2023/07/10 by Barlet, Daniel
#Algebraic Geometry (math.AG) #Complex Variables (math.CV) #FOS: Mathematics
paper · doi:10.48550/arxiv.2307.04395
The goal of this paper is to give a converse to the main result of my previous paper \cite[B.22], 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+10. 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: (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+10 in the (convergent) Brieskorn (a,b)-module Hn+10 associated to f, under the hypothesis 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' ∈ Ωn+10 ↦ (1)/(Γ(λ))∫ℂn+1 \vert f\vert2λ f-h ρω\wedge 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+10 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 \square \longrightarrow (1)/(Γ(λ))\vert f\vert2λ f-h\square.