A microlocal approach to the enhanced Fourier–Sato transform in dimension one

Abstract

Let M be a holonomic algebraic D-module on the affine line. Its exponential factors are Puiseux germs describing the growth of holomorphic solutions to M at irregular points. The stationary phase formula states that the exponential factors of the Fourier transform of M are obtained by Legendre transform from the exponential factors of M. We give a microlocal proof of this fact, by translating it in terms of enhanced ind-sheaves through the Riemann-Hilbert correspondence.

Perverse schobers and the Algebra of the Infrared

“…More precisely, the determination of singularities follows from [76] (XII.2.2). In fact, both this determination and the claim that the singularity at 0 is regular follow from the "stationary phase theorem" of [76] (see [23] Th. 1.3 for a concise formulation) which describes the exponential Puiseux factors of | M as the Legendre transforms of those of M.…”

“…This problem was studied before, first of all by Malgrange [76] in much greater generality (M is allowed to be irregular). The work of Malgrange was revisited by D'Agnolo-Kashiwara [23] from the point of view of their approach via enhanced ind-sheaves [22]. It was recognized in [63,21] that focusing specially on the case of regular M leads to important insights and further developments.…”

D-modules of pure Gaussian type and enhanced ind-sheaves

“…The theory has since been applied to the study of Stokes phenomena and Fourier-Laplace transforms (see e.g. [4,6,13,21]). Other recent approaches to the study of Fourier transforms of Stokes data have been developed in [24,27].…”

“…[10,30]), using a result of Mochizuki [25]. Finally, our method of computation needs less input in the following sense: By results like the stationary phase formula (see [6,28]), we could know a priori that the Fourier-Laplace transform of a D-module of pure Gaussian type is again of pure Gaussian type, and we can explicitly write down the exponential factors of the Fourier-Laplace transform. However, this a-priori-knowledge does not enter our arguments, but is rather obtained as a by-product of our computations automatically.…”

“…(Note that other authors usually define the category of enhanced sheaves as a certain subcategory of D b (k X ×R ), cf. [4,6]. We will not introduce it here, although we actually work in this subcategory).…”