On a reconstruction theorem for holonomic systems. (English) Zbl 1266.32012

Summary: Let \(X\) be a complex manifold. The classical Riemann-Hilbert correspondence associates to a regular holonomic system \(\mathcal{M}\) the \(\mathbb{C}\)-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-\(\mathbb{R}\)-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 \(G\).


32C38 Sheaves of differential operators and their modules, \(D\)-modules
35A20 Analyticity in context of PDEs
32S60 Stratifications; constructible sheaves; intersection cohomology (complex-analytic aspects)
34M40 Stokes phenomena and connection problems (linear and nonlinear) for ordinary differential equations in the complex domain
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