Communications in Number Theory and Physics

Volume 14 (2020)

Number 4

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

Pages: 739 – 783

DOI: https://dx.doi.org/10.4310/CNTP.2020.v14.n4.a2

Authors

Shinobu Hosono (Department of Mathematics, Gakushuin University, Mejiro, Toshima-ku, Tokyo, Japan)

Bong H. Lian (Department of Mathematics, Brandeis University, Waltham, Massachusetts, U.S.A.)

Hiromichi Takagi (Department of Mathematics, Gakushuin University, Mejiro, Toshima-ku, Tokyo, Japan)

Shing-Tung Yau (Department of Mathematics, Harvard University, Cambridge, Massachusetts, U.S.A.; and Yau Mathematical Science Center, Tsinghua University, Beijing, China)

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 covering 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 [20], where we introduce a K3 analogue of the elliptic lambda function in terms of genus two theta functions.

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S. Hosono was supported in part by Grant-in Aid Scientific Research (C20K03593, S17H06127, A18H03668).

B.H. Lian and S.-T. Yau were supported in part by the Simons collaboration grant on Homological Mirror Symmetry 2015–2019.

H. Takagi was supported in part by Grant-in Aid Scientific Research (C16K05090).

Received 1 December 2019

Accepted 20 March 2020

Published 2 October 2020