Iterated Vanishing Cycles, Convolution, and a Motivic Analogue of a Conjecture of Steenbrink

α∈Q nαt , nα in Z, which is constructed using the action of the monodromy on the mixed Hodge structure on the cohomology of the Milnor fiber at x. When f has an isolated singularity at x, all nα are in N, and the exponents of f , counted with multiplicity nα, are exactly the rational numbers α with nα not zero. Let us assume now that the singular locus of f is a curve Γ, having r local components Γl, 1 ≤ l ≤ r, in a neighborhood of x. We denote by ml the multiplicity of Γl. Let g be a generic linear form vanishing at x. For N large enough, the function f + g has an isolated singularity at x. In a neighborhood of the complement Γl to {x} in Γl, we may view f as a family of isolated hypersurface singularities parametrized by Γl . The cohomology of the Milnor fiber of this hypersurface singularity is naturally endowed with the action of two commuting monodromies: the monodromy of the function and the monodromy of a generator of the local fundamental group of Γl . We denote by αl,j the exponents of that isolated hypersurface singularity and by βl,j the corresponding rational numbers in [0, 1) such that the complex numbers exp(2πiβl,j) are the eigenvalues of the monodromy along Γ ◦ l .