Transformation de Fourier generalisee

TL;DR

This paper introduces a generalized geometric Fourier transformation for 1-motives, extending several classical transformations and establishing an equivalence between derived categories of D-Modules and O-Modules on abelian varieties.

Contribution

It constructs a new geometric transformation for 1-motives that unifies and extends existing Fourier-Mukai, Mellin, and Fourier transformations, and proves an equivalence of derived categories.

Findings

Established an equivalence between derived categories of D-Modules and O-Modules.

Unified several classical Fourier-type transformations under a generalized framework.

Extended the Fourier-Mukai transformation to generalized 1-motives.

Abstract

In this paper I construct a geometric transformation for generalized 1-motives which extends the Fourier-Mukai transformation for O-Modules on abelian varieties, the geometric Fourier transformation for D-Modules on vector spaces and the geometric Mellin transformation for D-Modules on tori. In particular, I construct an equivalence of triangulated categories between the derived category of quasi-coherent D-Modules on an abelian variety and the derived category of quasi-coherent O-Modules on the universal extension of the dual abelian variety. This equivalence has also been obtained by Mitchell Rothstein.