  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 T^6/Z_2 x Z_2, and an
orbifold of the quintic in CP_4, 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).
