# Topological computation of some Stokes phenomena on the affine line

Abstract : Let $\mathcal M$ be a holonomic algebraic $\mathcal D$-module on the affine line, regular everywhere including at infinity. Malgrange gave a complete description of the Fourier-Laplace transform $\widehat{\mathcal M}$, including its Stokes multipliers at infinity, in terms of the quiver of $\mathcal M$. Let $F$ be the perverse sheaf of holomorphic solutions to $\mathcal M$. By the irregular Riemann-Hilbert correspondence, $\widehat{\mathcal M}$ is determined by the enhanced Fourier-Sato transform $F^\curlywedge$ of $F$. Our aim here is to recover Malgrange's result in a purely topological way, by computing $F^\curlywedge$ using Borel-Moore cycles. In this paper, we also consider some irregular $\mathcal M$'s, like in the case of the Airy equation, where our cycles are related to steepest descent paths.
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Preprints, Working Papers, ...
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https://hal-polytechnique.archives-ouvertes.fr/hal-01727637
Contributor : Claude Sabbah <>
Submitted on : Friday, March 9, 2018 - 1:42:55 PM
Last modification on : Wednesday, March 27, 2019 - 4:10:22 PM

### Identifiers

• HAL Id : hal-01727637, version 1
• ARXIV : 1705.07610

### Citation

Claude Sabbah, Andrea d'Agnolo, Marco Hien, Giovanni Morando. Topological computation of some Stokes phenomena on the affine line. 2018. ⟨hal-01727637⟩

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