Maximal automorphisms of Calabi-Yau manifolds versus maximally unipotent monodromy

Let α be an automorphism of the local universal deformation of a Calabi-Yau 3-manifold X , which does not act by ±id on . We show that the bundle in the VHS of each maximal family containing X is constant in this case. Thus X cannot be a fiber of a maximal family with maximally unipotent monodromy, if such an automorphism α exists. Moreover we classify the possible actions of α on , construct examples and show that the period domain is a complex ball containing a dense set of CM points given by a Shimura datum in this case.