The Last Picard Rank 1 Double-Mirror Calabi-Yau Pair?

Abstract

We consider a certain pair of families of Picard rank 1 Calabi-Yau threefolds, that have appeared earlier in mathematical literature in unrelated contexts: the family $\mathcal{X}$ of degree 33 threefolds in $G(2,6)$ (constructed by Inoue-Ito-Miura) and the family $\mathcal{Y}$ of arithmetically Gorenstein degree 21 threefolds in $\mathbb{P}^8$ (constructed by Schenck-Stillman-Yuan). After establishing a natural geometric correspondence between their general members, we go on to show that any pair of corresponding threefolds in these families satisfy certain classical dualities. We moreover discover that their geometries may be related by a mathematical gauged linear sigma model, using which we prove that they are derived equivalent. This settles a conjecture of Miura, who predicted the existence of non-trivial Fourier-Mukai partners to members of $\mathcal{X}$, based on a study of its mirror moduli. This conjecture was also formulated later by Gerhardus-Jockers, in a physical context. In fact, starting from the other family $\mathcal{Y}$, we conjecturally arrive at the same mirror moduli. Thus, such a pair is a new, and quite possibly the last, addition to the small list of deformation families of non-birational, double-mirror Calabi-Yau threefolds having Picard rank 1.