On atypical values and local monodromies of meromorphic functions

Abstract

A meromorphic function on a compact complex analytic manifold defines a C∞ locally trivial bundle over the complement to a finite subset of the projective line CP1, the bifurcation set. The monodromy transformations of this bundle correspond to loops around the points of the bifurcation set. In this paper we show that the zeta functions of these monodromy transformations {reviewer's remark: the inverse of the one defined by A'Campo} can be expressed in local terms, namely as integrals of the zeta functions of meromorphic germs with respect to the Euler characteristic. A special case of a meromorphic function on the projective space CPn is a function defined by a polynomial in n variables. We describe some applications of our technique to polynomial functions.