# K3 surfaces from configurations of six lines in $\mathbb{P}^{2}$ and mirror symmetry II --- $\lambda_{K3}$-functions ---

## TL;DR

This paper explores the hypergeometric system related to K3 surfaces, constructing mirror maps using genus two theta functions, and discusses their implications for mirror symmetry and moduli space transformations.

## Contribution

It introduces two non-isomorphic lambda functions for K3 surfaces and analyzes their roles in mirror symmetry and moduli space flips.

## Key findings

- Constructed local solutions near boundary points in moduli space.
- Expressed mirror maps in terms of genus two theta functions.
- Identified two distinct lambda functions related to moduli space flips.

## Abstract

We continue our study on the hypergeometric system $E(3,6)$ which describes period integrals of the double cover family of K3 surfaces. Near certain special boundary points in the moduli space of the K3 surfaces, we construct the local solutions and determine the so-called mirror maps expressing them in terms of genus two theta functions. These mirror maps are the K3 analogues of the elliptic $\lambda$-function. We find that there are two non-isomorphic definitions of the lambda functions corresponding to a flip in the moduli space. We also discuss mirror symmetry for the double cover K3 surfaces and their higher dimensional generalizations. A follow up paper will describe more details of the latter.