Monodromy of a family of hypersurfaces

Di Gennaro, Vincenzo; Franco, Davide

Annales scientifiques de l'École Normale Supérieure, Série 4, Tome 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 $\delta$ be a positive integer such that ${ℐ}_{Z,Y}\left(\delta \right)$ is generated by global sections. Fix an integer $d\ge \delta +1$, and assume the general divisor $X\in |H^0(Y,{ℐ}_{Z,Y}\left(d\right))|$ is smooth. Denote by $H^m{(X;\mathbb{Q})}_{\perp Z}^{\mathrm{van}}$ the quotient of $H^m(X;\mathbb{Q})$ 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 $H^m{(X;\mathbb{Q})}_{\perp Z}^{\mathrm{van}}$ for the family of smooth divisors $X\in |H^0(Y,{ℐ}_{Z,Y}\left(d\right))|$ 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 $\delta$ un entier positif tel que ${ℐ}_{Z,Y}\left(\delta \right)$ soit engendré par ses sections globales. Fixons un entier $d\ge \delta +1$, et supposons que le diviseur général $X\in |H^0(Y,{ℐ}_{Z,Y}\left(d\right))|$ soit lisse. Désignons par $H^m{(X;\mathbb{Q})}_{\perp Z}^{\mathrm{van}}$ le quotient de $H^m(X;\mathbb{Q})$ 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 $H^m{(X;\mathbb{Q})}_{\perp Z}^{\mathrm{van}}$ pour la famille des diviseurs lisses $X\in |H^0(Y,{ℐ}_{Z,Y}\left(d\right))|$ est irréductible.

Classification : 14B05, 14C20, 14C21, 14C25, 14D05, 14M10, 32S55
Keywords: complex projective variety, linear system, Lefschetz theory, monodromy, isolated singularity, Milnor fibration
Mots-clés : variété projective lisse, système linéaire, théorie de Lefschetz, monodromie, singularité isolée, fibration de Milnor

[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. | Zbl | MR

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

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

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

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

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

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

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

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

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

[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. | Zbl | MR

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