Journal of Symplectic Geometry
Volume 9 (2011)
Toric geometry of convex quadrilaterals
Pages: 343 – 385
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 Kähler-Einstein and toric Sasaki-Einstein metrics constructed. As a byproduct, we obtain a wealth of extremal toric (complex) orbi-surfaces, including Kähler-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 Kähler metric with constant scalar curvature. Our results also provide explicit examples of relative K-unstable toric orbi-surfaces that do not admit extremal metrics.