Higher Bernstein Polynomials and Multiple Poles of $1/ Γ(λ) \int_X |f|^2λ \bar f^{ −h} ρω ∧ \barω′$ version 2
Résumé
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: $$ω′ ∈ Ω^n+1 \omega' → \frac{1}{\Gamma(\lambda}|f|^{2λ} \bar f^{−h} ρω ∧\bar ω ̄$$ 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 form ω ∈ Ωn+1 implies the existence of at least p roots (counting multiplicities) for 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 $$□ → \frac{1}{Γ(λ)} |f|^{2λ}\bar f^{−h}□.$$
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