Calcul du nombre de cycles évanouissants d'une hypersurface complexe

Tráng, Lê Dũng

doi:10.5802/aif.491

Cited by 84 publications

(3 citation statements)

References 3 publications

(4 reference statements)

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“…The following definitions of the relative polar varieties of f differ slightly from their more classical construction (see, for example [18,24,26]),…”

Section: Ipa-deformationsmentioning

confidence: 99%

Deformation Formulas for Parameterized Hypersurfaces

Hepler

2024

Annales De l'Institut Fourier

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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 [dim X] 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.Résumé. -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 [dim X] 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. Generalizing Milnor's Formula to Higher DimensionsSuppose that U is an open neighborhood of the origin in C 2 . Let f 0 : (U, 0) → (C, 0) be a complex analytic function which has an isolated critical point at the origin. Thus, f 0 defines a plane curve V (f 0

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“…The following definitions of the relative polar varieties of f differ slightly from their more classical construction (see, for example [18,24,26]),…”

Section: Ipa-deformationsmentioning

confidence: 99%

Deformation Formulas for Parameterized Hypersurfaces

Hepler

2024

Annales De l'Institut Fourier

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“…Let µ(f ) and µ(f | Hx 1 ) denote the Milnor number of f and that of a hyperplane section f | Hx 1 of f , where f | Hx 1 is the restriction of f to the hyperplane H x 1 = {x ∈ X | x 1 = 0}. Then, the classical Lê-Teissier formula [17,43] and the Grothendieck local duality imply the following:…”

Section: Local Cohomology and Dualitymentioning

confidence: 99%

Computing Regular Meromorphic Differential Forms via Saito's Logarithmic Residues

Tajima

Nabeshima²

2021

SIGMA

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Logarithmic differential forms and logarithmic vector fields associated to a hypersurface with an isolated singularity are considered in the context of computational complex analysis. As applications, based on the concept of torsion differential forms due to A.G. Aleksandrov, regular meromorphic differential forms introduced by D. Barlet and M. Kersken, and Brieskorn formulae on Gauss-Manin connections are investigated. (i) A method is given to describe singular parts of regular meromorphic differential forms in terms of non-trivial logarithmic vector fields via Saito's logarithmic residues. The resulting algorithm is illustrated by using examples. (ii) A new link between Brieskorn formulae and logarithmic vector fields is discovered and an expression that rewrites Brieskorn formulae in terms of non-trivial logarithmic vector fields is presented. A new effective method is described to compute non trivial logarithmic vector fields which are suitable for the computation of Gauss-Manin connections. Some examples are given for illustration.

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“…Since σ : Z → (Z, z) is the minimal embedded resolution of {f g = 0} in Z, the map π • ϕ factorizes uniquely through σ: Recall that the Milnor number of f at z (introduced in [12]) is a topological invariant of the germ of f = 0 at z (see [6] p. 261). It is the number of vanishing cycles of f = 0 at z and equals the first Betti number of {f = t} ∩ B ε (z), where B ε (z) is a sufficiently small ball centered at z and ε |t| > 0.…”

Section: On the Other Handmentioning

confidence: 99%

Linear systems and Multiplicity of ideals

Tráng

2009

São Paulo J. Math. Sci.

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Calcul du nombre de cycles évanouissants d'une hypersurface complexe

Cited by 84 publications

References 3 publications

Deformation Formulas for Parameterized Hypersurfaces

Deformation Formulas for Parameterized Hypersurfaces

Computing Regular Meromorphic Differential Forms via Saito's Logarithmic Residues

Linear systems and Multiplicity of ideals

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