On the Brieskorn (a,b)-module of an hypersurface singularity

Original · EN

We show in this note that for a germ g of holomorphic function with an isolated singularity at the origin of Cⁿ there is a pole for the meromorphic extension of the distribution equation* 1/Γ(λ) ∫ₓ | g |²λg⁻ⁿ * equation* at - n - α when α is the smallest root in its class modulo Z of the reduce Bernstein-Sato polynomial of g. This is rather unexpected result comes from the fact that the self-duality of the Brieskorn (a,b)-module Eg associated to g exchanges the biggest simple pole sub-(a,b)-module of Eg with the saturation of Eg by b⁻¹a. In the first part of this note, we prove that the biggest simple pole sub-(a,b)-module of the Briekorn (a,b)-module E of g is "geometric" in the sense that it depends only on the hypersurface germ {g = 0 } at the origin in Cⁿ and not on the precise choice of the reduced equation g, as the poles of (*). By duality, we deduce the same property for the saturation E of E. This duality gives also the relation between the "dual" Bernstein-Sato polynomial and the usual one, which is the key of the proof of the theorem.