Holomorphic bundles on 2-dimensional noncommutative toric orbifolds

Abstract

We define the notion of a holomorphic bundle on the noncommutative toric orbifold $T_{\theta}/G$ associated with an action of a finite cyclic group $G$ on an irrational rotation algebra. We prove that the category of such holomorphic bundles is abelian and its derived category is equivalent to the derived category of modules over a finite-dimensional algebras $\Lambda$. As an application we finish the computation of $K_0$-groups of the crossed product algebras describing the above orbifolds initiated by Kumjian, Walters and Buck. Also, we describe a torsion pair in the category of $\Lambda$-modules, such that the tilting with respect to this torsion pair gives the category of holomorphic bundles on $T_{\theta}/G$.