K3 surfaces from configurations of six lines in $\mathbb{P}^2$ and mirror symmetry I

Shinobu Hosono; Bong H. Lian; Hiromichi Takagi; Shing-Tung Yau

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arXiv · 1810.00606

K3 surfaces from configurations of six lines in $\mathbb{P}^2$ and mirror symmetry I

Shinobu Hosono·Bong H. Lian·Hiromichi Takagi·Shing-Tung Yau

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Abstract

From the viewpoint of mirror symmetry, we revisit the hypergeometric system $E(3,6)$ for a family of K3 surfaces. We construct a good resolution of the Baily-Borel-Satake compactification of its parameter space, which admits special boundary points (LCSLs) given by normal crossing divisors. We find local isomorphisms between the $E(3,6)$ systems and the associated GKZ systems defined locally on the parameter space and cover the entire parameter space. Parallel structures are conjectured in general for hypergeometric system $E(n,m)$ on Grassmannians. Local solutions and mirror symmetry will be described in a companion paper \cite{HLTYpartII}, where we introduce a K3 analogue of the elliptic lambda function in terms of genus two theta functions.

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Shinobu Hosono, Bong H. Lian, Hiromichi Takagi, Shing-Tung Yau. 2019-03-22. K3 surfaces from configurations of six lines in $\mathbb{P}^2$ and mirror symmetry I. https://arxiv.org/abs/1810.00606

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