On a reconstruction theorem for holonomic systems

Abstract

Let $X$ be a complex manifold. The classical Riemann-Hilbert correspondence associates to a regular holonomic system $\mathcal{M}$ the $\mathbf{C}$-constructible complex of its holomorphic solutions. Let $t$ be the affine coordinate in the complex projective line. If $\mathcal{M}$ is not necessarily regular, we associate to it the ind-$\mathbf{R}$-constructible complex $G$ of tempered holomorphic solutions to $\mathcal{M}\boxtimes\mathcal{D} e^{t}$. We conjecture that this provides a Riemann-Hilbert correspondence for holonomic systems. We discuss the functoriality of this correspondence, we prove that $\mathcal{M}$ can be reconstructed from $G$ if $\dim X=1$, and we show how the Stokes data are encoded in $G$.