On a desingularization of the moduli space of noncommutative tori

It is shown that the moduli space of the noncommutative tori ${\mathbb{A}}_{\theta}$ admits a natural desingularization by the group $Ext~ ({\mathbb{A}}_{\theta},{\mathbb{A}}_{\theta})$. Namely, we prove that the moduli space of pairs $({\mathbb{A}}_{\theta}, Ext~ ({\mathbb{A}}_{\theta},{\mathbb{A}}_{\theta}))$ is homeomorphic to a punctured two-dimensional sphere. The proof is based on a correspondence (a covariant functor) between the complex and noncommutative tori.