### Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 13 (2017), 067, 25 pages      arXiv:1301.7632      https://doi.org/10.3842/SIGMA.2017.067

### Minuscule Schubert Varieties and Mirror Symmetry

Makoto Miura
Korea Institute for Advanced Study, 85 Hoegiro, Dongdaemun-gu, Seoul, 130-722, Republic of Korea

Received August 23, 2016, in final form August 16, 2017; Published online August 23, 2017

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 ${\boldsymbol{\Sigma}}$ of the Cayley plane ${\mathbb{OP}}^2$. We calculate topological invariants and BPS numbers of this Calabi-Yau 3-fold and conjecture that it has a non-trivial Fourier-Mukai partner.

Key words: Calabi-Yau; mirror symmetry; minuscule; Schubert variety; toric degeneration.

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