标题： E^3/Z3 类 orbifold、镜像对称性与 Calabi-Yau 型 Hodge 结构 显示英文标题

标题： The E^3/Z3 orbifold, mirror symmetry, and Hodge structures of Calabi-Yau type

摘要： 从刚性 orbifold Z = E^3 /\mathbb{Z}_3 的 Kähler 模空间开始，人们会期望广义镜像的上同调具有 Calabi-Yau 型 Hodge 结构 (1,9,9,1)。 我们证明了这种结构可以自然地从 \Lambda ^3, \mathbb{K}^6, \mathbb{K}=\mathbb{Q}[\omega ] 上的有理 Hodge 结构中得出，其中 \omega 是单位根的本原三次根。 我们并未试图识别潜在的几何结构，而是展示了特殊几何如何在我们的抽象构造中出现。 我们还展示了这种 Hodge 结构如何作为更大 Hodge 结构的极化子结构被恢复，该 Hodge 结构由 Weil 型六维阿贝尔簇的第三上同调群给出。 显示英文摘要

摘要： Starting from the K\"ahler moduli space of the rigid orbifold Z=E^3/\mathbb{Z}_3 one would expect for the cohomology of the generalized mirror to be a Hodge structure of Calabi-Yau type (1,9,9,1). We show that such a structure arises in a natural way from rational Hodge structures on \Lambda^3 \mathbb{K}^6, \mathbb{K}=\mathbb{Q}[\omega], where \omega is a primitive third root of unity. We do not try to identify an underlying geometry, but we show how special geometry arises in our abstract construction. We also show how such Hodge structure can be recovered as a polarized substructure of a bigger Hodge structure given by the third cohomology group of a six-dimensional Abelian variety of Weil type.