A closer look at mirrors and quotients of Calabi-Yau threefolds

Abstract

Let X be the toric variety (P 1 ) 4 associated with its four-dimensional polytope 1. Denote byX the resolution of the singular Fano variety X o associated with the dual polytope 1 o . Generically, anticanonical sections Y of X and anticanonical sectionsỸ ofX are mirror partners in the sense of Batyrev. Our main result is the following: the Hodge-theoretic mirror of the quotient Z associated to a maximal admissible pair (Y, G) in X is not a quotientZ associated to an admissible pair inX. Nevertheless, it is possible to construct a mirror orbifold for Z by means of a quotient of a suitableỸ . Its crepant resolution is a Calabi-Yau threefold with Hodge numbers (8, 4). Instead, if we start from a non-maximal admissible pair, in the same case, its mirror is the quotient associated to an admissible pair.