Let $𝒰$ be an open neighborhood of the origin in ${ℂ}^{n+1}$ and let $f:(𝒰,\mathbf{0})\to (ℂ,0)$ be complex analytic. Let $z_0$ be a generic linear form on ${ℂ}^{n+1}$. If the relative polar curve ${\Gamma }_{f,z_0}^1$ at the origin is irreducible and the intersection number $({\Gamma }_{f,z_0}^1{·V\left(f\right))}_{\mathbf{0}}$ 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 $({\Gamma }_{f,z_0}^1{·V\left(f\right))}_{\mathbf{0}}$ is not prime.

We define open book structures with singular bindings. Starting with an extension of Milnor’s results on local fibrations for germs with nonisolated singularity, we find classes of genuine real analytic mappings which yield such open book structures.

Soit $k$ un corps de caractéristique nulle et $f$ une fonction non constante définie sur une variété lisse. Nous définissons dans cet article unefibre de Milnor motivique à l’infiniqui appartient à un anneau de Grothendieck des variétés. Elle est définie en termes d’une compactification choisie, non nécessairement lisse, mais est indépendante de ce choix. Lorsque $k$ est le corps des nombres complexes, en utilisant le morphisme de réalisation de Hodge, elle se réalise en le spectre à l’infini de $f$. Nous...

For a holomorphic function on a complex manifold, we show that the vanishing cohomology of lower degree at a point is determined by that for the points near it, using the perversity of the vanishing cycle complex. We calculate this order of vanishing explicitly in the case the hypersurface has simple normal crossings outside the point. We also give some applications to the size of Jordan blocks for monodromy.