On Principal Value and Standard Extension of Distributions

TL;DR

This paper studies the extension of distributions associated with holomorphic functions, showing their meromorphic extension properties and implications for D-module theory, including non-torsion results and module generators.

Contribution

It establishes the equivalence between distribution limits and meromorphic extensions for holomorphic functions, leading to new insights in D-module theory and applications to algebraic equations.

Findings

Distribution limits coincide with meromorphic extensions.

Distributions have the Standard Extension Property.

Results apply to generators of conjugate modules of holonomic D-modules.

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].