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

Abstract

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 $S$ is a K3 surface, $E$ is an elliptic curve, ${\alpha }_S\in \mathrm{Aut}(S)$ and ${\alpha }_E\in \mathrm{Aut}(E)$ 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 $(S\times E)/({\alpha }_S\times {\alpha }_E)$ 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 ${\mathcal{E}}_n:X\to S/{\alpha }_S$ whose generic fiber is isomorphic to $E$.

arXiv comment34 pages, 3 figures