Algebraic deformations of toric varieties I. General constructions

TL;DR

This paper introduces a method to create noncommutative deformations of toric varieties by deforming the algebraic torus while preserving the fan structure, using techniques from noncommutative geometry and toric geometry.

Contribution

It develops a new framework for noncommutative deformations of toric varieties, including explicit examples like Grassmann and flag varieties, and establishes a sheaf theory in this context.

Findings

Constructed noncommutative deformations of toric varieties.

Provided explicit examples including Grassmann and flag varieties.

Laid groundwork for studying instantons on noncommutative toric varieties.

Abstract

We construct and study noncommutative deformations of toric varieties by combining techniques from toric geometry, isospectral deformations, and noncommutative geometry in braided monoidal categories. Our approach utilizes the same fan structure of the variety but deforms the underlying embedded algebraic torus. We develop a sheaf theory using techniques from noncommutative algebraic geometry. The cases of projective varieties are studied in detail, and several explicit examples are worked out, including new noncommutative deformations of Grassmann and flag varieties. Our constructions set up the basic ingredients for thorough study of instantons on noncommutative toric varieties, which will be the subject of the sequel to this paper.