Abstract
Differential systems of pure Gaussian type are examples of D-modules on the complex projective line with an irregular singularity at infinity, and as such are subject to the Stokes phenomenon. We employ the theory of enhanced ind-sheaves and the Riemann–Hilbert correspondence for holonomic D-modules of D’Agnolo and Kashiwara to describe the Stokes phenomenon topologically. Using this description, we perform a topological computation of the Fourier–Laplace transform of a D-module of pure Gaussian type in this framework, recovering and generalizing a result of Sabbah.
Highlights
The study of D-modules with irregular singularities has recently experienced new impulses by a remarkable result of D’Agnolo and Kashiwara, the Riemann–Hilbert correspondence for holonomic D-modules
The theory has since been applied to the study of Stokes phenomena and Fourier–Laplace transforms
Other recent approaches to the study of Fourier transforms of Stokes data have been developed in [24,27]. In their original article [5, Sect. 9.8], the authors give an outlook on a topological study of the Stokes phenomenon of a D-module
Summary
The study of D-modules with irregular singularities has recently experienced new impulses by a remarkable result of D’Agnolo and Kashiwara, the Riemann–Hilbert correspondence for holonomic D-modules (see [5]). Laplace transform of this kind of system has again a formal type with exponential factors of pole order 2 Studying these connections is a natural step further, given that the theory of enhanced ind-sheaves has already proved to be useful in the case of exponents of pole order 1 4–7, we introduce the notion of D-modules of pure Gaussian type in the language of D-modules and describe step-by-step the topological object of enhanced solutions SolPE(M) of such a D-module M: Starting from the Stokes phenomenon, which yields a direct sum decomposition on small sectors, we discuss how large the radius and angular width of these sectors may be, introducing notions like Stokes multipliers in this framework It will turn out (Theorem 7.2) that. We treat a more general case (Theorem 11.2), illustrating how the methods of the above theorem are naturally adapted to other situations
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