The E³/Z3 orbifold, mirror symmetry, and Hodge structures of Calabi-Yau type

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.