最后的Picard秩1双镜像Calabi-Yau对？
The Last Picard Rank 1 Double--Mirror Calabi--Yau Pair?
- IMPAN（波兰科学院数学研究所）
- Tsinghua University（清华大学）
机构由 AI 辅助整理，请以论文原文为准。
AI总结：
本文证明两个Picard秩1的Calabi-Yau三维簇族（$\mathcal{X}$与$\mathcal{Y}$）之间存在导出等价，通过数学规范线性sigma模型建立联系，解决了Miura关于Fourier-Mukai伙伴的猜想，并指出这可能是最后的双镜像对。
AI中文摘要：
我们考虑一对Picard秩1的Calabi-Yau三维簇族，它们早前已在数学文献中以不相关的背景出现：$G(2,6)$中33次三维簇族$\mathcal{X}$（由Inoue-Ito-Miura构造）和$\mathbb{P}^8$中算术Gorenstein 21次三维簇族$\mathcal{Y}$（由Schenck-Stillman-Yuan构造）。在建立它们一般成员之间的自然几何对应后，我们进一步证明这些族中任何对应的三维簇对满足某些经典对偶性。此外，我们发现它们的几何可能通过一个数学规范线性sigma模型相关联，利用该模型我们证明它们是导出等价的。这解决了Miura的一个猜想，该猜想基于对镜像模空间的研究，预测$\mathcal{X}$的成员存在非平凡的Fourier-Mukai伙伴。该猜想后来也被Gerhardus-Jockers在物理背景下提出。实际上，从另一族$\mathcal{Y}$出发，我们推测性地得到相同的镜像模空间。因此，这样的一对是Picard秩1的非双有理双镜像Calabi-Yau三维簇形变族小列表中的一个新成员，且很可能是最后一个。
英文摘要：
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 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.
