Open Access
September 2011 Toric geometry of convex quadrilaterals
Eveline Legendre
J. Symplectic Geom. 9(3): 343-385 (September 2011).


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.


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Eveline Legendre. "Toric geometry of convex quadrilaterals." J. Symplectic Geom. 9 (3) 343 - 385, September 2011.


Published: September 2011
First available in Project Euclid: 11 July 2011

zbMATH: 1233.14032
MathSciNet: MR2817779

Rights: Copyright © 2011 International Press of Boston

Vol.9 • No. 3 • September 2011
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