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Journal of Topology Advance Access originally published online on September 17, 2009
Journal of Topology 2009 2(3):517-526; doi:10.1112/jtopol/jtp021
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© 2009 London Mathematical Society

On ‘maximal’ poles of zeta functions, roots of b-functions, and monodromy Jordan blocks

A. Melle-Hernández

Facultad de Matemáticas, Universidad Complutense, Plaza de Ciencias 3, E-28040, Madrid, Spain amelle@mat.ucm.es

T. Torrelli

Laboratoire Jean Alexandre Dieudonné, Université de Nice-Sophia Antipolis, Faculté des Sciences, Parc Valrose, 06108 Nice Cedex 02, France tristan.torrelli@laposte.net

Willem Veys

University of Leuven, Department of Mathematics, Celestijnenlaan 200B, B-3001 Leuven (Heverlee), Belgium wim.veys@wis.kuleuven.be

The main objects of this study are the poles of several local zeta functions: the Igusa, topological, and motivic zeta function associated to a polynomial or (germ of) holomorphic function in n variables. We are interested in poles of maximal possible order n. In all known cases (curves, non-degenerate polynomials) there is at most one pole of maximal order n, which is then given by the log canonical threshold of the function at the corresponding singular point. For an isolated singular point we prove that if the log canonical threshold yields a pole of order n of the corresponding (local) zeta function, then it induces a root of the Bernstein–Sato polynomial of the given function of multiplicity n (proving one of the cases of the strongest form of a conjecture of Igusa–Denef–Loeser). For an arbitrary singular point, we show under the same assumption that the monodromy eigenvalue induced by the pole has ‘a Jordan block of size n on the (perverse) complex of nearby cycles’.

Received February 25, 2009.

2000 Mathematics Subject Classification Primary: 14B05, 32S25. Secondary: 11S80, 32S45

The first author is partially supported by Spanish Contract MTM2007-67908-C02-02. The third author is partially supported by the Fund of Scientific Research–Flanders (G.0318.06).

Dedicated with admiration to C. T. C. Wall on the occasion of his seventieth birthday



References

  1. Arnold V. I., Gussein-Zade S. M., Varchenko A. N. Singularities of differentiable maps (1988) 83. Boston, MA: Birkhaüser. vol. II: Monodromy and asymptotics of integrals, Monographs in Mathematics.
  2. Bernstein J. N. Analytic continuation of generalized functions with respect to a parameter. Funkts. Anal. (1972) 6:26–40.
  3. Bernstein I., Gel’fand S. Meromorphic property of the function P{lambda}. Funct. Anal. Appl. (1969) 3:68–69.[CrossRef]
  4. Björk J.-E. Rings of differential operators (1979) North-Holland: Amsterdam.
  5. Deligne P. Théorie de Hodge, II. Publ. Math. Inst. Hautes Études Sci. (1971) 40:5–57.[CrossRef]
  6. Deligne P. Le formalisme des cycles évanescents. In: SGA7 XIII and XIV (1973) Berlin: Springer. 82–115. Lecture Notes in Mathematics 340. 116–164.
  7. Denef J. Poles of p-adic complex powers and Newton polyhedra. Nieuw Arch. Wiskd. (1995) 13:289–295.
  8. Denef J., Loeser F. Caractéristiques d’Euler-Poincaré, fonctions zêta locales et modifications analytiques. J. Amer. Math. Soc. (1992) 5:705–720.[CrossRef]
  9. Dimca A., Saito M. Monodromy at infinity and the weights of cohomology. Compos. Math. (2003) 138:55–71.[CrossRef]
  10. van Doorn M. G. M., Steenbrink J. H. M. A supplement to the monodromy theorem. Abh. Math. Sem. Univ. Hamburg (1989) 59:225–233.[CrossRef]
  11. Igusa J. Complex powers and asymptotic expansions I. In: J. Reine Angew. Math. (1974) 110–130. 268/269.
  12. Igusa J. Complex powers and asymptotic expansions II. In: J. Reine Angew. Math. (1975) 307–321. 278/279.
  13. Igusa J. An introduction to the theory of local zeta functions (2000) 14. Providence, RI: American Mathematical Society. AMS/IP Studies in Advanced Mathematics.
  14. Kashiwara M. Vanishing cycle sheaves and holonomic systems of differential equations. In: Algebraic geometry (Tokyo/Kyoto, 1982) (1983) Berlin: Springer. 134–142. Lecture Notes in Mathematics 1016.
  15. Kollár J. Singularities of pairs. In: Algebraic geometry, Santa Cruz 1995 (1997) Providence, RI: American Mathematics Society. Proceedings of Symposia in Pure Mathematics 62.
  16. Laeremans A., Veys W. On the poles of maximal order of the topological zeta function. Bull. Lond. Math. Soc. (1999) 31:441–449.[CrossRef]
  17. Loeser F. Fonctions d’Igusa p-adiques et polynômes de Bernstein. Amer. J. Math. (1988) 110:1–21.[CrossRef]
  18. Loeser F. Fonctions d’Igusa p-adiques, polynômes de Bernstein, et polyèdres de Newton. J. Reine Angew. Math. (1990) 412:75–96.
  19. Saito M. Modules de Hodge polarisables. Publ. Res. Inst. Math. Sci. (1988) 24:849–995.[CrossRef]
  20. Saito M. Mixed Hodge modules. Publ. RIMS, Kyoto Univ. (1990) 26:221–333.[CrossRef]
  21. Saito M. Multiplier ideals, b-function, and spectrum of a hypersurface singularity. Compos. Math. (2007) 143:1050–1068.
  22. Sato M., Shintani T. On zeta functions associated with prehomogeneous vector spaces. Proc. Natl. Acad. Sci. USA (1972) 69:1081–1082.[Abstract/Free Full Text]
  23. Steenbrink J. H. M. Mixed Hodge structure on the vanishing cohomology. In: Real and complex singularities (1977) 1976. Alphen aan den Rijn: Sijthoff & Noordhoff. 525–563. Proceedings of Nordic Summer School, Oslo.
  24. Steenbrink J. H. M. Applications of Hodge theory to singularities. 569–576. Proceedings of the International Congress of Mathematicians, Kyoto, 1990 (Mathematical Society of Japan, Tokyo, 1991).
  25. Varchenko A. N. Asymptotic Hodge structure in the vanishing cohomology. Russian Math. (1982) 18(Iz. VUZ):469–512.
  26. Veys W. Determination of the poles of the topological zeta function for curves. Manuscripta Math. (1995) 87:435–448.[CrossRef]

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