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TOMOGRAPHIE DES VARIÉTÉS SINGULIÈRES ET THÉORÈMES DE LEFSCHETZ

  • CHRISTOPHE EYRAL (a1)

Abstract

{\frenchspacing Nous \'etudions l'homotopie d'une vari\'et\'e quasi-projective dans un espace projectif complexe selon la m\'ethode de Lefschetz, c'est-\`a-dire en consid\'erant ses sections par les hyperplans d'un pinceau (tomographie). En particulier, nous aboutissons \`a un th\'eor\`eme du type de Lefschetz qui g\'en\'eralise dans une certaine direction les meilleurs r\'esultats connus dus \`a Hamm, L\^e, Goresky et MacPherson. Ce th\'eor\`eme est d\'emontr\'e par r\'ecurrence sur la dimension de l'espace projectif ambiant \`a partir d'un th\'eor\`eme sur les pinceaux d'axe g\'en\'erique qui constitue le r\'esultat principal de l'article. Ce dernier compare la topologie de la vari\'et\'e \`a celle de sa section par un hyperplan g\'en\'erique du pinceau sur la base des comparaisons (section hyperplane g\'en\'erique -- section par l'axe du pinceau) et (sections hyperplanes exceptionnelles -- section par l'axe); l'incidence des singularit\'es est mesur\'ee par un invariant appel\'e `profondeur homotopique rectifi\'ee globale' (analogue global de la notion de profondeur homotopique rectifi\'ee de Grothendieck).} \vspace{6mm} \noindent We study the homotopy of a quasi-projective variety in a complex projective space following Lefschetz's method, that is, by considering its sections by the hyperplanes of a pencil (tomography). Specifically, we obtain a theorem of Lefschetz type which generalizes in a certain direction the best-known results due to Hamm, L\^e, Goresky and MacPherson. This theorem is proved by induction on the dimension of the ambient projective space with the help of a theorem on pencils with generic axis which is the main result of the paper. The latter compares the topology of the variety with that of its section by a generic hyperplane of the pencil, on the basis of the following comparisons: section by a generic hyperplane with section by the axis of the pencil; and sections by the exceptional hyperplanes with section by the axis. The effect of the singularities is measured by an invariant called `global rectified homotopical depth' (a global analogue of the notion of rectified homotopical depth of Grothendieck). E-mail: eyral@cmi.univ-mrs.fr 2000 Mathematics Subject Classification: 32S50, 14F35, 14F17.

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Keywords

TOMOGRAPHIE DES VARIÉTÉS SINGULIÈRES ET THÉORÈMES DE LEFSCHETZ

  • CHRISTOPHE EYRAL (a1)

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