Realizations of exponential sheaves and Fourier transform

Abstract

This note concerns exponential sheaves, their realizations, and Fourier transforms, in the setting of mixed Hodge modules and D-modules. For relative categories of exponential sheaves constructed out of regular/tame objects, we study realization functors to holonomic (not necessarily regular) D-modules and show that they are t-exact, faithful on the heart, and compatible with the usual functors (including Verdier duality). We also develop a "universal" Fourier transform: it is invertible, satisfies a Fourier miracle (commuting with Verdier duality up to a twist), and recovers classical Fourier transforms under realizations. The categories considered also admit weight structures that satisfy the standard formalism, and the "universal" Fourier transform preserves purity. The motivation is N. Katz's "analogies" between exponential sums over finite fields and differential equations.