  Batyrev constructed a family of Calabi-Yau hypersurfaces dual to the first
Chern class in toric Fano varieties. Using this construction, we introduce a
family of Calabi-Yau manifolds whose SU-bordism classes generate the special
unitary bordism ring
$\varOmega^{SU}\otimes\mathbb{Z}[\frac{1}{2}]\cong\mathbb{Z}[\frac{1}{2}][y_{i}\colon
i\ge 2]$. We also describe explicit Calabi-Yau representatives for
multiplicative generators of the SU-bordism ring in low dimensions.
