Semi-simple Carrousels and the Monodromy

Massey, David B.

Massey, David B.

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

Abstract
Résumé

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

Soit $U$ un voisinage ouvert de l’origine dans $ℂ^{n + 1}$ et soit $f : (𝒰, 0) \to (ℂ, 0)$ une fonction analytique complexe. Soit $z_{0}$ une forme linéaire générale sur $ℂ^{n + 1}$ . Si la courbe polaire relative $Γ_{f, z_{0}}^{1}$ 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 $(Γ_{f, z_{0}}^{1} {\cdot V (f))}_{0}$ n’est pas premier.

MR 2228681 | Zbl 1102.32013

DOI : https://doi.org/10.5802/aif.2173
Classification: 32B99, 32A27, 14E99
Keywords: Carrousel, polar curve, monodromy, Milnor fiber

@article{AIF_2006__56_1_85_0,
     author = {Massey, David B.},
     title = {Semi-simple Carrousels and the Monodromy},
     journal = {Annales de l'Institut Fourier},
     publisher = {Association des Annales de l'institut Fourier},
     volume = {56},
     number = {1},
     year = {2006},
     pages = {85-100},
     doi = {10.5802/aif.2173},
     mrnumber = {2228681},
     zbl = {1102.32013},
     language = {en},
     url = {http://www.numdam.org/item/AIF_2006__56_1_85_0}
}

Massey, David B. Semi-simple Carrousels and the Monodromy. Annales de l'Institut Fourier, Volume 56 (2006) no. 1, pp. 85-100. doi : 10.5802/aif.2173. http://www.numdam.org/item/AIF_2006__56_1_85_0/

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