Deformation Formulas for Parameterized Hypersurfaces
[Formules de déformation pour les hypersurfaces paramétrées]
Hepler, Brian1
1 University of Wisconsin-Madison (USA)
Annales de l'Institut Fourier, Online first, 36 p.

Nous étudions les déformations à un paramètre de fonctions sur un espace affine qui définissent des hypersurfaces paramétrables. Avec l’hypothèse d’une activité polaire isolée à l’origine, nous pouvons exprimer complètement les nombres Lê de la fibre spéciale en fonction des nombres Lê de la fibre générique et des multiplicités polaires caractéristiques de la complexe comparaison, un faisceau pervers naturellement associé à tout espace analytique complexe réduit sur lequel le faisceau constant ℚ X • [dimX] est pervers. Cela généralise la formule classique du nombre de Milnor d’une courbe plane en termes de points doubles ainsi que le nombre de Milnor de l’image de Mond. Nous récupérons également les résultats de Gaffney et Bobadilla en utilisant ce cadre. Nous obtenons des formules de déformation similaires pour les cartes de ℂ 2 à ℂ 3 , et fournissons un ansatz pour obtenir des formules de déformation pour toutes les dimensions dans les dimensions agréables de Mather.

We investigate one-parameter deformations of functions on affine space which define parameterizable hypersurfaces. With the assumption of isolated polar activity at the origin, we are able to completely express the Lê numbers of the special fiber in terms of the Lê numbers of the generic fiber and the characteristic polar multiplicities of the comparison complex, a perverse sheaf naturally associated to any reduced complex analytic space on which the constant sheaf ℚ X • [dimX] is perverse. This generalizes the classical formula for the Milnor number of a plane curve in terms of double points as well as Mond’s image Milnor number. We also recover results of Gaffney and Bobadilla using this framework. We obtain similar deformation formulas for maps from ℂ 2 to ℂ 3 , and provide an ansatz for obtaining deformation formulas for all dimensions within Mather’s nice dimensions.

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Révisé le :
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DOI : 10.5802/aif.3613
Classification : 32B10, 32S25, 32S30, 32S55, 32S60, 14B07
Keywords: Milnor fiber, perverse sheaf, intersection cohomology, non-isolated singularities.
Mot clés : Les faiscieaux pervers, la fibre de Milnor, la cohomologie d’intersection, les singularités non isolées.
Hepler, Brian 1

1 University of Wisconsin-Madison (USA) 
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Hepler, Brian. Deformation Formulas for Parameterized Hypersurfaces. Annales de l'Institut Fourier, Online first, 36 p.

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