Toric geometry of convex quadrilaterals

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 Khler-Einstein and toric Sasaki-Einstein metrics constructed in As a byproduct, we obtain a wealth of extremal toric (complex) orbi-surfaces, including Khler-Einstein ones, and show that for a toric orbi-surface with four fixed points of the torus action, the vanishing of the Futaki invariant is a necessary and sufficient condition for the existence of Khler metric with constant scalar curvature. Our results also provide explicit examples of relative K-unstable toric orbi-surfaces that do not admit extremal metrics.