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Period- and mirror-maps for the quartic \(K3\). (English) Zbl 1303.14044
The paper gives a detailed description of mirror symmetry, in terms of Hodge structures, between quartic \(K3\) surfaces in projective space and the Dwork family of \(K3\) surfaces. More precisely, given a \(K3\) surface, we can endow integral cohomology with two different Hodge structures. The first one \(H_{A}\) depends only on the choice of Kähler form on X; the second \(H_{B}\) depends on its complex structure. P. S. Aspinwall and D. R. Morrison [AMS/IP Stud. Adv. Math. 1, 703–716 (1997; Zbl 0931.14020)] have described how mirror symmetry between two varieties \(X\) and \(X'\) can be seen as a Hodge isometry between \(H_A(X)\) and \(H_B(X')\). In this paper, \(X\) and \(X'\) are as follows: \(X_p\) is a family of quartic surfaces in projective space whose volume form is parametrized by a multiplicative factor \(p\); \(X'_t\) is the \(K3\) surface obtained as a fiber of the Dwork family \[ \left\{\sum_{i=0}^3 z_i^4 - 4 t z_0 z_1 z_2 z_3=0\right\}_{t\in\mathbb{C}} \] after taking a suitable resolution of a quotient by \(\mu_4\).
The author provide an explicit condition on \(t\) and \(p\) for \(X_{p}\) and \(X'_{t}\) to be mirrors, namely \[ \exp(2\pi p) = w + 104 w ^2 + 15188 w ^3 + 2585184 w ^4 + 480222434 w ^5 + \ldots \] where \(w = \frac{1}{(4t)^{4}}\). To establish the correspondence, the author computes the period maps for both families of surfaces and picks isomorphism between the Hodge structures at mirror points. He then proceeds to show that near points of maximal unipotent monodromy, this map is compatible with the Morrison mirror map [D. R. Morrison, in: Essays on mirror manifolds. Cambridge, MA: International Press. 241–264 (1992; Zbl 0841.32013)].

MSC:
14J28 \(K3\) surfaces and Enriques surfaces
14J33 Mirror symmetry (algebro-geometric aspects)
53D37 Symplectic aspects of mirror symmetry, homological mirror symmetry, and Fukaya category
14D05 Structure of families (Picard-Lefschetz, monodromy, etc.)
14D07 Variation of Hodge structures (algebro-geometric aspects)
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