Generalized Brieskorn Modules I: Convergent (a,b)-modules

Abstract

This paper is the first one of two papers whose goal is to give a converse to the main result of my previous paper [6], so to prove the existence of multiple poles for the distribution |f|2$λ$ with an hypothesis on a Higher Bernstein Polynomial of the (a,b)-module generated by the germ $ω$$\in$$Ω$n+1 0 of a given holomorphic volum form. Note that, even for the existence of a simple pole this converse is already new. One difficulty to prove such a result comes from the use of the formal completion in f of the Brieskorn module of the holomorphic germ f\,: (Cn+1 ,0) $\rightarrow$(C,0) which does not give access to the cohomology of the Milnor's fiber of f, which by definition, is outside {f = 0}. This leads to introduce generalized Brieskorn modules (convergent geometric (a,b)-modules) which allow this passage. The first aim of this part I is to give a solid basis of the theory of convergent (a,b)-modules. In order to take in account Jordan blocs of the monodromy in our results we introduce the semi-simple filtration of a generalized Brieskorn module (convergent (a,b)-module) and we shall use it to define in part II the higher order Bernstein polynomials in this context. They correspond to a decomposition of the ``standard'' Bernstein polynomial of a generalized Brieskorn module, taking in account the nilpotent order of the monodromy. In this part I we obtain also a full description of generalized Brieskorn-modules in terms of (convergent) asymptotics expansions of Nilsson class which will be used as a starting point in part II. We conclude this part I by making explicite the relationship between the semi-simple filtration of a generalized Brieskorn module E and the nilpotent filtration of the monodromy on its saturation E___ .