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Motivic Serre invariants, ramification, and the analytic Milnor fiber

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References

  1. Grothendieck, A., Raynaud, M., Rim, D.S.: Groupes de monodromie en géométrie algébrique. I. Séminaire de Géométrie Algébrique du Bois-Marie 1967–1969 (SGA 7 I), Lect. Notes Math. 288, Springer, Berlin (1972)

  2. Deligne, P., Katz, N.: Groupes de monodromie en géométrie algébrique. II. Séminaire de Géométrie Algébrique du Bois-Marie 1967–1969 (SGA 7 II), Lect. Notes Math. 340, Springer, Berlin (1973)

  3. A’Campo, N.: Le nombre de Lefschetz d’une monodromie. Indag. Math., New Ser. 35, 113–118 (1973)

    MathSciNet  Google Scholar 

  4. A’Campo, N.: La fonction zêta d’une monodromie. Comment. Math. Helv. 50, 233–248 (1975)

    Article  MATH  MathSciNet  Google Scholar 

  5. Ayoub, J.: Motives of rigid varieties. Paper presented at the Conference on Hodge Theory, Venice, 19–24 June 2006

  6. Berkovich, V.G.: Étale cohomology for non-Archimedean analytic spaces. Publ. Math., Inst. Hautes Étud. Sci. 78, 5–171 (1993)

    MATH  MathSciNet  Google Scholar 

  7. Berkovich, V.G.: Vanishing cycles for formal schemes II. Invent. Math. 125(2), 367–390 (1996)

    Article  MATH  MathSciNet  Google Scholar 

  8. Berthelot, P.: Cohomologie rigide et cohomologie rigide à supports propres. Prepublication, Inst. Math. de Rennes (1996)

  9. Bosch, S.: Lectures on formal and rigid geometry. Preprint, http://www.math.uni-muenster.de/sfb/about/publ/bosch.html. Cited 20 Dec. 2006

  10. Bosch, S., Güntzer, U., Remmert, R.: Non-Archimedean analysis. A systematic approach to rigid analytic geometry. In: Grundlehren Math. Wiss., vol. 261. Springer, Berlin (1984)

    MATH  Google Scholar 

  11. Bosch, S., Lütkebohmert, W., Raynaud, M.: Néron models. Ergeb. Math. Grenzgeb., vol. 21. Springer, Berlin (1990)

    MATH  Google Scholar 

  12. Bosch, S., Lütkebohmert, W., Raynaud, M.: Formal and rigid geometry. III: The relative maximum principle. Math. Ann. 302(1), 1–29 (1995)

    Article  MATH  MathSciNet  Google Scholar 

  13. Bosch, S., Schlöter, K.: Néron models in the setting of formal and rigid geometry. Math. Ann. 301(2), 339–362 (1995)

    Article  MATH  MathSciNet  Google Scholar 

  14. Bravo, A., Encinas, S., Villamayor, O.: A simplified proof of desingularization and applications. Rev. Mat. Iberoam. 21(2), 349–458 (2005)

    MATH  Google Scholar 

  15. Cluckers, R., Loeser, F.: Constructible motivic functions, and motivic integration. Preprint 2005

  16. Conrad, B.: Irreducible components of rigid spaces. Ann. Inst. Fourier 49(2), 473–541 (1999)

    MATH  MathSciNet  Google Scholar 

  17. de Jong, A.J.: Crystalline Dieudonné module theory via formal and rigid geometry. Publ. Math., Inst. Hautes Étud. Sci. 82, 5–96 (1995)

    MATH  Google Scholar 

  18. Deligne, P.: La conjecture de Weil. I. Publ. Math., Inst. Hautes Étud. Sci. 43, 273–307 (1973)

    MATH  Google Scholar 

  19. Denef, J.: Report on Igusa’s local zeta function. In: Séminaire Bourbaki, vol. 1990/91, Exp. No.730–744, Astérisque vol. 201–203, pp. 359–386. SMF, Paris (1991)

  20. Denef, J.: Degree of local zeta functions and monodromy. Compos. Math. 89, 207–216 (1993)

    MATH  MathSciNet  Google Scholar 

  21. Denef, J., Loeser, F.: Motivic Igusa zeta functions. J. Algebr. Geom. 7, 505–537 (1998). arxiv:math.AG/9803040

    Google Scholar 

  22. Denef, J., Loeser, F.: Germs of arcs on singular algebraic varieties and motivic integration. Invent. Math. 135, 201–232 (1999) arxiv:math.AG/9803039

