Mirror symmetry for lattice polarized \(K3\) surfaces
In the last years a considerable number of papers was devoted to the mirror symmetry. This yields a certain duality between some families of Calabi-Yau 3-folds made up by a group of theoretical physicists. The paper under review aims to supply the necessary details which support the idea (more or less known to some experts) that the mirror symmetry is a beautiful and non-trivial generalization to 3-folds of a certain duality for K3 surfaces, discovered in the late seventies independently by \textit{H. Pinkham} [C. R. Acad. Sci., Paris, Sér. A 284, 615-618 (1977; Zbl 0375.14004)], and by I. Dolgachev and V. Nikulin [see \textit{I. Dolgachev} and \textit{V. Nikulin}, ``Exceptional singularities of V. I. Arnol'd and K3 surfaces'', Proc. USSR Topological Conference (Minsk 1977), \textit{I. Dolgachev}, Sémin. Bourbaki, 35e année, Vol. 1982/83, Exp. No. 611, Astérisque 105/106, 251-275 (1985; Zbl 0535.10018), and \textit{V. V. Nikulin}, Math. USSR, Izv. 14, 103-167 (1980); translation from Izv. Akad. Nauk SSSR, Sér. Mat. 43, 111-177 (1979; Zbl 0408.10011)]. The results of this excellent survey are due to several people such as: P. Aspinwall and D. Morrison, M. Kobayashi, B. Lian and S.-T. Ya, M. Nagura and K. Sugiyama, S.-S. Roan, A. Todorov, as well as to V. Batyrev, V. Nikulin and the author. This duality for K3 surfaces was inspired from Arnold's strange duality. The contents of the paper is as follows: 1. Lattice polarized K3 surfaces; 2. Local deformations; 3. The period map; 4. Tube domain realization of the period space; 5. Some arithmetical conditions on \(M\); 6. Mirror symmetry, 7. Mirrors for the family of degree \(2n\) polarized K3 surfaces; 8. Toric hypersurfaces; and 9. Other examples. All in all, the paper is an excellent account of the mirror symmetry in the 2-dimensional case, which is presented in such a way to better understand the underlying ideas even in the 3-dimensional case.
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