A topological approach to the Fourier transform of an elementary D-module

Let V be a complex affine line, and M a holonomic D-module on it. The Fourier-Laplace transform M^ of M is a holonomic D-module on the dual affine line V*. Even if M is regular, M^ is irregular in general. It is natural and important to try to describe the Stokes structure of M^ in terms of the Stokes structure of M. In the literature dealing with this problem, let us mention in particular the work by Malgrange, Mochizuki, Hien-Sabbah, D'Agnolo-Hien-Morando-Sabbah. Malgrange gave a comprehensive treatment, Mochizuki has given a recipe for a complete description of the Fourier-Laplace transform of a general M using the rapid decay homology theory introduced by Bloch-Esnault. For a particular kind of D-module, so-called elementary, Hien-Sabbah gave a more explicit description. Using the Riemann-Hilbert correspondence of Deligne-Malgrange, they introduced a topological local Laplace transformation at the level of Stokes-filtered local systems, and computed it in terms of Cech cohomology. A different point of view to the study of the Stokes phenomena is given by the Riemann-Hilbert correspondence, as stated by D'Agnolo-Kashiwara. This associates to a holonomic D-module the enhanced ind-sheaf of its enhanced solutions. Moreover, by functoriality, such correspondence interchanges Fourier-Laplace transform for holonomic D-modules with Fourier-Sato transform for enhanced ind-sheaves. Using this point of view, D'Agnolo-Hien-Morando-Sabbah explicitly computed the Stokes structure of M^, for M regular holonomic. In this thesis, using this same point of view, our aim is to get a description of the Fourier-Laplace transform of an elementary D-module. Unlike Hien-Sabbah, our approach is purely topological. Like D'Agnolo-Hien-Morando-Sabbah, it is based on computations in terms of Borel-Moore homology classes. For that, we choose the most natural classes, namely those attached to steepest descent cycles.