Calabi--Yau quotients of hyperk\"ahler four-folds

The aim of this paper is to construct Calabi-Yau 4-folds as crepant resolutions of the quotients of a hyperk"ahler 4-fold $X$ by a non symplectic involution $\alpha$. We first compute the Hodge numbers of a Calabi-Yau constructed in this way in a general setting and then we apply the results to several specific examples of non symplectic involutions, producing Calabi-Yau 4-folds with different Hodge diamonds. Then we restrict ourselves to the case where $X$ is the Hilbert scheme of two points on a K3 surface $S$ and the involution $\alpha$ is induced by a non symplectic involution on the K3 surface. In this case we compare the Calabi-Yau 4-fold $Y_S$, which is the crepant resolution of $X\mathrm{/}\alpha$, with the Calabi-Yau 4-fold $Z_S$, constructed from $S$ through the Borcea–Voisin construction. We give several explicit geometrical examples of both these Calabi–Yau 4-folds describing maps related to interesting linear systems as well as a rational $2:1$ map from $Z_S$ to $Y_S$.