Toric geometry of convex quadrilaterals

Eveline Legendre (1)

1 IMT - Institut de Mathématiques de Toulouse UMR5219

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 in [6,22,14]. 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 4 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.

Domaines: Géométrie différentielle [math.DG]; Géométrie symplectique [math.SG]

Eveline Legendre. Toric geometry of convex quadrilaterals. Journal of Symplectic Geometry, 2011, 9, pp.343-385. ⟨hal-00715610⟩