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 (S x E)/ (alpha(S) x alpha(E)), where S is a K3 surface, E is an elliptic curve, alpha(S) is an element of Aut(S) and alpha(E) is an element of 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 crepant resolution, X, of (S x E)/ (alpha(S) X alpha(E)) 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 epsilon(n) : X -> S/alpha(S) whose generic fiber is isomorphic to E.