Noncommutative K3 surfaces

We consider deformations of a toroidal orbifold T4/Z2 and an orbifold of quartic in CP3. In the T4/Z2 case, we construct a family of noncommutative K3 surfaces obtained via both complex and noncommutative deformations. We do this following the line of algebraic deformation done by Berenstein and Leigh for the Calabi–Yau threefold. We obtain 18 as the dimension of the moduli space both in the noncommutative deformation as well as in the complex deformation, matching the expectation from classical consideration. In the quartic case, we construct a 4×4 matrix representation of noncommutative K3 surface in terms of quartic variables in CP3 with a fourth root of unity. In this case, the fractionation of branes occurs at codimension two singularities due to the presence of discrete torsion.