On a reconstruction theorem for holonomic systems

TL;DR

This paper proposes a conjectural extension of the Riemann-Hilbert correspondence to irregular holonomic systems, demonstrating reconstruction in dimension one and encoding of Stokes data within the associated solution complexes.

Contribution

It introduces a new conjectural framework for the Riemann-Hilbert correspondence for irregular systems and proves reconstruction in one-dimensional cases.

Findings

Reconstruction of holonomic systems from tempered solutions in dimension one

Encoding of Stokes data within the solution complex G

Proposal of a conjectural extension of the classical correspondence

Abstract

Let X be a complex manifold. The classical Riemann-Hilbert correspondence associates to a regular holonomic system M the C-constructible complex of its holomorphic solutions. Denote by t the affine coordinate in the complex projective line. If M is not necessarily regular, we associate to it the ind-R-constructible complex G of tempered holomorphic solutions to the exterior product of M with the D-module associated with the exponential e^t. We conjecture that this provides a Riemann-Hilbert correspondence for holonomic systems. We discuss the functoriality of this correspondence, we prove that M can be reconstructed from G if X has dimension 1, and we show how the Stokes data are encoded in G.