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

TL;DR

This paper explores the Hodge structures associated with a specific orbifold and its mirror, revealing how Calabi-Yau type structures naturally emerge from algebraic constructions without explicit geometric identification.

Contribution

It demonstrates the emergence of Calabi-Yau type Hodge structures from rational Hodge structures on algebraic modules related to the orbifold, linking abstract algebraic structures to special geometry.

Findings

Hodge structure of Calabi-Yau type (1,9,9,1) arises naturally

Hodge structures can be recovered from Abelian varieties of Weil type

Special geometry appears in the abstract algebraic framework

Abstract

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.