A classification of scalar-flat toric K\"ahler instantons in dimension 4

TL;DR

This paper classifies all scalar-flat toric K"ahler 4-manifolds with specific asymptotic behaviors, providing a complete description of their metrics and momentum functions using degenerate elliptic equations.

Contribution

It offers a comprehensive classification of scalar-flat toric K"ahler instantons in four dimensions under certain decay conditions, utilizing a Liouville theorem for degenerate elliptic equations.

Findings

Complete classification of scalar-flat toric K"ahler 4-manifolds with specified asymptotics

Identification of the image of the moment map as closed under these conditions

Explicit description of momentum functions and metrics

Abstract

We classify all scalar-flat toric K\"ahler 4-manifolds under either of two asymptotic conditions: that the action fields decay slowly (or at all), or that the curvature decay is quadratic; for example we fully classify instantons that have any of the ALE-F-G-H asymptotic types. The momentum functions satisfy a degenerate elliptic equation, and under either asymptotic condition the image of the moment map is closed. Using a recent Liouville theorem for degenerate-elliptic equations, we classify all possibilities for the momentum functions, and from this, all possible metrics.