Périodes évanescentes et $(a,b)$-modules monogènes

Daniel Barlet (1, 2)

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

In order to describe the asymptotic behaviour of a vanishing period in a one parameter family we introduce and use a very simple algebraic structure : regular geometric (a,b)-modules generated (as left \ $\A-$modules) by one element. The idea is to use not the full Brieskorn module associated to the Gauss-Manin connection but a minimal (regular) differential equation satisfied by the period integral we are interested in. We show that the Bernstein polynomial associated is quite simple to compute for such (a,b)-modules and give a precise description of the exponents which appears in the asymptotic expansion which avoids integral shifts. We show a couple of explicit computations in some classical (but not so easy) examples.

Mots clés: vanishing cycles; vanishing periods; (a; b)-modules; Brieskorn modules; Gauss-Manin systems

Domaines: Géométrie algébrique [math.AG]

Daniel Barlet. Périodes évanescentes et $(a,b)$-modules monogènes. Bolettino UMI, 2009, Bollettino U.M.I. (9) II (II), p.651-697. ⟨10.48550/arXiv.0901.1953⟩. ⟨hal-00352700⟩