Asian Journal of Mathematics
Volume 8 (2004)
Microlocalization and Stationary Phase
Pages: 747 – 768
In , G. Laumon defined a set of functors, called by him local Fourier transformations, which allow to analyze the local structure of the l-adic Fourier transform of a constructible l-adic sheaf G on the affine line in terms of the local behaviour of G at infinity and at the points where it is not lisse. These local Fourier transforms play a major role in his cohomological interpretation of the local constants and in his proof of the product formula (see loc. cit. and ). In this article, we are concerned with differential systems defined over a field K of characteristic zero. We define a set of functors which allow to prove a stationary phase formula, expressing the formal germ at infinity of the D-module theoretic Fourier transform of a holonomic K[t]h.ti-module M in terms of the formal germs defined by M at its singular points and at infinity. These functors might therefore be regarded as formal analogues of Laumon's local Fourier transformations ... We will use some results from D-module theory for which we refer e.g. to , . Our proof of the formal stationary phase formula follows the leitfaden of the one given by C. Sabbah in  for modules with regular singularities. I thank C. Sabbah, B. Malgrange, G. Christol and W. Messing for their useful remarks. I thank also G. Lyubeznik and S. Sperber for their invitation to the University of Minnesota, during which part of this work was done.