Classification of holomorphic vector bundles on noncommutative two-tori

TL;DR

This paper demonstrates that all holomorphic vector bundles on a noncommutative two-torus can be constructed from standard bundles, establishing an equivalence with a category derived from an elliptic curve.

Contribution

It shows that the category of holomorphic bundles on a noncommutative two-torus is equivalent to a heart of a t-structure on the derived category of an elliptic curve, providing a classification framework.

Findings

All holomorphic bundles are successive extensions of standard bundles.

The category is equivalent to a t-structure heart on an elliptic curve.

Provides a classification of bundles on noncommutative tori.

Abstract

We prove that every holomorphic vector bundle on a noncommutative two-torus $T$ can be obtained by successive extensions from standard holomorphic bundles considered in math.QA/0211262. This implies that the category of holomorphic bundles on $T$ is equivalent to the heart of certain $t$-structure on the derived category of coherent sheaves on an elliptic curve.