All toric Kahler surfaces with twistor 2-forms

TL;DR

This paper classifies all smooth 4-dimensional Kahler geometries with a specific symmetry property, identifying six families and analyzing their curvature, while proposing a conjecture about the norm of twistor 2-forms.

Contribution

It completes the classification of toric Kahler surfaces with twistor 2-forms and reveals a quadratic relation involving moment maps, suggesting broader applicability.

Findings

Six distinct families of geometries identified

Explicit formulas and curvature computations provided

Quadratic norm of twistor 2-form in moment maps

Abstract

We complete the classification of all smooth 4-dimensional Kahler geometries admitting a twistor (conformal Killing-Yano) 2-form invariant under a 2-torus action. We establish that there are six geometrically distinct families, and we provide them in a simple form amenable to calculations and compute their curvature. We also find that for toric geometries the square norm of the twistor 2-form is quadratic in moment maps, and we are led to conjecture that this holds when less symmetry is present.