Calabi-Yau 3-folds of Borcea-Voisin type and elliptic fibrations

We consider Calabi-Yau 3-folds of Borcea-Voisin type, i.e. Calabi-Yau 3-folds obtained as crepant resolutions of a quotient (5 × E)/(αS × αE), where S is a K3 surface, E is an elliptic curve, αS ∈ Aut(S) and αE ∈ Aut(E) act on the period of S and E respectively with order n = 2,3,4,6. The case n = 2 is very classical, the case n = 3 was recently studied by Rohde, the other cases are less known. First, we construct explicitly a crêpant resolution, X, of (S × E)/(αS × αE) and we compute its Hodge numbers; some pairs of Hodge numbers we found are new. Then, we discuss the presence of maximal automorphisms and of a point with maximal unipotent monodromy for the family of X. Finally, we describe the map ϵn: X → S/αS whose generic fiber is isomorphic to E.