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 $\hat{\mathcal{M}}$, 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, $\hat{\mathcal{M}}$ is determined by the enhanced Fourier-Sato transform $F^{⋏}$ of $F$. Our aim here is to recover Malgrange's result in a purely topological way, by computing $F^{⋏}$ 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.

50 pages, to appear at Annales de l'Institut Fourier, v3: some minor (editorial) corrections