Non-commutative Calabi - Yau manifolds

We discuss aspects of the algebraic geometry of compact non-commutative Calabi - Yau manifolds. In this setting, it is appropriate to consider local holomorphic algebras which can be glued together into a compact Calabi - Yau algebra. We consider two examples: A toroidal orbifold T⁶/Z₂ x Z₂, and an orbifold of the quintic in CP₄, each with discrete torsion. The non-Commutative geometry tools are enough to describe various properties of the orbifolds. First, one describes correctly the fractionation of branes at singularities. Secondly, for the first example we show that one can recover explicitly a large slice of the moduli space of complex structures which deform the orbifold. For this example we also show that we get the correct counting of complex structure deformations at the orbifold point by using traces of non-commutative differential forms (cyclic homology).