On Principal Value and Standard Extension of Distributions

DOI10.7146/MATH.SCAND.A-134458arXiv2204.01309OpenAlexW4280588321MaRDI QIDQ6395578

Daniel Barlet

Publication date: 4 April 2022

Abstract: For a holomorphic function f on a complex manifold M we explain in this article that the distribution associated to |f | 2$alpha$ (Log|f | 2) q f --N by taking the corresponding limit on the sets {|f | $ge$ $epsilon$} when $epsilon$ goes to 0, coincides for ($alpha$) non negative and q, N $in$ N, with the value at $lambda$ = $alpha$ of the meromorphic extension of the distribution |f | 2$lambda$ (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($sigma$) $lambda$ , the power $lambda$, where $lambda$ is any complex number, of the (multivalued) root of the universal equation of degree k, z k + k j=1 (--1) h $sigma$ h z k--h = 0 whose structure is studied in [4].

Full work available at URL: https://doi.org/10.7146/math.scand.a-134458

Mathematics Subject Classification ID

Complex singularities (32Sxx) Analytic spaces (32Cxx) (Co)homology theory in algebraic geometry (14Fxx)

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