Maximal automorphisms of Calabi-Yau manifolds versus maximally unipotent monodromy

Abstract

Let α be an automorphism of the local universal deformation of a Calabi-Yau 3-manifold X, which does not act by ±id on $${H^3(X,\mathbb{C})}$$ . We show that the bundle $${F^2(\mathcal{H}^3)}$$ 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 $${H^3(X,\mathbb{C})}$$ , 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.