Semi-simple Carrousels and the Monodromy

Massey, David B.

Annales de l'Institut Fourier, Tome 56 (2006) no. 1, pp. 85-100.

Soit $U$ un voisinage ouvert de l’origine dans ${ℂ}^{n+1}$ et soit $f:(𝒰,\mathbf{0})\to (ℂ,0)$ une fonction analytique complexe. Soit $z_0$ une forme linéaire générale sur ${ℂ}^{n+1}$. Si la courbe polaire relative ${\Gamma }_{f,z_0}^1$ à l’origine est irréductible et le nombre d’intersection est premier, alors cela impose des contraintes très fortes sur la valeur du rang de la $n$-ième cohomologie de la fibre de Milnor à l’origine. Nous obtenons aussi des résultats intéressants, mais plus faibles quand $({\Gamma }_{f,z_0}^1{·V\left(f\right))}_{\mathbf{0}}$ n’est pas premier.

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.

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