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

Let X be the toric variety (P1)4 associated with its four-dimensional polytope Δ. Denote by X the resolution of the singular Fano variety X° associated with the dual polytope A°. Generically, anticanonical sections Y of X and anticanonical sections Y of X 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 quotient Z associated to an admissible pair in X. Nevertheless, it is possible to construct a mirror orbifold for Z by means of a quotient of a suitable Y. 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.