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arXiv · 1308.2915

A Strange Family of Calabi-Yau 3-folds

Patrick Devlin·Howard J. Nuer
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Abstract

We study the predictions of mirror symmetry for the 1-parameter family of Calabi-Yau 3-folds $\tilde{X}$ with hodge numbers $h^{11}=31,h^{21}=1$ constructed in \cite{BN}. We calculate the Picard-Fuchs differential equation associated to this family, and use it to predict the instanton numbers on the hypothetical mirror. These exhibit a strange vanishing in odd degrees. We also calculate the monodromy action on $H^3(\tilde{X},\QQ)$ and find that it strangely predicts a positive Euler characteristic for its mirror. From a degenerate fiber of our family we construct a new rigid Calabi-Yau 3-fold. In an appendix we prove the expansion of the conifold period conjectured in \cite{ES} to hold for all 1-parameter families.

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Patrick Devlin, Howard J. Nuer. 2013-08-13. A Strange Family of Calabi-Yau 3-folds. https://arxiv.org/abs/1308.2915

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