A rigid Calabi–Yau three-fold

The aim of this paper is to analyze some geometricproperties of the rigid Calabi–Yau three-fold$\mathcal{Z}$ obtained by a quotient of $\EEE$, where $E$is a specific elliptic curve. We describe the cohomology of$\mathcal{Z}$ and give a simple formula for the trilinearform on $\mathrm{Pic}(\mathcal{Z})$. We describe someprojective models of $\mathcal{Z}$ and relate these to itsgeneralized mirror. A smoothing of a singular model is aCalabi–Yau three-fold with small Hodge numbers which wasnot known before.

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Published 24 October 2012