Higher Bernstein Polynomials and Multiple Poles

Abstract The goal of this paper is to give a converse to the main result of my previous paper [12], so to prove the existence of a pole with an hypothesis on the Bernstein polynomial of the (a,b)-module generated by the germ ω ∈ Ωn+1. A 0 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) → (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 convergent (a,b)-modules which allow this passage. In order to take in account Jordan blocs of the monodromy in our result we introduce the semi-simple filtration of a (convergent) geometric (a,b)-module and define the higher order Bernstein polynomials in this context which corresponds to a decomposition of the ”standard” Bernstein polynomial in the case of frescos. Our main result is to show that the existence of a root in −α − N for the p-th Bernstein polynomial of the fresco generated by a holomorphic form ω ∈ Ωn+1 in 0 the (convergent) Brieskorn (a,b)-module Hn+1 associated to f, under the hypothesis 0 that f has an isolated singularity at the origin relative to the eigenvalue exp(2iπα) of the monodromy, produces poles of order at least p for the meromorphic extension of the (conjugate) analytic functional, for some h ∈ Z: 1???? ̄ω′ ∈ Ωn+1 ????→ |f|2λf−hρω ∧ ω ̄′ 0 Γ(λ) Cn+1 at points −α − N for N and h well chosen integers. This result is new, even for p = 1. As a corollary, this implies that in this situation the existence of a root in −α − N for the p-th Bernstein polynomial of the fresco generated by a holomorphic ∗Barlet Daniel, Institut Elie Cartan UMR 7502Universit ́e de Lorraine, CNRS, INRIA et Institut Universitaire de France, BP 239 - F - 54506 Vandoeuvre-l`es-Nancy Cedex.France.e-mail : daniel.barlet@univ-lorraine.fr 1 form ω ∈ Ωn+1 implies the existence of at least p roots (counting multiplicities) for 0 the usual reduced Bernstein polynomial of the germ (f, 0).In the case of an isolated singularity we obtain that for each α ∈]0, 1] ∩ Q the biggest root −α − m of the reduced Bernstein polynomial of f in −α − N produces a pole at −α − m for some h ∈ Z for the meromorphic extension of the distribution 1 ̄□ −→ Γ(λ)|f|2λf−h□. AMS classification. 32 S 25; 32 S 40 ; 34 E 05