  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.
