Comptes Rendus
no. 3-4
Analytic Geometry/Topology
A remark on vanishing cycles with two strata
[Une remarque sur les cycles évanescents à deux strates]

Présenté par : Bernard Malgrange

Dũng Tráng Lê1 ; David B. Massey2
1 CMI, Université de Provence, 13453 Marseille cedex 13, France
2 Department of Mathematics, Northeastern University, Boston, MA 02115, USA
Comptes Rendus. Mathématique, Volume 350 (2012) no. 3-4, pp. 217-220.

Supposons que le lieu critique Σ dʼune fonction analytique complexe f sur un espace affine soit un espace avec un point singulier isolé à lʼorigine 0, et que le nombre de Milnor de la fonction f restreinte à des sections transverses à Σ−{0} soit constant. Alors, la théorie générale des faisceaux pervers impose des conditions strictes sur la cohomologie de la fibre de Milnor de f en 0 et, de façon encore plus surprenante, des restrictions sur la cohomologie de la fibre de Milnor dʼune section hyperplane générique.

Suppose that the critical locus Σ of a complex analytic function f on affine space is, itself, a space with an isolated singular point at the origin 0, and that the Milnor number of f restricted to normal slices of Σ−{0} is constant. Then, the general theory of perverse sheaves puts severe restrictions on the cohomology of the Milnor fiber of f at 0, and even more surprising restrictions on the cohomology of the Milnor fiber of generic hyperplane slices.

Reçu le :
Accepté le :
Publié le :
DOI : 10.1016/j.crma.2012.01.008
Dũng Tráng Lê 1 ; David B. Massey 2

1 CMI, Université de Provence, 13453 Marseille cedex 13, France
2 Department of Mathematics, Northeastern University, Boston, MA 02115, USA 
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Dũng Tráng Lê; David B. Massey. A remark on vanishing cycles with two strata. Comptes Rendus. Mathématique, Volume 350 (2012) no. 3-4, pp. 217-220. doi : 10.1016/j.crma.2012.01.008. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2012.01.008/

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[12] M. Shubladze Singularities with critical locus an complete intersection and transversal type A1 http://www.mpim-bonn.mpg.de/preblob/4141 (Preprint 2010-16, Max Planck Institute for Mathematics preprints)

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☆ This Note was written with the help of the Research fund of Northeastern University.

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