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\(K3\) surfaces from configurations of six lines in \(\mathbb{P}^2\) and mirror symmetry. II: \(\lambda_{K3}\)-functions. (English) Zbl 1486.14054

The paper under review continures discussion on double covers of \(\mathbb{P}^2\) branched along six lines in general poistion. A smooth resolution defines a \(K3\) surface over configuration space \({\mathcal{M}}_6\), called double cover family of \(K3\) surfaces. This family is a natural generalization of the Legendre family of elliptic curves over \({\mathcal{M}}_4\). This article is a sequel to the paper [S. Hosono et al., “\(K3\) surfaces from configurations of six lines in \(\mathbb{P}^2\) and mirror symmetry I”, Preprint, arXiv:1810.00606]. This article presents the two definitions of \(K3\) analogues of the elliptic lambda function.
The moduli space \({\mathcal{M}}_6\) is singular, and there are two natural resolutions \(\widetilde{{\mathcal{M}}_6}\) and \(\widetilde{{\mathcal{M}}_6^+}\). Accordingly, there are two \(K3\) analogues of the elliptic lambda function, denoted by \(\lambda_{K3}\) and \(\lambda_{K3}^+\), and called K3 lambda functions (which are not isomorphic) corresponding to a flip in the moduli space. The \(K3\) lambda functions are identified with the mirror maps (the inverse maps of period maps) of the \(K3\) family, and are expressed in terms of genus \(2\) theta functions.
The elliptic lambda function is known to have an explicit form in terms of the hypergeometric series. This paper determines the two \(K3\) lambda functions as generalizations of the elliptic lambda function in terms of the hypergeometric series. This is done by introducing the master equation and find the mirror map and the hypergeometric series near the LCSL satisfying the master equation, which in turn determine the K3 lambda functions.
Mirror symmetry of the \(K3\) family is also briefly discussed.

MSC:

14J28 \(K3\) surfaces and Enriques surfaces
14F10 Differentials and other special sheaves; D-modules; Bernstein-Sato ideals and polynomials
14J10 Families, moduli, classification: algebraic theory
14J33 Mirror symmetry (algebro-geometric aspects)
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