Kashiwara conjugation and the enhanced Riemann–Hilbert correspondence

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 (notably the Stokes matrices), generalizing and simplifying an argument given in our previous work.