Enhanced nearby and vanishing cycles in dimension one and Fourier transform (Q6081846)

Summary: 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.