The Fourier Transform and Characteristic Cycles of Monodromic $\ell$-adic Sheaves

TL;DR

This paper extends a classical result relating characteristic cycles and Fourier transforms from complex D-modules to the setting of $\\ell$-adic sheaves, establishing a canonical identification.

Contribution

It proves the analogue of Brylinski and Malgrange's theorem for monodromic $\\ell$-adic sheaves, linking characteristic cycles and Fourier transforms in this context.

Findings

Established the canonical identification of characteristic cycles with Fourier transforms for monodromic $\\ell$-adic sheaves.

Extended classical complex D-module results to the $\\ell$-adic sheaf setting.

Provided a foundational result for the study of $\\ell$-adic sheaves and their Fourier transforms.

Abstract

Brylinski and Malgrange proved in 1986 that, for a monodromic algebraic D-module on a finite dimensional vector space over the complex numbers, its characteristic cycle is canonically identified with the characteristic cycle of its Fourier transform. We prove the exact analogue of this in the $\ell$-adic context.