New mirror pairs of Calabi-Yau orbifolds

Abstract

We prove a representation-theoretic version of Borisov-Batyrev mirror symmetry, and use it to construct infinitely many new pairs of orbifolds with mirror Hodge diamonds, with respect to the usual Hodge structure on singular complex cohomology. We conjecture that the corresponding orbifold Hodge diamonds are also mirror. When X is the Fermat quintic in P 4 , and X * is a Sym 5 -equivariant, toric resolution of its mirror X * , we deduce that for any subgroup Γ of the alternating group A5, the Γ-Hilbert schemes Γ-Hilb(X) and Γ-Hilb( X * ) are smooth Calabi-Yau threefolds with (explicitly computed) mirror Hodge diamonds.

The Expanding Zoo of Calabi-Yau Threefolds

“…In the case that the group action has fixed points, it is often possible to resolve the resulting orbifold singularities in such a way as to again obtain a Calabi-Yau manifold. Examples can be found in [27,28,29]. The second technique is to vary either the complex structure or Kähler moduli of a known space until it becomes singular, and then desingularise it by varying the other type of moduli.…”