Calabi--Yau quotients of hyperk\"ahler four-folds

TL;DR

This paper constructs Calabi-Yau 4-folds via crepant resolutions of hyperk"ahler quotients by non symplectic involutions, computes their Hodge numbers, and compares different geometric constructions including Borcea--Voisin types.

Contribution

It introduces a method to produce Calabi-Yau 4-folds from hyperk"ahler quotients and analyzes their properties, including explicit examples and comparisons with Borcea--Voisin constructions.

Findings

Computed Hodge numbers for the constructed Calabi-Yau 4-folds.

Produced explicit examples with different Hodge diamonds.

Established a rational 2:1 map between two types of Calabi-Yau 4-folds.

Abstract

The aim of this paper is to construct Calabi-Yau 4-folds as crepant resolutions of the quotients of a hyperk\"ahler 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 $Y_S$.