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Calabi–Yau Quotients of Hyperkähler Four-folds
- Part of
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- Journal:
- Canadian Journal of Mathematics / Volume 71 / Issue 1 / February 2019
- Published online by Cambridge University Press:
- 15 February 2019, pp. 45-92
- Print publication:
- February 2019
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- Article
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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 
$Y_{S}$.
