Le th\`eme d'une p\'eriode \'evanescente

TL;DR

This paper introduces a new subclass of monogenic (a,b)-modules called themes, studies their properties, and constructs versal families for holomorphic deformations of Gauss-Manin systems related to vanishing period integrals.

Contribution

It defines themes with good functorial properties, establishes conditions for universal families, and explores their role in holomorphic deformations of Gauss-Manin systems.

Findings

Themes have canonical order on Bernstein polynomial roots.

Existence of finite dimensional versal families for given invariants.

Universal family may not exist for some invariant values.

Abstract

In this article we study holomorphic deformations of the filtered Gauss-Manin systems associated to a vanishing period integral. For that purpose we introduce a new sub-class of the class of monogenic (a,b)-modules (Brieskorn modules) which was studied in our previous article [B. 09]. We show that these new objects, called ?themes?, have good functorial properties and that there exists a canonical order on the roots of the corresponding Bernstein polynomial. We construct, for given fundamental invariants, a finite dimensional versal holomorphic family and we show that, when all themes with these fundamental invariants are ?stable?, this versal family is in fact universal. We also give a sufficient condition on the roots of the Bernstein polynomial in order that the previous condition is satisfied. We show with an example that a universal family may not exist for some values of the fundamental invariants.