Toric geometry of convex quadrilaterals

TL;DR

This paper explicitly solves the Abreu equation for convex quadrilaterals, confirming a conjecture and classifying toric Kähler-Einstein and Sasaki-Einstein metrics, while exploring extremal toric orbi-surfaces and stability conditions.

Contribution

It provides an explicit resolution of the Abreu equation on convex quadrilaterals, confirming Donaldson's conjecture and classifying related extremal toric metrics.

Findings

Confirmed a conjecture of Donaldson for convex quadrilaterals.

Classified explicit toric Kähler-Einstein and Sasaki-Einstein metrics.

Identified conditions for the existence of constant scalar curvature metrics.

Abstract

We provide an explicit resolution of the Abreu equation on convex labeled quadrilaterals. This confirms a conjecture of Donaldson in this particular case and implies a complete classification of the explicit toric K\"ahler-Einstein and toric Sasaki-Einstein metrics constructed in [6,22,14]. As a byproduct, we obtain a wealth of extremal toric (complex) orbi-surfaces, including K\"ahler-Einstein ones, and show that for a toric orbi-surface with 4 fixed points of the torus action, the vanishing of the Futaki invariant is a necessary and sufficient condition for the existence of K\"ahler metric with constant scalar curvature. Our results also provide explicit examples of relative K-unstable toric orbi-surfaces that do not admit extremal metrics.