A Strange Family of Calabi-Yau 3-folds

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.

23 pages

References in corpus (4)

• Mirror Symmetry and Integral Variations of Hodge Structure Underlying One Parameter Families of Calabi-Yau Threefolds
• Integral Cohomology and Mirror Symmetry for Calabi-Yau 3-folds
• The Brauer group is not a derived invariant
• Determinantal Quintics and Mirror Symmetry of Reye Congruences

Cited by in corpus (1)

• Calabi--Yau Operators