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

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We consider Calabi--Yau 3-folds of Borcea--Voisin type, i.e. Calabi--Yau 3-folds obtained as crepant resolutions of a quotient (S× 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 crepant 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 Eₙ: X → S/αₛ whose generic fiber is isomorphic to E.