On principal value and standard extension of distributions
For a holomorphic function f on a complex manifold M we explain in this article that the distribution associated to f vert^2α (Log f vert^2)^q f^-N by taking the corresponding limit on the sets { f vert ≥ 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 vert^2λ (Log f vert^2)^qf^-N. This implies that any distribution in the D_M -module generated by such a distribution has the standard extension property. This implies a non O_M 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 introduced and studied in [Barlet, D., On symmetric partial differential operators, Math. Z. 302 (2022), no. 3, 1627–1655] and [Barlet, D., On partial differential operators which annihilate the roots of the universal equation of degree k, arXiv:2101.01895] associated to the roots of universal equation of degree k, z^k + _h=1^k (-1)^hσ _hz^k-h = 0. , number=2, journal=Mathematica Scandinavica, author=Barlet, Daniel, year=2023, month=Jun.