On principal value and standard extension of distributions - Barlet, Daniel (Mathematica Scandinavica, 2023)
For a holomorphic function $f$ on a complex manifold $\mathscr {M}$ we explain in this article that the distribution associated to $\lvert f\rvert^{2α } (Log \lvert f\rvert^2)^q f^{-N}$ by taking the corresponding limit on the sets $\{ \lvert f\rvert ≥ ε \}$ when $ε $ goes to $0$, coincides for $\Re (α ) $ non negative and $q, N \in \mathbb {N}$, with the value at $λ = α $ of the meromorphic extension of the distribution $\lvert f\rvert^{2λ } (Log \lvert f\rvert^2)^qf^{-N}$. This implies that any distribution in the $\mathcal {D}_{\mathscr {M}}$-module generated by such a distribution has the standard extension property. This implies a non $\mathcal {O}_\mathscr {M}$ torsion result for the $\mathcal {D}_{\mathscr {M}}$-module generated by such a distribution. As an application of this result we determine generators for the conjugate modules of the regular holonomic $\mathcal {D}$-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 $k$, $z^k + \sum _{h=1}^k (-1)^hσ _hz^{k-h} = 0$.