A rigid Calabi–Yau three-fold

Abstract

The aim of this paper is to analyze some geometric properties of the rigid Calabi–Yau three-fold $\mathcal{Z}$ obtained by a quotient of $E_3$, where $E$ is a specific elliptic curve. We describe the cohomology of $\mathcal{Z}$ and give a simple formula for the trilinear form on $\mathrm{Pic}(\mathcal{Z})$. We describe some projective models of $\mathcal{Z}$ and relate these to its generalized mirror. A smoothing of a singular model is a Calabi–Yau three-fold with small Hodge numbers which was not known before.