On the Riemann-Hilbert problem for hyperplane arrangements with a good line

TL;DR

This paper investigates a variant of the Riemann-Hilbert problem for hyperplane arrangement complements, introducing a generalized Katz's middle convolution that preserves the problem's solvability.

Contribution

It generalizes Katz's middle convolution as a functor for local systems on hyperplane complements, maintaining the solvability of the Riemann-Hilbert problem.

Findings

Generalization of Katz's middle convolution to hyperplane arrangements

Preservation of problem solvability under the convolution

Insights into local systems on hyperplane complements

Abstract

We study a variant of the Riemann-Hilbert problem on the complements of hyperplane arrangements. This problem asks whether a given local system on the complement can be realized as the solution sheaf of a logarithmic Pfaffian system with constant coefficients. In this paper, we generalize Katz's middle convolution as a functor for local systems on hyperplane complements and show that it preserves the solvability of this problem.