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

TL;DR

This paper studies the extension of monodromy representations and the infinitesimal Torelli theorem for families of hypersurfaces with simple singularities, leading to results on the Steinness of their universal covers.

Contribution

It constructs a Deligne-Mumford stack for hypersurface families with A-D-E singularities and proves an infinitesimal Torelli theorem along isosingular strata.

Findings

Monodromy representation extends to the stack.

Infinitesimal Torelli theorem holds under transversality.

Universal covers are Stein spaces.

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 $\bar Z$.