Asymptotic geometry of toric Kahler instantons

TL;DR

This paper investigates the asymptotic geometry of toric Kähler instantons, establishing criteria for closed reductions and classifying possible geometries, while providing examples of non-closed reductions.

Contribution

It introduces geometric criteria for closed reductions and classifies asymptotic geometries of scalar-flat instantons with closed reductions, contrasting with examples of non-closed reductions.

Findings

Criteria for closed symplectic reductions in toric Kähler manifolds.

Classification of asymptotic geometries for scalar-flat instantons.

Examples of instantons with non-closed, non-polygon reductions.

Abstract

The symplectic reduction of a complete toric K\"ahler manifold need not be closed or even be a polygon. Sharp differences in behavior occur between those complete toric K\"ahler 4-manifolds with closed and with non-closed reductions. This paper establishes geometric criteria for these reductions to be closed, and classifies all asymptotic geometries possible in the case of scalar-flat instantons with closed reductions. Contrasting this, we provide examples of complete instantons with non-closed and non-polygon reductions.