On Principal Value and Standard Extension of Distributions
Résumé
For a holomorphic function f on a complex manifold M we explain in this article that the distribution associated to |f | 2α (Log|f | 2) q f −N by taking the corresponding limit on the sets {|f | ≥ ε} when ε goes to 0, coincides for (α) non negative and q, N ∈ N, with the value at λ = α of the meromorphic extension of the distribution |f | 2λ (Log|f | 2) q f −N. This implies that any distribution in the D Mmodule generated by such a distribution has the Standard Extension Property. This implies a non torsion result for the D M-module generated by such a distribution. As an application of this result we determine generators for the conjugate modules of the regular holonomic D-modules associated to z(σ) λ , the power λ, where λ is any complex number, of the (multivalued) root of the universal equation of degree k, z k + k j=1 (−1) h σ h z k−h = 0 whose structure is studied in [4].
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