Calabi–Yau Quotients of Hyperkähler Four-folds
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 th...
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Published in | Canadian journal of mathematics Vol. 71; no. 1; pp. 45 - 92 |
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Main Authors | , , |
Format | Journal Article |
Language | English |
Published |
Canada
Canadian Mathematical
01.02.2019
Cambridge University Press |
Subjects | |
Online Access | Get full text |
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Summary: | 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
$Y_{S}$
. |
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ISSN: | 0008-414X 1496-4279 |
DOI: | 10.4153/CJM-2018-025-1 |