Collapsing of K3 Surfaces and Special K\"ahler Structures

Abstract

We study the structure of $\mathfrak{M}_2$, the set of half-dimensional collapsing spaces of hyperk\"ahler metrics on K3 surfaces. We show that $\mathfrak{M}_2$ consists precisely of those underlying metric spaces of integral singular special K\"ahler structures (SKSs) on $ \mathbb{P}^1 $. Furthermore, we establish a bijection between integral singular SKSs on $\mathbb{P}^1 $ and Jacobian elliptic K3 surfaces with a marked holomorphic volume form. Additionally, we compute the number of Jacobian elliptic K3 surfaces that correspond to a given metric on $ \mathbb{P}^1 $.