Minuscule Schubert varieties and mirror symmetry

Abstract

We consider smooth complete intersection Calabi–Yau 3-folds in minuscule Schubert varieties, and study their mirror symmetry by degenerating the ambient Schubert varieties to Hibi toric varieties. We list all possible Calabi–Yau 3-folds of this type up to deformation equivalences, and find a new example of smooth Calabi–Yau 3-folds of Picard number one; a complete intersection in a locally factorial Schubert variety Σ of the Cayley plane OP2. We calculate topological invariants and BPS numbers of this Calabi–Yau 3-fold and conjecture that it has a non-trivial Fourier–Mukai partner.