Semi-simple Carrousels and the Monodromy

Let U be an open neighborhood of the origin in C and let f : (U ,0) → (C, 0) be complex analytic. Let z0 be a generic linear form on C. If the relative polar curve Γ1f,z0 at the origin is irreducible and the intersection number ( Γ f,z0 · 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 ( Γ f,z0 · V (f))0 is not prime. §0. Introduction In [Lê2] and [Lê3], Lê introduces his carrousel as a tool for analyzing the relative monodromy of the Milnor fiber of a function, f , modulo a hyperplane slice. In [T1] and [T2], Tibăr gives a careful presentation of Lê’s carrousel and uses it to obtain interesting results. Outside of the work of Lê and Tibăr, the carrousel seems to be a largely unused device. This is due in part to the complicated nature of the carrousel description. In this short paper, we look at some interesting special cases that occur and, in particular, look at the case where the relative polar curve, Γ1f,z0 , has a single component such that the intersection number ( Γ1f,z0 · V (f) ) 0 is prime. In this case, we show, in Theorem 2.3, how Lê’s carrousel tells one a great deal about the middle-dimensional homology/cohomology groups of the Milnor fiber of f , regardless of the dimension of the critical locus. §1. Lê’s Playground Let U be an open neighborhood of the origin in C and let f : (U ,0) → (C, 0) be a complex analytic function which has a critical point at the origin. Recall that a good stratification for f is a stratification S of V (f) which contains V (f) − Σf , and such that, for all S ∈ S, the pair (U − V (f), S) satisfies the af condition. After a linear change of coordinates, we may assume that the first coordinate, z0, is a prepolar form (or coordinate) for f at 0 (see [M1]); this means that there exists a neighborhood, W ⊆ U , of 0 such that, inside W − {0}, V (z0) transversely intersects all of the strata of a good stratification of V (f) (we do not need the condition of the frontier here – we could simply use a good partition). Then, at the origin, the relative polar curve Γ1f,z0 (see [M1]) is purely one-dimensional (or empty), and Γ 1 f,z0 properly intersects both V (f) and V (z0) (again, see [M1]). We always consider Γ 1 f,z0 with its cycle structure (see [M1]). We assume that U is small enough so that every component of Γ1f,z0 passes through the origin. Let D be a component of the cycle Γ1f,z0 (with either its reduced structure or its cycle structure). We have the following well-known formula, originally due to Teissier, ( D · V (f) ) 0 = ( D · V (z0) ) 0 + ( D · V ( ∂f ∂z0 ))