Calabi–Yau Quotients of Hyperkähler Four-folds

Camere, Chiara ; Garbagnati, Alice ; Mongardi, Giovanni

Canadian journal of mathematics, Tome 71 (2019) no. 1, pp. 45-92

Résumé

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 $Y_{S}$.

Détail

DOI : 10.4153/CJM-2018-025-1

Mots-clés : irreducible holomorphic symplectic manifold, Hyperkähler manifold, Calabi–Yau 4-fold, Borcea–Voisin construction, automorphism, quotient map, non-symplectic involution