Kashiwara conjugation and the enhanced Riemann-Hilbert correspondence

TL;DR

This paper explores the compatibility of Kashiwara's conjugation functor with the enhanced Riemann-Hilbert correspondence, extending Galois descent results and analyzing how local decompositions influence monodromy data.

Contribution

It proves the compatibility of Kashiwara's conjugation with the enhanced De Rham functor and generalizes Galois descent results for enhanced ind-sheaves.

Findings

Kashiwara's conjugation functor is compatible with the enhanced De Rham functor

Galois descent results for enhanced ind-sheaves are extended

Decomposition of enhanced ind-sheaves affects monodromy data over subfields

Abstract

We study some aspects of conjugation and descent in the context of the irregular Riemann-Hilbert correspondence of D'Agnolo-Kashiwara. First, we give a proof of the fact that Kashiwara's conjugation functor for holonomic D-modules is compatible with the enhanced De Rham functor. Afterwards, we work out some complements on Galois descent for enhanced ind-sheaves, slightly generalizing results obtained in previous joint work with Barco, Hien and Sevenheck. Finally, we show how local decompositions of an enhanced ind-sheaf into exponentials descend to lattices over smaller fields. This shows in particular that a structure of the enhanced solutions of a meromorphic connection over a subfield of the complex numbers has implications on its generalized monodromy data (in particular, the Stokes matrices), generalizing and simplifying an argument given in our previous work.