CY Manifolds with Locally Symmetric Moduli Spaces
  There is a natural sequence of CY manifolds that are double covers of the projective g dimensional spaces ramified over 2g+2 hyperplanes. We observe that some of them are obtained as quotient of the action of the semi-direct product of g-1 copies of cyclic Z/2Z groups with the symmetric g group on the product of g-copies of hyper-elliptic curves of genus g. The quotient is a double cover of the projective g space ramified over 2g+2 hyperplanes. This construction generalizes the construction of a Kummer surface. The Kodaira-Spencer classes on the Jacobian are invariant under the action of the group. Thus they form a basis of Kodaira-Spencer classes on the CY manifold. Since the bracket of any Kodaira-Spencer classes on the Jacobian are zero, then they will be zero on the CY manifold. This implies that the moduli space of those CY manifolds is a locally symmetric space. 
Andrey N. Todorov