Collapsing of K3 Surfaces and Special Kähler Structures
Zexuan Ouyang
TL;DR
The work determines the 2–dimensional collapsing limits of hyperkähler K3 metrics: these limit spaces are precisely the metric realizations of integral singular SKS on $\mathbb{P}^1$ and correspond bijectively to Jacobian elliptic K3 surfaces with a marked holomorphic volume form. It introduces twist transformations that preserve the metric while acting on the holomorphic data, yielding a base–fiber correspondence where SKS on the base encode torus fibrations with $\Omega$-data, and global models arise from Kodaira-type singular fibers. A complete picture is obtained by distinguishing isotrivial and non-isotrivial fibrations: the isotrivial case yields flat base metrics and, in the Kum$^+$er scenario, a one-parameter family of preimages, while the non-isotrivial case leads to a finite preimage count $N(G)$ governed by monodromy groups $G \le SL_2(\mathbb{Z})$. The paper also provides practical tools to compute $N(G)$ and describes the resulting stratification of the Jacobian elliptic K3 moduli via the map $\mathcal{F}$. Altogether, these results give a rigorous, globally consistent classification of GH limit spaces for collapsing hyperkähler K3 metrics and illuminate their deep links to integrable systems and elliptic fibrations.
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 $.