Differential operators on toric varieties and Fourier transform

TL;DR

This paper explores the geometric interpretation of Fourier transforms on Weyl algebras as isomorphisms between twisted differential operator rings on toric varieties, revealing new connections in algebraic geometry.

Contribution

It establishes a link between Fourier transforms on Weyl algebras and isomorphisms of differential operator rings on toric varieties, expanding the understanding of their geometric and algebraic interplay.

Findings

Fourier transforms induce isomorphisms between twisted rings of differential operators on toric varieties.

Reflections of cones in fans relate different toric varieties through these isomorphisms.

Includes examples like blow-ups and resolutions of singularities related to projective spaces.

Abstract

We show that Fourier transforms on the Weyl algebras have a geometric counterpart in the framework of toric varieties, namely they induce isomorphisms between twisted rings of differential operators on regular toric varieties, whose fans are related to each other by reflections of one-dimensional cones. The simplest class of examples is provided by the toric varieties related by such reflections to projective spaces. It includes the blow-up at a point in affine space and resolution of singularities of varieties appearing in the study of the minimal orbit of sl(n+1).