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.

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.

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.

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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