Calabi-Yau varieties of large index

Call a projective variety $X$ Calabi-Yau if its canonical divisor is ${\bf Q}$-linearly equivalent to zero. The smallest positive integer $m$ with $mK_X$ linearly equivalent to zero is called the index of $X$. We construct Calabi-Yau varieties with the largest known index in high dimensions. In our examples, the index grows doubly exponentially with dimension. We conjecture that our examples have the largest possible index, with supporting evidence in low dimensions. The examples are obtained by mirror symmetry from our Calabi-Yau varieties with an ample Weil divisor of small volume. We also give examples for several related problems, including Calabi-Yau varieties with large orbifold Betti numbers or small minimal log discrepancy.