Calabi–Yau Quotients of Hyperkähler Four-folds

Abstract

The aim of this paper is to construct Calabi-Yau 4-folds as crepant resolutions of the quotients of a hyperkähler 4-fold X by a non symplectic involution α. 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 α 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/α, 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 .One can directly check that there are no mirror pairs in the previous table, except for the self-mirror Calabi-Yau Y S associated to the values N ′ = N + 1 for N = 1, . . . , 5.