Enhanced Nearby and Vanishing Cycles in Dimension One and Fourier Transform

Enhanced ind-sheaves provide a suitable framework for the irregular Riemann–Hilbert correspondence. In this paper, we give some precision on nearby and vanishing cycles for enhanced perverse objects in dimension one. As an application, we give a topological proof of the following fact. Let $\mathcal{M}$ be a holonomic algebraic $\mathcal{D}$-module on the affine line, and denote by ${^{\mathsf{L}}}\mathcal{M}$ its Fourier–Laplace transform. For a point $a$ on the affine line, denote by $\ell\_a$ the corresponding linear function on the dual affine line. Then the vanishing cycles of $\mathcal{M}$ at $a$ are isomorphic to the graded component of degree $\ell\_a$ of the Stokes filtration of ${^{\mathsf{L}}}\mathcal{M}$ at infinity.