Another proof of the Riemann-Hilbert Correspondence for Regular Holonomic D-Modules

TL;DR

This paper provides a new proof of the Riemann-Hilbert correspondence for regular holonomic D-modules, leveraging the irregular correspondence and enhanced ind-sheaves, and extends the proof to the algebraic setting.

Contribution

It offers a novel proof of the classical Riemann-Hilbert correspondence using modern irregular theory and enhanced ind-sheaves, also establishing the algebraic case with the same approach.

Findings

Reproves the Riemann-Hilbert correspondence for regular holonomic D-modules.

Extends the proof to algebraic D-modules.

Utilizes the irregular Riemann-Hilbert correspondence and enhanced ind-sheaves.

Abstract

In this paper, we reprove the Riemann-Hilbert correspondence for regular holonomic D-modules of [M. Kashiwara, Publ. Res. Inst. Math. Sci., 1984] (see also [Z. Mebkhout, Compositio Math., 1984.]) by using the irregular Riemann-Hilbert correspondence of [A. D'Agnolo and M. Kashiwara, Publ. Math. Inst. Hautes Etudes Sci., 2016]. Moreover, we also prove the algebraic one by the same argument. For this purpose, we study C-constructible enhanced ind-sheaves of [Y. Ito, Tsukuba journal of Mathematics, 2020, Rend. Sem. Mat. Univ. Padova., 2021.] in more detail.