Generalized Brieskorn module I. Convergent (a,b) modules Version 2

TL;DR

This paper develops a theory of generalized convergent Brieskorn modules to analyze the poles of distributions related to powers of a holomorphic function, extending previous results with new algebraic and asymptotic tools.

Contribution

It introduces the semi-simple filtration of generalized Brieskorn modules and establishes their description via Nilsson class asymptotics, enabling analysis of higher Bernstein polynomials.

Findings

Full description of generalized Brieskorn modules in terms of Nilsson class expansions

Introduction of semi-simple filtration to account for monodromy Jordan blocks

Explicit relationship between semi-simple filtration and monodromy nilpotent filtration

Abstract

This paper is the first one of three papers whose goal is to give a converse to the main result of my previous paper, so to prove the existence of multiple poles for the distribution associated to powers of f with an hypothesis on a Higher Bernstein Polynomial of the Brieskorn module generated by the germ of a given holomorphic volume form. Note that, even for the existence of a simple pole this converse is already new. A 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 which does not give access to the cohomology of the Milnor fiber of f, which by definition, is outside the zero set of f. This leads to introduce generalized Brieskorn modules which allow this passage. The first aim of this part I is to give a solid basis of the theory of generalized (convergent) Brieskorn modules. In order to take into account Jordan blocks of the monodromy in our results we introduce the semi-simple filtration of a generalized Brieskorn 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 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 and the nilpotent filtration of the monodromy on its saturation