THE TOPOLOGICAL ZETA FUNCTION ASSOCIATED TO A FUNCTION ON A NORMAL SURFACE GERM

We associate to a regular function f on a normal surface germ (S, 0) an invariant, called the topological zeta function, which generalizes the same invariant for a plane curve germ; by definition it is a rational function in one variable.We study its poles and their relation with the local monodromy of f, in particular, we prove the “generalized holomorphy conjecture”. We give a formula for this topological zeta function in terms of the log canonical modelof (S, f -1,{0}), and we also introduce a still more general invariant.