Calabi-Yau quotients of hyperkähler four-folds

The aim of this paper is to construct Calabi-Yau 4-folds as crepant resolutions of the quotients of a hyperkähler 4-fold by a non symplectic involution . 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 is the Hilbert scheme of two points on a K3 surface and the involution is induced by a non symplectic involution on the K3 surface. In this case we compare the Calabi-Yau 4-fold , which is the crepant resolution of , with the Calabi-Yau 4-fold , constructed from 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 map from to .