Mirror symmetry for singular double cover Calabi--Yau varieties: quantum test

We continue our study on the pairs of singular Calabi–Yau varieties arising from double covers over semi-Fano toric manifolds. In this paper, we first investigate singular CY double covers of ${\mathbb{P}}^3$ branched along (1) a union of eight hyperplanes in general position, and (2) a union of four hyperplanes and a quartic in generation. Our previous construction produces hypothetical singular mirror partners. We prove that they are mirror pairs in the sense that the $B$-model of one (variation of Hodge structure) is equivalent to the $A$-model of another (the untwisted part of the genus zero orbifold Gromov–Witten invariants). The technique can be generalized and applied to the case when the nef-partition is trivial. As a byproduct, we also verify Morrison’s conjecture in certain circumstances.