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.

substantially revised, 54 pages