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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].

##### MSC:
 14J32 Calabi-Yau manifolds (algebro-geometric aspects) 14J28 $$K3$$ surfaces and Enriques surfaces
##### Keywords:
mirror symmetry; Calabi-Yau manifolds; strange duality