A strange family of Calabi-Yau 3-folds

摘要

We study the predictions of mirror symmetry for the 1-parameter family of Calabi-Yau 3-folds X with hodge numbers h~(11) = 31, h~(21) = 1 constructed by Borisov and Nuer. We calculate the Picard-Fuchs differential equation associated to this family, and use it to predict the instant on numbers on the hypothetical mirror. These exhibit a strange vanishing in odd degrees. We also calculate the monodromy action on H3(X,Q) 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 to hold for all 1-parameter families.