2006/01/10 by Daniel Barlet, D. Barlet, Barlet, D.
Computer Science · Mathematics · #32S05 #32S25 #32S40 #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #Complex Variables (math.CV) #FOS: Mathematics #Meromorphic and Entire Functions #Polynomial and algebraic computation #math.AG #math.CV #msc:32S05 #msc:32S25 #msc:32S40
paper · pdf · doi:10.48550/arxiv.math/0601210
arxiv created 2006/01/10 · openalex publication_date 2006/01/10 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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 (1)/(Γ(λ)) ∫X | g |2λg-n \square 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.