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On a class of non-simply connected Calabi-Yau 3-folds. (English) Zbl 1165.14032

Motivated by applications in string phenomenology, the authors construct and classify a new family of non-simply connected Calabi-Yau threefolds by taking quotients of Schoen threefolds.
In [Math. Z. 197, 177–199 (1988; Zbl 0631.14032)], C. Schoen studied Calabi-Yau threefolds which are fiber products of two rational elliptic surfaces \(\beta :B\rightarrow\mathbb{P}^1\), \(\beta ':B'\rightarrow\mathbb{P}^1\) over \(\mathbb{P}^1\) whose discriminant loci do not intersect. The authors consider finite groups \(G\) admitting two compatible actions on the rational elliptic surfaces which lift to a free action on \(X\), i.e. given \(\phi:G\rightarrow B\) and \(\phi':G\rightarrow B'\), one requires that \(\beta\circ\phi=\beta'\circ\phi'\) and that the action on \(X\) is free. Since Schoen Calabi-Yau threefolds are simply connected, the quotient varieties, which are also Calabi-Yau, have fundamental group \(G\).
The paper concludes with a neat list of all possible cases and a description of the moduli space of such quotients. In particular, the authors give the possible Hodge numbers and fundamental groups that one can obtain via this procedure.

MSC:

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

Citations:

Zbl 0631.14032
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