  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.
