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A rigid Calabi-Yau three-fold. (English) Zbl 1435.14036

Summary: 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 \(\text{Pic}(\mathcal{Z})\). 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.

MSC:

14J32 Calabi-Yau manifolds (algebro-geometric aspects)
14J30 \(3\)-folds
14J50 Automorphisms of surfaces and higher-dimensional varieties
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