Mirror symmetry for double cover Calabi-Yau varieties (Q6562510)

By studying the relationship between Calabi-Yau manifolds and certain conformal field theories called Gepner models, \textit{B. R. Greene} and \textit{M. R. Plesser} [Nucl. Phys., B 338, No. 1, 15-37 (1990)] first found a mirror pair, the quintic and the (orbifold) Fermat quintic threefold. Further evidence for this relationship came from the work of physicists [\textit{P. Candelas} et al., Nucl. Phys., B 359, No. 1, 21--74 (1991; Zbl 1098.32506)], who first predicted the correct numbers of rational curves on quintic by using mirror symmetry. These successful predictions shocked the math community. Utilizing reflexive polytopes, \textit{V. V. Batyrev} [J. Algebr. Geom. 3, No. 3, 493--535 (1994; Zbl 0829.14023)] gave a recipe for constructing mirror pairs for Calabi-Yau hypersurfaces in Gorenstein toric varieties. Soon after, \textit{V. V. Batyrev} and \textit{L. A. Borisov} [Invent. Math. 126, No. 1, 183--203 (1996; Zbl 0872.14035)] generalized the construction to Calabi-Yau complete intersections in Gorenstein toric varieties via nef-partitions. All these examples focused on smooth Calabi-Yau manifolds. \textit{K. Matsumoto} et al. [Int. J. Math. 3, No. 1, 1--164 (1992; Zbl 0763.32016)] studied the family of \(K3\) surfaces arising from double covers branched along six lines in \(\mathbb{P}^2\) in general positions and its mirror.\N\NThe authors extend this previous work to a general situation. Consider a nef-partition on \(X\) and its dual nef-partition on \(\tilde{X}\) in the sense of \textit{V. V. Batyrev} and \textit{L. A. Borisov} [loc. cit.], and let \(E_1,\ldots, E_r\) and \(F_1,\ldots, F_r\) be the sum of toric divisors representing nef-partitions on \(X\) and \(\tilde{X}\), respectively. Now let \(S_j\in \mathrm{H}^{0}(X, 2E_j)\) be a smooth section and \(Y\) be the double cover over X branched along \(s_1 \cdots s_r\). This is a non-smooth Calabi-Yau variety. A parallel construction can be applied on the dual side \(\tilde{Y}\). What the authors observe via computation of their cohomologies is that \(Y\) and \(\tilde{Y}\) form a topological mirror pair. So they conjecture that they should be a mirror pair. Since those are singular Calabi-Yau varieties, this conjecture should serve as an extension of the classical mirror correspondence to singular varieties.