Noncommutative Two-tori with Real Multiplication as Noncommutative Projective Varieties

We define analogues of homogeneous coordinate algebras for noncommutative two-tori with real multiplication. We prove that the categories of standard holomorphic vector bundles on such noncommutative tori can be described in terms of graded modules over appropriate homogeneous coordinate algebras. We give a criterion for such an algebra to be Koszul and prove that the Koszul dual algebra also comes from some noncommutative two-torus with real multiplication. These results are based on the techniques of [14] allowing to interpret all the data in terms of autoequivalences of the derived categories of coherent sheaves on elliptic curves.