Monodromy eigenvalues and zeta functions with differential forms

For a complex polynomial or analytic function f, one has been studying intensively its so-called local zeta functions or complex powers; these are integrals of |f|^{2s}w considered as functions in s, where the w are differential forms with compact support. There is a strong correspondence between their poles and the eigenvalues of the local monodromy of f. In particular Barlet showed that each monodromy eigenvalue of f is of the form exp(a2i\pi), where a is such a pole. We prove an analogous result for similar p-adic complex powers, called Igusa (local) zeta functions, but mainly for the related algebro-geometric topological and motivic zeta functions.Comment: To appear in Advances in Mathematics. 17 page

Zeta functions and monodromy for surfaces that are general for a toric idealistic cluster

In this article we consider surfaces that are general with respect to a 3- dimensional toric idealistic cluster. In particular, this means that blowing up a toric constellation provides an embedded resolution of singularities for these surfaces. First we give a formula for the topological zeta function directly in terms of the cluster. Then we study the eigenvalues of monodromy. In particular, we derive a useful criterion to be an eigenvalue. In a third part we prove the monodromy and the holomorphy conjecture for these surfaces