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A Closer Look ar Mirrors and Quotients of Calabi-Yau Threefolds

Let X be the toric variety (P^1)^4 associated with its four-dimensional polytope. Consider a resolution of the singular Fano variety associated with the dual polytope of X. Generically, anti-canonical sections Y of X and anticanonical sections of the resolution 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 associated to an admissible pair in the resolution. Nevertheless, it is possible to construct a mirror orbifold of Z by means of a quotient of a suitable anticanonical section of the resolution. The crepant resolution of Z is a Calabi-Yau threeefold with Hodge numbers (8, 4). Instead, if we start from a non-maximal admissible pair, in same cases, its mirror is the quotient associated to an admissible pair.