14J33 11S40 



I would like to thank the organizers of the VIASM Annual Meeting 2017 for the invitation to deliver a lecture at the meeting and to write this survey article for the proceedings. The results presented here are a result of joint work with Mircea Mustaţă and Chenyang Xu, and it is a pleasure to thank them both for the pleasant and interesting collaboration. I am also indebted to Wim Veys for sharing his ideas on the conjecture that constitutes the main subject of this text.

Funding Information

The author is supported by the ERC Starting Grant MOTZETA (project 306610) of the European Research Council.


  1. 1.
    Berkovich, V.G.: Spectral Theory and Analytic Geometry over Non-Archimedean Fields. Volume 33 of Mathematical Surveys and Monographs. American Mathematical Society, Providence (1990)Google Scholar
  2. 2.
    Borevich, A.I., Shafarevich, I.R.: Number Theory. Volume 20 of Pure and Applied Mathematics. Academic Press, New York (1966)zbMATHGoogle Scholar
  3. 3.
    Bories, B., Veys, W.: Igusa’s p-adic local zeta function and the monodromy conjecture for non-degenerate surface singularities. Mem. Am. Math. Soc. 242, vii+ 131p (2016)MathSciNetzbMATHGoogle Scholar
  4. 4.
    Boucksom, S., Favre, C., Jonsson, M.: Valuations and plurisubharmonic singularities. Publ. Res. Inst. Math. Sci. 44(2), 449–494 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    de Fernex, T., Kollár, J., Xu, C.: The dual complex of singularities. In: Higher Dimensional Algebraic Geometry, in Honour of Professor Yujiro Kawamatas 60th Birthday. Volume 74 of Advanced Studies in Mathematics, pp 103–130. American Mathematical Society, Providence (2017)Google Scholar
  6. 6.
    Denef, J.: Local zeta functions and Euler characteristics. Duke Math. J. 63(3), 713–721 (1991)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Gross, M.: Mirror symmetry and the Strominger-Yau-Zaslow conjecture. In: Current Developments in Mathematics 2012, pp 133–191. International Press, Somerville (2013)Google Scholar
  8. 8.
    Halle, L.H., Nicaise, J.: Motivic zeta functions of degenerating Calabi-Yau varieties. To appear in Math. Ann., arXiv:
  9. 9.
    Igusa, J.: Complex powers and asymptotic expansions. II. J. Reine Angew. Math. 278/279, 307–321 (1975)zbMATHGoogle Scholar
  10. 10.
    Kollár, J.: Singularities of pairs. In: Algebraic Geometry – Santa Cruz 1995. Volume 62 of Proc. Sympos. Pure Math., Part 1, pp 221–287. American Mathematical Society, Providence (1997)Google Scholar
  11. 11.
    Kollár, J., Xu, C.: The dual complex of Calabi-Yau pairs. Invent. Math. 205 (3), 527–557 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Kontsevich, M., Soibelman, Y.: Affine structures and non-archimedean analytic spaces. In: Etingof, P., Retakh, V., Singer, I.M. (eds.) The Unity of Mathematics. in Honor of the Ninetieth Birthday of I. M. Gelfand. Volume 244 of Progress in Mathematics, pp 312–385. Birkhäuser Boston, Inc, Boston (2006)Google Scholar
  13. 13.
    Laeremans, A., Veys, W.: On the poles of maximal order of the topological zeta function. Bull. London Math. Soc. 31, 441–449 (1999)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Lemahieu, A., Van Proeyen, L.: Monodromy conjecture for nondegenerate surface singularities. Trans. Amer. Math. Soc. 363(9), 4801–4829 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Loeser, F.: Fonctions d’Igusa p-adiques et polynômes de Bernstein. Am. J. Math. 110(1), 1–21 (1988)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Mustaţă, M., Nicaise, J.: Weight functions on non-archimedean analytic spaces and the Kontsevich-Soibelman skeleton. Alg. Geom. 2(3), 365–404 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Nicaise, J.: An introduction to P-adic and motivic zeta functions and the monodromy conjecture. In: Bhowmik, G., Matsumoto, K., Tsumura, H. (eds.) Algebraic and Analytic Aspects of Zeta Functions and L-Functions. Volume 21 of MSJ Memoirs, pp. 115–140. Mathematical Society of Japan (2010)Google Scholar
  18. 18.
    Nicaise, J.: Berkovich Skeleta and Birational Geometry. In: Baker, M., Payne, S. (eds.) Nonarchimedean and Tropical Geometry. Simons Symposia, pp 173–194. Springer, Cham (2016)Google Scholar
  19. 19.
    Nicaise, J., Xu, C.: The essential skeleton of a degeneration of algebraic varieties. Am. Math. J. 138(6), 1645–1667 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Nicaise, J., Xu, C.: Poles of maximal order of motivic zeta functions. Duke Math. J. 165(2), 217–243 (2016)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Institute of Mathematics, Vietnam Academy of Science and Technology (VAST) and Springer Nature Singapore Pte Ltd. 2017

Authors and Affiliations

  1. 1.South Kensington CampusImperial College LondonLondonUK
  2. 2.Department of MathematicsKU LeuvenHeverleeBelgium

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