  Based on the recent developments in the irregular Riemann-Hilbert
correspondence for holonomic D-modules and the Fourier-Sato transforms for
enhanced ind-sheaves, we study the Fourier transforms of some irregular
holonomic D-modules. For this purpose, the singularities of rational and
meromorphic functions on complex affine varieties will be studied precisely,
with the help of some new methods and tools such as meromorphic vanishing cycle
functors. As a consequence, we show that the exponential factors and the
irregularities of the Fourier transform of a holonomic D-module are described
geometrically by the stationary phase method, as in the classical case of
dimension one. A new feature in the higher-dimensional case is that we have
some extra rank jump of the Fourier transform produced by the singularities of
the linear perturbations of the exponential factors at their points of
indeterminacy. In the course of our study, not necessarily homogeneous
Lagrangian cycles that we call irregular characteristic cycles will play a
crucial role.
