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Calabi-Yau threefolds with large \(h^{2,1}\). (English) Zbl 1333.81384
Summary: We carry out a systematic analysis of Calabi-Yau threefolds that are elliptically fibered with section (‘EFS’) and have a large Hodge number \(h^{2,1}\). EFS Calabi-Yau threefolds live in a single connected space, with regions of moduli space associated with different topologies connected through transitions that can be understood in terms of singular Weierstrass models. We determine the complete set of such threefolds that have \(h^{2,1}\geq 350\) by tuning coefficients in Weierstrass models over Hirzebruch surfaces. The resulting set of Hodge numbers includes those of all known Calabi-Yau threefolds with \(h^{2,1} \geq 350\), as well as three apparently new Calabi-Yau threefolds. We speculate that there are no other Calabi-Yau threefolds (elliptically fibered or not) with Hodge numbers that exceed this bound. We summarize the theoretical and practical obstacles to a complete enumeration of all possible EFS Calabi-Yau threefolds and fourfolds, including those with small Hodge numbers, using this approach.

81T45 Topological field theories in quantum mechanics
14J81 Relationships with physics
14J32 Calabi-Yau manifolds (algebro-geometric aspects)
14J30 \(3\)-folds
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