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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Contributor : Claude Sabbah <>
Submitted on : Friday, March 9, 2018 - 1:42:55 PM
Last modification on : Monday, October 21, 2019 - 3:50:00 PM

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


Andrea d'Agnolo, Marco Hien, Giovanni Morando, Claude Sabbah. Topological computation of some Stokes phenomena on the affine line. Annales de l'Institut Fourier, Association des Annales de l'Institut Fourier, In press. ⟨hal-01727637⟩



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