Monodromy of a family of hypersurfaces
Annales scientifiques de l'École Normale Supérieure, Serie 4, Volume 42 (2009) no. 3, pp. 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: 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
@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},
     pages = {517--529},
     publisher = {Soci\'et\'e math\'ematique de France},
     volume = {Ser. 4, 42},
     number = {3},
     year = {2009},
     doi = {10.24033/asens.2101},
     mrnumber = {2543331},
     zbl = {1194.14016},
     language = {en},
     url = {http://www.numdam.org/articles/10.24033/asens.2101/}
}
TY  - JOUR
AU  - Di Gennaro, Vincenzo
AU  - Franco, Davide
TI  - Monodromy of a family of hypersurfaces
JO  - Annales scientifiques de l'École Normale Supérieure
PY  - 2009
SP  - 517
EP  - 529
VL  - 42
IS  - 3
PB  - Société mathématique de France
UR  - http://www.numdam.org/articles/10.24033/asens.2101/
DO  - 10.24033/asens.2101
LA  - en
ID  - ASENS_2009_4_42_3_517_0
ER  - 
%0 Journal Article
%A Di Gennaro, Vincenzo
%A Franco, Davide
%T Monodromy of a family of hypersurfaces
%J Annales scientifiques de l'École Normale Supérieure
%D 2009
%P 517-529
%V 42
%N 3
%I Société mathématique de France
%U http://www.numdam.org/articles/10.24033/asens.2101/
%R 10.24033/asens.2101
%G en
%F 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/articles/10.24033/asens.2101/

[1] E. Arbarello, M. Cornalba, P. A. Griffiths & J. Harris, Geometry of algebraic curves. Vol. I, Grund. Math. Wiss. 267, Springer, 1985. | Zbl

[2] V. Di Gennaro & D. Franco, Factoriality and Néron-Severi groups, Commun. Contemp. Math. 10 (2008), 745-764. | MR | Zbl

[3] A. Dimca, Sheaves in topology, Universitext, Springer, 2004. | MR | Zbl

[4] H. Flenner, L. O'Carroll & W. Vogel, Joins and intersections, Monographs in Mathematics, Springer, 1999. | MR | Zbl

[5] W. Fulton, Intersection theory, Ergebnisse Math. Grenzg. 2, Springer, 1984. | MR | Zbl

[6] R. Hartshorne, Algebraic geometry, Graduate Texts in Math. 52, Springer, 1977. | MR | Zbl

[7] K. Lamotke, The topology of complex projective varieties after S. Lefschetz, Topology 20 (1981), 15-51. | MR | Zbl

[8] E. J. N. Looijenga, Isolated singular points on complete intersections, London Mathematical Society Lecture Note Series 77, Cambridge University Press, 1984. | MR | Zbl

[9] A. Otwinowska & M. Saito, Monodromy of a family of hypersurfaces containing a given subvariety, Ann. Sci. École Norm. Sup. 38 (2005), 365-386. | Numdam | MR | Zbl

[10] A. N. Parshin & I. R. Shafarevich (éds.), Algebraic geometry. III, Encyclopaedia of Mathematical Sciences 36, Springer, 1998. | MR | Zbl

[11] E. H. Spanier, Algebraic topology, McGraw-Hill Book Co., 1966. | MR | Zbl

[12] J. H. M. Steenbrink, On the Picard group of certain smooth surfaces in weighted projective spaces, in Algebraic geometry (La Rábida, 1981), Lecture Notes in Math. 961, Springer, 1982, 302-313. | MR | Zbl

[13] C. Voisin, Théorie de Hodge et géométrie algébrique complexe, Cours Spécialisés 10, Soc. Math. France, 2002. | MR | Zbl

Cited by Sources: