Calabi–Yau hypersurfaces and SU-bordism

V. V. 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 $\mathrm {SU}$-bordism classes generate the special unitary bordism ring $\varOmega ^{\mathrm {SU}}\bigl [\tfrac 12\bigr ]\cong \mathbb {Z}\bigl [\tfrac 12\bigr ][y_i\colon i\ge 2]$. We also describe explicit Calabi–Yau representatives for multiplicative generators of the $\mathrm {SU}$-bordism ring in low dimensions.