The monodromy conjecture for zeta functions associated to ideals in dimension two

Abstract

The monodromy conjecture states that every pole of the topological (or related) zeta function induces an eigenvalue of monodromy. This conjecture has already been studied a lot; however, in full generality it is proven only for zeta functions associated to a polynomial in two variables. In this article we consider zeta functions associated to an ideal. First we work in arbitrary dimension and obtain a formula (like the one of A'Campo) to compute the 'Verdier monodromy' eigenvalues associated to an ideal. Afterwards we prove a generalized monodromy conjecture for arbitrary ideals in two variables.