Riemann-Hilbert correspondence for mixed twistor D-Modules

TL;DR

This paper extends the Riemann-Hilbert correspondence to regular relative holonomic D-modules, establishing an equivalence with relative constructible complexes, especially for those underlying regular mixed twistor D-modules.

Contribution

It introduces a notion of regularity for relative holonomic D-modules and constructs a quasi-inverse functor to establish an equivalence with relative constructible complexes.

Findings

Solution functor is essentially surjective for regular relative holonomic modules.

Constructs a right quasi-inverse functor to the solution functor.

Establishes an equivalence for modules underlying regular mixed twistor D-modules.

Abstract

We introduce the notion of regularity for a relative holonomic $\mathcal D$-module in the sense of arXiv:1204.1331. We prove that the solution functor from the bounded derived category of regular relative holonomic modules to that of relative constructible complexes is essentially surjective by constructing a right quasi-inverse functor. When restricted to relative $\mathcal D$-modules underlying a regular mixed twistor $\mathcal D$-module, this functor satisfies the left quasi-inverse property.