Calabi–Yau varieties of large index

Let us call a normal projective variety $X$ Calabi–Yau if its canonical divisor $K_X$ is $\mathbb{Q}$-linearly equivalent to zero. The smallest positive integer $m$ such that $m K_X$ is linearly equivalent to zero is called the index of $X$. It is expected that, under suitable assumptions on singularities, the index is bounded in each dimension. Following Esser–Totaro-Wang, we construct terminal Calabi-Yau varieties with large index in each dimension. The key idea of the construction is to apply mirror symmetry to Calabi–Yau varieties with an ample Weil divisor of small volume, which were previously obtained by the same authors.