Collapsing of K3 Surfaces and Special K\"ahler Structures

TL;DR

This paper characterizes the collapsing limits of hyperk"ahler metrics on K3 surfaces as integral singular special K"ahler structures on the sphere, establishing a bijection with Jacobian elliptic K3 surfaces and counting their correspondence.

Contribution

It identifies the precise structure of collapsing K3 surfaces as singular special K"ahler structures and links them to Jacobian elliptic K3 surfaces with a marked volume form.

Findings

$rak{M}_2$ consists of metric spaces from integral singular SKSs on $P^1$

A bijection exists between these SKSs and Jacobian elliptic K3 surfaces

The number of Jacobian elliptic K3 surfaces for a given metric on $P^1$ is computed

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 $.