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Monodromy Jordan blocks, b-functions and poles of Zeta functions for germs of plane curves

Melle Hernández, Alejandro and Torrelli , Tristan and Veys, Willen (2010) Monodromy Jordan blocks, b-functions and poles of Zeta functions for germs of plane curves. Journal of Algebra, 324 (6). 1364 -1382. ISSN 0021-8693

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Abstract

We study the poles of several local zeta functions: the Igusa, topological and motivic zeta function associated to a germ of a holomorphic function in two variables. It was known that there is at most one double pole for (any of) these zeta functions which is then given by the log canonical threshold of the function at the singular point. If the germ is reduced Loeser showed that such a double pole always induces a monodromy eigenvalue with a Jordan block of size 2. Here we settle the non-reduced situation, describing precisely in which case such a Jordan block of maximal size 2 occurs. We also provide detailed information about the Bernstein-Sato polynomial in the relevant non-reduced situation, confirming a conjecture of Igusa, Denef and Loeser.

Item Type:Article
Uncontrolled Keywords:Igusa and topological zeta function; Bernstein-Sato polynomial; Monodromy; Log canonical threshold; Plane curve singularities
Subjects:Sciences > Mathematics > Algebraic geometry
ID Code:13914
Deposited On:18 Nov 2011 08:27
Last Modified:06 Feb 2014 09:55

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