# Existence and compactness theory for ALE scalar-flat K\"ahler surfaces

## TL;DR

This paper proves a compactness theorem for sequences of ALE scalar-flat K"ahler metrics on minimal K"ahler surfaces and demonstrates the existence of moduli spaces for these metrics.

## Contribution

It establishes a compactness result for ALE scalar-flat K"ahler metrics and constructs global moduli spaces for several families of such metrics.

## Key findings

- Noncollapsed sequences have convergent subsequences.
- Existence of moduli spaces for scalar-flat K"ahler ALE metrics.
- Application to infinite families of K"ahler ALE spaces.

## Abstract

Our main result in this article is a compactness result which states that a noncollapsed sequence of asymptotically locally Euclidean (ALE) scalar-flat K\"ahler metrics on a minimal K\"ahler surface whose K\"ahler classes stay in a compact subset of the interior of the K\"ahler cone must have a convergent subsequence. As an application, we prove the existence of global moduli spaces of scalar-flat K\"ahler ALE metrics for several infinite families of K\"ahler ALE spaces.