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

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.