Monodromy of a family of hypersurfaces
[Monodromie d'une famille d'hypersurfaces]
Annales scientifiques de l'École Normale Supérieure, Série 4, Tome 42 (2009) no. 3, pp. 517-529.

Soit Y une variété projective complexe lisse irréductible de dimension m+1, plongée dans un espace projectif. Soit Z un sous-schéma fermé de Y, et soit δ un entier positif tel que Z,Y (δ) soit engendré par ses sections globales. Fixons un entier dδ+1, et supposons que le diviseur général X|H 0 (Y, Z,Y (d))| soit lisse. Désignons par H m (X;) Z van le quotient de H m (X;) par la cohomologie de Y et par les classes des composantes irréductibles de Z de dimension m. Dans cet article, nous prouvons que la représentation de monodromie sur H m (X;) Z van pour la famille des diviseurs lisses X|H 0 (Y, Z,Y (d))| est irréductible.

Let Y be an (m+1)-dimensional irreducible smooth complex projective variety embedded in a projective space. Let Z be a closed subscheme of Y, and δ be a positive integer such that Z,Y (δ) is generated by global sections. Fix an integer dδ+1, and assume the general divisor X|H 0 (Y, Z,Y (d))| is smooth. Denote by H m (X;) Z van the quotient of H m (X;) by the cohomology of Y and also by the cycle classes of the irreducible components of dimension m of Z. In the present paper we prove that the monodromy representation on H m (X;) Z van for the family of smooth divisors X|H 0 (Y, Z,Y (d))| is irreducible.

DOI : 10.24033/asens.2101
Classification : 14B05, 14C20, 14C21, 14C25, 14D05, 14M10, 32S55
Keywords: complex projective variety, linear system, Lefschetz theory, monodromy, isolated singularity, Milnor fibration
Mot clés : variété projective lisse, système linéaire, théorie de Lefschetz, monodromie, singularité isolée, fibration de Milnor
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     title = {Monodromy of a family of hypersurfaces},
     journal = {Annales scientifiques de l'\'Ecole Normale Sup\'erieure},
     pages = {517--529},
     publisher = {Soci\'et\'e math\'ematique de France},
     volume = {Ser. 4, 42},
     number = {3},
     year = {2009},
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Di Gennaro, Vincenzo; Franco, Davide. Monodromy of a family of hypersurfaces. Annales scientifiques de l'École Normale Supérieure, Série 4, Tome 42 (2009) no. 3, pp. 517-529. doi : 10.24033/asens.2101. http://www.numdam.org/articles/10.24033/asens.2101/

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