A Morse theoretical approach to Fourier transforms of holonomic D-modules in dimension one

Abstract

We study Fourier transforms of holonomic D-modules on the complex affine line and show that their enhanced solution complexes are described by a twisted Morse theory. We thus recover and even strengthen the well-known formula for their exponential factors i.e. the stationary phase method. Moreover, we define a Lagrangian cycle that we call the irregular characteristic cycle and describe the enhanced solution complex of the Fourier transform by it. In this way, we obtain a new perspective, from which we can geometrically see how the standard properties of holonomic D-modules are transformed via the Fourier transform. In the course of our study, a formula for the (classical) characteristic cycles of the Fourier transforms will be also obtained and natural bases of their holomorphic solutions will be constructed via rapid decay homology cycles.