Monodromy of a family of hypersurfaces
Annales scientifiques de l'École Normale Supérieure, Serie 4, Volume 42 (2009) no. 3, p. 517-529

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.

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.

DOI : https://doi.org/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
@article{ASENS_2009_4_42_3_517_0,
     author = {Di Gennaro, Vincenzo and Franco, Davide},
     title = {Monodromy of a family of hypersurfaces},
     journal = {Annales scientifiques de l'\'Ecole Normale Sup\'erieure},
     publisher = {Soci\'et\'e math\'ematique de France},
     volume = {Ser. 4, 42},
     number = {3},
     year = {2009},
     pages = {517-529},
     doi = {10.24033/asens.2101},
     zbl = {1194.14016},
     mrnumber = {2543331},
     language = {en},
     url = {http://www.numdam.org/item/ASENS_2009_4_42_3_517_0}
}
Di Gennaro, Vincenzo; Franco, Davide. Monodromy of a family of hypersurfaces. Annales scientifiques de l'École Normale Supérieure, Serie 4, Volume 42 (2009) no. 3, pp. 517-529. doi : 10.24033/asens.2101. http://www.numdam.org/item/ASENS_2009_4_42_3_517_0/

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