Aspinwall–Morrison conjecture on non-birational fibres of a Calabi–Yau family

Let $\mathcal{Y}\to B$ be the one-parameter family of smooth, multiply connected Calabi–Yau threefolds over

$$B=\mathbb{P}^1\setminus\{1,\xi,\ldots,\xi^4,\infty\},$$

where $\xi$ is a primitive fifth root of unity, and write $Y_t$ for the fibre over $t\in B$. Aspinwall–Morrison's conjecture. For general $t\in B$, the threefolds $Y_{\xi^i t}$ for $i=0,\ldots,4$ are not birationally equivalent to one another. This conjecture predicts that the family gives a counterexample to global Torelli for Calabi–Yau threefolds: the relevant fibres have isomorphic rational polarized Hodge structures, while the conjectured non-birationality distinguishes them even more strongly than non-isomorphism. The paper's main theorem establishes the corresponding non-isomorphism statement, but the stronger non-birationality assertion is presented here as conjectural.