Semi-simple Carrousels and the Monodromy

Abstract

Let $\Cal U$ be an open neighborhood of the origin in ${\mathbb{C}}^{n+1}$ and let $f:(\Cal U, \bold 0)\to(\Bbb C, 0)$ be complex analytic. Let $z_0$ be a generic linear form on ${\mathbb{C}}^{n+1}$. If the relative polar curve ${\Gamma }_{f,z_0}^1$ at the origin is irreducible and the intersection number $({\Gamma }_{f,z_0}^1\cdot V(f){)}_0$ is prime, then there are severe restrictions on the possible degree $n$ cohomology of the Milnor fiber at the origin. We also obtain some interesting, weaker, results when $({\Gamma }_{f,z_0}^1\cdot V(f){)}_0$ is not prime.

13 pages