Let \(X\) be a Kähler manifold of dimension \(n\). \(X\) is a Calabi-Yau manifold if the canonical bundle of \(X\) is trivial, and Hodge numbers \(h^{i,0}(X)=0\) for \(i=1, \cdots, n-1\). If \(X\) is simply connected and admits a holomorphic symplectic form, then \(X\) is called a hyperkähler manifold. There are examples of automorphisms of hyperkähler manifolds preserving holomorphic volume forms but not holomorphic symplectic forms. The quotients would be Calabi-Yau varieties with orbifold singularities, and certain resolutions would be Calabi-Yau manifolds if they exist.
The authors use this method to construct Calabi-Yau 4-folds via hyperkähler 4-folds with non-symplectic automorphisms. More precisely, the abstract of this paper describes the contents well as the following: ‘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 \(\alpha\). 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 \(X/\alpha\), 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 \(Y_S\).’