Skip Navigation


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
This Article
Right arrow Full Text (PDF)
Right arrow All Versions of this Article:
2/3/517    most recent
jtp021v1
Right arrow References
Right arrow Alert me when this article is cited
Right arrow Alert me if a correction is posted
Services
Right arrow Email this article to a friend
Right arrow Similar articles in this journal
Right arrow Alert me to new issues of the journal
Right arrow Add to My Personal Archive
Right arrow Download to citation manager
Right arrowRequest Permissions
Google Scholar
Right arrow Articles by Melle-Hernández, A.
Right arrow Articles by Veys, W.
Right arrow Search for Related Content
Social Bookmarking
Add to CiteULike   Add to Connotea   Add to Del.icio.us  
What's this?

© 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


  Abstract

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


Add to CiteULike CiteULike   Add to Connotea Connotea   Add to Del.icio.us Del.icio.us    What's this?




Disclaimer: Please note that abstracts for content published before 1996 were created through digital scanning and may therefore not exactly replicate the text of the original print issues. All efforts have been made to ensure accuracy, but the Publisher will not be held responsible for any remaining inaccuracies. If you require any further clarification, please contact our Customer Services Department.