Le thème d'une période évanescente

Daniel Barlet (1, 2)

1 IECN - Institut Élie Cartan de Nancy
2 IUF - Institut universitaire de France

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.

Mots clés: Brieskorn module; Vanishing period; Berstein polynomial; filtered Gauss-Manin system; (a; b)-module; Brieskorn module.

Domaines: Variables complexes [math.CV]; Géométrie algébrique [math.AG]

Daniel Barlet. Le thème d'une période évanescente. 2009. ⟨hal-00437634⟩