The Berglund-Hübsch-Chiodo-Ruan mirror symmetry for \(K3\) surfaces. (English) Zbl 1315.14051
A. Chiodo and Y. Ruan [Adv. Math. 227, No. 6, 2157–2188 (2011; Zbl 1245.14038)], using the transposition methods of P. Berglund and T. Hübsch [Nucl. Phys., B 393, No. 1–2, 377–391 (1993; Zbl 1245.14039)], provided a way of obtaining pairs of Calabi-Yau manifolds whose Hodge diamonds are symmetric. The paper under review is devoted to applying this method to a class of \(K3\) surfaces with a non-symplectic involution and as a result producing pairs of lattice mirror \(K3\) surfaces. To be more precise, let \(W\) be a \(K3\) surface defined in a weighted projective space by a specific non-degenerate Delsarte type polynomial whose matrix of exponents \(A\) is invertible over \(\mathbb{Q}\). Let \(G\) be a subgroup of the diagonal symmetries of \(W\) and \(J\) be the monodromy group of the affine Milnor fiber. Suppose that \(J\subset G\) is of determinant 1. Denote by \(W^T\) the hypersurface associated to the transpose \(A^T\), and let \(\widetilde{G}=G/J\) and \(\widetilde{G^T}=G^T/J^T\). The main result of the paper under review proves that the two orbifolds \([W/\widetilde{G}]\) and \([W^T/\widetilde{G^T}]\) belong to the lattice mirror families. These mirror orbifolds are not necessarily Gorenstein. This is the main difference with Batyrev mirror symmetry.
Reviewer: Amin Gholampour (College Park)
MSC:
| 14J28 | \(K3\) surfaces and Enriques surfaces |
| 14J33 | Mirror symmetry (algebro-geometric aspects) |
| 14J50 | Automorphisms of surfaces and higher-dimensional varieties |
| 14J10 | Families, moduli, classification: algebraic theory |
Software:
MagmaReferences:
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