Singularities at Infinity and their Vanishing Cycles, II. Monodromy

Abstract

Let $f:C^n\to C$ be any polynomial function. By using global polar methods, we introduce models for the fibers of $f$ and we study the monodromy at atypical values of $f$, including the value infinity. We construct a geometric monodromy with controlled behavior and define global relative monodromy with respect to a general linear form. We prove localization results for the relative monodromy and derive a zeta-function formula for the monodromy around an atypical value. We compute the relative zeta function in several cases and emphasize the differences to the “classical” local situation.