Categories of holomorphic line bundles on higher dimensional noncommutative complex tori

TL;DR

This paper constructs explicit noncommutative deformations of categories of holomorphic line bundles on higher dimensional tori using Heisenberg modules, revealing how noncommutativity influences their structure.

Contribution

It introduces a method to deform categories of holomorphic line bundles on higher dimensional tori via noncommutative geometry, utilizing Heisenberg modules and complex structures.

Findings

Explicit construction of noncommutative deformations

Differential graded categories derived from Heisenberg modules

Composition formulas depend on noncommutativity

Abstract

We construct explicitly noncommutative deformations of categories of holomorphic line bundles over higher dimensional tori. Our basic tools are Heisenberg modules over noncommutative tori and complex/holomorphic structures on them introduced by A. Schwarz. We obtain differential graded (DG) categories as full subcategories of curved DG categories of Heisenberg modules over the complex noncommutative tori. Also, we present the explicit composition formula of morphisms, which in fact depends on the noncommutativity.