The 24-Cell and Calabi-Yau Threefolds with Hodge Numbers (1,1)

TL;DR

This paper constructs specific Calabi-Yau threefolds with equal Hodge numbers using the 24-cell and analyzes their fundamental groups, contributing to the classification of these complex geometric structures.

Contribution

It introduces new Calabi-Yau threefolds with Hodge numbers (1,1) derived from the 24-cell and details their fundamental groups, expanding the known examples in the field.

Findings

Constructed Calabi-Yau threefolds with h^{11}=h^{21}=1

Identified fundamental groups as SL(2,3), Z_3 ⋉ Z_8, and Z_3 × Q_8

Demonstrated free quotients of hypersurfaces in toric varieties

Abstract

Calabi-Yau threefolds with h^11(X)=h^21(X)=1 are constructed as free quotients of a hypersurface in the ambient toric variety defined by the 24-cell. Their fundamental groups are SL(2,3), a semidirect product of Z_3 and Z_8, and Z_3 x Q_8.