Sur l'application des p\'eriodes d'une Variation de Structure de Hodge attach\'ee aux familles d'hypersurfaces \'a singularit\'es simples

Abstract

Let $n$ be a positive even integer and $d$ a positive integer . To every complete family $Z$ of n dimensional degree d hypersurfaces in the projective space with isolated A-D-E singularities we construct according to an idea of Carlson-Toledo a Deligne-Mumford stack $\bar Z$ whose moduli space is $Z$ such that the monodromy representation extends. We study the corresponding periods mapping and establish an infinitesimal Torelli theorem along the isosingular strata of lZ$ under transversality assumptions. We apply this result to prove Steiness of the universal covering space of $\bar Z$.