51. Calabi–Yau Quotients of Hyperkähler Four-folds
- Author
-
Alice Garbagnati, Chiara Camere, Giovanni Mongardi, Camere, Chiara, Garbagnati, Alice, and Mongardi, Giovanni
- Subjects
irreducible holomorphic symplectic manifold, Hyperkähler manifold, Calabi-Yau 4-fold, Borcea-Voisin construction, automorphism, quotient map, non symplectic involution ,automorphism ,Pure mathematics ,quotient map ,General Mathematics ,010102 general mathematics ,Hyperkähler manifold ,irreducible holomorphic symplectic manifold ,Calabi-Yau 4-fold ,Borcea-Voisin construction ,non symplectic involution ,Automorphism ,01 natural sciences ,Mathematics::Algebraic Geometry ,0103 physical sciences ,Calabi–Yau manifold ,Mathematics::Differential Geometry ,010307 mathematical physics ,0101 mathematics ,Mathematics::Symplectic Geometry ,Quotient ,Mathematics - 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 $\unicode[STIX]{x1D6FC}$. 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 $Y_{S}$.
- Published
- 2019