# Relative Riemann-Hilbert correspondence in dimension one

## TL;DR

This paper establishes a generic left quasi-inverse property of the relative Riemann-Hilbert functor on Riemann surfaces, enhancing the understanding of the correspondence between regular relative holonomic modules and constructible complexes.

## Contribution

It proves that the relative Riemann-Hilbert functor is a left quasi-inverse of the solution functor in a generic setting on Riemann surfaces, extending previous work.

## Key findings

- The functor $ ext{RH}^S$ is a left quasi-inverse in a generic sense.
- The correspondence holds in the context of regular relative holonomic modules.
- The result applies to Riemann surfaces, broadening the scope of the Riemann-Hilbert correspondence.

## Abstract

We prove that, on a Riemann surface, the functor $\mathrm{RH}^S$ constructed in a previous work as a right quasi-inverse of the solution functor from the bounded derived category of regular relative holonomic modules to that of relative constructible complexes satisfies the left quasi-inverse property in a generic sense.