The harmonic $2$-forms on $K3$ surfaces converging to a flat $4$-dimensional orbifold

TL;DR

This paper investigates how harmonic 2-forms on K3 surfaces behave asymptotically as the metrics approach a flat orbifold, revealing a decomposition into forms converging to flat and Chern forms.

Contribution

It demonstrates the decomposition of harmonic 2-forms on K3 surfaces into two converging subspaces in the limit of flat orbifold metrics.

Findings

One subspace converges to flat 2-forms on the quotient torus.

The other converges to Chern forms of anti-self-dual connections on ALE spaces.

Abstract

In this article, we study the asymptotic behavior of harmonic $2$-forms on $K3$ surfaces with Ricci-flat K\"ahler metrics, where metrics converge to the quotient of a flat $4$-torus by a finite group action. We can show that the space of anti-self-dual harmonic $2$ forms decomposes into two subspaces: one converges to the flat $2$-forms on the quotient of the torus, while the other converges to the first Chern forms of anti-self-dual connections on ALE spaces.