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 $(S\times E)/(\alpha_S\times \alpha_E)$,where$S$isaK3surface,$E$ is an elliptic curve, ${\alpha }_S\in \mathrm{A}\mathrm{u}\mathrm{t}(S)$ and ${\alpha }_E\in \mathrm{A}\mathrm{u}\mathrm{t}(E)$ 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 $(S\times E)\mathrm{/}({\alpha }_S\times {\alpha }_E)$ 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 $$\mathcal{E}$_n: X \rightarrow S/\alpha_S$whosegenericfiberisisomorphicto$E$.