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Calabi-Yau quotients of hyperkähler four-folds

  • Chiara Camere,
    Dipartimento di Matematica, Università degli Studi di Milano, via Cesare Saldini 50, 20133 Milano, Italy
  • Alice Garbagnati,
    Dipartimento di Matematica, Università degli Studi di Milano, via Cesare Saldini 50, 20133 Milano, Italy
  • Giovanni Mongardi,
    Dipartimento di Matematica, Alma Mater Studiorum Università di Bologna, Piazza di porta san Donato 5 , 40126 Bologna, Italy
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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 $\alpha$. 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 $Y_S$.
Keywords: irreducible holomorphic symplectic manifold, Hyperkähler manifold, Calabi-Yau 4-fold, Borcea-Voisin construction, automorphism, quotient map, non symplectic involution irreducible holomorphic symplectic manifold, Hyperkähler manifold, Calabi-Yau 4-fold, Borcea-Voisin construction, automorphism, quotient map, non symplectic involution
MSC Classifications: 14J32, 14J35, 14J50, 14C05 show english descriptions Calabi-Yau manifolds
$4$-folds
Automorphisms of surfaces and higher-dimensional varieties
Parametrization (Chow and Hilbert schemes)
14J32 - Calabi-Yau manifolds
14J35 - $4$-folds
14J50 - Automorphisms of surfaces and higher-dimensional varieties
14C05 - Parametrization (Chow and Hilbert schemes)
 

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