Calabi–Yau Quotients of Hyperkähler Four-folds

• Published in: Canadian journal of mathematics Vol. 71; no. 1; pp. 45 - 92
• Main Authors: Camere, Chiara, Garbagnati, Alice, Mongardi, Giovanni
• Format: Journal Article
• Language: English
• Published: Canada Canadian Mathematical 01.02.2019; Cambridge University Press
• Subjects: Geometry; Linear equations; Mathematics; 14J35; Hyperkähler manifold; Borcea–Voisin construction; quotient map; 14J50; automorphism; irreducible holomorphic symplectic manifold; 14J32; 14C05; Calabi–Yau 4-fold; non-symplectic involution
• ISSN: 0008-414X; 1496-4279
• DOI: 10.4153/CJM-2018-025-1

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 $\unicode[STIX]{x1D6FC}$ . 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 $\unicode[STIX]{x1D6FC}$ 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 $X/\unicode[STIX]{x1D6FC}$ , 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 $Y_{S}$ .