Périodes évanescentes et (a,b)-modules monogènes

Barlet, Daniel

Bollettino della Unione matematica italiana, Série 9, Tome 2 (2009) no. 3, pp. 651-697

Résumé

In order to describe the asymptotic behaviour of a vanishing period in the degeneration of a one parameter family of complex manifolds, we introduce and use a very simple algebraic structure encoding the corresponding filtered Gauss-Manin connection: regular geometric (a,b)-module generated (as left $\widetilde{A}$-modules) by one element. The idea is to use not the full Brieskorn module associated to the Gauss-Manin connection but the minimal (regular) filtered 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 the efficiency of this tool on a couple of explicit computations in some classical (but not so easy) examples.