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K3 surfaces with involution and mirror pairs of Calabi-Yau manifolds. (English) Zbl 0939.14021
Greene, B. (ed.) et al., Mirror symmetry II. Cambridge, MA: International Press, AMS/IP Stud. Adv. Math. 1, 717-743 (1997).
Given a K3 surface \(X_2\) with an involution \(\sigma\) acting by \(-1\) on \(H^{2,0}(X_2)\), and an elliptic curve \(E\), the natural desingularization of the variety \(X_2\times E/\sigma\times\iota\) (where \(\iota\) is the standard involution of \(E\)) is a Calabi-Yau threefold \(X_3\) equipped with an involution acting by \(-1\) on \(H^{3,0}(X_3)\). K3 surfaces of this kind have been classified by V. Nikulin [Proc. Int. Congr. Math., Berkeley 1986, Vol. I, 654-671 (1987; Zbl 0671.22006)] and they come in families parametrized by three integers \((r,a,\delta)\), where \(r\) is the rank of the \(\sigma\)-invariant sublattice of the K3 lattice. For each such family, the family \((20-r,a,\delta)\) can be regarded as a “mirror family.” By means of the construction previously sketched this yields a notion of mirror duality for families of Calabi-Yau threefolds which is consistent with the so-called topological mirror symmetry: The Hodge numbers \(h^{2,1}\) and \(h^{1,1}\) are interchanged by going from a family to its mirror family. \smallskip The author then compares his construction with Batyrev’s mirror construction in terms of reflexive polyhedra [V. V. Batyrev, J. Algebr. Geom. 3, No. 3, 493-535 (1994; Zbl 0829.14023)] finding agreement in a number of cases. The author also discusses the relation between Nikulin’s duality and Arnold’s strange duality, the case where the Calabi-Yau threefolds are fibred products of rational elliptic surfaces, and some higher-dimensional examples.
Further Hodge-theoretic properties of this construction have been studied by C. Voisin [in: Journées de Géometrie Algébrique d’Orsay, Astérisque 218, 273-323 (1993; Zbl 0818.14014)].
For the entire collection see [Zbl 0905.00079].

14J32 Calabi-Yau manifolds (algebro-geometric aspects)
14J28 \(K3\) surfaces and Enriques surfaces