On Atypical Values and Local Monodromies of Meromorphic Functions

A meromorphic function on a compact complex analytic manifold defines a $C^\infty$ locally trivial fibration over the complement to a finite set in the projective line $\mathbb{CP}^1$ – the bifurcation set. Loops around points of the bifurcation set give rise to corresponding monodromy transformations of this fibration. We show that the zeta-functions of these monodromy transformations can be expressed in local terms, namely, as integrals of zeta-functions of meromorphic germs with respect to the Euler characteristic. A particular case of meromorphic functions on the projective space $\mathbb{CP}^n$ are those defined by polynomial functions of $n$ variables. We describe some applications of this technique to polynomial functions.