    Google Scholar 

  23. Denef, J., Loeser, F.: Geometry on arc spaces of algebraic varieties. Prog. Math. 201, 327–348 (2001) arxiv:math.AG/0006050

    Google Scholar 

  24. Denef, J., Loeser, F.: Lefschetz numbers of iterates of the monodromy and truncated arcs. Topology 41(5), 1031–1040 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  25. Eisenbud, D., Harris, J.: The geometry of schemes. Grad. Texts Math., vol. 197. Springer, New York (2000)

    MATH  Google Scholar 

  26. Grothendieck, A., Dieudonné, J.: Eléments de Géométrie Algébrique, iii. Publ. Math., Inst. Hautes Étud. Sci. 11, 5–167 (1961)

    Google Scholar 

  27. Grothendieck, A., Dieudonné, J.: Eléments de Géométrie Algébrique, iv, première partie. Publ. Math., Inst. Hautes Étud. Sci. 20, 5–259 (1964)

    Google Scholar 

  28. Grothendieck, A., Dieudonné, J.: Eléments de Géométrie Algébrique, iv, quatrième partie. Publ. Math., Inst. Hautes Étud. Sci. 32, 5–361 (1967)

    Google Scholar 

  29. Guibert, G., Loeser, F., Merle, M.: Iterated vanishing cycles, convolution, and a motivic analogue of a conjecture of Steenbrink. Duke Math. J. 132(3), 409–457 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  30. Igusa, J.: An introduction to the theory of local zeta functions. Stud. in Advanced Mathematics. AMS, Providence, RI (2000)

    Google Scholar 

  31. Illusie, L.: Théorie de Brauer et caractéristique d’Euler–Poincaré (d’après P. Deligne). The Euler–Poincaré characteristic. Astérisque 82, 161–172 (1981)

    MathSciNet  Google Scholar 

  32. Kempf, G., Knudsen, F., Mumford, D., Saint-Donat, B.: Toroidal embeddings 1. Lect. Notes Math., vol. 339. Springer, Berlin (1973)

    Google Scholar 

  33. Kontsevich, M.: Lecture at Orsay. 7 December 1995

  34. Kulikov, V.: Mixed Hodge Structures and Singularities. Camb. Tracts Math., vol. 132. Cambridge University Press, Cambridge (1998)

    MATH  Google Scholar 

  35. Loeser, F., Sebag, J.: Motivic integration on smooth rigid varieties and invariants of degenerations. Duke Math. J. 119, 315–344 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  36. Milnor, J.: Singular points of complex hypersurfaces. Ann. Math. Stud., vol. 61. Princeton University Press, Princeton, NJ (1968)

    MATH  Google Scholar 

  37. Nicaise, J., Sebag, J.: Rigid geometry and the monodromy conjecture. Paper presented at the Conference Proceedings of Singularités, CIRM, Luminy, 24 January–25 February 2005

  38. Nicaise, J., Sebag, J.: Invariant de Serre et fibre de Milnor analytique. C. R., Math., Acad. Sci. Paris 341(1), 21–24 (2005)

    MATH  MathSciNet  Google Scholar 

  39. Nicaise, J., Sebag, J.: Motivic Serre invariants and Weil restriction. Preprint (2006)

  40. Nicaise, J., Sebag, J.: Motivic Serre invariants of curves. To appear in Manuscr. Math.

  41. Poonen, B.: The Grothendieck ring of varieties is not a domain. Math. Res. Lett. 9(4), 493–498 (2002). arxiv:math.AG/0204306

    Google Scholar 

  42. Rodrigues, B., Veys, W.: Poles of Zeta functions on normal surfaces. Proc. Lond. Math. Soc. 87(3), 164–196 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  43. Sebag, J.: Intégration motivique sur les schémas formels. Bull. Soc. Math. Fr. 132(1), 1–54 (2004)

    MATH  MathSciNet  Google Scholar 

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Authors and Affiliations

  1. Laboratoire Painlevé, CNRS – UMR 8524, Université Lille 1, Cité Scientifique, 59655, Villeneuve d’Ascq Cédex, France

    Johannes Nicaise

  2. IMB, Laboratoire A2X, Université Bordeaux 1, 351 cours de la libération, 33405, Talence Cédex, France

    Julien Sebag

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  1. Johannes Nicaise
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Nicaise, J., Sebag, J. Motivic Serre invariants, ramification, and the analytic Milnor fiber. Invent. math. 168, 133–173 (2007). https://doi.org/10.1007/s00222-006-0029-7

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  • DOI: https://doi.org/10.1007/s00222-006-0029-7

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