Collapsing of K3 Surfaces and Special Kähler Structures

Abstract

We study the structure of ${\mathfrak{M}}_2$, the set of half-dimensional collapsing spaces of hyperkähler metrics on K3 surfaces. We show that ${\mathfrak{M}}_2$ consists precisely of those underlying metric spaces of integral singular special Kähler 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 ${\mathbb{P}}^1$.

21 pages, Improved exposition and added remarks